Application of the PITM Method using Inlet Synthetic Turbulence Generation for the Simulation of the Turbulent Flow in a Small Axisymmetric Contraction Bruno Chaouat∗ ONERA , 92322 Chˆatillon, France Abstract We investigate the turbulence modeling of second moment closure used both in RANS and PITM methodologies from a fundamental point of view and its capacity to predict the flow in a low turbulence wind tunnel of small axisymmetric contraction designed by Uberoi and Wallis. This flow presents a complex phenomenon in physics of fluid turbulence. The anisotropy ratio of the turbulent stresses τ11 /τ22 initially close to 1.4 returns to unity through the contraction, but surprisingly, this ratio gradually increases to its pre-contraction value in the uniform section downstream the contraction. This point constitutes the interesting paradox of the Uberoi and Wallis experiment. We perform numerical simulations of the turbulent flow in this wind tunnel using both a Reynolds stress model developed in RANS modeling and a subfilter scale stress model derived from the partially integrated transport modeling method. With the aim of reproducing the experimental grid turbulence resulting from the effects of the square-mesh biplane grid on the uniform wind tunnel stream, we develop a new analytical spectral method of generation of pseudo-random velocity fields in a cubic box. These velocity fields are then introduced in the channel using a matching numerical technique. Both RANS and PITM simulations are performed on several meshes to study the effects of the contraction on the mean velocity and turbulence. As a result, it is found that the RANS computation using the Reynolds stress model fails to reproduce the increase of anisotropy in the centerline of the channel after passing the contraction. In the contrary, the PITM simulation predicts fairly well this turbulent flow according to the experimental data, and especially, the “return to anisotropy ” in the straight section of the channel downstream the contraction.
1
Introduction
Turbulent flow subjected to an axisymmetric contraction in the streamwise direction is encountered in many engineering applications in the field of wind tunnels, turbomachinery, water turbines, but ∗
Senior Scientist, Department of Computational Fluid Dynamics. E-mail address:
[email protected]
1
also industrial processes including alumina refinery plant, combined-cycle power plant and others systems. As known, the effect of contractions on the mean flow and on the turbulence is mainly to accelerate the flow and to reduce the turbulence activity. But as a subtle mechanism, it appears that the normal turbulent stress in the streamwise direction decreases more rapidly than the two other lateral stresses leading to a more isotropic state of turbulence downstream the contraction [1, 2, 3], while the flow structures are appreciably modified when passing through the contraction due to vortex stretching and rotation [4, 5]. Obviously, in practice, these effects must be accounted for optimizing the performance of engineering processes. From a physical point of view, wind tunnel flows are of interest in physics of fluid turbulence because they can be viewed locally as anisotropic homogeneous flows on the centerline of the channel and constitute therefore the starting point for theoretical studies aiming to understand the mechanisms involved in contraction [6, 7, 8, 9, 10]. Experimental studies of axisymmetric strained turbulence began with the measurements of Uberoi [1, 2], Uberoi and Wallis [3], Comte-Bellot and Corrsin [11], pursued afterwards by several authors such as Bennet and Corssin [12], Sj¨ogren and Johansson [10], more recently, Ertunc and Durst [13], Antonia et al. [14]. The authors Uberoi [1, 2], Sj¨ogren and Johansson [10], Ertunc and Durst [13], investigated the influence of axisymmetric high contraction shapes whereas Uberoi and Wallis [3], Comte-Bellot and Corrsin [11], Bernett and Corrsin [12], as well as Antonia et al. [14], studied the effects of grid-generated turbulence in a small contraction shape ranging from 1.19 to 1.41. In this framework, the latter experiment of Uberoi and Wallis [3] deserves a particular interest because it presents a complex phenomena that still raises some questions in physics of fluid turbulence. In this experiment, initial fluid particles are convected into a channel through a square-mesh biplane grid and are then subjected to a sudden contraction C = 1.25. The effect of the square mesh biplane grid made of round wooden rods on the uniform stream flow was to generate an anisotropic turbulence just downstream the grid [3]. This effect was also obtained for grids of different geometries [15]. Just behind the grid, the streamwise fluctuating velocity correlation τ11 was found higher than the two other lateral fluctuating velocity correlations, τ22 and τ33 , more precisely, τ22 = τ33 ≈ τ11 /1.4. As expected, the measurements indicated that this ratio τ11 /τ22 initially found to be close to 1.4 returned to unity through the contraction because of the rapid deformation of the mean flow. But as a surprising result, this ratio gradually increased to its pre-contraction value in the straight duct section downstream the contraction. This phenomena constitutes a paradox which is not clearly elucidated even at this time. The objective is then to investigate this complex flow in a small contraction by performing numerical simulations with an appropriate turbulence model. Obviously, direct numerical simulation (DNS) solving all the scales on a mesh with a grid-size at least of order of magnitude of the Kolmogorov scale is the best tool for investigating turbulent flows. But as shown in Appendix A, this approach remains out of reach for the wind tunnel flow of Uberoi and Wallis [3]. Large eddy simulation [16] (LES) which consists in modeling the more universal small scales while the large scales motions are explicitly computed is a promising route but is extremely costly in term of computer resources for domain of large dimension or at large Reynolds numbers, even with the rapid increase of super-computer power [17, 18, 19]. Moreover, from a physical standpoint, LES using eddy viscosity models assuming a direct constitution rela-
2
tion between the turbulence stress and strain components, cannot calculate the subgrid turbulent turbulent energy and the subgrid stresses because these subgrid energies are not attainable in such models although they may be an appreciable part of the total energies. A modeled transport equation for the subgrid energy is necessary to compute this energy. Furthermore, the stresses associated with the large scales are often overpredicted for coarse LES simulations. This point has been verified for the fully developed turbulent channel flow [20, 21, 22]. On the other hand, models based on Reynolds averaged Navier-Stokes (RANS) equations using eddy viscosity models or second moment closure (SMC) [23, 24, 25, 26, 27, 28, 29] are still often used in industry to simulate practical engineering and geophysical flows without requiring excessive computational cost. But in the case of the Uberoi and Wallis experiment [3], we will see that single-scale RANS models are unable to reproduce the “return to anisotropy ” occurred in the straight section of the channel downstream the contraction, even if applying advanced turbulence modeling such as Reynolds stress models (RSM) [23, 26] developed in the framework of SMC. Only multiple-scale models [30, 31, 32, 33] succeed in reproducing this strange phenomena. But they return only averaged results and cannot give information about the instantaneous flow. As the LES methodology is not suitable for simulating this wind-tunnel flow because of its prohibitive computational resources, the solution is then to consider hybrid RANS/LES methods that combine advantages of both RANS and LES methods [34, 35]. In this framework, Schiestel and Dejoan [36], Chaouat and Schiestel [37] have developed the partially integrated transport modeling (PITM) method with seamless coupling between the RANS and LES regions. Actually, according to the classification of Ref. [34], PITM is a unified LES/RANS method. This method is particularly well suited for performing simulations at high Reynolds numbers on coarse meshes while retaining some beneficial aspects of large eddy simulation and RANS modeling using transport equations. As a result of the modeling based on a mathematical physics formalism [37, 38, 39], a new formulation of the dissipation rate equation has been developed which constitutes the main ingredient of this approach. Relatively to hybrid zonal RANS/LES models which are applied in two distinct domains that are separated by an artificial interface, subfilter models derived by PITM have the property to continuously vary from RANS to LES in the whole domain with respect to a parameter which is the ratio of the turbulent length-scale to the grid-size. Among these models [36, 37], the subfilter scale (SFS) stress transport model developed in second moment closure accounting for seven equations for the subfilter stresses and dissipation-rate constitutes one of the most advanced turbulence models used in LES [20, 21, 22, 37]. This one has been proposed for simulating non-equilibrium unsteady flows on relatively coarse grids with seamless coupling between the RANS and LES regions considering that the cutoff wave number can be placed before the inertial zone within the energy spectrum as far as the grid size is however sufficient to describe correctly the mean flow. So, thanks to the complex subfilter scale model allowing numerical resolution of the large scales, a reduction of the computational cost can be obtained in comparison with the one required for performing highly resolved LES on fine grids [21, 40, 41]. This model allows to take into account more precisely the turbulent processes of production, transfer, pressure redistribution effects and dissipation of turbulence, in a better way than eddy viscosity models. Because of the pressure-strain correlation term, it accounts for some history effects in the turbulence interactions and reproduces fairly well the flow anisotropy. In that sense, it is well appropriate for simulating the wind-tunnel flow of Uberoi and Wallis [3].
3
The main objective of this paper is to elucidate the mechanism behind the return to anisotropy in the Uberoi and Wallis experiment [3] by means of the PITM method which has been developed for performing simulation of turbulent flows with a small computational cost. To do that, we will consider a subfilter scale stress model developed in the framework of second moment closure. Prior to the PITM simulation, we will first discuss results of single and multi-scale RSM/RANS models to explain the physical mechanism of return to anisotropy in the straight section of the channel downstream the contraction by means of turbulence scale separation. A special attention will be paid to the role of the pressure-strain correlation term that redistributes turbulent energy among the Reynolds stress components. For the PITM simulation, we will develop a specific analytical spectral method to generate a pseudo-random turbulent field instead of computing the fine grained turbulence around the experimental grid which is too costly in term of computing time and memory resources. This synthetic field is created in the spectral space and afterward computed from its inverse Fourier transform in the physical space leading to a turbulent field in a cubic box that is pushed into the wind tunnel to produce the inlet condition of the computational domain. We will perform PITM simulations on several meshes and we will investigate this turbulent flow with a special attention devoted to the effect of the contraction on the mean velocity and turbulence. We will analyze the mean velocity, the turbulent stresses, the flow anisotropy along the centerline of the channel and finally, the flow structures.
