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Axel Rohde July 26, 1999 08:30

TVD scheme at low Mach number
Has anyone ever programmed a TVD scheme, either on a Cartesian or curvilinear grid, and then run the code for a low free stream Mach number, say M=0.1 with a low value for the numerical viscosity epsilon, say eps=0.001? I have programmed a TVD solver based on Harten's UTL1C scheme for both Cartesian and Curvilinear coordinates. At low Mach number M<0.3, the code only gives physically correct results for numerical viscosity values epsilon near eps=0.5. This is satisfactory for inviscid solutions, however, if I use the TVD scheme in a full Navier Stokes solver (to compute the inviscid part), eps=0.5 is to high! For higher Mach numbers (M=0.5), my N-S solver works fine though (setting eps=0).

Hongjun Li July 26, 1999 11:03

Re: TVD scheme at low Mach number
TVD schemes were originaly developed for density based solvers, and later were applied to pressure based solvers for compressible flows. If you are using density based solver, there is a lower limit in free stream Mach number. Generally, a well programmed code can run Mach 0.1 for inviscid flows. For viscous flows, the Mach limit is much higher (0.3 - 0.5). So, in yours case, the problem does not come from TVD scheme but from the solver itself. If you need to solve low Mach compressible flows, you may consider using pressure based method (originaly developed for incompressible flows but now works for compressible flow as well). And, you may not neet to use the time consuming TVD scheme which was developed for capturing sharp shock waves.

John C. Chien July 26, 1999 12:59

Re: TVD scheme at low Mach number
(1). TVD was originally developed to supress the oscillations across a captured shock wave (shock wave is a discontinuity. it is simulated or captured by finite-difference or finite-volume methods), to eliminate the over-, under-shoot of the solution profiles. Shock is basically inviscid behavior (although it creates total pressure loss). (2). For a shock to occur, the flow must be transonic, or supersonic locally.

Paulo Zandonade July 26, 1999 21:09

Re: TVD scheme at low Mach number
I am not familiar with Harten's UTL1C scheme, but the Roe scheme (actually the numerical viscosity term in this scheme) has to be altered at low Mach numbers to ensure accuracy. This modification is normally used in preconditioning methods for convergence aceleration to a steady state, but can be used only to guarantee accuracy at a low mach number (view the papers at INRIA for time dependent calculations). You can find more information about preconditioning at the following internet sites: (Go to the reports page and search for "low mach")

Hope I was helpful.

Alexey July 27, 1999 04:51

Re: TVD scheme at low Mach number
The numerical viscosity called epsilon (or entropy correction) is dimensional value. It is of velocity dimension - m/sec. What kind form of eps do you use? What does it mean eps=0.001 m/sec or eps=0.5 m/sec? Usually dimensionless form (eps_dim = 0..0.1) is used so that eps = eps_dim*(abs(velocity) + speed_of_sound). But many different forms for eps are used also, for example eps=eps_dim*dx/dt and others. I had some problems (non physical effects) with TVD (or TVNI) schemes and now I use more robust and exact ENO schemes.

Best regards.

clifford bradford August 6, 1999 02:01

Re: TVD scheme at low Mach number
yeah TVD schemes are for higher mach # flows. for slow flows at the speeds you're talking about a simple preconditioned central difference scheme should work adequate. also i regions with low flow gradients (ie most/all of a mach 0.1 flow) the artificial dissipition should be small/nonexistent see jameson's 1981 paper on the 4 step runge kutta scheme. a pressure based scheme would also be good (as Honjun said) and are easy to code

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