E. Romenski , D. Drikakis

1. CONSERVATIVE FORMULATION AND NUMERICAL METHODS FOR MULTIPHASE COMPRESSIBLE MEDIA E. Romenski , D. Drikakis Fluid Mechanics & Computational Science Group, Cranfield University, UK The financial support from the EU Marie Curie Incoming International Fellowship Programme (contract MIF1-CT-2005-021368) is acknowledged.

2. The problem of multiphase flow modeling lies in the mathematical and numerical formulation of the problem - there is not yet a widely accepted formulation for the governing equations of multiphase flows -------------------------------------------------------------------------- The challenge is associated with the development of a mathematical model that satisfies three important properties: . . hyperbolicity(symmetric hyperbolic system in particular) . fully conservative form of the governing equations consistency of the mathematical model with thermodynamic laws -------------------------------------------------------------------------

3. Two-phase compressible flow models with different velocities, pressures and temperatures • Bayer-Nunziato-type model (Baer&Nunziato,1986; Saurel&Abgrall,1999). • Governing equations are based on the mass, momentum, • and energy balance laws for each phase in which • interfacial exchange terms (differential and algebraic) included. • Equations are hyperbolic (non-symmetric), but non-conservative • We propose extended thermodynamics approach and • thermodynamically compatible systems formalism • (Godunov-Romenski) to develop multiphase model. • Governing equations are written in terms of parameters • of state for the mixture and are taking into account • a two-phase character of a flow. • Equations arehyperbolic (symmetric) and conservative

4. Class of thermodynamically compatible systems of hyperbolic conservation laws ------------------------------------------------------------------------------------- Thermodynamically compatible system is formulated in terms of generating potential and variables All equations of the system are written in a conservative form and the system can be transformed to a symmetric hyperbolic form Many well-posed systems of mathematical physics and continuum mechanics can be written in the form of thermodynamically compatible system. Examples: gas dynamics, magneto-hydrodynamics, nonlinear and linear elasticity, electrodynamics of moving media, etc.

5. Thermodynamically compatible system in Lagrangian coordinates is determined by the generating potentialM and variables: The system is symmetric and hyperbolic if is a convex function. Flux terms are formed by invariant operators grad, div, curl _______________________________________________________________________________ Energy conservation law:

6. Thermodynamically compatible system in Eulerian coordinates can be obtained by passing to the new coordinates and corresponding transformation of generating potential and variables The system can be transformed to the symmetric one and hyperbolic if Lis a convex function __________________________________________________________________________________ -- energy conservation law

7. Development of the two-phase flow model consists of several closely interrelated steps: 1. Introduction a new physical variables in addition to the classical variables (velocity, density, entropy) characterizing two-phase flow. 2. Formulation of new conservation laws for these new variables in addition to the mass, momentum and energy conservation laws. 3. Introduction a source terms modelling phase interaction and dissipation. 4. Formulation of closing relationships, such as Equation of State for the mixture and functional dependence of source terms on the parameters of state

8. Physical variables characterizing two-phase flow - the phase number, - volume fraction of i-th phase, - mass densityof i-th phase - velocity vector of i-th phase - specific entropy of i-th phase - thermal impuls of i-th phase *********************************************************************************** Physical variable is connected with the heat flux vector by the relation - heat flux relaxation time, temperature and thermal conductivity coefficient

9. Governing equations for compressible two-phase flow with different pressures and temperatures of phases Derivation is based on the extended irreversible thermodynamics laws and thermodynamically compatible system formalism (Godunov-Romenski). -- total mass conservation law, -- volume fraction balance, -- 1st phase mass balance law -- total momentum conservation law -- relative velocity balance law -- phase heat flux balance laws -- phase entropy balance laws 17 equations, 13 algebraic source terms

10. SOURCE TERMS are responsible for phase interaction and dissipation. REQUIREMENTS: The total energy conservation laws for the mixture must be fulfilled The total mixture entropy production must be non-negative The partial phase entropy production must be non-negative Onsager’s principle of dissipative coefficients symmetry is held _______________________________________________________________________________________ The following source terms are introduced in the governing equations: - Phase pressure relaxation to the common value through the process of pressure waves propagation - Phase to phase transition - Interfacial friction force (the Stokes drag force) - Heat flux relaxation to the steady Fourier heat transfer process - Phase temperatures relaxation to the common value through the heat transfer between phases - Phase entropies productioncaused by phase interaction

11. SOURCE TERMS definition | | | | | | | | | | | | | | | | | | | | | | | | --------------------------------------------- - Drag coefficient • thermal conductivity • coefficients ---------------------------------------------- - mass fraction of the 1st phase ---------------------------------------------- The total entropy production is positive

12. Simplification of the model applicable in the case of small dispersed phase particles, if the phase temperatures equalizing processis fast 1D equations for single temperature model a consequence of the general model under assumption -- volume fraction balance law -- mass conservation for the 1st phase -- total mass conservation -- total momentum conservation -- relative velocity balance law -- total energy conservation ______________________________________________________________________ - pressure relaxation - interfacial friction

13. Further simplification applicable in the case of negligible thermal variations 1D conservative equations for isentropic model -- total mass conservation -- total momentum conservation -- volume fraction balance law with pressure relaxation source term -- mass conservation for the 1st phase -- relative velocity balance law with drag force source term ____________________________________________________________________________________ -- pressure relaxation -- interfacial friction (drag)

