1. Figure 24 shows the computed pressure field at t=2 and t=6, obtained byFR-LADp3, FVM BJ limiter, and FVM VK limiter. For making comparison easier, we define the simulation time t to be 0 when the vortex center proceeds to (,) = (2.0,0.0) that is the initial vortex center for the corresponding case in Inoue et al. [39] In the computation using FR-LAD, the vortex moves toward the shock wave with keeping the original shape, and interacts with the shock without generating spurious oscillations. The obtained pressure field is in good agreement with one of the corresponding case "C" in Inoue et al. [39] For the case of FVM BJ limiter, the smooth vortex turns into jagged profiles by the convection even before interacting with the shock, and the pressure field after the interaction is not smooth and contains many small wiggles. For FVM VK limiter, which is more dissipative than BJ limiter, the obtained pressure field is smooth but apparently dissipative than the other results. We also observe that the shape of vortex deformed rapidly and spurious pressure waves were generated in the upstream region of the shock in the FVM VK limiter. In Fig. 25, distributions of the sound pressure are plotted against the distance r from the vortex center for a fixed value of = -45° for different schemes. The profiles obtained by FR-LAD p2 and p3 agree well withthe reference result. A computation usingFR-limiter withp2 iscarried out for thiscase, but the obtained profile turns out to be quite erroneous with considerable oscillations. As we have seen the pressure field, FVM BJ limiter shows the non-smooth profile, and FVM VK limiter shows the smeared profile. In the comparison,FR-LADshows its superior performance to accurately simulate the shock-vortex interaction and the associated acoustic wave propagation.
2. Fig. 21. Mach number contours and artificial bulk viscosity contours with the grid computed by FR-LAD withp3for the transonic flowpastNACA0012.
3. The Runge–Kutta Discontinuous Galerkin Method for Conservation Laws V
4. A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier–Stokes Equations