TEST-CASES FOR INTERFACE TRACKING METHODS Test-cases Editors: Didier Jamet, DER/SSTH/LMDL, CEA/Grenoble, 38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 45 42, Fax: +33 (0)4 38 78 50 36, E-Mail: [email protected] Olivier Lebaigue, DER/SSTH/LMDL, CEA/Grenoble, 38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 36 70, Fax: +33 (0)4 38 78 50 36, E-Mail: [email protected] Herv´e Lemonnier, DER/SSTH/LIEX, CEA/Grenoble, 38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 45 40, Fax: +33 (0)4 38 78 50 45, E-Mail: [email protected]

Preliminary version of 19th April 2004

Contents Introduction 9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1 Test-case number 1: Rise of a spherical cap bubble in (PN) 1.1 Practical significance and interest of the test-case . . . . 1.2 Definitions and model description . . . . . . . . . . . . . 1.3 Summary of the requested calculations . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Test-case number 2: Free rise of a liquid inclusion (PN, PE) 2.1 Practical significance and interest of the test-case . . 2.2 Definitions and model description . . . . . . . . . . . 2.3 A series of six numerical test-cases . . . . . . . . . . 2.4 An experimental test-case . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

a stagnant liquid . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

13 13 14 15 17

in a stagnant liquid . . . . .

19 19 20 21 23 27

3 Test-case number 3: Propagation of pure capillary standing waves (PA) 3.1 Practical significance and interest of the test-case . . . . . . . . . . . . . . . 3.2 Definitions and model description . . . . . . . . . . . . . . . . . . . . . . . . 3.3 A series of test-cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 30 32 34

4 Test-case number 4: Rayleigh-Taylor instability for isothermal, incompressible and non-viscous fluids (PA) 4.1 Practical significance and interest of the test-case . . . . . . . . . . . . . . . 4.2 Definitions and physical model description . . . . . . . . . . . . . . . . . . . 4.3 Test-case description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Analytical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Test comparison criteria . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 36 36 37 38 40

5 Test-case number 5: Oscillation of an inclusion immersed fluid (PA) 5.1 Practical significance and interest of the test-case . . . . . . 5.2 Definitions and model description . . . . . . . . . . . . . . . 5.3 Numerical settings, initial and boundary conditions . . . . . 5.4 Requested calculations . . . . . . . . . . . . . . . . . . . . . 5.5 An illustrative example . . . . . . . . . . . . . . . . . . . .

41 41 41 44 45 45

3

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

in a quiescent . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

4/283

Test-Cases for interface tracking methods

5.6 Additional information for 2D calculations. . . . . . . . . . . . . . . . . . . 46 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6 Test-case number 6: Two-dimensional droplet pining on an inclined wall (PC) 6.1 Practical significance and interest of the test-case . . . . . . . . . . . . . . . 6.2 Description of the model for the contact angle hysteresis and definition of the test-case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Test procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Comparison criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49 50 51 51 52

7 Test-case number 7a: One-dimensional phase-change of a vapor phase in contact with a wall (PA) 7.1 Practical significance and interest of the test-case . . . . . . . . . . . . . . . 7.2 General definitions and model description . . . . . . . . . . . . . . . . . . . 7.2.1 Bulk balance equations . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Interfacial balance equations . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Interface thermodynamic equilibrium . . . . . . . . . . . . . . . . . . 7.3 Steady state model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Test-case description . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Unsteady model for a phase initially uniformly superheated or undercooled 7.4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Test-case description . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 54 54 55 56 56 56 59 59 60 60 65 69

8 Test-case number 7b: Isothermal vaporization due to piston aspiration (PA) 8.1 Practical significance and interest of the test-case . . . . . . . . . . . . . . . 8.2 Definitions and model description . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Incompressible model . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Inviscid compressible model . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Physical relevance of an isothermal phase-change problem . . . . . . 8.3 Test-case description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 72 74 74 76 76 77

9 Test-case number 10: Parasitic currents induced by surface tension (PC) 9.1 Practical significance and interest of the test-case . . . . . . . . . . . . . . . 9.2 Definitions and physical model description . . . . . . . . . . . . . . . . . . . 9.3 Test-case description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Example of comparison exercise . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 80 81 81 84

10 Test-case number 11a: Translation and rotation (N) 10.1 Practical significance and interest of the test-case 10.2 Definitions and physical model description . . . . 10.3 Test-case description . . . . . . . . . . . . . . . . 10.4 Example of comparison exercise . . . . . . . . . .

85 85 85 86 87

of a concentration disk . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

Test-Cases for interface tracking methods

5/283

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 11 Test-case number 11b: Stretching of a circle in a (N) 11.1 Practical significance and interest of the test-case . . 11.2 Definitions and physical model description . . . . . . 11.3 Test-case description . . . . . . . . . . . . . . . . . . 11.4 Example of comparison exercise . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

vortex velocity field . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

12 Test-case number 12: Filling of a cubic mould by a viscous jet (PN, 12.1 Practical significance and interest of the test-case . . . . . . . . . . . . 12.2 Definitions and physical model description . . . . . . . . . . . . . . . . 12.3 Test-case description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Figures, tables, captions and references . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Test-case number 13: Shock tubes (PA) 13.1 Introduction . . . . . . . . . . . . . . . . . . . 13.2 The mathematical model and the solution of Problem . . . . . . . . . . . . . . . . . . . . 13.3 The shock tube . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

. . the . . . . . .

89 89 90 90 90 92

PE) . . . . . . . . . . . . . . .

93 93 94 94 95 95

. . . . .

. . . . . . . . . . . . . . corresponding Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 Test-case number 14: Poiseuille two-phase flow (PA) 14.1 Practical significance and interest of the test-case . . . . 14.2 Definitions and physical model description . . . . . . . . 14.3 Test-case description . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Test-case number 15: Phase inversion in a closed 15.1 Practical significance and interest of the test-case . 15.2 Definitions and physical model description . . . . . 15.3 Test-case description . . . . . . . . . . . . . . . . . 15.4 Illustrations of the problem . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

box (PC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16 Test-Case number 16: Impact of a drop on a thin film of (PE, PA) 16.1 Practical significance and interest of the test case . . . . . 16.2 Definitions and physical model description . . . . . . . . . 16.3 Test-case description . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Test-case number 17: Dam-break flows on dry PA, PE) 17.1 Practical significance and interest of the test-case 17.2 Definitions and physical model description . . . . 17.3 Test-case description . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

. . . .

. . . . .

99 . 99 . 100 . 102 . 108 109 109 110 110 111

. . . .

. . . .

. . . . .

113 . 113 . 114 . 114 . 115 . 115

the same liquid 117 . . . . . . . . . . 117 . . . . . . . . . . 117 . . . . . . . . . . 119 . . . . . . . . . . 121

and wet surfaces (PN, 123 . . . . . . . . . . . . . . . 123 . . . . . . . . . . . . . . . 124 . . . . . . . . . . . . . . . 125 . . . . . . . . . . . . . . . 128

6/283

18 Test-case number 19: 18.1 Introduction . . . . 18.2 Description . . . . References . . . . . . . .

Test-Cases for interface tracking methods

Shock-Bubble Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(PN) 129 . . . . . . . . . . . . . . . 129 . . . . . . . . . . . . . . . 129 . . . . . . . . . . . . . . . 132

19 Test-case number 21: Gas bubble bursting at a formation (PN-PE) 19.1 Practical significance and interest of the test-case . 19.2 Definitions and physical model description . . . . . 19.3 Test-case description . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

free surface, with jet 133 . . . . . . . . . . . . . . 133 . . . . . . . . . . . . . . 134 . . . . . . . . . . . . . . 135 . . . . . . . . . . . . . . 136

20 Test-case number 22: Axisymmetric body emerging through a free surface(PE) 139 20.1 Practical significance and interest of the test-case . . . . . . . . . . . . . . . 139 20.2 Experimental setup description . . . . . . . . . . . . . . . . . . . . . . . . . 139 20.3 Test-case description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 21 Test-case number 23: Relative trajectories and collision a simple shear flow (PA) 21.1 Practical significance and interest of the benchmark . . . 21.2 Definitions and physical model description . . . . . . . . . 21.2.1 Description, notation and assumptions . . . . . . . 21.2.2 Formulation . . . . . . . . . . . . . . . . . . . . . . 21.3 The description of the benchmark . . . . . . . . . . . . . 21.3.1 Input parameters and physical properties . . . . . 21.3.2 Initial conditions . . . . . . . . . . . . . . . . . . . 21.3.3 Results and comparisons . . . . . . . . . . . . . . . 21.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

of two drops in 145 . . . . . . . . . . 145 . . . . . . . . . . 146 . . . . . . . . . . 146 . . . . . . . . . . 147 . . . . . . . . . . 148 . . . . . . . . . . 148 . . . . . . . . . . 149 . . . . . . . . . . 150 . . . . . . . . . . 150 . . . . . . . . . . 152

22 Test-case number 24: Growth of a small bubble immersed in a heated liquid and its collapse in a subcooled liquid (PE,PA) 22.1 Practical significance and interest of the test-case . . . . . . . . . . . 22.2 Model and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Bubble collapse: case 24-1 (PA) . . . . . . . . . . . . . . . . . . . . . 22.4 Initial stage of the growth of a vapor bubble, case 24-2 (PA) . . . . . 22.5 Thermally controlled growth of a vapor bubble (24-3) . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

super153 . . . . 153 . . . . 154 . . . . 155 . . . . 156 . . . . 160 . . . . 164

23 Test-case number 26: Droplet impact on hot walls 23.1 Practical significance and interest of the test-case . . 23.2 Definitions and physical model description . . . . . . 23.3 Test-case description . . . . . . . . . . . . . . . . . . 23.4 Relevant results for comparison . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

169 169 169 170 170 174

Test-Cases for interface tracking methods

7/283

24 Test-case number 27: Interface tracking based on field in a convergent-divergent channel (PN) 24.1 Practical significance and relevance of the test-case . 24.2 Definitions and model description . . . . . . . . . . . 24.3 Test-case description . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Test-case number 28: The Lock-Exchange Flow 25.1 Practical significance and interest of the test-case 25.2 Definitions and physical model description . . . . 25.3 Test-case description . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

an imposed velocity 175 . . . . . . . . . . . . . 175 . . . . . . . . . . . . . 176 . . . . . . . . . . . . . 177 . . . . . . . . . . . . . 181

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

26 Test-case number 29a: The velocity and shape of 2D long inclined channels or in vertical tubes (PA, PN) Part I : in liquid 26.1 Practical significance and interest of the test-case . . . . . . . . 26.2 Definitions and model description . . . . . . . . . . . . . . . . . 26.3 Motion in horizontal channel . . . . . . . . . . . . . . . . . . . 26.4 Motion in inclined channel . . . . . . . . . . . . . . . . . . . . . 26.5 Motion in vertical channel and in tube . . . . . . . . . . . . . . 26.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 26.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

183 . 183 . 184 . 185 . 187

bubbles in a stagnant 189 . . . . . . . 189 . . . . . . . 190 . . . . . . . 191 . . . . . . . 192 . . . . . . . 194 . . . . . . . 195 . . . . . . . 197 . . . . . . . 202

27 Test-case number 29b: The velocity and shape of 2D long bubbles in inclined channels or in vertical tubes (PA, PN) Part II: in a flowing liquid 205 27.1 Practical significance and interest of the test-case . . . . . . . . . . . . . . . 205 27.2 Definitions and model description . . . . . . . . . . . . . . . . . . . . . . . . 206 27.3 Motion in horizontal and inclined channel . . . . . . . . . . . . . . . . . . . 208 27.4 Motion in vertical channel and in tube . . . . . . . . . . . . . . . . . . . . . 209 27.5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 27.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 28 Test case number 30: Unsteady cavitation in (PN) 28.1 Practical significance and interest of the test-case 28.2 Definitions and physical model description . . . . 28.2.1 Physical model of cavitation . . . . . . . . 28.2.2 Numerical resolution . . . . . . . . . . . . 28.2.3 Turbulence model . . . . . . . . . . . . . 28.3 Geometry and boundary conditions . . . . . . . . 28.3.1 Grid . . . . . . . . . . . . . . . . . . . . . 28.3.2 Initial conditions . . . . . . . . . . . . . . 28.3.3 Calculations . . . . . . . . . . . . . . . . . 28.4 Comparison with experiments . . . . . . . . . . . 28.4.1 Overall behavior . . . . . . . . . . . . . . 28.4.2 Flow field inside the sheet of cavitation . References . . . . . . . . . . . . . . . . . . . . . . . . .

a Venturi type section 221 . . . . . . . . . . . . . . . 221 . . . . . . . . . . . . . . . 222 . . . . . . . . . . . . . . . 222 . . . . . . . . . . . . . . . 223 . . . . . . . . . . . . . . . 223 . . . . . . . . . . . . . . . 223 . . . . . . . . . . . . . . . 224 . . . . . . . . . . . . . . . 225 . . . . . . . . . . . . . . . 225 . . . . . . . . . . . . . . . 227 . . . . . . . . . . . . . . . 227 . . . . . . . . . . . . . . . 230 . . . . . . . . . . . . . . . 231

29 Test-case number 31: Reorientation of a Free Liquid Interface in a Partly Filled Right Circular Cylinder upon Gravity Step Reduction (PE) 233 29.1 Practical significance and interest of the test-case . . . . . . . . . . . . . . . 233 29.2 Definitions and model description . . . . . . . . . . . . . . . . . . . . . . . . 235 29.3 Experimental setup and procedure . . . . . . . . . . . . . . . . . . . . . . . 237 29.3.1 Apparatus and test case parameters . . . . . . . . . . . . . . . . . . 237 29.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 29.4.1 Test No. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 29.4.2 Test No. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 29.5 Proposed calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 30 Test-case number 33: Propagation of solitary waves over horizontal beds (PA, PN, PE) 30.1 Practical significance and interest of the test-case . . . 30.2 Definitions and model description . . . . . . . . . . . . 30.3 A series of three test-cases . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

in constant depths 251 . . . . . . . . . . . . 251 . . . . . . . . . . . . 252 . . . . . . . . . . . . 254 . . . . . . . . . . . . 262

31 Test-case number 34: Two-dimensional sloshing in cavity solution (PA) 31.1 Practical significance and interest of the test-case . . . . . . . . 31.2 Definitions and physical model description . . . . . . . . . . . . 31.2.1 Assumptions and model equations . . . . . . . . . . . . 31.2.2 Interface boundary conditions . . . . . . . . . . . . . . . 31.2.3 Exact solutions . . . . . . . . . . . . . . . . . . . . . . . 31.2.4 Interface shape for two longitudinal accelerations . . . . 31.3 Test-case description . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Test-case number 35: Flow rate limitation in open (PE) 32.1 Practical significance and interest of the test case . . . 32.2 Definitions and model description . . . . . . . . . . . . 32.3 The Experimental Test Case . . . . . . . . . . . . . . . 32.3.1 Experimental setup and procedure . . . . . . . 32.3.2 Test case parameter and boundary conditions . 32.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 Acknowledgement . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- an exact 263 . . . . . . . 263 . . . . . . . 263 . . . . . . . 263 . . . . . . . 264 . . . . . . . 265 . . . . . . . 266 . . . . . . . 267 . . . . . . . 270

capillary channels 271 . . . . . . . . . . . . 271 . . . . . . . . . . . . 272 . . . . . . . . . . . . 274 . . . . . . . . . . . . 274 . . . . . . . . . . . . 276 . . . . . . . . . . . . 277 . . . . . . . . . . . . 281 . . . . . . . . . . . . 283

Introduction Why collecting test-cases for interface tracking methods? Several years ago, Hewitt et al. (1986) identified that refined modeling of two-phase flow was a major key to take up some complex industrial challenges associated with nuclear energy production. In particular, they reported that the understanding of the intricate heat transfer and fluid mechanics phenomena that control the critical heat flux or the pressurized thermal shock could not be easily reached within the frame of area-averaged or time-averaged models. In addition, the understanding of probes interaction with two-phase flows or the local conditions controlling wall boiling could benefit from a refined analysis of two-phase flows describing both the motion of each phase and that of the interfaces. Beyond the energy field, the oil industry is also facing new challenges related to two-phase production of oil and gas. Sizing pipelines, separators and predicting the hydrate formation requires a flow description at a scale which is not covered by the existing 1D area-averaged models. From these examples issued from nuclear engineering and the oil industry, there is clearly a definite need to refine the models scale of analysis of two-phase flow with or without phase change. Modeling requires characterizing flow and heat transfer at a scale consistent with the scales described by the models. When refining the scales of observation, instrumentation may become unacceptably intrusive or mere observation may become impossible without hampering the flow features by modifying the boundary conditions. An example of this situation is forced convective boiling. Details of the high pressure water flows close to the walls are probably beyond the reach of existing experimental techniques for years and refined modeling was identified as a possible breakthrough to progress towards an in-depth physical understanding (Delhaye & Garnier, 1991). Another objective advocating the development and the use of local models of twophase flows is the tremendous difficulty to provide appropriate models (closure relations) to local-time-averaged models. Solving at a refined scale and analyzing the results by averaging them at the scale relevant to the time-averaged model is a possible way to suggest appropriate closure relations to these averaged models. However, local modeling of two-phase flow must not be confused with simulation as it is for example understood when solving single-phase flow Navier-Stokes equations. Indeed, wetting phenomena, coalescence physics or heat transfer along a moving contact line cannot be described and must be modeled. As a consequence and as usual when modeling is involved, validation is necessary to gain confidence in the model predictions.

10/283

Introduction

A historical view In 1994, CEA started studying new methods to describe local two-phase flows and heat transfer with phase change. At that time several two-phase CFD modeling methods or computational multiphase flow dynamics models, or CMFD after Yadigaroglu, were developed and the inclusion of phase change into them was still a real challenge. Among all those which were identified, two main paths were followed to account for phase change phenomena and were explored: improving front tracking algorithms to account for the normal velocity discontinuity at the interface and modifying the thermodynamic description of the interface within a single fluid approach. It was readily identified that various physical and numerical problems were to be solved and that the evaluation of the potential solutions required reference situations where both the physics and the numerical techniques were precisely controlled. This was the basic idea of the test-cases. In France and in Europe, several groups were also interested in these problems and during two meetings on January and June 2000 in Grenoble, France, it was realized that the need for test-cases was merely universal in this community and that exchanging, or better sharing, a common set of well tried and tested ”recipes” could benefit to the progress of CMFD development. A first gathering of nearly 20 test-cases were then decided. Originally written in French, it was thought useful to invite European colleagues to contribute and to select English as a common language. Next, collecting these test-cases into a book was the sound and logical followup of the basic idea to provide worldwide developers with this up to now scattered information . Multiphase Science and Technology traditionally fosters this type of activity. Hewitt et al. (1986) edited a selection of reference data sets for validating 1D areaaveraged two-phase flow and later, Hewitt et al. (1991) provided a forum to discuss the merits of various systems codes based on their ability to describe the physical situations relative to the collected data. The editors of this book thank Multiphase Science and Technology for constantly supporting them when suggesting authors to contribute and organize the internal review of the herein proposed test-cases.

Organization of the test-cases collection Test-cases were initially collected rather randomly and as they became more and more numerous, it was deemed important to provide the reader with a minimum quick information on the interest of the test-case and on the targeted part of the CMFD model. This is indicated by capital letters directly following the title of the test-case. Two main categories are proposed: • N: Purely numerical test-case to check for example some discretization methods, • P: Physical test-case to verify a selected physical model or phenomenon controlled by the balance of selected effects. In this latter category, further subdivision was considered: • PN: Physical test-case compared to a reference numerical method (regarded as more accurate) • PA: Physical test-case compared to an analytical solution possibly evaluated numerically

Introduction

11/283

• PE: Physical test-case compared to an experiment • PC: Test of coherence Each test-case is self supporting and focused. The interest and emphasis are developed in a first section. Next, the theory necessary to understand the physical situation and the reference model is shortly explained with a discussion of its validity domain. Necessary references are provided. Next, the details of the test-case are provided i.e. the definition of the computation domain, the boundary conditions and the physical properties. Finally some results in a form that allows an easy handling (analytical formula, arrays of figures) are proposed with a common method for evaluating the errors between the calculated results and the reference. Finally to detect incomplete data sets, a referee has been selected within the group of contributors to play the role of a potential user of the test-case. When the input of this internal referee was deemed significant by the authors they usually included him as the last author of their test-case. D. Jamet, O. Lebaigue, H. Lemonnier CEA/Grenoble, France

References Delhaye, J.-M., & Garnier, J. (eds). 1991. Multiphase Science and Technology. Vol. 11. Chap. Fastnet: a proposal for a ten-year effort in thermal-hydraulic research, pages 79–145. Hewitt, G. F., Delhaye, J. M., & Zuber, N. (eds). 1986. Multiphase Science and Technology. Vol. 3. Hemisphere Publishing Corporation and Springer. Hewitt, G. F., Delhaye, J. M., & Zuber, N. (eds). 1991. Multiphase Science and Technology. Vol. 6. Taylor & Francis Inc.

12/283

Introduction

Test-case number 1 by O. Lebaigue, C. Duquennoy and S. Vincent

13/283

Chapter 1

Test-case number 1: Rise of a spherical cap bubble in a stagnant liquid (PN) By Olivier Lebaigue, DER/SSTH/LMDL, CEA/Grenoble, F-38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 36 70, Fax: +33 (0)4 38 78 50 36, E-Mail: [email protected] Christophe Duquennoy, EDF-SEPTEN, 12-14 av. Dutrievoz, 69628 Villeurbanne cedex, France, E-Mail: [email protected] St´ephane Vincent, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 27 07, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected]

1.1

Practical significance and interest of the test-case

This test-case could usually be considered as a very preliminary one for a new numerical method. An extensive tester may want to reproduce most parts of the Clift, Grace and Weber map (Clift et al. , 1978). However, this selected case deserves special attention for the result not only consists in a final shape of the bubble (that is nevertheless a real criteria of comparison) but also in a precise build-up of the bubble velocity, starting from rest, exhibiting an overshoot before reaching its final asymptotic value. To get the proper results, mainly the correct terminal velocity, and to reproduce the overshoot, a numerical method has to take accurately into account buoyancy, viscous stresses and surface tension effects. In particular, this test allows validating the numerical model that takes care of jump conditions at the interface (see e.g. (Scardovelli & Zaleski, 1999)). However, the test is less severe than the ”Free rise of a liquid inclusion in a stagnant liquid”, a test-case proposed by Lemonnier and Hervieu, presented in this volume.

14/283

1.2

Test-case number 1 by O. Lebaigue, C. Duquennoy and S. Vincent

Definitions and model description

The situation of the test-case is relative to a fluid inclusion rising in another fluid. The inclusion and the surrounding fluid are initially at rest. Gravity induced buoyancy is the only force inducing the motion. The test-case consists in the computation of the transient build-up of the velocity of the rising inclusion that finally reaches a constant value. The physical model is reduced to Navier-Stokes equation in both phases, a constant surface tension at the interface. No phase-change takes place at the interface. As the solution does not depend on a possible compressibility of one or both of the phases, the test-case can be conducted in both cases (compressible / incompressible), depending on specific features of the numerical method to be evaluated. Reference calculations are available in non-dimensional units; however, a typical set of dimensional physical parameter is suggested. The length scale of the problem is the initial diameter de of the inclusion. The velocity scale for speed of displacement of the center of mass U is, p (1.1) Uc = gde , where g is the acceleration of the gravity. The time scale is therefore p tc = de /g.

(1.2)

Reduced parameters are τ = t/tc and u = U/Uc . According to these definitions, the non-dimensional reference calculation consist in the reduced time evolution of the speed of displacement of the center of mass: u = U/Uc = f (τ ) = f (t/tc ).

(1.3)

The computation can be conducted either for an axisymmetrical domain or in a true three-dimensional domain. As the limited extend of the domain has an impact on the terminal velocity of the inclusion (see e.g. Harmathy (1960)), the size of the computational domain has to be increased as long as an effect on the results is noted. As a rough first estimate, we suggest that that the computational domain has a minimal extend equal to ten diameters in all directions. According to the work of Harmathy (1960), the shape of the bubble is not affected by the domain extension whereas the terminal rising velocity modification can be estimated through the semi-empirical relation  2 de Ucconf ined ≈1− , (1.4) ∞ Uc D where D is a characteristic dimension of the domain in a plan perpendicular to the gravity direction. The physical parameters, namely ρL and ρV , respectively the density of the surrounding fluid and the fluid of the inclusion, µL and µV the dynamic viscosities and σ the surface tension, are chosen to get proper values of the non-dimensional quantities for which reference computations are available: the Morton number M o, the Bond number Bo and the ratio of densities ρL /ρV and viscosities µL /µV . The Morton number and the Bond number are defined as usually by, Mo =

g µ4L , ρL σ 3

(1.5)

Test-case number 1 by O. Lebaigue, C. Duquennoy and S. Vincent

15/283

Figure 1.1: Reduced time evolution of the speed of displacement for two mesh sizes. After a figure of Blanco-Alvarez (1995).

and Bo =

1.3

ρL g d2e . σ

(1.6)

Summary of the requested calculations

• Compute the displacement of an inclusion with the following non-dimensional properties M o = 0.056, Bo = 40, ρL /ρV = 100 and µL /µV = 100. • As an example, we suggest the following physical properties: ρL = 1000 kg.m−3 , ρV = 10 kg.m−3 , µL = 0.273556 P a.s−1 , µV = 0.00273556 P a.s−1 , σ = 0.1 N.m, g = 10 kg.s−2 , de = 0.02 m. • Extract the position of the center of mass of the inclusion and then deduce its speed of displacement. The first point of comparison is the value of the reduced asymptotic velocity. This value can be obtained even with a peculiar point of the interface, such as the apex. Of course, in this later case, the temporal evolution around the overshoot (Figure 1.1) cannot be capture. • In addition to the main result, additional features consist in comparisons of the non-dimensional values of the over-shoot in the build-up of velocity (Figure 1.1). • Further comparisons are the shape of lines of current, the equilibrium shape of the inclusion and the size of the recirculation zone (Figure 1.2). This late characteristic requires that the inclusion has risen a length of more than ten diameters (Hnat & Buckmaster, 1976).

16/283

Test-case number 1 by O. Lebaigue, C. Duquennoy and S. Vincent

Figure 1.2: Recirculation zone and steam lines at reduced time τ = 2.86. After a figure of Benkenida (1999).

Test-case number 1 by O. Lebaigue, C. Duquennoy and S. Vincent

17/283

References Benkenida, A. 1999. D´eveloppement et validation d’une m´ethode de simulation d’´ecoulements diphasiques sans reconstruction d’interfaces. Application la dynamique des bulles de Taylor. Ph.D. thesis, Institut National Polytechnique de Toulouse, France. Blanco-Alvarez, A. 1995. Quelques aspects de l’´ecoulement d’un fluide visqueux autour d’une bulle d´eformable : une analyse par simulation directe. Ph.D. thesis, Institut National Polytechnique de Toulouse, France. Clift, R., Grace, J.R., & Weber, M.E. 1978. Bubbles, drops and particles. Academic Press. Harmathy, T.Z. 1960. Velocity of large drops and bubbles in media of infinite or restricted extent. American Institute of Chemical Engineers J., 6, 281–288. Hnat, J.G., & Buckmaster, J.D. 1976. Spherical cap bubbles and skirt formation. Physics of Fluids, 19, 182–194. Scardovelli, R., & Zaleski, S. 1999. Direct Numerical Simulation of free-surface and interfacial flow. Annual Review of Fluid Mechanics, 31, 567–603.

18/283

Test-case number 1 by O. Lebaigue, C. Duquennoy and S. Vincent

Test-case number 2 by H. Lemonnier and E. Hervieu

19/283

Chapter 2

Test-case number 2: Free rise of a liquid inclusion in a stagnant liquid (PN, PE) By Herv´e Lemonnier, DER/SSTH/LIEX, CEA/Grenoble, 38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 45 40, Fax: +33 (0)4 38 78 50 45, E-Mail: [email protected] Eric Hervieu, DER/SSTH/LIEX, CEA/Grenoble, 38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 45 33, Fax: +33 (0)4 38 78 50 45, E-Mail: [email protected]

2.1

Practical significance and interest of the test-case

A numerical solution of a free boundary problem and a related experiment are provided here. The experiment is devoted to the stability of an inclusion rising freely under the sole action of gravity, a situation which is known to depend critically on the initial conditions in a non-linear way. In particular, there exists a critical capillary number beyond which a rising inclusion becomes unstable and this critical value depends non linearly on the shape of the inclusion. When the instability occurs, very high curvatures are experienced by the interface the motion of which results only from the equilibrium between surface tension effects and viscous shear stresses. This situation represents a very severe test of the description of surface tension and viscous stresses at the interface for solvers dealing with interfaces. The challenge consists here in predicting accurately the critical capillary number beyond which an initially distorted inclusion does not recover its equilibrium spherical shape. An analytical description of the initial inclusion shape of the related experiments is provided to ease further comparison of the results. When the motions are very slow such as those occurring in very viscous fluids, the inner and outer flows may be described by the steady restriction of the Stokes equations. As a result of the disappearance of non-linear terms in the equation of motion, there exists a boundary element method (BEM) formulation of the problem which only requires to discretize the common interface of the two fluids (Stone & Leal, 1989, Pozrikidis, 1990,

20/283

Test-case number 2 by H. Lemonnier and E. Hervieu

Hervieu et al. , 1992, Tavares, 1992). When in addition the flows own a cylindrical symmetry further reduction of the mathematical burden results and the model describes very accurately the experiments (Koh & Leal, 1990) at a relatively low computing cost.

2.2

Definitions and model description

The situations to be considered are relative to a liquid inclusion moving in another liquid. The reference calculations to be proposed are made in non-dimensional units. The length scale of the problem is the radius of the sphere the volume of which is equal to that of the initial shape of the inclusion. For a prolate ellipsoid the equivalent radius R is related to the two principal axis of the ellipsoid by, q R = 3 az a2R , (2.1) where az is the length of the axis in the z direction which is the symmetry axis of the cylindrical coordinate system and ar is the length of the axis in the radial direction. For a spherical inclusion rising freely under the action of buoyancy forces, the rise velocity has been calculated exactly by Hadamard and Rybczynski within the Stokes approximation (White, 1991, p. 184). It reads, U=

2 R2 |ρO − ρI |g 1 + λ , 3 µO 3 + 2λ

(2.2)

where ρ is the density, µ is the dynamic viscosity, g is the acceleration of the gravity, the subscripts I and O denote respectively the inner and the outer fluid and λ is the ratio of the outer to the inner fluid viscosity given by, λ=

µO . µI

(2.3)

The Hadamard rise velocity is the velocity scale of the problem (Koh & Leal, 1989). The non-dimensional problem is therefore completely determined by the ratio of the fluids viscosity and a capillary number measuring the effect of viscous shear on the interface which induces the deformation of the inclusion the scale of which is µ0 U/R and the restoring effect of the surface tension which resists to its distortion. The scale of the latter is σ/R (Koh & Leal, 1989). This capillary number is defined by, Ca =

µO U , σ

(2.4)

where σ is the surface tension. It is reminded that the Stokes approximation is valid when the Reynolds number relative to each fluid is very small, Re =

ρU R  1. µ

(2.5)

It must be noted that the time scale of the problem, τ , is therefore, τ=

R . U

(2.6)

During an experiment the volume of the inclusion can be controlled and is measured. Therefore, the equivalent radius R is given. The initial shape can be measured or controlled. If a prolate ellipsoid is considered as an initial shape, then the ratio of its two axis, az /ar is given. Finally the physical properties of each fluid and the interface are known, i. e. ρO , ρI , µO , µI and σ.

Test-case number 2 by H. Lemonnier and E. Hervieu

2.3

21/283

A series of six numerical test-cases

It is proposed to calculate the evolution of the shape of an ellipsoid initially at rest rising in another liquid of infinite extent according to the parameters of table 2.1. It is required to compare the shapes at the non-dimensional times shown in the caption of the figures 2.1 and 2.2 selected from the simulations presented by Koh & Leal (1990). The shapes must be plotted every 2 non-dimensional time units and the simulation stops when the spherical shape is recovered or when the instability is fully developed. Test-case 2.1 2.2 2.3 2.4 2.5 2.6

az /aR 2 2 2 0.5 0.5 0.5

Ca 1.25 1.5 2 3.5 4 10

λ 0.5 0.5 0.5 0.5 0.5 0.5

Table 2.1: Values of the non-dimensional parameters describing the simulations by Koh & Leal (1990) presented in figures 2.1 and 2.2. The definitions of the non-dimensional parameters are given in section 2.2.

It is also required to compare the non-dimensional time when the instability is fully developed, i. e. the time when some of the inner fluid is entirely encapsulated in the inclusion (see figure 2.2) or when a fragment detaches from the main body (see figure 2.1). These times are indicated in the figures captions. The reference solution has been calculated by utilizing a boundary elements method derived for the Stokes equation. It is to be noted that these results have been reproduced by Tavares (1992) and Hervieu et al. (1992) who reported only insignificant differences at the latest times with the original calculations published by Koh & Leal (1990). The proposed simulations can therefore be regarded as exact solutions to the problem. The precision of the simulation can also be appreciated by comparing the free rise of a spherical drop to the theoretical value given by Hadamard (2.2). Koh & Leal (1989) reported that in their calculations, the velocity of all the points on the interface was within ±0.05% of this theoretical value. The mean value and standard deviation on the interface values of the velocity are acceptable criteria.

Fluid outer inner #1 inner #2 inner #3 inner #4

viscosity (Pa s) 39.1 1.02 10.1 29.1 102

density (kg/m3 ) 1021 972 972 972 975

superficial tension (N/m) 5.8 6.0 5.8 5.8

10−3 10−3 10−3 10−3

Table 2.2: Values of the physical properties in S. I. units of the experiments by Koh & Leal (1989). The superficial tension is to be considered between the outer fluid and each inner fluid.

It is acknowledged that most of the simulation tools do not utilize non-dimensional units. As a result, some preprocessing of the data may be necessary to find a physical

22/283

Test-case number 2 by H. Lemonnier and E. Hervieu

Figure 2.1: Numerical simulation of the evolution of an initially elongated (prolate) ellipsoid. Cases 2.1 to 2.3 of table 2.1. After the figure 1 of Koh & Leal (1989).

Figure 2.2: Numerical simulation of the evolution of an initially blunt (oblate) ellipsoid. Cases 2.4 to 2.6 of table 2.1. After the figure 5 of Koh & Leal (1989).

Test-case number 2 by H. Lemonnier and E. Hervieu

23/283

situation corresponding to the non-dimensional test-cases of the figures 2.1 and 2.2. Some physical properties extracted from the experimental work of Koh & Leal (1990) may help in identifying these situations. It must be noted that with these values and a drop with an initial volume ranging from 1 to 10 cm3 the Reynolds number is always less than 10−2 and the capillary number ranges from 1 to 5.5. Summary of the required calculations • The mean and standard deviation of the velocity at the interface of a spherical inclusion rising in Stokes regime for λ = 10, 1 and 0.1. • The shapes of inclusions according to table 2.1 every 2 non-dimensional time units. • The time at which the instability is fully developed for the cases of table 2.1.

2.4

An experimental test-case

Zagustin (1992) made a series of experiments in conditions close to those of Koh & Leal (1990). Among them, prolate cases have been selected since they do not cause any interpretation difficulties. Oblate cases ends up with some inner fluid encapsulation where refraction distorts the inner fluid boundary. The prolate inclusions are not prone to this visualization artifact. It is required to calculate the evolution of the shape of the inclusion as described in figure 2.3 at time t=0, at the later times t=2.9, 5.4, 8.5, 12.3 and 14.8 s. The physical properties relative to this experiment are given in table 2.3. The experiments where conducted in silicone oil (1000 cS) the viscosity of which is comparable with the inner fluid number 1 of Koh & Leal (1990). The inner fluid was a blend of castor oil and 3% in volume of methanol. The initial shape is given in table 2.4 and an analytical approximation is proposed in the appendix for the sake of simplicity.

Fluid outer inner

viscosity (Pa s) 0.988 0.593

density (kg/m3 ) 970 953

superficial tension (N/m) 3.8 10−3

Table 2.3: Values of the physical properties in S. I. units of the experiments by Zagustin (1992). The superficial tension is to be considered between the outer fluid and the inner fluid.

It must be noted there exist some sources of uncertainty in the data. The physical properties including the surface tension have been measured in the laboratory by using conventional techniques (Zagustin, 1992). The most uncertain quantity is the density difference between the two fluids. Moreover, the initial velocity of the inclusion is unknown and it may be assumed that it is at rest which is not a too unrealistic assumption owing to the experimental procedure utilized to create elongated inclusions. Summary of the requested calculations • The shapes of the inclusion at the times shown in figure 2.3.

24/283

Test-case number 2 by H. Lemonnier and E. Hervieu

Figure 2.3: Experiment on the free rise of a liquid inclusion rising in another liquid. Physical properties listed in table 2.3. λ = 0.6, Ca = 1.4 ± 0.2 and ∆ρ = 17 kg/m3 . After Zagustin (1992)

Test-case number 2 by H. Lemonnier and E. Hervieu

Z (mm) 20.09 19.31 18.32 17.33 16.35 15.36 14.38 13.39 12.41 11.42 10.43 9.45 8.46 7.48

R (mm) 0.00 3.75 5.47 6.61 7.53 8.16 8.66 9.08 9.42 9.71 9.87 9.92 9.94 9.97

Z (mm) 6.49 5.39 4.52 3.54 2.55 1.56 0.58 -0.41 -1.39 -2.62 -3.52 -4.35 -5.33 -6.26

R (mm) 9.97 9.91 9.74 9.54 9.31 9.08 8.84 8.60 8.39 8.06 7.79 7.57 7.32 7.01

25/283

Z (mm) -7.31 -8.29 -9.37 -10.26 -11.25 -12.23 -13.22 -14.20 -15.20 -16.15 -17.74 -18.69 -20.07 -20.90

R (mm) 6.65 6.33 5.97 5.73 5.40 5.08 4.75 4.32 3.98 3.59 2.98 2.49 1.55 0.00

Table 2.4: Coordinates of the initial shape of figure 2.3. Lengths are in mm.

Appendix: an analytical approximation of the initial shape The data shown in table 2.4 describing the initial shape of the inclusion shown in figure 2.3 may be uneasy to handle. Hervieu et al. (1992) used cubic spline functions fitted on nodes sampled from the snapshots of the experiments. The spline functions interpolates respectively the axial and radial coordinate as a function of the curvilinear abscissa along the shape. This method provides the best input for the boundary element method. It has been thought that this procedure may deemed rather cumbersome and unnecessary for most of the solvers. This is the reason why an analytical description of the initial shape is proposed. The points of table 2.4 have been transformed by an affine transformation of ratio 2 in the r direction according to Mathieu (2002). This provides a rather spherical object which can be easily described by trigonometric expansion in polar coordinates according to, r0 (θ) =

k=8 X

ak cos(kθ)

(2.7)

k=0

where the ak ’s are given in table 2.5. This fit has been performed by the nonlinear least-squares Marquardt-Levenberg algorithm programmed in the gnuplot fit function. The original coordinates are recovered from the trigonometric expansion (2.7) by the following,  z = r0 (θ) cos(θ) (2.8) r = 12 r0 (θ) sin(θ) where the angle θ ranges from 0 to 2π. A comparison of the data and the approximation is shown in figure 2.4. A reasonable agreement is noted.

26/283

Test-case number 2 by H. Lemonnier and E. Hervieu

order 0 1 2 3 4 5 6 7 8

value (mm) 18.9478 2.3417 1.0630 -2.4794 0.0268 -0.1311 0.4327 -0.0661 -0.0117

Table 2.5: List of the ak ’s in equation (2.7). Unit is mm.

radial coordinate in mm (r)

Curve fit to the initial position 14

Analytical approximation Data points

12 10 8 6 4 2 0 −25

−20

−15

−10

−5 0 5 10 Axial coordinate in mm (z)

15

20

25

Figure 2.4: Comparison between the data describing the initial shape in table 2.4 and the analytic approximation (2.7) and (2.8).

Test-case number 2 by H. Lemonnier and E. Hervieu

27/283

References Hervieu, E., Coutris, N., & Tavares, M. 1992. Deformation of a fluid particle falling freely in an infinite medium. Pages 201–207 of: ANS Proc. of 28th National Heat Transfer Conference. San Diego, CA. Koh, C. J., & Leal, L. G. 1989. The stability of drop shapes for translation at zero Reynolds number through a quiescent liquid. Physics of Fluids A, 1(8), 1309–1313. Koh, C. J., & Leal, L. G. 1990. An experimental investigation on the stability of viscous drops translating through a quiescent liquid. Physics of Fluids A, 2(12), 2103–2109. Mathieu, B. 2002. Fit of an axisymmetrical body shape by trigonometric functions. Private communication. Pozrikidis, C. 1990. The instability of a moving viscous drop. J. Fluid Mechanics, 210, 1–21. Stone, H. A., & Leal, L. G. 1989. Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid. J. Fluid Mechanics, 198, 399–427. Tavares, M. 1992. Simulation de la d´eformation d’un domaine fluide en mouvement dans un autre fluide par la m´ethode des ´el´ements de fronti`ere. Ph.D. thesis, Institut National Polytechnique de Grenoble, France. White, F. M. 1991. Viscous fluid flow. McGraw Hill. Zagustin, K. 1992. An experimental investigation on the deformation of a fluid particle translating in an infinite viscous medium. Internal note, DTP/SETEX, CEA/Grenoble, France.

28/283

Test-case number 2 by H. Lemonnier and E. Hervieu

Test-case number 3 by P. Lubin and E. Canot

29/283

Chapter 3

Test-case number 3: Propagation of pure capillary standing waves (PA) By Pierre Lubin, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 33 07, Fax: +33 (0)5 40 00 66 68 E-Mail: [email protected] Edouard Canot, IRISA, Campus de Beaulieu, 35042 Rennes cedex, France Phone: +33 (0)2 99 84 74 89, Fax: +33 (0)2 99 84 71 71, E-Mail: [email protected]

3.1

Practical significance and interest of the test-case

Analytical solutions are provided here, developed for standing small-amplitude water waves. It provides a basis for applications to a series of numerical experiments. The interest consists here in predicting accurately the evolution of the interface of capillary waves in order to evaluate the coupling between inertial and viscous effects and estimating the effect of the numerical viscosity. When simulating two-phase flows, it is important to evaluate the general accuracy and the validity of the numerical methods and numerical schemes used and the conservation laws of mass and energy in the computing domain. In particular, it is important to check that the behavior of the interface between two media is well taken into account, considering surface tension and viscous effects. As a matter of fact, capillary waves are similar to gravity waves but, firstly, they involve smaller scales, both in length and time. Secondly, They require a more difficult computation, because surface tension forces are based on the interface curvature, which needs to be accurately described. Thus, the results provided for pure capillary waves are considered, as initial conditions to simulate their propagations in constant depths over horizontal beds. The precision of the simulation is checked by comparing the free-surface shapes to theoretical values, including the predicted decay rate

30/283

Test-case number 3 by P. Lubin and E. Canot

due to viscous effects.

