Block (2D Space-Time Compressible NS) Derivation:
I. Coefficient Matrices
M. Behr
http://www.cats.rwth-aachen.de/library/research/technotes
Abstract
This note describes the derivation of the generalized advective-diffusive equation system
from the 2D space-time compressible Navier-Stokes governing equations, in conservation form.
1. Introduction
Starting with the governing equations, we derive the coefficient matrices used to express
these equations as a generalized advective-diffusion system. Large portion of the material
follows Aliabadi [1] and Mittal and Tezduyar [2]. See also a separate technical note:
http://www.cats.rwth-aachen.de:8080/pdf/tn-blkcns3dst-derivation.pdf
which describes the derivation of the generalized advective-diffusive equation system from
the 3D space-time compressible Navier-Stokes equations.
2. Governing Equations
Let Ωt ⊂ R2 and (0, T ) be the spatial and temporal domains respectively, and let Γt denote
the boundary of Ωt . The spatial and temporal coordinates are denoted by x and t. The
Navier-Stokes equations of compressible flows in conservation form are:
∂ρ
+ ∇ · (ρu) = 0
∂t
∂ (ρu)
+ ∇ · (ρuu) + ∇p − ∇ · T = 0
∂t
∂ (ρe)
+ ∇ · (ρeu) + ∇ · (pu) − ∇ · (Tu) + ∇ · q = 0
∂t
on Ωt
∀t ∈ (0, T ),
(1)
on Ωt
∀t ∈ (0, T ),
(2)
on Ωt
∀t ∈ (0, T ).
(3)
Here ρ(x, t), u(x, t), p(x, t), T(x, t), e(x, t), and q(x, t) are the density, velocity, pressure,
viscous stress tensor, total energy per unit mass, and the heat flux vector, respectively. The
viscous stress tensor is defined as:
T = µ ∇u + (∇u)T + λ (∇·u) I,
(4)
where µ and λ are the viscosity coefficients, assumed to be related by:
2
λ = − µ.
3
(5)
Block (2D Space-Time Compressible NS) Derivation I
M. Behr
Pressure is related to the other variables via the equation of state. For ideal gases, the
equation of state assumes the special form:
p = (γ − 1)ρi,
(6)
where γ is the ratio of specific heats and i is the internal energy per unit mass. The internal
energy is related to the total energy and kinetic energy as:
1
i = e − u2 .
2
(7)
q = −κ∇θ,
(8)
The heat flux vector is defined as
where κ is the heat conductivity and θ is the temperature. The temperature is related to
the internal energy with the following expression:
θ=
i
,
Cv
(9)
where Cv is the specific heat of the fluid at constant volume. For an ideal gas
Cv =
R
,
γ−1
(10)
where R is the ideal gas constant. Prandtl number (Pr), assumed to be specified, relates the
heat conductivity of the fluid to its viscosity according to:
κ=
µCp
,
Pr
(11)
where Cp is the specific heat of the fluid at constant pressure. For an ideal gas
Cp =
γR
.
γ−1
(12)
Equations (1), (2), and (3) may be written in conservation variables as:
∂U ∂Fi ∂Ei
+
−
=0
∂t
∂xi
∂xi
on Ωt
∀t ∈ (0, T ),
(13)
where U = (ρ, ρu1 , ρu2 , ρe) is the vector of conservation variables, and Fi and Ei are,
respectively, the Euler and viscous flux vectors defined as:


ui ρ
 uiρu1 + δi1 p 

(14)
Fi = 
 uiρu2 + δi2 p  ,
ui (ρe + p)
2
Block (2D Space-Time Compressible NS) Derivation I

0


τi1
.
Ei = 


τi2
−qi + τik uk
M. Behr

(15)
Here ui, qi , and τik are the components of the velocity, heat flux, and viscous stress tensor,
respectively. Equation (13) can be re-written as
∂U
∂U
∂U
∂
+ Ai
−
(16)
Kij
=0
on Ωt ∀t ∈ (0, T ),
∂t
∂xi ∂xi
∂xj
where
∂Fi
∂U
is the Euler Jacobian matrix, and Kij is the diffusivity matrix satisfying:
Ai =
Kij
∂U
= Ei .
∂xj
(17)
(18)
Corresponding to Equation (16), the boundary and initial conditions are represented as:
U = g on Γg ∀t ∈ (0, T ),
n · E = h on Γh ∀t ∈ (0, T ),
U(x, 0) = U0 on Ω0 .
(19)
(20)
(21)
3. Coefficient Matrices
The coefficient matrices of the generalized advective-diffusive system (16) use the following
additional parameters for simplicity:
γ̄ = γ − 1,
κ
,
µ̄ = µ −
Cv
c1 = γ̄u2 − γe,
γ̄
c2 = γe − u2 ,
2
κ 1
2
u − i − µu2
c3 =
Cv 2
κ u2 − e − µu2 .
=
Cv
3
(22)
(23)
(24)
(25)
(26)
(27)
Block (2D Space-Time Compressible NS) Derivation I
M. Behr
In order to derive the matrices A1 and A2 one has to express the fluxes F1 and F2 solely in
terms of the independent variables U:


