Numerical Simulation of Complex and Multiphase Flows Porquerolles, April 2005 thanks to: ERCOFTAC, Conseil Général Var, Région PACA, USTV and Stana. copy of the students passports (ERCOFTAC grants). extra nights have to be paid today. lectures (morning) and advanced communications (afternoon).

Applications of the finite volumes method for complex flows: from the theory to the practice Philippe HELLUY, ISITV, Université de Toulon, France. Spring school « numerical simulations of multiphase and complex flows », April 2005, Porquerolles.

I) Introduction to finite volumes for hyperbolic systems of conservation laws. II) An industrial application. III) Mixtures thermodynamics and numerical schemes for phase transition flows.

I Introduction to finite volumes for hyperbolic systems of conservation laws

Example: let u(x,t) be a solution to Characteristic curve The characteristic has thus the equation That implies that u is an arbitrary function of x-ct. A characteristic is a curve along which a regular solution of a first order Partial Differential Equation (PDE) is constant u(x(t),t)=Cst 

Other example: Burgers equation A regular solution satisfies The characteristic equation is thus Consider now an initial condition of the form If u were regular, it would imply u(x,t)=1=0 if x>1 and t>1 or

Admissible shock wave A shock wave is a discontinuous solution of a first order PDE, w t +f(w) x =0, emerging from an intersection of characteristics. If the shock curve is parametrized by (x(t),t), the normal vector to the shock is n=(1,-s), with s=x’(t). We denote by [w]=w R -w L the jump of the quantity w in the shock. According to distribution theory, we must have the Rankine-Hugoniot relation The characteristic intersection condition gives Lax’s condition (for Burgers: u L >u R ) (For Burgers: s=1/2(u L +u R ) )

Entropy condition More practical criterion: often, a supplementary conservation law can be deduced from the conservation law w t +f(w) x =0 If U is convex (Lax entropy), we require that In the sense of distributions. Example: Burgers

The Riemann problem Example of the Burgers equation + entropy condition The solution is noted R(x/t,u L,u R ) and is Rarefaction waveShock wave

Hyperbolic systems In order to generalize to a System of Conservation Laws (SCL), w t +f(w) x =0, w in R m, we must suppose that f’(w) has only real eigenvalues. The solution of Riemann problem has then the form (for m=2) x/t= 1 – x/t= 1 + w 0 =w L w m =w R x/t= 2 + x/t= 2 – w1w1 x t (m+1 constant states separated by shock or rarefaction waves in the (x,t) plane)

Finite volume schemes In two dimensions the SCL reads w t +f(w) x +g(w) y =0, and one solves a Riemann problem for each edge between two finite volumes in the normal direction to the edge. CiCi CjCj Godunov flux "Exact" in some sense, and satisfies a discrete entropy inequality In one dimension: let  be a time step, h a space step, x i =ih, t n =n , The cell (or finite volume) C i is the interval ]x i-1/2,x i+1/2 [

II Industrial application

Multifluid model Gas generator: industrial tool to eject device Two phases (air and water) Compressible (pressure up to 300 atm) Duration: 50 ms, thus evaporation and viscosity neglected.

The unknowns are the density  (x,t), the velocity u(x,t), the internal energy  (x,t) the pressure p(x,t) and the mass fraction of gas y(x,t) Euler equations (in 1D in order to simplify) Pressure law:

Hyperbolicity In order to prove hyperbolicity, we consider the change of variables w  ( ,u,s,y), where s is the physical entropy satisfying Tds=d  +pd(1/  ). One gets The eigenvalues are u-c, u, u, u+c with With a good pressure law, the system is thus hyperbolic. We have also that U(w)= –  s is a Lax entropy

Practical pressure law We use a pressure law that allows the exact resolution of the Riemann problem and acceptable precision: the stiffened gas equation of state (EOS) For numerical reasons, it appears that the last conservation law should be replaced by a non-conservative transport equation (Abgrall-Saurel 1996) then

Numerical results Rouy, 2000

III Mixtures thermodynamics and numerical schemes for phase transition flows.

Cavitation boiling

Demonstration

Liquid area heated at the center by a laser pulse Bubble collapse near a rigid wall Ambient liquid (1atm) Heated liquid (1500 atm) 2.0 mm, 70 cells 2.4 mm, 70 cells 1.4 mm 0.15 mm0.45 mm Wall

Mixture pressure (from 0 to 2ns) Bubble close to a rigid wall

Volume Fraction of Vapor (from 0 to 66ns) Bubble close to a rigid wall

1) Thermodynamics of a single fluid and of an immiscible mixture of two fluids; 2) Relaxation scheme for flow with phase transition (coupling the hydrodynamics and the thermodynamics); 3) Miscible mixtures and super-critical fluids. Mixtures thermodynamics and numerical schemes for phase transition flows T. Barberon and Helluy. Finite volume simulation of cavitating flows. Computers and Fluids, P. Helluy, N. Seguin. A simple model for super-critical fluids, 2005, preprint.

Single fluid thermodynamics S(W) is the entropy. It is extensive and concave We define the temperature T byThe pressure p byThe chemical potential This gives Single fluid of mass M, volume V and energy E. W=(M,V,E). An extensive variable X is a function of W that is Positively Homogeneous of degree 1 (PH1) : X is an intensive variable if it is PH0:

The gradient of a PH1 function is PH0 thus pressure, temperature and chemical potential are intensive. The euler relation for PH1 functions gives Definition of the specific volume, energy and entropy , s.

