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Department of Flow, Heat and Combustion Mechanics. 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.

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a mach uniform pressure correction algorithm

Department of Flow, Heat and Combustion Mechanics

A Mach-uniform pressure correction algorithm

Krista Nerinckx, J. Vierendeels and E. Dick

Dept. of Flow, Heat and Combustion Mechanics

GHENT UNIVERSITY

Belgium

introduction

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

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
mach uniform efficiency

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

mach uniform efficiency1

Department of Flow, Heat and Combustion Mechanics

Mach-uniform efficiency
  • Remedy stiffness problem:
  • Remove acoustic CFL-limit
  • Treat acoustic information implicitly

?Which terms contain acoustic information ?

mach uniform efficiency2

Department of Flow, Heat and Combustion Mechanics

Mach-uniform efficiency
  • Acoustic terms ?
  • Consider Euler equations (1D, conservative) :
mach uniform efficiency3

Department of Flow, Heat and Combustion Mechanics

Mach-uniform efficiency
  • Expand to variables p, u, T
  • Introduce fluid properties : r=r(p,T) e=e(T) or re=re(p,T)
mach uniform efficiency4

Department of Flow, Heat and Combustion Mechanics

Mach-uniform efficiency
  • Construct equation for pressure and for temperature from equations 1 and 3:

quasi-linear system in p, u, T

c: speed of sound (general fluid)

with

mach uniform efficiency6

Department of Flow, Heat and Combustion Mechanics

Mach-uniform efficiency
  • Go back to conservative equations to see where these terms come from:

momentum eq.

mach uniform efficiency7

continuity eq.

Department of Flow, Heat and Combustion Mechanics

Mach-uniform efficiency

write as

energy eq.

mach uniform efficiency8

Department of Flow, Heat and Combustion Mechanics

Mach-uniform efficiency
  • Special cases: - constant density: r=cte

0

energy eq. contains no acoustic information

continuity eq.determines pressure

mach uniform efficiency9

Department of Flow, Heat and Combustion Mechanics

Mach-uniform efficiency

- perfect gas:

0

continuity equation contains no acoustic information

energy eq. determines pressure

mach uniform efficiency10

Department of Flow, Heat and Combustion Mechanics

Mach-uniform efficiency

- 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

Department of Flow, Heat and Combustion Mechanics

Implementation
  • Successive steps:
  • Predictor (convective step): - continuity eq.:r*- momentum eq.:ru*
  • Corrector (acoustic / thermodynamic step):
implementation1

Department of Flow, Heat and Combustion Mechanics

Implementation
  • Momentum eq. :
  • Continuity eq. :
  •  p’-T’ equation
implementation2

Department of Flow, Heat and Combustion Mechanics

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
results

Department of Flow, Heat and Combustion Mechanics

Results
  • Test cases
  • (perfect gas)
  • Adiabatic flow p’-eq. based on the energy eq. (NOT continuity eq.)
results1

Department of Flow, Heat and Combustion Mechanics

Results
  • Low speed:
  • Subsonic converging-diverging nozzle
  • Throat Mach number: 0.001
  • Mach number and relative pressure:

-4

-8

x 10

x 10

10

5

9.8

0

9.6

M

9.4

-5

DeltaP

9.2

-10

9

8.8

-15

0

20

40

60

80

100

0

20

40

60

80

100

x

x

results2

Department of Flow, Heat and Combustion Mechanics

Results

0

10

  • No acoustic CFL-limit
  • Convergence plot:

Cont, CFL=10 (+UR)

-5

10

Cont, CFL=1 (+UR)

Max(Residue)

Energy, CFL=1

Energy, CFL=10

-10

10

-15

10

0

100

200

300

400

Number of time steps

Essential to use the energy equation!

results3

Department of Flow, Heat and Combustion Mechanics

Results
  • High speed:
  • Transonic converging-diverging nozzle
  • Normal shock
  • Mach number and pressure:

1.4

0.75

0.7

1.2

0.65

0.6

M

p

1

0.55

0.5

0.8

0.45

0.6

0.4

0

20

40

60

80

100

0

20

40

60

80

100

x

x

results4

Department of Flow, Heat and Combustion Mechanics

  • 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
Results
results6

Department of Flow, Heat and Combustion Mechanics

Results

Thermal driven cavity

  • Ra = 103
  • e = 0.6
  • very low speed: Mmax=O(10-7) no acoustic CFL-limit allowed
  • diffusion >> convection in wall vicinity: very high DT  no diffusive Ne-limit allowed
slide27

Department of Flow, Heat and Combustion Mechanics

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