u1 ρ
 u ρu + p 

 1 1
F1 = 
(28)


 u1 ρu2



= 




= 

u1 (ρe + p)
u1 ρ
u1 ρu1 + γ̄ρ e − 12 u2
u ρu
1 12
u1 ρ γe − 2 γ̄u2





(ρu1 )
2
(ρu1 ) /(ρ) + γ̄(ρe) − γ̄2 (ρu1 )2 /(ρ) − γ̄2 (ρu2 )2 /(ρ)
(ρu1 )(ρu2 )/(ρ)
γ̄
γ(ρu1 )(ρe)/(ρ) − 2 (ρu1 )3 /(ρ)2 − γ̄2 (ρu1 )(ρu2 )2 /(ρ)2
(29)



.

(30)
Similarly,



F2 = 




= 

u2 ρ
u2 ρu1
u2 ρu2 + p
u2 (ρe + p)





(31)
(ρu2 )
(ρu2 )(ρu1 )/(ρ)
(ρu2 )2 /(ρ) + γ̄(ρe) − γ̄2 (ρu1 )2 /(ρ) − γ̄2 (ρu2 )2 /(ρ)
γ(ρu2 )(ρe)/(ρ) − γ̄2 (ρu2 )(ρu1 )2 /(ρ)2 − γ̄2 (ρu2 )3 /(ρ)2
The advective coefficient matrices can now be clearly derived as:

0
1
0
0
 γ̄ ||u||2 − u u
(3 − γ)u1
−γ̄u2
γ̄
1 1

A1 =  2

−u1 u2
u2
u1
0
c2 − γ̄u1 u1
−γ̄u1 u2
γu1
c1 u 1

0
0
1
0

−u2 u1
u2
u1
0

A2 =  γ̄
2
 2 ||u|| − u2 u2
−γ̄u1
(3 − γ)u2
γ̄
−γ̄u2 u1
c2 − γ̄u2 u2
γu2
c1 u 2
4



.

(32)



,

(33)



.

(34)
Block (2D Space-Time Compressible NS) Derivation I
M. Behr
The diffusive coefficient matrices are obtained by expanding the diffusive flux terms in terms
of derivatives of the variables U:


0


τ11


E1 = 

τ12


−q1 + τ11 u1 + τ12 u2


0


1
1
2


+ λ ∂u
+ ∂u
2µ ∂u
∂x
∂x
∂x
1
1
2


= 

∂u2
1

+
µ ∂u

∂x
∂x
2
1


∂u1
∂u1
∂u2
∂u1
∂u2
∂θ
κ ∂x1 + 2µ ∂x1 u1 + λ ∂x1 + ∂x2 u1 + µ ∂x2 + ∂x1 u2


0
4µ ∂u1
∂u2


− 2µ


3 ∂x1
3 ∂x2
= 

∂u1
∂u2
µ
+
µ


∂x2
∂x1
4µ
2µ
∂u1
∂u2
∂u1
∂u2
κ ∂e
κ ∂u2
− 2Cv ∂x1 + 3 u1 ∂x1 − 3 u1 ∂x2 + µu2 ∂x2 + µu2 ∂x1
Cv ∂x1


0
∂((ρu2 )/(ρ))
4µ ∂((ρu1 )/(ρ))


− 2µ


3
∂x1
3
∂x2


∂((ρu2 )/(ρ))
µ ∂((ρu∂x1 )/(ρ))
+
µ


∂x
2
1

2 /(ρ)2
2 /(ρ)2
= 
∂
(ρu
)
∂
(ρu
)
( 1
)
( 2
)


κ ∂((ρe)/(ρ))
κ
κ
−
−


Cv
∂x1
2Cv
∂x1
2Cv
∂x1

∂((ρu1 )/(ρ))
∂((ρu2 )/(ρ)) 
2µ


. . . + 4µ
u
−
u
3 1
∂x1
3 1
∂x2
∂((ρu1 )/(ρ))
∂((ρu2 )/(ρ))
. . . + µu2
+ µu2
∂x2
∂x1


0


∂(ρu2 )
4µ ∂(ρu1 )
− 4µ
u ∂(ρ) − 2µ
+ 2µ
u ∂(ρ)