2 Description of the wind tunnel experiment of Uberoi and Wallis 2.1
Experimental results
The wind tunnel used in the Uberoi and Wallis experiment described in detail in Refs. [1, 2, 3] is sketched in Fig. 1. The dimension of the square-mesh biplane grid is D1 = 60.96 cm whereas the dimension of the wind tunnel in the longitudinal direction is L1 = 254 cm. At 76.2 cm, downstream of the grid, the four tunnel walls are fully lined. The thickness of the liner increases from 0 to 3.2258 cm producing a small axisymmetric contraction of the wind tunnel cross-section of ratio C = A1 /A2 = (D1 /D2 )2 = 1.25, where A1 and A2 denote the section areas before and after the contraction, Di is the side length of Ai . The Reynolds number Re = Ub D1 /ν based on the bulk velocity Ub , the height D1 and the kinematic viscosity ν takes on the value Re ≈ 4.47 × 105 . The Mach number of the wind-tunnel is very low, Ma = 0.0317, the flow being fully incompressible. Third set of experiments [3] were performed with a first square-mesh biplane grid of 5.08 cm made of round wooden rods of 1.27 cm diameter, a second square-mesh grid of 2.54 cm made of rods of 0.635 cm diameter and also a third square-mesh grid of 1.27 cm made of rods of 0.3175 cm diameter corresponding to the mesh Reynolds numbers RM = Ub M/ν = 9300, 18600 and 37200, respectively, where M denotes the mesh size of the grid (5.08 cm, 2.54 cm and 1.27 cm). As a result, it was found that the effect of a fixed contraction on the turbulence decreases with decreasing mesh size of the grid and that for all cases, the initial ratio of the stress components τ11 /τ22 close
4
0 0.5
Z
1 c1
Y
1.5 2
X
2.5
Figure 1: View in real ratio of the wind-tunnel axisymmetric contraction C = (D1 /D2 )2 = 1.25 of the Uberoi and Wallis experiment [1] including two planar cross-sections. L1 = 2.54 m, D1 = 60.96 cm (upstream the contraction) and D2 = 54.52 cm (downstream the contraction).
to 1.4 behind the grid gradually decreases in the axisymmetric contraction. After passing the axisymmetric contraction, this ratio reincreases to its pre-contraction value in the uniform straight section of the channel until the exit. This outcome was observed for all grids of three mesh sizes M.
2.2
Grid-turbulence
It is a hard task to simulate the contracting flow at one specific mesh Reynolds number RM of the experiment because the interaction between the uniform stream flow and the square mesh biplane grid involving all the flow details cannot be reproduced easily by numerical simulation, even by DNS. However, whatever this value of the mesh Reynolds number, the important point to note is that the anisotropy ratio just behind the grid was γ = τ11 /τ22 = 1.4 and that the return to anisotropy was more or less marked versus the size M of the mesh. That being said, we will try to simulate the case that is closest to RM = 37200 because of its higher contraction effects on turbulence [3]. The issue to address is then to determine the inlet state of turbulence characterized itself by its turbulent energy and dissipation rate that corresponds to this case. In this experiment, the largest scales in the grid-generated turbulence are essentially determined by the mesh size of the grid M whereas the smallest scales are fixed by the Kolmogorov law. Unfortunately, there are no measurements available just behind the grid. The only source of information is found on Figs. 2, 3 and 4 of Ref. [3] showing the evolution of the ratio of the mean square flow velocity to the turbulent stress, α = 10−3 Ub2 /τ11 and β = 10−3 Ub2 /τ22 , as well as the ratio of the streamwise turbulent stress to the spanwise turbulent stress γ = τ11 /τ22 , for the flows at the mesh Reynolds
5
numbers RM = Ub M/ν = 9300, 18600 and 37200. For each Reynolds number, one can see that the dimensionless variable α = 10−3 Ub2 /τ11 just after the grid at x1 /D1 = 0.6 is close to 0.3, or equivalently, τ11 /Ub2 = 0.33 %. The turbulent energy is then extrapolated from Fig. 2 of Ref. [3] considering in a first approximation and for simplification purposes the data at x1 /D1 = 0.6. As the ratio γ = τ11 /τ22 is close to 1.4 just behind the grid, the turbulent energy is easily computed by 2 + γ 10−3 Ub2 k0 = (1) 2γ α In the following, the parameter α is set to 0.3 leading to the value of the turbulent energy k0 = 0.49 m2 /s for Ub = 11.0 m/s. The dissipation-rate is determined by the turbulent Reynolds number from the relation Rt = k2 /νǫ. If referring to wind-tunnel experiments, we can mention the work of Comte-Bellot and Corrsin [11] as well as Sj¨ogren and Johansson [10] that provide a rough order of magnitude of the Reynolds number. For the experiment of Comte-Bellot and Corrsin [11], the turbulent Reynolds numbers was set to Rt = 800 whereas for the experiment of Sj¨ogren and Johansson [10], the Reynolds number was varied from 250 to 2500. In the present case, as an intermediate value, the Reynolds number is set to 1600 that is twice the Reynolds number value of Comte-Bellot and Corrsin implying that the dissipation rate computed as ǫ0 = k02 /νRt takes on 3/2 the value 10.0 m2 /s3 for ν = 1.50 × 10−5 m2 /s. The large scale computed as L0 = k0 /ǫ0 is close to 3.43 cm which is of the same order of the mesh size M of the experimental biplane grid (5.08 cm, 2.54 cm and 1.27 cm). This concordance between the large scale value L and the mesh size M confirms the good estimate of the dissipation-rate ǫ. The Kolmogorov scale η0 computed as η0 = (ν 3 /ǫ0 )1/4 is 0.135 mm. The frequency is then ǫ0 /k0 = 20.4 s−1 .
3 Reynolds averaged Navier Stokes equations (RANS) method 3.1
Motion equation
Turbulent flow of a viscous incompressible fluid is considered. Any variable φ(x, t) of the flow as a function in time and space can be decomposed into an ensemble average part hφi and a fluctuating part that embodies all the turbulence scales φ′ such as φ = hφi + φ′ . The motion equation reads 1 ∂ hpi ∂ 2 hui i ∂τij ∂ hui i ∂(hui i huj i) + =− +ν − ∂t ∂xj ρ ∂xi ∂xj ∂xj ∂xj
(2)
where ui , p, ν, τij , are the velocity vector, the pressure, the molecular viscosity and the Reynolds stress tensor, respectively. The closure of this equation requires to model the Reynolds stress turbulent stress τij = hui uj i − hui i huj i that is made in the framework of SMC, by its transport equation.
6
3.2
Single-scale Reynolds stress models
In a compact form, the modeled transport equation for the stress τij reads [23, 24, 26, 42] ∂τij ∂ + (huk i τij ) = Pij + Πij + Jij − ǫij ∂t ∂xk
(3)
where huk i denotes the mean statistical velocity, and the terms appearing in the right-hand side of this equation are identified as production, redistribution, diffusion and dissipation, respectively. The production term is defined by Pij = −τik
∂ huj i ∂ hui i − τjk ∂xk ∂xk
(4)
The redistribution term Πij is composed by the slow distribution term Π1ij modeled using the Rotta’s hypothesis as a function of the anisotropy stress 1 ǫ 1 τij − τmm δij (5) Πij = −c1 k 3 where c1 is a constant coefficient and by the rapid redistribution term Π2ij modeled assuming the isotropization hypothesis 1 2 (6) Πij = −c2 Pij − Pmm δij 3 where c2 is a constant coefficient. In absence of velocity gradient, the return to isotropy is then governed by Π1ij and so the frequency ǫ/k whereas in presence of velocity gradient, it is dominated by Π2ij taking into account the production of turbulence Pij . The tensorial dissipation rate ǫij is approached by 2/3ǫδij and the transport equation for the dissipation-rate is modeled as ǫ ǫ2 ∂ ∂ǫ + (huk i ǫ) = cǫ1 P − cǫ2 + Jǫ ∂t ∂xk k k
(7)
where P denotes the production term of the turbulent energy P = Pmm /2, Jǫ is the diffusion term and cǫ1 and cǫ2 are constant coefficients used in RANS modeling. This turbulence modeling based on second moment closure is described in its basic formulation in the pioneer paper of Launder et al. [42, 43]. Taking into account the initial turbulence state defined in Sec. 2.2, simulations are performed on several meshes. To check the grid independence solution, the anisotropy ratio τ11 /τ22 . is investigated for three meshes accounting for 100, 200 and 400 grid-points in the streamwise direction x1 and 100 grid-points in the lateral directions x2 , x3 . Figure 2 indicates that the RSM computations performed on the medium and refined meshes return both same results while a slight deviation is observed for the coarse mesh. Since the change is negligible between the medium and refined grids, only results associated with the medium grid are presented in the following. Figure 3 shows the evolution of the dimensionless bulk velocity Ub∗ = Ub /Ub (x0 ) as well as the anisotropy ratio τ11 /τ22 obtained by the RSM computation. As expected, one can see that the anisotropy ratio decreases in the contraction zone. But it remains unchanged downstream the contraction in the straight section of the channel. This result can be easily explained if assuming in a first
7
approximation that the flow is homogeneous in the straight section of the channel. In the first part of the channel upstream the contraction, the mean velocity gradient ∂ hu1 i /∂x1 is zero implying that the production term Pij as well as the rapid redistribution term Π2ij are zero. The anisotropy of turbulence is then governed by the action of the slow redistribution term Π1ij and in particular, the return to isotropy is more or less rapid according to the frequency ǫ/k of the turbulence scale. This term depends on the initial frequency ǫ0 /k0 at the inlet. In the contraction zone of the channel, both the redistribution terms Π1ij and Π2ij are activated. The rapid distribution term Π2ij is activated because of the streamwise velocity gradient ∂ hu1 i /∂x1 which is non zero due to the increase of the mean velocity. In the second part of the channel, downstream the contraction, the production term Pij again is zero because of the absence of mean velocity gradients. So that the flow anisotropy is governed the slow redistribution term Π2ij . If we assume now that the turbulence state is isotropic just after the contraction τ11 = τ22 = τ33 , as it is observed in practice in the wind tunnel flow of Uberoi and Wallis, as a result of the experimental conditions (value of the contraction ratio (D1 /D2 )2 = 1.25, Reynolds number Re = Ub D1 /ν = 4.47 × 105 , characteristics of the square-mesh biplane grid, mesh Reynolds number RM = Ub M/ν etc ...) [3], then, both the slow redistribution terms Π1ij as a function of the anisotropy tensor aij = (τij − 2/3kδij )/k and the rapid redistribution term Π2ij as a function of the production of turbulence reduce to zero. Consequently, the turbulent stress τii remains unchanged in the uniform section downstream the contraction. Note that this reasoning is formally true only if the diffusion process is zero or negligible. So, single-scale Reynolds stress model is not able to reproduce the increase of anisotropy downstream the contraction in which there is no vortex stretching of the mean flow because the channel is again straight. Physically, we will see that this flow is governed by scale effects and not by wall effects. Overall, any RANS model based on single scale, whatever its degree of sophistication, including for instance a quadratic model for the pressure-strain model [44] will return the same result.
