NUMERICAL SIMULATION OF COMPRESSIBLE TWO-PHASE FLUID...

NUMERICAL SIMULATION OF COMPRESSIBLE

TWO-PHASE FLUID FLOW :

Ghost fluid method vs Saurel Abgrall approach 1

Mathieu Bachmann

Institut für Geometrie und Praktische Mathematik,RWTH Aachen University

Joint work with :

Josef Ballmann, Siegfried Müller, RWTH Aachen University.

Mohsen Alizadeh, Dennis Kröninger, Thomas Kurz, Werner Lauterborn, Universität Göttingen.

Philippe Helluy, Hélène Mathis, Université de Louis Pasteur Strasbourg.

1DFG-CNRS FOR 563: Micro-Macro Modelling and Simulation of Liquid-Vapor Flows

RWTH Aachen IGPM Mathieu Bachmann 1

Outline

• Motivation

• Mathematical Model

• Numerical Discretization

– Saurel-Abgrall Approach– "Real" Ghost Fluid Method

• Numerical Results

– Shock-Interface Interaction (Convergence Study, 1D)– Shock-Bubble Interaction (Application, 2D)

• Outlook


Motivation

Cavitation near a Solid 2

rapid collapse, jet formation near boundaries

high-speed and short-time photography

(75000 fps) (1 million fps)

2Courtesy of Lauterborn et al.


Mathematical Model

• Assumption :

high speed flows and short observation times two phases are immiscible no phase transition

• Compressible 1D Euler equations :

∂

∂ t(ρ) +

∂

∂ x(ρ v) = 0,

∂

∂ t(ρ v) +

∂

∂ x(ρ v2 + p) = 0,

∂

∂ t(ρ E) +

∂

∂ x(ρ v(E + p/ρ)) = 0

• Stiffened gas law :

p(ρ, e, ϕ) = (γ(ϕ)− 1) ρ e− γ(ϕ) π(ϕ)


Mathematical Model

Evolution of Phase Boundary by Gas Fraction

• Gas fraction : ϕ = 0 (liquid), 0 < ϕ < 1 (mixture), ϕ = 1 (gas),

• Evolution equation : (no mass transfer)

D ϕ

D t≡ ∂ ϕ

∂ t+ v

∂ ϕ

∂ x= 0

• Mixture pressure : Linear interpolation

p(ρ, e, ϕ) = (γ(ϕ)− 1) ρ e− γ(ϕ) π(ϕ) =

pliquid , ϕ = 0pmixture , 0 < ϕ < 1pgas , ϕ = 1

β1(ϕ) :=1

γ(ϕ)− 1

β2(ϕ) :=γ(ϕ)π(ϕ)γ(ϕ)− 1

⇔

γ(ϕ) = 1 +

1β1(ϕ)

π(ϕ) =β2(ϕ)

1 + β1(ϕ)

β1(ϕ) = ϕβ1(1) + (1− ϕ)β1(0), β2(ϕ) = ϕβ2(1) + (1− ϕ)β2(0)


Mathematical Model

Evolution of Phase Boundary by Level Set Function

• Level Set Function : Φ < 0 (liquid), Φ = 0 (interface), Φ > 0 (gas),

• Evolution equation : (no mass transfer)

D ΦD t

≡ ∂ Φ∂ t

+ v∂ Φ∂ x

= 0

• Initialization : Distance function

Φ(0, x) = sgn(x− xI) |x− xI|

• Reinitialization : (Sussman et al.)

∂Φ∂τ

= sgn(Φ)(

1−∣∣∣∣∂ Φ∂ x

∣∣∣∣) resp.

∂ Φ∂ τ

+ a(Φ)∂ Φ∂ x

= sgn(Φ), a = sgn(Φ) ∂ Φ∂ x

/ ∣∣∣∂ Φ∂ x

∣∣∣RWTH Aachen IGPM Mathieu Bachmann 6

Numerical Discretization

I. Saurel-Abgrall Approach

II. "Real" Ghost Fluid Method (Wang, Liu, Khoo)



Saurel-Abgrall Approach

• Finite Volume Discretization :

– Time Evolution of conserved quantities:

vn+1i = vn

i −∆t

∆x

(F n

i+12− F n

i−12

)– Numerical Flux at the cell interface xi+1

2:

∗ Compute 2nd order reconstruction of the primitive variables and thegas fraction: W

±i+1

2= (ρ±, v±, p±, ϕ±)T

i+12

∗ Solve two-phase Riemann problem: W i+12

= W(ξ = 0, W

−i+1

2, W

+

i+12

)• Non-conservative Upwind Discretization :

– Idea: Preserve contact discontinuity avoid pressure oscillations– Time Evolution of gas fraction:

ϕn+1i = ϕn

i −∆t

∆x

(v n

i+12(ϕn

i+12− ϕn

i )− v ni−1

2(ϕn

i−12− ϕn

i ))



"Real" Ghost Fluid Method (Wang, Liu, Khoo)

• Finite Volume Discretization :

– Time Evolution of conserved quantities:

vn+1i = vn

i −∆t

∆x

(F n,−

i+12− F n,+

i−12

)– Numerical Flux at the cell interface xi+1

2with ϕi ϕi+1 > 0:

