1. 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

2. 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

3. 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

4. Mach-uniform efficiency Convergence rate Low Mach: stiffness problem Highly disparate values of u and c Acoustic CFL-limit: breakdown of convergence Department of Flow, Heat and Combustion Mechanics

5. 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 ?

6. Mach-uniform efficiency Acoustic terms ? Consider Euler equations (1D, conservative) : Department of Flow, Heat and Combustion Mechanics

7. 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

8. Mach-uniform efficiency Construct equation for pressure and for temperature from equations 1 and 3: Department of Flow, Heat and Combustion Mechanics with c: speed of sound (general fluid) quasi-linear system in p, u, T

9. Mach-uniform efficiency Identify acoustic terms: Department of Flow, Heat and Combustion Mechanics

10. Mach-uniform efficiency Go back to conservative equations to see where these terms come from: Department of Flow, Heat and Combustion Mechanics momentum eq.

11. Mach-uniform efficiency Department of Flow, Heat and Combustion Mechanics continuity eq. energy eq. write as

12. Mach-uniform efficiency Department of Flow, Heat and Combustion Mechanics continuity eq. determines pressure 0 Special cases: - constant density:  =cte energy eq. contains no acoustic information

13. Mach-uniform efficiency Department of Flow, Heat and Combustion Mechanics 0 - perfect gas: energy eq. determines pressure continuity equation contains no acoustic information

14. 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

15. Implementation Successive steps: Predictor (convective step): - continuity eq.:  * - momentum eq.:  u* Corrector (acoustic / thermodynamic step): Department of Flow, Heat and Combustion Mechanics

16. Implementation Momentum eq. : Continuity eq. :  p’-T’ equation Department of Flow, Heat and Combustion Mechanics

17. 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

18. 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.)

19. Results Low speed: Subsonic converging-diverging nozzle Throat Mach number: 0.001 Mach number and relative pressure: 020406080100 8.8 9 9.2 9.4 9.6 9.8 10 x 10 -4 x M 020406080100 -15 -10 -5 0 5 x 10 -8 x DeltaP Department of Flow, Heat and Combustion Mechanics

20. Results No acoustic CFL-limit Convergence plot: Essential to use the energy equation! 0100200300400 10 -15 10 -10 10 -5 10 0 Number of time steps Max(Residue) Cont, CFL=10 (+UR) Cont, CFL=1 (+UR) Energy, CFL=1 Energy, CFL=10 Department of Flow, Heat and Combustion Mechanics

21. Results High speed: Transonic converging-diverging nozzle Normal shock Mach number and pressure: 020406080100 0.6 0.8 1 1.2 1.4 x M 020406080100 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 x p Department of Flow, Heat and Combustion Mechanics

22. 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

23. Results Department of Flow, Heat and Combustion Mechanics 1D nozzle flow

24. Results Department of Flow, Heat and Combustion Mechanics Thermal driven cavity Ra = 10 3  = 0.6 very low speed: M max =O(10 -7 )  no acoustic CFL-limit allowed diffusion >> convection in wall vicinity: very high  T  no diffusive Ne-limit allowed

25. Results Department of Flow, Heat and Combustion Mechanics log

26. Results Department of Flow, Heat and Combustion Mechanics Temperature distribution:Streamlines:

27. 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