3.3
Multiple-scale Reynolds stress models (m)
In this approach, the transport equation for the partial stress τij zone [κm−1 , κm ] reads [23, 31] (m)
∂τij
∂t
+
in the spectral wavenumber
∂ (m) (m−1) (m) (m) (m) (m) (m) = Pij + Fij − Fij + Πij + Jij − ǫij huk i τij ∂xk (m)
(m)
(m)
where in this equation, Pij , Πij , Jij and dissipation terms and
(m) Fij
(m)
ǫij
(8)
are the partial production, redistribution, diffusion
denotes the tensorial flux transfer of energy at the wave number (m)
κm . These terms are defined in Refs. [31, 45, 46]. The production term Pij (m)
Pij
(m) ∂ huj i
= −τik
∂xk
(m) ∂ hui i
− τjk
∂xk
is given by (9)
The closure proposed in multiple-scale models consists in applying locally equations (5) and (6) in the particular spectral slice [κm−1 , κm ] leading to (m) 1 (m) 1,(m) (m) F (m) (10) Πij = −c1 τij − τmm δij 3 k(m)
8
(m)
where c1
is a coefficient dependent on the spectral slice and 1 (m) 2,(m) (m) (m) Πij = −c2 Pij − Pmm δij 3
(m)
(m)
where c2 is still taken as a constant. The tensorial flux Fij computed by
(11)
is as a function of F (m) that is
(m) 2 (m) F (m−1) (m) ǫ(m) (m) P (m) dF (m) (m) F (m) F (m) F (m) F + C + C + C = C1 2 3 4 dt k(m) k(m) k(m) k(m)
(12)
(m)
where Ci are coefficients depending on the spectral slice and k(m) denotes the partial turbulent energy. Calculations have been performed by Schiestel [31] using a three-scale model based on the numerical resolution of equations (8) and (12). Figure 2 shows the evolution of the ratio τ11 /τ22 returned by Schiestel’s computation [31]. As a result, one can see that the increase of 1.3
1.1
U*b
τ11/τ22
1.2
1
0.9 0.8
1
1.2 x1/D1
1.4
1.6
Figure 2: Effect of grid density. RSM computations. Dimensionless bulk velocity Ub∗ (x1 ) and the ratio τ11 /τ22 . Enlargement view in the contraction zone. - - -, 100 × 100 × 100; . . . , 200 × 100 × 100; -.-.-., 400 × 100 × 100. anisotropy downstream the contraction is well recovered. As suggested in Refs. [3, 31], the increase of anisotropy in the straight section of the channel could be explained by a spectral distribution of the turbulent energy into some spectral range [κm−1 , κm ] considering that the deformation due to the contraction cannot remove the anisotropy in every spectral range. Multiple-scale models succeed in reproducing the return to anisotropy because equations (10) and (11) act locally in the spectral slice [κm−1 , κm ] and therefore can see the partition of the energy spectrum. In particular, 1,(m) (m) the slow term Πii for the normal stress τii given by equation (10) is activated everywhere in (m)
(m)
is different from zero, the channel due to the fact that the partial anisotropy tensor τii − 31 τjj P (m) (m) even if τii − 13 τjj = m τii − 31 τjj = 0 just at the end of the contraction zone (the Einstein summation convention is here suspended for indice i). Like for the rapid term Π2ii of the single
9
1.6
1.4
1.2
1
0.8
0
1
2
3
4
5
x1/D1
Figure 3: Evolution of the dimensionless bulk velocity Ub∗ (x1 ) and the ratio τ11 /τ22 on the centerline of the channel in the streamwise direction. • Experiment of Uberoi and Wallis [3]; - - - Single-scale RSM model; -.-.- Three-scale RSM model (Schiestel’s computation [31]) Re = 4.47 × 105 2,(m)
(m)
scale model, the rapid term Πii given by equation (11) associated with the normal stress τii is activated only in the contraction zone because of the mean velocity gradient that is non-zero.
4 4.1
Partially integrated transport modeling (PITM) method Motion equation
In PITM, as for large eddy simulation, any flow variable φ is decomposed into a large or resolved scale φ¯ and a subfilter or modeled scale φ> . The instantaneous fluctuation φ′ includes in fact the large scale fluctuating part φ< and the small scale fluctuating part φ> such that φ′ = φ< + φ> . So that φ can then be rewritten as the sum of a mean statistical part hφi, a large scale fluctuating part φ< and a small scale fluctuating part φ> as follows φ = hφi + φ< + φ> . In a first approach [47, 48], the resolved scale motion equation is ∂(τij )sf s ui u ¯j ) ∂u ¯i ∂(¯ 1 ∂ p¯ ∂2u ¯i + =− +ν − ∂t ∂xj ρ ∂xi ∂xj ∂xj ∂xj
(13)
where the subfilter-scale tensor (τij )sf s appearing in the right-hand side of this equation is defined by the mathematical relation ¯i u ¯j (14) (τij )sf s = ui uj − u The resolved scale tensor is (τij )les = u ¯i u ¯j − hui i huj i
10
(15)
The Reynolds stress tensor τij including the small and large scale fluctuating velocities can be computed in a first approximation [38] as the sum of the subfilter and resolved stress tensors τij = h(τij )sf s i + h(τij )les i
(16)
whereas the statistical turbulent kinetic energy is obtained as the half-trace of the stress tensor τij . The presence of the turbulent contribution (τij )sf s in equation (13) indicates the effect of the subfilter scales on the resolved field. Closure of the momentum equation (13) requires to model the subfilter turbulent stress tensor (τij )sf s that is made here in the framework of SMC.
4.2
Subfilter scale turbulence model
4.2.1
General properties
The PITM method is developed in the spectral space considering the Fourier transform of the twopoint fluctuating velocity correlation equations with an extension to non-homogeneous turbulence. This equation formally reads [36, 37, 38] Dϕij (X, κ) = Pij (X, κ) + Tij (X, κ) + Ψij (X, κ) + Jij (X, κ) − Eij (X, κ) Dt
(17)
where D denotes the mean material derivative, ϕij is the spectral tensor of the two-point velocities, X is the local position, κ is the wave number, the different terms appearing in the right-hand side of equation (17) are respectively the production, transfer, redistribution, diffusion and dissipation contributions. This spectral equation constitutes the cornerstone of the PITM method. The subfilter PITM equation in the physical space is then obtained by integrating this equation over spectral zones of wave numbers [37, 38]. Generally speaking, the PITM method based on equation (17) allows to convert any RANS model to its corresponding subfilter scale model. In regards with conventional LES [16], the PITM method enables to simulate turbulent flows on relatively coarse grids when the cutoff wave number can be placed before the inertial zone as far as the grid size is however sufficient to describe correctly the mean flow [21, 22, 37, 40]. In the present work, we naturally apply a subfilter scale stress transport model [22, 40] based on second moment closure including the transport equations for the subfilter-scale stresses (τij )sf s and dissipation rate ǫsf s . As a result of integration of equation (17) in the spectral space of wave numbers [37, 38, 39], the transport equations for the subfilter scale stress and dissipation rate equations look formally like the corresponding RANS equations but the coefficients used in this model are now some functions of the dimensionless cutoff parameter ηc = κc Le involving the cutoff wave number κc = π/∆ and the integral turbulent length scale Le = k3/2 /ǫ. As for multiple-scale models acting within the range [κm−1 , κm ], this subfilter scale model describes the turbulent processes of production, dissipation and flux transfer of energy within the spectral zones [0, κc ], [κc , κd ] and [κd , ∞[ where κc and κd denote the cutoff wave number and dissipative wave number. For the limiting condition when the parameter ηc goes to zero, the subfilter-scale model behaves like a RANS/RSM model whereas when ηc goes to infinity, the computation switches to DNS if the grid-size is enough refined.
11
4.2.2
Subfilter scale stress transport equation
The transport equation for the subfilter stress tensor can be written in the simple compact form as [37] ∂(τij )sf s ∂ + (¯ uk (τij )sf s ) = (Pij )sf s + (Πij )sf s + (Jij )sf s − (ǫij )sf s (18) ∂t ∂xk where the terms appearing in the right-hand side of this equation are identified as subfilter production, redistribution, diffusion and dissipation, respectively. The transport equation for the subfilter turbulent energy is obtained as half the trace of equation (18). The production term (Pij )sf s accounts for the interaction between the subfilter stresses and the filtered velocity gradients (Pij1 )sf s = −(τik )sf s
∂u ¯j ∂u ¯i − (τjk )sf s ∂xk ∂xk
(19)
The redistribution term [20] (Πij )sf s appearing in equation (18) is modeled into a slow part (Π1ij )sf s that characterizes the return to isotropy due to the action of subgrid turbulence on itself ǫsf s 1 (20) (τij )sf s − (τmm )sf s δij , (Π1ij )sf s = −c1sf s ksf s 3 and a rapid part, (Π2ij )sf s that describes the action of the filtered velocity gradients 1 (Π2ij )sf s = −c2 (Pij )sf s − (Pmm )sf s δij , 3
(21)
where c1sf s plays the same role as the Rotta coefficient c1 but is no longer constant whereas c2 is the same coefficient used in RANS modeling. In practice, the function c1sf s is modeled as c1sf s = c1 α(η) where α is an increasing function of the parameter η to strengthen the return to isotropy for large wave numbers. The diffusion terms (Jij )sf s is modeled assuming a well-known gradient law [40].
