Artificial Selective Damping
The DRP scheme is a central difference scheme and, therefore, has no intrinsic dissipation. For the purpose of eliminating spurious short waves and to improve numerical stability, artificial selective damping terms are added to the discretized finite difference equations. For the present problem, artificial selective damping is also critically needed for shock-capturing purposes.
In the interior region, the 7-point damping stencil with a half-width of a = 0.2n is used. An inverse mesh Reynolds number (Яд = va/(amAx)), where ua is the artificial kinematic viscosity, of 0.05 is prescribed over the entire computation domain. This is to provide general background damping to eliminate possible propagating spurious waves. Near the boundaries of the computational domain where a 7-point stencil does not fit, the 5- and 3-point damping stencils given in Section 7.2 are used.
Spurious numerical waves are usually generated at the boundaries of a computational domain. The boundaries are also favorite sites for the occurrence of numerical instability. This is true for both exterior and interior boundaries such as the nozzle walls and buffer regions where there is a change of mesh size. To suppress both the generation of spurious numerical waves and numerical instability, additional artificial selective damping is imposed along these boundaries. Along the inflow, radiation, and outflow boundaries, a distribution of inverse mesh Reynolds number in the form of a Gaussian function with a half-width of 4 mesh points (normal to the boundary) and a maximum value of 0.1 right at the outmost mesh points is incorporated into the time marching scheme. Adjacent to the jet axis, a similar addition of artificial selective damping is implemented with a maximum value of the inverse mesh Reynolds number at the jet axis set equal to 0.35. On the nozzle wall, the use
of a maximum value of additional inverse mesh Reynolds number of 0.2 has been found to be very satisfactory.
The two sharp corners of the nozzle lip and the transition point between the use of the outflow and the radiation boundary condition on the right side of the computational domain are locations requiring stronger numerical damping. This is carried out by adding a Gaussian distribution of damping around these special points.
As shown in Figure 15.54, the four computational subdomains are separated by buffer regions. Here, additional artificial selective damping is added to the finite difference scheme. In the supersonic region downstream of the nozzle exit, a shock cell structure develops in the jet flow. In order to provide the DRP scheme with shock-capturing capability, the variable stencil Reynolds number method discussed in Section 8.3 is used. The jet mixing layer in this region has a very large velocity gradient in the radial direction. Because of this, the Ustencil of the variable stencil Reynolds number method is determined by searching over the 7-point stencil in the axial direction. In the radial direction only the 2 mesh points immediately adjacent to the computational point are included in the search. An inverse stencil mesh Reynolds number distribution of the following form:
1, 0 < x < 9
is used. It is possible to show, based on the damping curve (a = 0.3n), that the variable damping has minimal effect on the instability wave of the feedback loop. Also, extensive numerical experimentations indicate that the method used can, indeed, capture the shocks in the jet plume and that the time-averaged shock cell structure compares favorably with experimental measurements.