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Numerical Viscosity and Reynolds number



Here we compute the fluid flow around a rigid body, for relatively weak shocks, with different resolutions. The PPM solves the Euler equations of inviscid flow. Therefore, the method's viscosity is actually due to the approximation of calculating the equations as finite differences performed on a computational grid.

In particular, the behaviour of the numerical viscosity is very similar to a true viscosity and dissipates the kinetic energy of small-scale fluid motions into heat. This dissipation is discussed by Porter and Woodward (1994, hereafter PW94) and by Edgar and Woodward (1992) for the case of a shock hitting a wedge. For some well representative case, the effective kinematic viscosity nueff was found to be approximately :

nueff/(csD)= (D/dx)-3 (M +0.25)3.15 )

where cs is the local sound speed, M is the adiabatic Mach number of the flow, dx the resolution of the computational mesh and D the scale length - i.e. the diameter D=2*R - of the rigid body

The above expression was obtained by PW94 by studying the evolution of a sinusoidal wave perturbation in a hydrodynamical flow at different Mach numbers, using the PPM hydro scheme. Given the statistical shape of the pulsations in a developed turbulence medium, the perturbation studied by PW94 is very reasonably applicable to a wide manifold of flows in the conditions studied by our paper. We carefully checked the results of PW94, reproducing exactly the same behaviour. Furthermore, we applied the same perturbation to other hydro schemes (Van Leer upwind scheme with linear and parabolic approximation -- e.g. the "ZEUS-2D" code (by U.I.-NCSA, Urbana-Champaign) -- the first order Godunov scheme and the second order TVD scheme of Harten "UNO"), obtaining the same qualitative dependence at a given Mach number, but with exponents less than 3, ranging in the interval [3,1].

We chose the latter, because it is much less viscous than the other methods tested and is certainly among the less numerically viscous methods in the literature for 2D calculations. This characteristic of having a substantially lower numerical viscosity is very important, as we are forced to compute runs with lower and lower minimum mesh size, in order to be able to calculate cases with higher and higher Reynolds numbers. Of course, if the chosen method's numerical viscosity were higher, one could obtain only lower Reynolds numbers per given computation time costs.

The Reynolds number Re is defined as :

Re= (u* D) /(nueff)

where nueff is the kinematic viscosity, while in our case it is the numerical viscosity.