HLLC solver for hyperbolic non-equilibrium two-phase flows

Damien Furfaro and Richard Saurel Comp. & Fluids, 2015

Motivations and importance Drops

Gas Slugs

Liquid

Bubbles

Build a 3D // unstructured code to solve multiphase mixture flows and interfaces. Design a very robust scheme able to deal with very strong shocks (mega-bar) and vacuum (1 Pa) for cavitating flows. Design a method simpler than the DEM (Abgrall and Saurel, 2003) and more robust.

2

Two-phase flow model  1  1  ui  (p1  p2 ) t x

Baer and Nunziato (1986)

( ) 1 ( u) 1  0 t x  1 ( u)1 ( u²  p)1   pi   (u2  u1 ) t x x ( E) 1  u( E  p)  1  1   pi' (p1  p2 )   u'i (u2  u1 )   pi ui t x x

+ Symmetric system for phase 2 Z u  Z 2u2  (p  p1 ) ui  1 1  sgn( 1 ) 2 Z1  Z 2 x Z1  Z 2

pi 

Z1p2  Z 2p1  (u  u1 )Z1Z 2  sgn( 1 ) 2 Z1  Z 2 x Z1  Z 2

These symmetric estimates are given in Saurel et al. 2003 and extended in Saurel et al. 2014 to granular mixtures. The symmetric variant involves 7 waves instead of 6  very important for the two-phase RP.

3

Conventional Riemann problem t

-

x

6 or 7 waves, many states to compute, linear solver not robust enough, Roe solver useless (non conservative terms) and difficult to build (shocks), regularization needed for the volume fraction wave (Schendeman, Kapila), this regularization is not robust enough (shock interface interaction), extension to N phases: 10 waves (3 fluids), 14 waves (4 fluids), …

Discrete Equations Method (DEM) Abgrall and Saurel, JCP, 2003 Saurel, Gavrilyuk and Renaud, JFM, 2003 Chinnayya, Daniel and Saurel, JCP, 2004 Le Metayer, Massoni and Saurel, JCP, 2005 Berry, Saurel, Le Metayer, NED, 2010 Solves non conservative products (even with shocks) Computes accurately two-phase fluxes with Euler Riemann solvers

(but complicate for a 3D unstructured code) 5

Characteristic function

‘1’  ‘2’ 

‘1’ 

 1 if the pointbelongsto the phasek X k  otherwise 0

X k   X k obeys  ui .X k 0 t ‘1’ 

 u i represents the local interface velocity

6

The method starts as conventional averaging methods We start from : X k

t

u

I

X k x

v

U F G 0 t x y Fluid selection

X k Uk t

I

X k y

0

(pure fluid =Euler for example)

X Euler 0 k

X k Fk  x

X kG k  y

X k X k (Fk  ui Uk )  (G k  v i Uk ) x y

The equations are then integrated over volume and time.

7

The input with this method is the flow topology j+1/2

j+1/2 Fluid 2

Fluid 2

Fluid 2

Cell ij

Fluid 2

Fluid 2

Cell ij

Fluid 1

Fluid 1

Fluid 1

Fluid 1 Fluid 2

Fluid 1 Fluid 1

b

b Fluid 2 j-1/2 i-1/2

Stratified flow

c

c

Fluid 2

j-1/2 i+1/2

i-1/2

i+1/2

Bubbly flow

Given the flow topology we are going to integrate over space and time the Riemann problem solutions at each interface inside the 2D control volume and at its boundaries 8

Finite volume type integration 9 X k Uk t

 X

Fk x k

 X

Gk y k

(F

k

t

t

    X k Fk X kG k   X k Uk   dxdydt         t x x   0  Cij   0  Cij

 ui Uk )

X k X k  (G k  v i Uk ) x y

 X k X k    Fk  ui Uk   dxdydt   G k  v i Uk   x  y  

For the sake of time restrictions and importance we focus on the non-conservative term t

  X k     F  u U   dxdydt k i k   x   0  Cij 

At a given cell boundary 10

- The contact lenght for each pair of fluids are computed Fluid 2

Fluid 2

i

Fluid 2

i-1

S22

Cell i

Fluid 1

S12

Fluid 1

i-1

Fluid 1

H

i+1

i

S11 i-1/2

- In this example

S11  H min( ,  1,i 1,  1,i )

S22  H min(  2,i 1,  2,i )

i+1/2

,

S12  H  S11  S22

-At each contact (11), (12), (22) the Riemann problem is solved -The fluxes and non conservative terms are integrated along each contact

Focus on non-conservative terms t

  X k     F  uU   dxdydt   x   0  Cij 

We have to compute Xk

is discontinuous at the interfaces

But at the interfaces the Lagrangian flux is precisely locally

constant: pI cst . , uI cst .

t

F Lagrange

 0     F  uU  p   pu   

The product ( F  uU )

X x

is thus well defined !

Shock

Interface

x 11

Non conservative terms are thus integrated easily Fluid 2

Fluid 2

i

Fluid 2

i-1

S22

Cell i

Fluid 1

S12

Fluid 1

i-1

Fluid 1

i+1

i

S11 i-1/2

i+1/2

t

  X k   * *Lagrange * *Lagrange * *Lagrange     F  u U dxdydt   t S X F  S X F  S X F   i 11 1 , 11 11 22 1 , 22 22 12 1 , 12 12   x   0  Cij 





1-1=0





0-0=0





0-1=-1

t

* Lagrange *  tS12F12   if u12 0 X k     F  u U dxdydt     i   x  0 otherwise  0  Cij   12

DEM = set of discrete equations

 n1Un1  n Un  XF i,1 i,1 i,1 i,1  1,i1/ 2   XF 1,i 1/ 2 (F u U)X   Relaxation   i t x  x   i,1

 XF 1,i 1/ 2  segmentlengthcharact.func. 

