Simulation of phase transition in ... - Université de Nantes

Simulation ofphase transition

in avan-der-Waals

fluid

Vincent PERRIER

Modelling andnumerical issues

Numerical scheme

Numerical results

Conclusion

Simulation of phase transition in acompressible isothermal fluid governed by

the van-der-Waals equation of state

Vincent PERRIER

INRIA, Concha Team and Université de Pau et des Pays de l’Adour

Castro-Urdiales, 7-11 September 2009

INRIA Concha Vincent PERRIER Simulation of phase transition in a van-der-Waals fluid 7-11 September 2009 1/23


in avan-der-Waals

fluid

Vincent PERRIER


Numerical scheme

Numerical results

Conclusion

1 Modelling and numerical issues

2 Numerical scheme

3 Numerical results



in avan-der-Waals

fluid

Vincent PERRIER


Numerical scheme

Numerical results

Conclusion

Modelling and numerical issues



in avan-der-Waals

fluid

Vincent PERRIER


Numerical scheme

Numerical results

Conclusion

Dynamic of phase transition

Isothermal Euler system{∂tρ+ div(ρu) = 0

∂t(ρu) + div(ρu⊗ u + P) = 0

unknowns: density ρ and velocity uA Closure is necessary : P = P(ρ).Modelling of phase transition: (adimensioned) van-der-Waalsequation of state

P(ρ) =8Tρ3− ρ

− 3ρ2 0 < ρ < 3



in avan-der-Waals

fluid

Vincent PERRIER


Numerical scheme

Numerical results

Conclusion

Hyperbolicity of the system

Eigenvalues for this system are u · n, u · n±√

P ′(ρ)

two densities, ρg and ρl between which P ′(ρ) decreases.⇐= The system is not hyperbolic. The Cauchy problem is ill

posed.

ρg ρl



in avan-der-Waals

fluid

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Numerical scheme

Numerical results

Conclusion

Regularization of the system

Add viscosity and capillarity terms (see the talk by P.LeFloch)

∂tρ+ div(ρu) = 0∂t(ρu) + div(ρu⊗ u + P) = µdiv(τ)−∇ (ε∇ρ⊗∇ρ)

Numerical Method

Solve the regularized system

Closer of the mathematicaltheory.can be costly if the flow isdominated by theconvection.

Solve the limit system

Use the regularization fordefining the phase transitionwaves.How to remain hyperbolic?



in avan-der-Waals

fluid

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Numerical scheme

Numerical results

Conclusion

Basic problem for diffuse interface method

A phase transition wave between two states ρ1 < ρg and

ρ2 > ρl . define α =ρ− ρ2

ρ1 − ρ2.

the pressure inside the wave should not be P(ρ), butαP(ρ1) + (1− α)P(ρ2).Divide the initial condition between two sets: ρ < ρg (gasphase), and ρ > ρl (liquid phase).⇐= Use a multiphase numerical method.



in avan-der-Waals

fluid

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Numerical scheme

Numerical results

Conclusion

New formulation of the problem

coupling of hyperbolic systems.

Solve

for k = 1,2 χk

(∂tρ+ div(ρu)

∂t(ρu) + div(ρu⊗ u + P)

)= 0

∂tχk + σ · ∇χk = 0

where σ is the velocity of the phase transition wave.

σ

σ



in avan-der-Waals

fluid

Vincent PERRIER


Numerical scheme

Numerical results

Conclusion

Numerical scheme



in avan-der-Waals

fluid

Vincent PERRIER


Numerical scheme

Numerical results

Conclusion

Finite volume scheme (1/3)

Ci Ci+1

α1ρ1u1α2ρ2u2

i

α1ρ1u1α2ρ2u2

i+1

In the cells i and i + 1, each of the cells have their ownvolume fraction, velocity, and density.



in avan-der-Waals

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Numerical scheme

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Conclusion


Ci Ci+1

(ρ2,u2)i

(ρ1,u1)i

(ρ2,u2)i+1

(ρ1,u1)i+1

The volume fraction are used for filling the cells, with eitherthe red or the blue fluid.Solve the four Riemann problems: red-red, blue-red,red-blue, blue-blue.



in avan-der-Waals

fluid

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Numerical scheme

Numerical results

Conclusion


Ci Ci+1

(ρ2,u2)i

(ρ1,u1)i

(ρ2,u2)i+1

(ρ1,u1)i+1

Integrate the different Riemann problems.Add or substract the Eulerian Fluxes F (Ui) or F (Ui).Add on both of the phases F (U)− σU where σ is the velocityof the phase transition, if a phase transition occurs.



in avan-der-Waals

fluid

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Numerical scheme

Numerical results

Conclusion


Ci Ci+1

(ρ2,u2)i

(ρ1,u1)i

(ρ2,u2)i+1

(ρ1,u1)i+1

Take the mathematical expectancy E()

Total Flux = Pi+1/2(�,�)F (Ui ,Ui+1) + Pi+1/2(�,�)F (Ui ,Ui+1)Pi+1/2(�,�)F (Ui ,Ui+1) + Pi+1/2(�,�)F (Ui ,Ui+1)



in avan-der-Waals

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Numerical scheme

Numerical results

Conclusion


Ci Ci+1

(ρ2,u2)i

(ρ1,u1)i

(ρ2,u2)i+1

(ρ1,u1)i+1

See [Abgrall-Saurel-2004] and [Lemétayer-Saurel-2006] forthe details of the integration.



in avan-der-Waals

fluid

Vincent PERRIER


Numerical scheme

Numerical results

Conclusion

Finite volume scheme (2/3)Choice of the weights

Suppose that the weights Pi+1/2(�,�) depend only on thetopography (not on the local velocities or densities).Consider a simple case: a phase transition wave.