3.2

Definitions and model description

The important parameters to describe waves are their length and height, and the water depth d over which they are propagating. The length of the wave, L, is the horizontal distance between two successive wave crests or two successive wave troughs. H is the height between the trough and the crest of the wave. The wave period, T , is the time required for two successive crests or troughs to pass a particular point. The speed of the wave, called the celerity c, is then defined as c = L/T . The water surface elevation η is the distance between the water surface and the mean water depth h. Let us consider a standing small-amplitude wave with water surface displacement given by:

η(x, t) =

H cos(kx) cos(ωt), 2

(3.1)

with ω = 2π/T being the angular frequency of the wave, calculated from the dispersion relationship, ω 2 = gk tanh(kd), and k = 2π/L being the wave number. At t = 0, the water wave has a cosine shape, as shown in figure 3.1. This is known as the linear wave theory, developed under the following assumptions. The fluid is supposed to be homogeneous and incompressible (density is constant), ideal or inviscid (lacks viscosity), the wave form is invariant in time and space (except its amplitude), the waves are two-dimensional and the sea bed is an horizontal, fixed, impermeable boundary which implies that the vertical velocity at the sea bed is zero. The restriction to small-amplitude implies that the ratio of the maximum elevation to the wavelength H/L 1/2 or kd > π/2), the trajectories are circles decaying exponentially with depth. According to the theoretical prediction for small-amplitude capillary waves (Lamb, 1932)[sec. 266], a generalized analytical value of the frequency ωth , for finite depth, is given by: 2 ωth =

σk 3 tanh(kd), ρl + ρg

(3.2)

with σ being the constant surface tension, ρl and ρg being the densities. Moreover, in the case where ν = νl = νg is the kinematic viscosity of both fluids, an analytical solution has developed by Prosperetti (1981) to calculate the evolution of the amplitude of a capillary wave. This solution takes into account the effects of the viscosity and the initial condition. In addition to the analytical value of ωth (3.2), a dimensionless viscosity  is defined:

=

νk 2 ωth

(3.3)

Prosperetti (1981) gives the solution for the shape of the interface: η(x, t) = a(t), η(x, 0)

(3.4)

with a(t) being the amplitude of the considered capillary wave. This amplitude is expressed as: √ a(τ ) 4(1 − 4β)2 = erfc( τ ) a0 8(1 − 4β)2 + 1 4   r τ  X zi ωth τ  2 + exp (z − ω ) erfc zi , i th Zi zi 2 − ωth ωth ωth

(3.5)

i=1

with τ = ωth t, and erfc being the complementary error function. zi are solutions of the following equation: 1

z 4 − 4β(ωth ) 2 (z 3 + 2(1 − 6β)(ωth )z 2 (3.6) 3 2

2

2

+4(1 − 3β)(ωth ) z + (1 − 4β)(ωth ) + ωth = 0, The coefficient Z1 is given by Z1 = (z2 − z1 )(z3 − z1 )(z4 − z1 ), and the other coefficients Z2 , Z3 and Z4 are obtained by circular permutation of the subscripts. The parameter β is defined as: β=

ρl ρg , (ρl + ρg )2

(3.7)

In the case where νl and νg are being chosen with different values, the analytical solution (3.5) is no longer valid.

32/283

3.3

Test-case number 3 by P. Lubin and E. Canot

A series of test-cases

It is proposed to evaluate the numerical diffusion by simulating pure capillary waves (g = 0) propagating on the interface between two viscous fluids in a two-dimensional domain of length equal to the wavelength L, and to compare the numerical results with the analytical solutions developed previously. The proposed numerical configuration is to consider an initial wave computed from the theory detailed before. The crest is located on both sides of the numerical domain (x = 0 and x = L, as shown in figure 3.1, symmetry boundary conditions being imposed on the lateral boundaries. Thus, at the instant t = 0, for 0 < x < L, we have (Lamb, 1932)[sec. 250]: η(x, 0) = a0 cos(kx), with a0 being the amplitude of the wave a0 = being at rest.

H 2.

(3.8)

There is no initial velocity, the fluid

Non-viscous case In the case of two fluids which viscosities are negligible (ν = νl = νg = 0), capillary waves should not be damped and should oscillate with a constant frequency (3.2). It is so proposed to evaluate the variation of the ratio ωnum over ωth as a function of the mesh size. The computation should then converge to this value of the frequency. However, the limit values obtained numerically will not be exactly equal to the theoretical ones: the effect of a numerical diffusion will then be highlighted. As we are in the case where ν = νl = νg , the amplitude of the oscillations a(t) should also be plotted as a function of time and should be compared with the analytical solution given in (3.5). Duquennoy (2000) proposed the following parameters: • d/L = 0.5, H/L = 1, a/L = 2.7 .10−2 ; • ρl /ρg = 1, νl /νg = 1, with kνg /ωth a(t) 0) g : gravity (g > 0) σ : surface tension between the two liquids (σ ≥ 0) By choosing the following scales, we turn the problem into a nondimensional form : • length scale : L (half length of the box in the 2D case, or radius of the box in the 3D case) • pressure scale : (ρu − ρl )gL √ • velocity scale : U = 2A gL, where A is the Atwood number defined below (note that this choice has been found by requiring the forcing term of the problem, i.e. the gravity, to be of the order of inertia : (ρu − ρl )gL = 21 (ρu + ρl )U 2 ) The non dimensional parameters are then : • Atwood number : A =

ρu − ρl (which appears only in the time scale) ρu + ρl

• E¨otv¨ os number : Eo =

(ρu − ρl )gL2 (which is sometimes also called Bond number) σ

Hereafter, all variables are under the dimensionless form.

4.3

Test-case description

The following cases are studied : (4.1) 2D plane case – rectangular box : −1 < x < 1, − H L m > 0, (5.3) 1  e  Ymn (θ, ϕ) rn+1 o , and Y e are respectively the odd and even surface spherical harmonics of first where Ymn mn kind as defined by Morse & Feshbach (1953, p. 1264, eq. 10.3.25) and θ is the angle with the z axis. These harmonics are related to the Legendre functions, Pnm , by the following relations, o Ymn = sin(mϕ)Pnm (cos θ),

(5.4)

e Ymn

(5.5)

=

cos(mϕ)Pnm (cos θ).

The spherical harmonics (5.4) and (5.5) are also the eigenmodes of deformation of the inclusion. The interesting result is that their frequency of oscillation does not depend on the azimuthal mode number m but only on n. This test-case can therefore be used for

Test-case number 5 by H. Lemonnier and O. Lebaigue

43/283

axisymmetric cases and really 3D flows as well. In particular, the axisymmetric modes are obtained for m = 0 where the Legendre functions becomes the Legendre polynomials denoted by Pn . The angular frequency of the oscillations is obtained for each mode by using the linearized momentum jump at the interface; it is given by, ωn2 =

n(n + 1)(n − 1)(n + 2) σ (n + 1)ρI + nρO R3

(5.6)

where ρI and ρO are respectively the inner and outer fluid density. It is reminded that the period, Tn , and the frequency, fn , are related to the angular frequency, ωn , by: 2π , ωn ωn fn = . 2π

Tn =

(5.7) (5.8)

Some additional intermediate results are of interest to set the size of the calculation domain and appreciate the necessary smallness of the oscillations. Let us consider a harmonic oscillation of the inclusion along one of its eigenmodes, r = R + Ymn sin(ωt + u),

(5.9)

where  is the amplitude of the oscillation, u is an arbitrary phase and Ymn is one of the spherical harmonics (5.4) or (5.5). The analysis shows that the corresponding potentials have the following form, ωR rn Ymn n cos(ωt + u), n R ωR Rn+1 φO = Ymn n+1 cos(ωt + u). n+1 r φI = −

(5.10) (5.11)

The corresponding radial velocities are given by, rn−1 cos(ωt + u), Rn Rn+1 vO = ωRYmn n+2 cos(ωt + u), r

vI = −ωRYmn

(5.12) (5.13)

whereas the corresponding pressure fields are given by, ρI ω 2 R rn Ymn n sin(ωt + u), n R ρO ω 2 R Rn+1 pO = − Ymn n+1 sin(ωt + u). n+1 r pI =

(5.14) (5.15)

Since the fluids are assumed to be incompressible, there is no point in comparing a pressure fluctuation to an absolute value of the pressure. However, a calculation domain sizing argument could be that the pressure fluctuation at a distance a of the inclusion center is much smaller than its maximum value reached near the interface: pO (r = a) Rn+1 = n+1  1. pO (r = R) a

(5.16)

44/283

Test-case number 5 by H. Lemonnier and O. Lebaigue

This ratio represents the order of magnitude of the error on the pressure induced by a calculation domain of size a at the boundary of which the pressure is imposed uniform. For the second order deformation, n = 2, which represents an ellipsoidal-spherical oscillation, the error decreases as the cube of the domain size. If the calculation domain is limited by the available computing capacity, it is recommended to select a mode compatible with the size of the domain according to (5.16). On the other hand, if the domain is assumed impermeable the same reasoning with the velocity leads to the following condition: vO (r = a) Rn+2 = n+2  1. vO (r = R) a

(5.17)

Therefore, it seems more favorable to work with a zero velocity than a uniform pressure at the outer boundary. The linearization of the problem introduces some limitations on the amplitude of the oscillations. The first one comes from the transfer of the boundary condition on the undistorted shape of the inclusion and the second from discarding the kinetic term in the Bernoulli equation used to express the momentum jump at the interface. Intuitively, the second source of error should be the most important. It requires that the corresponding terms in (5.12), (5.13) and (5.14), (5.15) are such that, 1 ρk vk2  pk . 2

(5.18)

1 (n + m)! Pnl (z)Pnm (z) dz = δlm (n + )−1 , 2 (n − m)! −1

(5.19)

Using the following identity, Z

1

where δlm is the Kronecker symbol, the following condition on the oscillation amplitude must be satisfied, 1 ρk vk2 n+1 . 2 pk 4



1 n+ 2

− 1 2

 . R

(5.20)

It is recommended to select a value of  which corresponds to an error in the pressure of the same magnitude than that induced by the finite size of the calculation domain. For example, if one considers the dynamical pressure perturbation induced by the interface motion to be small with respect to the static pressure perturbation when it is 100 times smaller for the axisymmetric mode n = 2, then (5.20) shows that  should be smaller than 1/30 of the initial radius R ( 21 ρk vk2 /pk < 1/100 is achieved when /R < 1/30 according to equation 5.20) .

5.3

Numerical settings, initial and boundary conditions

It is recommended to start the calculation with a distorted spherical shape by adding several perturbations of the kind of (5.9) with m = 0 for the axisymmetric case. It is further recommended to consider a similar amplitude for all the selected modes. In this latter case, the Legendre functions values at selected values of the parameter z = cos θ

Test-case number 5 by H. Lemonnier and O. Lebaigue

45/283

are easy to calculate by using the following recurrence formulas (Abramovitch & Stegun, 1965, eq. 8.5.1)
−1 m (n − m)zPnm (z) − (n + m)Pn−1 (z) , 1 − z2 m m (n − m + 1)Pn+1 (z) = (2n + 1)zPnm (z) − (n + m)Pn−1 (z), Pnm+1 (z) = √

(5.21) (5.22)

where the following initial terms are given by, P00 = 1 P10

= cos θ,

P11

(5.23) = sin θ

3 3 1 P20 = (3 cos 2θ + 1), P21 = sin 2θ, P22 = (1 − cos 2θ) 4 2 2

(5.24) (5.25)

The test-case does not specify the shape nor the dimensions of the outer boundary of the domain. The error estimates (5.16), (5.20) or (5.17), (5.20) must be used to justify the considered choices.

5.4

Requested calculations

It is proposed to calculate the time evolution of the shape of an initially non spherical inclusion. A possible postprocessing method could be to Fourier-analyze the position, z(t), of the top intersection of the interface with the z axis and to consider the relative deviation of the eigenmodes frequency, ∆fn /fn , from (5.6) and (5.8). Another suggestion consists in determining the instantaneous amplitude of the eigenmodes which can be calculated by projecting the instantaneous shape, re(θ, t), or r(z, t) on the eigenmodes according to, Z π  1 ηn (t) = n + re(θ, t)Pn (cos θ) sin θ dθ (5.26) 2 0 where the orthogonality of the Legendre functions (5.19) has been utilized. If the calculation is correct, the amplitude time trace, ηn (t), should be a perfect cosine curve with an amplitude, n , and phase, u, consistent with the mode selected (5.9) and a frequency equal to that given by (5.6) and (5.8). The time evolution of the eigenmodes amplitude must be provided to assess the diffusive nature of the method. In addition to this results analysis the following is proposed. Considering that the volume of the inclusion should remain strictly constant, it is requested to compute its time evolution.

5.5

An illustrative example

This section provides an illustration of a possible initial shape to start with and shows the time trace of the north pole displacement which is expected. The first significant eigenmode for this test case is n = 2. Indeed, it is easy to show that the eigenmodes n = 0, 1 are, in the frame of our approximations, two equilibrium states. The n = 0 mode does not preserve the volume of the inclusion and it is therefore irrelevant; whereas the the mode n = 1 does it though the frequency relation (5.6) shows clearly that it is another equilibrium state. The figure 5.1 shows the two modes n = 2 and n = 3.

Test-case number 5 by H. Lemonnier and O. Lebaigue

1.5

1.5

1

1

0.5

0.5 z−axis

z−axis

46/283

0

0

−0.5

−0.5

−1

−1

−1.5 −1.5

−1

−0.5

0 r−axis

0.5

1

1.5

−1.5 −1.5

−1

(a) n = 2.

−0.5

0 r−axis

0.5

1

1.5

(b) n = 3.

Figure 5.1: Plot of the two eigenmodes n = 2 and n = 3. The length unit is the radius, R, and the relative amplitude of each mode is set to /R = 0.1. The equation of the surface is r = 1 + 0.1Pn (cos θ) where P2 = 1/4(3 cos 2θ + 1) and P3 = 1/8(5 cos 3θ + 3 cos θ).

The figure 5.2(a) shows the superposition of the two modes of figure 5.1; whereas the figure 5.2(b) shows the expected evolution of the inclusion north pole position. The initial shape is given in non-dimensional form by,   r 1 1 = 1 + 0.1 (3 cos 2θ + 1) + (5 cos 3θ + 3 cos θ) . (5.27) R 4 8 The position of the north pole of the inclusion is given by, z(t) = 1 + 0.1(cos 2πτ + cos 2Ωπτ ), R

(5.28)

where τ = t/T is a non dimensional time, and T the period of oscillation of the second eigenmode of an empty bubble (ρI = 0) surrounded by a liquid of density ρO . The period of oscillation of mode 2 is given by, T =

2π σ , ω22 = 12 , ω2 ρO R 3

(5.29)

and Ω is the ratio of the two considered eigenmodes angular frequency obtained by, r ω3 10 Ω= = . (5.30) ω2 3 As shown by figure 5.2(b), the time evolution of the position of the north pole is not periodic as suggested by the non rational value of Ω given by (5.30).

5.6

Additional information for 2D calculations.

It is of interest to mention that analogous developments are available for purely 2D cases (Toutant, 2003). The analogous of equation (5.6) in two dimensions is the following, ωn2 =

n(n + 1)(n − 1) σ (ρI + ρO ) R3

(5.31)

Test-case number 5 by H. Lemonnier and O. Lebaigue

1.5

47/283

1.25 1.2

1 Position of the north pole

1.15

z−axis

0.5

0

−0.5

1.1 1.05 1 0.95 0.9

−1 0.85 −1.5 −1.5

0.8 −1

−0.5

0 r−axis

0.5

1

(a) Superposition of modes 2 and 3.

1.5

0

1

2 3 Non−dimensional time

4

5

(b) Time trace of the north pole position.

Figure 5.2: Initial shape of the inclusion obtained by adding the two eigenmodes of figure 5.1. Time trace of the position z(t) of the inclusion north pole. This configuration is relative to a bubble (ρI = 0). The length scale is the bubble radius R and the time scale is the period of oscillation of the mode n = 2.

References Abramovitch, M., & Stegun, I. A. 1965. Handbook of mathematical functions. Dover. Lamb, H. 1975. Hydrodynamics. sixth edn. Cambridge University Press. Morse, P. M., & Feshbach, H. 1953. Methods of theoretical physics. McGraw-Hill. Toutant, A. 2003. Interaction interfaces et turbulence : application une colonne ` a bulles isotherme. Rapport de stage fin d’´etudes, ENSTA-DTP/SMTH/LDTA, CEA/Grenoble.

48/283

Test-case number 5 by H. Lemonnier and O. Lebaigue

Test-case number 6 by O. Lebaigue, C. Duquennoy, D. Jamet and F. Feuillebois

49/283

Chapter 6

Test-case number 6: Two-dimensional droplet pining on an inclined wall (PC) By Olivier Lebaigue, DER/SSTH/LMDL, CEA/Grenoble, F-38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 36 70, Fax: +33 (0)4 38 78 50 36, E-Mail: [email protected] Christophe Duquennoy, EDF-SEPTEN, 12-14 av. Dutrievoz, 69628 Villeurbanne cedex, France, E-Mail: [email protected] Didier Jamet, DER/SSTH/LMDL, CEA/Grenoble, 38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 45 42, Fax: +33 (0)4 38 78 50 36, E-Mail: [email protected] F. Feuillebois, PMMH, UMR 7636 CNRS, ESPCI, 10 rue Vauquelin 75231 Paris cedex 05, France Phone: +33 (0)1 40 79 45 53, Fax: +33 (0)1 40 79 47 95 E-Mail: [email protected]

6.1

Practical significance and interest of the test-case

In many experimental situations, see e.g. (Dussan, 1976) or (Carey, 1992), the actual value of the static contact angle is a consequence of the past motion of the contact line, before its pining: this phenomenon is called the hysteresis of the contact angle, for the value is usually higher for a (previously) advancing contact line than for a (previously) receding contact line. The amplitude of the hysteresis, that is the difference between the advancing and the receding contact angles, is mainly due to effects that arise at scales smaller than the micrometer, especially effects of wall roughness and effects of non-homogeneous chemical composition of the solid surface or contamination. These small-scale physicochemical phenomena only affect the macroscopic scales through the apparent contact angle and its hysteresis. Taking into account the effects of the unresolved physics with a macroscopic model is a classical approach in Direct Numerical Simulation

50/283

Test-case number 6 by O. Lebaigue, C. Duquennoy, D. Jamet and F. Feuillebois

techniques. This test-case is only qualitative and is primarily dedicated to test the ability of a numerical method to simulate contact line effects and the contact line hysteresis in particular. The original test-case can be found in (Duquennoy, 2000) and (Duquennoy et al. , 2000). The main goal of the test-case is to qualify and illustrate the ability of a numerical method to cope with the contact line hysteresis and to take into account the values of the advancing and receding contact angles. The main idea of the test-case is to check the onset of sliding of a droplet on an inclined surface. In a two-dimensional simulation, balance between gravity (droplet weight) and surface tension (contact angles) effects results in a limiting value for the angle of inclination of the surface on which the droplet is standing. This simplicity is apparent, because the onset conditions are usually not reached on both contacts for the same value of the inclination (Feuillebois, 2000). However, making the assumption of a simultaneous onset on both contacts provides a reasonable estimate (Duquennoy, 2000). Moreover, the double procedure proposed to determine the limiting angle, and explained in the last section of this test-case, gives an estimate of the validity of this assumption. This test-case is therefore not an accurate validation of the implementation of the contact line effects, but is nevertheless easy to perform and can provide a decent qualitative qualification.

6.2

Description of the model for the contact angle hysteresis and definition of the test-case

The simplest way to describe a macroscopic contact angle which undergoes a hysteretical behavior is   θapp = θA if V > 0 V =0 while θR < θapp < θA (6.1)  θapp = θR if V < 0 where θapp is the macroscopic contact angle, θA and θR are the advancing and receding contact angles respectively and V is the speed of displacement of the contact line. The test-case consists in computing the conditions for which of a two-dimensional droplet sticks or slides on an inclined wall. Let us denote α the angle between the horizontal direction and the inclined wall. For a given hysteresis, that is for a given set (θA , θR ), the inclination has a limit value αlim so that the droplet may stay at rest on the wall. A balance between gravity and surface tension forces characterizes this equilibrium. Since we consider the two-dimensional case, the marginal equilibrium is approximately given by   theory sin αlim =−

σ (cos (θA ) − cos (θR )) , (ρL − ρG ) gVdroplet

(6.2)

where Vdroplet is the volume (surface) of the droplet, ρL and ρG are respectively the density of the droplet liquid and the density of the surrounding gas, σ is the surface tension between the gas and the liquid and g is the gravity intensity. The physical fluid model is reduced to the Navier-Stokes equations in both phases, with a constant surface tension at the interface. The simplest choice for the other physical properties is constant

Test-case number 6 by O. Lebaigue, C. Duquennoy, D. Jamet and F. Feuillebois

51/283

densities ρL and ρG , constant dynamic viscosities µL and µG . Another basic assumption states that no phase-change takes place at the interface. Since the solution does not depend on a possible compressibility of one or both of the phases, the test-case can be conducted in both cases (compressible / incompressible), depending on specific features of the numerical method to be tested. A further simplification is even possible (if needed) by neglecting the gas phase (e.g. ρG = 0 and µG = 0). A simple initial shape is a half circle. Both fluids may also be initialized at rest. The test may be conducted with any convenient set of physical parameters, because equation (6.2) provides a good estimate in most situations. However, a typical set is suggested, which approximately corresponds to the values for air and water at room temperature: ρL = 958 kg.m−3 , µL = 282 10−5 Pa.s, ρG = 0.59 kg.m−3 , µG = 12.3 10−5 Pa.s, σ = 59 10−3 N.m−1 , g = 9.81 m.s−2 . In addition to those values, the hysteresis defined in (Duquennoy, 2000) is: θA ≈ 85.94˚ (1.5 rad), θR ≈ 22.92˚ (0.4 rad). The last parameter is the radius of the initial half circle de and therefore the value of the surface Vdroplet . A value of de = 2 mm (Vdroplet ≈ 6.28 mm2 ) is suggested. For this set of values, theory an approximate solution for the limit angle of equilibrium is αlim ≈ 58.2˚ (≈ 1.01 rad).

6.3

Test procedure

The aim of the test is to find an estimate of the limit angle αlimit . The test has to be conducted in two steps: • In a first step of the procedure, the equilibrium shape of a droplet is found for an inclination α smaller than θR . A transient step-by-step increase of this angle allows lower . For all the values of the to determine a lower bound for the limit angle αlim : αlim lower test angle α lower than αlim , the droplet is able to reach an equilibrium state, the shape of the interface depending on the value of α. Stepping should be performed carefully, especially in the case of low viscosities and when approaching the limit lower . The result of a rough stepping is to under-estimate the value of αlower . value αlim lim Additional attention should be paid to the mesh size used for the computation: When the mesh is too coarse, the result is usually erratic for a slight change in the value of the angle α. upper • In the second step of the procedure, the upper limit for αlim is searched: αlim . upper Computations are performed with values of α decreasing down to αlim . This limit lower , because it is not a succession of quasiis more difficult to determine than αlim steady states. However, its allows to get an estimate of the uncertainty that subsists on the determination of αlim .

6.4

Comparison criteria

lower and The reduced difference 1 between the numerically estimated limit for the angle αlim theory its theoretical value αlim given by equation (6.2), is a good measure of the quality of the numerical implementation of the hysteresis:

1 =

lower theory αlimit − αlimit (θA − θR )

.

(6.3)

52/283

Test-case number 6 by O. Lebaigue, C. Duquennoy, D. Jamet and F. Feuillebois

The reduced difference 2 between the numerically estimated upper and lower limits of the angle αlim is a good measure of the quality of the assumption of a simultaneous onset of both contact lines: 2 =

upper lower − αlimit αlimit . (θA − θR )

(6.4)

According to (Duquennoy, 2000), 1 = 1% and 2 = 5% is a very good result.

References Carey, V.P. 1992. Liquid-vapour phase change phenomena. Hemisphere Publishing Corporation. Duquennoy, C. 2000. D´eveloppement d’une approche de simulation num´erique directe de l’´ebullition en paroi. Ph.D. thesis, Institut National Polytechnique de Toulouse, France. Duquennoy, C., Lebaigue, O., & Magnaudet, J. 2000. A numerical model of gas-liquidsolid contact line. In: Fluid Mechanics and its Applications, vol. 62. Kluwer Academic Publishers, A.C. King and Y.D. Shikhmurzaev Eds. IUTAM Symposium of Free Surface Flows, Birmingham, UK, 10-14 July 2000. Dussan, V.E.B. 1976. The moving contact line: the slip boundary condition. J. Fluid Mech., 77(4), 665–684. Feuillebois, F. 2000. Accrochage d’une goutte bidimensionnelle sur un plan inclin´e - Solution analytique approch´ee. Private communication.

Test-case number 7a by D. Jamet and C. Duquennoy

53/283

Chapter 7

Test-case number 7a: One-dimensional phase-change of a vapor phase in contact with a wall (PA) By Didier Jamet, DER/SSTH/LMDL, CEA/Grenoble, 38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 45 42, Fax: +33 (0)4 38 78 50 36, E-Mail: [email protected] Christophe Duquennoy, EDF-SEPTEN, 12-14 av. Dutrievoz, 69628 Villeurbanne cedex, France, E-Mail: [email protected]

7.1

Practical significance and interest of the test-case

In numerical methods dedicated to two-phase flows and for which interfaces are well defined in space (local instantaneous models rather than averaged models), phase-change phenomena are particularly difficult to simulate. The main reason is that the velocity field is discontinuous at an interface, which speed of displacement is different from the fluid velocities. Another difficulty comes from the local thermodynamic equilibrium condition which must be satisfied at an interface and which governs mass and energy transfer and therefore the local speed of displacement of the interface. In this paper, analytical solutions of one-dimensional liquid-vapor phase-change problems are provided. The general problem is that of a phase in contact with a heated wall, as sketched in figure 7.1. Due to this heating, the temperature of the fluid in contact with the wall rises up and the temperature at the interface, located at the upper boundary of this phase, also tends to increase. However, the interface tends to keep at local thermodynamic equilibrium. Therefore, if the pressure of the system is given, for instance if the upper boundary of the upper phase is maintained at a constant pressure, say P0 , the pressure at the interface is approximately P0 and, due to the local thermodynamic equilibrium condition, the interface temperature T i is equal to the

54/283

Test-case number 7a by D. Jamet and C. Duquennoy

saturation temperature corresponding to the pressure P0 : T i = T sat (P0 ). Thus, the interface temperature remains constant. Since the heating of the lower phase tends to increase the interface temperature, heat must be absorbed at the interface to keep the interface temperature constant, which is achieved through vaporization. The rate of phase-change is related to the energy supplied by the heated wall through the phase in contact with the latter.

liquid

vapor

















 

















 






























heated 


wall

 Figure 7.1: Sketch of a one-dimensional phase-change problem.

The proposed test-case gives analytical solutions in one dimension which can be compared to numerical results and therefore tests the ability of a numerical method to account for interfacial liquid-vapor phase-change dominated by thermal effects. Indeed, in the physical situation described above, it is clear that the interfacial phase-change is due to a thermal equilibrium condition and not to a pressure equilibrium condition. Two different solutions are provided. The first one is a steady state solution in the frame of reference linked to the interface. As will be discussed in section 7.3, this solution corresponds to a physical problem in which heat transfer is dominated by heat conduction. This first test-case aims at testing the ability of a numerical method to account for interfacial mass and energy balance equations accurately. The second solution corresponds to an unsteady problem for which the initial temperature of one of the phases is uniformly superheated (its temperature is larger than the saturation temperature) or subcooled (its temperature is lower than the saturation temperature). This test-case aims at testing the ability of a numerical method to accurately solve highly unsteady phase-change problems dominated by thermal effects. Indeed, in this test-case, the two key issues are the development of the thermal boundary layers close to the interface and the interfacial mass and energy balance equations. In both cases, the pressure of the system is assumed to be imposed and not to vary much in time, which is relevant essentially if the phases can be approximated as incompressible.

7.2

General definitions and model description

Let us consider a plane interface and assume that all the physical variables are functions of the only space coordinate normal to the interface, say z.

7.2.1

Bulk balance equations

Let us assume that both the liquid and vapor phases can be modeled as incompressible fluids. If gravity forces are neglected, the general equations of motions of the bulk phases

Test-case number 7a by D. Jamet and C. Duquennoy

55/283

are the following: ∂ρ ∂(ρV ) + =0 ∂t ∂z

ρ

ρCp





∂V ∂V +V ∂t ∂z

∂T ∂T +V ∂t ∂z





∂ = ∂z

(7.1)



∂P ∂ =− + ∂z ∂z



∂T k ∂z



4 ∂V µ 3 ∂z



dP 4 + T κT + µ dt 3

(7.2)



∂V ∂z

2

(7.3)

where ρ is the density, V is the velocity, P is the pressure, µ is the viscosity, T is the temperature, Cp is the heat capacity at constant pressure, k is the thermal conductivity and κT is the coefficient of thermal expansion at constant pressure. If the density is assumed to be constant, which means that the density does not depend on the temperature, these equations are simplified as follows: ∂V =0 ∂z

ρCp

ρ





∂T ∂T +V ∂t ∂z

(7.4)

∂V ∂V +V ∂t ∂z



=−



=

∂ ∂z

∂P ∂z



k

∂T ∂z

(7.5)



(7.6)

Equation (7.4) shows that the velocity is uniform within each bulk phase.

7.2.2

Interfacial balance equations

The interfacial balance equations are the following (Delhaye et al. , 1981):

ρv V v − V i = ρl V l − V i = m ˙ Pli



−k

∂T ∂z

i l

−

Pvi

=m ˙

2



1 1 − ρv ρl



     ∂T i m ˙2 1 1 − −k =m ˙ L+ − ∂z v 2 ρ2v ρ2l

(7.7)

(7.8)

(7.9)

where the subscripts l and v denote the liquid and vapor phases respectively, the superscript i denotes the interface, V i is the speed of displacement of the interface, m ˙ is the interfacial mass flux, L is the latent heat of vaporization. Note that the term in m ˙ 2 in equation (7.9) accounts for kinetic energy effects and is generally negligible compared to L.

56/283

7.2.3

Test-case number 7a by D. Jamet and C. Duquennoy

Interface thermodynamic equilibrium

The unknowns of the problem are, the two bulk velocities, the two bulk pressure fields, the two bulk temperature fields and the speed of displacement of the interface. Therefore, an equation must be added to the system (7.4)-(7.9). This equation is a condition of local thermodynamic equilibrium of the interface, which reads: T i = T sat (P i )

(7.10)

where P i is the pressure at the interface given by (Ishii, 1975) Pi =

Pvi + Pli 2

(7.11)

Of course, boundary and initial conditions must be added to the above system of PDEs. In the models considered here, different thermal boundary conditions are considered. However, it is gnerally considered that the pressure at the upper boundary is imposed, say P0 .

7.3

Steady state model q=0

P = P0 liquid

vapor q = qvp > 0

V =0

Figure 7.2: One-dimensional phase-change problem with imposed boundary heat fluxes.

7.3.1

Model

Let us consider that the system sketched in figure 7.2 is initially at rest and at a uniform temperature equal to the saturation temperature corresponding to the pressure P0 : T (z, t = 0) = T sat (P0 ). Let us then impose a heat flux on the lower boundary, which leads to a phase-change at the interface as described in section 7.1. It can be shown that, in certain conditions (see hereafter), the interface reaches a constant speed. In this case, in the frame of reference linked to the interface, the problem is stationary. The equations of motion of the bulk phases (7.4)-(7.6) then read: dV =0 dz

(7.12)

dP =0 dz

(7.13)

Test-case number 7a by D. Jamet and C. Duquennoy

dT d m ˙ Cp = dz dz

57/283



dT k dz



(7.14)

Since the pressure of the liquid phase is imposed at the upper boundary, equation (7.13) can be integrated and one gets: Pl (z) = P0

2

Pv (z) = P0 − m ˙



(7.15)

1 1 − ρv ρl



(7.16)

If Cp and k are assumed to be constant, equation (7.14) can be integrated to get   B m ˙ Cp + T (z) = A exp k m ˙ Cp where A and B are two constants of integration. Let us denote z p the position of the phase boundaries and z i the position of the interface; since the system is supposed to be in a steady state, these are constant. If we assume that the conductive heat flux is imposed for z = z p and is denoted q p , and since the interface temperature is known, the two constants of integration can be determined and one gets:     
qp m ˙ Cp p m ˙ Cp p i i T (z) = T + exp − z −z − exp − (z − z) (7.17) m ˙ Cp k k where the interface temperature T i is determined by equations (7.10) and (7.11)    m ˙2 1 1 i sat T =T P0 − − 2 ρv ρl

(7.18)

The temperature profile (7.17) is valid for each phase. It can be shown, using the Clapeyron relation, that the interface temperature given by (7.18) can be approximated by T i ' T sat (P0 )

(7.19)

if the following condition is satisfied m ˙2 L 2



1 1 − ρv ρl

2

(7.20)

It is worth noting that this condition is generally satisfied. From the temperature profiles (7.17), the interfacial energy equation (7.9) implies       

m ˙ Cpl p m ˙ Cpv p m ˙2 1 1 p i p i ql exp − zl − z − qv exp − zv − z =m ˙ L+ − 2 kl kv 2 ρ2v ρl (7.21) Since the position of the boundaries zlp and zvp are given, this equation can be seen as a relation between the heat fluxes qlp and qvp , the position of the interface z i

58/283

Test-case number 7a by D. Jamet and C. Duquennoy

(which is an unknown of the problem) and the interfacial mass flux m. ˙ For instance, if the heat fluxes are imposed as well as the position of the interface (in the middle of the domain for instance z i = (zlp + zvp )/2), this relation gives the interfacial mass flux m. ˙ Let us introduce the bulk Peclet numbers Pe =

m ˙ Cp Lp k

(7.22)

where Lp is the length of the considered phase. If P e  1, the general temperature profile (7.17) is greatly simplified: T (z) ' T i +

qp p (z − z) k

(7.23)

This equation shows that, when the convective heat flux can be neglected compared to the conductive heat flux (i.e. P e  1), the temperature profile within each phase is linear. This means that the information concerning the temperature T sat at the interface has the time to diffuse within the phases without being perturbed by convection effects. In this case, the interfacial energy balance equation (7.21) simplifies as follows:    m ˙2 1 1 p p ql − qv = m ˙ L+ − 2 (7.24) 2 ρ2v ρl Note that this relation is different from the general condition (7.9) since the fluxes are those imposed on the boundaries and not those at the interface, which is a great simplification. This is because the condition P e  1 implies, as shown by equation (7.14), that the conductive heat flux is constant within each phase. Note also that equation (7.24) can generally be simplified by neglecting the kinetic energy compared to the latent heat:   1 m ˙2 1 − 2 (7.25) L 2 ρ2v ρl Note that, since ρv < ρl , this condition is more restrictive than the condition (7.20). If the condition (7.25) is satisfied, one can return very easily to the absolute frame of reference and one finds that the position of the interface is given by: z i (t) = z i (0) + V i t where V i = Vl −

qlp − qvp ρl L

or qlp − qvp V = Vv − ρv L i

and the corresponding interfacial mass flux is given by m ˙ =

qlp − qvp L

Test-case number 7a by D. Jamet and C. Duquennoy

59/283

For instance, for the system sketched in figure 7.2, Vv = 0, qlp = 0 and qvp > 0, one gets m ˙ =−

qvp L

(7.26)

and the corresponding speed of displacement of the interface is given by Vi =−

m ˙ ρv

The speed of the liquid phase is therefore given by   1 1 Vl = −m ˙ − ρv ρl

7.3.2

(7.27)

(7.28)

Comments

The model described in section 7.3.1 shows that the speed of displacement of the interface, i.e. a macroscopic characteristic of the system, depends on the interfacial mass and energy balance equations (7.27) and (7.26). Therefore, this model can be used as a test-case to verify how well a numerical method accounts for the interfacial mass and energy balance equations. In the general interfacial energy balance equation (7.9), which is valid even for an unsteady motion of the interface, only the conductive heat fluxes at the interface appear; the convective fluxes appear as a correction of the latent heat and can generally be neglected, which can be checked through the condition (7.25). The reason why the energy balance equation (7.9) gives rise to the relatively complex equation (7.21) is because the conductive heat fluxes at the interface are different from the imposed heat fluxes at the boundaries because of convection effects within the bulk phases, which have been integrated. The general difficulty of two-phase systems, and in particular when phase-change exists, comes from interfacial characteristics and not from bulk phase characteristics. Therefore, it is fair to assume that the issue concerning the ability of a method to accurately account for the competition between the conductive and convective heat fluxes within the bulk phases can be seen as secondary compared to the issue of knowing how accurately the interfacial mass and energy balance equations are accounted for by a numerical method. That’s the reason why it is proposed that the simplified model corresponding to the condition P e  1 should be used as a first test-case.

7.3.3

Test-case description

In this section, we describe how the previous analytical results should be used as a numerical test to verify how well a numerical method accounts for the interfacial mass and energy balance equations. The physical problem considered is that sketched in figure 7.2, on which the boundary conditions imposed are indicated, and one seeks for a state in which the interface moves at a constant speed. The main physical parameters are • the densities ρl and ρv ; • the specific heat capacities Cpl and Cpv ; • the thermal conductivities kl and kv ; • the corresponding thermal diffusivities αl and αv ;

60/283

Test-case number 7a by D. Jamet and C. Duquennoy

• the pressure P0 and the corresponding saturation temperature T sat and latent heat of vaporization L; • the heat flux qvp ; • the length L of the system. Their values should be provided. The stationary speed of displacement of the interface V i and interfacial mass flux m ˙ can be approximated by equations (7.27) and (7.26) respectively. It should be verified that the Peclet numbers of the phases defined by (7.22) are always small compared to one, so that the approximation (7.24) is valid. Note that the Peclet numbers calculated using the total length of the system L instead of the larger length of one phase during the calculation Lp is an upper bound for the validity of the approximation. It should also be checked that the condition (7.25) is satisfied so that the simplified solution (7.26) can be used. Note that if the condition (7.25) is not satisfied, equation (7.26) can still be used but L must be replaced by [L + m ˙ 2 /2 (1/ρ2v − 1/ρ2l )], in which case equation (7.26) is a non-linear equation in m. ˙ No particular initial state is imposed since only the stationary state is of interest for this test-case. However, it is highly recommended that the initial length of the vapor phase should be as small as possible to shorten the unsteady stage of the simulation (in the frame linked to the interface). Moreover, it is recommended that the initial temperature profile should be that given by equation (7.23) (where it is recalled that the position of the interface is taken as the origin z i = 0). The main physical parameter which must be compared to the analytical solution is the speed of displacement of the interface V i . It is recommended that V i (t) should be provided and compared to its theoretical value given by equation (7.27) through the L1 norm of the error. Likewise, the speed of the liquid velocity should be compared to its theoretical value given by equation (7.28). Moreover, the pressure within the phases should also be compared to their theoretical values given by equation (7.15) and (7.16).

7.4 7.4.1

Unsteady model for a phase initially uniformly superheated or undercooled Model

In this section, we consider the same system as the one considered in section 7.3, which is governed by the general system of equations (7.4)-(7.11). The difference between the case considered in this section and that treated in section 7.3 comes essentially from the initial conditions. Indeed, in section 7.3, the system was supposed to be stationary in the frame of reference linked to the interface; if the Peclet number in each phase is small enough, the initial conditions are not important because the temperature field within each phase reaches a quasi-steady state. On the contrary, in the case considered in this section, the initial conditions are essential. Indeed, the initial state is such that both phases are at rest and the temperature of one of the phases is uniform and equal to the saturation temperature corresponding to the imposed pressure, whereas the temperature of the other phase is uniform but different from the saturation temperature. This system has been studied analytically and numerically by

Test-case number 7a by D. Jamet and C. Duquennoy

61/283

Welch & Wilson (2000) and the developments given here are inspired by this work. However, a fully analytical solution is provided here whereas Welch & Wilson (2000) provided only a semi-analytical solution. This fully analytical solution was initially proposed by Duquennoy (2000) in the particular case for which the liquid phase is initially superheated. In the following, we consider the system sketched in figure 7.3 for which the vapor is in contact with a fixed wall, whereas the liquid is free to flow through the upper boundary. Note that equation (7.4) implies that the vapor phase is always at rest: Vv = 0

(7.29)

In section 7.4.1 and 7.4.1 we consider the cases for which the vapor and the liquid temperature respectively is initially different from the saturation temperature. Vapor initially not at thermodynamic equilibrium

T = T sat (P0 )

P = P0

liquid

vapor T = T sat + ∆T

V =0

Figure 7.3: One-dimensional phase-change problem for a vapor initially superheated (∆T > 0).

In this section we consider the case for which the initial temperature of the liquid phase is the saturation whereas the initial temperature of the vapor phase is uniform but different from the saturation temperature: Tl (z, 0) = T sat (P0 )

(7.30)

Tv (z, 0) = T sat (P0 ) + ∆T

(7.31)

If ∆T > 0, the phase is said to be superheated and if ∆T < 0, the phase is said to be subcooled. Since the liquid is initially at thermodynamic equilibrium at the interface and if the condition (7.20) is satisfied, no temperature gradient will appear in the liquid phase; equation (7.6) then shows that the temperature of the liquid phase does not evolve in time. Therefore, the temperature of the liquid phase is constant and uniform and equal to the saturation temperature T sat . The situation is different for the vapor since this phase is initially not at thermodynamic equilibrium at the interface. Since the temperature of each phase is supposed to be

62/283

Test-case number 7a by D. Jamet and C. Duquennoy

always equal to the saturation temperature at the interface, the initial temperature field in the vapor at the interface varies from T sat to (T sat + ∆T ) on a zero distance which creates an infinite temperature gradient which tends to diffuse in time. In the following, the vapor temperature field T (z, t) is determined analytically. Let us first make the following change of variable: ζ =z−

Z

t

V i (τ ) dτ

0

The energy balance equation (7.6) then reads    
∂Tv ∂ ∂Tv i ∂Tv ρv Cpv + Vv − V = kv ∂t ∂ζ ∂ζ ∂ζ

(7.32)

The initial conditions (7.31) therefore reads: Tv (ζ, 0) = T sat + ∆T

(7.33)

The boundary condition at the interface (7.10) reads:

Tv (ζ = 0, t) = T sat

(7.34)

To simplify the analysis, it is assumed that the vapor phase is infinite so that the boundary condition on the wall is transformed as follows:

Tv (ζ → −∞, t) = T sat + ∆T

(7.35)

This approximation is valid as long as the initial perturbation of the temperature field has a small influence on the temperature on the real boundary, which is measured by the Fourier number F o = k/(ρCp) τ /L2 where L is a characteristic length and τ is a characteristic time. The above approximation is therefore valid for times t < τ with τ such that F o  1. This issue is analyzed in detail in section 7.4.2. Since the liquid phase is at a constant temperature, equation (7.9) implies   ∂T k (ζ = 0, t) = m ˙ L ∂ζ v

(7.36)

in which kinetic energy effects have been neglected for the sake of simplicity. Since the vapor phase is always at rest, equation (7.7) implies m ˙ = −ρv V i

(7.37)

Equations (7.36) and (7.37) show that the speed of displacement of the interface is given by   kv ∂T i V (t) = − (ζ = 0, t) (7.38) ρv L ∂ζ v

Test-case number 7a by D. Jamet and C. Duquennoy

63/283

The energy balance equation (7.32) then reads ∂Tv ∂Tv 1 ∂ −Vi = ∂t ∂ζ ρv Cpv ∂ζ



∂Tv kv ∂ζ



(7.39)

If the thermal conductivity kv is assumed to be constant, we can introduce the following variables: r 1 ζ √ η= (7.40) 2 αv t

θ(η) =

Cpv Tv (ζ, t) L

(7.41)

where αv =

kv ρv Cpv

(7.42)

Equation (7.39) can then be transformed, to get the following ordinary differential equation on θ(η):
θ00 + η − θ0 (0) θ0 = 0 (7.43) where θ0 ≡ dθ/dη. The boundary conditions (7.34) and (7.35) imply the following boundary conditions for θ(η): θ(0) =

θ(η → −∞) =

Cpv sat T L

(7.44)


Cpv T sat + ∆T L

(7.45)

Equation (7.43) can be integrated once to get  2  η 0 0 0 θ = θ (0) exp − + η θ (0) 2

(7.46)

From (7.38) and (7.46), the speed of displacement of the interface is then given by r αv θ0 (0) i √ V =− (7.47) 2 t Since θ0 (0) is constant (which is still to be determined), √ equation (7.47) shows that the speed of displacement of the interface V i varies as 1/ t. This means that the initial infinite temperature gradient at the interface leads to an initial infinite value for V i and that, as thermal diffusion develops, the temperature gradient at the interface decreases as well as V i . Let θ0 (0)2 Θ0 = θ (0) exp 2 0





(7.48)

64/283

Test-case number 7a by D. Jamet and C. Duquennoy

Λ=

η − θ0 (0) √ 2

(7.49)

Integrating (7.46) and accounting for (7.48) and (7.49), one gets √ π Cpv sat θ(Λ) = Θ0 [erf(Λ) − erf(Λ0 )] + T 2 L

(7.50)

where 2 erf(x) = √ π

Z

x

2

e−t dt

(7.51)

0

Equation (7.50) gives the temperature field. However, two constants Θ0 and Λ0 appear in the expression for θ which still have to be determined. The definitions (7.48) and (7.49) show that these two constants depend only on θ0 (0), which is therefore the only constant to be determined. The value of θ0 (0) is determined thanks to the boundary condition (7.35). Indeed, given the expression (7.50) for θ, the boundary condition (7.35) implies:  0 2 √   0  Cpv θ (0) π θ (0) 0 ∆T + θ (0) exp 1 − erf √ =0 (7.52) L 2 2 2 Equation (7.52) is a non-linear equation in θ0 (0). Equations (7.52) and (7.50) therefore fully determine the temperature field in the vapor phase. Turning back to the dimensional variables, one gets the following temperature field in the frame of reference linked to the interface: q   ζ erf(λv ) − erf λv + α1v 2√ t Tv (ζ, t) = T sat + ∆T (7.53) 1 + erf(λv ) √ where λv = −θ0 (0)/ 2 satisfies the following non-linear equation r π Cpv 2 ∆T = λv eλv (1 + erf(λv )) (7.54) L 2 Note that, for “small” values of ∆T , the following approximation for λv can be used r 2 Cpv λv ' ∆T π L

From equation (7.47), one gets the following expression for the speed of displacement of the interface: V i (t) =

√

λv αv √ t

Equation (7.7) allows to determine the liquid phase velocity:   ρv i Vl (t) = V (t) 1 − ρl

(7.55)

(7.56)

Test-case number 7a by D. Jamet and C. Duquennoy

T = T sat + ∆T

65/283

P = P0

liquid

vapor T = T sat

V =0

Figure 7.4: One-dimensional phase-change problem for a liquid initially undercooled (∆T < 0).