4.2.3
Subfilter scale dissipation-rate transport equation
The subfilter tensorial transfer rate (ǫij )sf s is approached by 2/3ǫsf s δij at high Reynolds number and is modeled by means of its transport equation which is obtained from the PITM method using spectral splitting techniques and partial integration in the spectral space. As a result, the transfer-rate equation reads [36, 37, 39] ǫ2sf s ǫsf s ∂ǫsf s ∂ + (¯ uk ǫsf s ) = csf sǫ1 Psf s − csf sǫ2 + (Jǫ )sf s ∂t ∂xk ksf s ksf s
(22)
where (Jǫ )sf s denotes the diffusion term. In equation (22), the coefficient csf sǫ1 appearing in the source term of the transfer-rate equation is the same as the one used in the corresponding RANS dissipation equation csf sǫ1 = cǫ1 but the coefficient csf sǫ2 appearing in the destruction term of the transfer-rate equation is now given by csf sǫ2 = cǫ1 +
hksf s i (cǫ2 − cǫ1 ) k
12
(23)
where cǫ1 and cǫ2 are the constant coefficients used in the statistical RANS well-known dissipationrate equation (7). The ratio hksf s i /k appearing in equation (23) is evaluated by reference to an analytical energy spectrum E(κ) inspired from a Von K´ arm´ an spectrum [20] leading to the result [20] cǫ2 − cǫ1 csf sǫ2 (ηc ) = cǫ1 + (24) [1 + βη ηc3 ]2/9 Equation (24) indicates that the parameter ηc acts like a dynamic parameter which depends on the location of the cutoff wave number κc within the energy spectrum. Then, the value of the function csf sǫ2 controls the relative amount of turbulence energy contained in the subfilter range. The diffusion term (Jǫ )sf s is modeled by a gradient tensorial law [40]. The subfilter model relying on equations (18) and (22) is defined at low Reynolds number in Ref. [40]. In particular, the coefficients c1 and c2 are some functions depending on the Reynolds number and on the subgrid anisotropy tensor, the second and third subgrid-scale invariants and the flatness parameter.
5 Modeling of the inlet condition using a spectral method with an imposed energy spectrum The flow conditions are nominally atmospheric air. The simulation of the flow around the squaremesh biplane grid made on round wooden rods [3] is a tedious task to perform numerically because of the three dimensional geometry of the grid that requires a specific work in mesh generation. In addition, the simulation of the flow details around the fine grid is extremely costly in term of computational resources, both for the number of grid-points and computer time. Taking into account these constraints, we prefer to create an anisotropic pseudo-random turbulence field characterized by its anisotropy condition τ22 = τ33 ≈ τ11 /1.4 by means of an analytical spectral method to produce the inlet conditions. This procedure consists in generating a complete tridimensional flow in a cubic box at a given time that is then progressively moved at the inlet of the wind-tunnel with a constant convection velocity corresponding to the uniform stream flow of the experiment. From a practical point of view, in spite of its artificial character, this method allows to mimic the grid turbulence. It is an useful alternative for simulation of turbulent flows with, firstly, the advantage to save a large amount of computational time, secondly, the possibility to generate anisotropic fields while preserving spatial and temporal correlations of the flow that are not always attainable in experimental devices or even by direct numerical simulation and, thirdly, the guarantee to get analytical solution by avoiding numerical problems arising from the solving of equations. But as accurate as possible, this artificial turbulence field cannot obviously reproduce all the characteristics of turbulence that is solution of the Navier-Stokes equations, even if a pseudo-random field is a good approximation that is akin to real turbulence. The generation of the synthetic turbulence is inspired from the method of Roy [49] initially developed for isotropic fields and is extended here to account for anisotropic fields by applying a transformation vector on the vector stream function [50, 51]. The motivation to develop this method is to easily generate any anisotropic turbulence state from an isotopic state with an imposed spectrum E(κ) while controlling the ratio between
13
the resolved and subgrid scales directly in the spectral space by means of the cutoff wave number κc . This method is of low computational cost and relies on three random operators. Note that other routes are possible to follow and in particular, a thorough review on these methods has been conducted recently by Shur et al. [52]. In the present case, the first step of the method consists in generating an homogeneous isotropic field, uniformly distributed on a sphere of radius unity in ˆ the spectral space by means of a random vector stream function ψ(κ). The fluctuating velocity is ˆ ˆ (κ) = κ ∧ ψ(κ) with specific conditions imposed on then obtained from the stream function as u the stream function ψ to satisfy the energy equation given by its energy spectrum density E(κ). The second step consists in applying a second order tensor β operating on the isotropic vector field to produce the anisotropic flow field. As a result, the velocity in the spectral space is then given by ˆ ˆ (κ) = κ ∧ β ψ(κ) u (25) ensuring that the continuity equation is still preserved. We show in Appendix B that it is possible to determine the tensor β by means of algebra calculus in the spectral space to get the desired anisotropic resolved field and in particular the turbulent field verifying the condition τ11 /τ22 = 1.4, provided some approximations are conceded. The resolved scale stress as a function of the cutoff wave number κc given by the grid-size of the cubic box is then computed from integration of the spectral stress tensor as follows ZZZ
u ˆi (κ)ˆ u∗j (κ) d3 κ (26) (τij )les = |κ|≤κc
ˆ ∗ denotes the conjugate of u ˆ . The subfilter scale stresses are computed by a viscosity model where u using the Boussinesq hypothesis verifying also the anisotropy condition. So that, the resolved and subgrid turbulent fields are built in order to satisfy the anisotropy condition from a statistical point of view in the whole cubic domain at a given time.
6 6.1
Numerical framework Numerical code
The present numerical simulations are performed with the research code developed by Chaouat [53] that solves the filtered Navier-Stokes equations and the transport equations for the turbulence including the subfilter scale stress tensor (τij )sf s and dissipation rate ǫsf s . The equations are integrated in time using an explicit Runge-Kutta scheme of fourth-order accuracy in time and solved in space using a quasi-centered scheme of fourth-order accuracy in space. A numerical procedure is also applied on the turbulence model to accelerate the numerical convergence towards the numerical solution [39]. As the question of CPU time requirement is of first importance in LES, note that the additional cost resulting from the solving of seven transport equations requires only roughly 30 % to 50 % more CPU time than viscosity models [40] so that the benefit to apply this advanced turbulence modeling is greatly appreciated.
14
6.1.1
Fully developed turbulent channel flow
The PITM method has been calibrated on the well known fully developed turbulent channel flow [37] as well as the decaying of homogeneous turbulence [20, 39]. For illustrative purpose to indicate how PITM works in practice, we briefly present the results associated with the simulation of the fully turbulent channel flow performed on a medium grid 32 × 64 × 84 at the Reynolds number Rτ = uτ δ/2ν = 395 where δ and uτ denote the channel half width and the friction velocity, respectively. Figure 4 shows the sharing out of turbulent energy among the sufbilter and resolved energies of the turbulent flow. The flow is dominated by the large scales in the center of the channel whereas it is governed by the small scales in the immediate vicinity of the wall. Figure 1/2 5 exhibits the turbulence intensities τii /uτ computed as the sum of the subfilter and resolved scale energies versus the channel height of the channel. One can see that the PITM results agree perfectly with the data of direct numerical simulation [54]. In particular, the flow anisotropy as well as the turbulent peaks near the walls are well reproduced by the PITM simulation. 3
τii
SFS−LES
2
1
0
0
0.2
0.4
0.6
0.8
1
X3/δ
Figure 4: Subfilter and resolved turbulence intensities in wall unit. h(τii )sf s i1/2 /uτ and h(τii )les i1/2 /uτ (32 × 64 × 84). SFS : N, i=1; ◭, i =2; ◮, i =3. LES : N, i =1; ◭, i =2; ◮, i =3. Rτ = 395.
6.1.2
A large variety of internal and external flows
Previous simulations have shown that the PITM method was able to reproduce fairly well a large variety of internal and external flows such as rotating flows encountered in turbomachinery [21], pulsed turbulent flows [36], flows with wall mass injection [37], flows with separation and reattachment of the boundary layer [22, 40] a mixing of two turbulent flows of different scales [50], and airfoil flows [55].
6.2
Numerical simulation
The computational domain is of dimension of the experimental wind tunnel. The simulations are performed on coarse curvilinear meshes with the aim to appreciably reduce the computational
15
3
τii
2
1
0
0
0.2
0.4
0.6
0.8
1
X3/δ
1/2
Figure 5: Turbulence intensities in wall unit. τii /uτ (32 × 64 × 84). PITM : N, i =1; ◭, i =2; ◮, i =3. DNS : —. Rτ = 395.
cost while obtaining satisfying accurate solution. Non-uniform grid distributions are used in the streamwise and lateral directions. In particular, the grids are refined in the wall regions to capture the boundary layers although this is not crucial in the present case. The numerical solution of the flow returned by PITM at the head end of the channel must be consistent with the numerical solution of the flow generated in the box by the analytical spectral method (see Appendix B). This means that the energy spectrum density E(κ) given by the simulation should be close to the one defined in the cubic box. To recover the prescribed ratio value ksf s /k by the numerical simulation, the grids are then refined in the head end of the channel to adjust the dynamical parameter ηc to its fair value. Indeed, we know that this parameter controls the sharing out of turbulence energy among the subfilter and resolved turbulence scales. The statistics are obtained by averaging in time the instantaneous flow accounting for roughly six convective time scale T = D1 /Ub where Ub is the bulk velocity computed at the entrance of the channel. Because of the long time calculations that are necessary to perform to get the convergence of the statistics, several cubes are in fact independently generated from each other and introduced at inlet with a constant convection velocity using a matching numerical technique to prevent discontinuities between two adjacent cubes.
