for the Euler equations
Friedemann Kemm BTU Cottbus
kemm@tu-cottbus.de
-0.001 0 0.001 0.002 0.003 0.004 0.005
0 5 10 15 20 25 30 35 40
0
100
200
300
400 0 5
10 15
20 0
2 4 6 8 10 12
There is no unique splitting of the Euler flux into an advective and a pressure flux
H-Cusp E-Cusp VCT-Cusp
ρ ρ
ρ
H
+
0 p 0 0
ρ ρ
ρ
E
+
0 p 0 p
ρ ρ
ρ
1
2 ρ2
+
0 p
γ 0
γ−1 p
AUSM etc. Zha-Bilgen, AUFS etc. Vázquez Cendón and Toro
There is no splitting of the Euler flux into advection and acoustics
Eigenvalues of split fluxes:
advective part pressure part H-cusp γ, deficient 0, 0, −(γ − 1)
E-cusp , deficient 0, ±
Èγ − 1 γ
c
VCT-cusp 0, deficient 0,
2 ±
È2
4 + γp ρ
The splitting can be used directly or with additional upwinding for the pressure part
General
F = U + P
Direct application
Fnum = ¯ Uup + Pnum
With additional upwinding (e. g. AUFS)
Fnum = ¯ Uup + MPup + (1 − M)Pnum
How to obtain the numerical viscosity for Pnum?
One of the numerical viscosities provided by Sun and Takayama for AUFS vanishes for entropy waves
Steger-Warming
1 2c
p
p
p Hp
− 1 2cr
pr
rpr
rpr Hrpr
AUFS
1 2 ¯c
p − pr p − prr p − prr
¯ c2
γ−1(p − pr) + 12(p2
− pr2
r )
Rusanov
¯ c 2
ρ − ρr ρ − ρrr ρ − ρrr
E − Er
Exact resolution of entropy waves leads to loss of positivity
-0.001 0 0.001 0.002 0.003 0.004 0.005
0 5 10 15 20 25 30 35 40
at time 1.6 initial condition
Pressure in left half of steady shock test
The AUFS-approach with the different viscosities can be directly transferd from E-cusp to VCT-cusp
Compute viscosity as for original AUFS Multiply viscosity with
mx
2 ± s
2
4 + γp ρ
,
r 2 ±
s
2
r
4 + γpr ρr
¯ c
In last flux component replace p by γ−1γ p
In advective flux replace total energy by total kinetic energy
Using VCT-cusp stabilizes the original AUFS-scheme
-0.001 0 0.001 0.002 0.003 0.004 0.005
0 5 10 15 20 25 30 35 40
at time 1000 initial condition
Pressure in left half of steady shock test
After changing the viscosity or transition to VCT the resolution of entropy waves is still reasonable
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
with E-cusp
LLF SW original reference scheme
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
with VCT-cusp
LLF SW original reference scheme
Sod shock-tube with different AUFS-type schemes
Due to the viscosity on the shear wave AUFS-schemes don’t show the carbuncle
but overshoots near strong shocks
0 10 20 30 40 50
0 10 20 30 40 50 60
with E-cusp
SW LLF reference scheme
0 10 20 30 40 50
0 10 20 30 40 50 60
with VCT-cusp
original SW LLF
The viscosity on the shear wave may be reduced by a reverse carbuncle fix
Simple fix
Multiply 3rd component of viscosity by
|p − pr| p + pr
α
·
| − r|
|| + |r|
β
with α, β ∈ [0,1]
Elaborate fix
Compute norm krk of residual of RH-condition for contact Apply monotone increasing function g : [0,∞] → [0,1]
Multiply 3rd component of viscosity by g(krk)
For α = 1/4, β = 0 and VCT with Rusanov-type viscosity, no carbuncle is found in most tests . . .
0 15 30 45 60
0 15 30
0 10 20 30 40
0 20 40 60 80 100 0
20 40
0 1 2 3 4 5 6
Colliding flow (t = 30) and steady shock with shear correction
. . . but not all
0
100
200
300
400 0 5
10 15
20 0
2 4 6 8 10 12
0
100
200
300
400 0 5
10 15
20 0
2 4 6 8 10 12
Quirk test without and with shear correction
The full potential, especially of VCT, is not yet exploited
Direct application of splitting
ongoing work of Vázquez Cendón and Toro
different standard Riemannsolvers for pressure part for E-cusp already many failed schemes
With additional upwinding
for VCT take into account the asymmetry in wave speeds . . .