Liquid initially not at thermodynamic equilibrium If the temperature of the liquid phase is initially uniform at a value different from the saturation temperature, whereas the vapor phase is initially at equilibrium, a similar analysis to that developed in the previous section can be developed and one finds that the temperature field within the liquid phase, in the frame of reference linked to the interface, is the following   q ζ erf λl + α1l 2√ − erf(λl ) t Tl (ζ, t) = T sat + ∆T (7.57) 1 − erf(λl ) where Cpl ∆T = L

r

π 2 λl eλl (1 − erf(λl )) 2

(7.58)

In this case, the speed of displacement of the interface is the following V i (t) =

ρl √ λ l αl √ ρv t

(7.59)

The expression for the liquid phase velocity is given by equation (7.56).

7.4.2

Test-case description

About the boundary conditions One of the difficulties of this test-case comes from the fact that the analytical solution provided in section 7.4.1 is valid for an infinite length, which is impossible to achieve numerically. The solution recommended is to impose a constant temperature on the boundaries of the system as sketched in figures 7.3 and 7.4. The analytical solutions (7.53) and (7.57) are therefore only approximations of the exact solutions of the problems sketched in figures 7.3 and 7.4, which will be actually simulated numerically. In this section, we study the conditions under which the analytical solutions (7.53) and (7.57) are good approximations of the solutions of the problems actually simulated. In other words, we study the conditions under which the approximation of an infinite length of

66/283

Test-case number 7a by D. Jamet and C. Duquennoy

the phases approaches the solution for a finite length. First, we study the case for which the vapor is initially not at thermodynamic equilibrium. Integrating equation (7.55), one gets the position of the interface as a function of time: z i (t) = z0i + 2 λv

√

αv t

(7.60)

Let Lv be the length of the vapor phase. From (7.60), one gets Lv (t) = Lv0 + 2 λv

√

αv t

(7.61)

The lower boundary is therefore located at a distance Lv (t) from the interface. If the vapor phase had an infinite length, the temperature at this location T pv (t) would be given by equation (7.53) for which ζ = −Lv (t):

T pv (t) = T sat + ∆T

erf(λv ) + erf



Lv0 √ 2 αv t



1 + erf(λv )

(7.62)

We seek for the conditions under which the approximation T pv ' T sat + ∆T is satisfactory. Let ε be an approximation considered as reasonable, for instance ε = 10−2 : T pv = T sat + ∆T (1 − ε)

(7.63)

From equation (7.62), one gets: Lv0 = 2 erf−1 [1 − ε (1 + erf(λv ))] √ αv τv

(7.64)

where erf−1 represents the inverse of the error function and τv is the time at which the condition (7.63) is satisfied. Note that the left-hand-side of equation (7.64) can be expressed with a Fourier √ number based on the initial length of the vapor phase: 1/ F ov . This shows that the approximation of a infinite length for the vapor phase is valid only for a certain period of time: beyond t = τv the analytical solution (7.53) is no longer an acceptable approximation of the problem for which the temperature on the lower boundary is kept constant. From a computationally point of view, if one aims at comparing his numerical results to the analytical solution (7.53) with a precision ε up to a time τv , the initial length of the vapor phase must be that given by the condition (7.64). A similar analysis can be developed in the case for which the liquid is initially not at thermodynamic equilibrium and one finds that the condition which must be satisfied is the following:   ρl Ll0 = −2 λl 1 − + 2 erf−1 [1 − ε (1 + erf(λl ))] √ αl τl ρv

(7.65)

Test-case number 7a by D. Jamet and C. Duquennoy

67/283

About the initial condition In the analytical solutions developed in section 7.4.1, the initial temperature field is not continuous at the interface: it varies from T sat to T sat +∆T on a zero distance. This condition cannot be imposed numerically. This initial condition must therefore be modified. It is recommended to use the analytical solution at a time t0 > 0 as the initial condition. No particular value for t0 is recommended; however, it is recommended to justify the choice for t0 on physical or numerical arguments. For instance, the number of grid points within the initial thermal boundary lower is considered as a good indication. Test-case The physical parameters used for the simulation should be provided. These are • the densities ρl and ρv ; • the specific heat capacities Cpl and Cpv ; • the thermal conductivities kl and kv ; • the corresponding thermal diffusivities αl and αv ; • the pressure P0 and the corresponding saturation temperature T sat and latent heat of vaporization L; • the initial temperature difference ∆T ; • the length of the system and the initial lengths of the phases Ll0 and Lv0 . The time t0 for which the analytical solutions are used to initialize the temperature, velocity and pressure field should be provided. Since the parameters λl or λv are key parameters of the analytical solutions (7.53) and (7.57) respectively, their value, determined through the numerical resolution of the non-linear equations (7.54) and (7.58), should be provided. The precision ε at which the analytical solution is supposed to approach the solution of the finite length solution should be provided, as well as the times τl or τv , determined through the resolution of the non-linear equations (7.64) and (7.65) respectively, until which a reasonable comparison is still relevant. It is recommended that numerical temperature fields should be compared to the analytical ones given by equations (7.53) or (7.57) at different times from t0 to τv or τl . At least one velocity and pressure field should be provided and compared to the corresponding analytical fields through the L1 norm of the error for instance. The position of the interface z i (t) as well as its speed of displacement V i (t) should also be provided and compared to the analytical solutions (7.55) or (7.59). Example Let us consider water at a pressure P0 = 160 105 Pa; the corresponding saturation temperature is T sat = 620 K. The physical characteristics of the bulk phases are the following: ρl = 586.5 kg/m3 , ρv = 106.4 kg/m3 , Cpl = 9.35 103 J/kg K, Cpv = 15.4 103 J/kg K, kl = 0.444 W/m K, kv = 0.114 W/m K, αl = 0.8096 10−7 m2 /s, αv = 0.6958 10−7 m2 /s

68/283

Test-case number 7a by D. Jamet and C. Duquennoy

and L = 941. J/kg; note that the latter value for L is not the experimental value but is divided by a factor 103 so that the interface moves at a speed comparable to the speed at which the thermal boundary layer develops. Let us assume that the initial superheat of the vapor is ∆Tv = 5 K. The solution of mined by equation by equation (7.55) mined by equation 7.7 respectively.

equation (7.54) is λv = 1.71814. The interface position z i (t) is deter(7.55), the speed of displacement of the interface V i (t) is determined and the temperature profile within the vapor phase Tv (ζ, t) is deter(7.53); their graphical representations is given in figures 7.5, 7.6 and

0.0025 0.002 0.0015 0.001 0.0005 0 0

1

2

3

4

5

Figure 7.5: Time evolution of the interface position z i (t) (position is in m and time is in s).

0.0022 0.002 0.0018 0.0016 0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002 0

1

2

3

4

5

Figure 7.6: Time evolution of the speed of displacement of the interface V i (t) (velocity is in m/s and time is in s).

Note that, from equations (7.37) and (7.55) we can determine that the condition (7.25) is satisfied as long as t  10−10 s, which is not a restrictive condition. Therefore, kinetic effects can be neglected compared the latent heat of vaporization. If the initial length of the vapor phase is Lv0 = 5 mm and if the precision of the comparison between the numerical and the analytical temperature fields is ε = 10−2 , simulations can be run up to t = 33 s; this time has been determined by solving equation (7.64). Equation (7.61) shows that at this particular time, the length of the vapor phase is Lv = 10.216 mm. This means if the simulation is run up to t = 33 s, the initial length of

Test-case number 7a by D. Jamet and C. Duquennoy

69/283

625 624 623 622 621 620 -0.01

t=0.5s t=1s t=2s t=4s t=8s t=16s -0.008

-0.006

-0.004

-0.002

0

Figure 7.7: Time evolution of the temperature profile Tv (ζ, t) (temperature is in K and distance is in m).

the liquid phase should be such that Ll0 > 5.2 mm. Note that if the value of L were taken equal to its experimental value, at t ' 30 s, the interface would be only about 0.1 mm away from its initial position, and the numerical and theoretical positions of the interface would be more difficult to compare.

References Delhaye, J. M., Giot, M., & Riethmuller, M.L. 1981. Thermohydraulics of two-phase systems for industrial design and nuclear engineering. Hemisphere Publishing Corporation. Duquennoy, C. 2000. D´eveloppement d’une approche de simulation num´erique directe de l’´ebullition en paroi. Ph.D. thesis, Institut National Polytechnique de Toulouse. Ishii, M. 1975. Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrolles. Welch, S.W.J., & Wilson, J. 2000. A volume of fluid based method for fluid flows with phase change. J. Comp. Phys., 160(2), 662–682. doi: 10.1006/jcph.2000.6481.

70/283

Test-case number 7a by D. Jamet and C. Duquennoy

Test-case number 7b by D. Jamet

71/283

Chapter 8

Test-case number 7b: Isothermal vaporization due to piston aspiration (PA) By Didier Jamet, DER/SSTH/LMDL, CEA/Grenoble, 38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 45 42, Fax: +33 (0)4 38 78 50 36, E-Mail: [email protected]

8.1

Practical significance and interest of the test-case

An analytical solution of a one-dimensional isothermal liquid-vapor phase-change problem is provided. In models dedicated to the description of multiphase flows, an important part is the modeling of the interfaces which separate the phases of the system. The equations of motion of the interfaces consist in balance equations (mass, momentum, energy and entropy) and closure relations. However, in general, these equations of motion are simplified by assuming some sort of equilibrium. For instance, it is generally assumed that the Laplace relation, which relates the pressure of the bulk phases to the curvature of the interface at equilibrium, is satisfied locally all along the interfaces. In this case, the general momentum balance equation at the interface is simplified by assuming local equilibrium at the interface. If the system under consideration is a two-phase system made of liquid and vapor phases of a pure substance, phase-change might occur at an interface. This phasechange (evaporation or condensation) corresponds to a reaction of the system placed in a non-equilibrium state to recover an equilibrium state. For instance, let us consider a liquidvapor system for which the pressure is assumed to be constant and uniform and let us heat up the vapor phase. Due to thermal conduction in the vapor phase, the temperature will increase, in particular at the interface. If local thermodynamic equilibrium is assumed at the interface, the temperature of the interface must remain constant and equal to the saturation temperature corresponding to the imposed pressure. To keep the interface at a constant temperature (i.e. at local thermodynamic equilibrium), evaporation occurs: the latent heat absorbed during evaporation prevents the interface temperature from rising up.

72/283

Test-case number 7b by D. Jamet

The local thermodynamic equilibrium condition is generally imposed in terms of the interface temperature as described above. However, the relation Ti = T sat (Pi )1 , where Ti is the local interface temperature, Pi is the local interface pressure and T sat is the corresponding saturation temperature, can be inverted: Pi = P sat (Ti ), where P sat is the saturation pressure. Even though these two conditions are equivalent, they correspond to very different situations: in the first case, it is assumed that the pressure is imposed and that phase-change is mainly driven by thermal effects, whereas in the second case, it is assumed that the temperature is imposed and that phase-change is mainly driven by dynamic or pressure effects. Such a situation corresponds to the physical system sketched in figure 8.1: different velocities are imposed to the liquid and vapor phases, which creates a pressure gradient within the bulk phases, which tends to let the interface pressure be different from the saturation pressure, which then triggers phase-change.


         








  
        







  


      






  
 
 
piston 
 
 
 
  vapor





































































wall



 liquid

Figure 8.1: Sketch of a one-dimensional isothermal phase-change problem.

The present test-case gives an analytical solution of a problem for which phase-change is driven by dynamic effects and not by thermal effects. This solution might be used to test the capability of a numerical method to simulate liquid-vapor phase-change due to dynamic effects and not thermal effects.

8.2

Definitions and model description

Let us consider a liquid-vapor system of a pure substance. The system is assumed to be one-dimensional and the interface to be flat; therefore only the direction normal to the interface, say z, is relevant. The system is also assumed to be isothermal. Physically, this means that the thermal conductivity is very large and that energy is supplied by thermal conduction where needed (mainly at the interface) instantly. No volumetric force is applied to the system. The equations of motion of the system are the following. 1

Curvature effects are neglected here for the sake of simplicity.

Test-case number 7b by D. Jamet

73/283

• Within the bulk phases ∂ρ ∂(ρV ) + =0 ∂t ∂z ∂(ρV ) ∂(ρV 2 ) ∂P ∂ + =− + ∂t ∂z ∂z ∂z

(8.1) 

4 ∂V µ 3 ∂z



(8.2)

where ρ is the density, V is the velocity, P is the pressure and µ is the dynamic viscosity. Note that, if the bulk phases are assumed to be incompressible, which is a fair approximation as long as the velocity is small compared to the sound speed, the system (8.1)-(8.2) reads ∂V =0 ∂z   ∂V ∂V ∂P ρ +V =− ∂t ∂z ∂z

(8.3) (8.4)

• At the interface (superscript i)

ρiv Vvi − V i = ρil Vli − V i Pli

Pvi

(8.5)

=P

sat

m ˙i + 2


2 

1 1 − i i ρv ρl



(8.6)

=P

sat

m ˙i − 2


2 

1 1 − i i ρv ρl



(8.7)

where V i is the speed of displacement of the interface and m ˙ i is the mass flux across the interface, which is defined by

m ˙ i = ρiv Vvi − V i = ρil Vli − V i (8.8) The subscripts v and l represent the vapor and liquid phases respectively. Equations (8.6) and (8.7) correspond to the local thermodynamic condition at the interface (e.g. Ishii, 1975, p. 39 and Jamet et al. , 2001, p. 636). Note that the terms
2 in m ˙ i correspond to the effect of the kinetic energy. • The boundary conditions are the following: ρV (z = 0, t) = m ˙l

(8.9)

ρV (z = L, t) = m ˙v

(8.10)

∂V (z = 0, t) = 0 ∂z

(8.11)

∂V (z = L, t) = 0 (8.12) ∂z where L is the length of the system and m ˙ l and m ˙ v are two different constants which represent respectively the liquid mass flux entering the system and the vapor mass flux exiting the system. Figure 8.1 corresponds to the case m ˙ l = 0 and m ˙ v > 0. Note that the boundary conditions (8.11) and (8.12) are necessary only if the bulk phases are compressible.

74/283

8.2.1

Test-case number 7b by D. Jamet

Incompressible model

As a first approximation, let us consider the incompressible model. In this case, the density is uniform within each bulk phase by definition of the incompressibility, and it is straightforward to show that the velocity and the pressure are also uniform within each bulk phase. The value of the bulk velocities depends on the imposed mass fluxes on the system boundaries (8.9) and (8.10). The value of the bulk pressures is determined by the interface local equilibrium conditions (8.6) and (8.7). For general boundary conditions m ˙l and m ˙ v , one gets: Vl =

m ˙l ρl

(8.13)

Vv =

m ˙v ρv

(8.14)

Pl = P sat +

ρl ρv (Vv − Vl )2 2 (ρl − ρv )

(8.15)

Pv = P sat −

ρl ρ v (Vv − Vl )2 2 (ρl − ρv )

(8.16)

The speed of displacement of the interface V i and the interfacial mass flux m ˙ i are given by m ˙ l−m ˙v ρl − ρv

(8.17)

ρl m ˙ v − ρv m ˙l ρl − ρv

(8.18)

Vi =

m ˙i=

Equation (8.17) shows that the speed of displacement of the interface is constant, which implies that, in the frame of reference linked to the interface, the problem is stationary. Equation (8.18) shows that the interfacial mass flux m ˙ i is constant and that its value depends only on the dynamic boundary conditions imposed through m ˙ v and m ˙l and not on any thermal effect since the temperature is assumed to be constant and uniform. It is worth noting that the value of the pressure within the bulk phases depend on the condition of local thermodynamic equilibrium at the interface as shown by equations (8.15) and (8.17).

8.2.2

Inviscid compressible model

We showed in the previous section that, when the bulk phases are incompressible, the interface moves at a constant speed and that the problem is therefore stationary in the framework linked to the interface. Let us therefore consider this framework and assume that the flow in the bulk phases is stationary. For the sake of simplicity, the bulk phases are assumed to be inviscid. The equations of motion (8.1)-(8.2) read ∂(ρV ) =0 ∂z

Test-case number 7b by D. Jamet

75/283

∂(ρV 2 ) ∂P =− ∂z ∂z The constants which appear after integration are determined using the conditions (8.5), (8.6), (8.7), (8.9) and (8.10). One gets (ρV )l (z) = m ˙l

(8.19)

(ρV )v (z) = m ˙v

(8.20)



2
2 1 ρil ρiv P + ρV 2 l (z) = P sat + Vli − Vvi + ρil Vli i i 2 ρl − ρv

(8.21)



1 ρil ρiv i i 2 i i 2 V V − V + ρ v v v l i 2 ρl − ρiv

(8.22)

P + ρV 2




(z) = P sat − v

In equations (8.19)-(8.22), the densities and the velocities of the liquid and vapor phases at the interface are unknown. Since the bulk phases are compressible, the densities ρil and ρiv are solution of the following system of equations Pl (ρil ) + ρil Vli


2

= P sat +


2
2 1 ρil ρiv Vli − Vvi + ρil Vli i i 2 ρl − ρv

(8.23)


2

= P sat −


2
2 1 ρil ρiv Vli − Vvi + ρiv Vvi i i 2 ρl − ρv

(8.24)

Pv (ρiv ) + ρiv Vvi

in which the velocities Vli and Vvi are expressed by Vli =

m ˙l ρil

(8.25)

Vvi =

m ˙v ρiv

(8.26)

It is straightforward to show that the expressions for the speed of displacement of the interface and the interfacial mass flux are the following m ˙ l−m ˙v i ρl − ρiv

(8.27)

ρil m ˙ v − ρiv m ˙l i i ρl − ρv

(8.28)

Vi =

m ˙i=

76/283

8.2.3

Test-case number 7b by D. Jamet

Physical relevance of an isothermal phase-change problem

The physical situation sketched in figure 8.1 corresponds to a situation in which the liquidvapor phase-change is imposed through a dynamic effect and not through a thermal effect. However, the fact that phase-change can occur at a constant temperature is less intuitive. Indeed, interfacial mass transfer is possible only if sufficient energy is provided at the interface, since the interfacial energy balance equation reads qli − qvi = m ˙ iL where q is the conductive heat flux and L is the latent heat of vaporization (in which kinetic energy effects are taken into account). Given this interfacial energy balance equation, the present model is such that any energy needed at the interface for mass transfer is provided instantly. This corresponds to infinite values of the bulk thermal conductivities in which case the conductive heat fluxes can be finite with a zero temperature gradient. More precisely (see Jamet, 1998, pp. 213-230), the conditions which must be satisfied are: P ev  1 P el  1

βv

qv Lv 1 kv

βl

ql Ll 1 kl

where P e is the Peclet number, β is the thermal expansion coefficient, L is a characteristic length and k is the thermal conductivity. Note that, for finite thermal conductivities, these conditions impose a limitation on the interfacial mass flux m ˙ i through the interfacial energy balance equation.

8.3

Test-case description

In this section, we describe how the previous analytical results should be used as a numerical test. The problem is one dimensional, the length L of the domain being large enough so that a steady state in the framework linked to the interface can be achieved during the simulation. The main physical parameters of the problem are the densities of the bulk phases at saturation ρl and ρv . These parameters are the only ones for the incompressible model and the equations of state of the phases Pl (ρl ) and Pv (ρv ) are two other characteristics necessary for the compressible model. No restriction is imposed on any of these physical characteristics but they should be provided. For the incompressible model, viscosity does not play any role. For the compressible model, viscous effects should be weak compared to inertial effects (see equation (8.2)). The boundary conditions consist in imposing constant and in general different mass fluxes m ˙ l and m ˙ v on the liquid and vapor boundaries.

Test-case number 7b by D. Jamet

77/283

The equations solved should be, in general, unsteady, which means that the interface position should vary in time 2 . However, one should seek for a solution in which the speed of displacement of the interface is constant. The initial conditions are not imposed in this test-case; however, to decrease the unsteady stage of the computation (in terms of the speed of displacement of the interface), it is recommended to use the analytical solution given in section 8.2, namely equations (8.13)-(8.18) for an incompressible model and equations (8.23)-(8.28) for a compressible model. The main numerical results that should be compared to the analytical solution are 1. the speed of displacement of the interface as a function of time V i (t) which must reach an asymptotic value given by equation (8.17) for an incompressible model and equation (8.27) for a compressible model; 2. the bulk pressures Pl and Pv and in particular their values at the interface Pli and Pvi which should also reach asymptotic values given by equations (8.15) and (8.16) for an incompressible model and by equations (8.23) and (8.24) for a compressible model; 3. the interfacial mass flux m ˙ i; 4. the bulk velocities Vl and Vv and in particular their values at the interface Vli and Vvi . It is recommended that the L1 -norm of the error (difference between the numerical results and the analytical results once the steady-state is reached) should be provided as a function of the mesh size and for all the above physical quantities. Acknowlegment on the paper.

The author would like to thank J.-M. H´erard for his useful remarks

References Ishii, M. 1975. Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrolles. Jamet, D. 1998. Etude des potentialit´es de la th´eorie du second gradient pour la simulation num´erique directe des ´ecoulements liquide-vapeur avec changement de phase. Ph.D. thesis, Ecole Centrale Paris. Jamet, D., Lebaigue, O., Coutris, N., & Delhaye, J. M. 2001. The second gradient method for the direct numerical simulation of liquid-vapor flows with phase-change. J. Comp. Phys., 169(2), 624–651.

2 The only case for which the interface position is stationnary corresponds to the particular boundary conditions m ˙l=m ˙ v.

78/283

Test-case number 7b by D. Jamet

Test-case number 10 by S. Vincent, J.-P. Caltagirone and J.-M. Le Gouez

79/283

Chapter 9

Test-case number 10: Parasitic currents induced by surface tension (PC) By St´ephane Vincent, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 27 07, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected] Jean-Paul Caltagirone, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 66 80, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected]

9.1

Practical significance and interest of the test-case

The assessment of the consistency of a numerical model is proposed by comparison to theoretical results. The surface tension modeling is tested thanks to two analytical test cases. The first problem consists in verifying the equilibrium of a cylindrical drop initially at rest. The pressure in the drop is defined analytically (Laplace law). In the second test, an initially square cylindrical drop is oscillating under surface tension forces. The viscous damping of drop oscillations around a cylindrical shape is studied in order to reach a steady cylindrical final state corresponding to the Laplace problem. The oscillation frequency is known theoretically. The aim of the the two cases is to estimate the discretization error on the curvature of a free surface leading to the generation of parasitic numerical currents. • in the first test 10a, the non-conforming of the numerical free-surface shape with the theoretical one induces local pressure variations in the drop. These pressure gradients lead to local velocities and to the propagation of surface waves of small amplitude. This spurious behavior is refereed to as parasitic currents. • in the second test 10b, the oscillation frequency is well recovered by numerical methods with almost several percent error. However, the steady cylindrical state exhibits parisitc currents as in the previous test.

80/283

9.2

Test-case number 10 by S. Vincent, J.-P. Caltagirone and J.-M. Le Gouez

Definitions and physical model description

Two-dimensional configurations are considered where a viscous liquid drop is initially centered in a square cavity of characteristic length L, full of air. In test 10a, the radius of the circular shape drop is R0 whereas l is the typical length scale of the square drop of test 10b. A zero-gravity field is imposed, the flow is assumed isothermal and the surface tension coefficient is constant. The tests can be easily extended to three dimensions. With respect to the length scale of the problems, analytical results can be obtained on the pressure and oscillation frequency: • for test10.a, the droplet keeps a cylindrical (or spherical) shape during time such that R(t) = R0 and the pressure jump between the inside and outside drop pressures pl and pg is given by the Laplace equation pl − pg =

2σ (3D case) R0

(9.1)

σ pl − pg = (2D case) R0 • for test 10.b, the initial square drop shape corresponds to a mode n = 2 perturbation. We compare the initial square drop to a cylindrical drop of same volume, radius of which is defined by  1/3 3 R1 = l(3D case) 4π (9.2) R1 = π −1/2 l(2D case) The oscillation period T0 can be estimated by T0 =

2π ω0

(9.3)

σ . (ρl + ρg )R13

(9.4)

where ω0 = (n3 − n)

This period remains constant during time. Only the magnitude of the oscillations are diminishing under viscous effects to converge to a circular (or spherical) shape of radius R1 . Numerical methods for front tracking or interface capturing are demonstrated to generate artificial numerical flows instead of keeping steady cylindrical drops (or spherical shapes in 3D). Following the work of Lafaurie et al. (1994), the order of magnitude of the spurious velocities up can be estimated according to the surface tension coefficient σ and dynamic viscosity µ of the drop, σ up = Cp (9.5) µ where Cp is a numerical constant characteristic of the quality of the numerical modeling of surface tension forces (a non-dimensional number similar to a capillary number). The optimal value of Cp is zero. Typical values of Cp are found between 10−3 and 10−10 .

Test-case number 10 by S. Vincent, J.-P. Caltagirone and J.-M. Le Gouez

81/283

R0 /L 0.1 0.175 0.25 0.325 0.4 ∆pR 0.95 1.015 1.02 1.01 0.985 σ Theoretical value 1 1 1 1 1 R0 /∆x 3 6 10 13 16 20 ∆pR 0.925 0.98 1.02 1.01 0.99 0.995 σ Theoretical value 1 1 1 1 1 1 Table 9.1: Two-dimensional VOF-PLIC simulation of Laplace equation. Convergence test according to drop radius (top) and mesh size (bottom)

9.3

Test-case description

The fluid characteristics are ρl = 797.88 kg.m−3 and µl = 1.2 10−3 Pa.s for ethanol and ρg = 1.1768 kg.m−3 and µg = 10−5 Pa.s for air. The surface tension coefficient between ethanol and air is σ = 0.02361 N.m−1 . Initially, the velocity field is zero in the whole domain. Wall boundary conditions are considered in the two problems. The parasitic currents are observed in numerical simulations whatever the grid type and size. For test 10.a, the geometrical parameters are • R0 = 210−3 m • L = 7.510−3 m whereas for test 10.b, we choose • l = 410−2 m • L = 7.510−2 m

9.4

Example of comparison exercise

The Navier-Stokes equations in their single-fluid formulation for multiphase flows presented by Vincent & Caltagirone (2000) are implemented on a fixed cartesian grids for the two proposed problems. A Piecewise Linear Interface Construction of Youngs (1982) associated to a Volume Of Fluid (VOF) function C is used to track the free surface and the surface tensions are modeled thanks to the Continuum Surface Force (CSF) of Brackbill et al. (1992). In addition, test case 10.a is computed with an Eulerian-Lagrangian front-tracking method of Shin & Juric (2002) with the surface tension forces modeled by using the Fresnet relation and Peskin approximation. The table 9.1 and figure 9.1 relative to test 10.a show that the VOF-PLIC approach induces the generation of spurious currents which disturb the convergence of the solution towards the known equilibrium. On the contrary, the front tracking method gives a better approximation of the Laplace equation on the same grid and reduces the velocities in the drop. For the case presented on figure 9.1, Cp = 5 · 10−5 for the VOF-PLIC method and Cp = 5 · 10−7 for the front tracking method, corresponding to up = 10−3 m.s−1 and up = 10−5 m.s−1 respectively inside ethanol. The VOF-PLIC numerical simulation of problem 10.b is presented on figure 9.2. The 0.4s period of drop oscillation presented by Brackbill et al. (1992) is recovered. As in the previous reference simulations, spurious currents are observed for long calculation times which prevent the drop from converging to a cylindrical shape under viscous effects.

82/283

Test-case number 10 by S. Vincent, J.-P. Caltagirone and J.-M. Le Gouez

0.8

0.4

0.2

PRESSURE

0.6

0

0 0.25 0.5

-0.2 0.75 1

1

0.8

0.6

0.4

0.2

0

0.8

0.4

0.2

PRESSURE

0.6

0

0 0.25 0.5

-0.2 0.75 1

1

0.8

0.6

0.4

0.2

0

Figure 9.1: Two-dimensional simulation of Laplace equation with VOF-PLIC (top) and front-tracking (bottom) methods on a 30 x 30 grid. The pressure is plotted on the left whereas the velocity field and the free surface are shown on the right. The reference pressure jump across interface given by equation (9.1) is 1 Pa.

Test-case number 10 by S. Vincent, J.-P. Caltagirone and J.-M. Le Gouez

83/283

Figure 9.2: Simulation of square drop oscillation under surface tension force with VOF-PLIC and CSF methods on a 30 x 30 grid. The velocity field and the free surface are plotted for t = 0, 0.05, 0.1, 0.2, 1 et 5 s (from left to right and from top to bottom). Figure at time t = 5 s emphasizes the presence of spurious currents in a near equilibrium drop state.

84/283

Test-case number 10 by S. Vincent, J.-P. Caltagirone and J.-M. Le Gouez

References Brackbill, J.U., Kothe, D.B., & Zemach, C. 1992. A continuum method for modeling surface tension. J. Comput. Phys., 100, 335–354. Lafaurie, B., Nardone, C., Scardovelli, R., Zaleski, S., & Zanetti, G. 1994. Modelling merging and fragmentation in multiphase flows with SURFER. J. Comput. Phys., 113, 134–147. Shin, S., & Juric, D. 2002. Modeling three-dimensional multiphase flow using a level contour reconstruction method for front tracking without connectivity. J. Comput. Phys., 180, 427–470. Vincent, S., & Caltagirone, J.-P. 2000. A One Cell Local Multigrid method for solving unsteady incompressible multiphase flows. J. Comput. Phys., 163, 172–215. Youngs, D.L. 1982. Time-dependent multimaterial flow with large fluid distortion. K.W. Morton and M.J. Baines (eds), New-York, U.S.A.

Test-case number 11a by S. Vincent, S. Jay and B. Despr`es

85/283

Chapter 10

Test-case number 11a: Translation and rotation of a concentration disk (N) By St´ephane Vincent, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 27 07, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected] Jean-Paul Caltagirone, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 66 80, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected]

10.1

Practical significance and interest of the test-case

The aim of the test is to validate the advection algorithm for volume fraction for analytical velocity field involving only translation and rotation. In this way, no deformations of the initial interface shape are induced. The interest of advection problems is that they can be handled by all numerical interface tracking methods, whatever their type Lagrangian or Eulerian, with a low computer cost. Two configurations are proposed concerning the advection of a circular concentration field in a single vortex (case 11a.a) and the rotation of a hollow disk in the same velocity field. The numerical diffusion, mass conservation and advection accuracy can be estimated to demonstrate the overall quality of several interface tracking techniques. More details are available in the work of Unverdi & Tryggvason (1992) and Rudman (1997).

10.2

Definitions and physical model description

In two-phase flow modeling, the volume fraction or concentration C is a local function defined between 0 and 1 which is characteristic of the presence of one of the two fluids. The interface between the two phases is defined as C = 0.5. In an analytical velocity field

86/283

Test-case number 11a by S. Vincent, S. Jay and B. Despr`es

u, the advection of the volume fraction is solved through a passive scalar equation ∂C + u · ∇C = 0 ∂t

(10.1)

where t denotes time.

10.3

Test-case description

In a square box with a side lenght of 1 meter, a concentration zone is initialized as follows: • Case 11a.a: A cylindrical interface is placed at point (0.25, 0.5) with radius R0 = 0.15 m. • Case 11a.b: A cylindrical interface is placed at point (0.5, 0.5) with radius R0 = 0.25 m. This disc is cut down by a rectangle whose two opposite corners are (0.5, 0.45) and (0.75, 0.55). The same velocity field is used for the two test cases:   0.5 − y u(x, y) = x − 0.5

(10.2)

Figure 10.1: Topology of interface calculating the solution of tests 11b.a (top) and 11a.b (bottom) for n=0 (left) and (right). Initial and numerical solutions are plotted in solid and dashed lines respectively.

The estimate of the accuracy of the numerical advection methods is carried out according to the following criteria:

Test-case number 11a by S. Vincent, S. Jay and B. Despr`es

87/283

• the capacity to advect the field C without numerical diffusion, i.e. keeping a well defined interface between the two fluids, • the conservation of mass M ∗ . The quantity ∗

M =

y −1 NX x −1 N X

i=2

Ci,j

(10.3)

j=2

must be perfectly conserved during numerical time. The indices i and j are related to the discretization points in x and y directions whereas Nx and Ny are the number of grid points in these directions. The relative variation can be calculated through ∆M ∗ =

y −1 NX x −1 N X

i=2

0 ) (Ci,j − Ci,j

j=2

0 Ci,j

(10.4)

0 is the initial concentration field. where Ci,j

• the accuracy of interface advection. The relative error EC between the initial and final interface position, in the case when the concentration field is advected until it recovers its initial position, can be calculated through the L1 norm of the difference between the two solutions: y −1 NX x −1 N X 1 n 0 (Ci,j − Ci,j ) EC = (Nx − 2)(Ny − 2)

i=2

(10.5)

j=2

The superscript n refers to the last iteration of numerical solving.

10.4

Example of comparison exercise

On a 128 x 128 grid, the test cases 11a.a and 11a.b are solved by using an Eulerian Volume of Fluid approach, called VOF PLIC, based on a Piecewise Linear Interface Construction in each grid cell Youngs (1982). Figure 10.1 illustrates the initial and final solutions so obtained after one turn. A constant time step δt is chosen equal to 0.01 s corresponding to a final number of iterations, n = 628. After one turn, the mass conservation M ∗ equals to 10−16 for the two test cases whereas the relative error is Ec = 0.001 for test 11a.a and Ec = 0.5 for test 11a.b.

88/283

Test-case number 11a by S. Vincent, S. Jay and B. Despr`es

References Rudman, M. 1997. Volume tracking methods for interfacial flow calculations. Int. J. Numer. Meth. Fluids, 24, 671–691. Unverdi, S., & Tryggvason, G. 1992. A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys., 100, 25–37. Youngs, D. L. 1982. Numerical Methods for Fluid Dynamics. Academic Press, New York. K. W. Morton & M. Baines Eds. Chap. Time-dependent multimaterial flow with large fluid distortion.

Test-case number 11b by S. Vincent, J.-P. Caltagirone and D. Juric

89/283

Chapter 11

Test-case number 11b: Stretching of a circle in a vortex velocity field (N) By St´ephane Vincent, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 27 07, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected] Jean-Paul Caltagirone, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 66 80, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected] Damir Juric, CNRS/LIMSI, BP 133, 91403 Orsay cedex, France Phone: +33 (0)1 69 85 80 89, E-Mail: [email protected]

11.1

Practical significance and interest of the test-case

The accuracy and overall quality of front tracking and front capturing methods is of major importance for fundamental and industrial research simulations devoted to multiphase flows. Two advection numerical tests dedicated to strong interface stretching are proposed here. Our objective is to estimate the sensitivity of interface tracking methods to free surface deformations and to quantify mass conservation in strongly varying interface shape problems. The first test-case considers a single vortex centered in a square box whereas in the second problem, a periodic multi-vortex velocity field is generated in a square cavity. In an analytical velocity field, an initially circular concentration shape of radius R0 is distorted during n time iterations until interface structures of characteristic length less than R0 /20 are generated. Then, the flow field is reversed and a same calculation is performed during an equal duration to recover the initial cylindrical shape. After 2n iterations, the theoretical solution of the scalar advection problem is the initial interface condition. The works of Rider & Kothe (1995) or Rudman (1997) scrutinize numerical solutions provided by Volume Of Fluid, Level-Set or Front Tracking approaches on these

90/283

Test-case number 11b by S. Vincent, J.-P. Caltagirone and D. Juric

two tests.

11.2

Definitions and physical model description

In two-phase flow modeling, the volume fraction or concentration C is a local function defined between 0 and 1 which is characteristic of the presence of one of the two fluids. The interface between the two phases is defined as C = 0.5. In an analytical velocity field u, the advection of the volume fraction is solved through the passive scalar equation, ∂C + u · ∇C = 0, ∂t

(11.1)

where t is the time variable.

11.3

Test-case description

The same initial conditions are used for the two test-cases. A cylindrical concentration of radius R0 is initially centered at point (xc , yc ) in a square cavity of length L with the following characteristics • R0 = 0.15 m, • L = 1 m, • xc = L/2 , yc = 3L/4 for test 11b.a • xc = L/2 , yc = L/2 for test 11b.b The velocity fields of the problems are respectively: • Case 11b.a: ua (x, y) =



cos[π(x − L/2)] sin[π(y − L/2)] −sin[π(x − L/2)] cos[π(y − L/2)]



(11.2)

ub (x, y) =



cos[4π(x + L/2)] cos[4π(y + L/2)] sin[4π(x + L/2)] sin[4π(y + L/2)]



(11.3)

• Case 11b.b:

11.4

Example of comparison exercise

The front tracking method of Shin & Juric (2002) is implemented to illustrate an almost analytical behavior of the test-cases thanks to markers put on the interface. With this approach, equation (11.1) is solved by advecting the markers in a lagrangian way. If xi is the position of marker i, u the interface velocity and ni the local unit normal to interface, the method reads as follows: dxi · ni = u · ni dt

(11.4)

At each time step, the local volume fraction is then obtained by solving a Poisson equation: Z ∇2 C = ∇ · ni δ(x − xi )dγ, (11.5) Γ(t)

Test-case number 11b by S. Vincent, J.-P. Caltagirone and D. Juric

91/283

where δ(x − xi ) is a delta function that is non zero where x = xi and Γ(t) is a time varying interface parameterized by the markers. To illustrate the ability of the numerical method, a 100 x 100 grid is chosen to run the simulation with a time step ∆t = 0.005 s. 2000 iterations are calculated for both problems using equations (11.2) and (11.3) and 2000 further time steps with a reverse velocity field choosing u = −ua or u = −ub (x, y). In this way, after 4000 iterations, the initial cylindrical shape should be recovered, as presented in figure 11.1. Several authors have demonstrated that these test-cases are very difficult to achieve, in particular for Eulerian methods such as VOF-PLIC, VOF-TVD or Level-Set techniques (Rider & Kothe, 1995, Rudman, 1997, Vincent & Caltagirone, 2000).

Figure 11.1: Front tracking simulation of test 11b.a (top) and 11b.b (bottom) after 2000 (left) and 4000 (right) iterations. The marker positions are plotted.

92/283

Test-case number 11b by S. Vincent, J.-P. Caltagirone and D. Juric

References Rider, W.J., & Kothe, D.B. 1995. Stretching and tearing interface tracking methods. AIAA paper, 95, 1–11. Rudman, M. 1997. Volume tracking methods for interfacial flow calculations. Int. J. Numer. Meth. Fluids, 24, 671–691. Shin, S., & Juric, D. 2002. Modeling Three-Dimensional Multiphase Flow using a Level Contour Reconstruction Method for Front Tracking without Connectivity. J. Comput. Phys., 180, 427–470. Vincent, S., & Caltagirone, J.-P. 2000. A One Cell Local Multigrid method for solving unsteady incompressible multiphase flows. J. Comput. Phys., 163, 172–215.

Test-case number 12 by S. Vincent, J.P. Caltagirone and O. Lebaigue

93/283

Chapter 12

Test-case number 12: Filling of a cubic mould by a viscous jet (PN, PE) By St´ephane Vincent, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 27 07, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected] Jean-Paul Caltagirone, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 66 80, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected] Olivier Lebaigue, DER/SSTH/LMDL, CEA/Grenoble, F-38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 36 70, Fax: +33 (0)4 38 78 50 36, E-Mail: [email protected]

12.1

Practical significance and interest of the test-case

The interest of the injection of a viscous fluid in a cubic cavity is to estimate the consistency and the physical meaning of the numerical solutions of multiphase flows modeled by interface tracking methods. The considered problem emphasizes the competition between the inertia of the jet, the viscous effects and the gravity. Even if the surface tension exists and can be taken into account, it is negligible in the filling process with a viscous fluid. Under certain velocity and geometrical conditions, the jet fills the mould regularly whereas it can oscillate in a three dimensional manner under other assumptions. This test is interesting because as it induces strong interface deformations and tests the ability of the numerical method to capture 3D instabilities. The three-dimensional character of the problem makes axisymmetric simulations impossible to be leaded.

94/283

12.2

Test-case number 12 by S. Vincent, J.P. Caltagirone and O. Lebaigue

Definitions and physical model description

A cubic cavity of side L, initially full of air, is filled with a viscous fluid by a cylindrical injector of radius R centered at point (xc , yc , zc ) on the upper boundary of the mould. The injection velocity U0 is considered constant over time and in the jet. The density and the viscosity of the fluids are ρl and µl for the liquid and ρg and µg for air. The flow is assumed isothermal with a constant surface tension between the two fluids. The flow is submitted to inertia, gravity and viscous effects. The relevant dimensionless number for the problem are the Reynolds number Re and the Weber number W e: We =

ρl U02 D σ (12.1)

ρl U0 D Re = µl where D = 2R. Following the work of Cruickshank (1988), it is observed that the jet is unstable during the injection for Reynolds numbers in the range Re < 0.56 and for cavity over injector aspect ratios L/D > (2n + 1)π, where n is the instability mode of the jet. Depending on the problem configuration (Reynolds number and aspect ratio), the oscillations can be two dimensional (from left to right) or toroidal (three-dimensional rotation), corresponding to instability mode n = 0 and n = 1 respectively. The first instability which appears is n = 0 and it evolves generally towards the n = 1 mode.