7 7.1
Numerical results Inlet turbulence field in the cubic box
The isotropic and anisotropic turbulence fields are generated on a cubic box of dimension D1 = 0.6096 m accounting for N = 683 grid-points. The wave numbers are defined by κm = 2π/m where m varies in the range [−N/2 + 1, N/2] leading to a minimum wave number κmin = 10.13 m−1 and κmax = π/∆ = 344.5 m−1 . However, the effective wave number retained in the present
16
case is κc = 272 m−1 to suppress high frequencies of the fluctuating velocities. Figures 6 displays the energy spectrum density E(κ) of the flow which is generated by the spectral method given in Appendix B by E(κ) = χκm for κ < κ0 and E(κ) = Cκ ǫ2/3 κ−5/3 for κ ≥ κ0 where κ0 denotes the matching wave number. It is found that the parameter γ introduced in the matrix β defined in Appendix B takes on the value γ = 0.20 to get the prescribed value τ11 /τ22 = 1.4. The ratio of the subfilter energy to the total energy is hksf s i /k = 0.35. The isotropic and anisotropic turbulent stress fields are characterized by τ11 /k = 0.666 τ22 /k = 0.666 τ33 /k = 0.666 τ11 /k = 0.823 τ22 /k = 0.588 τ33 /k = 0.583
(27)
The second and third invariants of the anisotropy tensor aij are defined as A2 = aij aji and 4
10
3
3
2
E(cm /s )
10
2
10
1
10
0.1
1.0 −1 κ (cm )
10
Figure 6: Inlet energy spectrum E(κ) of the turbulent field defined in Appendix B. A3 = aij ajk aki where aij = (τij − 2/3kδij )/k. For the anisotropic field, the values of the second and third invariants are respectively A2 = 0.0369 and A3 = 0.0029 with a11 = 0.156, a22 = a33 = −0.078, leading to G = [A2 /2]/[(A2 /3)3/2 ] = 1. The values of the Reynolds stress tensor including one strong component and two weak components correspond to rod-like turbulence, the Reynolds stress tensor forms a prolate spheroid shape according to the definition of Ref. [56]. Figure 7 shows the vortical activity of the synthetic turbulence both for the isotropic and anisotropic fields illustrated by the Q criterion defined as Q = 21 (Ωij Ωij − Sij Sij ) accounting for the balance between the local rotation rate Ω and the strain rate S of the velocity [57]. Although the turbulent fields are artificial, i.e., not solution of the Navier-Stokes equations, one can observe at a first sight that they look like “real ” turbulence with the presence of vortical structures. Figure 8 shows the random velocity components in a plane section [x2 , x3 ] of the cubic box both for the isotropic and anisotropic fields. The vortices are well visible for both cases even if they seem more pronounced for the isotropic field than for the anisotropic field. Once the flow has been generated in the physical space, it has been checked that the flow anisotropy is well recovered by computing the averaged resolved scales stresses h(τii )les i = h¯ ui u ¯i ic − h¯ ui ic h¯ ui ic where h.ic denotes here the space average in the cubic box.
17
(a)
(b) Figure 7: View of the turbulent field generated in a cube. Vortical activity illustrated by the Q isosurfaces. (a) isotropic field. (b) anisotropic field. Computation 68 × 68 × 68.
18
(a)
(b) Figure 8: Velocity field in a plane [x2 , x3 ] of the cubic box. (a) isotropic field. (b) anisotropic field. The flow (b) serves as an inlet field of the channel flow with axisymmetric contraction. Computation 68 × 68 × 68.
19
Figure 9: Cross-section of the curvilinear grid 200 × 100 × 100 of the channel with axisymmetric contraction.
7.2 Turbulent flow in the channel of small axisymmetric contraction Several PITM simulations have been performed on different meshes 100×100×100, 200×100×100 and 400×100×100 to explore the effect of the grid-size on the solution, especially in the streamwise direction. The refinement in the lateral directions was found sufficient to describe the velocity profile. As a result, it has been found in practice that the PITM simulation performed on the mesh 200 × 100 × 100, respectively in the streamwise x1 and lateral directions (x2 , x3 ), provides sufficient accurate results that not change from one grid to another, both for the mean flow and the second moment statistics, even if the grid is not as so fine as the one used in conventional LES. Figure 9 shows the cross-section of the curvilinear grid 200 × 100 × 100 of the channel with axisymmetric contraction. In the normal direction to the wall, the grid points are distributed in different spacings with a refinement near the wall according to the transformation x3j =
1 tanh [ξj F (ξj ) atanh a] 2
(28)
where ξj = −1 + 2(j − 1)/(N3 −q 1) (j = 1, 2, · · · N3 ), F is a function introduced to adjust the refinement near the wall, F (ξj ) = (1 + ξj2 )/2, a is a constant coefficient set to 0.992 for N3 = 100. The grid in is also refined in the streamwise direction at the head end of the channel. Although the contraction of the channel is of a constant value (D1 /D2 )2 = 1.25, note that the grids in the contraction zone are analytically generated by the method of natural cubic spline interpolations to ensure a regular flow solution. Several attempts to simulate this flow with different inlet values of the dissipation-rate ǫ0 have been performed. As expected, it has been found that the value ǫ0 chosen in Section 2.1 is adequate for reproducing the right level of turbulence intensity τii /Ub2 along the centerline of the wind-tunnel as it will be seen later in Fig. 17. Consequently, for sake of clarity and simplification, only the results associated with the mesh 200× 100× 100 using k0 = 0.49
20
m2 /s for Ub = 11.0 m/s and ǫ0 = 10.0 m2 /s3 are presented and discussed in the following. Note also that it has been verified that the contraction value C = 1.25 is here too small to induce a detachment of the velocity boundary layer from the lateral walls.
7.2.1 Mean and instantaneous velocity profiles at different cross sections of the channel To illustrate the mean features of the average flow, we examine first the velocities at two locations of the channel. Figure 10 exhibits the mean streamwise velocity profiles at x1 /D1 = 1.0 and x1 /D1 = 1.5 just upstream and downstream the contraction zone versus the channel height at several computational times. At first sight, one can see that the flatness of the mean velocity is well marked due to the turbulence effects. The velocities in the immediate vicinity of the wall go to zero because of the no-slip condition. According to the contraction value C = 1.25, the mean dimensionless velocity goes from unity to 1.25. Figure 11 displays the mean spanwise velocities at x1 /D1 = 1.0 and x1 /D1 = 1.5 at several computational times. In addition, one can see that the mean streamwise velocity profiles are perfectly symmetric whereas the spanwise velocity profiles are anti-symmetric. This section shows that the flow can be considered as homogeneous in the straight section of the channel because the mean streamwise velocity u1 is almost constant in a cross section, the ratio of the magnitude velocity u2 to the magnitude velocity u1 is about 0.04/1 = 4% and 0.02/1.2 = 1.6% from x1 /D1 = 1 to 1.5.
7.2.2
Turbulent stress profiles at different cross sections of the channel
Figures 12, 13 and 14 display the evolution of the streamwise, spanwise and normal subfilter stresses (τii )sf s /Ub2 , resolved stresses (τii )les /Ub2 , and total turbulent stresses τii /Ub2 , for i = 1, 2, 3 normalized by the bulk velocity versus the channel height at the same locations as the ones chosen for the velocities at x1 /D1 = 1 and 1.5. The total stresses are computed as the sum of the subfilter and resolved stresses. Figure 12 shows that the subfilter scale stresses are almost uniform in both sections except in the immediate vicinity of the wall that reveals the presence of turbulent peaks. At x1 /D1 = 1, the subfilter scale stresses appear quasi isotropic (τ11 )sf s ≈ (τ22 )sf s ≈ (τ33 )sf s but at x1 /D1 = 1.5, the stresses are slightly anisotropic (τ22 )sf s ≈ (τ33 )sf s > (τ11 )sf s . Figure 13 shows that the resolved stresses are anisotropic in both sections at x1 /D1 = 1 and 1.5, (τ22 )les ≈ (τ33 )les < (τ11 )les . When comparing Fig. 12 with Fig. 13, one can see that the resolved stresses appear less uniform than the subfilter stresses versus the channel height. On overall, one can see that the PITM model behaves like a RANS model in the wall region because of the presence of the peaks of turbulence whereas it switches into an hybrid RANS/LES mode in the core flow. In the center of the channel, the sharing out of the turbulence energy among the subfilter and resolved scales is about 50 %. As it was indicated in Sec. 4.2, the sharing out of energy is governed by the parameter η = Le /∆. In the near wall region, η ≈ 0 because the refinement of the grid-size ∆ is not as high as the variation of the turbulence length-scale Le . Just upstream and downstream the contraction zone, the ratio (τ11 )sf s /(τ22 )sf s goes roughly from unity to 0.88 whereas (τ11 )les /(τ22 )les goes from 1.48 to 1.22 indicating that the reduction of anisotropy is higher for the resolved scales than for the subfilter scales. When analyzing Fig. 14, one can see that the total turbulence stresses are anisotropic upstream the contraction τ22 ≈ τ33 < τ11 but downstream the contraction, the stresses appear more
21
1.4 1.2 1
u1/Ub
0.8 0.6 0.4 0.2 0 −0.8
−0.4
0 x3/D1
0.4
0.8
−0.4
0 x3/D1
0.4
0.8
(a) 1.4 1.2 1
u1/Ub
0.8 0.6 0.4 0.2 0 −0.8
(b)
Figure 10: Mean streamwise velocities at (a) x1 /D1 = 1.0 and (b) x1 /D1 = 1.5. -•-, hu1 i /Ub . Re = 4.47 × 105 or less isotropic τ11 ≈ τ22 ≈ τ33 = 2/3k. As expected, the turbulence intensity is reduced when the flow passes through the contraction zone. The decrease of turbulence activity is roughly of 50 % that is relatively high for the contraction value C = 1.25. Apart from the wall region that shows the presence of turbulent peaks characterized by a strong flow anisotropy, the turbulent stress profiles are found to be quasi uniform in the cross sections, this flatness being however more marked for the lateral stresses than for the streamwise stress, so that the flow can be considered in a first approximation as locally homogeneous in a statistical sense. It can be interesting to appreciate the intensity of the subfilter, resolved and total stresses and these contribution change when passing from x1 /D1 = 1 to x1 /D1 = 1.5. For the streamwise stresses, one can see that the subfilter stress intensity is of the same order as the one of the resolved stress and that this ratio is kept from x1 /D1 = 1 to 1.5, while for the normal stresses, the subfilter stress intensity is roughly twice as high as the one of the resolved stress but this ratio slightly increases from x1 /D1 = 1 to 1.5.