3 ∂x1
3 1 ∂x1
3 ∂x2
3 2 ∂x2


∂(ρu1 )
∂(ρu2 )
∂(ρ)
∂(ρ)

µ
−
µu
+
µ
−
µu
1
1
2
∂x
∂x
∂x
∂x
2
2
1
1


=
∂(ρu1 )
∂(ρu2 )
κ ∂(ρe)
κ ∂(ρ)
κ
κ
κ
2 ∂(ρ) 

ρ  Cv ∂x1 − Cv e ∂x1 − Cv u1 ∂x1 − Cv u2 ∂x1 + Cv u ∂x1 


. . . + 4µ
u ∂(ρu1 ) − 4µ
u2 ∂(ρ) − 2µ
u ∂(ρu2 ) + 2µ
u u ∂(ρ)

3 1 ∂x1
3 1 ∂x1
3 1 ∂x2
3 1 2 ∂x2 
1)
2)
. . . + µu2 ∂(ρu
− µu1u2 ∂(ρ)
+ µu2 ∂(ρu
− µu22 ∂(ρ)
∂x2
∂x2
∂x1
∂x1


0
∂(ρu2 )
4µ ∂(ρu1 )


− 4µ
u ∂(ρ) − 2µ
+ 2µ
u ∂(ρ)


3 ∂x1
3 1 ∂x1
3 ∂x2
3 2 ∂x2


∂(ρ)
∂(ρ)
∂(ρu1 )
∂(ρu2 )
µ ∂x2 − µu1 ∂x2 + µ ∂x1 − µu2 ∂x1 
1


∂(ρe)
∂(ρ)
µ
=
(35)
κ
κ

,
2
2
+
(u
−
e)
−
µu
−
u
u
1
1


ρ
Cv ∂x1
Cv
3
∂x1


∂(ρu1 )
∂(ρu2 ) 
4µ
κ
κ

.
.
.
+
−
+
µ
−
u
u
1 ∂x1
2 ∂x1


3
Cv
Cv
∂(ρu2 )
∂(ρ)
∂(ρu1 )
2µ
µ
. . . − 3 u1 ∂x2 − 3 u1 u2 ∂x2 + µu2 ∂x2
with terms coded blue-green contributing to the K11 coefficient matrix, and terms coded
purple contributing to the K12 matrix.
5
Block (2D Space-Time Compressible NS) Derivation I
Similarly, second diffusive flux can be expanded as:


0


τ21


E2 = 

τ22


−q2 + τ21 u1 + τ22 u2


0


2
1


+ ∂u
µ ∂u
∂x1
∂x2


= 

∂u2
∂u1
∂u2
2µ ∂x2 + λ ∂x1 + ∂x2




∂u2
∂u1
∂u2
∂u1
∂u2
∂θ
+
µ
+
+
2µ
u
+
λ
+
κ ∂x
u
u
1
2
∂x1
∂x2
∂x2 2
∂x1
∂x2
2


0


2
1
+ µ ∂u
µ ∂u


∂x1
∂x2
= 

4µ ∂u2
2µ ∂u1
−


3 ∂x2
3 ∂x1
2
4µ
2µ
∂u2
∂u1
∂u2
∂u1
κ ∂e
κ ∂u
− 2Cv ∂x2 + µu1 ∂x1 + µu1 ∂x2 + 3 u2 ∂x2 − 3 u2 ∂x1
Cv ∂x2


0


µ ∂((ρu∂x2 )/(ρ))
+ µ ∂((ρu∂x1 )/(ρ))


1
2


)/(ρ))
)/(ρ))
∂((ρu
∂((ρu
4µ
2µ
2
1
− 3


3
∂x
∂x
2
1

2 /(ρ)2
2 /(ρ)2
= 
∂
(ρu
)
∂
(ρu
)
(
)
(
)
1
2


κ ∂((ρe)/(ρ))
κ
κ
−
−


Cv
∂x2
2Cv
∂x2
2Cv
∂x2

∂((ρu2 )/(ρ))
∂((ρu1 )/(ρ)) 


. . . + µu1
+ µu1
∂x1
∂x2
∂((ρu2 )/(ρ))
∂((ρu1 )/(ρ))
4µ
2µ
. . . + 3 u2
− 3 u2
∂x2
∂x1


0


2)
1)
− µu2 ∂(ρ)
+ µ ∂(ρu
− µu1 ∂(ρ)
µ ∂(ρu


∂x1
∂x1
∂x2
∂x2


∂(ρu2 )
∂(ρu1 )
∂(ρ)
∂(ρ)
4µ
4µ
2µ
2µ

− 3 u2 ∂x2 − 3 ∂x1 + 3 u1 ∂x1
1
3 ∂x2


=
∂(ρu1 )
∂(ρu2 )
κ ∂(ρe)
κ ∂(ρ)
κ
κ
κ
2 ∂(ρ) 

