An accurate and robust HLLC‐type Riemann solver for the compressible Euler system at various Mach numbers

Abstract

Summary A simple, robust, and accurate HLLC‐type Riemann solver for the compressible Euler equations at various Mach numbers is built. To cure shock instability of the HLLC solver at strong shocks, a pressure‐control technique, which plays a role in limiting the propagation of erroneous pressure perturbation, is proposed. With an all Mach correction method for the compressible Euler system, the proposed method is further extended to compute flow problems at low Mach numbers. The proposed all Mach HLLC‐type scheme has been implemented and used to compute a variety of flow problems ranging from hypersonic compressible to low Mach incompressible flow regimes. Various numerical results demonstrate that the obtained all Mach HLLC‐type scheme is both accurate and stable for all speed ranges.

A Review of Riemann Solvers for Hypersonic Flows

“…For the HLLEM scheme presented in Sect. 3 , Xie and his coworkers conducted numerical analyses based on the odd–even decoupling case and indicated that the HLLEM scheme can not ensure the consistency of mass flux across the normal shock, which makes it easy to produce unphysical shock anomalies [ 73 , 74 ]. Similar studies can also be found in Refs [ 75 , 76 ].…”

“…In terms of the low-speed issues, Qu carried out asymptotic analysis of the HLLEMS scheme’s behavior at low speeds in Ref [ 77 ]. Results indicate that both HLLEM and HLLEMS are incapable of obtaining physical solutions at low speeds [ 73 , 121 ]. Based on such analyses, the HLLEMS-AS scheme for all speeds is proposed to control the numerical dissipation in the momentum equations at low speeds.…”

On Asymptotic Behavior of HLL-Type Schemes at Low Mach Numbers

“…Following the simplification in [16], namely, assuming that there is no net influx over the boundary due to u (0) in the ghost cells and there is no gradient in p (1) across the boundary, equation (55) can be further transformed into…”

“…We use this test case to assess whether the LM-HLLEM scheme and the LM-HLLEM2 scheme can accurately simulate transonic turbulent flows and thus make a smooth transition between the calculations of incompressible and compressible flows. e setting of flow conditions can be found in [16]. e inlet Mach number is 0.729, and the angle of attack is 2.31°.…”

“…eir further research showed that this method can be used in other numerical schemes and can also be used in the verification of other low Mach number schemes, like the AUSM-type methods [15] and the LMRoe scheme [5]. Based on the work of Dellacherie et al [13,14], Xie et al [16,17] managed to develop an all-speed HLLC-type Riemann solver and an all-speed Roe-type Riemann solver, which are not only endowed with a high resistance against shock anomalies but also provide accurate solutions in low Mach number systems. Based on the simple and robust AUSM family schemes [15,18], Shima and Kitamura developed all-speed flux functions, namely, SLAU [19], SLAU2 [20], and SD-SLAU [21].…”