A Two Fluid Model for Two-Phase A Two Fluid Model for Two-Phase - PowerPoint PPT Presentation

A Two Fluid Model for Two-Phase A Two Fluid Model for Two-Phase Flows with Free Interface Flows with Free Interface G. Chanteperdrix P. Villedieu J.P. Vila ONERA/CNES ONERA/MIP MIP/ONERA Workshop “Numerical Methods for Multimaterial Compressible Fluid Flows”, Paris, September 23-25, 2002

2 Context and Applications Fluids behaviour in tanks : • Sloshing effects due to transverse accelerations • Surface tension effects due to possible gravity reduction and interface curvature • Thermal effects due to external heat fluxes WNMMCFF, Paris, September 23-25, 2002

3 Main Features of our approach • Eulerian method on a fixed grid, with no interface reconstruction or interface tracking, existence of an artificial mixture zone → need of a mixture closure law leading to a well-posed problem (hyperbolic system, thermodynamic consistency) → need of a low diffusive numerical scheme • able to deal with a wide range of applications relative to fluid behaviour in a launcher tank (including coupling between thermal and hydrodynamic effects) → compressible two fluid model, eventually with “pseudo” sound speeds, to overcome low Mach number difficulties WNMMCFF, Paris, September 23-25, 2002

✄ ✁ ✄ ✄ � ✄ ✄ ✂ 4 Modeling Hypothesis • Gas of density ρ g , liquid of density ρ , supposed to be compressible and inviscid • no thermal effects : linearized EOS ρ = + ρ − ρ ρ = + ρ − ρ p p c p p c 2 2 ( ) ( ) , ( ) ( ) g g g g g 0 0 0 0 • the two fluid “fictitiously” coexist everywhere, presence characterised by volume fraction α g , α : α g + α =1 • only one velocity field WNMMCFF, Paris, September 23-25, 2002

☎ ✆ ☎ ☎ ✆ ✆ 5 The Equilibrium Model ~ ∂ ρ  ∂ ∂ ~ ~ g + ρ + ρ = u v  0 g g ∂ ∂ ∂ t x y  ~ ∂ ρ ∂ ∂  ~ ~ + ρ + ρ = u v 0  ∂ ∂ ∂ t x y  ( ) E  ∂ ∂ ∂ ( )  ρ + ρ + + ρ = u u P uv 2  x ∂ ∂ ∂ t x y  ∂ ∂ ∂ ( )  ρ + ρ + ρ + = v uv v P 2  y ∂ ∂ ∂ t x y � ~ ~ ~ ~ ρ = αρ ρ = − α ρ ρ = ρ + ρ with : , ( 1 ) , g g g WNMMCFF, Paris, September 23-25, 2002

✝ ✟ ✝ ✝ ✞ ✟ 6 The Equilibrium Model : closure laws mixture pressure Mixture pressure P : ~ ρ ~   ρ ( ) ( )   ~ ~ g ρ ρ = α   + − α P p p   * * ,   1 g g α − α *  *    1 where : ( ) ~ ~ α = α ρ ρ * * , g is the equilibrium gas volume fraction insuring : ~ ρ ~   ρ   g =   p p     g α − α   * *   1 WNMMCFF, Paris, September 23-25, 2002

7 The Equilibrium Model : closure laws source terms Source terms : • acceleration in the local reference frame (gravity + transverse acceleration) F = σκ ∇ α • surface tension, CSF way : c (Brackbill et al, JCP 1992 )   ∇ σ the surface tension coefficient   ∇ = −∇ ⋅ ∇   κ the interface curvature with ∇   WNMMCFF, Paris, September 23-25, 2002

8 The Equilibrium Model : closure laws “ conservative” capillary force joint work with D. Jamet, CEA   ∇   ∇ ⋅ ∇ F = − σ   The capillary force : c ∇     ∇ ( )   ∇ ⋅ ⊗ ∇ + ∇ ∇ F = − σ σ also reads :   c ∇   → yields a completely conservative formulation of momentum equation → numerically, ∇α needs only to be computed at the faces center WNMMCFF, Paris, September 23-25, 2002

✠ ✡ ✡ ✡ ✡ ✡ ☛ ☛ ✠ ✠ ✠ The Equilibrium Model : 9 mathematical properties (without surface tension) • The mixture pressure law leads to an hyperbolic system with mixture sound speed c * such that :   ∂ α ∂ α K K  +    * *   ~ ~ ~ ~ ρ = ρ + ρ − = − ρ + ρ c c c K     2 2 2 1 1 with   g g g ~ ~ α − α ∂ ρ ∂ ρ *     * * 1   g ( ) ( ) F * F ~ ~ ~ ~ ρ ρ ρ = α ρ ρ ρ u u * • Existence of a Lax entropy , , , , , with : g g ( ) F 1 p ∫ ∫ ~ ~ ~ ~ p α ρ ρ ρ = ρ + ρ + ρ u u g 2 , , , ρ ρ d d g g g 2 2 ρ ρ 2 g kinetic free energy energy • Difficulties with the explicit solution of the Riemann problem due to the complexity of the mixture pressure law → Relaxation model (in the spirit of Coquel & Perthame, SIAM 1998) WNMMCFF, Paris, September 23-25, 2002

