Some Aspects of the Godunov Method Applied to Multimaterial Fluid Dynamics

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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 2011September 5-9, 2011, Arcachon,
France
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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
Fs - numerical flux discrete analog that models
the interaction between parcels of fluid.
s
ns
j
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.
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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.

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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 bn is the total volume
fraction of the dispersive phase. a a1 an
represents the total volume of the continuous
phase or porosity, with a b 1.
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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 describe
the motion of the CP and DP components,
respectively.
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
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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
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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
S1m0 (no interaction between phases)
Stage 2. Integration of the DP-system under the
assumption that CP-parameters are frozen and
exchange term S2m0 (no interaction between
phases)
Stage 3. Integration of the full system to take
into account phase exchange term Sm
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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
b is the close-packed structure volume fraction.
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.
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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 r1, u1, p1
a2 r2, u2, p2
x
When a1a2 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
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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 X0 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 X0. The gap at this
point closely depends on the wave configurations
on the left and on the right.
C
t
W1
W2
X
2
1
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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
with W the Jacobian of the
extended flux. The matrix W has 4 eigenvalues

Corresponding eigenvectors denote
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DP
CP
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OBJECTIVE
(3/ 3)
These results tempt us to question reliability of
Powells 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.
• Mathematical model of the base flow to be
  studied.
• Results of the linear-stability analysis
  (LSA) and comparison with
• experimental data.
• Non-linear development of unstable modes found by
  the LSA.
• Summary.

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BASE FLOW MODEL
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