Description and Analysis of a Multiphase Model of Saurel and Abgrall
This document describes a multiphase model proposed by Saurel and Abgrall in the
article “A Multiphase Godunov Method for Compressible Multifluid and Multiphase
flows” [1]. The model is based on the formal theory of Drew and Passman [2] for a fluid
with individual densities, pressures, and velocities for each fluid species in the mixture.
The Saurel and Abgrall model is based on pressure and velocity relaxation to equilibrium
values. The model imposed conservation of individual species mass, total momentum,
and total energy. The description here is based on a simple generalization of the model
discussed in [1] to multiple space dimensions and more than two fluid species.
Notation
The basic notation for the description of the fluid motion is given in Table 1. The flow
consists of a mixture of N separate fluid species.
Table 1
Variable
N
d
t
xi , 1 i  d
Description
Number of fluid species.
Spatial dimension.
The solution time.
The spatial coordinate in direction i .
Volume fraction of fluid species k. In
Units
None
None
Time
Length
N
k , 1  k  N
general 0  k  1 and  k  1 . Here we
k 1
assume the flow is saturated so
None
N
that  k  1 .
k 1
N
A
A=


k
=1
k , 1  k  N
k
Mass fraction of fluid species k .
None
None
N
w
w   k
None
k , 1  k  N
Micro-density of fluid species k. The
quantity k is defined as the ratio of the
mass of species k and the volume
occupied by species k . It is assumed to
be non-negative.
Mass/Volume
k 1
1
Variable
Vk , 1  k  N
Description
The thermodynamic specific volume of
species k given byV k  1  .
Units
Volume/Mass
k
Total density of the fluid
N
mixture   k k . The volume
k 1

fractions and mass fractions are related
by the formula k k  k  so that
Mass/Volume
N
0  k  1 and  k  1.
k 1
vk , 1  k  N
v k2
v ki , 1  k  N , 1  i  d
Velocity of species k .
Square of the velocity of species k
defined by v k2  v k  v k
The i th spatial component of the velocity
vector v k .
The center-of-mass velocity
Velocity
Energy/Mass
Velocity
N
 v  k k v k or
v
k 1
Velocity
N
equivalently v   k v k .
k 1
v i , 1 i  d
 vk , 1  k  N
v ki , 1  k  N , 1  i  d
bk , 1  k  N
bki , 1  k  N , 1  i  d
The i spatial component of the velocity
vector v .
The relative velocity or slip of species k
defined by  v k  v k  v .
th
The i spatial component of the velocity
slip vector  v k .
The body force per unit mass acting on
species k .
th
The i spatial component of the body
force per unit mass of vector b k .
The center-of-mass body force per unit
th
Velocity
Velocity
Velocity
Force/Mass
Force/Mass
N
mass b  k k bk or
b
k 1
Force/Mass
N
equivalently b   k bk .
k 1
bi , 1 i  d
The i spatial component of the body
force per unit mass vector b .
The energy body force due to fluid
th
Force/Mass
N
r
interaction given by r  k k  v k  bk
k 1
N
or equivalently r  k k v kj bkj .
k 1
2
Work/Mass
Variable
Pk , 1  k  N
Description
The thermodynamic pressure of
species k .
The volume averaged
N
P
pressure P  k Pk .
Units
Pressure
Pressure
k 1
σ   ij

The composite Cauchy stress for the
mixture defined by
N
 ij  P  ij  k k v ki v kj .
Pressure
k 1
Tk , 1  k  N
The pressure fluctuation of species k ,
defined by  Pk  Pk  P .
The thermodynamic specific internal
energy of species k .
The temperature of species k .
Sk , 1  k  N
The specific entropy of species k .
ck , 1  k  N
k , 1  k  N
The sound speed of species k .
The Grüneisen exponent of species k .
The mass averaged specific internal
 Pk , 1  k  N
ek , 1  k  N
Pressure
Energy/Mass
Temperature
Energy/Mass/
Temperature
Velocity
None
N
energy e  k k ek or
e
k 1
Energy/Mass
N
equivalently e   k ek .
k 1
The composite material specific internal
energy given by
N
e  e  21 k k  v k   v k or
e
k 1
Energy/mass
N
equivalently e  e  21 k k v ki v ki .
k 1
Hk , 1  k  N
The specific enthalpy for species k
given by H k  ek  PkVk .
The heat flux due to fluid interaction
defined
Energy/mass
N
q  q j

by q j  k k( H k  21  v k   v)k v kj
k 1
or equivalent
Energy*Velocity/
Volume
N
q j  k k( H k  21 v ki v )ki v kj
k 1
Tk , 1  k  N
Sk , 1  k  N

The thermodynamic temperature of
species k .
The thermodynamic specific entropy of
species k .
The dynamic compaction viscosity.
3
Temperature
Entropy
Energy/Temperature
Time*Length/Mass
Variable
Description
Velocity relaxation coefficient.