12.3

Test-case description

The operating conditions of the filling process for a typical oscillating jet are the following. R = 0.08 m L=1m u0 = (0, 0, −0.8) and U0 = 0.8 in m.s−1 0.0143 m ≤ ∆x ≤ 0.033 m xc = L/2, yc = L/2, zc = L g = (0, 0, −9.81) in m.s−2

(12.2)

where g is the gravity vector. The characteristics of the two phases are: ρl = 1800 kg m−3 ρg = 1.1768 kg m−3 µl = 5.102 Pa s µg = 10−5 Pa s

(12.3)

The surface tension of the liquid-air interface is σ = 0.03 N.m−1 . The dimensionless numbers (12.1) of the problem have the values: W e = 6144 Re = 0.4608

(12.4)

No slip conditions are imposed on all the boundaries of the cavity except on the open z upper one which is described by a free outlet boundary condition ( ∂u ∂z = 0) except on the the injector outlet where a uniform velocity u0 in set.

Test-case number 12 by S. Vincent, J.P. Caltagirone and O. Lebaigue

12.4

95/283

Figures, tables, captions and references

The numerical simulations are leaded with the VOF-PLIC method of Youngs (1982) using the CSF method of Brackbill et al. (1992) for the treatment of the surface tension. Following the works of Vincent (1999) and Vincent & Caltagirone (1999), a regular Cartesian 70 x 70 x 70 grid is implemented to illustrate a possible numerical simulation (see figure 12.1). This grid was prooved sufficient to obtain a converged numerical solution for an unstable jet assuming H/D = 8.33 and Re = 0.4608. With respect to several values of the aspect ratrio, H/D, and the Reynolds number, Re, (see figure 12.2), the numerical stability of the jet is compared to the reference experimental and theoretical results of Cruickshank (1988). The transition limit on Re is well described by the numerical solutions. However, the calculated transitional aspect ratio H/D is found to be in the range of 2nπ whereas the asymptotic analysis of Cruickshank (1988) on an axisymetric jet shows H/D = (2n + 1)π. The gap between the numerical and theoretical results can be explained by the three-dimensional character of the instability in the numerical simulation.

References Brackbill, J.U., Kothe, D.B., & Zemach, C. 1992. A continuum method for modeling surface tension. J. Comput. Phys., 100, 335–354. Cruickshank, J.O. 1988. Low-Reynolds-number instabilities in stagnating jet flow. J. Fluid Mech., 193, 111–127. Vincent, S. 1999. Modeling incompressible flows of non-miscible fluids. Ph.D. thesis, Speciality: Mechanical Engineering, Bordeaux 1 University, France. Vincent, S., & Caltagirone, J.-P. 1999. Efficient solving method for unsteady incompressible interfacial flow problems. Int. J. Numer. Meth. Fluids, 30, 795–811. Youngs, D.L. 1982. Time-dependent multimaterial flow with large fluid distortion. K.W. Morton and M.J. Baines (eds), New-York, U.S.A.

96/283

Test-case number 12 by S. Vincent, J.P. Caltagirone and O. Lebaigue

Figure 12.1: Three-dimensional numerical simulation of the filling of a square cavity by a viscous jet. The results correspond to a space scale ∆x = 0.0143 m and time t = 1, 2, 4, 5, 6 et 8 s (from left to right and from top to bottom).

Test-case number 12 by S. Vincent, J.P. Caltagirone and O. Lebaigue

97/283

Theoretical transition curve to instability Direct numerical simulation - non oscilating jets Direct numerical simulation - oscilating jets

Figure 12.2: Instability transition diagram for a stagnating viscous jet in a square mould. Comparisons between theoretical and experimental results of Cruickshank (1988) and numerical simulations.

98/283

Test-case number 12 by S. Vincent, J.P. Caltagirone and O. Lebaigue

Test-case number 13 by S. Rouy and P. Helluy

99/283

Chapter 13

Test-case number 13: Shock tubes (PA) By Sandra Rouy, CIG Ecole des Mines de Paris, 35 rue Saint-Honor´e, 77305 Fontainebleau cedex, France Phone: +33 (0)1 64 69 47 33, Fax: +33 (0)1 64 69 47 03, E-Mail: [email protected] Philippe Helluy, ANAM, ISITV, BP 56, 83162 La Valette du Var cedex, France Phone: +33 (0)4 94 14 25 59, Fax: +33 (0)4 94 24 48, E-Mail: [email protected] Samuel Kokh, DEN/DM2S/SFME/LETR, CEA/Saclay, 91191 Gif-sur-Yvette, France E-Mail: [email protected]

13.1

Introduction

We are interested here in test-cases about two-fluid compressible flows. An analytical solution is given by the exact solution of the Riemann Problem associated with the system of balance equations describing the behavior and the motion of the flow (Rouy, 2000, Barberon et al. , 2003b,a). The hypotheses considered are the following: • both fluids of the flow are separated by a topologically simple interface, • the flow is compressible, inviscid, with no chemical reaction and no heat exchange, • the surface tension and the gravity effects are neglected. First, we propose one-dimensional analytical results for a gas-gas flow, then onedimensional analytical results for a gas-liquid flow are also shown.

100/283

13.2

Test-case number 13 by S. Rouy and P. Helluy

The mathematical model and the solution of the corresponding Riemann Problem

The mixture of both fluids is supposed to be a continuous medium: density, momentum and energy can be defined at each point of the domain. These functions are not necessarily continuous. According to the physical hypotheses described below, the equations to take into consideration are the balance equations for a compressible flow: balance of mass ρ − → (13.1), momentum ρ V (13.2) and specific total energy ρE (13.3) ∂ρ − → + div(ρ V ) = 0, ∂t − → −−→ ∂(ρ V ) −→ − → − → + div(ρ V ⊗ V ) + gradP = 0, ∂t ∂ − → (ρE) + div((ρE + P ) V ) = 0. ∂t

(13.1) (13.2) (13.3)

In order to solve this system, a closure relation must be used. When dealing with a single fluid, it is classical to use an equation of state P = P (ρ, e) which links − → pressure to density and to internal energy e with E = e + 1/2|| V ||2 . The perfect gas equation P = (γ − 1)ρe where γ = cp /cv denotes the ratio of the specific heat capacities, is one of these. Other relations exist. For example, the Stiffened-gas equation P = (γ − 1)ρe − γP∞ can be used for different types of materials. The constants γ and P∞ depend on the considered material and are determined by experimental measurements. This relation has been chosen because it is relevant for liquids and gases. The coefficients γ and P∞ must be in such a way that they respect the nature of the fluid. The pressure of the medium does not only depend here on the density and the internal energy but also depends on the location of the interface Γ between the two fluids. In fact the interface is seen as a discontinuity of the physical properties of the continuous medium. Consequently to determine the pressure and to close the system, we need to track the interface. The interface moves at the velocity of the continuum. If we define a quantity denoted φ which determines its location, this quantity verifies the following equation, ∂φ − → −−→ + V .gradφ = 0. ∂t

(13.4)

Thanks to the mass balance (13.1), this convection equation can be rewritten in a conservative form, ∂ (ρφ) − → + div(ρ V φ) = 0. ∂t

(13.5)

Several choices of φ are possible (volume fraction, level set function, etc.). They correspond to different types of numerical methods for tracking interfaces: VOF methods (Hirt & Nichols, 1979), front tracking methods (Sethian et al. , 1992, Glimm, 1986) or multifluid methods (Saurel & Abgrall, 1999). Finally, we obtain the following system, ∂U ∂f (U ) + = 0, x ∈ R, t > 0. ∂t ∂x

(13.6)

Test-case number 13 by S. Rouy and P. Helluy

101/283

with U = (ρ, ρu, ρE, ρφ)T , x ∈ R, t > 0, f (U ) = (ρu, ρu2 + P, (ρE + P )u, ρφu)T , E = e + 12 u2 and the closure relation P = (γ(φ) − 1)ρe − γ(φ)P∞ (φ) Definition 1 The one-dimensional Riemann Problem associated to the system (13.6) is the following Cauchy Problem, ∂U ∂f (U ) + = 0, x ∈ R, t > 0 ∂t ∂x U , x 0

(13.7)

It is well known that this system admits a weak auto-similar solution (function of x/t), physically admissible which consists in a succession of constant states separated by entropic shock waves, rarefaction waves or contact discontinuities. This solution is unique and is shown in figure 13.1.

Figure 13.1: The solution of the Riemann problem

According to Rouy (2000), solving the Riemann Problem (13.7) is equivalent to solve the above equation, uL − uR − ΞL (P ) − ΞR (P ) = 0,

(13.8)

where i

Ξ (P ) =



Φi (P ), if P > Pi (shock wave) Ψi (P ), if P < Pi (rarefaction wave)

(13.9)

with, √ (P − Pi ) 2τi Φ (P ) = p (γi + 1)P + (γi − 1)Pi + 2γi P∞,i i

(13.10)

and, √

Ψi (P ) = 2

i −1 γi τi p P + P∞,i γ2γ Pi + P∞,i (( ) i − 1) γi − 1 Pi + P∞,i

(13.11)

τ denoting the covolume i.e. the inverse of the density, i = L for the left-hand side wave and i = R for the right-hand side wave. The solution of the equation (13.8) for P is obtained by Newton’s method. Then, we determine τ1 and τ2 with the following relations:

102/283

Test-case number 13 by S. Rouy and P. Helluy

• if P > PL , we have a 1-shock, τ1 = hL s (P ) • if P < PL , we have a 1-rarefaction wave, τ1 = hL r (P ) • if P > PR , we have a 3-shock, τ2 = hR s (P ) • if P < PR , we have a 3-rarefaction wave, τ2 = hR r (P ), with Pi + P∞,i γ1 ) i P + P∞,i

(13.12)

(γi + 1)Pi + (γ − 1)P + 2γi P∞,i (γi + 1)P + (γi − 1)Pi + 2γi P∞,i

(13.13)

hir (P ) = τi ( and his (P ) = τi

Finally, the velocity of the contact discontinuity, u, can be obtained for example by the following relation, u(P ) = uL − ΞL (P )

(13.14)

The indexes L or R show us from which constant state (Left or Right) the functions Ξ and h are depending. Remark 1 In Rouy (2000), Barberon et al. (2003a), we have demonstrated that the Riemann Problem admits a global solution i.e. even in the case of vacuum apparition. Physically, this corresponds to a null pressure (in the particular case of vacuum apparition in gas) or to a negative pressure (in the particular case of cavitation in water).

13.3

The shock tube

This is a one-dimensional test-case. We consider a tube with an arbitrary length (here L=8 m) initially filled with two fluids separated by a membrane (see figure 13.2). We suppose that the membrane breaks and we observe the evolution. The interface between both fluids moves. The thermodynamical and kinematical quantities change. This solution is given by the solution of the Riemann Problem presented above.

Figure 13.2: The shock tube

We propose 4 test-cases : 1. Both parts of the tube are initially filled with the same gas, with different physical characteristics detailed in table 13.1.

Test-case number 13 by S. Rouy and P. Helluy

ρ u P γ P∞

Left state 1 kg/m3 0 m s−1 1.2 bar 1.4 0 bar

103/283

Right State 2 kg/m3 0 m s−1 1 bar 1.4 0 bar

Table 13.1: Parameters for test-case 1

ρ u P γ P∞

Left state 0.192 kg/m3 0 m s−1 1 bar 1.667 0 bar

Right State 1.156 kg/m3 0 m s−1 1 bar 1.4 0 bar

Table 13.2: Test-case 2

2. The left part contains helium and the right part contains air. Both gases have null velocity and have the same pressure (test-case of stationary contact discontinuity, see table 13.2). 3. The previous test is modified in such a way that the ratio of the two pressures increases, the velocity of helium is now equal to 10 m/s (see table 13.3).

ρ u P γ P∞

Left state 0.192 kg/m3 10 m s−1 10 bar 1.667 0 bar

Right State 1.156 kg/m3 0 m s−1 1 bar 1.4 0 bar

Table 13.3: Test-case 3

4. The last test deals with a gas-liquid flow with a ratio of pressure equal to 100, the initial conditions are shown in table 13.4. ρ u P γ P∞

Left state 10 kg/m3 10 m s−1 100 bars 1.4 0 bar

Right State 1000 kg/m3 0 m s−1 1 bar 5.5 4900 bars

Table 13.4: Test-case 4

The calculated results to be considered as the reference solution to these test-cases are shown in figures 13.3,13.4, 13.5 and 13.6.

104/283

Test-case number 13 by S. Rouy and P. Helluy

Figure 13.3: Shock tube 1, gas-gas, time=5 ms, CFL=0.18, 400 cells.

Test-case number 13 by S. Rouy and P. Helluy

Figure 13.4: Shock tube 2, helium-air, time=5 ms, CFL=0.39, 400 cells.

105/283

106/283

Test-case number 13 by S. Rouy and P. Helluy

Figure 13.5: Shock tube 3, helium-air, time=1 ms, CFL=0.28, 400 cells.

Test-case number 13 by S. Rouy and P. Helluy

Figure 13.6: Shock tube 4, gas-liquid, time=1 ms, CFL=0.14, 400 cells.

107/283

108/283

Test-case number 13 by S. Rouy and P. Helluy

References Barberon, T., Helluy, P., & Rouy, S. 2003a. Apparition du vide - M´ethode `a deux flux. Computers and Fluids to appear. Barberon, T., Helluy, P., & Rouy, S. 2003b. Practical computation of axisymmetrical multifluid flows. International Journal of Finite Volumes. Glimm, J. 1986. Front tracking applied to Rayleigh-Taylor instability. Physics of fluids, 10. Hirt, C., & Nichols, B. 1979. VOF Method for the dynamics of free boundaries. Journal of Computational Physics, 39, 201–225. Rouy, S. 2000. Mod´elisation math´ematique et num´erique d’´ecoulements diphasiques compressibles - Application au cas industriel d’un g´en´erateur de gaz. Ph.D. thesis, Universit´e de Toulon et du Var, France. Saurel, R., & Abgrall, R. 1999. A simple method for compressible multifluid flows. SIAM Journal of Scientific Computing, 21, 1115–1145. Sethian, J., Mulder, W., & Osher, S. 1992. Computing interface motion in compressible gas dynamics. Journal of Computational Physics, 100, 209–228.

Test-case number 14 by S. Vincent, J.P. Caltagirone and O. Lebaigue

109/283

Chapter 14

Test-case number 14: Poiseuille two-phase flow (PA) By St´ephane Vincent, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 27 07, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected] Jean-Paul Caltagirone, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 66 80, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected] Olivier Lebaigue, DER/SSTH/LMDL, CEA/Grenoble, F-38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 36 70, Fax: +33 (0)4 38 78 50 36, E-Mail: [email protected]

14.1

Practical significance and interest of the test-case

The two-phase Poiseuille flow is a simple interfacial flow that permits to estimate accurately the time and space convergence order of the numerical resolution of the Navier-Stokes equations in their Eulerian two-phase flow formulation. Moreover, this test case allows characterizing the sensitivity of the numerical solution with respect to the averages implemented on the density and the viscosity at the interface in the discretization of the motion equations. To finish with, the Poiseuille flow allows calculating analytically the viscous stress tensor to verify if the continuity of its tangential component is verified numerically at the interface.

The present test case is interesting because it possesses a theoretical solution. However, no interface deformation is induced in this problem. Therefore, it represents a necessary test, but certainly not a sufficient reference.

110/283

Test-case number 14 by S. Vincent, J.P. Caltagirone and O. Lebaigue

Figure 14.1: Two-phase Poiseuille flow between to parallel walls

14.2

Definitions and physical model description

The horizontal stratified flow of a two fluid between two parallel walls is considered (see figure 14.1). The gravity and the surface tension forces are neglected. For long times, a steady solution is obtained for the two-phase Poiseuille flow problem which can be described by an analytical solution. If L is the length of the horizontal walls, d is the distance between the bottom horizontal boundary and the interface and H the distance between the two horizontal boundaries, the velocity field, u = (ux , uy ), and the pressure, p, can be calculated by assuming the velocity to be parallel to the x axis and by considering the continuity of the velocity and the viscous stress tensor at the interface. In this way, the velocity field, u1 (ux,1 , uy,1 ) and u2 (ux,2 , uy,2 ), respectively in each fluid are given by the following equations (Coutris et al. , 1989, Vincent, 1999) ∆p(d[µ2 − µ1 (1 − d)]y 2 − H[µ2 − µ1 (1 − d2 )]y) 2µ1 d(µ2 − µ1 [1 − d]) uy,1 (x, y) =0

ux,1 (x, y) =

ux,2 (x, y) =

∆p(d[µ2 − µ1 (1 −

d)]y 2

d2 )]y

− H[µ2 − µ1 (1 − − 2µ2 d(µ2 − µ1 [1 − d])

(14.1) (14.2) H 2 [µ2 d

+ µ1 (1 − d)]) (14.3)

uy,2 (x, y) =0 (pr − pl ) p(x, y) = x + pl L

(14.4) (14.5)

where ∆p = pr − pl is the pressure difference between the outlet and the inlet boundary pressure.

14.3

Test-case description

The flow characteristics are defined by • ∆p = −0.212435 P a,

Test-case number 14 by S. Vincent, J.P. Caltagirone and O. Lebaigue

111/283

• H/d = 0.5, • 1 ≤ ρ2 /ρ1 ≤ 1000, • 1 ≤ µ2 /µ1 ≤ 5000, • ∆H/512 ≤ x ≤ H/2. No-slip boundary conditions are imposed on the horizontal walls whereas Neumann conditions are assumed on the velocity on the inlet and outlet boundaries. The pressures pl and pr are imposed on the left and right limits of the calculation domain to ensure the pressure difference ∆p. As an example, the comparison between the analytical and the numerical velocity fields implemented with a Volume Of Fluid (VOF) numerical model (Vincent & Caltagirone, 2000) is presented in figure 14.2 with H = 0.02m, µ1 = 5 · 10−4 P a.s and µ2 = 1.85 · 10−5 P a.s. Whatever the grid resolution N in each direction (from 2 to 512 discretization points in the cross flow direction), the L2 absolute error is almost equal to the truncation error (see table 1).

0.175 Theoretical Numerical

Velocity (m/s)

0.15 0.125 0.1 0.075 0.05 0.025 0

0

0.005

0.01

0.015

0.02

Height (m) Figure 14.2: VOF simulation of the two-phase Poiseuille flow between to parallel walls on a 32 x 32 grid for H = 0.02m, µ1 = 5 · 10−4 P a.s and µ2 = 1.85 · 10−5 P a.s - Comparison between numerical and theoretical solutions.

References Coutris, N., Delhaye, J.M., & Nakach, R. 1989. Two-phase flow modelling: the closure issue for a two-layer flow. Int. J. Multiphase Flow, 15, 977–983.

112/283

Test-case number 14 by S. Vincent, J.P. Caltagirone and O. Lebaigue

N 2 32 512

Absolute error 0.2262 · 10− 15 0.6921 · 10− 15 0.7679 · 10− 13

Table 14.1: Evolution of the absolute error on velocity with respect to the grid resolution.

Vincent, S. 1999. Modeling incompressible flows of non-miscible fluids. Ph.D. thesis, Speciality: Mechanical Engineering, Bordeaux 1 University, France. Vincent, S., & Caltagirone, J.-P. 2000. A One Cell Local Multigrid method for solving unsteady incompressible multiphase flows. J. Comput. Phys., 163, 172–215.

Test-case number 15 by S. Vincent, J.P. Caltagirone and D. Jamet

113/283

Chapter 15

Test-case number 15: Phase inversion in a closed box (PC) By St´ephane Vincent, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 27 07, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected] Jean-Paul Caltagirone, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 66 80, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected] Didier Jamet, DER/SSTH/LMDL, CEA/Grenoble, 38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 45 42, Fax: +33 (0)4 38 78 50 36, E-Mail: [email protected]

15.1

Practical significance and interest of the test-case

The flow induced by the action of the buoyancy force on an oil inclusion in a cavity full of water is considered. While the density difference effect is counteracted by the viscous and surface tension forces, an unsteady turbulent two-phase flow develops leading to strong interface shearing and stretching with drop extraction and collapse. The interest of this test-case is the simplicity of its initial condition and the complexity of the interface structures generated in terms of strong deformations or shear instabilities. In this regard, this test-case is aimed at testing the ability of a numerical method to simulate turbulent two-phase flows with large and complex interface deformations. Due to the complexity of the flow, the detailed structure of the flow cannot be analyzed and compared to any reference solution. The second asset of the problem is the existence of a theoretical solution for sufficiently long times concerning the exact position of the interface, which corresponds to a horizontal oil layer in the top part of the cavity and a horizontal water layer in the bottom part of the square box. At equilibrium, the two fluids are separated by a horizontal interface whose position only depends on the initial volumes of the fluids. In this regard, this test-case is particularly relevant to test the ability of a numerical to conserve mass and volume even for very large and complex interface deformations. To

114/283

Test-case number 15 by S. Vincent, J.P. Caltagirone and D. Jamet

finish with, the independence of the steady solution of the phase inversion problem on the physical parameters such as density, viscosity or surface tension allows to treat various numerical methods and physical configurations.

15.2

Definitions and physical model description

A square oil inclusion of height H is initially placed in the left corner of a square cavity full of water, whose characteristic length is L. Gravity and surface tension effects are taken into account. The problem can be solved in two- or three-dimensions without loss of generality. H = L/2, so that for long time simulations, the equilibrium heights of the oil and water layers are respectively H/4 and 3H/4 in two dimensions and H/8 and 7H/8 in three-dimensions. p Based on the characteristic velocity u0 = gH, where g is the gravity, the characteristic dimensionless numbers of the problem, are the Weber and the Reynolds numbers defined by (ρw − ρo )u20 L σ ρw u 0 L • Rew = µw • We =

• Reo =

15.3

ρo u 0 L µo

Test-case description

The physical parameters chosen as an interesting and characteristic example of turbulent and unsteady two-phase flow are the following: • H = 0.1 m, • L = 0.2 m, • 0.0002 m ≤ ∆x ≤ 0.001 m, where ∆x is the mesh spacing supposed to be regular and equal in all directions, • g = 9.81 m.s−2 . The corresponding characteristic velocity is u0 = 0.99 m.s−1 . The physical characteristics of oil and water are the following: • densities: ρw = 1000 kg.m−3 , ρo = 900 kg.m−3 • dynamic viscosities: µw = 5. 10−3 Pa.s, µo = 10−1 Pa.s • surface tension: σ = 0.045 N.m−1 The corresponding characteristic dimensionless numbers are the following: • W e = 436 • Rew = 3960 • Reo = 891 Both phases are initially at rest and no slip boundary conditions are imposed on the walls of the closed cavity.

Test-case number 15 by S. Vincent, J.P. Caltagirone and D. Jamet

15.4

115/283

Illustrations of the problem

The numerical simulations are run with a VOF-PLIC method of Youngs (1982) using the CSF method Brackbill et al. (1992) for the treatment of the surface tension. The results shown in figure 15.1 correspond to an example due to Caltagirone & Vincent (2003) when a regular Cartesian 256 x 256 grid is implemented.

References Brackbill, J.U., Kothe, D.B., & Zemach, C. 1992. A continuum method for modeling surface tension. J. Comput. Phys., 100, 335–354. Caltagirone, J.P., & Vincent, S. 2003. Validation of the numerical fluid dynamics library Aquilon. Internal report, 1, 1–250. Youngs, D.L. 1982. Time-dependent multimaterial flow with large fluid distortion. K.W. Morton and M.J. Baines (eds), New-York, U.S.A.

116/283

Test-case number 15 by S. Vincent, J.P. Caltagirone and D. Jamet

Figure 15.1: Direct numerical simulation of the phase inversion problem in two-dimensions for oil and water. With ∆x = 0.390625 mm, the results show the interface profiles corresponding to times t = 0, 1.2, 2.4, 4.8, 10, 20, 30, 36 and 44 s (from left to right and from top to bottom). The simulations are stopped after thousands of iterations. For longer times, a steady state with two horizontal layers should be obtained.

Test-case number 16 by F. Pigeonneau and F. Feuillebois

117/283

Chapter 16

Test-Case number 16: Impact of a drop on a thin film of the same liquid (PE, PA) By F. Pigeonneau, Saint-Gobain Recherche, BP 135 - 39 quai Lucien Lefranc F-93303 Aubervilliers cedex, France Phone: +33 (0)1 48 39 59 99, Fax: +33 (0)1 48 39 58 78 E-Mail:[email protected] F. Feuillebois, PMMH, UMR 7636 CNRS, ESPCI, 10 rue Vauquelin 75231 Paris cedex 05, France Phone: +33 (0)1 40 79 45 53, Fax: +33 (0)1 40 79 47 95 E-Mail: [email protected]

16.1

Practical significance and interest of the test case

Various physical phenomena may occur following the impact of a drop, depending on the target and physical conditions. This test case is limited to the impact of a drop on a thin film of the same liquid. This problem has a number of applications, e.g. in chemical engineering (coating, cooling by drops, erosion by drops, painting, etc.), material processing (welding, etc.), agriculture (dispersion of products in fields, etc.). The purpose of the present test case is to compare numerical results to existing experimental ones. As described by Levin & Hobbs (1971), Cossali et al. (1977) and Cossali et al. (1991), the impact of a drop on a thin film consists of several steps: formation of a crown and jetting, instability of the crown and formation of jets which themselves form secondary droplets. Pictures reproduced from Cossali et al. (1977) are presented in figure 16.1. We will limit ourselves here to study the formation of the crown and its growth with time, as shown essentially by the first three pictures in figure 16.1.

16.2

Definitions and physical model description

Let d be the drop diameter and dc be the crown diameter. Let U be the drop impact velocity. Let h be the height of the liquid film. Consider the following experiments from

118/283

Test-case number 16 by F. Pigeonneau and F. Feuillebois

Figure 16.1: Evolution of the crown for a drop impact onto a thin liquid film. From Cossali et al. (1977).

Levin & Hobbs (1971) and Cossali et al. (1977), using water in both cases:

Levin & Hobbs (1971) Cossali et al. (1977)

d (mm) 2.9 5.1

h (mm) 0.5 0.5

U (m/s) 4.8 2.14

Let ρ be the mass density of the liquid and σ its surface tension. Note that the Weber number defined by: We =

ρ Ud σ

is of the same order of magnitude for both cases. The value of U for Cossali et al. (1977) was derived from their data that We = 320 in their case, using the standard values ρ = 103 kg/m3 and σ = 7.3 × 10−2 N/m. The Ohnesorge number Oh = √

µ , dσρ

where µ is the fluid dynamic viscosity, is also of the same order for both cases, so that as explained in Cossali et al. (1977) the phenomena are expected to be similar for both experiments. Moreover, the relative height of the film is also of the same order for both cases. Based on these ideas, we plotted the results for the evolution of dc with time τ from Levin & Hobbs (1971) and Cossali et al. (1977) in dimensionless variables, using: Dc =

dc , d

t=

τU . d

It is observed in figure 16.2 that both sets of experiments superimpose. We also

Test-case number 16 by F. Pigeonneau and F. Feuillebois

119/283

7

Nondimensional crown diameter Dc

6

5

4

3

2

1

0

Levin & Hobbs (1971) experiments Cossali et al (1997) experiments Yarin & Weiss (1995) formula Power law with exponent n= 0.38 0

1

2

3 4 Nondimensional time t

5

6

7

Figure 16.2: Time evolution of the crown diameter, in dimensionless variables. Experiments by Levin & Hobbs (1971) and Cossali et al. (1977); square root law from the model of Yarin & Weiss (1995) and fitted power law Dc ∼ t0.38 .

√ plotted in the same figure the square root law Dc ∼ t proposed by Yarin & Weiss (1995) from their numerical simulation assuming an inviscid fluid. Note that the numerical simulation of Josserand & Zaleski (2003) which includes viscosity effects also gives a √ Dc ∼ t behavior. As seen in figure 16.2, a power law of the form Dc ∼ t0.38 fits better all experimental results. In later experimental results, Cossali et al. (1991) defined several crown diameters, viz. upper, lower, inner, outer and measured their time evolution. They also remarked that attempts to fit the whole experimental data with the model of Yarin & Weiss (1995) were not successful. Instead, they found for all crown diameters a law of the form Dc ∼ tn , where n is around 0.4, that is similar to the results mentioned above. Correlating some experimental point of Thoroddsen (2003) for the radial deceleration of the sheet (his figure 3(b)) gives dDc /dt ∼ t(−0.65) , consistent the preceding results. There are not so many results concerning the height of the crown. The maximum height of the crown was measured for various liquids by Macklin & Metaxas (1976) and the ratio of the crown height H to the crown radius Rc = dc /2 is plotted versus the Weber number based on the drop radius Wb=We/2 in figure 16.3. The case of a shallow liquid is of interest here. Cossali et al. (1991) find a law H/Rc ∼ W bm , where m is about 0.65 to 0.75.

16.3

Test-case description

The calculation will concern the impact of a drop of liquid on a thin film. Typically, water will be considered, with drop diameters and velocities of the order of the experimental data given in the table. A challenge is to model the experimental results for the time

120/283

Test-case number 16 by F. Pigeonneau and F. Feuillebois

Figure 16.3: Evolution of the dimensionless crown height from Macklin & Metaxas (1976). Here, D∗ = h/Rc .

evolution of the crown diameter which follows an empirical law of the type Dc ∼ tn , where n is close to 0.4 rather than the value 0.5 obtained in Yarin & Weiss (1995) and Josserand & Zaleski (2003). Then comparison with the experimental data for the crown height (figure 16.3 from Macklin & Metaxas (1976)) and Cossali et al. (1991)) may be tried, for various liquids. Note that this test case also suggests that more experiments are needed.

Test-case number 16 by F. Pigeonneau and F. Feuillebois

121/283

References Cossali, G.E., Coghe, A., & Marengo, M. 1977. The impact of a single drop on a wetted solid surface. Experiments in fluids, 22, 463–472. Cossali, G.E., Brunello, G., Coghe, A., & Marengo, M. 1991. Impact of a single drop on a liquid film: experimental analysis and comparison with empirical models. Proc. Italian Congress of Thermofluid Dynamics UIT, Ferrara, 30 June - 2 July. Josserand, C., & Zaleski, S. 2003. Droplet splashing of a thin liquid film. Phys. Fluids, 15(6), 1650–1657. Levin, Z., & Hobbs, P.V. 1971. Splashing of water drops on solid and wetted surfaces: Hydrodynamics and charge separation. Phil. Trans. R. Soc. A, 269, 555–585. Macklin, W.C., & Metaxas, G.J. 1976. Splashing of drops on liquid layers. J. Appl. Phys., 47(9), 3963–3970. Thoroddsen, S.T. 2003. The ejecta sheet generated by the impact of a drop. J. Fluid Mech., 451, 373–381. Yarin, A.L., & Weiss, D.A. 1995. Impact of drops on solid surfaces: Self-similar capillary waves and splashing as a new type of kinematic discontinuity. J. Fluid Mech., 283, 141–173.

122/283

Test-case number 16 by F. Pigeonneau and F. Feuillebois

Test-case number 17 by S. Vincent, J.P. Caltagirone and E. Canot

123/283

Chapter 17

Test-case number 17: Dam-break flows on dry and wet surfaces (PN, PA, PE) By St´ephane Vincent, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 27 07, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected] Jean-Paul Caltagirone, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 66 80, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected]

17.1

Practical significance and interest of the test-case

The dam-break problem has been widely studied in the literature by many different experimental, theoretical and numerical methods (see Harlow & Welch (1965) for example). Under the assumptions of isothermal and incompressible flow, two configurations are considered in this test-case (see figure 17.1): • Problem 17.a: the dam-break on a wet ground where initially, a water layer of height hl and length L/2 is considered, to the right hand side whereas we consider a water layer of height hr on the other side. The dam is supposed to break instantaneously and we want to predict the free surface evolutions, hi , and the hydrodynamics at every instant after the dam breaks. • Problem 17.b: the dam-break on a dry ground with identical characteristics, except that a dry bottom exists downstream from the dam (hr = 0). The two test cases are very interesting because they allow to validate the numerical simulations compared to experimental data and unsteady theoretical solutions. The problem emphasizes the influence of the gravity, the surface tension and the viscosity. Shock and rarefaction waves appear during the wave breaking over the downstream water layer.

124/283

Test-case number 17 by S. Vincent, J.P. Caltagirone and E. Canot

In certain cases, it is observed the developmental of a water jet leading to free surface shearing, stretching and droplet ejection. The strong deformations of the interface and the unsteady character of the flow confer on the two test cases a reference point of view to validate front tracking methods as well as numerical approximation of the two-phase Navier-Stokes equations.

17.2

Definitions and physical model description

Dam-break simulations in two dimensions are considered here. The two problems 17.a and 17.b were studied experimentally by Martin & Moyce (1952) on the dry case and P.K. Stansby & Barnes (1998) for the two cases. Accurate experimental data are available in these previous references. The dam-break flow admits in addition analytical solutions under the hydrostatic pressure and perfect fluid assumptions (see the works of S. Vincent & Caltagirone, 2001, Vincent, 1999, for example), through the Saint-Venant or Shallow-Water model and the non-linear characteristic theory (see figures 17.2 and 17.3). The theoretical solution of test case 17.a is given by the following procedure where the notations are defined in figure 17.2. √ • from x = −L/2 to OA (OA is the curve x = − ghl t), the analytical solution is um = 0 and hi = hr . √ • from OA to OB (OB is the curve x = hs t), we have u = ( 23 ghl + 23 xt ) and 2p x ( ghl − )2 3 3t . hi = g • from OB to OS (OS corresponds to the curve x = st, where s is the shock velocity), the free surface hs and the velocity us , as well as s, are obtained by solving the following system p p us + 2 ghs − 2 ghl = 0 (17.1) s(hr − hs ) + hs us = 0

shs us − (

gh2r (hs us )2 gh2s )+( + )=0 2 hs 2

(17.2)

(17.3)

• from OS to x = L/2, the solution is um = 0 and hi = hr where um is the mean velocity of the flow deduced from the Saint-Venant equations, t is time and x is the horizontal space coordinate. For test case 17.b, we obtain, • from x = −L/2 to OA,√a steady state corresponding to the initial state is obtained. OA is the curve x = − ghl t. The analytical solution is um = 0 and hi = hl • from OA to OS, a rarefaction wave develops containing a sonic √ point at x = 0. √ OS is the curve x = 2 ghl t. The theoretical solution is um = ( 32 ghl + 23 xt ) and √ x )2 ( 32 ghl − 3t hi = g • from OS to x = L/2, the steady initial state is kept. The solution is then um = 0 and hi = hr

Test-case number 17 by S. Vincent, J.P. Caltagirone and E. Canot

125/283

√ Considering the celerity of the surface waves cg = ghl as the reference velocity of the problem, we define the following dimensionless Froude and Reynolds numbers : • Fr =

u0 , cg

• Re =

ρl (hl − hr )cg . µl

where u0 is a characteristic velocity of the flow defined as the averaged velocity in the flow domain.

17.3

Test-case description

In cases 17.a and 17.b, the fluid characteristics (water and air, referred by the subscripts l and g for liquid and gas) are the following. • density: ρl = 1000 kg.m−3 , ρg = 1.1768 kg.m−3 • dynamic viscosity: µl = 10−3 Pa.s, µg = 10−5 Pa.s Paying attention to the space scale of the problem, the surface tension effects are neglected as the space scale of the problem is very large. The gravity norm is g = 9.81 m.s−2 . The calculation domain is described by the length L and the height H. The geometrical parameters reads for case 17.a • L/2 = 0.6 m, • H = 0.14 m, • hg = 0.1 m, • hr < hl , • 0.0005 m ≤ ∆x ≤ 0.002 m whereas for test 17.b, we choose • L/2 = 0.6 m, • H = 0.14 m, • hl = 0.1 m, • hr = 0 m, • 0.0005 m ≤ ∆x ≤ 0.002 m The mesh size used for the simulations is referred by ∆x. Neumann homogeneous boundary conditions are imposed on the upper, left and right limits whereas no-slip conditions are implemented on the lower boundary of the calculation domain.

126/283

Test-case number 17 by S. Vincent, J.P. Caltagirone and E. Canot

g

Initial dam

position

hl

hr

L

Figure 17.1: Definition sketch

t

C+

S

B

A

S

C-

C-

C+ CC-

-L/2

0

L/2

x -L/2

0

x

L/2

√ Figure 17.2: Dam-break flow on wet bottom. Left: description of the characteristic curves C+ = u+ ghs in the plane(x, t). Initially, the dam-break occurs at x = 0. A shock wave S appears. Right: description √ of the characteristic curves C− = u − ghs in the plane (x, t). A rarefaction wave is generated between A et B.

t

C+

S

A

S

C-

C-

C+ C-

-L/2

0

L/2

x -L/2

C-

0

L/2

x

√ Figure 17.3: Dam-break flow on dry bottom. Left: description of the characteristic curves C− = u − ghs in the plane (x, t). Initially, the dam-break is situated at x = 0.√ A rarefaction wave appears between A and B. Right: description of the characteristic curves C+ = u + ghs in the plane (x, t). A shock wave is generated in S.

Test-case number 17 by S. Vincent, J.P. Caltagirone and E. Canot

127/283

Figure 17.4: Direct numerical simulation of the dam-break flow on wet bottom with hl = 0.1 m et hr = 0.01 m. The free surface and the velocity field are plotted. The physical parameters are the following : ∆x = 0.002 m, t = 0.06, 0.24, 0.3 and 0.42 s (from top to bottom), Re = 9 · 103 and 0 ≤ F r ≤ 3.

128/283

Test-case number 17 by S. Vincent, J.P. Caltagirone and E. Canot

References Harlow, F.H., & Welch, J.E. 1965. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluid, 8, 2182–2189. Martin, J.C., & Moyce, W.J. 1952. An experimental study of the collapse of liquid columns on a rigid horizontal plane. Phys. Trans. Serie A, Math. Phys. Sci., 244, 312–325. P.K. Stansby, A. Chegini, & Barnes, T.C.D. 1998. The initial stages of dam-break flow. J. Fluid Mech., 374, 407–424. S. Vincent, P. Bonneton, & Caltagirone, J.-P. 2001. Numerical modelling of bore propagation and run-up on sloping beaches using a MacCormack TVD scheme. J. Hydraulic Res., 39, 41–49. Vincent, S. 1999. Modeling incompressible flows of non-miscible fluids. Ph.D. thesis, Speciality: Mechanical Engineering, Bordeaux 1 University, France.

Test-case number 19 by G. Allaire and S. Kokh

129/283

Chapter 18

Test-case number 19: Shock-Bubble Interaction (PN) By Gr´egoire Allaire, CMAP, Ecole Polytechnique, 91128 Palaiseau cedex, France Phone: +33 (0)1 69 33 46 11, Fax: +33 (0)1 69 33 30 11 E-Mail: [email protected] Samuel Kokh, DEN/DM2S/SFME, CEA/Saclay, 91191 Gif-sur-Yvette cedex, France E-Mail: [email protected]

18.1

Introduction

We describe here a test proposed by Quirk & Karni (1996) based on the experiments of Haas & Sturtevant (1987). Its goal is to simulate the propagation of a shock through a helium bubble in air. Haas and Sturtevant initial purpose was to get a better understanding of the Richtmeyer-Meshkov instabilities. More generally they wanted to understand how pressure waves in heterogeneous media can generate turbulent phenomena which tend to mix the fluids. From a numerical point of view the goal of this test is to validate compressible multifluid flows models as well as numerical methods used for solving these models. This test has been performed at least in the following studies by Abgrall (1996), Fedkiw et al. (1999), Karni (1996), Kokh & Allaire (2001) and Saurel & Abgrall (1999).

18.2

Description

Geometry. The test is two-dimensional. The computational domain is a rectangular box which is 890 mm long (horizontal axis) and 89 mm high (vertical axis). At time t = 0, the bubble has a 50 mm diameter and its center is located at (xc , yc ), xc = 420 mm, yc = 44.5 mm (the origin being the low left corner of the domain). The initial location of the shock is a vertical line which is 222.5 mm away from the right side of the domain. Physical Model. The behavior of each fluid is governed by the gas dynamics compressible Euler equations (without any diffusion term, neither surface tension, nor gravity).

130/283

Test-case number 19 by G. Allaire and S. Kokh

89mm

50mm

222.5mm

222.5mm

890mm

Figure 18.1: Computational domain.

Gas Air Helium

γ 1.4 1.648

R (kJ.kg−1 .K−1 ) 0.287 1.578

Cv (kJ.kg−1 .K−1 ) 0.72 2.44

Table 18.1: Equation of state parameters.

Each fluid is assumed to obey the perfect gas equation of state. Thus the fluid i = 1, 2 is modeled by the following equations,  ∂t ρi + div (ρi ui ) = 0      ∂t (ρi ui ) + div (ρi ui ⊗ ui + pi I) = 0 (18.1)      ∂t (ρi ei ) + div [(ρi ei + pi )ui ] = 0 where ρi is the density, ui the velocity (it is a two-component vector), ei the specific total energy such that ei = εi + |ui |2 /2 with εi the specific internal energy, the pressure pi being provided by, pi = (γi − 1)ρi εi ,

(18.2)

where γi is the ratio of the heat capacities of the ith gas. The interface modeling and its numerical treatment are free choices (most of the simulation references use an isothermalisobaric mixture law in order to thicken the interface which is then captured on an Eulerian mesh). Numerical Data. The equation of state parameters for air and the helium bubble are provided in table 18.1 (data for R and Cv are, a priori, not required). Let us note that the parameters for the helium bubble describe indeed a 28% mass mixture between helium and air. The shock travels from the right side of the domain to the left side with a 1.22 Mach velocity. This means its velocity is 1.22 times higher than the sound velocity in the −3 pre-shock air at rest (atmospheric pressure and density equal p to 1 kg.m ). Let us recall that for a perfect gas, the sound velocity is given by c = γp/ρ and that the horizontal shock velocity is −1.443523 × 103 m.s−1 . The helium bubble is supposed to be initially

Test-case number 19 by G. Allaire and S. Kokh

zone units post-shock air (right side) pre-shock air (left side) helium bubble

131/283

density

pressure

internal energy

velocity

(kg.m−3 )

(bar)

(105 J.kg−1 )

(103 m.s−1 )

1.376363 1.0 0.181875

1.569800 1.0 1.0

2.851355 1.0 8.48500

(-0.394728 ; 0.0) (0.0 ; 0.0) (0.0 ; 0.0)

Table 18.2: Initial state.

at mechanical equilibrium with the surrounding air. Thanks to the Rankine-Hugoniot conditions it is possible to find the initial values for air after the shock. Using compatible units with those used in the equation of state, the initial state is defined in table 18.2 (let us recall that: 1 bar = 105 Pa, and Pa = 1 J m−3 ). Boundary conditions. The horizontal boundaries of the domain are solid walls where “mirror” boundary conditions are to be applied (i.e. non-penetration conditions). The right vertical boundary is set to be an “inflow” boundary condition equal to the initial data for air to the right of the shock. The left vertical boundary is treated as a free “outflow”, which means a zero order extrapolation of the variables has to be performed out of the computational domain. Measures and Comparisons. It is required to compare the shape of the bubble with Haas and Sturtevant experimental results at the following time steps: 32, 52, 62, 72, 82, 102, 245, 427, and 674 µs. We shall also plot the pressure evolution in time downstream from the position of the bubble. Remark. Data provided in Fedkiw et al. (1999) are different from those in Quirk & Karni (1996) which are being used here. There exists a similar test where the helium is replaced with a gas heavier than air (refrigerant fluid R22, see Quirk & Karni, 1996)) Remark. Let us note that similar tests are available (see e.g. Allaire et al. , 2002, Shyue, 1999), however no direct comparison is possible as these tests deal with different equations of state than those which are proposed here.