14. Comparison of Conservative Model with Baer-Nunziato-Type Model Isentropic one-dimensional case Conservative model B-N -type model (Saurel&Abgrall) _______________________________________________________________________________________________ Systems are similar if to denote Definition of interfacial velocity is the same: Definition of interfacial pressure is different:

15. Comparison of Conservative Model with Baer-Nunziato-Type Model Isentropic multidimensional case The difference between two models becomes more significant - The momentum equations in conservative model can be written as follows: Extra terms appear which are not presented in the B-N-type model: These are forces arising for the flow with nonzero relative velocity, caused by the phase vorticities and are called as lift forces

16. NUMERICAL METHOD Standard finite-volume method is employed for solving the system of conservation laws ___________________________________________________________________ Since the Riemann problem for general equations can not be easily solved because the eigenstructure can not be obtained explicitly we apply recently proposed GFORCE method for flux evaluation (E.F. Toro, V.A. Titarev, 2006). ______________________________________________________________________________________ GFORCE is a convex average of the Lax-Friedrichs flux and Lax-Wendroff flux: - Lax-Friedrichs flux - Lax-Wendroff flux Here Δt is the local time step chosen without any relation to the global time step In Toro&Titarev (2006) it is reported that the GFORCE flux is upwind and reproduces the Godunov upwind flux for linear advection equation.

17. WATER FAUCET PROBLEM: water column flow in air annulus in a tube under the effect of gravity (Ransom,1987) Initial data: Tube length – 10 m - air volume fraction - water velocity - air velocity - uniform pressure ______________________________________ gravity Boundary conditions: Inlet: Outlet: initial state steady state t = 0.5 s _______________________________________ Exact solution:

18. Numerical solution of water faucet problem using isentropic model equations Instantaneous pressure relaxation is assumed, drag force is neglected Linearized Riemann Solver (1st order Godunov method) GFORCE flux (Toro&Titarev,2005,2006 ) (MUSCL-Hancock 2nd order method) Blue - exact solution Black - 200 mesh cells Red - 400 mesh cells Green - 800 mesh cells Purple - 1600 mesh cells GFORCE is comparable with the 1st order linearized solver

19. WATER/AIR SHOCK TUBE Two-phase flow test case (Saurel&Abgrall, 1999) Numerical solution with the use of single temperature model (GFORCE flux) Modelling of the moving water/air interface – instantaneous pressure relaxation, infinite drag coefficient water + small amount of air air + small amount of water Riemann problem, initial discontinuity at x=0.7 m Initial data: left: right: Black – exact solution Red -- 200 cells Blue --- 800 cells, gives a very good agreement with the exact solution

20. 2D Test case Shock– bubble interaction (single temperature model) Interaction of shock wave propagating in air with a cylindrical bubble Shock wave with the Mach number 1.23 Region: 225 x 44.5 (in millimeter), Bubble radius: 25 (in millimeter) Mesh: 300 x 100 cells Light (Helium) and Heavy (Freon R22) bubbles have been considered [experiments: Haas&Sturtevant (1987)]

21. Shock –bubble interaction (single temperature model) Light (Helium) bubble Perfect gas EOS withγ=1.4for air and γ =1.648forHelium Both gases are initially at atmospheric pressure ___________________________________________ Shock wave with the Mach number 1.23. The pressure behind the wave is 1.68 atmosphere ___________________________________________ Instantaneous pressure relaxation is assumed. Drag coefficient is Mixture density

22. Shock –bubble interaction (single temperature model) Heavy (Freon R22) bubble Perfect gas EOS with γ=1.4 for air and γ =1.249 for R22 Both gases are initially at atmospheric pressure ___________________________________________ Shock wave with the Mach number 1.23. The pressure behind the wave is 1.68 atmosphere ___________________________________________ Instantaneous pressure relaxation is assumed. Drag coefficient is Mixture density

23. Conclusions A new approach in multiphase flow modelling based on thermodynamically compatible systems theory is proposed. A hierarchy of conservative hyperbolic models for two-phase compressible flow is presented and robust numerical method for solving equations of the models is developed. Further Developments • Implement a phase to phase transition kinetics • Include dispersed phase coalescence and breakdown • Develop high-accuracy numerical methods for 3D flows

24. REFERENCES 1. Godunov S.K., Romenski E., Elements of continuum mechanics and conservation laws. Kluwer Academic/Plenum Publishers, New York (2003). 2. GodunovS.K., Romenski E., Thermodynamics,conservation laws and symmetric forms of differential equations inmechanics of continuous media, in Computational FluidDynamics Review 1995. John Wiley & Sons, New York, (1995), 19--31. 3. GodunovS.K. , Mikhailova T.Yu., RomenskiE.I.,Systems of thermodynamically coordinated laws of conservationinvariant under rotations. Siberian Math. J., V. 37 (1996),690--705. 4. Romensky E., Thermodynamics and hyperbolic systemsof balance laws in continuum mechanics, in GodunovMethods: theory and applications, Kluwer Academic/Plenumpublishers, (2001), 745--761. 5. Romenski E., Toro E.F., Compressibletwo-phase flow models: two-pressure models and numerical methods, Computational Fluid Dynamics Journal, V.13 (2004), 403--416. 6. Romenski E., Resnyansky A.D., Toro E.F., Conservativehyperbolic formulation for compressible two-phase flow with differentphase pressures and temperatures, Quarterly of Applied Mathematics V. 65 (2007), 259-279. 7. Romenski E., Drikakis D., Compressible two-phase flow modelling based on thermodynamically compatible systems of hyperbolic conservation laws, submitted to Int. J. for Numerical Methods in Fluids.