Units
Mass/(Volume*Time)
Equations of Motion
The equations of motion are based on the conservation of individual species mass,
conservation of total momentum and energy, and relaxation to uniform strain. The system
is given by
 k

 v  ∇ k   k  Pk
t
A
 k k
 ∇   k  k v k   0
t
 k k v k

 ∇   k k v k  v k   ∇  k Pk    k  v k  P ∇ k   k k b k
t
W

  k k e k  21 v k2 
t
∇ 

k

k v k e k  21 v k2   ∇   k Pk v k   
P v  ∇ k   k k v k  b k .
k
W
v   vk  
k
A
(1)
P  Pk 
It is sometimes convenient to write this system using index notation (with no summing
over the species index k)
 k
 k

v j
  k  Pk
t
x j
A
 k k  k kv kj

0
t
x j
 k kv ki  k kv ki v kj k Pk

 k


  k v ki  P
  k k bki
t
x j
x i
W
x i

  k k e k  21 v ki v ki 
t
  
k
kv kj e k  21 v ki v ki 
x j
Pv j

 k Pkv kj


  k v j v kj   k P  Pk 
x j
W
A
 k
  k kv kj bkj .
x j
Summing the advection equation for the volume fractions over all species we obtain
4
(2)
N
   k
 k 1
t


N
  v ∇    0 ,
 k 
 k 1 
(3)
N
since by definition   Pk  0 . Thus is the flow is initially saturated it will remain
k 1
N
saturated for all time. In the following we thus assume that  k  1 .
k 1
Summing the second equation corresponding to mass conservation for systems (1) or (2)
we obtain the equation for conservation of total mass

 ∇   v   0 .
t
(4)
This in summation form becomes
 kv

t
x j
j
0.
(5)
The conservation of total momentum equation is obtained by summing the third
equations in systems (1) and (2):
 v
 ∇    v  v   ∇  σ  b
t
(6)
v i v iv j  ij


 b i .
t
x j
x j
(7)
or in summation form:
The conservation of total energy law is obtained by added the fourth equations from
systems (1) and (2).
 e  21 v  v 
 ∇   v e  21 v  v   ∇  σv  ∇  q   v  b  r
t
(8)
or in summation form
 e  21v jv
t
j
  v e 
i
x i
1
2
v jv j 
5

 ijv
x i
j

q i
 v j b j  r .
x i
(9)
It is important to note that even when each individual species is inviscid, the composite
flow has a non-isotropic stress tensor and non-trivial heat conduction due to the
interactions between species at the subgrid level.
Characteristic Analysis
The system (1) consists of a set of N *(3  d) coupled partial differential equations in the
N *(4  d) unknowns k , k ,v ki 1  i  d ,ek , and Pk . The system is closed through the
individual species thermodynamic equations of state ek  ek Vk ,Sk  where Vk  1/ k is the
specific volume, and S k is the specific entropy of species k . The pressure and
temperature of the species are then given byTk 
e k
S k
,Pk  
Vk
e k
V k
. This system consists
Sk
of blocks of equations corresponding to the propagation of volume fractions and each
fluid species that are only coupled via source terms, thus we can perform the
characteristic analysis on each block separately.
The first block of equations corresponding to the time advection of the volume fraction
k
k

v j
  k  Pk
t
x j
A
(10)
is clearly in characteristic form so that the composite velocity field form a multiplicity
N linearly degenerate characteristic field.
The derivation of the remaining characteristics proceeds in the usual manner. First it is
convenient to rewrite equation (10) in the form
k
k
k

 v kj
 v kj
  k  Pk
t
x j
x j
A
(11)
which emphasizes that changes in the k th species volume fraction are driven by velocity
and pressure fluctuations from the mean values. Combining equation (11) with the
continuity equation for that species we can derive the k th species specific volume update
equation
 Vk
k k 
 t
 v kj
Vk
x j


v kj
j k
  k  Pk  k
  v k
x j
A
x j

that reduces to the standard equality between the velocity divergence and logarithm
derivative of specific volume as the velocity and pressure fluctuations approach zero.
6
(12)
Expanding out the momentum species equations we can derive
 v ki
k k 
 t
 v kj
v ki
x j

P
k

  k v ki  k k bki
  k k   Pk

x

x
W
i
i

(13)
which again reduces to the standard single component equations as the pressure and
velocity fluctuations approach zero.
A somewhat long but straightforward computation using the energy equations and
equations (11), (12), and (13) allow us to derive the entropy equation
 Sk
k kTk 
 t
 v kj
Sk
x j