132/283

Test-case number 19 by G. Allaire and S. Kokh

References Abgrall, R. 1996. How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach. J. Comp. Phys., 125, 150–160. Allaire, G., Clerc, S., & Kokh, S. 2002. A five-equation model for the simulation of interfaces between compressible fluids. J. Comp. Phys., 181, 577–616. Fedkiw, R., Aslam, T., Merriman, B., & Osher, S. 1999. A non-oscillatory eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comp. Phys., 152, 457–492. Haas, J.-F., & Sturtevant, B. 1987. Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J. Fluid Mech., 181, 41–76. Karni, S. 1996. Hybrid multifluid algorithms. SIAM J. Sci. Comput., 17, 1019–1039. Kokh, S., & Allaire, G. 2001. Numerical simulation of 2-D two-phase flows with interface. In: Godunov Methods: Theory and Applications. E.F. Toro ed., Kluwer Academic/Plenum Publishers. Quirk, J., & Karni, S. 1996. On the dynamics of a shock-bubble interaction. J. Fluid Mech., 318, 129–163. Saurel, R., & Abgrall, R. 1999. A simple method for compressible multifluid flows. SIAM J. Sci. Comput., 21, 1115–1145. Shyue, K.M. 1999. A fluid-mixture type algorithm for compressible multicomponent flow with van der Waals equation of state. J. Comp. Phys., 156, 43–88.

Test-case number 21 by E. Canot and S.-C. Georgescu

133/283

Chapter 19

Test-case number 21: Gas bubble bursting at a free surface, with jet formation (PN-PE) By Edouard Canot, IRISA, Campus de Beaulieu, 35042 Rennes cedex, France Phone: +33 (0)2 99 84 74 89, Fax: +33 (0)2 99 84 71 71, E-Mail: [email protected] Sanda-Carmen Georgescu, Hydraulic Dept., Polytechnical University of Bucharest, 313 Spl. Independentei, S6, code 060032, Bucharest, Romania Phone: +40 (0)21 402 97 05, Fax: +40 (0)21 402 98 65 E-Mail: [email protected] St´ephane Vincent, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 27 07, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected]

19.1

Practical significance and interest of the test-case

The bursting of a gas bubble, initially in equilibrium under a gas/liquid surface, is numerically studied. Computation takes place in a 3D axisymmetric geometry. Gravity and surface tension forces lead to a focus of surface waves and, depending on the problem parameters, produce an upward liquid jet. Usually, numerical simulation is stopped when the pinching of this jet arises, just before creating the first liquid droplet; however, in the present case, the code is able to follow few ejected droplets. The present test-case can be used to verify the precision of the interface position as well as the treatment of surface tension forces, which are preponderant here. Extensive simulations show a great sensitivity to initial conditions. Initial shape, as described in table 19.1, must be scrupulously respected; if not, different results (especially in the shape and speed of the upward liquid jet) would be obtained.

134/283

19.2

Test-case number 21 by E. Canot and S.-C. Georgescu

Definitions and physical model description

A potential (irrotational) viscous flow model is used. Inertia, gravity and surface tension forces are computed without any approximation, while viscous forces are reduced to normal stress on the interface (no shear stress at the interface). More details can be found in Georgescu et al. (2002). Before the bursting (i.e. when t < 0), the gas bubble is in equilibrium and its shape is computed from a theoretical model (Ivanov et al. , 1986), where the upper liquid film is supposed to be thin. In the case where the bubble radius is small (e.g. less than 1 mm for the air/water couple), this shape is well approximated by a sphere (see figure 19.1).

1 0.5

z*

0 −0.5 −1 −1.5 −2 −4

−3

−2

−1

0 r*

1

2

3

4

Figure 19.1: Theoretical initial shape of the bubble : air/water. R0 = 0.75 mm, F r = 13.2, Re = 233

At time t = 0, the upper liquid film is removed, leading to an open cavity, whose shape is smoothed by a spline interpolation (cubic piecewise splines), as shown in figure 19.2. The (r, z) coordinates of this initial shape are shown in table 19.1. r∗ 0.000000E+00 1.716977E-01 3.371046E-01 4.908729E-01 6.315874E-01 7.584025E-01 8.619770E-01 9.326885E-01 9.762858E-01 9.990170E-01 9.701360E-01 9.288620E-01 8.934681E-01 8.496407E-01 7.747112E-01 6.841768E-01

z∗ -1.881678 -1.866767 -1.823116 -1.752881 -1.656978 -1.533295 -1.388082 -1.242522 -1.101206 -9.331290E-01 -6.522152E-01 -5.128708E-01 -4.235851E-01 -3.393725E-01 -2.401656E-01 -1.582818E-01

r∗ (follow) 5.951954E-01 5.235801E-01 4.946865E-01 5.301410E-01 6.248325E-01 7.899529E-01 9.444048E-01 1.081222 1.305878 1.640423 2.035410 2.428478 2.821837 3.215174 3.608510 4.000000

z ∗ (follow) -9.232008E-02 -3.798701E-02 1.861030E-04 1.853348E-02 1.969254E-02 8.371505E-03 -1.975520E-03 -4.671268E-03 3.186445E-05 3.852850E-04 -6.959044E-05 3.810910E-06 -1.183168E-07 1.673474E-07 1.122199E-06 -1.647956E-07

Table 19.1: Nondimensional coordinates of points on figure 19.2.

Test-case number 21 by E. Canot and S.-C. Georgescu

135/283

0.5

0

z*

−0.5

−1

−1.5

−2

−2.5

0

0.5

1

1.5

r*

Figure 19.2: Practical initial shape of the bubble : initial condition for the computation.

19.3

Test-case description

Let’s denote R0 the initial radius of the bubble, σ the liquid/gas surface tension, g the gravity, ρL the liquid density and µL the viscosity. In order to obtain a nondimensional framework, the following scales are chosen : • length : R0 r • velocity : • pressure :

σ R 0 ρL

σ R0

The nondimensional parameters are then : • Weber : W e = 1 (always, due to the choice of reference scales) σ ρL g R02 √ R 0 σ ρL • Reynolds : Re = µL • Froude : F r =

l Computational results will then use the dimensionless length l∗ = and the dimenR 0 r σ sionless time t∗ = t . The size of the computational domain is defined by : ρL R03 0 < r∗ < 4,

z ∗ > −4

136/283

Test-case number 21 by E. Canot and S.-C. Georgescu

Comparison may be done on the upward jet speed (fig. 19.3), on the interface shape at selected time values (fig. 19.4), on the radius of the first ejected droplet, and on the critical bubble radius for which the liquid jet decays without ejecting a liquid droplet (cf. Georgescu et al. (2002)). For experimental comparisons, see Kientzler et al. (1954), and Suzuki & Mitachi (1995). Photographic sequences taken from Kientzler et al. (1954) are depicted on figures 19.5 and 19.6 for an air bubble bursting in fresh water. (N.B. The three small bubbles visible on each frame of fig. 19.6 don’t depend on the bursting process; it seems that they are attached to the glass wall through which photos have been taken.) Boulton-Stone & Blake (1993) also present numerical results with the same kind of method as used here. 15

10

v*

5

0

−5

−10

−15

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t*

Figure 19.3: Upward liquid jet velocity vs time. R0 = 0.75 mm, F r = 13.2, Re = 233.

References Boulton-Stone, J. M., & Blake, J. R. 1993. Gas bubbles bursting at a free surface. J. Fluid Mech., 254, 437–466. ´ 2002. Jet drops ejection in bursting gas Georgescu, S.-C., Achard, J.-L., & Canot, E. bubble processes. Eur. J. Mech., B-Fluids, 21, 265–280. Ivanov, I. B., Kralchevsky, P. A., & Nikolov, A. D. 1986. Film and line tension effects on the attachment of particles to an interface: I. Conditions for mechanical equilibrium of fluid and solid particles at a fluid interface. J. Colloid Interf. Sci., 112(1), 97–107. Kientzler, C. F., Arons, A. B., Blanchard, D. C., & Woodcock, A. H. 1954. Photographic investigation of the projection of droplets by bubbles bursting at a water surface. Tellus, 6(1), 1–7.

Test-case number 21 by E. Canot and S.-C. Georgescu

4 *

t =0

z*

2 0

z*

−2

0

4

4

2

2

2

0

0

0

2

−2

0

−2 t* = 0.489 2

−2

0

−2 t* = 0.734 2

−2

4

4

4

4

2

2

2

2

0

0

0

0

−2 t* = 0.871 −2

z*

4

−2 t* = 0.244

−2

137/283

0

−2 t* = 1.098 2

−2

0

−2 t* = 1.151 2

−2

0

2

−2

4

4

4

2

2

2

2

0

0

0

0

−2

0 *

r

−2 t* = 1.239 2

−2

0 *

r

−2 t* = 1.346 2

−2

0 *

r

2

−2 t* = 1.181

4

−2 t* = 1.203

0

0

2

−2 t* = 1.403 2

−2

0

2

*

r

Figure 19.4: Free-surface time evolution after bursting bubble. R0 = 0.75 mm, F r = 13.2, Re = 233.

Suzuki, T., & Mitachi, K. 1995. Experimental observation of the droplet ejection due to gas bubble bursting at gas liquid interface. Pages IP2.3–IP2.9 of: Proc. 2nd Int. Conf. on Multiphase Flow, Kyoto, Japan.

138/283

Test-case number 21 by E. Canot and S.-C. Georgescu

Figure 19.5: Bursting bubble (after Kientzler et al. (1954), fig. 1, p.3), R0 = 0.5 mm. View angle is 10 degrees above horizontal plane. The time interval between frames is about 0.3 ms.

Figure 19.6: Bursting bubble (after Kientzler et al. (1954), fig. 2, p.4), R0 = 0.75 mm. Photos taken through a glass wall. The time interval between frames is about 0.3 ms.

Test-case number 22 by E. Canot, A. Cartellier and E. Hervieu

139/283

Chapter 20

Test-case number 22: Axisymmetric body emerging through a free surface(PE) By Edouard Canot, IRISA, Campus de Beaulieu, 35042 Rennes cedex, France Phone: +33 (0)2 99 84 74 89, Fax: +33 (0)2 99 84 71 71, E-Mail: [email protected] Alain Cartellier, LEGI, BP 53, 38041 Grenoble cedex 9, France Phone: +33 (0)4 76 82 50 48, Fax: +33 (0)4 76 82 52 71 E-Mail: [email protected] Eric Hervieu, DER/SSTH/LIEX, CEA/Grenoble, 38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 45 33, Fax: +33 (0)4 38 78 50 45, E-Mail: [email protected]

20.1

Practical significance and interest of the test-case

The current test-case concerns the free surface deformation due a body that emerges from a liquid; the study is restricted only to the surge effect. While this problem has been numerically studied for the large scales case where capillary effects are negligible, the present work provides experimental results when small scales (of the order of 1 cm) are considered. Therefore, it would certainly be a good case for testing the full Navier-Stokes numerical simulations, including capillary effects : enclosed photos and experimental data (which systematically include uncertainties) may be used to compare interface position and qualify the implementation of surface tension forces.

20.2

Experimental setup description

A circular cylindrical body, equipped with an hemispherical head, vertically submerged in a liquid tank, rises upwardly and uniformly towards the free-surface.

140/283

Test-case number 22 by E. Canot, A. Cartellier and E. Hervieu

All the geometry is axisymmetric. The body is only characterized by its radius R, because its length is very large in comparison with R (see fig. 20.1). The tank size is considered as infinite. Two different radius values have been used : • R = 5 mm, R = 10 mm Gravity reads, as usual : • g = 9.81 m s−2 The upward body velocity is denoted by V : • 0.1 m s−1 < V < 1 m s−1 Lastly, the physical properties of liquids that were used (water and aqueous glucose solution) are the following : • water : density : ρ = 1000 kg m−3 viscosity : µ = 10−3 Pa s surface tension : σ = 72 · 10−3 N m−1 • glucose : density : ρ = 1210 kg m−3 viscosity : µ = 26.3 · 10−3 Pa s surface tension : σ = 59 · 10−3 N m−1

gas liquid

η

111111 000000 000000 111111 000000 111111 00000 11111 V 00000 11111 R 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111

Figure 20.1: Definition sketch of surging interface problem.

The interface deformation is determined from digital video frames taken from a CCD camera. The present test-case shows only the surge effect, excluding the liquid exit, piercing and wetting-off stages; it is described in detail by Liju (1997) and Liju et al. (2001). Reported sequences are limited to weakly distorted interface, when the residual film on the top of the body is not too thin; for this reason, viscous effects are negligible for Reynolds numbers larger than about 200. Indeed, for large Reynolds numbers, viscous effects are confined in a thin layer surrounding the body head. However, for small Reynolds numbers, momentum convective and diffusive times are of the same order and then, interface position is slightly perturbed by viscous forces.

Test-case number 22 by E. Canot, A. Cartellier and E. Hervieu

20.3

141/283

Test-case description

Initially, the shape of the gas/liquid interface is horizontal, and the body is far under the free-surface. The following non dimensional parameters, Froude, Weber and Reynolds, are introduced : ρ V 2R ρR V V2 We = Re = F r2 = gR σ µ In all experiences provided by the attached literature, the ranges of the previous parameters are : 0.1 < F r2 < 20

0.6 < W e < 200

46 < Re < 6000

Experimental results are presented under a dimensionless form and make systematically use of uncertainties. The main reference scales are, for length and time, respectively : Lref = R

tref =

R V

The dimensionless time origin t∗ = 0 corresponds to the instant where the top of the body head reaches the position of the non deformed free surface.

Figure 20.2: glucose/air, V = 0.2 m/s, R = 10 mm F r2 = 0.4, W e = 8.2, Re = 92 t∗ = 0.41

Figure 20.3: t∗ = 0.58 (same parameters as in figure 20.2)

142/283

Test-case number 22 by E. Canot, A. Cartellier and E. Hervieu

Two kinds of results are presented : • interface shape η ∗ (r∗ ) at different instants (figures 20.2 and 20.3). The y ∗ origin corresponds to the initial position of the free surface (the dashed line shape is the contour of the body head). • position of the interface apex ηr∗∗ =0 (t∗ ) versus time (figures 20.4 to 20.7). Two series of measurements are superimposed on each figure, while uncertainties are reported by thin cross bars.

0.7

0.7 η

η∗

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 W

W

0.0 -1.0

-0.5

0.0

0.5

1.0

0.0 -1.5

-1.0

Figure 20.4: glucose/air V = 0.2 m/s 2

F r = 0.4

Re = 92

0.0

0.5

1.0

Figure 20.5: water/air

R = 10 mm

W e = 8.2

-0.5

V = 0.2 m/s 2

F r = 0.8

R = 5 mm

W e = 2.8

Re = 1000

References Liju, P.-Y. 1997. Caract´erisation exp´erimentale de la travers´ee d’une interface fluide/fluide par un corps solide. DEA M´ecanique des Fluides et Transferts, Institut National Polytechnique de Grenoble, France. Liju, P.-Y., Machane, R., & Cartellier, A. 2001. Surge effect during the water exit of an axisymmetric body traveling normal to a plane interface : experiments and BEM simulation. Exp. in Fluids, 31(3), 241–248.

Test-case number 22 by E. Canot, A. Cartellier and E. Hervieu

0.6

143/283

1.0 η

η

0.9

0.5

0.8 0.7

0.4

0.6 0.5

0.3

0.4 0.2

0.3 0.2

0.1

0.1

W

0 -1.5

-1.0

-0.5

0.0

0.5

1.0

W

0.0 -2.0

-1.5

Figure 20.6: water/air V = 0.2 m/s 2

F r = 0.4

Re = 2000

-0.5

0.0

0.5

Figure 20.7: water/air

R = 10 mm

W e = 5.6

-1.0

V = 0.4 m/s 2

F r = 1.6

R = 10 mm

W e = 22.2

Re = 4000

1.0

144/283

Test-case number 22 by E. Canot, A. Cartellier and E. Hervieu

Test-case number 23 by by F. Pigeonneau, F. Feuillebois and N. Coutris

145/283

Chapter 21

Test-case number 23: Relative trajectories and collision of two drops in a simple shear flow (PA) By F. Pigeonneau, Saint-Gobain Recherche, BP 135, 39 quai Lucien Lefranc 93303 Aubervilliers cedex, France Phone: +33 (0)1 48 39 59 99, Fax: +33 (0)1 48 39 58 78 E-Mail: [email protected] F. Feuillebois, PMMH, UMR 7636 CNRS, ESPCI, 10 rue Vauquelin 75231 Paris cedex 05, France Phone: +33 (0)1 40 79 45 53, Fax: +33 (0)1 40 79 47 95 E-Mail: [email protected] N. Coutris, INPG/ENSPG and DER/SSTH/LMDL CEA/Grenoble, 38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 91 91, Fax: +33 (0)4 38 78 50 36 E-Mail: [email protected]

21.1

Practical significance and interest of the benchmark

Suspensions in gases, or aerosols, have applications in various domains like meteorology, nuclear and chemical engineering. In the Stokes regime of fluid motion, a moving drop in a flow creates hydrodynamic perturbations which interact with other drops. These interactions may lead to collisions and coalescence, thereby modifying the aerosol histogram. For small values of the volume fraction of the dispersed phase, the only significant hydrodynamic interactions are those between pairs of drops. This incentive to study the hydrodynamic interactions between two inclusions in a flow provides the aim of this test case. The problem is restricted here to the motion of drops of equal size with initially

146/283

Test-case number 23 by by F. Pigeonneau, F. Feuillebois and N. Coutris

spherical shape embedded in a simple shear flow. This kind of problem is essential for the formation of clouds Pruppacher & Klett (1978), Jonas (1996). The purpose is to study the relative trajectories and collision of the two drops in the Stokes flow regime. The problem described in this paper has been solved analytically for non-deformable drops in the Stokes regime Pigeonneau. (1998), Pigeonneau & Feuillebois (2002). It will be of interest to compare these results with numerical simulations. Some numerical techniques may also go further and look at the deformation of drops (see e.g. Zapryanov & Tabakova, 1999), but this is outside the scope of the present benchmark.

21.2

Definitions and physical model description

21.2.1

Description, notation and assumptions

Consider two identical drops of radius a of the same material, embedded in a simple shear flow with velocity: u∞ (x) = γzi,

(21.1)

at a point x. Here, γ denotes the shear rate and we use a system of rectangular coordinates (x, y, z), with i the unit vector along x. The unperturbed flow velocity and the two drops are represented in figure 21.1. z Drop #1 y

Drop #2

x

O

Figure 21.1: Schematic representation of two drops moving in simple shear flow.

The dispersed phase is denoted with a subscript L (for liquid) and the continuous phase with a subscript G (for gas). Let ρ be the density, η and ν the dynamic and kinematic viscosities respectively, with the relevant subscript for each phase. All these physical quantities are assumed to be constant. Let also ρˆ =

ρL ρG

(21.2)

ηˆ =

ηL ηG

(21.3)

be the density ratio and

Test-case number 23 by by F. Pigeonneau, F. Feuillebois and N. Coutris

147/283

be the viscosity ratio. The perturbed flows for the two phases and the motions of the two drops interfaces are described by Navier-Stokes equations. However here, we assume that the Reynolds numbers associated to the external and internal flows are low: ReG =

γa2  1; νG

ReL =

γa2 1 νL

so that fluid inertia is negligible and Stokes equations apply. Note that the inertia in the drops motion may nevertheless be relevant, since the Stokes number St, defined by: 2 St = ρˆReG , 9

(21.4)

may be of order unity. On the other hand, gravity will be ignored in the motion of the drops. The physical consistency of these assumptions is discussed in full detail in Pigeonneau & Feuillebois (2002). The Reynolds numbers being low, the relevant non-dimensional number for the interfacial momentum balance is the capillary number: Ca =

ηG γa , σ

(21.5)

The surface tension σ is assumed to be constant here. Provided that the capillary forces are much larger than the viscous forces along the interface, viz. Ca  1, the drops remain spherical during their motion. Note √ that for close drops, Chesters (1991) showed that their deformation is proportional to Ca.

21.2.2

Formulation

Dimensionless quantities are defined in terms of a reference length a and a reference velocity γa. The dimensionless pressure pG is defined in terms of the dimensional one p∗G as: pG =

p∗G γηG

and the dimensionless dynamic pressure pL in terms of the dimensional pressure p∗G and the pressure jump across the interface as: pL =

p∗ 2 + L . Ca γηG

With these notations, the Stokes equations are written in dimensionless form in the continuous phase: divuG = 0, 2

(21.6a)

∇ uG = grad pG ,

(21.6b)

divuL = 0,

(21.7a)

and in the dispersed phase :

2

ηˆ∇ uL = grad pL .

(21.7b)

148/283

Test-case number 23 by by F. Pigeonneau, F. Feuillebois and N. Coutris

Note that the time derivative of the fluid velocity is removed as the Reynolds numbers are vanishingly small in the Navier-Stokes equations. Thus, the equations are quasi-steady. The question of an initial condition for the flow field is irrelevant here, but the initial positions of the two drops have to be specified. The boundary conditions on each drop interface are the continuity conditions for the fluid velocity uG = uL ,

(21.8)

the continuity of the tangential stresses and the jump condition for the normal stresses: ηˆτ L · DL · nL = τ L · DG · nL ,

(21.9a)

pL − pG = ηˆnL · DL · nL − nL · DG · nL ,

(21.9b)

where D denotes the strain rate tensor, nL is a unit vector normal to the interface, outwardly directed and τ L is a unit vector tangent to the interface. Finally, the two flow fields outside and inside the moving drops are the solutions of the equations (21.6a-b), (21.7a-b), with the interfacial conditions (21.8) and (21.9a). In Pigeonneau. (1998), Pigeonneau & Feuillebois (2002), the motion of the two interacting drops is determined in two steps. First, the drag forces on the two drops embedded in an unperturbed general linear flow field are obtained by using an exact solution of the Stokes equations with the bispherical coordinates system. Then, the study of the motions of their centers leads to the paths of the drops. Scaling the forces with the quantity 6πηG a2 γ, the dimensionless drag forces on drops 1 and 2 are: F 1 = − {A11 · [V 1 − u∞ (x1 )] + A12 · [V 2 − u∞ (x2 )] + G1 : D∞ } , ∞

∞

∞

F 2 = − {A21 · [V 1 − u (x1 )] + A22 · [V 2 − u (x2 )] + G2 : D } .

(21.10a) (21.10b)

where x1 and x2 are the positions of the drops centers, V 1 and V 2 are their velocities. The detailed expressions of the second rank tensors A11 , A12 , A21 , A22 and the third rank tensors G1 and G2 are detailed in Pigeonneau. (1998), Pigeonneau & Feuillebois (2002). Scaling the time t with γ −1 , the dimensionless equations leading to the motion of the drops can be written as: dx1 (t) = V 1, dt dx2 (t) = V 2, dt

(21.11a) (21.11b)

dV 1 (t) = F 1, dt dV 2 (t) St = F 2. dt

St

21.3

The description of the benchmark

21.3.1

Input parameters and physical properties

(21.11c) (21.11d)

The method described above can be used to study the motion of water drops in air. The value of the shear rate of the unperturbed flow velocity (21.1) is taken as γ = 103 s−1 .

Test-case number 23 by by F. Pigeonneau, F. Feuillebois and N. Coutris

Shear rate Radius Water

Air Surface tension

Quantity γ a ρL ηL νL ρG ηG νG σ

Value 103 10−5 1000 1.787 · 10−3 1.787 · 10−6 1.29 1.71 · 10−5 1.325 · 10−5 7.61 · 10−2

149/283

Unity s−1 m kg/m3 Pa.s m2 /s kg/m3 Pa.s m2 /s N/m

Table 21.1: Selected values of the physical properties of water and air.

The drops radius is a = 10 µm. Selected values of the physical properties of water and air and of the surface tension between air and water are given in table 21.1. Values of the dimensionless numbers for this set of data are given in table 21.2. Both the Reynolds numbers and capillary number are small. Consequently, the Stokes regime for each phase is relevant. The assumption of non-deformable drops is also justified by the small value of the capillary number.

Number ReG ReL Ca St ηˆ ρˆ

Value 7.55 · 10−3 5.59 · 10−2 2.25 · 10−6 1.3 104.5 775.19

Table 21.2: Values of the dimensionless numbers.

21.3.2

Initial conditions

Initially, the drops are embedded in the flow field, the distance between their centers being equal to ten times their radius a. The velocities of the drops are initialized to zero. These initial conditions are summarized in table 21.3, where u, v, z denote the components of the velocity in the Cartesian reference frame (x, y, z).

Drop 1 Drop 2

Position x y z -5 0 0 5 0 0

Velocity u v w 0 0 0 0 0 0

Table 21.3: Initial positions and velocities of the two drops.

150/283

Test-case number 23 by by F. Pigeonneau, F. Feuillebois and N. Coutris

Drop 1 t 0 9.161 91.793 173.62 184.01 185.08 186.39 186.45 186.49

x1 −5 −4.997 −4.436 −2.010 −1.005 −0.914 −0.710 −0.695 −0.681

z1 0 1.050 · 10−3 1.393 · 10−2 7.981 · 10−2 2.797 · 10−1 0.415 0.704 0.719 0.732

x2 5 4.997 4.436 2.010 1.005 0.914 0.710 0.695 0.681

Drop 2 z2 0 −1.050 · 10−3 −1.393 · 10−2 −7.981 · 10−2 −2.797 · 10−1 −0.415 −0.704 −0.719 −0.732

Table 21.4: Drop coordinates at various times.

21.3.3

Results and comparisons

The two centers remain in the plane Oxz. Nevertheless, the flow problem is threedimensional. A very important point concerns the forces on the drops in their initial positions: the only non-zero component is along the z axis. This can be proven as in Bretherton (1962) using the linearity and reversibility of the Stokes equation. The forces on drops 1 and 2 are opposite, the force on drop 1 being z-directed, their values are deduced from the results obtained in Pigeonneau. (1998), Pigeonneau & Feuillebois (2002) are F1z = −F2z = 1.4221 · 10−4 .

(21.12)

Using the analytical solution, it is shown in Pigeonneau. (1998), Pigeonneau & Feuillebois (2002) that the drops collide at the positions defined in table 21.4 at the dimensionless time t = 186.49. The trajectories of the drops are shown in figure 21.2. These trajectories are symmetric with respect to the origin of the Cartesian reference frame. One of the reasons is that the drops have the same radius. The dimensionless coordinates at different times are given in table 21.4.

21.4

Conclusion

The challenge is to recover the drops trajectories for the physical data mentioned above. There are various ways to compute a solution. One of them could be e.g. the boundary integral method for Stokes flow.

Test-case number 23 by by F. Pigeonneau, F. Feuillebois and N. Coutris

4.0

Drop #1 trajectory Drop #2 trajectory

151/283

Final position of drops

2.0

z

0.0

−2.0 Initial position of drops −4.0 −6.0

−4.0

−2.0

0.0 x

2.0

4.0

Figure 21.2: Trajectories of the drops from initial to final positions.

6.0

152/283

Test-case number 23 by by F. Pigeonneau, F. Feuillebois and N. Coutris

References Bretherton, F. P. 1962. The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech., 14, 284–304. Chesters, A. K. 1991. The modelling of coalescence processes in fluid-liquid dispersions: A review of current understanding. Trans. Instn. Chem. Engrs., 69, 259–270. Jonas, P. R. 1996. Turbulence and cloud microphysics. Atmospheric Research, 40, 283– 306. Pigeonneau., F. 1998. Mod´elisation et calcul num´erique des collisions de gouttes en ´ecoulements laminaires et turbulents. Ph.D. thesis, Universit´e Pierre et Marie Curie, Paris VI. Pigeonneau, F., & Feuillebois, F. 2002. Collision of drops with inertia effects in strongly sheared linear flow fields. J. Fluid Mech., 455, 359–386. Pruppacher, H. R., & Klett, J. D. 1978. Microphysics of clouds and precipitation. D. Reidel publishing company. Zapryanov, Z., & Tabakova, S. 1999. Dynamics of bubbles, drops and rigid particles. Kluwer academic publishers, Dordrecht.

Test-case number 24 by H. Lemonnier and O. Lebaigue

153/283

Chapter 22

Test-case number 24: Growth of a small bubble immersed in a superheated liquid and its collapse in a subcooled liquid (PE,PA) By Herv´e Lemonnier, DER/SSTH/LIEX, CEA/Grenoble, 38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 45 40, Fax: +33 (0)4 38 78 50 45, E-Mail: [email protected] Olivier Lebaigue, DER/SSTH/LMDL, CEA/Grenoble, F-38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 36 70, Fax: +33 (0)4 38 78 50 36, E-Mail: [email protected]

22.1

Practical significance and interest of the test-case

This test-case describes an analytical solution of a series of simple free boundary problems. Firstly, the growth of a vapor bubble initially at rest, in mechanical and local thermal equilibrium with the superheated liquid. Next, the collapse of a vapor bubble immersed in a subcooled liquid is considered. In this latter situation, the bubble is initially at rest and in mechanical equilibrium only with the liquid phase. The theory that provides the reference solution is that of Plesset & Zwick (1954) who discussed thoroughly its validity domain. An experiment is also proposed to strengthen the confidence to be put in their model. It is claimed by these authors that experimental conditions are consistent with the theory. This is confirmed by their analysis of the problem scales. It is therefore proposed three test-cases selected from their work. • Inertia controlled collapse of a bubble, dubbed the Rayleigh regime, where the heat flux from the liquid to the bubble is irrelevant to the problem. • The initial stage of the growth of a vapor bubble where both surface tension and transient heat flux to the bubble interface governs the dynamics of the phenomenon.

154/283

Test-case number 24 by H. Lemonnier and O. Lebaigue

• The long-term growth of the same bubble, which is only controlled by the rate of heat transfer from the liquid to the interface. The analytical solutions to be described here have been obtained by coding an improved version (variable time step) of the original algorithm proposed by Zwick & Plesset (1954). The heat transfer model in the liquid phase describes both the convection and the conduction. It is solved in closed form with a slight approximation (Plesset & Zwick, 1952). The main assumption to get this solution is the spherical symmetry, which might be questionable for the final stage of a bubble collapse. Finally, coupling the heat transfer problem to the motion of the interface results in a non-linear integral-differential problem for which Zwick & Plesset (1954) proposed a solution algorithm. In their original paper there are few misprints which make the material useless for practical calculations. Lemonnier (2001) revisited this work and proposed a corrected version of the theory validated by a comparison of numerical calculations with the results of the original paper. Moreover, the original work of Plesset & Zwick (1954) compared the original model with experiments. One of them is selected as an experimental test-case.

22.2

Model and assumptions

The assumptions of the model are the following. H1. The liquid and its vapor are not compressible. H2. The liquid and vapor viscosity are neglected. H3. The vapor enclosed by the bubble is assumed to have uniform thermodynamic properties and is in thermodynamic equilibrium with the liquid at the interface. The only exception to this assumption is relative to the density of the vapor, which is allowed to vary with time but however remains in saturated state corresponding to the interface temperature. H4. The physical and transport properties of the liquid are uniform and constant. H5. Convection that would be expected from the bubble buoyancy is neglected (no gravity). H6. At any time, the system remains spherically symmetric. Under these circumstances, it can be shown that the time evolution of the bubble radius is given by,     d2 R 3 dR 2 1 2σ R 2 + = pv (T ) − pL∞ − , (22.1) dt 2 dt ρL R where R is the bubble radius, t is time, ρL is the liquid density, T is the interface temperature, pv (T ) is the saturation pressure of the liquid evaluated at the temperature of the interface, pL∞ is the pressure far from the bubble and σ is the liquid-vapor superficial tension. The mechanical evolution equation of the bubble (22.1) degenerates to the well known Rayleigh-Plesset equation when the interface temperature remains constant and equal to that of the liquid.

Test-case number 24 by H. Lemonnier and O. Lebaigue

155/283

On the contrary, when the vapor bubble expansion produces significant interface cooling, equation (22.1) is no longer closed. An additional equation must be provided to calculate it. It is given by the solution of a convection diffusion equation with the following boundary conditions. The temperature at infinity is kept constant whereas the heat flux at the interface is deduced from the enthalpy balance at the interface. In addition, initial conditions must be provided. They are as follows, • The pressure in the liquid is uniform and set to p0 . • The temperature is uniform and set to T0 . Depending on particular circumstances, it may be larger or less than the saturation temperature corresponding to p0 . ˙ • The initial value of the time derivative of the bubble radius is set to zero (R(0) = 0). The initial radius of the bubble, R0 , can be different from the unstable equilibrium radius given by, Req =

2σ . pv (T0 ) − p0

(22.2)

The evolution of the interface temperature has been solved by Plesset & Zwick (1952) and is based on the only assumption that the thermal boundary layer that develops beyond the bubble interface is thin with respect to the bubble radius. The analytical solution to this problem is given by, L   1 Z t R2 (x) ∂T ∂r r=R(x) DL 2 T (t) = T0 − (22.3) hR i1/2 dx, π t 4 0 x R (y)dy where DL is the thermal diffusivity of the liquid and the temperature gradient at the interface in the liquid phase is deduced from the enthalpy balance of the interface by assuming consistently no heat flux into the vapor,  ∂TL hlv ρv (T )R˙ = , (22.4) ∂r r=R(t) kL where hlv is the heat of vaporization of the liquid, kL is the heat conductivity of the liquid and ρv (T ) is the vapor density at the temperature of the interface for saturation conditions.

22.3

Bubble collapse: case 24-1 (PA)

The initial conditions of this problem are those proposed by Zwick & Plesset (1954) and are relative to previous experiments by Plesset. The liquid is initially subcooled so that the bubble shrinks continuously. The initial conditions are as follows, p0 = 0.544 atmosphere ≈ 0, 544 × 101.3 kPa, o

(22.5)

T0 = 22 C,

(22.6)

R0 = 2.5 mm.

(22.7)

The transport and thermodynamic properties of the liquid and the vapor for the initial conditions are given in Table 22.1.

156/283

Test-case number 24 by H. Lemonnier and O. Lebaigue

σ N/m 0.0724

ρL kg/m3 997.8

hlv MJ/kg 2.448

pv0 kPa 2.65

ρv0 g/m3 19.4

kL W/m/K 0.602

DL m2 /s 1.44 10-7

Table 22.1: Transport and thermodynamic properties of the liquid and vapor for the collapse test case.

It is requested to calculate the variations of the bubble radius, the temperature of the interface and the velocity of the interface with time. The numerical solution of (22.1) and (22.3) is shown in figure (22.1). The bubble evolution is clearly similar to the Rayleigh regime since the heating of the interface only appears at the later stage of the collapse. This results from the very low saturation pressure of the vapor and is related to the high initial subcooling of the liquid. The numerical solution is provided as a text file and tables extracted from Lemonnier (2001). A selection of numerical results is also shown in table 22.2. The numerical solution in the Rayleigh regime is shown in table 22.3.

22.4

Initial stage of the growth of a vapor bubble, case 24-2 (PA)

The initial conditions for this test-case are from Zwick & Plesset (1954) and correspond roughly to the condition of the experimental test-case to be described later. The liquid is initially slightly superheated by a few Kelvin to remain consistent with the model assumptions. There exists in these conditions an unstable equilibrium radius satisfying both the thermodynamic and mechanical equilibrium conditions (22.2). These conditions are the following: p0 = 1 atmosphere ≈ 1.013 bar, o

(22.8)

T0 = 103 C,

(22.9)

R0 = (1 + )Req ≈ 10.27 µm.

(22.10)

The thermodynamic and transport properties of the liquid and the vapor at the initial temperature and pressure are given in Table 22.4. The initial conditions correspond to an unstable equilibrium state. The evolution of the system from these conditions is however ”frozen” since they corresponds to a stationary point of (22.2) and (22.3). It is therefore necessary to initiate the instability either by heating up the liquid at small rate either by starting the calculation with a slightly larger radius than the equilibrium value. Zwick & Plesset (1954) have shown that the growth proceeds along three phases. Each steps has been solved by an asymptotic analysis including the matching of them. This is a very cumbersome procedure and a numerical algorithm was proposed by these authors and was implemented by Lemonnier (2001). The three steps of the bubble growth are the following: • A latent period, the details and length of which depends on the particular way to destabilize the bubble, such as the heating rate of the liquid or the relative excess, , of the initial radius with respect to the equilibrium radius (22.2). • The initial growth, which can be described by a linearized version of (22.1) and (22.3).

Test-case number 24 by H. Lemonnier and O. Lebaigue

3

157/283

Rayleigh Zwick and Plesset

Bubble radius (mm)

2.5

2

1.5

1

0.5

0 0

0.05

0.1

0.15 0.2 Time (ms)

60

0.25

0.3

0.35

Rayleigh Zwick and Plesset

55

Temperature (Celsius)

50 45 40 35 30 25 20 0

0.05

200

0.1

0.15

0.2 Time (ms)

0.25

0.3

0.35

0.4

0.2 Time (ms)

0.25

0.3

0.35

0.4

Rayleigh Zwick and Plesset

−dR/dt (m/s)

150

100

50

0 0

0.05

0.1

0.15

Figure 22.1: From top to bottom, time variations of the radius, the interface temperature and the negative of the interface velocity for a bubble of initial radius R0 = 2.5 mm immersed in a liquid at an initial temperature of 22o C and a pressure of 0.544 atmosphere. Solution by the Zwick and Plesset algorithm and the Rayleigh model. Calculation: Zwick et Plesset01.for, plot: zw01.plt. Data files : zw01.txt and zw01-R.txt. Values at selected values of time are given in Table 22.2 and 22.3.

158/283

Test-case number 24 by H. Lemonnier and O. Lebaigue

tn (ms) .00000 .01001 .02005 .03015 .04037 .05047 .06048 .07058 .08066 .09068 .10074 .11082 .12085 .13087 .14092 .15092 .16095 .17099 .18100 .19100 .20101 .21102 .22105 .23109 .24111 .25113 .26116 .27118 .28119 .29122 .30126 .31131 .31510

Rn (mm) 2.5000 2.4989 2.4958 2.4904 2.4828 2.4730 2.4611 2.4468 2.4302 2.4114 2.3900 2.3660 2.3395 2.3102 2.2780 2.2428 2.2043 2.1623 2.1166 2.0668 2.0126 1.9532 1.8881 1.8164 1.7374 1.6491 1.5496 1.4359 1.3026 1.1396 .9247 .5696 .2250

Tn (o C) 22.00 22.01 22.03 22.06 22.09 22.14 22.18 22.24 22.29 22.36 22.43 22.51 22.59 22.68 22.78 22.89 23.01 23.14 23.28 23.44 23.61 23.81 24.03 24.29 24.59 24.95 25.39 25.94 26.68 27.76 29.65 35.23 55.11

R˙ n (m/s) .000 -.232 -.434 -.644 -.863 -1.081 -1.303 -1.532 -1.766 -2.004 -2.254 -2.512 -2.781 -3.063 -3.360 -3.674 -4.011 -4.374 -4.766 -5.195 -5.667 -6.198 -6.803 -7.502 -8.323 -9.323 -10.581 -12.231 -14.572 -18.284 -25.634 -54.615 -246.093

Table 22.2: Collapse of a steam bubble. p0 = 0.544 atmosphere, T0 = 22 o C, R0 = 2.5 mm, Model of Zwick & Plesset (1954), data file: ZW01.txt.

Test-case number 24 by H. Lemonnier and O. Lebaigue

tn (ms) .00000 .01001 .02005 .03015 .04037 .05047 .06048 .07058 .08066 .09068 .10074 .11082 .12085 .13088 .14092 .15093 .16096 .17101 .18103 .19104 .20105 .21109 .22109 .23111 .24112 .25114 .26117 .27119 .28121 .29122 .30127 .31128 .31462

Rn (mm) 2.5000 2.4989 2.4958 2.4904 2.4828 2.4730 2.4611 2.4468 2.4302 2.4114 2.3899 2.3659 2.3394 2.3101 2.2778 2.2427 2.2041 2.1620 2.1161 2.0662 2.0118 1.9522 1.8870 1.8153 1.7360 1.6474 1.5474 1.4330 1.2987 1.1346 .9167 .5524 .2514

Tn (o C) 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00

159/283

R˙ n (m/s) .000 -.232 -.434 -.644 -.863 -1.081 -1.303 -1.532 -1.767 -2.005 -2.255 -2.514 -2.784 -3.066 -3.363 -3.679 -4.016 -4.380 -4.774 -5.206 -5.681 -6.216 -6.823 -7.526 -8.353 -9.362 -10.633 -12.306 -14.689 -18.473 -26.082 -57.201 -201.231

Table 22.3: Collapse of a steam bubble in the Rayleigh regime (constant interface temperature). p0 = 0.544 atmosphere, T0 = 22 o C, R0 = 2.5 mm, Model of Zwick & Plesset (1954), data file: ZW01-R.txt.

160/283

Test-case number 24 by H. Lemonnier and O. Lebaigue

σ N/m 0.0583

ρL kg/m3 956.2

hlv MJ/kg 2.248

pv0 MPa 0.1127

ρv0 kg/m3 0.660

kL W/m/K 0.680

DL m2 /s 1.68 10-7

Table 22.4: Transport and thermodynamic properties for the simulation of the initial stage of the growth of a vapor bubble.

• The fully developed growth solved by expanding the solution for long times. Plesset & Zwick (1954) have shown that the time origin for the asymptotic solution for long times was arbitrary. Therefore, to compare the model solution to experiments, these authors shifted the asymptotic solution in time to get the best agreement with the data. This procedure would have been unnecessary if the full 3-zone solution would have been used. However, the solution depends critically on the initial process that triggers the instability. Lemonnier (2001) shows that  plays he same role as the initial heating of the liquid to start the bubble growth: for different values of , the radius evolves along parallel paths. To get the same latent period than Zwick & Plesset (1954) who used a heating rate of the liquid of 0.01o C/s, it is sufficient to select an initial radius slightly larger that the equilibrium radius by a relative amount (see equation 22.10)  = 5 10−8 . The results of the numerical solution of (22.2) and (22.3) for the above mentioned initial conditions are shown in Figure 22.2. In these figures, the symbols represents the asymptotic solution for long times by Zwick & Plesset (1954). The continuous line represents the numerical solution of (22.1) and (22.3) according to the original developments by Zwick & Plesset (1954, Appendix 2) while the dashed line represents the solution obtained with a constant vapor density to be consistent with the asymptotic approach of Zwick & Plesset (1954). The variable density assumption has obviously a direct impact on the onset of the instability. It is requested to calculate the time variations of the bubble radius, interface temperature and velocity. Results are shown in Figure 22.2 and available as text files and arrays in Lemonnier (2001). Values at selected times of these solutions are given in Tables 22.5 and 22.6.