(ρ2,u2) (ρ1,u1)σ

⇐= For the scheme, this case is similar as an advection on α.For being consistent with the usual discretization of anadvection equation, the weights must be

Pi+1/2(�,�) = min(αi , αi+1)Pi+1/2(�,�) = min(αi , αi+1)Pi+1/2(�,�) = max(αi − αi+1,0)Pi+1/2(�,�) = max(αi − αi+1,0)



in avan-der-Waals

fluid

Vincent PERRIER


Numerical scheme

Numerical results

Conclusion

Finite volume scheme (3/3)Solution of the Riemann problem

[Merkle-Rohde-2007] C. Merkle, C.Rohde Thesharp-interface approach for fluids with phase change:Riemann problems and ghost fluid techniques M2AN 41(2007), no. 6, 1089–1123.The complexity of the solution of the Riemann problem is notonly due to the phase transition, but also to the fact that thefields are not genuinely nonlinear.A “simple” left or right wave can be rarefaction wave

shock waveattached wave

+phase transition+{

attachedrarefaction wave

}non uniqueness of the solution is solved

explicit kinetic relationone phase solutions are prefered to two phase solutions.reject solutions with constant pressure phase transition.



in avan-der-Waals

fluid

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Numerical scheme

Numerical results

Conclusion

High Order extension

The high order accuracy is achieved by a discontinuousGalerkin method.Difficulty is on the non conservative formulation of thescheme: formally the scheme reads

∂t (E(χk U)) + div((E(χk F (U)))) = E ((F − σU)∇χk )

⇐= Going back to the discrete formulation allows to perform adiscrete integration by part of the non conservative product.

The slope limiter is applied as follows:If α 6= 0 or α 6= 1, project on DG(0) the densities and thevelocities. Use the slope limiter on α.If α = 0 or α = 1, use the slope limiter on the phase whichexists.



in avan-der-Waals

fluid

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Numerical scheme

Numerical results

Conclusion

Numerical results



in avan-der-Waals

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Numerical scheme

Numerical results

Conclusion

Computing conditions

The scheme is implemented in the Concha Library (C++,python).Explicit time integration (TVB).The spectral minmod slope limiter is used ([Krividonova,Flaherty]).In all the simulations, the adimensioned temperature is equalto 0.98.



in avan-der-Waals

fluid

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Numerical scheme

Numerical results

Conclusion

One dimensional shock tube (1/3)

Same test as in [Merkle-Rohde-2007]The densities are equal to the Maxwell points

ρ= 1.28943u = 0

ρ= 0.726691u =−3.5



in avan-der-Waals

fluid

Vincent PERRIER


Numerical scheme

Numerical results

Conclusion




in avan-der-Waals

fluid

Vincent PERRIER


Numerical scheme

Numerical results

Conclusion




in avan-der-Waals

fluid

Vincent PERRIER


Numerical scheme

Numerical results

Conclusion




in avan-der-Waals

fluid

Vincent PERRIER


Numerical scheme

Numerical results

Conclusion

One dimensional shock tube (3/3)Comparison dg0/dg1



in avan-der-Waals

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Numerical scheme

Numerical results

Conclusion

One dimensional shock tube (3/3)Comparison dg0/dg1



in avan-der-Waals

fluid

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Numerical scheme

Numerical results

Conclusion

Collapse of a Bubble (1/2)A metastable bubble inside a stable liquid

ρ= 0.835u = 0

ρ= 1.31u = 0

<FILM>



in avan-der-Waals

fluid

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Numerical scheme

Numerical results

Conclusion


Phase transition induces sonic waves



in avan-der-Waals

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Focalisation of the sonic wave in the center of the bubble



in avan-der-Waals

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densification induces nucleation inside the bubble. Formationof a ring.



in avan-der-Waals

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Final collapse of the bubble.



in avan-der-Waals

fluid

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Numerical scheme

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Conclusion



in avan-der-Waals

fluid

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Numerical scheme

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Conclusion

Conclusion

ResultsThe scheme is conservative in the original variables ρ and u.High order can be achieved by the Discontinuous Galerkinmethod.Can be extended to any dimension.The method can be extended to any coupling of hyperbolicsystems, provided the heterogeneous Riemann problem canbe solved.

To doTest other slope limiters.Implement realistic kinetic closures; take into account the localcurvature for the surface tension in 2d.extend it to the non isothermal system.



in avan-der-Waals

fluid

Vincent PERRIER


Numerical scheme

Numerical results

Conclusion

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Simulation of phase transition in ... - Université de Nantes