☞ ☞ ✌ ✌ ☞ ☞ The Relaxation Model 10 −  p p ∂ α ∂ α ∂ α + + = g , < ε << u v  (0 1 ) ∂ ∂ ∂ ε  t x y R  ( ) ε ∂ ∂ ∂ w  + + =  ∂ ∂ ∂ t x y � ~ ~     ~ ~ g , = α   + (1 − α ) P p p   with ( , )   g g α − α     1 similar models in : • Saurel & Abgrall (JCP 1999) , Saurel & Lemetayer (JFM 2001) • Dellacherie & Rency (preprint 2001) Remark : the relaxation source term is linked to F by : F F F   − p p ∂ ∂ ∂ 1 1   g = − = − − thermodynamic potential   ε ε ∂ α ε ∂ α ∂ α 2   g WNMMCFF, Paris, September 23-25, 2002

✍ ✍ 11 The Relaxation Model : mathematical properties (without surface tension) • Also hyperbolic with mixture sound speed c such that : ~ ~ ρ = ρ + ρ c c c 2 2 2 g g and verifying the sub-characteristic condition : ≤ c 2 2 c * → good mathematical frame for relaxation a Chapman-Enskog like expansion of the relaxation model can formally be derived (as in Coquel & Perthame, SIAM 1998) • F is an entropy, compatible with the relaxation source term : the relaxation source term contributes to the decrease in entropy WNMMCFF, Paris, September 23-25, 2002

12 Numerical Method • Finite Volume method : – RK2 in time (2nd order) – Godunov- MUSCL in space (2nd order) • Each RK2 stage divided in two parts – hyperbolic step ( exact Riemann solver ) – pressure relaxation step WNMMCFF, Paris, September 23-25, 2002

13 Numerical Method : hyperbolic step ∂ ρα ∂ ρα ∂ ρα u v  + + = 0  ∂ ∂ ∂ t x y  Given α n and w n , we solve :  ∂ ∂ ∂ w  + + =  ∂ ∂ ∂ t x y � n n ρα * ρα − g g      −  f f j + j −   =   − ∆  +  + ∆ t + − t i i 1 / 2 1 / 2 1 / 2 1 / 2       by ij ∆ ∆ w w x y       ij ij ( ) n α w * , to obtain intermediate state ∆ t is fixed by a CFL condition WNMMCFF, Paris, September 23-25, 2002

14 Numerical Method : hyperbolic step numerical scheme Numerical flux functions f and g based on exact resolution of the associated Riemann problem : Godunov Scheme t u-c u u+c u shock contact shock u-c u+c Nature or discontinuity or rarefaction rarefaction x WNMMCFF, Paris, September 23-25, 2002

✎ Numerical Method : 15 pressure relaxation step ( ) n α w * Given , we solve with ε→0 : , − p g p  ∂ α =   ∂ τ ε only α change  ∂ w  =  � ∂ τ to obtain α n+1 and w n+1 : ( ) no numerical scheme is needed for the  ~ ~ + α n = α ρ n ρ n  1 * * * , volume fraction with this approach 1 2   n + = n w w Remark : α * can be explicitly computed 1 * � for linearized EOS WNMMCFF, Paris, September 23-25, 2002

16 Numerical Results : Sloshing in a 2D wedge high Froude number, high Bond number mesh size : 150 × 75 WNMMCFF, Paris, September 23-25, 2002

17 Numerical Results : Sloshing in a 2D wedge high Froude number, high Bond number WNMMCFF, Paris, September 23-25, 2002

18 Numerical Results : Sloshing in a 2D wedge high Froude number, high Bond number WNMMCFF, Paris, September 23-25, 2002

19 Numerical Results : Sloshing in a 2D wedge high Froude number, high Bond number WNMMCFF, Paris, September 23-25, 2002

20 Physically Relevant Quantities V ∫ = α d V • Total gas volume : g Ω F F ∫ ∫ = + σ ∇ α d V d V • Energies : tot Ω Ω pressure part of the free energy (F ) remaining part of p + the free energy : capillary energy (F ) kinetic energy (F ) ts ec This total energy F tot is decreasing for the relaxation model (up to a slight modification of the pressure closure law, current work with D. Jamet, CEA) WNMMCFF, Paris, September 23-25, 2002

21 Numerical Results : square to circular bubble mesh size : 80 × 80 WNMMCFF, Paris, September 23-25, 2002

✏ 22 Numerical Results : square to circular bubble energy balance Bubble oscillations in a inviscid fluid (Lamb, 1932), deformation modes frequencies : n=2 σ 1 = + − + f n n n n ( 1 )( 1 )( 2 ) π ρ R 3 n=4 WNMMCFF, Paris, September 23-25, 2002