22.5

Thermally controlled growth of a vapor bubble (24-3)

This last proposed test-case corresponds to an experiment described by Plesset. The initial conditions are the known values of the pressure and the liquid temperature. However, the initial radius of the bubble is unknown in the experiment since it is hardly measurable. To analyze this situation, Plesset & Zwick (1954) have proposed to shift the real time origin such that the time evolution of the solution agrees with the observed results. In their asymptotic analysis, Plesset & Zwick (1954) have chosen the equilibrium radius as an initial condition whereas for solving the evolution equations (22.1) and (22.3) we have again chosen  = 10−8 . The latent period of the bubble growth is almost negligible with this parameter value with respect to the overall simulation time and there is therefore no need to take care of the physical time origin of the problem. The initial conditions of the

Test-case number 24 by H. Lemonnier and O. Lebaigue

161/283

0.07 0.06

Radius (mm)

0.05

0.04 0.03 0.02

0.01

Variable density Constant density Long term sol. by Zwick and Plesset (1954)

0 0

0.05

104

0.1

0.15

0.2 Time (ms)

0.25

0.3

0.35

0.4

0.35

0.4

0.35

0.4

Variable density Constant density Long term sol. by Zwick and Plesset (1954)

103.5

Temperature (Celsius)

103 102.5 102 101.5 101 100.5 100 0

0.05

0.5

0.1

0.15

0.2 Time (ms)

0.25

0.3

Variable density Constant density Long term sol. by Zwick and Plesset (1954)

0.45 0.4

dR/dt en m/s

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

0.05

0.1

0.15

0.2 Time (ms)

0.25

0.3

Figure 22.2: From top to bottom, time variations of the radius, the interface temperature and the interface velocity for a bubble of initial radius R0 = 10.27 µm immersed in a liquid at an initial temperature of 103o C and a pressure of 1 atmosphere. Solution by the Zwick and Plesset algorithm with constant or variable vapor density. Symbols: sample points extracted from the asymptotic solution of Zwick & Plesset (1954). Calculation: Zwick et Plesset02.for, plot: zw02.plt. Data files : zw02.txt and zw02r1.txt, columns number 5,6,7 and 8. Files: zwr-ref.txt, zwt-ref.txt and zwrd-ref.txt. Values at selected times of the reference solution are given in Table 22.5 and 22.6.

162/283

Test-case number 24 by H. Lemonnier and O. Lebaigue

tn (ms) .00000 .02600 .05200 .07800 .10396 .12957 .15552 .18140 .20741 .23275 .25870 .28508 .31119 .33643 .36253 .38799 .41428 .43933 .46470 .49050 .51602 .54211 .56815 .59467 .62047 .64556 .67108 .69766 .72307 .74904 .77527 .80030 .82547 .85064 .87669 .90242 .92783 .95290 .97905 1.00074

Rn (mm) .0103 .0103 .0103 .0103 .0103 .0104 .0112 .0153 .0235 .0328 .0420 .0508 .0588 .0660 .0730 .0795 .0858 .0915 .0970 .1024 .1075 .1125 .1174 .1222 .1267 .1310 .1352 .1395 .1435 .1475 .1515 .1552 .1588 .1624 .1660 .1695 .1730 .1763 .1797 .1824

Tn (o C) 103.00 103.00 103.00 103.00 103.00 102.97 102.78 102.09 101.40 101.01 100.79 100.65 100.56 100.50 100.45 100.41 100.38 100.35 100.33 100.32 100.30 100.29 100.27 100.26 100.25 100.24 100.24 100.23 100.22 100.22 100.21 100.20 100.20 100.20 100.19 100.19 100.18 100.18 100.18 100.17

R˙ n (m/s) .000 .000 .000 .000 .001 .009 .072 .253 .358 .366 .345 .319 .296 .277 .260 .246 .233 .222 .213 .204 .197 .190 .184 .178 .173 .168 .164 .160 .156 .152 .149 .146 .143 .141 .138 .135 .133 .131 .129 .127

Table 22.5: Growth of a steam bubble. p0 = 1 atmosphere, T0 = 103 o C, R0 = (1 + 5 10−8 )Req , Model of Zwick & Plesset (1954), data file: ZW02.txt.

Test-case number 24 by H. Lemonnier and O. Lebaigue

tn (ms) .00000 .02600 .05200 .07800 .10397 .12973 .15553 .18151 .20721 .23229 .25813 .28418 .30949 .33512 .36119 .38675 .41237 .43761 .46363 .48872 .51392 .53967 .56612 .59133 .61662 .64269 .66819 .69387 .71949 .74490 .77003 .79581 .82194 .84740 .87375 .89931 .92542 .95073 .97640 1.00028

Rn (mm) .0103 .0103 .0103 .0103 .0103 .0103 .0108 .0134 .0198 .0278 .0361 .0440 .0511 .0578 .0642 .0701 .0758 .0810 .0862 .0910 .0956 .1002 .1047 .1088 .1129 .1170 .1208 .1246 .1283 .1319 .1354 .1388 .1423 .1456 .1489 .1521 .1553 .1583 .1614 .1641

Tn (o C) 103.00 103.00 103.00 103.00 103.00 102.98 102.87 102.36 101.64 101.18 100.91 100.74 100.64 100.56 100.51 100.46 100.43 100.40 100.37 100.35 100.34 100.32 100.31 100.29 100.28 100.27 100.26 100.26 100.25 100.24 100.23 100.23 100.22 100.22 100.21 100.21 100.20 100.20 100.20 100.19

163/283

R˙ n (m/s) .000 .000 .000 .000 .001 .005 .040 .178 .301 .326 .313 .292 .272 .254 .238 .225 .213 .204 .195 .187 .180 .174 .168 .163 .158 .154 .150 .146 .142 .139 .136 .133 .131 .128 .126 .123 .121 .119 .117 .116

Table 22.6: Growth of a steam bubble under the assumption of constant vapor density. p0 = 1 atmosphere, T0 = 103 o C, R0 = (1 + 5 10−8 )Req , Model of Zwick & Plesset (1954), data file: ZW02r1.txt.

164/283

Test-case number 24 by H. Lemonnier and O. Lebaigue

calculation and the experiment are the following, p0 = 1 atmosphere ≈ 101.3 kPa,

(22.11)

o

T0 = 103.1 C,

(22.12)

R0 = (1 + )Req ≈ 9.92 µm.

(22.13)

The physical and transport properties of the liquid and the vapor are given in Table 22.7. Figure 3 shows that the model of Zwick & Plesset (1954) is in good agreement σ N/m 0.0583

ρL kg/m3 956.1

hlv MJ/kg 2.248

pv0 MPa 0.1131

ρv0 kg/m3 0.620

kL W/m/K 0.680

DL m2 /s 1.685 10-7

Table 22.7: Physical and transport properties of the liquid and the vapor for the growth of a vapor bubble.

with the data. The time evolution of the radius slightly differs depending on the assumption of constant or variable vapor density. However, it is clear that neglecting the cooling of the interface by the liquid evaporation induces a much faster growth of the bubble (the Rayleigh regime), which disagrees with the experiment. It is requested to calculate the time evolution of the bubble radius, the interface temperature and its velocity. The numerical values of the to be used as a reference are given by Lemonnier (2001) and are provided at selected values of time in Tables 22.8, 22.9 and 22.10.

References Lemonnier, H. 2001. Croissance et effondrement d’une bulle de vapeur selon le mod`ele de Plesset et Zwick (1954). SMTH/LDTA/2001-025, CEA/Grenoble. Plesset, M. S., & Zwick, S. A. 1952. A nonsteady heat diffusion problem with spherical symmetry. J. of Applied Physics, 23(1), 95–98. Plesset, M. S., & Zwick, S. A. 1954. The growth of vapor bubbles in superheated liquids. J. of Applied Physics, 25(4), 493–500. Zwick, S. A., & Plesset, M. S. 1954. On the dynamics of small vapor bubbles in liquid. J. Math. Phys, 33, 308–330.

Test-case number 24 by H. Lemonnier and O. Lebaigue

4

165/283

Variable density Constant density Rayleigh Solution Data after Plesset et Zwick (1954) Fig.2

3.5

Radius (mm)

3 2.5 2 1.5 1 0.5 0 0

2

4

6 8 Time (ms)

10

103.5

12

14

Variable density Constant density

103

Temperature (C)

102.5

102

101.5 101

100.5

100 0

2

4

6

8 Time (ms)

10

0.45

12

14

16

14

16

Variable density Constant density

0.4 0.35

dR/dt (m/s)

0.3 0.25 0.2 0.15 0.1 0.05 0 0

2

4

6

8

10

12

Time (ms)

Figure 22.3: From top to bottom, growth of a vapor bubble with an initial radius equal to 9.92 µm immersed in water at 103.1o C and a pressure equal to 1 atmosphere. Solution after Zwick & Plesset (1954) considering the vapor density constant or variable with the temperature, Rayleigh solution (constant interface temperature), symbols: experimental data. Numerical model: Zwick et Plesset03.for, plot: zw03.plt. Numerical results : zw03fig2.txt, zwr1fig2.txt et zwrafig2.txt

166/283

Test-case number 24 by H. Lemonnier and O. Lebaigue

tn (ms) .00000 .30110 .60209 .90238 1.20323 1.50384 1.80442 2.10485 2.40522 2.70547 3.00604 3.30702 3.60770 3.90834 4.21002 4.51154 4.81227 5.11253 5.41363 5.71432 6.01475 6.31501 6.61516 6.91524 7.21528 7.51530 7.81532 8.11534 8.41536 8.71540 9.01619 9.31690 9.61819 9.91933 10.21976 10.52016 10.82103 11.12277 11.42387 11.72486 12.02533 12.32653 12.62797 13.23010 13.83057 14.43201 15.00001

Rn (mm) .0099 .0643 .1323 .1792 .2173 .2502 .2795 .3062 .3309 .3540 .3757 .3964 .4161 .4349 .4531 .4705 .4874 .5036 .5195 .5348 .5497 .5643 .5785 .5923 .6059 .6191 .6321 .6449 .6574 .6696 .6817 .6936 .7053 .7169 .7282 .7393 .7503 .7612 .7719 .7825 .7929 .8032 .8134 .8334 .8528 .8719 .8895

Tn (o C) 103.10 100.51 100.24 100.18 100.14 100.12 100.11 100.10 100.09 100.09 100.08 100.08 100.07 100.07 100.07 100.06 100.06 100.06 100.06 100.05 100.05 100.05 100.05 100.05 100.05 100.05 100.04 100.04 100.04 100.04 100.04 100.04 100.04 100.04 100.04 100.04 100.04 100.04 100.04 100.03 100.03 100.03 100.03 100.03 100.03 100.03 100.03

R˙ n (m/s) .000 .303 .179 .138 .117 .103 .093 .085 .079 .075 .070 .067 .064 .061 .059 .057 .055 .053 .052 .050 .049 .048 .047 .046 .045 .044 .043 .042 .041 .041 .040 .039 .039 .038 .037 .037 .036 .036 .035 .035 .034 .034 .034 .033 .032 .031 .031

Table 22.8: Growth of a steam bubble. p0 = 1 atmosphere, T0 = 103.1 o C, R0 = 9.92 µm, under the assumption of variable density.

Test-case number 24 by H. Lemonnier and O. Lebaigue

tn (ms) .00000 .30085 .60226 .90229 1.20296 1.50316 1.80327 2.10374 2.40438 2.70455 3.00521 3.30600 3.60670 3.90810 4.20903 4.50954 4.81022 5.11045 5.41070 5.71202 6.01231 6.31271 6.61283 6.91345 7.21505 7.51546 7.81546 8.11565 8.41641 8.71683 9.01736 9.31833 9.61900 9.92066 10.22168 10.52326 10.82478 11.12577 11.42584 11.72601 12.32737 12.92777 13.52972 14.00044

Rn (mm) .0099 .0564 .1184 .1608 .1952 .2249 .2514 .2755 .2979 .3187 .3384 .3570 .3748 .3919 .4082 .4239 .4391 .4538 .4681 .4820 .4955 .5086 .5214 .5339 .5462 .5582 .5699 .5814 .5928 .6039 .6148 .6255 .6360 .6465 .6567 .6668 .6767 .6865 .6962 .7057 .7244 .7426 .7604 .7740

Tn o ( C) 103.10 100.58 100.27 100.20 100.16 100.14 100.12 100.11 100.10 100.10 100.09 100.08 100.08 100.08 100.07 100.07 100.07 100.06 100.06 100.06 100.06 100.06 100.06 100.05 100.05 100.05 100.05 100.05 100.05 100.05 100.05 100.05 100.04 100.04 100.04 100.04 100.04 100.04 100.04 100.04 100.04 100.04 100.04 100.04

167/283

R˙ n (m/s) .000 .277 .162 .125 .106 .093 .084 .077 .072 .067 .064 .061 .058 .055 .053 .051 .050 .048 .047 .045 .044 .043 .042 .041 .040 .039 .039 .038 .037 .037 .036 .035 .035 .034 .034 .033 .033 .032 .032 .031 .031 .030 .029 .029

Table 22.9: Growth of a steam bubble. p0 = 1 atmosphere, T0 = 103.1 o C, R0 = 9.92 µm, under the assumption of constant density.

168/283

Test-case number 24 by H. Lemonnier and O. Lebaigue

tn (ms) .00000 .30060 .60140 .90261 1.20284 1.50383 1.80422 2.10498 2.40607 2.70660 3.00030

Rn (mm) .0099 .6901 1.5450 2.4040 3.2613 4.1213 4.9800 5.8399 6.7010 7.5605 8.4006

Tn (o C) 103.10 75.31 61.38 51.01 42.40 34.84 28.06 21.82 16.03 10.61 5.60

R˙ n (m/s) .000 2.830 2.849 2.854 2.857 2.858 2.859 2.860 2.860 2.860 2.861

Table 22.10: Growth of a steam bubble. p0 = 1 atmosphere, T0 = 103.1 o C, R0 = 9.92 µm , under the assumption of constant interface temperature (Rayleigh regime).

Test-case number 26 by J.-L. Estivalezes and O. Lebaigue

169/283

Chapter 23

Test-case number 26: Droplet impact on hot walls By Jean-Luc Estivalezes, DMAE, ONERA, 2 Av. Edouard Belin , BP 4025, F-31055 Toulouse cedex Phone: +33 (0)5 62 25 28 32, Fax: +33 (0)5 62 25 25 83, E-Mail: [email protected] Olivier Lebaigue, DER/SSTH/LMDL, CEA/Grenoble, F-38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 36 70, Fax: +33 (0)4 38 78 50 36, E-Mail: [email protected]

23.1

Practical significance and interest of the test-case

Fuel droplet impingement on a hot surface is encountered in many practical processes, ranging from various types of internal combustion engines. Droplets impacting on a hot wall in a diesel combustor engine usually experience wall temperature in the range 200 − 300o C, which means that they exhibit typical Leidenfrost phenomenon. Evaporation and hydrodynamic deformations are accompanied by heat, mass and momentum transfers that still require some fundamental investigations. Here we present experimental results for the thermal and dynamical behavior of the droplets before and after impact. To reproduce the experimental data is a challenge for any numerical method as it means taking into account properly the impact of a droplet with the Leidenfrost phenomenon that usually rely on pressure building in thin vapor layer ranging down to the micrometer. In addition to this scale, the phase change in a non-condensable gas, the effect of roughness on the vapor flow during the Leidenfrost, etc., may also request small-scale description for the mass diffusion layers.

23.2

Definitions and physical model description

The experiment consists on a droplet generator based on Rayleigh instability that generates perfectly calibrated droplets at given frequency and velocity. This generator is vertical and flowing droplets downward as can be see on figure 23.1, for some other experiments,

170/283

Test-case number 26 by J.-L. Estivalezes and O. Lebaigue

the generator can flow droplet upward (figure 23.2). The droplets are then impacting on an inclined hot plate. Thanks to this device, all droplets have the same history. The relevant parameters are: • Incident normal Weber number: W e =

2 D ρl Vi,n l σ

ρl Vinc,n Dl µl µl • Ohnesorge number: Oh = √ ρl Dl σ • Reynolds number: Re =

All quantities with subscript l are related to the liquid. We give in the table 23.1 the physical properties of the liquid, namely ethanol in our experiments.

23.3

Test-case description

For this test case, the droplet chain is ascending according to figure 23.2. Θ is the angle between the plate and the vertical. We summarize in the table 23.2 the initial conditions for this case. For that case, we observe perfect rebound of the droplet. However, due to hot wall, part of the droplet is evaporating. We give in the table 23.3, the surface temperature of the droplet measured by infrared techniques as described in detail in (LeClercq et al. , 1999b), the diameter of the droplet measured by image processing and the velocity measured by Phase-Doppler techniques at a location 12 mm after rebound. The duration of impact is of order 5 10−4 s. It should be noticed for this test case, that the roughness of the hot wall is around 1 micrometer. The numerical simulation for this case must take into account evaporation. It will be supposed that physical properties like surface tension, dynamic viscosity do not depend of the temperature. In order to get the proper gas temperature field in the neighborhood of the heated wall, a preliminary computation of the thermal boundary layer must be done with an imposed temperature on the wall given by table 23.2. As a suggestion, the computational domain for this calculation could be a rectangular box of 0.1 m by 0.04 m the length of the wall is 12 mm. We give on figure 23.3 an example of such calculation (LeClercq et al. , 1999a). In order to show the dynamic of the physical phenomenon, we show in figure 23.4 a snapshot of the chain droplets impact for a descending droplet stream. The droplets diameter is 174 µm, the wall temperature is 623 K. The wall angle is θ = 15˚ , the droplet velocity is 3.68 m/s.

23.4

Relevant results for comparison

This test-case is today a very difficult one. The main challenge of such a test is first to get results with a CFD code, and second to obtain these results with physical and numerical descriptions that stand the mesh refinement needed to achieve numerical convergence with respect to the spatial discretization. Once simulation results are successfully obtained, it is possible to compare with the experimental results. In this stage, the main features to be reproduced are the size and

Test-case number 26 by J.-L. Estivalezes and O. Lebaigue

171/283

velocity of droplets after the rebound. It is also possible to compare the other experimental values given in table 23.3, but the droplet surface temperature will be highly sensitive to the physical wall roughness effect on the impact feature. Therefore this last quantity is probably more an indication of the quality of the physical model than a test-case for the numerical method used. Liquid density ρl , kg/m3 Surface tension σ, kg/s2 Dynamic viscosity µl , kg/m s Boiling temperature Tb , K Critical temperature Tc , K Leidenfrost temperature TL , K

777.95 0.0221 0.001052 351.5 561.25 458

Table 23.1: Physical properties of liquid

Droplet diameter Dl , µm Frequency f , Hz Initial velocity V~l , m/s Initial normal velocity Vi,n , m/s Initial temperature Ti , K Wall temperature Tw , K Plate angle Θi , degree Incident normal Weber number

210 7500 4.9 1.268 297 623 14 11.88

Table 23.2: Initial conditions

Droplet diameter Drb 10 mm after rebound, µm Velocity after rebound V~rb , m/s Normal velocity after reboundVi,n rb , m/s Surface temperature after rebound Trb , K Angle after rebound Θrb , degree Table 23.3: Experimental results after bouncing

192 3.97 0.75 317 11

172/283

Test-case number 26 by J.-L. Estivalezes and O. Lebaigue

Droplet generator

g

Hot wall

Q Figure 23.1: Experimental set-up, descending droplets

Θ

Hot wall

g

Droplet generator

Figure 23.2: Experimental set-up, ascending droplets

Test-case number 26 by J.-L. Estivalezes and O. Lebaigue

173/283

Figure 23.3: Computed gas thermal field, the heated wall goes from y=0.04 m to y=0.052 m

Figure 23.4: Experimental result for descending droplets stream

174/283

Test-case number 26 by J.-L. Estivalezes and O. Lebaigue

References LeClercq, P., Estivalezes, J.L., & Lavergne, G. 1999a. Drop impact on a heated wall: Global models and experimental results comparison. In: ILASS Europe. Toulouse, France. LeClercq, P., Ravel, O., Estivalezes, J.L., & Farre, J. 1999b. Thermal and dynamical characteristics of droplets after impact on heated wall. In: ILASS Europe. Toulouse, France.

Test-case number 27 by H. Friess, D. Lakehal and S. Vincent

175/283

Chapter 24

Test-case number 27: Interface tracking based on an imposed velocity field in a convergent-divergent channel (PN) By Hansmartin Friess, Nuclear Engineering Laboratory, ETH Zurich ETH-Zentrum, CLT, CH-8092 Zurich, Switzerland Phone: +41 (0) 1 6324614, E-Mail: [email protected] Djamel Lakehal, Nuclear Engineering Laboratory, ETH Zurich ETH-Zentrum, CLT, CH-8092 Zurich, Switzerland Phone: +41 1 6324613; E-Mail: [email protected] St´ephane Vincent, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 27 07, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected]

24.1

Practical significance and relevance of the test-case

The purpose of the problem described here is to assess interface tracking methods in incompressible, two-dimensional two-phase flow on the basis of an analytically given velocity field. Imposing an analytical velocity field has the advantage of simple and well-defined test conditions. In particular, the assessment of tracking methods is not affected by numerical aspects of Navier-Stokes solvers. On the other hand, imposed velocity fields are often poorly related to realistic experimental situations, as, for instance, the velocity fields proposed by Zalesak (1979) and B. & Kothe (1995). By contrast, the test problem proposed here can be considered as a reasonable approximation to the physical process of a bubble advected by laminar flow in a contracted channel. A particular feature of the test problem is the curved boundary of the flow field.

176/283

Test-case number 27 by H. Friess, D. Lakehal and S. Vincent

Solving the Navier-Stokes equation in this situation would require particular measures, such as introducing obstacles in a rectangular mesh configuration, or using cuvilinear coordinates. For an imposed velocity field, however, no particular measures are necessary to solve the problem, even with Cartesian coordinates. In fact, the detailed problem description given below is entirely based on Cartesian coordinates. An important element of our test procedure is the construction of a discretely solenoidal velocity field, as an alternative to infinitesimally solenoidal fields used in the aforementioned references.

24.2

Definitions and model description

We consider a channel where the center line coincides with the x-axis of the coordinate system, and the function f (x) specifies the (positive) distance between the channel wall and the center line (see Figure 24.1). The velocity field is derived from the assumption that the component u parallel to the center line has a parabolic profile in every cross-section of the channel. This assumption is strictly true in the case of laminar, stationary flow and straight channel walls, characterized by f (x) = constant. For general channel shapes, the assumption can be expected to hold approximately, in particular if the variation of f (x) is weak. The only solenoidal velocity field (u, v) having a parabolic profile of u in every crosssection of the channel is given – up to a constant factor – by the equations,  2 ! y 1 u= 1− , f (x) f (x) (24.1)  2 ! y y d v = 1− f (x) . 2 f (x) (f (x)) dx Equations (24.1) apply, of course, to the region |y| ≤ f (x). Since the channel will be included in a rectangular computational domain (see Section 24.3), we extend the definition of u and v to the whole (x, y) plane by setting u = 0 and v = 0 in the region |y| ≥ f (x). The velocity field (u, v) defined in this way can be derived from the stream function   3  1 y y   − if |y| ≤ f (x)   3 f (x) f (x)      2 (24.2) Ψ= if y ≤ −f (x) +   3         −2 if y ≥ f (x) 3 according to the equations, u=− Thus the field (u, v) is solenoidal.

∂Ψ , ∂y

v=

∂Ψ . ∂x

(24.3)

Test-case number 27 by H. Friess, D. Lakehal and S. Vincent

177/283

However, a solenoidal velocity field does not necessarily provide suitable test conditions, since some interface tracking algorithms require discretely solenoidal fields in order to work properly. For a discussion of this issue we restrict ourselves to a discretization scheme with square computational cells and face-centered velocity nodes. In this case, a velocity field (u, v) is discretely solenoidal if every computational cell satisfies the condition u2 − u1 + v2 − v1 = 0,

(24.4)

where u1 , u2 , v1 , v2 are velocity components on the face centers as indicated in figure 24.2. However, if u1 , u2 , v1 , v2 are derived from (24.3), we get (from a Taylor expansion around the cell center)  3  ∂ u ∂3v h3 u2 − u1 + v2 − v1 = + + O(h5 ) (24.5) ∂x3 ∂y 3 cc 24 where h is the side length of computational cells, and the subscript “cc” refers to the cell center. We have actually identified the non-vanishing right-hand side in (24.5) as the reason for numerical artifacts in some VOF tests, as discussed below in Section 24.3. To associate a stream function Ψ with a discretely solenoidal velocity field, we propose to use face-centered velocities defined by, u1 = −

Ψ12 − Ψ11 , h

Ψ21 − Ψ11 v1 = , h

u2 = −

Ψ22 − Ψ21 , h

(24.6)

Ψ22 − Ψ12 v2 = , h

where Ψ11 , Ψ12 , Ψ21 , Ψ22 are the values of Ψ in the corners of the corresponding computational cell, as indicated in figure 24.2. The velocities defined by (24.6) satisfy (24.4) exactly. They differ from their infinitesimally solenoidal counterparts derived from (24.3) by a term of order h2 .

24.3

Test-case description

It is proposed to track the interface of a bubble using the following parameters: • Channel shape. The channel profile is given by   1  x 2 f (x) = 1 − a exp − 2 b

(24.7)

with shape parameters a = 0.75 and b = 0.5. The maximum velocity resulting from (24.1) and (24.7) is p 1 1 wmax ≡ max u2 + v 2 = = =4. (24.8) f (0) 1−a The maximum occurs at the point (0, 0). • Computational domain; discretization of space. The computational domain is the rectangle [−2.5, 2.5] × [−1.25, 1.25] divided into 256 × 128 computational cells, implying square computational cells of side length h = 0.01953125. • Initial state. In the initial state, at time t = 0, the bubble is circular. The center is located at the point (−1.95, 0), and the radius equals 0.5.

178/283

Test-case number 27 by H. Friess, D. Lakehal and S. Vincent

• Final states; discretization of time. Two final states are considered: at time t = t1 = 1220∆t, and at t = t2 = 2020∆t. The time step ∆t corresponds to the Courant number C = 0.3. This implies ∆t =

hC = 0.00146484375 . wmax

(24.9)

It is required to compare the interface of the two final states with “exact” solutions that are obtained by means of advected marker points. Marker point advection is governed by the ordinary differential equation d (x, y) = (u, v) dt

(24.10)

with (u, v) given by (24.1). The numerical error of the corresponding solution can easily be controlled, which justifies considering marker point solutions as an exact reference. Figure 24.1 shows the initial state of the bubble as well as the exact final states. The advection of 13 selected marker points on the interface (equally spaced in the initial state) is also shown. The coordinates of the marker points are listed in table 24.1. The error of the tabulated data is less than one unit of the last decimal place given. Figure 24.3 shows the result of the test procedure when applied to the following two interface tracking methods: (i) unsplit PLIC VOF (Meier et al. , 2000), and (ii) Level Set (Sussman et al. , 199). The results illustrate a characteristic complementarity between VOF and Level Set methods: VOF methods have superior volume conservation properties, but are liable to develop small-scale topological irregularities like the outward bend of the two “fin tips” in the state t = t2 . The VOF results presented in figure 24.3 are based on discretely solenoidal fields, with face-centered velocities derived from (24.6). When face-centered velocities are derived instead from (24.1), we obtain spurious contaminations of one phase by the other, as well as some additional small-scale irregularities of the interface. This finding indicates that discretely solenoidal velocity fields can be a prerequisite for obtaining meaningful test results.

Test-case number 27 by H. Friess, D. Lakehal and S. Vincent

t=0 xi yi -1.450 0.000 -1.467 0.129 -1.517 0.250 -1.596 0.354 -1.700 0.433 -1.821 0.483 -1.950 0.500 -2.079 0.483 -2.200 0.433 -2.304 0.354 -2.383 0.250 -2.433 0.129 -2.450 0.000

i 1 2 3 4 5 6 7 8 9 10 11 12 13

t = t1 xi yi 1.270 0.000 1.221 0.126 1.081 0.234 0.865 0.296 0.571 0.264 0.074 0.125 -0.423 0.238 -0.602 0.307 -0.660 0.297 -0.648 0.239 -0.599 0.159 -0.547 0.076 -0.525 0.000

179/283

t = t2 xi yi 2.447 0.000 2.380 0.131 2.193 0.252 1.929 0.355 1.641 0.432 1.384 0.476 1.201 0.479 1.117 0.453 1.133 0.408 1.219 0.340 1.327 0.244 1.414 0.128 1.447 0.000

Table 24.1: Coordinates of selected marker points on the interface in the initial state (t = 0) and the two final states considered in the test problem (t = t1 , and t = t2 ).

1.25 1 0.75 7

0.5

7 7

0

13

t=0

1

2f (x )

y

0.25

13

–1.5

–1

–0.5

t = t1

1

13

t = t2

1

–0.25 –0.5 –0.75 –1 –1.25 –2.5

–2

0

0.5

1

1.5

2

2.5

x Figure 24.1: Computational domain, with channel region (dark grey), channel width 2f (x), and the three states of the advected bubble (bright grey) considered in the test problem. The black dots are marker points advected by the flow. The labels 1, 7, and 13 refer to the numbering scheme used in table 24.1.

180/283

Test-case number 27 by H. Friess, D. Lakehal and S. Vincent

Ψ12

v2

Ψ22

u1

Ψ11

u2

h

y

h

v1

Ψ21

x Figure 24.2: Computational cell of size h, with stream function values Ψ11 , Ψ12 , Ψ21 , Ψ22 in the corners, and velocity components u1 , u2 , v1 , v2 at face centers.

VOF, t = t1

VOF, t = t2

LS, t = t1

LS, t = t2

Figure 24.3: The grey area represents final bubble shapes obtained with two different interface tracking methods, Volume of Fluid (VOF) and Level Set (LS). The dotted lines indicate exact interface curves obtained by means of marker point advection.

Test-case number 27 by H. Friess, D. Lakehal and S. Vincent

181/283

References B., W. J. Rider D., & Kothe. 1995. Stretching and Tearing Interface Tracking Methods. In: Proc. of the 12th AIAA CFD Conference, San Diego, June 19-22. AIAA Paper 95-1717, LANL Report LA-UR-95-1145. Meier, M., Yadigaroglu, G., , & Andreani, M. 2000. Numerical and Experimental Study of Large Steam-Air Bubbles Injected in a Water Pool. Nuclear Science and Engineering, 136, 363–375. Sussman, M., Almgren, A. S., Bell, J. B., L., L. H. Colella M., & Welcome. 199. An Adaptive Level Set Approach for Incompressible Two-Phase Flows. Journal of Computational Physics, 148, 81–124. Zalesak, S. T. 1979. Fully Multi-Dimensional Flux Corrected Transport Algorithms for Fluid Flow. Journal of Computational Physics, 31, 335–362.

182/283

Test-case number 27 by H. Friess, D. Lakehal and S. Vincent

Test-case number 28 by D. Lakehal and H. Lemonnier

183/283

Chapter 25

Test-case number 28: The Lock-Exchange Flow By Djamel Lakehal, Nuclear Engineering Laboratory, ETH Zurich ETH-Zentrum, CLT, CH-8092 Zurich, Switzerland Phone: +41 1 6324613; E-Mail: [email protected] Herv´e Lemonnier, DER/SSTH/LIEX, CEA/Grenoble, 38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 45 40, Fax: +33 (0)4 38 78 50 45, E-Mail: [email protected]

25.1

Practical significance and interest of the test-case

The lock flows belong to the category of large-scale, gravity-driven currents, in which surface tension can be neglected. Gravity driven flows are induced by density variations due to a difference in temperature, such as atmospheric fronts, or due to the presence of a dispersed solid phase or a heavier dense gas. These are simple flow configurations, which may, however, result in very complex flows characterized by physical processes such as the emergence of Kelvin-Helmholtz-like instabilities, the formation of lobes and clefts at the front leading edge, etc. The lock flow consists of two fluids initially separated by a gate. Mutual penetration develops after the gate is withdrawn; a pair of gravity-driven fronts propagates along the upper and lower surfaces of the channel. A basic configuration of the flow is shown in figure 25.1. For density ratios of the order of one, the penetrations proceed almost symmetrically. However, the situation changes appreciably for higher density ratios, in that the lighter phase travels at much smaller speed than the dense gas underneath. From a practical view point, the quantities of interest in these flows are (i) the front propagation velocity, Uf , and (ii) the run-out length, xf . Both quantities depend primarily on the density ratio between the involved phases, ρg /ρa , where ρg designates the density of the dense gas, and ρa the density of the lighter gas, the dimensions of the channel as well as the effect of the wall boundary conditions, i.e. slip and non-slip, may also play an important role.

184/283

Test-case number 28 by D. Lakehal and H. Lemonnier

The interest of the test case is twofold: Evaluating the numerical scheme employed for the transport of the interface, and comparing the numerical results to the analytical solution derived on the basis of the Boussinesq fronts theory. The objective of the test case is to assess the capability of the scheme to conserve mass, and to deliver the right front shape and propagation velocity. Interface tracking schemes such as VOF and Level Sets could be employed. The test case falls into the following categories: • N: Purely numerical test designed to assess numerical schemes or part of them. • P: Physical test-case where comparison to analytical solutions is required. It therefore belongs to the category: – PA: Comparison to a purely analytical solution. The test case is attractive for its simplicity, and also because of the possibility to compare the results to the analytical solution.

25.2

Definitions and physical model description

The lock-exchange flow is considered for this computational exercise. The flow is accelerated during a short transient phase after the gate is withdrawn. The developing fronts attain a steady propagation velocity immediately after the transition phase. The simplest wall +h u fa

h

dense gas

h

light gas

0

u fg

-h -L/2

y x

0 wall

+ L/2

Figure 25.1: Basic sketch of the lock-exchange flow. The domain extends from −L/2 to +L/2 and from −h to +h. The front velocities are denoted by Uf g and Uf a .

way to arrive at an analytical solution for the front propagation velocity is based on the following assumptions for the flow (Yih, 1965): (i) viscous dissipation is neglected, and (ii) the kinetic energy associated with the front motion is balanced by the loss by the system in potential energy. The scale of kinetic (∆Ek ) and potential energy (∆Ep ) is ∆Ek = ρm hUf3

and ∆Ep =

∆ρgh2 Uf , 2

(25.1)

respectively, where ρm = (ρg + ρa )/2 denotes the average density, ∆ρ = ρg − ρa , and g is the acceleration of gravity. In the limit of small density ratios, the above hypothesis leads to the following relationship between the front speed and the buoyancy velocity, Ub : Uf = Uf a = Uf g

1 = √ Ub ; 2

Ub =

s

∆ρ hg ρm

(25.2)

Test-case number 28 by D. Lakehal and H. Lemonnier

25.3

185/283

Test-case description

It is proposed to calculate the evolution of the mutual intrusions at the lower boundary and the top boundary of the channel according to the parameters of table 25.1. Specifically, the front propagation velocity and run-out length will be displayed as a function of time, according to table 25.1. Note that the front speed Uf is defined as the speed at which the foremost point of the front travels, i.e. Uf = dxf /dt, where xf denotes the position of the nose in the longitudinal direction. In a second step, it is required to compare the front propagation velocities of both phases, Uf g versus Uf a , and draw a 3D map showing the onset of the front speed asymmetry (quantified by the ratio Uf g /Uf a ) as a function of parameters R1 and R2 (c.f. table 25.1). This can be used to examine the validity of Yih’s theory, which assumes slip wall-boundary conditions, flow velocities being uniform over the respective heights and equal to the front speed, and small density ratios.

Test-case

R1 = ρg /ρa

CO2 /Argon CO2 /Argon R22/Air R22/Air Argon/Helium Argon/Helium R22/Helium R22/Helium Water/Air Water/Air

1.11 1.11 2.18 2.18 9.93 9.93 21.6 21.6 1000 1000

R2 =

q

ρg −ρa ρg +ρa

0.22 0.22 0.61 0.61 0.90 0.90 0.95 0.95 0.999 0.999

L/2h 5 20 5 20 5 20 5 20 5 20

Table 25.1: Gas combinations used in the simulations of the lock flow.

The length scale characteristic of the problem is represented by the half channel-height denoted by h. The flow is two dimensional, incompressible, Newtonian, and laminar. The Navier-Stokes equations, without surface tension effects, should be resolved in time. The computational domain consists of a channel of length L, i.e. [−L/2, L/2] × [2h]. The computational grid suggested depends on the aspect ratio L/2h. Based on our earlier grid-sensitivity studies (Lakehal et al. , 2002), we suggest to use a grid consisting of 250×50 nodes, at least, for the L/2h = 5 case; the L/2h = 20 configuration requires 1000×50 computational nodes. The gate is initially located at x/h = 0, and the flow is at rest. The upper and lower boundaries should be treated using non-slip conditions (with friction). The vertical (end wall) planes have to be treated as frictionless walls. This treatment is suggested here in order to be consistent with the DNS of Hartel et al. (2000) of the same flow. Summary of the required calculations • The front propagation velocity (normalized by by h) as a function of time.

√

2Ub ) and run-out length (normalized

• Compare the front propagation velocities of both phases, and establish a 3D map showing the onset of the front speed asymmetry as a function of parameters R1 and R2 .

186/283

Test-case number 28 by D. Lakehal and H. Lemonnier

• Study the effect of the end walls on the front speeds as a function of their distance from the propagating front. Results reported in Lakehal et al. (2002) obtained with the Level-Set method for the lock flow are shown in figure 25.2. The test case corresponds to the L/2h = 5 configuration. In Lakehal et al. (2002), attention was focused exclusively on the run-out length as a function of time. Figure 25.2 shows the interface evolution in the lock-exchange problem

Figure 25.2: Interface evolution in the lock flow obtained by Level Sets for R1 = 1.38; R2 = 0.4; L/2h = 5. After Lakehal et al. (2002)

for R1 = 1.38, where the front intrusions are clearly reproduced. The predicted run-out lengths of the dense and light gas for the CO2 /argon gas combination (R1 = 1.11) have shown that the fronts have nearly equal velocities, in line with the Boussinesq theory of Yih. For density ratios higher than two (e.g. R22/argon), the dense-gas fronts were found to travel appreciably faster than the fronts of the light gas.

Test-case number 28 by D. Lakehal and H. Lemonnier

187/283

References Hartel, C.J., Meiburg, E., & Necker, F. 2000. Analysis and Direct Numerical Simulation of the Flow at a Gravity-current Head. Journal of Fluid Mechanics, 418(9), 189–212. Lakehal, D., Meier, M., & Fulgosi, M. 2002. Interface tracking towards the direct simulation of heat and mass transfer in multiphase flows. Journal of Heat and Fluid Flow, 23(3), 242–257. Yih, C.S. 1965. Dynamics of Nonhomogeneous Fluids. Macmillan.

188/283

Test-case number 28 by D. Lakehal and H. Lemonnier

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

189/283

Chapter 26

Test-case number 29a: The velocity and shape of 2D long bubbles in inclined channels or in vertical tubes (PA, PN) Part I : in a stagnant liquid By Hien Ha-Ngoc, Institute of Mechanics, 264 Doi Can, Hanoi, Vietnam. E-Mail: [email protected] Jean Fabre, Institut de M´ecanique des Fluides de Toulouse, All´ee du Pr. Camille Soula, 31400 Toulouse, France. E-Mail: [email protected]

26.1

Practical significance and interest of the test-case

The motion of a long bubble in a channel or in a tube has a practical interest for modeling slug flow that occurs in various industrial applications. This motion is a result of two mechanisms (i) the action of gravity, (ii) transport by the liquid movement. In fully-developed flow, the motion of a long bubble is uniform and independent on its length. This means that the bubble moves with a constant velocity that does not depend on the types of wakes that are formed behind the bubble: they are slave of the bubble motion. The influence of the liquid movement can be therefore characterized by a velocity profile imposed far upstream from the bubble head. We provide in two articles (part I and part II) analytical and numerical solutions for the velocity and shape of 2D long bubbles (plane and axis-symmetrical) in stagnant liquid (part I) and in flowing liquid (part II). The solutions are considered for the case when viscous effects on the bubble motion are negligible i. e. when the motion is inertial. In this regime the influence of surface tension will be discussed. According to

190/283

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

White & Beardmore (1962) and Wallis (1969), the effect of viscosity is negligible when the inverse viscosity number N f =(gD 3 )1/2 /ν is greater than 300 or according to Zukoski (1966) when the bubble Reynolds number Re=VD/ ν is greater than 100. Neglecting the viscosity is justified by observing that the growth rate of the boundary layer created at the wall and at the bubble surface is slow. Moreover the acceleration of the liquid around the bubble nose prevents the boundary layer to grow. These results suggest that in the case of large enough Reynolds number potential flow theory can be applied. Such an inertia-controlled regime is often met in the industrial applications (H´eraud, 2002). In a moving frame attached to the bubble, the flow is steady. For an inviscid fluid and a two-dimensional flow, the liquid motion is described by a Poisson equation for the stream function with a source term depending on the vorticity. This vorticity is determined by the velocity profile given ahead of the bubble and remains constant along a streamline. As a result, a rotational flow can be described by only one equation for the stream function (Batchelor, 1967, pp. 507-593)(Lamb, 1932, pp. 244-245). This equation can be solved with corresponding boundary conditions. The bubble shape is determined from an equation obtained by combining the Bernoulli equation and the jump conditions at the interface. An algorithm based on a boundary element method (BEM) solving simultaneously the Poisson equation and the equation at the interface has been proposed by Ha-Ngoc (2003). It allows determining accurately the flow characteristics and the bubble shape. The results have been compared to different theoretical, experimental and numerical results (Zukoski, 1966, Collins et al. , 1978, Vanden-Bro¨eck, 1984a,b, Bendiksen, 1984, Cou¨et & Strumulo, 1987).

26.2

Definitions and model description

We consider the motion of a long gas bubble in a two-dimensional channel or in a vertical tube filled-up with a liquid. The gas density is relatively small compared to that of the liquid so that the gas motion can be neglected and the pressure in the bubble, considered as constant. Indeed the pressure difference in the gas due to its motion should be of the order of magnitude of ρG V 2 /2, considerably less than that in the liquid. The reference calculations to be proposed have been carried out in dimensionless form. The reference scales are: for the length, the half-width, a, of a channel or the radius R of a tube, for the velocity, (gD)1/2 , where D=2a. In stagnant liquid, the flow may be characterized by two dimensionless parameters: • The bubble Reynolds number, Re: VD . ν

(26.1)

ρgD2 , σ

(26.2)

Re = • The E¨ otv¨os number, Eo: Eo =

where V is the bubble velocity, ν, the kinematic viscosity of the liquid, ρ, its density, σ, the surface tension and g, the gravity.

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

191/283

The dimensionless bubble velocity is represented by the bubble Froude number, Fr, defined as : V Fr = √ . gD

(26.3)

Experiences have shown that there exists a critical bubble Reynolds number Re ∗ over which the bubble velocity does not depend on the viscosity. Zukoski (1966) has shown that for a vertical tube Re ∗ is about 100. In this regime, only inertia, gravity and possibly surface tension balance. Therefore, the liquid can be considered as inviscid and the bubble motion is characterized by only one dimensionless parameter, the E¨ otv¨os number, Eo. It can considered that the solutions of present study corresponds to: Re > 100.

(26.4)

For an inclined channel, two cases must be distinguished: (i) a case when the bubble touches the upper wall and (ii) a case when the bubble has no contact with a wall. For the case (i), a relation between the bubble velocity and the contact angle θ0 of the liquid with the wall should be given to complete the problem formulation. A sensitivity analysis on the influence of the contact angle on the bubble velocity can be done. In fact, the contact angle may vary between 0 and π, but for small surface tension values, Eo>100, it can be shown that the bubble velocity is not sensitive to θ0 . In further calculations the contact angle θ0 = π/2 will be always used. For the case (ii), the interface is a smooth line and there exists a stagnation point at the interface in the moving coordinates. The condition of smoothness at this point can be used for determining the bubble velocity. For vertical flow, when the symmetry is imposed, the problem can be treated as the case (i) with given interface slope angle at symmetry axis θ0 = π/2. It can be shown that there exist multiple solutions for this problem and the criterion of maximum velocity suggested by Garabedian (1957) is used for selecting the physically observable solution.