ρ  Cv ∂x2 − Cv e ∂x2 − Cv u1 ∂x2 − Cv u2 ∂x2 + Cv u ∂x2 


2)
1)
. . . + µu1 ∂(ρu
− µu1u2 ∂(ρ)
+ µu1 ∂(ρu
− µu21 ∂(ρ)

∂x1
∂x1
∂x2
∂x2 
. . . + 4µ
u ∂(ρu2 ) − 4µ
u2 ∂(ρ) − 2µ
u ∂(ρu1 ) + 2µ
u u ∂(ρ)
3 2 ∂x2
3 2 ∂x2
3 2 ∂x1
3 1 2 ∂x1


0
2)
1)


− µu2 ∂(ρ)
+ µ ∂(ρu
− µu1 ∂(ρ)
µ ∂(ρu


∂x1
∂x1
∂x2
∂x2


∂(ρ)
∂(ρ)
4µ ∂(ρu2 )
4µ
2µ ∂(ρu1 )
2µ
− 3 u2 ∂x2 − 3 ∂x1 + 3 u1 ∂x1

3
∂x
2
1


∂(ρ)
µ
=
κ ∂(ρe)
κ

,
2
2
+
(u
−
e)
−
µu
−
u
u
2
2

ρ
Cv ∂x2
Cv
3
∂x2


∂(ρu1 )
∂(ρu2 ) 
4µ
κ
κ

.
.
.
+
µ
−
+
−
u
u
1
2


Cv
∂x2
3
Cv
∂x2
∂(ρu2 )
∂(ρ)
∂(ρu1 )
µ
2µ
. . . + µu1 ∂x1 − 3 u1 u2 ∂x1 − 3 u2 ∂x1
M. Behr
(36)
with terms coded blue-green contributing to the K21 coefficient matrix, and terms coded
purple contributing to the K22 matrix.
6
Block (2D Space-Time Compressible NS) Derivation I
M. Behr
The diffusive coefficient matrices can now be derived as:

0
0
0
4µ
4µ

0
1  − 3 u1
3
K11 = 
ρ
−µu2
0
µ
µ
µ
µ̄u2
c3 − 3 u1 u1 3 + µ̄ u1

K12 =
1


ρ

K21 =
1


ρ

K22 =
1


ρ
0
0
0
0
0
µ
µu2
0
− 2µ
3
0
2µ
− 3 u1
0
0
0
0
0
−µu2
2µ
u
3 1
µ
− 3 u2 u1
0
0
− 2µ
3
− 2µ
u
3 2
0
µ
0
µu1
0
0
0
0
0
−µu1
− 4µ
u
3 2
µ
c3 − 3 u 2 u 2
0
µ
0
µ̄u1
0
0
0
0
0
µ
3
+ µ̄ u2


,

(37)
κ
Cv
0
2µ
u
3 2
−µu1
− µ3 u1 u2
4µ
3




,

(38)



,

(39)



.

(40)
κ
Cv
Given these coefficient matrices, the element-level matrices are derived in a standard fashion
as for any generalized advective-diffusion system.
4. Implementation
The above system is implemented in the standard generalized advective-diffusive framework
in the blkcns2dst.F subroutine. The following variables are used in the code:
7
Block (2D Space-Time Compressible NS) Derivation I
Variable name Quantity
dr
ρ
ru
ρu1
rv
ρu2
re
ρe
vels
||u||2
difq
µ/ρ
µ
di13
/ρ
3
4µ
di43
/ρ
3
c1
c1 /ρ
c3
c3 /ρ
M. Behr
Variable name
Quantity
du
dv
de
u1
u2
e
kapq
di23
dbar
c2
κ
/ρ
Cv
2µ
/ρ
3
µ̄
c2 /ρ
History
April 29, 1999
April 30, 1999
January 25, 2000
May 3, 2001
Written (a separate note about element-level matrices will follow).
Implemented.
Corrections in expressions for E1 .
Added implementation section.
References
[1] S.K. Aliabadi, Parallel Finite Element Computations in Aerospace Applications, Ph.D.
thesis, University of Minnesota, Department of Aerospace Engineering and Mechanics,
1994.
[2] S. Mittal and T.E. Tezduyar, “A unified finite element formulation for compressible
and incompressible flows using augmented conservation variables”, Computer Methods
in Applied Mechanics and Engineering, 161 (1998) 229–243.
8