Riemannflux)cellboundary

X   ( F  u U )    i  x   i,1 func.jump Lagrangian Riemann flux)  segmentlengthcharact.

cells boundaries

Need to store Eulerian fluxes and Lagrangian fluxes. Not exactly a finite volume method - complexity at first order, increasing complexity at second order (extra NC terms appear) etc. Stiff source terms, vacuum and complex thermodynamic are present too. A simpified method is needed with enhanced robustness (geniune positivity).

13

HLLC two-phase (Furfaro and Saurel, C&F 2015) Key points : -each subsystem of PDE’s for a given phase can be decoupled of the overall system - 4 waves only are present (instead of 7), - a locally conservative formulation is available.

Flow model for a given phase without source terms  1  1  ui 0 t x

( ) 1 ( u) 1  0 t x (u)1 (u²  p)1  1  pi t x x ( E) 1  u( E  p)  1  1   pi ui t x x

+ interfacial variables given by Euler-Euler Riemann solvers based on left and right Z p  Z 2p1  (u  u1 )Z1Z 2 Z1u1initial  Z 2u2data  1 (p2  p1 ) p 1 2  sgn( 1 ) 2 ui 

Z1  Z 2

 sgn( ) x Z1  Z 2

i

Z1  Z 2

x

Z1  Z 2

These two interfacial variables are thus locally constant. 15

The phases are coupled only through the NC terms. But pi and ui are local

Local conservative formulation  1 u i  1  0 t x

( ) 1 ( u) 1  0 t x (u)1 11u12  1 (p1  pi )  0 t x (E) 1   u(E  p)  1 pi u i  1  0 t x

HLLC with 4 waves (not so simple a priori). t S L ,k W

WL ,k

* L ,k

S M ,k ** k

W

uI WR*,k

WR ,k

S R ,k x

16

Wave speeds estimates ,

.

SR ,k Max  u L,k  c L,k , u R ,k  c R ,k  Davis (1988)

SL,k Min  u L,k  c L,k , u R ,k  c R ,k  SM , k 









 R ,k u 2  P R ,k   L ,k u 2  P L,k  SL,k   u  L,k  SR ,k   u  R ,k   L,k   R ,k PI

 u  R ,k   u  L,k  SL,k   L,k  SR ,k    R ,k

Combining the 4 RH systems across the 4 waves results in:

u *R ,k

 P  

*

R ,k



  P  L,k    L ,k   R ,k PI SR ,k     R ,k  u R ,k  SR ,k  u ISM ,k  SR ,k u R ,k 

  R ,k  u R ,k  SR ,k SR ,k  u I  SM ,k  u R ,k    P  R ,k   P *L,k   L,k   R ,k PI

And a miracle occurs …

u *R ,k SM ,k

(all velocities in star state are equal)

HLL approx

HLLC summary WR* ,k

WL*,k

;



*L,k  L,k * L,k



L,k

* R,k

** k  L,k

 R,k

;

;

u L,k  SL,k SM,k  SL,k

*R,k R,k  

u *R ,k

u*L,k SM,k E*L,k E L,k 

Wk**

;

 ;

 Pu  L,k  PL*,kSM,k  L , k  u L , k  SL , k 

PL*,k PL,k  L,k  u L,k  SL,k  u L,k  SM ,k  ** Wk

E; *R ,k

u R,k  SR,k SM,k  SR,k

SM ,k  E R ,k  ;

 Pu  R ,k  PR* ,kSM,k  R , k  u R , k  SR , k 

;

PR* ,k PR ,k  R ,k  u R ,k  SR ,k  u R ,k  SM ,k 

  *k*   *R ,k u ** k SM,k E*k* E*R ,k 

Pk** PL*,k

The solver is explicit.  ; Its extension to an arbitrary number of fluids is straightforward.  ; It is entropy preserving (Furfaro and Saurel, 2015) and genuinely positive.  ;  ;

 L,k   R ,k

 

* R ,k

PI

Godunov type scheme ,

k  1,2

 k       k  U k  ,    u  k     E k  

U k Fk H k   k 0 t x x

U nk ,i1

U nk ,i

i

H

* k,i 12

k u I   uI         u   0 k   H   Fk  k 2   PI   k  u  P    k PI k        E  P  u   P u   PI u I  k I I  k k k

t *  Fk ,i  1  Fk*,i  1   nk ,i H*k ,i  1  H*k ,i  1 2 2 2 2 x







,

   u I  k,i 1 , 0, 2

.

 PI  k,i

1 2

,

 PI u I  k,i 



T

1 2

This simple method merges with the more sophisticated DEM method. Warning: When dealing with higher order extensions, extra NC terms appear.

1D test problem: Shock tube with volume fraction discontinuity

Job done by NC terms. No pressure velocity relaxation is needed to fulfill interface conditions.

Double expansion

Lines = DEM results Symbols = HLLC

3D computations

96mm

52mm

160mm

Latter times

Thank you for your attention

Announce: There is an open permanent research position at RS2N (small private company) for a young doctor in CFD, see CFD ONLINE or www.rs2n.eu in France, close to Marseille and Aix-en-Provence.

Lagrangian fluxes at internal interfaces a

X  Lag,* Lag,*    tNa F ( 2 , 1 )  F (1,2)  ij (F  ui U)  internal  x  interfaces  The differences between interface variables inside the control volume result in relaxation terms. 26