Robustness versus accuracy in shock-wave computations

Abstract

Despite constant progress in the development of upwind schemes, some failings still remain. Quirk recently reported (Quirk JJ. A contribution to the great Riemann solver debate. International Journal for Numerical Methods in Fluids 1994; 18: 555-574) that approximate Riemann solvers, which share the exact capture of contact discontinuities, generally suffer from such failings. One of these is the odd -even decoupling that occurs along planar shocks aligned with the mesh. First, a few results on some failings are given, namely the carbuncle phenomenon and the kinked Mach stem. Then, following Quirk's analysis of Roe's scheme, general criteria are derived to predict the odd -even decoupling. This analysis is applied to Roe's scheme (Roe PL, Approximate Riemann solvers, parameters vectors, and difference schemes,

A matrix stability analysis of the carbuncle phenomenon

“…Unfortunately, this correction, even modified to account for directional independency and to reduce the dissipation addition on linear waves, is combined with a tunable pressure sensing function to become active only in the vicinity of shocks [16]. It must be pointed out that even using HartenÕs entropy fix on the eigenvalue associated to the shear wave, a finite amount of dissipation is needed to damp transverse perturbations since, for instance a typical value of HartenÕs parameter d ¼ 0:1 cannot damp perturbations in QuirkÕs problem while d ¼ 0:2 removes the odd-even decoupling problem [17]. Similarly, a transverse velocity component can damp transverse perturbations (see Section 4.3) provided that it has reached a sufficient value.…”

A parameter-free AUSM-based scheme for healing carbuncle phenomenon

“…The triple point structure never looks physical and density perturbations are distributed behind the incident shock. Such artifacts are caused due to insufficient dissipations that can suppress the transverse perturbation [6,21].…”

“…The carbuncle phenomenon is known to be highly griddependent but does not require a large number of grid points [21]. The computational domain for this problem consists of a structured triangular grid with 14 × 318 cells in r-and θ-directions, respectively.…”

Vortical regime of the flow behind the bow shock wave

“…However, schemes based on Riemann solvers are known to suffer from the carbuncle instability and oddeven decoupling (Quirk 1994). The schemes used here have been tested by an odd-even grid perturbation problem which was suggested by Quirk (1994) and used in many other papers (Robinet et al 2000;Gressier and Moschetta 2000;Liou 2000;Pandolfi and D'Ambrosio 2001). It is assumed that a shock wave propagates down a straight duct with a Mach number 6 and γ = 1.4.…”