Some Aspects of the Godunov Method Applied to Multimaterial Fluid Dynamics Igor MENSHOV 1,2 Sergey KURATOV 2 Alexander ANDRIYASH 2 1 Keldysh Institute.

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, Russia2 VNIIA, ROSATOM Corp., Moscow, Russia

MULTIMAT 2011September 5-9, 2011, Arcachon, France

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

jn

Discrete model:

ii

dV s

dt

qFF

F- numerical flux numerical flux = discrete analog

that models the interaction between parcels of fluid.

In the Godunov method F is treated through the Riemann problem Riemann problem solution.

In this sense, it seems to be a unique method that involves the physics of the physics of the phenomenonphenomenon of interest. As for mathematics, it is rather accurate possessing thelowest level of numerical dissipationlowest level of numerical dissipation.

Our talk will concern the benefit one can gain implementing the Riemann Riemann problem solution in numerical methods for complex multi-material problem solution in numerical methods for complex multi-material simulationssimulations.

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.

PHYSICAL MODEL

The model to be considered is represented by a heterogeneous mixture ofheterogeneous mixture ofdifferent materialsdifferent materials (components). In general the components (or some of them) can be contained in two phasestwo phases: continuous (CP) and/or dispersed (DP) 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.

PHYSICAL MODEL

Mass compositionMass 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; ;

0k 0

k k k 0k 0

k k k : k k k

1= n 0= 01= n

The velocity fieldsThe velocity fields describe the motion of the CP and DP components, respectively.

and u v

EOSEOS for each CP component: ;

The specific internal energy of the CP:

0,j j je e p 0,j jp p T 0

1 1 ... , ,N N j je e e e p

Unique (mixture) EOS for CPUnique (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 , , je e p T , , , je e p

PHYSICAL MODEL

The system of governing equationsThe system of governing equations: conservation laws of mass, momentum, and energy for the CP and the DP:

kp m

kt x

q f

S S

T0 0 0 0j k j j k u E v q

k k 1 2 3f , , , - fluxes of mass, momentum, and energy;

Sp - a vector of non-conservative terms due to Archimedes force

Sm – 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.

k

p

x

Splitting the system vectors into 2 sub-vectors related to CP and DP, respectively:

1 2 k 1k 2k p 1p 2 p q q q f f f S S S, ; , ; ,yields 2 sub-systems to determine CP and DP parameters:

1 11 1

kp m

kt x

q f

S SCP: DP: 2 22 2

kp m

kt x

q f

S S

NUMERICAL MODEL

Use splitting physical processes 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 S1m=0 (no interaction between phases):

1 11 2; k

pk

constt x

q f

S q

Stage 2.Stage 2. Integration of the DP-system under the assumption that CP-parameters are frozen and exchange term S2m=0 (no interaction between phases):

2 22 1; k

pk

constt x

q f

S q

Stage 3.Stage 3. Integration of the full system to take into account phase exchange term Sm:

m

d

dt

qS

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 ( )p Stage 2.Stage 2. Special consideration at this stage is intergranular pressure . Typically it has the form of degenerative function:

0

0,

1( ) 1

1

k

if

B otherwise

0.5 0.6 0.7 0.8 0.9 1.0-200

0

200

400

600

800

1000

1200

1400

1600

sigm

a,ba

r

beta

exp model: k=0.57, B=770.4 bar, 0=0.5

Model: =B((((1-0)/(1-))**k-1)

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.

RIEMANN PROBLEM FOR POROUS MEDIUM

The system of governing CP-equations: ,t x q f g

0 0 0 the vector of conservative variables,T

u E q2

0 0 conservative part of the flux vector,T

u u p uH f

0 0 non-conservative part of the flux vector.T

p x g

We use the Godunov methodthe Godunov method to discretize in space these system of equations.The key element of this method is the solution to the Riemann problem:

x

u1, p1

u2, p2

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 termsof two fundamental solutions – shock wave and rarefaction waveshock wave and rarefaction wave. This solution denote as

1 2, , , =R R x

t q q q q

RIEMANN PROBLEM FOR POROUS MEDIUM

When , there is no simple solution; the problem becomes no simple solution; the problem becomes

more involvedmore involved because in addition to the standard wave configuration an extraan extra

stationar discontinuity arises at X=0stationar discontinuity arises at X=0 related with the jump in the momentum the momentum

flux is not longer conserved due to the skeleton reactionflux 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

1 2

RIEMANN PROBLEM FOR POROUS MEDIUM ( )

In this case we follow the idea proposed by D. Rochette et.al.(2005): extend the extend the system of equationssystem of equations to one more adding .. It leads to a system to determine the vector

0 0 0 T

u E q

that can be written in quasi-linear form with = the Jacobian of the extended flux.The matrix has 4 eigenvalues Corresponding eigenvectors denote

0,t x

q q

1 2 3 4, 0, , .u c u u c 1 2 3 4, , , andR R R R

1 2

DPCP

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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.

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BASE FLOW MODELBASE FLOW MODELBASE FLOW MODELBASE FLOW MODEL