22
0.06
0.04
u3/Ub
0.02
0
−0.02
−0.04
−0.06 −0.8
−0.4
0 x3/D1
0.4
0.8
−0.4
0 x3/D1
0.4
0.8
(a) 0.04
u3/Ub
0.02
0
−0.02
−0.04 −0.8
(b)
Figure 11: Mean spanwise velocities at (a) x1 /D1 = 1.0 and (b) x1 /D1 = 1.5. -•-, hu3 i /Ub . Re = 4.47 × 105 7.2.3
Mean velocity, stress and anisotropy ratio on the centerline of the channel
In this section, we examine the effects of the small axisymmetric contraction on the mean velocity and turbulent stresses especially along the centerline of the channel and we compare the PITM results with the experimental data corresponding to the mesh Reynolds number Ub M/ν = 37, 200 (see Fig. 2 of Ref. [3]). Figure 15 shows the evolution of the dimensionless bulk velocity Ub∗ (x1 ) = Ub (x1 )/Ub (0) and the anisotropy ratio τ11 /τ22 returned by the PITM simulation and RSM computation (see Sec. 3.3). First at all, one can see that the bulk velocity passing from unity upstream the contraction to 1.25 downstream the contraction agrees perfectly with the experimental data. As expected, the anisotropy ratio τ11 /τ22 initially close to 1.4 in the entrance of the channel gradually decreases to unity in the contraction zone suggesting that the turbulence tends to an isotropic state. Just downstream the contraction zone, τ11 /τ22 ≈ 1. But as a result of interest, it is found that this ratio reincreases roughly to 1.08 in the straight section downstream the contraction according to the experimental data. As it was mentioned in Sec. 3, this behavior is not reproduced by the single scale Reynolds stress model that predicts a ratio τ11 /τ22 = 1 everywhere in the channel downstream the contraction. To understand the origin of this reincrease of anisotropy, Uberoi and Wallis have advanced some qualitative arguments in
23
0.002
0.0016
τii SFS/Ub2
0.0012
0.0008
0.0004
0 −0.8
−0.4
0 x3/D1
0.4
0.8
−0.4
0 x3/D1
0.4
0.8
(a) 0.002
0.0016
τii SFS/Ub2
0.0012
0.0008
0.0004
0 −0.8
(b)
Figure 12: Subfilter scale stress (τii )sf s /Ub2 at (a) x1 /D1 = 1.0 and (b) x1 /D1 = 1.5. (i=1) N; (i=2) ◭; (i=3) ◮. Re = 4.47 × 105 Ref. [3] based on the turbulence scales relaxation. They have suggested that the turbulence state just after the contraction is an apparent isotropy state such as the anisotropies of the small scales and large scales are in fact just compensating. After the contraction, as the small scales relax rapidly, while the large scale anisotropy overshoots or relaxes at a slower rate, depending on the strain history of the initial field, the anisotropy can reappear in the downstream straight section of the channel. This analysis in the large scales and small scales has been identified by Lee and Reynolds [58] in their numerical simulations of homogeneous turbulence from irrotational strains. Furthermore, Schiestel [59] performed LES simulation of decaying turbulence in a cubic box with an initial energy spectrum perturbed by introduction of anisotropy in Fourier modes. Physically, this simulation corresponds to the evolving of the flow from an initial state of turbulence that is similar to the one of the wind tunnel flow of Uberoi and Wallis [3] just downstream the contraction zone and assuming a compensation between small and large scales such that τ11 /τ22 = 1. As a result shown in Figure 3 of Ref. [59], Schiestel also observed an increase of anisotropy versus the time, highlighting the scale relaxation effect. The objective is now to clarify this issue by means of the PITM simulation. To do that, we analyze the subfilter and resolved turbulent stresses as functions of the dimensionless cutoff parameter ηc = κc Le involving itself the cutoff wave number
24
0.0006
τii LES/Ub2
0.0004
0.0002
0 −0.8
−0.4
0 x3/D1
0.4
0.8
−0.4
0 x3/D1
0.4
0.8
(a) 0.0006
τii LES/Ub2
0.0004
0.0002
0 −0.8
(b)
Figure 13: Resolved scale stress (τii )les /Ub2 at (a) x1 /D1 = 1.0 and (b) x1 /D1 = 1.5. (i=1) N; (i=2) ◭; (i=3) ◮. Re = 4.47 × 105 κc = π/∆ and the integral turbulent length scale Le = k3/2 /ǫ. Figure 16 shows the evolution of the subfilter and resolved turbulent stresses hτii isf s /Ub2 and hτii iles /Ub2 , respectively, returned by PITM along the centerline of the channel. One can see that the curves present the same regular decay with a change of curvature in the contraction zone. For both subfilter and resolved stresses, the decay rate for the streamwise stresses remains higher than for the two lateral stresses. The resolved stresses are however rapidly dissipated, especially at the head end of the channel, probably due to the fact that the injected fluctuations are not solutions of the Navier-Stokes equations but are artificial fluctuations. Whatever the method of turbulence generation used, an artificial field will never compete with a “real ” field. Roughly speaking, one can see that the curves associated with the subfilter stresses cross each other at x1 /D1 = 1 and that the curves associated with the resolved stresses are closer together at about x1 /D1 = 1.5 suggesting that the small scales return more rapidly to isotropy than the large scales. For information purposes, the values of the subfilter, resolved and total turbulent stresses returned by PITM are indicated in Table 1 at four different locations. As a result of interest, it is found that the ratio (τ11 )sf s /(τ22 )sf s goes from unity at x1 /D1 = 1 to 0.88 at x1 /D1 = 1.5. This evolution should be compared with the one returned by the single RSM model where this ratio τ11 /τ22 has varied from 1.2 to unity in the contraction zone
25
0.002
0.0016
τιι/Ub2
0.0012
0.0008
0.0004
0 −0.8
−0.4
0 x3/D1
0.4
0.8
−0.4
0 x3/D1
0.4
0.8
(a) 0.002
0.0016
τιι/Ub2
0.0012
0.0008
0.0004
0 −0.8
(b)
Figure 14: Reynolds stress τii /Ub2 at (a) x1 /D1 = 1.0 and (b) x1 /D1 = 1.5. (i=1) N; (i=2) ◭; (i=3) ◮. Re = 4.47 × 105 (see Fig. 3). This outcome was expected since the small scales return more rapidly to isotropy than the large scales. Indeed, in the case of single RSM/RANS model, the stress τii includes both the small and large scales. So that the decay of anisotropy for τ11 /τ22 in RANS associated with the whole spectrum E(κ) is slower than for the subfilter scales (τ11 )sf s /(τ22 )sf s associated with the modeled part of the spectrum downstream the cutoff wave number, E(κ) for κ ≥ κc . In addition, the ratios (τ11 )sf s /(τ22 )sf s ≈ 0.88 and (τ11 )les /(τ22 )les ≈ 1.22 just at the end of the contraction zone although τ11 /τ22 = 1.01. This result confirms the assumptions made by Uberoi and Wallis [3] considering that the anisotropy is distributed among spectral wave numbers. Moreover, this result highlights the compensating effect between the small scales and large scales of the flow leading to an apparent isotropy state of turbulence just after the contraction. This result complies with the physics of fluid turbulence considering also that in absence of contraction, once again, the small scales return more rapidly to isotropy than the large scales before cascading into smaller scales by non-linear interactions. The big eddies have large time scales and so their unsteadiness is not so high in frequency than the unsteadiness of the small eddies that have the time to go to isotropy. The reasoning made on the subfilter and resolved stresses gives some clues to understand the mechanism that leads to the return to anisotropy downstream the contraction. But strictly
26
speaking, one has to recall that in PITM, the scale separation process is worked out by means of the dynamical parameter ηc which is the ratio of the turbulence length scale to the grid-size of the mesh. From a physical point of view, to get turbulence scales dissociated from the grid-size, we should formally perform direct numerical simulation of turbulence but this approach is here out of reach as shown in Appendix A. In the present case, intermediate scales are probably incorporated in the resolved part of energy instead of being included in the modeled part of energy that may alter the discussion based on the scale separation. The ideal will be to place the cutoff wave number κc even more upstream within the energy spectrum to consider only very large scales of the flow. Finally, Figure 17 compares the evolution of the dimensionless coefficients α = 10−3 Ub2 /τ11 and β = 10−3 Ub2 /τ22 returned by PITM with the experimental data (see Fig. 2 of Ref. [3]). As a result, a qualitative agreement is obtained between the experimental data and PITM. The mean tendencies and orders of magnitude are fairly well recovered even if some deviations are visible in the turbulence intensity. At this step, this outcome should be due to the too high dissipation of the artificial large scales in the inlet region of the channel. To address this issue, one solution would be to shorten the channel section before the contraction to get the exact level of turbulence that develops in the channel. Figure 17 confirms however that the inlet turbulence values are well estimated (see Sec. 2.2). 1.8
1.6
U/Ub
τ11/τ22
1.4
1.2
1
0.8
0.6
0
1
2
3
4
5
x1/D1
Figure 15: Evolution of the dimensionless bulk velocity Ub∗ (x1 ), and the ratio τ11 /τ22 on the centerline of the channel in the streamwise direction. Ub∗ (x1 ) : N, experiment [3]; – –, PITM. τ11 /τ22 : •, experiment [3]; —, PITM; – - –, Single scale RSM model. Re = 4.47 × 105
7.2.4
Energy spectrum densities of velocities
The energy spectrum densities of one dimensional spectra of the three velocity components is computed to take further the analysis. The time evolution of the large scale fluctuating velocity < < components u< 1 , u2 and u3 is given in Fig. 18 just upstream and downstream the contraction zone in the center of the channel. As expected, this signal is smoothed because of the filtering produced by the coarse grid. At a first sight, it can be seen that the recorded signal associated with
27
0.30
τii /Ub2
0.20
0.10
0.00
0
1
2