26.3

Motion in horizontal channel

Benjamin (1968) obtained an exact analytical solution for the drift velocity and the bubble shape for the horizontal case with negligible surface tension (σ=0 i. e. Eo → ∞). For a bubble in a channel, the velocity is given by: p V = 0.5 gD (26.5) The coordinates of bubble shape are given in Table 26.1 (see Appendix) and the shape is presented in Figure 26.1 in comparison with a numerical solution by Ha-Ngoc (2003). The numerical solution is obtained by an iterative procedure. The calculation begins with initial bubble shape and bubble velocity V 0 . In the (x, y) coordinates, where the x -axis coincides with the axis of symmetry and the y-axis passes by the stagnation point (0, -1), the following initial bubble shape x=X (y) is used:   y+1 2 X(y) = −1 − A ln (1 − ( ) ) , 0 < d < 2, −1 6 y 6 d (26.6) d where A and d are the arbitrary parameters.

192/283

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

Figure 26.1: Shape of a long bubble moving in a horizontal channel at Eo → ∞ (numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8).

For the numerical solution, the conditions at infinity upstream the bubble and in the receding film are taken on lines perpendicular to the channel walls. The distances of these lines to the stagnation point have been chosen so that they do not affect the solution. The interface is discretized and approximated by a polygon. A non-uniform discretization was used with the mesh size changing from ds min near the stagnation point to ds max far downstream. The sensitivity analysis of the bubble length, l, and the discretization for Eo=40÷4000 has shown that the converged solution could be obtained with ds min 660.025 and l >5. The solution becomes more sensitive to the mesh size at small Eo. The numerical results presented here have been carried out with ds min =0.0125, ds max =0.2 and l =8. For fixed Eo, different converged solutions can be obtained from different initial conditions (bubble shape and bubble velocity V 0 ). For each Eo, there exists a solution whose velocity is the greatest: it is considered as the physical solution. This solution is not sensitive to the initial conditions. For example, for Eo=100, the solution with the greatest velocity, V = 0.44, can be obtained for a rather large range of the parameters A, d and V 0 : {A=1.5, d =1.2, V 0 =0.5}, {A=1.5, d =1.0, V 0 =0.7}, {A=2.5, d =1.5, V 0 =0.30} etc. The results of bubble velocity are presented in Figure 26.2 for E¨otv¨os number in the range [10, 1000]. The solution calculated by utilizing the boundary element method is in good agreement with the results of Cou¨et & Strumulo (1987) who used a method based on the conformal mapping technique. The calculated bubble shapes are presented in Figure 26.3 and in Table 26.2 (Appendix) for different E¨ otv¨os numbers (Eo=10, 100, 1000).

26.4

Motion in inclined channel

According to the experimental results of Maneri & Zuber (1974), three different shape regimes may be observed when the inclination angle varies: • The first regime corresponds to inclination angles from α =0◦ (horizontal position) to about 60◦ . The bubble touches the upper wall and the liquid drains in the film under the bubble. • In the second regime the bubble is vertical or slightly deviated from the vertical (80◦ 6α690◦ ). The liquid drains in two films around the bubble and the bubble velocity is nearly as in vertical channel.

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

193/283

Figure 26.2: Dimensionless bubble velocity versus E¨ otv¨ os number for horizontal channel (numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8).

Figure 26.3: Horizontal bubble shapes for different E¨ otv¨ os number (present numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8).

• For inclination between 60◦ and 80◦ , there is a change in both velocity and shape. This is a transition region that is generally unknown and may be characterized by the instability of the liquid film. It must be pointed out that the simulation converges well for 60◦ 6α680◦ , but it gives two different solutions according to the case under investigation: (i) or (ii). None of these two solutions seems to be a physical one. We provide here the numerical solutions for the first flow regime i.e. for 0◦ 6α660◦ . For the simulation, the initial bubble shape of the form (26.6) and interface discretization with ds min =0.0125, ds max =0.2 and l =8 have been used. The bubble velocity is presented in Figure 26.4 as a function of inclination for different surface tensions (Eo=10, 100, 1000). It should be noted that the numerical results are comparable with the experimental results of Maneri & Zuber (1974). The velocity increases with inclination from the horizontal position, reaches its maximum value at about α=30◦ ÷45◦ and then decreases.

194/283

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

Figure 26.4: Dimensionless bubble velocity: influence of channel inclination (present numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8).

Figure 26.5: Bubble shapes for different inclinations at Eo=100 (present numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8 ).

The calculated bubble shapes are presented in Figure 26.5 and in Table 26.3 for different inclination angle (α=0o , 15o , 45o ) at Eo=100.

26.5

Motion in vertical channel and in tube

The vertical case has been the most extensively investigated, theoretically, experimentally and numerically. For the purely inertial regime, the dimensionless bubble velocity is about 0.345 for an axis-symmetrical bubble and 0.23 for a plane bubble (Dumitrescu, 1943, Davies & Taylor, 1950, Vanden-Bro¨eck, 1984b, Collins, 1965, Zukoski, 1966, Bendiksen, 1984, 1985, Mao & Dukler, 1990, 1991). The influence of surface tension was studied by Vanden-Bro¨eck, Cou¨et and Strumolo for plane case and by Zukoski and Bendiksen for axis-symmetrical case. Vanden-Bro¨eck (1984b) has show that the case of zero surface tension is a mathematically degenerate case. The solution for this case can be obtained by letting σ→0 but not by setting σ=0. By studying the multiple solution problem, Cou¨et and Strumolo have noted that the solutions could densely cover the whole segment

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

195/283

[0, 0.23]. For the boundary element method used here, the solution converges well for Eo4. For the axis-symmetrical case, it requires also a domain discretization to take the surface integration on the flow domain. A simple triangulation procedure has been used (Ha-Ngoc, 2003).

Figure 26.6: Velocity of a plane bubble in vertical channel in function of surface tension (numerical results by BEM with ds min =0.025, ds max =0.1 and l =4).

The bubble velocity is presented in the Figure 26.6 for plane bubbles and in Figure 26.7 for axis-symmetrical bubbles as a function of surface tension for Eo∈[5, 4000]. Some examples of bubble shapes are presented in the Figures 26.8 and 26.9. Bubble shape coordinates are given in Tables 26.4 and 26.5 (Appendix). The accuracy of the simulation can be appreciated by the fact that the bubble velocity and shape are in very good agreement with the experimental results of Davies & Taylor (1950), Zukoski (1966) and numerical results of Vanden-Bro¨eck (1984b), Mao & Dukler (1990) for small surface tension cases (Ha-Ngoc, 2003).

26.6

Acknowledgements

This work was partly sponsored by the CNRS France in the frame of a Program of Scientific Cooperation with NCST Vietnam. The authors wish to thank Dr H. Lemonnier for his helpful comments and suggestions.

196/283

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

Figure 26.7: Velocity of an axis-symmetrical bubble in vertical tube in function of surface tension (numerical results by BEM with ds min =0.05, ds max =0.1 and l =4).

Figure 26.8: Bubble shapes in vertical flow for different surface tensions: plane bubble (present numerical results by BEM with ds min =0.025, ds max =0.1 and l =4).

Figure 26.9: Bubble shapes in vertical flow for different surface tensions: axis-symmetrical bubble (present numerical results by BEM with ds min =0.05, ds max =0.1 and l =4).

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

26.7

197/283

Appendix

Theoretical solution of Benjamin (1968) x/a y/a 0.0136 -0.978 0.0582 -0.9162 0.1362 -0.8264 0.2428 -0.7258 0.3662 -0.6296 0.4948 -0.5454 0.6202 -0.4754 0.7386 -0.4182 0.8488 -0.3714 0.9504 -0.3330 1.1310 -0.2744 1.3562 -0.2158 1.6480 -0.1584 1.8732 -0.1248 2.5632 -0.0602 4.1254 -0.0116 4.7906 -0.0058 -

Numerical solution of Ha-Ngoc (2003) x/a 0 0.0021 0.0068 0.0125 0.0186 0.0259 0.0342 0.0438 0.0548 0.0673 0.0815 0.0977 0.1160 0.1368 0.1604 0.1872 0.2174 0.2517 0.2905 0.3344 0.3841

y/a -1 -0.9877 -0.9761 -0.965 -0.9541 -0.9423 -0.9296 -0.9158 -0.9008 -0.8845 -0.8669 -0.8479 -0.8272 -0.805 -0.7810 -0.7552 -0.7276 -0.6981 -0.6666 -0.6331 -0.5978

x/a 0.4402 0.5035 0.5749 0.6554 0.7459 0.8477 0.9619 1.0897 1.2326 1.3920 1.5695 1.7667 1.9636 2.1611 2.3590 2.5571 2.7555 2.9539 3.1524 3.3509 3.5495

y/a -0.5607 -0.5219 -0.4817 -0.4403 -0.3980 -0.3554 -0.3129 -0.2712 -0.2309 -0.1926 -0.1571 -0.1248 -0.0989 -0.0780 -0.0611 -0.0477 -0.0369 -0.0284 -0.0214 -0.0161 -0.0117

x/a 3.7481 3.9467 4.1453 4.3439 4.5425 4.7411 4.9397 5.1383 5.3369 5.5355 5.7341 5.9327 6.1313 6.3300 6.5286 6.7272 6.9258 7.1244 7.3230 7.5216 7.7202

y/a -0.0083 -0.0056 -0.0034 -0.0018 -0.0004 0.0006 0.0015 0.0021 0.0027 0.0031 0.0034 0.0036 0.0038 0.0040 0.0039 0.0041 0.0040 0.0040 0.0038 0.0035 0.0039

Table 26.1: Interface coordinates of a long bubble moving in a horizontal channel at Eo → ∞ (numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8).

198/283

Test-case number 29a by Hien Ha-Hgoc and J. Fabre Eo = 10 x/a y/a 0 -1 0.0001 -0.9875 0.0006 -0.9750 0.0013 -0.9625 0.0023 -0.9501 0.0037 -0.9363 0.0057 -0.9212 0.0083 -0.9046 0.0117 -0.8864 0.0161 -0.8663 0.0218 -0.8444 0.0289 -0.8204 0.0378 -0.7943 0.0488 -0.7659 0.0623 -0.7351 0.0788 -0.7019 0.0989 -0.6661 0.1231 -0.6278 0.1521 -0.5872 0.1867 -0.5441 0.2276 -0.4990 0.2758 -0.4521 0.3321 -0.4038 0.3976 -0.3545 0.4733 -0.3049 0.5603 -0.2556 0.6595 -0.2073 0.7720 -0.1607 0.8991 -0.1166 1.0418 -0.0755 1.2013 -0.0380 1.3791 -0.0042 1.5767 0.0257 1.7739 0.0494 1.9716 0.0686 2.1696 0.0843 2.3677 0.0974 2.5660 0.1085 2.7644 0.1181 2.9628 0.1266 3.1613 0.1343 3.3598 0.1414 3.5583 0.1482 3.7568 0.1547 3.9553 0.1611 4.1538 0.1674 4.3523 0.1736 4.5508 0.1798 4.7494 0.1860 4.9479 0.1920 5.1464 0.1979 5.3449 0.2034 5.5435 0.2085 5.7420 0.2129 5.9406 0.2167 6.1392 0.2195

Eo = 100 x/a y/a 0 -1 0.0003 -0.9875 0.0011 -0.9750 0.0025 -0.9626 0.0044 -0.9502 0.0069 -0.9367 0.0103 -0.9218 0.0147 -0.9056 0.0202 -0.8879 0.0271 -0.8686 0.0356 -0.8476 0.0460 -0.8248 0.0585 -0.8002 0.0734 -0.7737 0.0912 -0.7451 0.1122 -0.7145 0.1368 -0.6817 0.1656 -0.6468 0.1992 -0.6097 0.2381 -0.5706 0.2830 -0.5295 0.3347 -0.4865 0.3942 -0.4420 0.4623 -0.3964 0.5400 -0.3499 0.6282 -0.3030 0.7281 -0.2562 0.8411 -0.2106 0.9681 -0.1666 1.1106 -0.1247 1.2699 -0.0862 1.4475 -0.0513 1.6449 -0.0203 1.8421 0.0033 2.0398 0.0221 2.2379 0.0367 2.4362 0.0477 2.6346 0.0566 2.8331 0.0633 3.0317 0.0683 3.2302 0.0726 3.4288 0.0755 3.6274 0.0778 3.8260 0.0798 4.0246 0.0810 4.2232 0.0822 4.4219 0.0831 4.6205 0.0835 4.8191 0.0842 5.0177 0.0844 5.2163 0.0846 5.4149 0.0851 5.6135 0.0850 5.8121 0.0852 6.0107 0.0854 6.2094 0.0851

Eo = 1000 x/a y/a 0 -1 0.0007 -0.9875 0.0026 -0.9752 0.0055 -0.9630 0.0091 -0.9510 0.0137 -0.9380 0.0196 -0.9240 0.0267 -0.9087 0.0352 -0.8922 0.0453 -0.8744 0.0572 -0.8551 0.0711 -0.8344 0.0873 -0.8120 0.1060 -0.7879 0.1276 -0.7621 0.1524 -0.7345 0.1808 -0.7050 0.2133 -0.6735 0.2505 -0.6401 0.2929 -0.6048 0.3412 -0.5676 0.3960 -0.5287 0.4583 -0.4882 0.5288 -0.4464 0.6085 -0.4035 0.6985 -0.3600 0.7998 -0.3165 0.9137 -0.2732 1.0415 -0.2312 1.1843 -0.1907 1.3438 -0.1528 1.5214 -0.1178 1.7187 -0.0864 1.9157 -0.0615 2.1133 -0.0415 2.3113 -0.0258 2.5095 -0.0131 2.7079 -0.0033 2.9064 0.0045 3.1049 0.0108 3.3034 0.0155 3.5020 0.0194 3.7006 0.0223 3.8992 0.0248 4.0978 0.0265 4.2964 0.0280 4.4950 0.0292 4.6936 0.0300 4.8922 0.0308 5.0908 0.0312 5.2894 0.0317 5.4881 0.0319 5.6867 0.0323 5.8853 0.0324 6.0839 0.0325 6.2825 0.0327

Table 26.2: Interface coordinates of long bubbles in horizontal channel for different E¨ otv¨ os numbers (present numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8).

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

x/a 0 0.0003 0.0011 0.0023 0.0041 0.0064 0.0096 0.0137 0.0189 0.0253 0.0332 0.0429 0.0545 0.0684 0.0850 0.1046 0.1276 0.1546 0.1860 0.2227 0.2651 0.3141 0.3705 0.4353 0.5095 0.5942 0.6905 0.7998 0.9234 1.0626

α = 15˚ y/a x/a -1 1.2188 -0.9875 1.3937 -0.9750 1.5887 -0.9626 1.7840 -0.9502 1.9802 -0.9366 2.1771 -0.9217 2.3744 -0.9054 2.5721 -0.8876 2.7699 -0.8681 2.9679 -0.8469 3.1660 -0.8238 3.3642 -0.7988 3.5624 -0.7717 3.7607 -0.7424 3.9591 -0.7109 4.1574 -0.6769 4.3558 -0.6406 4.5542 -0.6018 4.7527 -0.5605 4.9511 -0.5168 5.1496 -0.4707 5.3481 -0.4224 5.5466 -0.3722 5.7450 -0.3204 5.9436 -0.2673 6.1421 -0.2136 6.3406 -0.1598 6.5391 -0.1068 6.7377 -0.0550 6.9362

y/a -0.0055 0.0413 0.0848 0.1210 0.1517 0.1777 0.2003 0.2201 0.2374 0.2532 0.2672 0.2802 0.2922 0.3033 0.3138 0.3236 0.3329 0.3417 0.3501 0.3582 0.3658 0.3732 0.3802 0.3870 0.3935 0.3997 0.4057 0.4113 0.4166 0.4214

x/a 0 0.0002 0.0009 0.0020 0.0034 0.0054 0.0081 0.0116 0.0160 0.0216 0.0285 0.0369 0.0470 0.0593 0.0740 0.0914 0.1120 0.1364 0.1649 0.1983 0.2373 0.2826 0.3350 0.3958 0.4659 0.5464 0.6386 0.7441 0.8639 0.9999

199/283

α = 45˚ y/a x/a -1 1.1533 -0.9875 1.3258 -0.9750 1.5190 -0.9626 1.7129 -0.9502 1.9081 -0.9365 2.1043 -0.9215 2.3011 -0.9050 2.4983 -0.8870 2.6958 -0.8673 2.8936 -0.8457 3.0915 -0.8222 3.2896 -0.7965 3.4877 -0.7686 3.6859 -0.7383 3.8841 -0.7056 4.0824 -0.6701 4.2808 -0.6320 4.4791 -0.5909 4.6775 -0.5470 4.8760 -0.5002 5.0744 -0.4505 5.2729 -0.3979 5.4713 -0.3429 5.6698 -0.2856 5.8683 -0.2264 6.0668 -0.1659 6.2653 -0.1049 6.4639 -0.0439 6.6624 0.0157 6.8609

y/a 0.0735 0.1284 0.1795 0.2225 0.2587 0.2898 0.3166 0.3399 0.3607 0.3790 0.3957 0.4108 0.4247 0.4375 0.4493 0.4605 0.4708 0.4806 0.4897 0.4985 0.5067 0.5145 0.5220 0.5290 0.5359 0.5422 0.5485 0.5543 0.5599 0.5650

Table 26.3: Interface coordinates of long bubbles in inclined channel for different inclination angles at Eo=100 (present numerical results by BEM with ds min =0.0125, ds max =0.2 and l =8).

200/283

Test-case number 29a by Hien Ha-Hgoc and J. Fabre Eo = 10 x/a y/a 0 0 0.0005 0.0250 0.0020 0.0499 0.0045 0.0748 0.0080 0.0996 0.0129 0.1261 0.0193 0.1543 0.0278 0.1843 0.0384 0.2162 0.0517 0.2498 0.0681 0.2851 0.0880 0.3220 0.1119 0.3603 0.1402 0.3997 0.1736 0.4399 0.2125 0.4806 0.2575 0.5212 0.3090 0.5613 0.3674 0.6003 0.4332 0.6376 0.5066 0.6728 0.5879 0.7055 0.6774 0.7353 0.7748 0.7619 0.8733 0.7841 0.9726 0.8025 1.0723 0.8181 1.1724 0.8313 1.2727 0.8427 1.3732 0.8525 1.4738 0.8609 1.5745 0.8681 1.6753 0.8741 1.7762 0.8790 1.8770 0.8833 1.9779 0.8874 2.0788 0.8915 2.1797 0.8955 2.2806 0.8992 2.3815 0.9021 2.4824 0.9043 2.5834 0.9063 2.6843 0.9086 2.7852 0.9113 2.8862 0.9138 2.9871 0.9154 3.0881 0.9166 3.1890 0.9181 3.2899 0.9202 3.3909 0.9221 3.4919 0.9231 3.5928 0.9236

Eo = 100 x/a y/a 0 0 0.0005 0.0250 0.0020 0.0500 0.0044 0.0748 0.0079 0.0996 0.0127 0.1261 0.0191 0.1543 0.0274 0.1844 0.0380 0.2163 0.0513 0.2499 0.0677 0.2852 0.0876 0.3221 0.1117 0.3602 0.1404 0.3994 0.1744 0.4391 0.2141 0.4790 0.2602 0.5183 0.3131 0.5565 0.3731 0.5931 0.4405 0.6273 0.5156 0.6588 0.5986 0.6869 0.6896 0.7118 0.7882 0.7336 0.8875 0.7514 0.9875 0.7659 1.0876 0.7784 1.1881 0.7889 1.2886 0.7979 1.3893 0.8061 1.4900 0.8131 1.5907 0.8194 1.6915 0.8253 1.7924 0.8304 1.8932 0.8352 1.9941 0.8396 2.0950 0.8436 2.1958 0.8475 2.2967 0.8509 2.3977 0.8542 2.4986 0.8573 2.5995 0.8601 2.7004 0.8629 2.8013 0.8654 2.9023 0.8678 3.0032 0.8701 3.1042 0.8722 3.2051 0.8744 3.3060 0.8763 3.4070 0.8782 3.5079 0.8797 3.6089 0.8804

Eo = 1000 x/a y/a 0 0 0.0005 0.0250 0.0020 0.0500 0.0044 0.0748 0.0078 0.0996 0.0126 0.1261 0.0189 0.1544 0.0272 0.1844 0.0376 0.2163 0.0507 0.2500 0.0669 0.2855 0.0865 0.3225 0.1103 0.3608 0.1387 0.4002 0.1722 0.4403 0.2116 0.4806 0.2572 0.5204 0.3097 0.5592 0.3694 0.5962 0.4367 0.6308 0.5117 0.6623 0.5947 0.6906 0.6857 0.7155 0.7843 0.7369 0.8838 0.7543 0.9837 0.7686 1.0840 0.7806 1.1844 0.7907 1.2850 0.7995 1.3857 0.8072 1.4864 0.8141 1.5872 0.8202 1.6880 0.8258 1.7888 0.8308 1.8897 0.8355 1.9906 0.8398 2.0914 0.8437 2.1923 0.8474 2.2932 0.8508 2.3941 0.8540 2.4951 0.8570 2.5960 0.8599 2.6969 0.8625 2.7978 0.8650 2.8988 0.8674 2.9997 0.8697 3.1007 0.8718 3.2016 0.8739 3.3025 0.8758 3.4035 0.8776 3.5044 0.8791 3.6054 0.8798

Table 26.4: Interface coordinates of plane bubbles in vertical channel for different E¨ otv¨ os numbers (present numerical results by BEM with ds min =0.025, ds max =0.1 and l =4).

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

Eo = 10 x/R r/R 0 0 0.0019 0.0500 0.0075 0.0996 0.0169 0.1488 0.0299 0.1970 0.0474 0.2467 0.0698 0.2974 0.0976 0.3487 0.1313 0.4001 0.1711 0.4511 0.2175 0.5010 0.2705 0.5493 0.3304 0.5954 0.3971 0.6386 0.4707 0.6786 0.5510 0.7149 0.6380 0.7474 0.7314 0.7761 0.8265 0.8001 0.9226 0.8200 1.0192 0.8367 1.1163 0.8509 1.2136 0.8630 1.3111 0.8735 1.4087 0.8826 1.5065 0.8903 1.6044 0.8966 1.7023 0.9016 1.8003 0.9054 1.8984 0.9086 1.9964 0.9118 2.0944 0.9154 2.1924 0.9193 2.2904 0.9229 2.3885 0.9253 2.4865 0.9267 2.5846 0.9278 2.6827 0.9297 2.7807 0.9325 2.8787 0.9349 2.9768 0.9360 3.0749 0.9363 3.1730 0.9374 3.2710 0.9396 3.3691 0.9415 3.4672 0.9421 3.5652 0.9416

Eo = 100 x/R r/R 0 0 0.0018 0.0500 0.0071 0.0997 0.0160 0.1489 0.0284 0.1973 0.0451 0.2472 0.0668 0.2983 0.0939 0.3500 0.1269 0.4018 0.1663 0.4531 0.2125 0.5032 0.2658 0.5513 0.3263 0.5964 0.3943 0.6377 0.4697 0.6741 0.5520 0.7058 0.6406 0.7335 0.7354 0.7573 0.8317 0.7757 0.9286 0.7908 1.0258 0.8039 1.1233 0.8144 1.2210 0.8232 1.3187 0.8315 1.4166 0.8382 1.5145 0.8442 1.6124 0.8500 1.7103 0.8547 1.8083 0.8592 1.9063 0.8636 2.0043 0.8670 2.1023 0.8708 2.2003 0.8740 2.2984 0.8768 2.3964 0.8799 2.4944 0.8823 2.5925 0.8848 2.6905 0.8873 2.7886 0.8892 2.8866 0.8915 2.9847 0.8933 3.0828 0.8951 3.1808 0.8971 3.2789 0.8985 3.3769 0.9003 3.4750 0.9015 3.5731 0.9013

201/283

Eo = 1000 x/R r/R 0 0 0.0018 0.0500 0.0072 0.0997 0.0160 0.1489 0.0279 0.1974 0.0438 0.2476 0.0646 0.2991 0.0910 0.3511 0.1233 0.4034 0.1617 0.4554 0.2073 0.5061 0.2600 0.5547 0.3201 0.6005 0.3878 0.6421 0.4627 0.6797 0.5447 0.7121 0.6333 0.7398 0.7282 0.7631 0.8246 0.7814 0.9215 0.7965 1.0188 0.8085 1.1163 0.8188 1.2140 0.8274 1.3118 0.8349 1.4097 0.8416 1.5076 0.8472 1.6055 0.8527 1.7035 0.8572 1.8015 0.8617 1.8995 0.8656 1.9975 0.8692 2.0955 0.8726 2.1935 0.8756 2.2916 0.8786 2.3896 0.8812 2.4876 0.8839 2.5857 0.8862 2.6837 0.8884 2.7818 0.8906 2.8799 0.8925 2.9779 0.8945 3.0760 0.8962 3.1740 0.8981 3.2721 0.8996 3.3702 0.9012 3.4682 0.9024 3.5663 0.9021

Table 26.5: Interface coordinates of axis-symmetrical bubbles in vertical tube for different E¨ otv¨ os numbers (present numerical results by BEM with ds min =0.05, ds max =0.1 and l =4).

202/283

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

References Batchelor, G. K. 1967. An introduction to fluid dynamics. Cambridge University Press. Bendiksen, K. H. 1984. An experimental investigation of the motion of long bubbles in inclined tubes. Int. J. Multiphase Flow, 10, 467–483. Bendiksen, K. H. 1985. On the motion of long bubbles in vertical tubes. Int. J. Multiphase Flow, 11, 797–812. Benjamin, T. B. 1968. Gravity currents and related phenomena. J. Fluid Mech., 31, 209–248. Collins, R. 1965. A simple model of the plane gas bubble in a finite liquid. J. Fluid. Mech., 22, 763–771. Collins, R., De Moraes, F. F., Davidson, J. F., & Harrison, D. 1978. The motion of a large gas bubble rising though liquid flowing in a tube. J. Fluid. Mech., 89, 497–514. Cou¨et, B., & Strumulo, G. S. 1987. The effects of surface tension and tube inclination on a two-dimensional rising bubble. J. Fluids Mech., 184, 1–14. Davies, R. M., & Taylor, G. 1950. The mechanics of large bubbles rising through extended liquids and through liquids in tubes. Proc. Roy. Soc. Ser. A, 200, 375–390. Dumitrescu, D. T. 1943. Str¨ omung an einer Luftblase im Senkrechten Rohr. Z. Angew. Math. Mech., 23, 139. Garabedian, P. R. 1957. On steady-state bubbles generated by Taylor instability. Proc. Roy. Soc. Ser. A., 241, 423–431. Ha-Ngoc, H. 2003. Etude th´eorique et num´erique du mouvement de poches de gaz en canal et en tube. Ph.D. thesis, Institut National Polytechnique de Toulouse, France. H´eraud, P. 2002. Etude exp´erimentale de dynamique de bulles. Ph.D. thesis, Universit´e de Provence, France. Lamb, H. 1932. Hydrodynamics. Cambridge University Press. Maneri, C., & Zuber, N. 1974. An experimental study of plane bubbles rising at inclination. Int. J. Multiphase Flow, 1, 623–645. Mao, Z. S., & Dukler, A. E. 1990. The motion of Taylor bubbles in vertical tubes I. A numerical simulation for the shape and rise velocity of Taylor bubbles in stagnant and flowing liquids. J. Comp. Phys., 91, 132–160. Mao, Z. S., & Dukler, A. E. 1991. The motion of Taylor bubbles in vertical tubes II. Experimental data and simulations for laminar and turbulent flow. Chem. Eng. Sci., 46, 2055–2064. Vanden-Bro¨eck, J. M. 1984a. Bubbles rising in a tube and jets falling from a nozzle. Phys. Fluids, 27, 1090–1093. Vanden-Bro¨eck, J. M. 1984b. Rising bubble in two-dimensional tube with surface tension. Phys. Fluids, 27, 2604–2607. Wallis, G. B. 1969. One Dimensional Two-phase Flow. McGraw-Hill, New York.

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

203/283

White, E. T., & Beardmore, R. H. 1962. The velocity of rise of single cylindrical bubbles through liquids contained in vertical tubes. Chem. Eng. Sci., 17, 351–361. Zukoski, E. E. 1966. Influence of viscosity, surface tension, and inclination angle on motion of long bubbles in closed tubes. J.Fluid Mech., 25, 821–837.

204/283

Test-case number 29a by Hien Ha-Hgoc and J. Fabre

Test-case number 29b by Hien Ha-Hgoc and J. Fabre

205/283

Chapter 27

Test-case number 29b: The velocity and shape of 2D long bubbles in inclined channels or in vertical tubes (PA, PN) Part II: in a flowing liquid By Hien Ha-Ngoc, Institute of Mechanics, 264 Doi Can, Hanoi, Vietnam. E-Mail: [email protected] Jean Fabre, Institut de M´ecanique des Fluides de Toulouse, All´ee du Pr. Camille Soula, 31400 Toulouse, France. E-Mail: [email protected]

27.1

Practical significance and interest of the test-case

In flowing liquid, the motion of the long bubbles results from the complex influence of both buoyancy and mean motion of the liquid. Following Nicklin et al. (1962) the velocity of these long bubbles would result as the sum of two terms: V = C0 U + V∞

(27.1)

where V 8 is the bubble velocity in a stagnant liquid and U is the mean liquid velocity far from the bubble head. Such decomposition does not mean that V is a linear function of U since C 0 is a dimensionless number that may depend on U through the other dimensionless parameters of the problem. Because the motion of a long bubble does not depend on its length, C 0 represent therefore the influence of the liquid velocity profile imposed far from the bubble head – it’s called the distribution coefficient. The experimental results of Nicklin et al. (1962) in vertical pipe have shown that: C 0 increases from 0.9 for negative U (downward flow) to a maximum of 1.8 near U =0 and then decreases toward an asymptotic value of 1.2 for liquid Reynolds number Re L =UD/ ν

206/283

Test-case number 29b by Hien Ha-Hgoc and J. Fabre

greater than 8000. Therefore C 0 is not only sensitive to the velocity profile but also to the flow direction. In downward flow, bubble may become asymmetrical: such a situation is observed experimentally (Griffith & Wallis, 1961, Nicklin et al. , 1962, Martin, 1976) but it is neither explained theoretically nor reproduced by numerical simulation. An analytical investigation on C 0 for an axis-symmetrical bubble in upward flow in a vertical tube was carried out by Collins et al. (1978) using inviscid theory. Like in stagnant liquid, the viscosity effects is negligible when inertia dominates. The condition is satisfied provided Nf = D3/2 g 1/2 /ν > 300 (Fabre & Lin´e, 1992). In fact, viscosity acts essentially to develop the liquid velocity profile far ahead of the bubble, but it has a negligible influence near the bubble front. For an inviscid axis-symmetrical flow, the Stokes stream function satisfies a Poisson equation which is solved by applying the boundary conditions at the bubble surface. Collins et al. (1978) have obtained an approximate solution for the bubble velocity for both laminar and turbulent velocity profile upstream the bubble. In this approach, the condition of constant pressure was satisfied locally at the bubble nose. Bendiksen (1985) used the same approach to include the effect of surface tension. In this paper, we provide analytical and numerical solutions for the velocity and shape of 2D long bubbles (plane and axis-symmetrical) moving in a flowing liquid. The analytical solution is an extension of the Benjamin (1968) solution for the horizontal case without surface tension, whereas the numerical solutions are obtained for the inertial regime with surface tension. The Poisson equation for the stream function with a source term determined by the velocity profile given ahead of the bubble is solved by an algorithm based on the boundary element method (BEM). The bubble shape is determined by satisfying the condition of constant pressure all along the bubble surface (Ha-Ngoc, 2003). For plane bubbles, in contrast to the case of a stagnant liquid where the source term in the Poisson equation vanishes, the source term always exists in flowing liquid. Moreover, in most cases, this is a non-linear function of the stream function. The numerical algorithm requires, therefore, a domain discretization for the integration of the source term and an additional iterative loop for treating its non-linearity character. The results obtained by the method are in good agreement with different theoretical, experimental and numerical results of Zukoski (1966), Collins et al. (1978), Bendiksen (1985), Mao & Dukler (1990, 1991)(Ha-Ngoc, 2003).

27.2

Definitions and model description

The reference calculations to be proposed are made in dimensionless units. The length scale of the problem is the channel haft-width a or the tube radius R. The velocity scale is (gD)1/2 , where D is the channel width (2a) or the tube diameter (2R). In flowing liquid, the flow may be characterized by four dimensionless parameters: • The bubble Reynolds number, Re Re =

VD ν

(27.2)

ReL =

UD ν

(27.3)

• The liquid Reynolds number, Re L

Test-case number 29b by Hien Ha-Hgoc and J. Fabre

207/283

• The liquid Froude number, Fr L U F rL = √ gD

(27.4)

• The E¨ otv¨os number, Eo ρgD2 , (27.5) σ where ν is the kinematic viscosity of the liquid, ρ, the liquid density, σ, the surface tension and g, the gravity. The dimensionless bubble velocity is presented by the bubble Froude number, Fr, defined as : Eo =

V Fr = √ . gD

(27.6)

Like in the case of stagnant liquid, it is expected that there exists a critical bubble Reynolds number Re ∗ such as for Re>Re ∗ the bubble velocity does not depend on the viscosity. In this flow regime, the liquid can be considered as inviscid and the bubble motion is characterized by three dimensionless parameters: the liquid Reynolds number Re L , the liquid Froude number Fr L and the E¨otv¨os number Eo. Because the flow is assumed inviscid, the liquid Reynolds number Re L is used essentially to determine the velocity profile imposed far ahead of the bubble. For small Re L , the liquid flow is laminar and has a parabolic distribution. For turbulent flow, the velocity distribution is a function of Re L and can be determined by empirical correlation. For axis-symmetrical flow in tubes, in the cylindrical coordinate system the following correlation is employed (Collins et al. , 1978, Bendiksen, 1985):
(27.7) ux (r) = umax 1 − γr2 − (1 − γ) r2n , where u max is the maximum velocity on the tube axis, γ and n depend on the Reynolds number and can be determined by the wall friction law. According to Bendiksen (1985), using the following wall friction law: p 1/fw = 3.5 log10 (ReL ) − 2.6, (27.8) where f w is the friction factor, the following values of γ and n yields: 7.5 , 4.12 + 4.95(log10 (ReL ) − 0.743) γ log (ReL ) − 0.743 n = (γ − 1)[ 10 − (1 − )]−1 − 1. log10 (ReL ) + 0.31 2 γ=

(27.9) (27.10)

The parabolic distribution can be obtained from (27.7) by setting γ=1. For the case of plane bubbles, a turbulent velocity distribution similar to that of (27.7) is also used in the calculations. For the relation between the stream function and the vorticity far upstream the bubble to be uniform, the solutions have to be restricted to a specific range of Froude numbers [Fr L − , Fr L + ]. This restriction comes simply from the solution method which requires the elimination of vorticity by the stream function to obtain the equations of the problem. In the calculations, the contact angle, θ0 =π/2, is always used for case when the bubble touches the upper wall and the criterion of maximum velocity (Garabedian, 1957) is used to select the physically observable solution.

208/283

27.3

Test-case number 29b by Hien Ha-Hgoc and J. Fabre

Motion in horizontal and inclined channel

For horizontal channel, the exact solutions for bubble velocity, V, and the liquid film thickness, d, can be obtained by an extension of the Benjamin (1968) method for the case when surface tension is negligible (Fabre & Ha-Ngoc, 2003). The solutions are given under the form of Taylor series for small liquid Froude number (Fr L 1.8R to prevent interaction of the flow with the bottom boundary of the vessel. The total height of the entire liquid/gas domain was 70 mm. The gas phase in the atmosphere of the vessel was air for test case No. 1 (at T = 25.7◦ C) and argon in the case No. 2 (at T = 25.5◦ C) in each case under atmospheric pressure p = 105 Pa. The properties of air and argon are given in Table 29.1. The test liquids of the test cases are silicone fluids (abbreviated with SF) with a nominal viscosity between 0.65 cSt and 3.0 cSt (SF 0.65 is AK 0.65 supplied by Wacker GmbH and SF 3.0 is Baysilone M3 by Bayer AG). The properties of the test liquids are given in Table 29.1, which were measured with standard laboratory equipment over a temperature range of 15◦ C ≤ T ≤ 35◦ C in good agreement with the values given by the manufacturers.

238/283

Test-case number 31 by M. Michaelis and M. E. Dreyer

In order to vary the contact angle from γs = 0 for test No. 1, a thin film of a surface modifier (FC-732 by 3M Co.) with a low surface energy of (11 − 12) · 10−3 N/m was applied to the cylinder wall of the PMMA test vessel just prior to the drop of the capsule by filling the cylinder with FC-732 as a liquid and spilling it rapidly to achieve a homogenous film. The thin film of a thickness on the order of 1 µm was dried at ambient air and has a surface roughness of RM S = (176 ± 43) nm. Since for this solid/liquid system with partial wetting hysteresis of the contact angle has to be considered, the advancing and receding contact angles were measured with the Wilhelmy plate method. The different values of the static contact angle are given in Table 29.3. For test case No. 2, the solid/liquid/gas system consists of quartz glass/SF 0.65/argon. Due to the high surface energy σso of quartz glass and the low surface tension σ of SF 0.65 a static contact angle of γs = 0 arises for this system. A difference between advancing and receding contact angles is not taken into account. Thus test No. 1 represents a system with partial wetting, while test No. 2 stands for a complete wetting system.

29.4

Results

The evaluation of the liquid/gas reorientation was carried out by detecting the liquid interface using digital im age processing techniques programmed in Matlabr . On this way contour histories with very high resolution in space (approx. 0.04 mm/pixel for No. 1 and 0.08 mm/pixel for No. 2) and in time (0.002 s for No. 1 and 0.004 s for No. 2) were obtained. On the liquid interface h(r, t), the center point zc (t) = h(r = 0, t) and the contact point zw (t) = h(r = R, t) were defined to be characteristic to describe the surface reorientation (compare Figure 29.1). The accuracy of the detection of the liquid interface and thus of the contour histories is about ±1 pixel, which corresponds to ±0.04 mm and ±0.08 mm. The start of the experiment was defined with an accuracy of ±0.004 s, when the free liquid interface shows the first response, after the drop capsule is released.

29.4.1

Test No. 1

The series of video images for test No. 1 are given in Figure 29.2 for specific times, which in general correspond to extremal deflections of the center point and the contact point of the liquid interface. In the initial situation at t = 0 the liquid interface has a flat looking shape. A determination of the liquid interface for each video image was carried out by using digital images processing techniques. The interface was fitted by a polynomial of

Test case No. 1 No. 2

R [mm] 10 20

h0 /R [-] 2 1.8

Oh [10−3 ] 6.42 0.98

Boi [-] 47.69 194.1

Mo [10−3 ] 16.88 3.67

Table 29.2: Parameters for the test cases. The cylinder radius is denoted with R and the fill height with h0 . The dimensionless numbers Ohnesorge-number by Eq. (29.4), Bond-number by Eq. (29.1) and Morton-number by Eq. (29.3) are calculated with the properties given in Table 29.1.

Test-case number 31 by M. Michaelis and M. E. Dreyer

239/283

t=0

t = 0.034 s

t = tcp1 = 0.092 s

t = twp1 = 0.120 s

t = tcp2 = 0.174 s

t = twp2 = 0.216 s

t = tcp3 = 0.268 s

t = tcp4 = 0.340 s

t = tcp5 = 0.416 s

t = 4.702 s

Figure 29.2: Development of the liquid interface for test case No. 1. The liquid phase appears bright, while the meniscus of the liquid interface is dark due to total reflection. The coordinate system is located on the initial liquid interface in the cylinder axis, as depicted for t = 0 for which the contact line is marked as a dashed white line. The gas phase above the liquid is observed throw the lens type of vessel geometry, thus it is darkened close to the vessel wall at both sides due to refraction. The area below the liquid phase appears dark caused by a screen, which was mounted between the light source and the test vessel to enhance the contrast of the liquid interface. t = 0: initial configuration; t = 0.034 s: initial rise at the wall; t = tcp1 = 0.092 s: 1st maximum at the center; t = twp1 = 0.120 s: 1st maximum at the wall; t = tcp2 = 0.174 s: 1st minimum at the center; t = twp2 = 0.216 s: 1st minimum at the wall; t = tcp3 = 0.268 s: 2nd maximum at the center and fixing of contact point; t = tcp4 = 0.340 s: 2nd minimum at the center; t = tcp5 = 0.416 s: 3rd maximum at the center; t = 4.702 s: final equilibrium configuration.

240/283

Test-case number 31 by M. Michaelis and M. E. Dreyer

Figure 29.3: History of the center point with the given deflections, from Figure 29.2. After the contact point is pinned (at twf = 0.268 s), the oscillatory motion of the center point is approximately linear. The uncertainties for the specified deflections are ±0.038 mm.

Figure 29.4: History of the contact point zw = h(r = R, t) with the given deflections, from Figure 29.2. After the contact point reaches its maximum deflection zwp1 a following oscillation of the contact line can be observed until it fixes at twf = 0.268 s. The uncertainties for the specified deflections are ±0.152 mm.

Test-case number 31 by M. Michaelis and M. E. Dreyer

Test Case No. 1 No. 2

Liquid/solid/gas SF 3.0/FC-732/Air SF 0.65/SiO2 /Argon

γsa1 [◦ ] 56.4 ±1 0

241/283

γsr1 [◦ ] 46.7 ±2.1 0

γsa2 [◦ ] 51.7±1.6 0

γse [◦ ] 53.6±1.6 0

Table 29.3: Static advancing and receding contact angles of liquid/solid combinations at T = 25◦ C, which were measured using the Wilhelmy method. γsa1 denotes the first static advancing contact angle on a dry surface. γsr is the static receding contact angle and γsa2 defines the static advancing contact angle on a prewetted surface. The final static contact angle γse is given by the final deflection of the liquid interface, assuming a spherical shape.

