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

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

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 ?

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

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)

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

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

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

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

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

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

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

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

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

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

Department of Flow, Heat and Combustion Mechanics Results • Test cases • (perfect gas) • Adiabatic flow p’-eq. based on the energy eq. (NOT continuity eq.)

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

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!

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

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

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

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

Department of Flow, Heat and Combustion Mechanics Results log

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

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

A Mach-uniform pressure correction algorithm