3
4
5
x1/D1
Figure 16: Evolution of the subfilter and resolved stresses hτii isf s /Ub2 and hτii iles /Ub2 returned by PITM along the centerline of the channel. – –, hτ11 isf s ; -.-.-, hτ22 isf s ; . . . , hτ11 iles ; - -, hτ22 iles . Re = 4.47 × 105 the streamwise velocity u< 1 presents lower frequencies undulations than for the two other lateral < velocities u< and u . The frequency associated to the streamwise velocity is roughly of order 2 3 f ≈ 20 Hz, or in dimensionless variable normalized by the bulk velocity Ub and the channel height D1 , the dimensionless frequency f ∗ = νD1 /Ub ≈ 1.20. As usually made in signal processing, the one dimensional time spectrum is then obtained by taking the windowed fast Fourier transform (FFT) with a Welch function [60] of the large scale fluctuating velocity correlation tensor as follows
< Eii (ν) = FFT [ u< (29) i (t)ui (t + τ ) ]
for i = 1, 2, 3 (no summation). The three spectra E11 , E22 and E33 associated with the resolved stresses (τ11 )les , (τ22 )les and (τ33 )les are plotted in Fig. 19 versus the dimensionless frequency. Overall, one can observe that the spectra E11 , E22 and E33 present almost the same regular decay for roughly one decade of low frequencies that compare favorably with the slope decay −5/3 of the Kolmogorov law corresponding to the inertial zone of the spectrum. As the frequencies increase, after the cutoff of the mesh, the spectra undergo a rapid drop of energy because of the action of the subfilter scale stresses which are present in the right hand side of the motion equation (13). One can see that for the dimensionless frequency f ∗ ≈ 1, the curve associated with E11
Table 1: Subfilter, resolved and total turbulent stresses (m2 /s2 ) at different locations in the centerline of the channel. x1 /D1 (τ11 )sf s (τ22 )sf s (τ11 )les (τ22 )les τ11 τ22 0 0.204 0.146 0.246 0.178 0.451 0.324 1 0.0456 0.0457 0.0368 0.0246 0.082 0.070 1.5 0.0299 0.0338 0.0252 0.0206 0.0551 0.0542 4. 0.0128 0.0129 0.0074 0.0057 0.0202 0.0187 τii = (τii )sf s + (τii )les 28
10
8
−3
10 Ub2/τιι
6
4
2
0
0
1
2
3
4
5
x1/D1
Figure 17: Evolution of the dimensionless variables α = 10−3 Ub2 /τ11 and β = 10−3Ub2 /τ22 on the centerline of the channel in the streamwise direction. Symbols : experiment [3]. •, α; △, β. PITM : −−, α; . − .−, β. Re = 4.47 × 105 crosses the other curves associated with E22 and E33 . In particular, upstream and downstream the contraction, for f ∗ > 1, E11 > E22 ≈ E33 whereas for f ∗ < 1, E11 < E22 ≈ E33 . This result suggests that f ∗ = 1, denoted fa∗ , yields the value of the cutoff frequency that allows to separate the “large ” scales from the “small ” scales. For both locations at x1 /D1 = 1 and 1.5, at low frequencies, the energy spectrum density E11 is of higher intensity than the two other components E22 and E33 . This point confirms that the large scales are anisotropic. Moreover, at x1 /D1 = 1.5, at low frequencies, E11 still remains of higher intensity than E22 and E33 , although the deviation is smaller than at x1 /D1 = 1. At x1 /D1 = 1.5, at high frequencies, the decay of E11 is more pronounced than at x1 /D1 = 1 to compensate the anisotropy of the large scales. Assuming the Taylor hypothesis of a frozen turbulence that is convected with the bulk velocity Ub , the cutoff wave number can be calculated from the frequency by κ = 2πf /Ub leading to κ = 2πf ∗ /D1 or κa = f π/D1 for fa∗ = 1. As expected, this wave number κa is much smaller than the cutoff wave number κc = π/∆ associated with the frequency fc∗ = D1 /2∆ used in the PITM simulation since ∆ ≪ D1 confirming that the resolved part of energy takes into account not only the large scales of the flow but also intermediate scales.
7.2.5
Turbulent flow structures
Figures 20 and 21 show the Q isosurfaces of the flow performed on the coarse mesh illustrating the vortical activity and the instantaneous flow structures. As expected, one can see that the flow structures appear elongated in the streamwise direction when passing the contraction zone due to the contraction effects both on the mean flow and on the turbulence. Just downstream the contraction, some structures form a thin layer on the wall surface perhaps due to the streamline curvature effects. As known, the vorticity ω1 is increased by vortex stretching while ω2 and ω3 are attenuated. If the PITM simulation succeeds in qualitatively reproducing these dynamic structures on a such coarse grid, a more realistic description of the flow structures obviously requires a very
29
0.8
Ui
0.4
0
−0.4
−0.8
0
0.1
0.2
0.3
(a)
0.4
0.5
0.6
0.7
0.4
0.5
0.6
0.7
t 0.8
Ui
0.4
0
−0.4
−0.8
0
0.1
0.2
0.3
(b)
t
Figure 18: Evolution of the large scale fluctuating velocity u< ¯i − hui i at (a) x1 /D1 = 1 and (b) i = u x1 /D1 = 1.5 versus the time advancement. —, u1 ; - - -, u2 ; −. − ., u3 . Re = 4.47 × 105 fine grid in streamwise and lateral directions.
8
Concluding remarks
In the framework of second moment closure, we have established the link that exists between Reynolds stress models including single and multiple-scales developed in RANS and the subfilter scale stress model developed in PITM. We have then performed numerical simulations of the turbulent flow in the small axisymmetric contraction of the wind tunnel designed by Uberoi and Wallis [3] using both Reynolds stress model and subfilter scale stress model, respectively. For this application, a new analytical spectral method has been especially developed to generate pseudorandom velocity fields as an inflow condition. This method allows to mimic the effects of the square-mesh biplane grid on the uniform wind tunnel stream. As a result, it has been found that the PITM simulation performed on a relatively coarse mesh reproduces this turbulent flow in good agreement with the experiment. In particular, PITM returns qualitatively well the increase of
30
10
10
−2
−4
Eii(κ)
10
0
10
10
10
−6
−8
−10
0
1 f D1/Ub
10
0
1 f D1/Ub
10
(a) 10
10
−2
−4
Eii(κ)
10
0
10
10
10
−6
−8
−10
(b)
Figure 19: Energy spectrum density of one-dimensional spectra of the three velocity components computed in the centerline of the channel at (a) x1 /D1 = 1 and (b) x1 /D1 = 1.5 returned by PITM. —, E11 ; - - -, E22 ; −. − ., E33 ; − − −, (f D1 /Ub )−5/3 . Re = 4.47 × 105
Figure 20: Vortical activity illustrated by the Q isosurfaces at Re = 4.47 × 105 (enlargement view in the contraction zone). PITM simulation 200 × 100 × 100.
31
Figure 21: Vortical activity illustrated by the Q isosurfaces at Re = 4.47 × 105 (upper view). PITM simulation 200 × 100 × 100. the anisotropy ratio τ11 /τ22 along the centerline of the channel downstream the contraction zone while the RSM single-scale model fails to predict this strange phenomena. This point identified as “return to anisotropy ” constitutes the interesting paradox of the Uberoi and Wallis experiment. The flow was thoroughly investigated in detail through the mean velocities, turbulent stresses, energy spectrum densities, and some interesting insights into the structures of the turbulence were provided.
A
COMPUTATIONAL RESOURCES FOR DNS
It is worth evaluating, as a rough guide, the necessary computer resources for simulating the flow in the experimental wind tunnel of Uberoi and Wallis [3] at the Reynolds number Re ≈ 4.47 × 105 in terms of number of grid-points and computational times. DNS simulations require that the gridsize is at least of order of magnitude of the Kolmogorov scale η computed as η = (ν 3 /ǫ)1/4 . We consider here a simple model where the turbulent kinetic and the dissipation-rate evolve along the centerline of the channel with the coordinate x1 according to the power law decay of homogeneous turbulence leading to the equations −n ǫ0 x1 (30) k(x1 ) = k0 1 + nk0 Ub and ǫ(x1 ) = ǫ0
ǫ0 1+ x1 nk0 Ub
32
−(n+1)
(31)
where n = 1/(cǫ2 − 1) and k(x0 ) = k0 , ǫ(x0 ) = ǫ0 . In the case of an infinitesimal domain of the channel between x1 and x1 + dx1 , the number of grid-points dN1 (x1 )dN2 dN3 of the mesh is then given by 64 D12 dx1 (32) dN1 (x1 )dN2 dN3 = η 3 (x1 ) in order to describe a “minimal ” sine curve on a full period using at least four grid points. The number of grid-points is then computed by integration of equation (32). As a result, it is found that − 3(n+1) Z L1 Z 4 64 D12 L1 ǫ0 dx1 (33) N1 N2 N3 = N2 N3 dN1 dx1 = N2 N3 3 x1 1+ nk0 Ub η0 0 0 Hence, ǫ0 L1 64 D12 k0 Ub F N1 N2 N3 = (34) ǫ0 nk0 Ub η03 where # " 1−3n 4 4n ǫ0 F = −1 (35) L1 1+ 1 − 3n nk0 Ub The computational time is proportional to the number of grid points N1 N2 N3 and the number of temporal iterations Nit and the time required by the central processing unit tCPU per iteration and per grid-point, leading to the result t = N1 N2 N3 Nit tCPU . The number of iteration is given by Nit = T /δt where T is the convective time allowing the eddies to move towards the exit of the channel, whereas δt is given by the CFL condition δt = η0 /Ub that is here a more stringent constraint p than the Kolmogorov time-scale τ0 = ν/ǫ0 = η02 /ν by a factor about 100. The convective time is computed from T = L1 /Ub so that the computational time is therefore given by L1 t = N1 N2 N3 tCPU (36) η0 √ The turbulent Reynolds number is Rt = k2 /νǫ = Le k/ν = (Le /η)4/3 . Using the preceding values Ub = 11.0 m/s, k0 = 0.46 m2 /s and√ǫ0 = 10 m2 /s3 the turbulence length-scale and the Kolmogorov scale are computed by Le = νRt / k so that in the entrance of the channel, Le0 ≈ 3.43 cm and η0 ≈ 0.135 mm, respectively. From equations (34) and (35), it is found that the number of grid points is order of Nη ≈ 6 × 1012 , and that the dimensionless computational time t/tCPU ≈ 1017 . These numerical order of magnitudes clearly show that DNS and also by extension highly resolved LES are not at all affordable in term of computational resources even with the rapid increase of super computer power.
