A Mach-uniform pressure correction algorithm

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


Mach-uniform accuracy

Discretization

• Finite Volume Method (conservative)

• Flux: AUSM+

• OK for high speed flow

• Low speed flow: scaling + pressure-velocity coupling


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



Remedy stiffness problem:

• Remove acoustic CFL-limit

• Treat acoustic information implicitly


? Which terms contain

acoustic information ?


Acoustic terms ?

• Consider Euler equations (1D, conservative) :


0

0

0

u

t xu uu p

t x xE Hu

t x


• Expand to variables p, u, T

• Introduce fluid properties : =(p,T) e=e(T) or e=e(p,T)


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


• Construct equation for pressure and for temperature from equations 1 and 3:


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


• Identify acoustic terms:


2 0

0

0

1

T

p pu

t xu uu

t x

T T p

uc

xp

x

uu

t x e x


• Go back to conservative equations to see where these terms come from:

•


10

u uu

t x

p

x

0u uu

t x

p

x

momentum eq.



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

•



TT

u u

xe e p

x

continuity eq.determines pressure

0

• Special cases:

- constant density: =cte

energy eq. contains no acoustic information

. 0v



TT

u u

xe e p

x

0

- perfect gas:

energy eq. determines pressure

1

pe

continuity equation contains no acoustic information



- 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):


1

1

1

'

* '

* '

n n

n

n

p p p

u u u

T T T

Implementation

• Momentum eq. :

• Continuity eq. :

p’-T’ equation


' ( ')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


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


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


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


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


Results


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


1D nozzle flow

Results


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


6

1

(10 )

70

u

u c

CFL O

CFL O

Ne

log

Results


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


A Mach-uniform pressure correction algorithm

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