a mach-uniform pressure correction algorithm krista nerinckx, j. vierendeels and e. dick dept. of...
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A Mach-uniform pressure correction algorithm
Krista Nerinckx, J. Vierendeels and E. Dick
Dept. of Flow, Heat and Combustion Mechanics
GHENT UNIVERSITY
Belgium
Department of Flow, Heat and Combustion Mechanics
Introduction
Mach-uniform algorithm
• Mach-uniform accuracy• Mach-uniform efficiency• For any level of Mach number• Segregated algorithm:
pressure-correction method
Department of Flow, Heat and Combustion Mechanics
Mach-uniform accuracy
Discretization
• Finite Volume Method (conservative)
• Flux: AUSM+
• OK for high speed flow
• Low speed flow: scaling + pressure-velocity coupling
Department of Flow, Heat and Combustion Mechanics
Mach-uniform efficiency
Convergence rate
• Low Mach: stiffness problem
• Highly disparate values of u and c
• Acoustic CFL-limit:
breakdown of convergence
( ) MAXu c
u c tCFL
x
Department of Flow, Heat and Combustion Mechanics
Mach-uniform efficiency
Remedy stiffness problem:
• Remove acoustic CFL-limit
• Treat acoustic information implicitly
Department of Flow, Heat and Combustion Mechanics
? Which terms contain
acoustic information ?
Mach-uniform efficiency
Acoustic terms ?
• Consider Euler equations (1D, conservative) :
Department of Flow, Heat and Combustion Mechanics
0
0
0
u
t xu uu p
t x xE Hu
t x
Mach-uniform efficiency
• Expand to variables p, u, T
• Introduce fluid properties : =(p,T) e=e(T) or e=e(p,T)
Department of Flow, Heat and Combustion Mechanics
0
0
0
p T
p T
p p T T uu u
t x t x x
u u pu
t x xp p T T u
e u e u e pt x t x x
Mach-uniform efficiency
• Construct equation for pressure and for temperature from equations 1 and 3:
Department of Flow, Heat and Combustion Mechanics
2 0
10
0T
p p uu c
t x xu u pu
t x x
T T p uu
t x e x
with2 1
1 T
T p
pc
e
c: speed of sound (general fluid)
quasi-linear system in p, u, T
Mach-uniform efficiency
• Identify acoustic terms:
Department of Flow, Heat and Combustion Mechanics
2 0
0
0
1
T
p pu
t xu uu
t x
T T p
uc
xp
x
uu
t x e x
Mach-uniform efficiency
• Go back to conservative equations to see where these terms come from:
•
Department of Flow, Heat and Combustion Mechanics
10
u uu
t x
p
x
0u uu
t x
p
x
momentum eq.
Mach-uniform efficiency
Department of Flow, Heat and Combustion Mechanics
2 0p pu
t x
uc
x
0p T TT p T
p pe e u e
t x
u ue p
x x
continuity eq.
0u
xt
energy eq.
0E Hu
t x
21
2e p uHu u u
write as
•
Mach-uniform efficiency
Department of Flow, Heat and Combustion Mechanics
TT
u u
xe e p
x
continuity eq.determines pressure
0
• Special cases:
- constant density: =cte
energy eq. contains no acoustic information
. 0v
Mach-uniform efficiency
Department of Flow, Heat and Combustion Mechanics
TT
u u
xe e p
x
0
- perfect gas:
energy eq. determines pressure
1
pe
continuity equation contains no acoustic information
Mach-uniform efficiency
Department of Flow, Heat and Combustion Mechanics
- perfect gas, with heat conduction:
conductive terms in energy equation energy and continuity eq. together determine pressure and temperature
treat conductive terms implicitly to avoid diffusive time step limit
Implementation
Successive steps:
• Predictor (convective step): - continuity eq.: * - momentum eq.: u*
• Corrector (acoustic / thermodynamic step):
Department of Flow, Heat and Combustion Mechanics
1
1
1
'
* '
* '
n n
n
n
p p p
u u u
T T T
Implementation
• Momentum eq. :
• Continuity eq. :
p’-T’ equation
Department of Flow, Heat and Combustion Mechanics
' ( ')u f p
1
11
2
* ' '
* '
ni T p
n
i
T p
u u u
Implementation
• energy eq. :
p’-T’ equation
• coupled solution of the 2 p’-T’ equations
• special case: perfect gas AND adiabatic energy eq. becomes p’-equation no coupled solution needed
Department of Flow, Heat and Combustion Mechanics
1
1 21
2
11
2
* ' '
1* * * *
2* '
* '
n
i p T
n
i
n
i
E E e p e T
Hu e p uu u u
T Tq
x
Results
Department of Flow, Heat and Combustion Mechanics
Test cases
(perfect gas)
1. Adiabatic flow
p’-eq. based on the energy eq. (NOT continuity eq.)
Results
Low speed:
• Subsonic converging-diverging nozzle
• Throat Mach number: 0.001
• Mach number and relative pressure:
0 20 40 60 80 1008.8
9
9.2
9.4
9.6
9.8
10x 10-4
x
M
0 20 40 60 80 100-15
-10
-5
0
5x 10-8
x
Del
taP
Department of Flow, Heat and Combustion Mechanics
Results
• No acoustic CFL-limit
• Convergence plot:
Essential to use the energy equation!
0 100 200 300 40010
-15
10-10
10-5
100
Number of time steps
Max
(Res
idue
)
Cont, CFL=10 (+UR)
Cont, CFL=1 (+UR)
Energy, CFL=1
Energy, CFL=10
Department of Flow, Heat and Combustion Mechanics
Results
High speed:
• Transonic converging-diverging nozzle
• Normal shock
• Mach number and pressure:
0 20 40 60 80 1000.6
0.8
1
1.2
1.4
x
M
0 20 40 60 80 1000.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
x
p
Department of Flow, Heat and Combustion Mechanics
Results
Department of Flow, Heat and Combustion Mechanics
2. non-adiabatic flow
• if p’-eq. based on the energy eq.
explicit treatment of conductive terms diffusive time step limit:
• if coupled solution of p’ and T’ from continuity and energy eq.
no diffusive time step limit
2
1
2
tNe
x
Results
Department of Flow, Heat and Combustion Mechanics
1D nozzle flow
Results
Department of Flow, Heat and Combustion Mechanics
Thermal driven cavity
• Ra = 103
• = 0.6
• very low speed: Mmax=O(10-7) no acoustic CFL-limit allowed
• diffusion >> convection in wall vicinity: very high T no diffusive Ne-limit allowed
Results
Department of Flow, Heat and Combustion Mechanics
6
1
(10 )
70
u
u c
CFL O
CFL O
Ne
log
Results
Department of Flow, Heat and Combustion Mechanics
Temperature distribution: Streamlines:
Conclusions
Conclusions:
• Pressure-correction algorithm with- Mach-uniform accuracy- Mach-uniform efficiency
• Essential: convection (predictor) separated from acoustics/thermodynamics (corrector)
- General case: coupled solution of energy and continuity eq. for p’-T’
- Perfect gas, adiabatic: p’ from energy eq.
• Results confirm the developed theory
Department of Flow, Heat and Combustion Mechanics