1. Some Aspects of the Godunov Method Applied to Multimaterial Fluid Dynamics Igor MENSHOV 1,2 Sergey KURATOV 2 Alexander ANDRIYASH 2 1 Keldysh Institute for Applied Mathematics, RAS, Moscow, Russia 2 VNIIA, ROSATOM Corp., Moscow, Russia MULTIMAT 2011 September 5-9, 2011, Arcachon, France

2. WHY THE GODUNOV METHOD? Objective: Objective: Application of the Godunov approach to developing numerical models for problems of multi-material fluid dynamics, including dynamics of solids. i  j nn Discrete model: FF numerical flux F  - numerical flux = discrete analog that models the interaction between parcels of fluid. Riemann problem In the Godunov method F  is treated through the Riemann problem solution. physics of the phenomenon In this sense, it seems to be a unique method that involves the physics of the phenomenon of interest. As for mathematics, it is rather accurate possessing the lowest level of numerical dissipation lowest level of numerical dissipation. Riemann problem solution in numerical methods for complex multi-material simulations Our talk will concern the benefit one can gain implementing the Riemann problem solution in numerical methods for complex multi-material simulations.

3. OUTLINE The presentation is outlined as follows. Basic concepts of the physical model; Basic concepts of the numerical model; Riemann problem for fluid dynamics in porous medium; Riemann problem for granular (dispersed phase) flow; Motion of solids.

4. PHYSICAL MODEL heterogeneous mixture of The model to be considered is represented by a heterogeneous mixture of different materials two phasescontinuous (CP) and/or dispersed (DP) different materials (components). In general the components (or some of them) can be contained in two phases: continuous (CP) and/or dispersed (DP). Each CP component occupies a part of the domain; its distribution is described by the volume fraction  k ; k = 1,…, n, where n is the number of components. The DP component is characterized by the volume fraction  k, k = 1,…, n. The quantity  =  1 +···+  n is the total volume fraction of the dispersive phase.  =  1 +···+  n represents the total volume of the continuous phase or porosity, with  +  = 1.

5. PHYSICAL MODEL Mass composition Mass composition: DP: = density of a DP component, = average density, CP: = density of a CP component, = average density; averaged over porosity density - bulk density of the CP; ; The velocity fields The velocity fields describe the motion of the CP and DP components, respectively. EOS EOS for each CP component: ; The specific internal energy of the CP: Unique (mixture) EOS for CP Unique (mixture) EOS for CP. Assuming pressure p and temperature T to be the same for CP components, thermodynamics of the CP mixture is described by an unique EOS: and

6. PHYSICAL MODEL The system of governing equations The system of governing equations: conservation laws of mass, momentum, and energy for the CP and the DP: - fluxes of mass, momentum, and energy; S p - a vector of non-conservative terms due to Archimedes force S m – a vector of the interaction between CP and DP that models mass exchange (fragmentation of the CP or defragmentation of the DP), momentum exchange (drag forces), and energy exchange. Splitting the system vectors into 2 sub-vectors related to CP and DP, respectively: yields 2 sub-systems to determine CP and DP parameters: CP:DP:

7. NUMERICAL MODEL splitting physical processes Use splitting physical processes to divide the problem into more simple sub-problems. This is done in 3 stages (for each time step). Stage 1. Stage 1. Integration of the CP-system under the assumption that DP-parameters are frozen and exchange term S 1m =0 (no interaction between phases): Stage 2. Stage 2. Integration of the DP-system under the assumption that CP-parameters are frozen and exchange term S 2m =0 (no interaction between phases): Stage 3. Stage 3. Integration of the full system to take into account phase exchange term S m :

8. NUMERICAL MODEL What should be paid attention to considering discretization of the above equations? Stage 1. Stage 1. This is a typical problem of the flow in porous medium. The DP components make up a fixed in space granular skeleton, which the CP components move through. The porosity of this skeleton given by  has non-uniform in space distribution and might be in general discontinuous. The main issue should be paid attention to at this stage is how to treat a non-conservative term Stage 2. Stage 2. Special consideration at this stage is intergranular pressure . Typically it has the form of degenerative function: System of characteristics degenerates. The question is how to account for this peculiarity of the DP equations  * is the close-packed structure volume fraction. We solve these 2 issues by means of the solutions to appropriate Riemann problems.

9. RIEMANN PROBLEM FOR POROUS MEDIUM The system of governing CP-equations: the Godunov method We use the Godunov method to discretize in space these system of equations. The key element of this method is the solution to the Riemann problem: x      u 1, p 1      u 2, p 2 When  1 =  2 the equations are reduced to the standard gas dynamics equations. The Riemann problem solution is formulated and sought in this case in terms shock wave and rarefaction wave of two fundamental solutions – shock wave and rarefaction wave. This solution denote as

10. RIEMANN PROBLEM FOR POROUS MEDIUM no simple solution; the problem becomes When, there is no simple solution; the problem becomes more involvedan extra more involved because in addition to the standard wave configuration an extra stationar discontinuity arises at X=0the momentum stationar discontinuity arises at X=0 related with the jump in  the momentum flux is not longer conserved due to the skeleton reaction flux is not longer conserved due to the skeleton reaction, so that there is always discontinuity at X=0. The gap at this point closely depends on the wave configurations on the left and on the right. X t W2 1 2 W1 C

11. RIEMANN PROBLEM FOR POROUS MEDIUM ( ) extend the system of equations In this case we follow the idea proposed by D. Rochette et.al.(2005): extend the system of equations to one more adding.. It leads to a system to determine the vector that can be written in quasi-linear form with  = the Jacobian of the extended flux. The matrix  has 4 eigenvalues Corresponding eigenvectors denote

12. DPCP

13. 13 OBJECTIVE (3/ 3) These results tempt us to question reliability of Powell’s theory and put forward an alternative hypothesis concerning the screech mechanism: Jet screech = Sound associated with helical instability The purpose of the present paper is to investigate flow stability in a simple model of the real jet flow. Our presentation is outlined as follows. 1) Mathematical model of the base flow to be studied. 2) Results of the linear-stability analysis (LSA) and comparison with experimental data. 3)Non-linear development of unstable modes found by the LSA. 4)Summary.

14. 14 BASE FLOW MODEL