Realization of contact resolving approximate Riemann solvers for strong shock and expansion flows

Abstract

SUMMARYThe Harten-Lax-van Leer contact (HLLC) and Roe schemes are good approximate Riemann solvers that have the ability to resolve shock, contact, and rarefaction waves. However, they can produce spurious solutions, called shock instabilities, in the vicinity of strong shock. In strong expansion flows, the Roe scheme can admit nonphysical solutions such as expansion shock, and it sometimes fails. We carefully examined both schemes and propose simple methods to prevent such problems. High-order accuracy is achieved using the weighted average flux (WAF) and MUSCL-Hancock schemes. Using the WAF scheme, the HLLC and Roe schemes can be expressed in similar form. The HLLC and Roe schemes are tested against Quirk's test problems, and shock instability appears in both schemes. To remedy shock instability, we propose a control method of flux difference across the contact and shear waves. To catch shock waves, an appropriate pressure sensing function is defined. Using the proposed method, shock instabilities are successfully controlled. For the Roe scheme, a modified Harten-Hyman entropy fix method using Harten-Lax-van Leer-type switching is suggested. A suitable criterion for switching is established, and the modified Roe scheme works successfully with the suggested method.

Robust HLLC Riemann solver with weighted average flux scheme for strong shock

“…Hence, mass flux is only used in the continuity equation. Detailed expressions of the mass flux for the HLLC scheme is described in [8] and the resulting dissipation term of the continuity equation is expressed as…”

“…We have proposed a control method of flux difference across the contact for strong shock and expansion flows [8]. In this paper, we provide a simple and clear way of adding dissipation to the HLLC scheme to remedy the numerical shock instability problem.…”

Application of central differencing and low‐dissipation weights in a weighted compact nonlinear scheme

“…We obtain the relationship between the weights w k and optimal weights d k from Equation (22), that is,…”

“…The upper and lower boundaries were treated as periodic boundary conditions through the use of a virtual computational node. In this problem, the HLLC-HLL scheme [22] was adopted instead of the AUSMDV scheme because R-WCNS-CU6-Z-τ 6 -V provides a different computational result. Figure 16 indicates that the density distribution of the Richtmyer-Meshkov instability problem enlarges in the order 0.5 ⩽ x ⩽ 1.5, 0 ⩽ y ⩽ 1.…”

“…The upper and lower boundaries are treated as symmetrical boundary conditions through the use of a virtual computational node. In this problem, the HLLC-HLL scheme [22] was adopted instead of the AUSMDV scheme because the use of R-WCNS-CU6-Zτ 6 -V with the AUSMDV scheme blows up. Figure 19 indicates the density distribution of the shock-bubble interaction problem enlarged for the region 0.5 ⩽ x ⩽ 1.5, 0 ⩽ y ⩽ 1 and shows that our computational results, that is, the form of the bubble after the shock wave, are almost equivalent to those of the previous experimental result and numerical analysis.…”