∗ Compute 2nd order reconstruction of primitive variables: W±i+1

2

∗ Solve single-phase Riemann problem: Wi+1

2= W

“ξ = 0, W

−i+1

2, W

+

i+12

”∗ Evaluate flux: F n,−

i+12

= F n,+

i+12

= F (Wi+1

2)

– Numerical Flux at the cell interface xi+12

with ϕi ϕi+1 < 0:∗ solve a two-phase Riemann problem interfacial states at the phase boundary∗ determine states in the ghost cells and modify real fluid by interfacial states∗ compute the reconstruction of the primitive variables∗ solve two single-phase Riemann problems at the cell interface



Numerical Flux at Phase Boundary

• Solve a two-phase Riemann problem with the states UL:=U i−1 and UR:=U i+2

interfacial states: U IL:=(ρIL, uI , pI )T and U IR:=(ρIR,uI , pI)T

• Redefine real fluid in cells i and i + 1 by interfacial states: U i=U IL and U i+1=U IR

• Define states in ghost cells (boundary cells of fluid A and B) by interfacial states:ghost fluid A: UA

j :=U IL, j > i + 1

ghost fluid B: UBj :=U IR, j < i− 1

• Solve two single-phase Riemann problems for fluid A and B two numerical fluxes F n,±

i+12

at the cell interface next to the phase boundary


Numerical Results

I. Shock-Interface Interaction (Convergence Study, 1D)

II. Shock-Bubble Interaction (Application, 2D)


Shock-Interface Interaction

UW UWS UA

ρ [kg/m3] 1620.6 1000 1v [m/s] 1087.1 -100 -100p [Pa] 3.6801E+09 1E+05 1E+05

Ω=[-4;2]m, t ∈[0;1.5e-3]s

N0 = 100, 5 ≤ L ≤ 13, εL=1.e-5, cfl=0.9

γ [-] π [Pa]Water 3.0 7.499e+8Air 1.4 0

UW∗ UA∗ρ [kg/m3] 900 5.57v [m/s] 2361.4 2361.4p [Pa] 7.48506E+06 7.48506E+06



Results with L=7 at t=1.5 ms



Results with L=12 at t=1.5 ms



Convergence study : L1 error of density

Saurel-Abgrall rGFMLevels L1 error Order L1 error Order5 10.20 - 1.8 -7 4.81 0.54 0.51 0.918 3.27 0.56 0.31 0.729 2.24 0.54 0.17 0.8710 1.55 0.53 0.11 0.6711 1.09 0.51 6.90E-02 0.6212 0.77 0.51 4.54E-02 0.6013 0.54 0.50 3.05E-02 0.57

• Numerical order of convergence is 0.5

• Larger error for the S-A approach due to the smearing at the interface



Convergence Study: Error in Interface Position

• Interface position :rGFM (zero level set) and S-A approach (gas fraction of 0.5)

Saurel-Abgrall rGFMLevels hL [m] Error Order Error Order5 1.87E-03 7.12E-03 - 3.55E-03 -7 4.69E-04 3.51E-03 0.51 1.03E-03 0.898 2.34E-04 2.46E-03 0.51 5.25E-04 0.979 1.17E-04 1.72E-03 0.52 2.93E-04 0.8410 5.86E-05 1.20E-03 0.52 1.59E-04 0.8811 2.93E-05 8.42E-04 0.51 9.21E-05 0.7912 1.46E-05 5.92E-04 0.51 5.43E-02 0.7613 7.32E-06 4.22E-04 0.49 3.38E-05 0.68

• Numerical order of convergence is 0.5

• In case of rGFM: interface position is always shifted by 2-5 cells



Influence of Shock-Interface Interaction for S-A Approach



Conclusion

• Numerical order of convergence is 0.5 for both schemes

• Influence of shock-interface interaction:

– S-A approach: oscillations in water and wrong shock position in air– rGFM: interface velocity slightly over-predicted– Perturbations become weaker under grid refinement



Shock-Bubble Interaction

Material parametersVapor Water

γ 1.4 7.15π [Pa] 0 300000000Cv [J/kgK] 717.5 201.1

Initial conditions for Ms = 1.1.vapor pre-shocked post-shocked

water water

% [kg/m3] 0.026077 1000 1044.5vx [m/s] 0 0 68.6p [Pa] 2118 100000 110676724T [K] 283 243 318



Saurel-Abgrall approach Real ghost fluid method

t = 9.8 µs t = 11.6 µs

t = 24.6 µs t = 23.3 µs



Saurel-Abgrall approach Real ghost fluid method

t = 33.8 µs t = 46.7 µs

t = 41.9 µs t = 116.5 µs



Conclusion

Saurel-Abgrall:

• Significant smearing of interface→ Unphysical phase transition→ Wave propagation is affected

• Strongly grid-dependent

Real Ghost Fluid Method:

• No smearing of interfacebut : interface capturing requiressufficient resolution of the bubble

• Wave dynamics less grid-dependent

• Problem with higher-order reconstruction

• Bubble vanishes

• Computations are time-consuming


NUMERICAL SIMULATION OF COMPRESSIBLE TWO-PHASE FLUID...

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