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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
WHY THE GODUNOV METHOD?
Objective: Application of the Godunov approach to developing numerical
models for problems of multi-material fluid dynamics, including dynamics of
solids.
Discrete model:
Fs
i
s
ns
Fs - numerical flux = discrete analog
j
that models the interaction between
parcels of fluid.
In the Godunov method Fs is treated
through the Riemann problem solution.
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.
Our talk will concern the benefit one can gain implementing the Riemann problem
solution in numerical methods for complex multi-material simulations.
OUTLINE
The presentation is outlined as follows.
пѓј Basic concepts of the physical model;
пѓј Basic concepts of the numerical model;
пѓј Riemann problem for fluid dynamics in porous medium;
пѓј Riemann problem for granular (dispersed phase) flow;
пѓј Motion of solids.
PHYSICAL MODEL
The model to be considered is represented by a heterogeneous mixture of
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 ak;
k = 1,…, n, where n is the number of components.
The DP component is characterized by the volume
fraction bk, k = 1,…, n.
The quantity b= b1 +В·В·В·+ bn is the total volume
fraction of the dispersive phase.
a= a1 +В·В·В·+ an represents the total volume of the
continuous phase or porosity, with a + b = 1.
PHYSICAL MODEL
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
components, respectively.
;
00
describe the motion of the CP and DP
EOS for each CP component:
;
The specific internal energy of the 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
PHYSICAL MODEL
The system of governing equations: conservation laws of mass, momentum, and
energy for the CP and the DP:
- 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.
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:
NUMERICAL MODEL
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. Integration of the CP-system under the assumption that DP-parameters are
frozen and exchange term S1m=0 (no interaction between phases):
Stage 2. Integration of the DP-system under the assumption that CP-parameters are
frozen and exchange term S2m=0 (no interaction between phases):
Stage 3. Integration of the full system to take into account phase exchange term Sm:
NUMERICAL MODEL
What should be paid attention to considering discretization of the above equations?
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 a 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. Special consideration at this stage is
intergranular pressure s. Typically it has the form of
degenerative function:
1600
exp
model: k=0.57, B=770.4 bar, b0=0.5
1400
1200
sigma,bar
1000
Model: s=B((((1-b0)/(1-b))**k-1)
800
600
400
200
b* is the close-packed structure volume fraction.
0
-200
0.5
0.6
0.7
0.8
0.9
beta
System of characteristics degenerates. The question is how to account for this
peculiarity of the DP equations
We solve these 2 issues by means of the solutions to appropriate Riemann problems.
1.0
RIEMANN PROBLEM FOR POROUS MEDIUM
The system of governing CP-equations:
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:
a1
пЃІ1 , u 1 , p 1
a2
пЃІ2 , u 2 , p 2
x
When a1=a2 the equations are reduced to the standard gas dynamics equations.
The Riemann problem solution is formulated and sought in this case in terms
of two fundamental solutions – shock wave and rarefaction wave.
This solution denote as
RIEMANN PROBLEM FOR POROUS MEDIUM
When
, there is no simple solution; the problem becomes
more involved because in addition to the standard wave configuration an extra
stationar discontinuity arises at X=0 related with the jump in s; the momentum
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.
t
C
W1
W2
X
1
2
RIEMANN PROBLEM FOR POROUS MEDIUM (
)
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
of the extended flux.
The matrix W has 4 eigenvalues
Corresponding eigenvectors denote
with W = the Jacobian
CP
DP
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.
13
BASE FLOW MODEL
14
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