   Pk

 k
k 

j
j
 v kj

   k v v k

t

x
W
j 

(14)
We observe that equation (14) is in characteristic form so that each of the N species
velocities corresponds to a linearly degenerate characteristic field. We also observe that it
is fluctuations in pressure and velocity that break the thermal isolation of a fluid species.
If we use the thermodynamic relation dVk  
1
k c k
2
2
dPk 
k
k c k2
Tk dSk to replace the specific
volume derivative with pressure and entropy derivatives in equation (12) and use
equation (14) to replace entropy derivatives with volume fraction derivatives, we obtain
 Pk
 v kj
k 
 t
Pk
x j
j

 
k 

2 v k
j
j
  k c k2  k  Pk   k  v kj
  k k c k
  k  k v v k .
 t

x

x
W
j
j 


(15)
Combining equations(10),(13),(14), and (15) we get the quasi-linear system
 k
 k

v j
  k  Pk
t
x j
A
 S k
  k
 k 

j
j
 v kj

   k v v k

t

t

x
W
j



j
 Pk

 
Pk
 k
2 v k
 k 
 v kj
  k c k2  k  Pk   k  v kj
   k k c k
 t

t

x

x
x j
j 
j


 v i
v i 
P


 k k  k  v kj k    k k   Pk k   k v ki   k k bki .
x j 
x i
x i
W
 t
 k kTk 
 v kj
S k
x j

   Pk



j
j
  k  k v v k
W

(16)
System (16) is block structured for each material species, so the characteristics for the
complete multi-material equation consist of the union of the characteristics for each
block. Given a spatial direction ξ  1, ,d  and a characteristic speed  the symbol for
the system consists of a block diagonal matrix formed by the single species symbol given
by the 3  d   3  d  matrix
7
0
0
0

  v  ξ 

 Pk    v k  ξ 
 k kTk    v k  ξ 
0
0

Mk  
2
 c  k  Pk     v k  ξ 
0
k    v k  ξ 
 k k c k2 ξ
 k k

 Pk ξT
0d d
 k ξT
 k k    v k  ξ  Id d




 (17)



in terms of the state representation Sk  k ,Sk ,Pk , vTk  . It is easy to shown that
T
d
2
det Mk   k3 k2Tk    v  ξ     v k  ξ     v k  ξ   c k2ξ  ξ  and thus the total system is


hyperbolic with characteristic speeds   v  ξ of multiplicity N ,   v k  ξ of multiplicity
d for 1  k  N and   vk  ξ  ck ξ , 1  k  N .
The eigenvectors correspond to the characteristic field corresponding to the center of
mass velocity are then given by
  v ξ
r    v k  ξ 

2
  k kTk
 P
2
k
 c k2 ξ  
   kTk  Pk

0d



   v  ξ    c 2     1  P T
k
k
k  k
 k k



l  1 0 0 0d 
0




0


  k   v k  ξ  


ξT


(18)
In order to determine the characteristic type of this field (genuinely nonlinear or linearly
degenerate) we need to compute the derivative of the composite velocity with respect to
N
N
k 1
k 1
the state variable. Using   k k ,  v  k k , and k  k Pk ,Sk  we find that
v  ξ
 k
v  ξ
S k
v  ξ
Pk
v  ξ
v k
k
 v k  ξ 

T 
  k k2 k  v k  ξ 
ck

 k 2  v k  ξ 
k c k

(19)
 k ξ
Using (18) and (19) you can then calculate that for the center of mass velocity
field r    0 so that this characteristic family is linearly degenerate.
Just as for the single material Euler equations, we see that the eigenvectors corresponding
to the velocity of the k th material field are given by
8
  vk  ξ
 0   0  
    
 1 0 
r  span   ,   , ξη  0
 0   0  
 0   η  



l  span  0 1 0 0  ,  0 0 0 ηT
(20)
, ξη  0
Since
v k  ξ
 k
v k  ξ
S k
v k  ξ
Pk
v k  ξ
v k
0
0
(21)
0
ξ
we easily check that this characteristic family is also linearly degenerate.
Finally for the k th material sonic wave families with speeds   vk  ξ  ck ξ we get
eigenvectors
  vk  ξ  ck ξ
0




0


r
  k c k ξ 


T
 ξ

2

2  c    k  1  Pk
l   ck ξ k k
 vk  ξ  ck ξ

(22)
0
ξ

c k ξ 

Computing
  vk  ξ  ck ξ 
 k
  vk  ξ  ck ξ 
S k
  vk  ξ  ck ξ 
Pk
  vk  ξ  ck ξ 
v k
0

c k
S k
c
 k
Pk
ξ
9
ξ
Pk
ξ 
Sk
Gk  1
k c k
(23)
ξ
where Gk is the fundamental derivative of gas dynamics [3, 4] for the k th material field
and we compute for this family r    Gk ξ which will be genuinely non-linear for nonzero Gk .
1.
Saurel, R. and R. Abgrall, A Multiphase Godunov Method for Compressible
Multifluid and Multiphase Flows. Jour. Comp. Phys, 1999. 150: p. 425-467.
2.
Drew, D. and S. Passman, Theory of Multicomponent Fluids. 1999, New York:
Springer-Verlag.
3.
Thompson, P., Compressible-Fluid Dynamics. 1972, New York: McGraw-Hill.
4.
Menikoff, R. and B. Plohr, The Riemann Problem for Fluid Flow of Real
Materials. Rev. Mod. Phys., 1989. 61: p. 75-130.
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