Artificial Surface Tension to Stabilize Captured Shockwaves

Computationally captured shockwaves, especially at very high Mach number, often display anomalous behavior not encountered in nature. Figure 1 shows three of the most notorious examples. The passage of a normal shock over a slightly perturbed mesh was observed to be unstable by Quirk. However, it was subsequently discovered that neither the mesh perturbation nor the shock motion was always needed. Diffraction of a plane shock wave by a wedge often causes a kink in the Mach stem, which does not properly intersect the wedge surface. Perhaps most notorious is the so-called carbuncle phenomenon, first reported by Peery and Imlay. These anomalies are perplexing, not least because they are strongly related to genuine physical solutions of the Euler equations. When they appear in steady solutions, the residuals are frequently very small, indicating that a weak solution has been achieved, and moreover the shockwaves are compressive and entropysatisfying. Versions of the carbuncle have been demonstrated experimentally at high Reynolds number, and even proposed as practical devices to reduce drag and heat transfer. There are even anomalies in one dimension. Astonishingly, Godunov’s method has no solution for a steady one-dimensional shock at Mach numbers greater than 6.0 (if γ = 7/5) unless the shock resides within a certain range of locations relative to the mesh. It has been hypothesized that the failure is in part thermodynamic, associated with the fact that even the ”exact” solution of the Riemann problem does not strictly enforce the Second Law locally. However, Kitamura et aldemonstrated that while flux functions free of this defect may be satisfactory in one dimension, the carbuncle can return in higher dimensions. It seems that some intrinsically multidimensional approach needs to be taken, in addition to correcting the thermodynamics.