B GENERATION OF ANISOTROPIC TURBULENCE WITH AN IMPOSED ENERGY SPECTRUM B.1
Generation of isotropic turbulence in the spectral space
The first step consists in generating an homogeneous isotropic field in a cubic box of dimension L using the method developed by Roy [49]. In this aim, we define a random vector stream function
33
in the spectral space [61, 62] ˆ ˆ1 (κ) + jb(κ)ψ ˆ2 (κ) ψ(κ) = a(κ)ψ
(37)
ˆ1 (κ) and ψ ˆ2 (κ) are two real vectors uniformly distributed on the sphere (see Fig. 22) of where ψ radius unity in the half space κ3 ≥ 0, a and b are two functions defined as a(κ) = α1 (κ) cos(λκ) and b(κ) = α2 (κ) sin(λκ) , α1 and α2 are real functions and λ is a random number in the interval [0, 2π]. ˆ1 (κ) and ψ ˆ2 (κ) obtained by drawing lots in the Fourier space define The two real random fields ψ ˆ a complex vector stream function ψ(κ) in the spectral space. If working in spherical coordinates, κ3 ^ (κ) ψ 1
Ψ n,p
^ (κ) ψ 2
κ θn κ2 ϕp
κ1
Figure 22: Random vector fields in the spectral space on the sphere of radius unity. n,p the random vector Ψn,p on the sphere of radius unity is obtained by Ψn,p 1 = sin θn cos φp , Ψ2 = sin θn sin φp and Ψn,p 3 = cos θn , where θn and φp are the polar and azimuthal angles, respectively. The random angles are then given by φp = 2πp and θn = arccos(1 − 2n) where p and n are uniform random numbers in the interval [0, 1]. The stream function is then computed in the half space ˆ ˆ ∗ (κ), where ψ ˆ ∗ denotes the conjugate of ψ, ˆ to ensure that the κ3 < 0 by the relation ψ(−κ) =ψ velocity is real. The velocity is therefore obtained by the relation
ˆ ˆ1 (κ) + jα2 (κ) sin(λκ)(κ ∧ ψ ˆ2 (κ) ˆ (κ) = κ ∧ ψ(κ) u = α1 (κ) cos(λκ)(κ ∧ ψ
(38)
verifying automatically the continuity equation. In order to satisfy the energy equation, k=
X 1X hˆ um (κ)ˆ u∗m (κ)i = E(κ)δκ 2 κ κ
(39)
one finds that the spectral velocity correlation tensor must satisfy the spectral energy equation hˆ um (κ)ˆ u∗m (κ)i
=
34
2π L
3
E(κ) 2πκ2
(40)
It is straightforward to see that the two functions a(κ) and b(κ) which appear in equation (37) are given by 1 3 cos(λκ) 2π 2 E(κ) 2 a(κ) = (41) ˆ1 (κ)|| L 2πκ2 ||κ ∧ ψ 1 3 sin(λκ) 2π 2 E(κ) 2 b(κ) = (42) ˆ2 (κ)|| L 2πκ2 ||κ ∧ ψ where ||.|| denotes the norm. In equation (37), the energy spectrum density is modeled by equations E(κ) = χκm f or κ < κ0
(43)
E(κ) = Cκ ǫ2/3 κ−5/3 f or κ ≥ κ0
(44)
where the coefficient χ is given by (5+3m)/3
χ = Cκ ǫ2/3 κ0
and the matching wave number κ0 is determined by the energy condition k = to the value 3m + 5 3/2 ǫ κ0 = Cκ3/2 3/2 2(m + 1) k
(45) R∞ 0
E(κ)dκ leading (46)
For κc ≥ κ0 where κc is the cutoff wave number, the ratio value between the subgrid scale energy and the total energy is ksf s 3(m + 1) κc −2/3 = (47) k 3m + 5 κ0
B.2
Generation of anisotropic turbulence in the spectral space
The second step consists in applying a tensor transformation β on the isotropic field produced by ˆ the stream function ψ(κ) to generate an anisotropy field. As for the isotropic case, this procedure ˆ ˆ = κ ∧ β ψ(κ). allows to satisfy the continuity equation. The velocity is then computed by u It is possible to compute the coefficient βij by means of algebra calculus in the spectral space. The starting point consists in applying another tensor transformation α on the velocity itself leading ˆ ˆ = α(κ ∧ ψ(κ)). to u The issue to address is then to determine the tensor β as a function of α. The resulting equation which must be solved finally reads
or for the i component,
ˆ ˆ ˆ (κ) = α(κ ∧ ψ(κ)) u = κ ∧ β ψ(κ)
(48)
u ˆi = jκj αim ǫmjk ψˆk = jǫijk κj βkm ψˆm
(49)
Equation (49) allows to work on the velocity as well as on the stream function field. The spectral turbulent energy associated with the wave number κ is then given by E D E D (50) hˆ ui u ˆ∗i i = αim αin ǫmjk ǫnpq κj κp ψˆk ψˆq∗ = ǫijk ǫipn βkm βnq κj κp ψˆm ψˆq∗
35
For simplification purposes, we consider now that α and β are diagonal matrices. Moreover, as ˆ ψ(κ) is an isotropic vector, the right hand-side of equation (50) becomes E D ∗ ∗ ˆ ˆ (51) hˆ ui u ˆi i = ǫijk ǫipn βkk βkk κj κp ψk ψk
or in a developed form,
E D E D 2 2 ˆ ˆ∗ 2 2 ˆ ˆ∗ κ3 ψ2 ψ2 hˆ u1 u ˆ∗1 i = β33 κ2 ψ3 ψ3 + β22 E D E D hˆ u2 u ˆ∗ i = β 2 κ2 ψˆ1 ψˆ∗ + β 2 κ2 ψˆ3 ψˆ∗ 2
1
11 3
33 1
(52) (53)
3
E D E D 2 2 ˆ ˆ∗ 2 2 ˆ ˆ∗ κ2 ψ1 ψ1 hˆ u3 u ˆ∗3 i = β22 κ1 ψ2 ψ2 + β11
(54)
To reproduce the axisymmetric turbulence of the Uberoi and Wallis experiment [3] such as hu2 u2 i = hu3 u3 i, equations (53) and (54) imply that β22 = β33 . Moreover, to get hu1 u1 i > hu2 u2 i, these equations lead to β11 /β22 < 1. Taking into account these conditions, it is found that the matrix β can be written as 1 − 2γ 0 0 β= 0 1+γ 0 0 0 1+γ where γ is a numerical coefficient which must be determined to get the desired ratio anisotropy. Equation (50) allows to compute the coefficient γ and leads to the following equations E D E D Ei D E h D (55) hˆ u1 u ˆ∗ i = α2 κ2 ψˆ3 ψˆ∗ + κ2 ψˆ2 ψˆ∗ = β 2 κ2 ψˆ3 ψˆ∗ + β 2 κ2 ψˆ2 ψˆ∗ 1
11
2
3
3
2
33 2
3
22 3
2
E D E D Ei D E h D 2 2 ˆ ˆ∗ 2 2 ˆ ˆ∗ κ1 ψ3 ψ3 κ3 ψ1 ψ1 + β33 hˆ u2 u ˆ∗2 i = α222 κ23 ψˆ1 ψˆ3∗ + κ21 ψˆ3 ψˆ3∗ = β11 E D E D Ei D E h D hˆ u3 u ˆ∗ i = α2 κ2 ψˆ2 ψˆ∗ + κ2 ψˆ1 ψˆ∗ = β 2 κ2 ψˆ2 ψˆ∗ + β 2 κ2 ψˆ1 ψˆ∗ 3
33
1
2
2
1
22 1
2
11 2
1
(56) (57)
Now, with the aim to obtain equations which do not depend anymore on the wave number κ, we define the spherical mean of the Fourier transform M by the relation [61] ZZ 1 ψ(κ) = M(ψ(κ)) = ψ(κ) dA (58) A ∂A where A denotes the area element on the sphere of radius κ = |κ|. Applying this operator M on equations (55), (56) and (57) leads to the resulting equations that read 2 2 2α211 = β33 + β22 2 2 2α222 = β11 + β33 2 2 2α233 = β11 + β22
(59) ˆ ˆ ˆ (κ) = α κ ∧ ψ(κ) As κ ∧ ψ(κ) is an isotropic vector whereas u is an anisotropic vector, the anisotropy of the turbulence in the spectral space can be measured by the ratio α211 hˆ u1 (κ)ˆ u∗1 (κ)i = =ζ hˆ u2 (κ)ˆ u∗2 (κ)i α222
36
(60)
In this case, for a given value of ζ, the solving of the system (59) gives the results α11 = 1 + γ0 , α22 = α33 = ((5γ02 − 2γ0 + 2)/2)1/2 and the coefficient γ0 is solution of the second degree equation 5 γ02 (1 − ζ) + γ0 (2 + ζ) + (1 − ζ) = 0 2
(61)
Starting with this initial value γ0 , it is then possible to compute its exact value γ by successive approximations to get the prescribed value hu1 (x)u1 (x)i / hu2 (x)u2 (x)i = ζ.
B.3
Return to the physical space
As usually, the velocity in the physical space u(x) is then computed from its inverse Fourier transform given by X X X ui (x1 , x2 , x3 ) = u ˆi (κ1 , κ2 , κ3 ) exp (jκm xm ) (62) κ1 =
2πn1 L
κ2 =
2πn2 L
κ3 =
2πn3 L
for (n1 , n2 , n3 ) ∈ [− N2 + 1, N2 ].
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