6th order (k = 6) corresponding to z(r) =

i=k X

ai ri ,

(29.5)

i=0

where the values for the coefficients ai and their uncertainties ∆ai are given in Table 29.5 for the different points of time. The corresponding histories of the center point zc = h(r = 0, t) and the contact point zw = h(r = R, t) are depicted in Figure 29.3 and Figure 29.4, where the characteristic deflections are specified in comparison with the video still frames at the given times. For this system with partial wetting after step reduction in gravity, the liquid interface starts to rise along the cylinder wall, where a maximum velocity uwmax = (18.2±1.7) mm/s of the contact point can be evaluated at t = (0.024±0.006) s. The mean velocity to the first maximum of the contact point is u ¯w = zwp1 /twp1 = (11.33±1.8) mm/s, where zwp1 is the deflection and twp1 the time at the first peak of the contact line. After the arrival of the contact point at its maximum deflection, an oscillatory motion of the contact line can be observed with a receding and again advancing contact point until it fixes at twf = 0.268 s. In the following time regime, after the contact point is fixed, the oscillatory motion of the center point around the equilibrium position (with final contact angle γse = 53.6◦ ) shows a linear trend with a constant peak to peak frequency of approximately ωd = 2π/T¯ = (42.83±1.3) 1/s of the system, where T¯ is the mean period of the oscillation. The damping of the system can be determined in this period if the center point deflection is transformed and scaled by zc∗ =

zc + 1, zce

(29.6)

where zce = −1.34 mm is the final equilibrium position of the center point. According to the theory for a linear damped oscillator the logarithmic decrement of the damped oscillation is given by Λ=

∗ zcpi 1 ln ∗ , n zcpI

(29.7)

∗ is the first dimensionless peak deflection after the pinning of the contact point where zcpi ∗ at twf = 0.268 s and zcpI the last detectable peak deflection after n full cycles (with I = 2i). The logarithmic decrement Λ can be evaluated from the center point history for both the minimum peaks (with odd values of i, compare Figure 29.3) and the maximum peaks (even values of i), where a mean value Λ = 0.202 ± 0.018 was determined. The dimensionless damping ratio of the oscillation then becomes

D=√

Λ , − Λ2

4π 2

(29.8)

242/283

Test-case number 31 by M. Michaelis and M. E. Dreyer

Time t = 0.034 tcp1 = 0.092 twp1 = 0.120 tcp2 = 0.174 twp2 = 0.216 tcp3 = 0.268 tcp4 = 0.340 tcp5 = 0.416 t = 4.702

s s s s s s s s s

zc [mm] -0.51 -2.2 -1.9 -1.0 -1.18 -1.6 -1.18 -1.48 -1.34

zw [mm] 0.49 1.18 1.36 1.03 0.90 0.99 0.99 0.99 0.99

Description initial rise at the wall 1st maximum at the center 1st maximum at the wall 1st minimum at the center 1st minimum at the wall 2st maximum at the center and fixing of contact point 2st minimum at the center 3st maximum at the center final equilibrium configuration

Table 29.4: Deflections of the center point and the contact point for given points of time of test case No. 1. The errors for the deflection are ∆zw = 0.152 mm and ∆zc = 0.038 mm. The errors for the definition of the time are ∆t = 0.004 s.

which is D = 0.0322 ± 0.0029 for test case No. 1. With the damping ratio D the center point oscillation can be analytically described for the second main period by z ∗ (τ ) =

−√ D τ 1 e 1−D2 1 − D2    D cos τ − arctan √ , 1 − D2

√

(29.9)

where τ = tωd is the dimensionless time. Coefficient/ Error a0 a1 [10−3 ] a2 [10−3 ] a3 [10−5 ] a4 [10−4 ] a5 [10−7 ] a6 [10−7 ] ∆a0 [10−3 ] ∆a1 [10−4 ] ∆a2 [10−4 ] ∆a3 [10−5 ] ∆a4 [10−6 ] ∆a5 [10−7 ] ∆a6 [10−8 ]

0.0 s -0.011 1.220 3.630 0.566 -1.250 -0.665 16.735 1.330 4.609 1.602 1.801 4.356 1.572 3.125

0.034 s -0.506 4.620 0.077 -2.465 2.509 1.592 -7.661 2.870 9.916 3.447 3.874 9.371 3.382 6.722

0.092 s -2.216 4.920 62.850 4.234 -2.923 -3.562 7.434 1.300 4.496 1.563 1.757 4.249 1.533 3.048

Time t 0.120 s 0.174 s -1.903 -1.016 6.940 10.180 39.620 11.740 0.917 -6.887 0.906 2.295 -1.888 4.241 -9.330 -6.738 1.330 1.76 4.596 6.076 1.598 2.112 1.796 2.374 4.343 5.743 1.567 2.072 3.116 4.119

0.216 s -1.181 6.920 25.150 1.479 0.060 -1.590 3.322 1.16 4.023 1.398 1.572 3.802 1.372 2.727

0.268 s -1.575 6.520 41.480 -0.335 -1.607 0.859 8.509 1.350 4.667 1.623 1.824 4.411 1.592 3.164

0.340 s -1.192 4.000 23.020 1.799 0.572 0.577 1.006 1.220 4.218 1.467 1.648 3.987 1.439 2.860

Table 29.5: Coefficients ai and their errors ∆ai for the description of the liquid interface by Eq. (29.5) for test case No. 1 at different points of time. Compare also Figure 29.2 for the corresponding video images.

29.4.2

Test No. 2

For test case No. 2, representing a system with complete wetting, the series of video images is given in Figure 29.5 for the total view. The video images also specify characteristic deflections of the liquid interface during the capillary driven reorientation upon step

Test-case number 31 by M. Michaelis and M. E. Dreyer

243/283

t = 0.000 s

t = 0.200 s

t = tcp1 = 0.384 s

t = twp1 = 0.764 s

t = tcp2 = 1.140 s

t = tcp3 = 1.588 s

t = tcp4 = 2.288 s

t = tcp5 = 2.816 s

t = 4.704 s

Figure 29.5: Development of the liquid interface for test case No. 2; t = 0: initial configuration; t = 0.2 s: initial rise at the wall; t = tcp1 = 0.384 s: 1st maximum at the center; t = twp1 = 0.764 s: 1st maximum at the wall; t = tcp2 = 1.14 s: 1st minimum at the center; t = tcp3 = 1.588 s: 2nd maximum at the center and fixing of contact point; t = tcp4 = 2.288 s: 2nd minimum at the center; t = tcp5 = 2.816 s: 3rd minimum at the center; t = 4.704 s: final equilibrium configuration.

244/283

Test-case number 31 by M. Michaelis and M. E. Dreyer

t = 0.32 s

t = 0.40 s

t = 0.48 s

t = 0.56 s

t = 0.64 s

t = 0.72 s

t = 0.80 s

t = 0.88 s

t = 0.96 s

t = 1.04 s

t = 1.12 s

t = 1.2 s

t = 1.28 s

t = 1.36 s

t = 1.44 s

t = 1.52 s

Figure 29.6: Formation of the liquid layer at the cylinder wall, after the contact point reaches the first maximum zwp = 17.96 mm. The solid line denotes the final equilibrium position and the uncertainties are depicted with dashed lines. The plotted scale corresponds to 1 mm.

Test-case number 31 by M. Michaelis and M. E. Dreyer

245/283

change to microgravity. The contours of the liquid interface were determined and fitted with polynomials of 8th order (k = 8) and expressed by Eq. (29.5), where the coefficients of the polynomials and their errors are given in Table 29.7. The histories of the center point zc = h(r = 0, t) and the contact point zw = h(r = R, t) are depicted in Figure 29.7 and in Figure 29.8. For this system with complete wetting and a comparably small Ohnesorge number, especially the behavior of the contact point is different. With a much higher maximum velocity of uwmax = 52.7±4 mm/s at t = (0.044±0.008) s an overshoot of the contact point to the first peak above the equilibrium position (with a mean velocity u ¯w = zwp /twp1 = (23.4 ± 1.2) mm/s) can be observed. In comparison to experiments with partial wetting, a layer of liquid remains at the maximum position after the first peak (shown in detail in Figure 29.6). No receding of the contact line could be observed. In the following time period a contraction of the liquid interface is formed between the liquid bulk and the layer, which leads to a thinning of the liquid layer caused by a pressure gradient. This liquid layer gives reason to a lower mean peak to peak frequency ωd = (5.37 ± 0.47) 1/s (after the 3rd peak deflection). The damping ratio for the center point oscillation was determined with the same method described for test case No. 1, where the center point peaks for t > 1.588 s (starting with the third peak) where taken for the evaluation. As described above, a distinct pinning of the contact line could not be observed as for test case No. 1 (typical for partial wetting systems). For test case No. 2 a mean damping ratio D = 0.185 ± 0.03 and a logarithmic decrement Λ = 1.184 ± 0.203 was evaluated.

Figure 29.7: History of the center point for test case No. 2 with the given deflections, from Figure 29.5. The initial behavior of the center point can not be observed due to refraction at the liquid interface (marked with a grey bar). The uncertainties for the specified deflections are ±0.08 mm.

29.5

Proposed calculations

For the proposed calculations the following assumptions can be made for the system. It can be assumed, that there are no thermal gradients within the system thus it is treated to be isothermal. The test liquids can be considered to be non-volatile and Newtonian

246/283

Test-case number 31 by M. Michaelis and M. E. Dreyer

Figure 29.8: History of the contact point for text case No. 2 with the maximum deflection at t = 0, 764 s, from Figure 29.5. After the contact point arrives at the maximum deflection a further determination of the contact point deflection is possible, due to the lack of total reflection and the formation of the liquid layer. The uncertainty for the specified deflection is ±0.64 mm.

and have constant material properties. The gas phase above the liquid interface has the properties given above in Table 29.1. The cylinder wall can be treated to be rigid and homogenous with a surface roughness of RM S = (176 ± 43) nm (for test No. 1) and RM S = 10 nm (No. 2). Assuming an initial acceleration vector aligned with the cylinder axis and a constant initial static contact angle along the circumference, the system can be considered to be axisymmetric with the origin of the coordinate system located on the initial liquid interface at h0 . For times t ≤ 0 the system is dominated by a constant acceleration kz = kzi = g for which it is required to calculate the initial liquid interface for the static equilibrium. The general solution for the equilibrium shape of the free liquid interface in a right circular

Time t = 0.200 tcp1 = 0.384 twp1 = 0.764 tcp2 = 1.140 tcp3 = 1.588 tcp4 = 2.288 tcp5 = 2.816 t = 4.7004

s s s s s s s s

zc [mm] -5.1 -12.4 -8.2 -5.0 -8.6 -6.2 -7.4 -6.84

zw [mm] 7.0 11.5 17.9 11.6

Comment initial rise at the wall 1st maximum at the center 1st maximum at the wall 1st minimum at the center 2st maximum at the center 2st minimum at the center 3rd maximum at the center final equilibrium configuration

Table 29.6: Deflections of the center point and the contact point for given point of time of test case No. 2. The errors for the deflection are ∆zw = 0.64 mm and ∆zc = 0.08 mm. The errors for the definition of the time are ∆t = 0.004 s.

Test-case number 31 by M. Michaelis and M. E. Dreyer

Coefficient/ Error a0 a1 [10−3 ] a2 [10−3 ] a3 [10−4 ] a4 [10−4 ] a5 [10−7 ] a6 [10−7 ] a7 [10−10 ] a8 [10−10 ] ∆a0 [10−3 ] ∆a1 [10−3 ] ∆a2 [10−4 ] ∆a3 [10−5 ] ∆a4 [10−6 ] ∆a5 [10−7 ] ∆a6 [10−8 ] ∆a7 [10−10 ] ∆a8 [10−11 ]

0.0 s 0.063 -4.290 -2.110 0.178 0.400 -0.558 -2.149 -0.239 3.646 3.130 0.721 1.534 1.295 1.853 0.669 0.769 1.021 1.017

0.20 s -5.019 -4.010 -1.230 0.135 1.340 -3.145 -1.567 5.560 1.138 5.430 1.250 2.674 2.261 3.243 1.174 1.351 1.798 1.795

0.384 s -12.325 -10.060 39.710 -1.188 1.055 5.665 0.545 -6.896 -4.146 7.910 1.830 3.897 3.295 4.725 1.711 1.968 2.619 2.616

Time t 0.764 s 1.140 s -7.935 -4.919 -12.570 5.780 7.390 11.180 2.655 -0.748 3.751 0.956 -11.878 5.373 -21.653 -4.253 7.417 -12.380 41.107 7.746 56.250 6.910 12.970 1.600 27.600 3.402 23.283 2.876 33.320 4.125 12.036 1.494 13.821 1.718 18.349 2.287 18.289 2.284

247/283

1.588 s -8.458 -4.680 20.680 -1.706 1.644 11.574 -7.831 -24.530 14.802 10.920 2.520 5.378 4.547 6.521 2.360 2.715 3.613 3.608

2.288 s -6.080 -5.190 16.690 -1.922 1.049 13.042 -4.895 -26.175 9.047 8.680 2.010 4.276 3.616 5.184 1.877 2.159 2.873 2.869

2.816 s -7.365 3.750 19.110 -2.351 1.414 16.190 -6.632 -32.717 12.204 10.940 2.530 5.385 4.554 0.653 2.364 2.719 3.618 3.613

Table 29.7: Coefficients ai and their errors ∆ai for the description of the liquid interface by Eq. (29.5) for test case No. 2 at different points of time. Compare also Figure 29.5 for the corresponding video images and Table 29.4 for the center point and contact point deflections.

cylinder under a constant acceleration kzi is described by Concus (1968) 1 r∗ h

h∗r 1+

(h∗r )2

i1/2 + h

h∗rr 1+

(h∗r )2

∗ i3/2 = C + Boi h ,

(29.10)

with h = Rh∗ , r = Rr ∗ , h∗r = ∂h∗ /∂r∗ , h∗rr = ∂ 2 h∗ /∂r∗ 2 and the initial Bond number Boi by Eq. (29.1) (see Table 29.2). The differential equation (29.10) can be solved numerically (e.g. using the Runge-Kutta method, Kreyszig, 1999), where the constant C is varied until the boundary condition for the initial static liquid interface h∗r |r∗ =1 = cot γsa1 ,

(29.11)

is satisfied. In Eq. (29.11) γsa1 denotes the advancing static contact angle for test No. 1 or γsa1 = γs = 0 for test No. 2, given in Table 29.3. After having determined the initial liquid interface a step reduction of the dominating acceleration from kzi = g to kze = 10−6 g ≈ 0 must be applied to the system. The transition time for the change of the body forces can be assumed to be infinitesimal. The velocity field of the liquid after transition to microgravity is for times t > 0 governed by the continuity and the Navier-Stokes equations ∇ · v = 0,

(29.12)

∂v 1 + (v · ∇) v = − ∇p + ν∆v, ∂t ρ

(29.13)

where v = (u, w) is the velocity vector of the flow and p the pressure. The corresponding properties of the test liquids are given in Table 29.1. Equations (29.12) and (29.13) are

248/283

Test case 1 1 1 1 1 2 2 2 2 2

Test-case number 31 by M. Michaelis and M. E. Dreyer

Parameter ωd uwmax u ¯wr D Λ ωd uwmax u ¯wr D Λ

Value 42.83 1/s 18.2 mm/s 11.33 mm/s 0.032 0.202 5.37 1/s 52.7 mm/s 23.4 mm/s 0.185 1.184

Error ±1.3 ±1.7 ±1.85 ±0.003 ±0.018 ±0.47 ±4.0 ±1.2 ±0.03 ±0.203

Description natural frequency of damped oscillation maximum rise velocity of contact point mean rise velocity of contact point damping ratio logarithmic decrement natural frequency of damped oscillation maximum rise velocity of contact point mean rise velocity of contact point damping ratio logarithmic decrement

Table 29.8: Results of the test cases.

subject to the following boundary condition at the liquid/gas interface since there is no mass flow through the meniscus h (r, t) (v − vh ) · nl = 0,

(29.14)

which is the kinematic condition. Here vh is the velocity vector of the liquid interface h (r, t) and nl the normal unit vector on h pointing in direction of the gas phase (see Figure 29.1). The dynamic conditions for balancing normally and tangentially the stress components across the liquid/gas interface are given by Ferziger & Peri´c (1999) [nl · T]l · nl = [nl · T]g · nl + 2σK,

(29.15)

[nl · T]l · tl = [nl · T]g · tl + ∇σ = 0,

(29.16)

where tl denotes the tangential unit vector on the liquid interface, T the stress tensor and K the mean curvature of the liquid interface. The corresponding properties of the liquids and the gas phases are tabulated in Table 29.1. The curvature for the axisymmetric interface is given by Langbein (2002) " # rhr 1 d 2K = , (29.17) r dr (1 + h2r )1/2 with hr = dh/dr. Furthermore the condition for a smooth interface requires in the cylinder axis at r = 0 ∂h = 0. (29.18) ∂r r=0

In case of a moving contact line the contact line boundary condition may be formulated by ∂h = cot γd , (29.19) ∂r r=R where γd is the dynamic contact angle. For the behavior of the apparent dynamic contact angle γd at the unsteady advancing three phase contact line, evaluation of experiments with small static contact angles γs < 10◦ show a good agreement with the empirical calculation from Bracke et al. (1989) cos γsa − cos γd = 2Ca1/2 , cos γsa + 1

(29.20)

Test-case number 31 by M. Michaelis and M. E. Dreyer

249/283

where Ca =

µuw σ

(29.21)

is the Capillary-number with the velocity of the contact point uw and γsa the advancing static contact angle, which is γsa = γsa1 for test No. 1 and γsa = γs for test No. 2. In case of a receding contact line, as it was observed for test No. 1 after the first rise to the maximum deflection, Eq. (29.20) might also be applied using the receding static contact angle γsr1 with a negative right hand side of Eq. (29.20) considering the receding motion of the contact line. In the case of an oscillatory contact line (for test No. 1) contact angle hysteresis should be implemented in the numerical calculation to enable a temporary pinning of the contact line at the peak positions until a further movement of the contact line might appear. For an anchored/fixed contact line the liquid meets the cylinder wall at the three phase junction with the static contact angle ∂h = cot γs , (29.22) ∂r r=R

which lies inside the hysteresis range γsr ≤ γs ≤ γsa given by the static receding contact angle γsr and the static advancing contact angle γsa . As a result of the proposed calculations the liquid interface should be calculated for different points of time defined for the two test cases. The contours of the liquid interface can be quantitatively compared with the polynomials given for the different points of time (Table 29.5 for test No. 1 and Table 29.7 for test No. 2). The specific behavior of the liquid interface during the formation of a liquid layer at the cylinder wall for test No. 2 can be compared qualitatively with the video images in Figure 29.6. The deflections of the center point zc = h(r = 0) and the contact point zw = h(r = R) should be calculated with a higher resolution in time to be able to compare both the specific deflections and the points of time given for the two test cases (see Figure 29.3, Figure 29.4 and Table 29.5 for test No. 1; Figure 29.7, Figure 29.8 and Table 29.7 for test No. 2). Furthermore from the histories of the contact point the maximum and mean velocities (uwmax and u ¯wr ) can be evaluated and compared with the values given for the test cases in Table 29.8. Finally the center point histories enable the evaluation of the mean frequency ωd and the damping ratio D, which can be verified with the experimental results tabulated in Table 29.8.

Acknowledgement The funding of the research project by the Federal Ministry of Education and Research (BMBF) and the German Aerospace Center (DLR) under grant number 50 WM 9901 and 50 WM 0241 is gratefully acknowledged.

250/283

Test-case number 31 by M. Michaelis and M. E. Dreyer

References Bauer, H. F., & Eidel, W. 1990. Linear Liquid Oscillations in Cylindrical Container under Zero-Gravity. Appl. microgravity tech., 2, 212–220. Bracke, M., de Voeght, F., & Joos, P. 1989. The Kinetics of Wetting: The Dynamic Contact Angle. Progr. Colloid Polym. Sci, 79, 142–149. Brandrup, J., & Immergut, E. H. 1989. Polymer Handbook. 3rd edn. New York: Wiley. Concus, P. 1968. Static Menisci in a Vertical Right Circular Cylinder. J. Fluid Mech., 34.3, 481–495. Dussan, V. E. B., & Davis, S. H. 1974. On the Motion of a Fluid-Fluid Interface Along a Solid Surface. J. Fluid Mech., 65, 71–95. Ferziger, J. H., & Peri´c, M. 1999. Computational Methods for Fluid Dynamics. Berlin, Heidelberg: Springer Verlag. Gerstmann, J., Dreyer, M. E., & Rath, H. J. 2000. Surface Reorientation Upon Step Reduction in Gravity. Pages 847–853 of: El-Genk, M. S. (ed), Space Technology and Applications International Forum. Albuquerque, New Mexico, USA: AIP Conference 504. Kaukler, W. F. 1988. Fluid Oscillation in the Drop Tower. Metallurgical Transactions AIME, 19a, 2625–2630. Kreyszig, E. 1999. Advanced Engineering Mathematics. 8 edn. New York: Wiley. Langbein, D. 2002. Capillary Surfaces: Shape - Stability - Dynamics, in Particular under Weightlessness. Berlin, Heidelberg, New York: Springer Verlag. Michaelis, M. 2003. Kapillarinduzierte Schwingungen freier Fl¨ ussigkeitsoberfl¨ achen. Ph.D. thesis, University of Bremen. Michaelis, M., Dreyer, M. E., & Rath, H. J. 2002. Experimental Investigation of the Liquid Interface Reorientation Upon Step Reduction in Gravity. Pages 246–260 of: Sadhal, S. S. (ed), Microgravity Transport Processes in Fluid, Thermal, Biological, and Material Sciences, vol. 974. New York Academy of Sciences, New York. Overbury, S. H., Bertrand, P. A., & Somorjai, G. A. 1975. The Surface Composition of Binary Systems. Prediction of Surface Phase Diagrams of Solid Solutions. Chemical Reviews, 75(5), 547–560. Siegert, C. E., Petrash, D. A., & Otto, E. W. 1964. Time Response of Liquid-Vapor Interface After Entering Weightlessness. Tech. rept. NASA TN D-2458. Verein Deutscher Ingenieure (ed). 1988. VDI-W¨ armeatlas: Berechnungsbl¨ atter f¨ ur den W¨ arme¨ ubergang. 5 edn. D¨ usseldorf: VDI Verlag. Weislogel, M. M., & Ross, H. D. 1990. Surface Reorientation and Settling in Cylinders Upon Step Reduction in Gravity. Microgravity Sci. Technol., 3, 24–32. W¨olk, G., Dreyer, M. E., Rath, H. J., & Weisvogel, M. M. 1997. Damped Oscillations of a Liquid/Gas Surface upon Step Reduction in Gravity. J. Spacecr. Rockets, 34(1), 110–117.

Chapter 30

Test-case number 33: Propagation of solitary waves in constant depths over horizontal beds (PA, PN, PE) By Pierre Lubin, TREFLE - UMR CNRS 8508, ENSCPB Universit´e Bordeaux 1, 33607 Pessac cedex, France Phone: +33 (0)5 40 00 33 07, Fax: +33 (0)5 40 00 66 68, E-Mail: [email protected] Herv´e Lemonnier, DER/SSTH/LIEX, CEA/Grenoble, 38054 Grenoble cedex 9, France Phone: +33 (0)4 38 78 45 40, Fax: +33 (0)4 38 78 50 45, E-Mail: [email protected]

30.1

Practical significance and interest of the test-case

Analytical solutions and a related experiment are provided here. The experiment is devoted to the propagation of a solitary wave in a wave tank. In particular, the water surface profiles and corresponding water particle velocities of several solitary waves have been measured. These experimental results can be compared with existing analytical theories which follow different orders of approximation. Solitary waves are known for having some interesting properties: indeed, such a wave has a symmetrical form with a single hump and propagates with a uniform velocity without changing form. When simulating two-phase flows, it is important to evaluate the general accuracy of the numerical methods and numerical schemes used by checking for example the balance of mass and energy in the computing domain. Thus, the results of the different solitary wave theories can be used to compute the initial kinematic properties and simulate their propagation in constant depths over horizontal beds in periodic domains, the precision of the simulation being assessed by comparing the free-surface shapes and velocities to the theoretical values.

252/283

30.2

Test-case number 33 by P. Lubin and H. Lemonnier

Definitions and model description

Several analytical solutions can be found in the literature (Boussinesq, 1871, McCowan, 1891, Grimshaw, 1971, Fenton, 1972, Lee et al. , 1982). The reference variables of the initial wave are the celerity, c, the depth, d, of the water channel and the wave height, H. All the solutions detailed in the following sections will give solitary waves propagating from the left to the right end of the numerical domain. Boussinesq solution or 1st order solution The initial wave shape, and the shape at any time t, celerity and velocity field (u and v being the cartesian components) can be computed from the first order solitary wave theory (Boussinesq, 1871):

2

η(x, t) = Hsech

r 3H 4d

 (x − x − ct) , 0 3

(30.1)

where x0 is the initial position of the wave crest. The celerity is defined by:

c=

p g(d + H).

(30.2)

In the following, it will be considered that η is a sole function of the coordinate x0 which is the longitudinal coordinate in the frame attached to the wave. η = η(x0 (x, t)) with x0 (x, t) = x − x0 − ct.

(30.3)

Next the velocity field is given by: h u 1 d2 h 3 z 2 i ∂ 2 η∗ i √ =  η∗ − η∗ 2 + 2 1 − 4 3c 2 d2 ∂t2 gd (30.4) v  h 1  ∂η∗ d2  z 2  ∂ 3 η∗ i √ =z 1 − η∗ + 2 1− 2 c 2 ∂t 3c 2d ∂t3 gd with  = H/d, η∗ = η/H, and: q q  3H 3H 0 2c sinh x 4 d3 4 d3 ∂η∗ = q  ∂t 3H 0 cosh3 x 4 d3 ∂ 2 η∗ ∂t2

∂ 3 η∗ ∂t3

h q  i 3H 0 c2 32 dH3 2 cosh2 x − 3 3 4d = q  3H 0 cosh4 x 3 4d

=

6c3 dH3

q

3H 4 d3

sinh

A lower order solution can be given:

q

h

cosh2 q  3H 0 cosh5 x 4 d3 3H 0 4 d3 x

(30.5)

q

3H 0 4 d3 x



i −3

Test-case number 33 by P. Lubin and H. Lemonnier

253/283

u η √ = d gd (30.6) v z 1 ∂η √ = d c ∂t gd Note that this solution is a very rough one as the horizontal velocity component, u, is independent of depth, z. Grimshaw solution or 3rd order solution The initial wave shape, celerity and velocity field are then given by the following solution developed by Grimshaw (1971): 5 3 101 4 2  η = s2 − 2 s2 t2 + 3 s2 t2 − s t d 4 8 80

(30.7)

h 1  z 2  3 u 9 i √ = s2 − 2 − s2 + s4 + s2 − s4 4 d 2 4 gd  z 2  3 1 6 15 15  s2 + s4 − s6 + − s2 − s4 + s6 4 5 5 d 2 4 2  z 4  3 45 45 i − s2 + s4 − s6 + d 8 16 16 z  h h3  z 2  31 1 v 3 i √ = (3) 2 t − s2 + 2 s2 + 2s4 + s2 − s4 d 8 d 2 2 gd −3

h 19

(30.8)

h 49  z 2  13 17 18 25 15  s2 − s4 − s6 + − s2 − s4 + s6 640 20 5 d 16 16 2  z 4  3 9 27 ii − s2 + s4 − s6 + d 40 8 16

+3

1

with  = H/d, α = ( 34 ) 2 (1 − 58  +

71 2 128  ),

s =sech(αx0 ) = 1/ cosh(αx0 ) et t =tanh(αx0 ).

The pressure and celerity are then given by: z  h3  z 2  3 p 3 9 i =1− + s2 + 2 s2 − s4 + − s2 + s4 ρgd d 4 2 d 2 4 h 1 19 11 +3 − s2 − s4 + s6 2 20 5  z 2  3 39 33   z 4  3 2 45 4 45 6 i + s2 + s4 − s6 + s − s + s d 4 8 4 d 8 16 16

(30.9)

p  3  12 1 (30.10) gd 1 +  − 2 − 3 20 70 Some other solutions can be found in the literature, as McCowan’s analytical solution (McCowan, 1891), Fenton’s 9th order solution (Fenton, 1972) or Tanaka’s exact solution (Tanaka, 1986), the latter two being numerically obtained. c=

254/283

30.3

Test-case number 33 by P. Lubin and H. Lemonnier

A series of three test-cases

It is proposed to simulate the propagation of solitary waves in three different configurations, according to the parameters of table 30.1. Water particle velocities for various depths and wave profiles are measured by Lee et al. (1982) along the time, as shown in figures 30.1, 30.2, 30.3, 30.4 and 30.5. Results obtained with Tanaka’s algorithm (Tanaka, 1986) are also compared.

Test-case 1 2 3

 0.11 0.19 0.29

d(m) 0.302 0.4046 0.204

H (m) 0.03322 0.076874 0.05916

Table 30.1: Values of the parameters describing the experiments by Lee et al. (1982), with  = H/d.

As described in section 30.2, the solitary wave is completely defined for a given depth, d, and relative amplitude, . Therefore, it is possible to compare the p computed evolution of the wave profiles and velocities during the non-dimensional time t g/d, at various depths, z, to the results of the experiments and to the analytical solutions for the three values of . The proposed numerical configuration is to consider an initial solitary wave computed from a chosen analytical theory, any of those presented in section 30.2 giving good results, except the already mentioned low order solution, Eq. 30.6. The free-surface shapes are almost identical, as shown in figure 30.1, whatever the analytical theory. The differences between the analytical theories can be estimated from the tables 30.2, 30.3 and 30.4, where we give sample of the extremal value of the non-dimensional cartesian components of the velocities for given depths. The most appropriate case for an accurate comparison is the test-case 1, with d = 0.3020 m and  = H/d = 0.11, the prediction of the wave profile being guaranteed to less than 1.10−3 m. The crest is located in the middle of the numerical domain, periodic boundary conditions being imposed in the flow direction. All calculations should be made with the densities and the viscosities of air and water (ρa = 1.1768 kg.m−3 and ρw = 1000 kg.m−3 , µa = 1.85.10−5 kg.m−1 .s−1 and µw = 1.10−3 kg.m−1 .s−1 ). It is obvious that a sufficient number of grid points may be chosen in order to have enough accuracy in the free-surface description. The simulation time step is chosen to verify the stability criterion (CourantFriedrichs-Levy) less than one for the interface algorithm. It is required to check that the solitary wave maintains its original shape as it propagates. The differences can be calculated between the theoretical and numerical results. The free-surface profiles are to be plotted versus the non-dimensional time, as shown in figure 30.1, and compared to the analytical values with 30.1 and 30.7. It is also required to plot the velocity distributions along the depth, as the wave propagates, versus the non-dimensional time, as shown in figures 30.2, 30.3, 30.4 and 30.5. For an easier comparison, the main values to be checked are given in tables 30.2, 30.3 and 30.4. The analytical models are non dissipative. Therefore, the conservation of mass and energy may be checked during the numerical simulation, the kinetic energy, the potential

Test-case number 33 by P. Lubin and H. Lemonnier

255/283

energy and the total energy being: Z Z 1 ρu2 dxdy 2 Z Z Ep = ρzdxdy Ek =

(30.11)

Et = Ek + Ep As a matter of fact, an easy check to do is to consider the celerity of the initial wave, estimate the theoretical distance it has to propagate during the time of the simulation and compare it to the final position of the wave crest. 1st order c

(m.s−1 )

√ c/ gd z/d √ umax / gd √ vmax / gd (.10−1 ) c (m.s−1 ) √ c/ gd z/d √ umax / gd √ vmax / gd (.10−1 ) c (m.s−1 ) √ c/ gd z/d √ umax / gd √ vmax / gd (.10−1 )

1.8134 1.054 0.92 0.78 0.62 0.109 0.106 0.104 ± 0.202 ± 0.170 ± 0.133 3rd order 1.81288 1.053 0.92 0.78 0.62 0.107 0.106 0.104 ± 0.195 ± 0.163 ± 0.129 Tanaka 1.81284 1.053 0.92 0.78 0.62 0.106 0.105 0.103 ± 0.195 ± 0.165 ± 0.128

0.45 0.103 ± 0.0958

0.45 0.103 ± 0.0927

0.45 0.102 ± 0.0952

Table 30.2: Values of the maximum of the non-dimensional velocity components at given non-dimensional depths (z/d). Test case 1, d = 0.3020 m,  = 0.11.

Summary of the required calculations for propagations of solitary waves Three cases of solitary waves propagating in constant depths over horizontal beds are proposed. The test case control parameters are given in table 30.1, an initial analytical theory has to be chosen between those presented in 30.2. The two-phase flow should be simulated with the densities and the viscosities of air and water (ρa = 1.1768 kg.m−3 and ρw = 1000 kg.m−3 , µa = 1.85.10−5 kg.m−1 .s−1 and µw = 1.10−3 kg.m−1 .s−1 ). It is required to check the following results with the reference model: • The conservation of the shapes of the solitary waves. • The conservation of mass and energy during the simulation. • The total distance of propagation during the simulation.

256/283

Test-case number 33 by P. Lubin and H. Lemonnier

c (m.s−1 ) z/d √ umax / gd √ vmax / gd (.10−1 )

1.05 0.193 ± 0.500

c (m.s−1 ) z/d √ umax / gd √ vmax / gd (.10−1 )

1.05 0.182 ± 0.465

c (m.s−1 ) z/d √ umax / gd √ vmax / gd (.10−1 )

1.05 0.184 ± 0.469

1st order 2.1733 1.02 0.92 0.191 0.186 ± 0.484 ± 0.429 3rd order 2.1714 1.02 0.92 0.181 0.178 ± 0.450 ± 0.402 Tanaka 2.1713 1.02 0.92 0.183 0.179 ± 0.457 ± 0.408

0.45 0.168 ± 0.199

0.45 0.169 ± 0.189

0.45 0.170 ± 0.198

Table 30.3: Values of the maximum of the non-dimensional velocity components at given non-dimensional depths (z/d). Test case 2, d = 0.4046 m,  = 0.19.

c (m.s−1 ) z/d √ umax / gd √ vmax / gd (.10−1 )

1.05 0.296 ± 0.876

c (m.s−1 ) z/d √ umax / gd √ vmax / gd (.10−1 )

1.05 0.260 ± 0.786

c (m.s−1 ) z/d √ umax / gd √ vmax / gd(.10−1 )

1.05 0.273 ± 0.788

1st order 1.6067 1.03 0.92 0.294 0.280 ± 0.855 ± 0.745 3rd order 1.6035 1.03 0.92 0.259 0.253 ± 0.770 ± 0.680 Tanaka 1.6032 1.03 0.92 0.271 0.264 ± 0.770 ± 0.672

0.67 0.255 ± 0.522

0.67 0.245 ± 0.485

0.67 0.251 ± 0.471

Table 30.4: Values of the maximum of the non-dimensional velocity components at given non-dimensional depths (z/d). Test case 3, d = 0.204 m,  = 0.29.

Test-case number 33 by P. Lubin and H. Lemonnier

a.

b.

d = 0.4046 m,  = 0.19

257/283

d = 0.302 m,  = 0.11

c.

d = 0.204 m,  = 0.29

Figure 30.1: Solitary wave profiles: water surface elevation, η/d, is plotted versus non-dimensional time, p t g/d. Comparison of theories and experiments. ◦ 1st order, / 3rd order,  Tanaka (Tanaka, 1986), + Lee (Lee et al. , 1982).

258/283

Test-case number 33 by P. Lubin and H. Lemonnier

(a) z/d = 0.92

(b) z/d = 0.78

(c) z/d = 0.62

(d) z/d = 0.45

Figure 30.2: Horizontal velocities at various depths p z/d: non-dimensional water particle velocities, √ u/ gd, are plotted versus non-dimensional time, t g/d. Comparison of experiments and theories for d = 0.3020 m,  = 0.11. ◦ 1st order, / 3rd order,  Tanaka (Tanaka, 1986), + Lee (Lee et al. , 1982).

Test-case number 33 by P. Lubin and H. Lemonnier

259/283

(a) z/d = 1.05

(b) z/d = 1.02

(c) z/d = 0.92

(d) z/d = 0.45

Figure 30.3: Horizontal velocities at various depths p z/d: non-dimensional water particle velocities, √ u/ gd, are plotted versus non-dimensional time, t g/d. Comparison of experiments and theories for d = 0.4046 m,  = 0.19. ◦ 1st order, / 3rd order,  Tanaka (Tanaka, 1986), + Lee (Lee et al. , 1982).

260/283

Test-case number 33 by P. Lubin and H. Lemonnier

(a) z/d = 1.05

(b) z/d = 1.03

(c) z/d = 0.92

(d) z/d = 0.67

Figure 30.4: Horizontal velocities at various depths p z/d: non-dimensional water particle velocities, √ u/ gd, are plotted versus non-dimensional time, t g/d. Comparison of experiments and theories for d = 0.204 m,  = 0.29. ◦ 1st order, / 3rd order,  Tanaka (Tanaka, 1986), + Lee (Lee et al. , 1982).

Test-case number 33 by P. Lubin and H. Lemonnier

261/283

(a) z/d = 0.92

(b) z/d = 0.78

(c) z/d = 0.62

(d) z/d = 0.45

√ Figure 30.5: Vertical velocities at variousp depths z/d: non-dimensional water particle velocities, v/ gd, are plotted versus non-dimensional time, t g/d. Comparison of experiments and theories for d = 0.302 m,  = 0.11. ◦ 1st order, / 3rd order,  Tanaka (Tanaka, 1986), + Lee (Lee et al. , 1982).

262/283

Test-case number 33 by P. Lubin and H. Lemonnier

References Boussinesq, J. 1871. Th´eorie de l’intumescence liquide appel´ee onde solitaire ou de translation se propageant dans un canal rectangulaire. C. R. Acad. Sci. Paris, 755. Fenton, J. 1972. A ninth-order solution for the solitary wave. Journal of Fluid Mechanics, 53, 257–271. Grimshaw, R. 1971. The solitary wave in water of variable depth. Part 2. Journal of Fluid Mechanics, 46, 611–622. Lee, J.-J., Skjelbreia, J. E., & Raichlen, F. 1982. Measurements of velocities in solitary waves. J. of Waterway, Port, Coastal, and Ocean Eng., WW2(108), 200–218. McCowan, J. 1891. On the solitary wave. Philosophy Magazine, 32(5), 45–58. Tanaka, M. 1986. The stability of solitary waves. Physics of Fluids, 29, 650–655.

Test-case number 34 by G. Chanteperdrix and J.-L. Estivalezes

263/283

Chapter 31

Test-case number 34: Two-dimensional sloshing in cavity - an exact solution (PA) By Guilhem Chanteperdrix, DTIM,ONERA, 2 Av Edouard Belin, BP 4025, 31055 Toulouse cedex Phone: +33 (0)5 62 25 28 67, Fax: +33 (0)5 62 25 25 83, E-Mail: [email protected] Jean-Luc Estivalezes, DMAE, ONERA, 2 Av Edouard Belin, BP 4025, 31055 Toulouse cedex Phone: +33 (0)5 62 25 28 32, Fax: +33 (0)5 62 25 25 83, E-Mail: [email protected]

31.1

Practical significance and interest of the test-case

This test case is an analytical one. We provide an exact solution for the flow in a rectangular cavity of two non-miscible inviscid fluids of different densities ρ1 and ρ2 with ρ2 > ρ1 . The two layers of fluid are superimposed with the lighter one over the heavier one. Gravity is acting in the vertical downward direction. The description of the configuration is given in figure 31.1. At initial time, both fluids are at rest, then the cavity is submitted to an horizontal time dependant acceleration noted a(t).

31.2

Definitions and physical model description

31.2.1

Assumptions and model equations

We suppose that a(t) is small enough such that fluid oscillations remain in the linear regime. The two fluids are also supposed to be incompressible, and inertial terms associated with the Bernoulli equation (1/2v 2 ) are neglected compared to unsteady terms ( ∂ϕ ∂t ). Indeed, those assumptions imply that the amplitude of the oscillations are small compared

264/283

Test-case number 34 by G. Chanteperdrix and J.-L. Estivalezes

to the wavelength of the traveling wave. One can then assume that the flow of both fluids is potential. Let ϕ1 and ϕ2 be the potentials corresponding to fluid 1 and 2 respectively. Integration of the Euler equations for each phase implies:    p1 = −ρ1 gy + ρ1 a(t)x − ρ1 ∂ϕ1 ∂t (31.1) ∂ϕ   p2 = −ρ2 gy + ρ2 a(t)x − ρ2 2 ∂t The mass balance equations for each fluid are given by:  ∆ϕ1 = 0 ∆ϕ2 = 0

(31.2)

Since both fluids are inviscid, boundary conditions express that there is no flow across the wall by:    ∂ϕi   = 0    ∂x x=0,L (31.3)     ∂ϕ  i  = 0  ∂y y=−h2 ,h1 where L is the length of the domain as shown in figure 31.1. y

h1 ρ1

∆ϕ1 =0

ρ2 > ρ1

∆ϕ2 =0

0

-h2 0

L

x

Figure 31.1: Problem definition. Gravity is vertical downward and the acceleration a(t) is along the x axis.

31.2.2

Interface boundary conditions

Now, we have to specify interface boundary conditions. It is assumed that the effect of surface tension can be neglected. Therefore, the pressure is continuous across the interface. As a result of the mass balance at the interface, since there is no phase change, the normal component of the velocity at the interface must also be continuous.

Test-case number 34 by G. Chanteperdrix and J.-L. Estivalezes

265/283

If ξ denotes the vertical deviation of the interface from y = 0, for small amplitude oscillations and by linearizing he boundary conditions at the interface, one gets the following equations for the continuity of the pressure and the normal component of the velocity at the interface: gξ − ax +

ρ2 ∂ϕ2 ρ1 ∂ϕ1 − =0 ρ2 − ρ1 ∂t ρ2 − ρ1 ∂t ∂ϕ1 ∂ξ ∂ϕ2 = = ∂y ∂t ∂y

(31.4) (31.5)

From (31.4) and (31.5), after some algebra, one gets the following equation for ξ: g

31.2.3

∂ξ ρ2 ∂ 2 ϕ 2 ρ1 ∂ 2 ϕ 1 − a0 x + − =0 ∂t ρ2 − ρ1 ∂t2 ρ2 − ρ1 ∂t2

(31.6)

Exact solutions

We seek for the solutions of (31.2) by the method of separation of variables in the following form: X ϕ1 = φ1n (t) cosh(kn (y − h1 )) cos(kn x) n

(31.7) ϕ2 =

X

φ2n (t) cosh(kn (y + h2 )) cos(kn x)

n

where the wave number, kn , is defined by, kn =

2πn L

(31.8)

The interface conditions (31.5) et (31.6) give φin . By replacing in equation (31.6) ∂ξ ∂t ∂ϕ2 1 respectively by ∂ϕ ∂y and ∂y and projecting on the base cos(kn x), one obtains finally the following system of ODE:  0 2   ¨1 + ω 2 φ1 = − a Xn ωn φ  n n n   gkn sinh kn h1 (31.9)  0X ω2  a  n n   φ¨2n + ωn2 φ2n = + gkn sinh kn h2 where Xn is the projection of x on the base cos(kn x) and where the dot denotes the time derivative. To close the problem, we have to calculate the Xn . Those coefficients are given by: L 2

X0

=

X2n

= 0

X2n+1 = −

(31.10) 4L 4 =− 2 2 2 (2n + 1) π Lk2n+1

In (31.9) ωn is given by: ωn2 =

gkn ∆ρ ρ1 coth(kn h1 ) + ρ2 coth(kn h2 )

(31.11)

266/283

Test-case number 34 by G. Chanteperdrix and J.-L. Estivalezes

with ∆ρ = ρ2 − ρ1 . This last relation is the dispersion equation of the problem. One can clearly see that as h1 goes to infinity and if ρ1  ρ2 , the general dispersion relation for gravitational surface waves of depth h2 is recovered: ω 2 = gk tanh kh2 . To solve the system of ODE (31.9), one needs initial conditions for φin . Since potentials are defined to an additive constant, one can choose φin (0) = 0 with no loss of generality. Initial conditions for the time derivative of φin are furthermore given by the initial interface shape: φ1n (0) = 0

φ2n (0) = 0

a(0)Xn ωn2 φ˙ 1n (0) = − gkn sinh kn h1

a(0)Xn ωn2 φ˙ 2n (0) = + gkn sinh kn h2

(31.12)

Since the shape of the interface is only a function of φ˙ in , one only needs to calculate φ˙ in from the previous system of ODE:   Z t Xn ωn2 φ˙ 1n = − a(t) − a(τ )ωn sin ωn (t − τ )dτ gkn sinh kn h1 0 (31.13)   Z t 2 Xn ωn φ˙ 2n = + a(t) − a(τ )ωn sin ωn (t − τ )dτ gkn sinh kn h2 0

31.2.4

Interface shape for two longitudinal accelerations

Heaviside acceleration In this case, the acceleration a(t) is given by:  0 t