The sound speed c of the fluid computed from p=p(  ) must be real Hyperbolicity if The sign of p is not important… Hyperbolicity H. B. Callen. Thermodynamics and an introduction to thermostatistics, second edition. Wiley and Sons, J.-P. Croisille. Contribution à l’étude théorique et à l’approximation par éléments finis du système hyperbolique de la dynamique des gaz multidimensionnelle et multiespèces. PhD thesis, Université Paris VI, 1991.

Mixture of two immiscible fluids (1) and (2) Mixture entropy out of equilibrium Constraints Equilibrium entropy at equilibrium: (Isobaric law) M 1,V 1,E 1 M 2,V 2,E 2 M,V,E

out-of-equilibrium pressure and temperature: It is more practical to use intensive variables. Out-of-equilibrium specific entropy

Equilibrium specific entropy The maximum is under constraints: at equilibrium, is not necessarily zero…

Simple example (perfect gases mixture) The fractions  and z can then be eliminated. We suppose temperature and pressure equilibrium Out-of-equilibrium specific entropy (before mass transfer)

Pressure law out of equilibrium and saturation curve Out of equilibrium, we have a perfect gas law The saturation curve is thus a line in the (T,p) plane. On the other side,

Phase 2 is the most stablePhase 1 is the most stable Phases 1 and 2 are at equilibrium

Equilibrium pressure law Let We suppose (for a fixed temperature, fluid (2) is heavier than fluid (1))

Comparison between the real and simplified models

Partial conclusion: the temperature and pressure of the mixture are obtained from the entropy; the equilibrium entropy is solution of a constrained convex optimisation problem. Next step: coupling with hydrodynamics; numerical scheme.

One-velocity two-fluid models Instantaneous equilibrium No phase transition Other possible models…

Formal limit It is natural to study the weak entropy solutions of the formal limit system: R. Menikoff and B. J. Plohr. The Riemann problem for fluid flow of real materials. Rev. Modern Phys., 61(1):75–130, S. Jaouen. Étude mathématique et numérique de stabilité pour des modèles hydrodynamiques avec transition de phase. PhD thesis, Université Paris VI, November Generally, this system has several Lax solutions Liu entropy criterion is more adequate

Standard schemes as Rusanov's may converge towards different solutions CFL=0.9418CFL= density numerical entropy production

Relaxation scheme based on entropy optimisation When =0, the previous system can be written in the classical form 1) Finite volumes scheme (relaxation of the pressure law) 2) Projection on the equilibrium pressure law

Other works about relaxation schemes Yann Brenier. Averaged multivalued solutions for scalar conservation laws. SIAM J. Numer. Anal., B. Perthame. Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal., F. Coquel and B. Perthame. Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J. Numer. Anal., Saurel, Richard; Abgrall, Rémi A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys., G. Chanteperdrix, P. Villedieu, and Vila J.-P. A compressible model for separated two-phase flows computations. In ASME Fluids Engineering Division Summer Meeting. ASME, Montreal, Canada, July Stéphane Dellacherie. Relaxation schemes for the multicomponent Euler system. M2AN Math. Model. Numer. Anal., …

Numerical results with CFL=  comparison with the Liu solution for a simple Riemann problem Density

Velocity

Pressure

Mixture of stiffened gases We suppose only temperature equilibrium (elimination of z). Out of equilibrium, the mixture still satisfies a stiffened gas law: exact Riemann solver in the relaxation step. Equilibrium is obtained after optimization with respect to  and . The pressure law is not analytic.

Liquid area heated at the center by a laser pulse Bubble collapse near a rigid wall Ambient liquid (1atm) Heated liquid (1500 atm) 2.0 mm, 70 cells 2.4 mm, 70 cells 1.4 mm 0.15 mm0.45 mm Wall

Mixture pressure (from 0 to 2ns) Bubble close to a rigid wall

Volume Fraction of Vapor (from 0 to 66ns) Bubble close to a rigid wall

Partial conclusion: the relaxation scheme is based on entropy optimisation it seems to converge towards the Liu solution it can be used in practical configurations Next step: super-critical fluids

Mixture of two miscible fluids Constraints: Equilibrium entropy At equilibrium: (Dalton's law)

Inf-convolution and Legendre transform In the two cases, miscible or not, the equilibrium entropy is the inf-convolution of the two entropies S 1 and S 2 (for the miscible mixture, the inf-convolution is computed only with respect to W=(M,E), the volume V=V 1 =V 2 being fixed.) Legendre transform

A simple model for super-critical fluids We suppose that the mixture becomes more and more "miscible" when the energy increases

Optimisation problem in intensive variables constraints

Case of two perfect gases The elimination of the energy and volume fractions gives

Isothermal lines in the ( ,p) plane critical point saturation "liquid" "gas" critical isotherm super-critical fluid

In the saturation zone: modified isobaric pressure law In the "liquid" zone the two phases are present In the "gas" zone the two phases are present In the super-critical zone: Dalton's law

Conclusion the relaxation scheme expresses on the discrete level the physical entropy production. it seems to converge towards the Liu solution it is possible to implement the scheme in realistic configurations the qualitative features of super-critical fluids can be recovered with the entropy optimisation procedure Prospects obtaining a precise critical behavior is still a challenging difficulty