Assessment of Gradient-based Reconstruction and Artificial Diffusivity Methods in Simulating High-Speed Compressible Flows

TL;DR

The paper compares local artificial diffusivity (LAD) and centralised gradient-based reconstruction (C-GBR) for simulating high-speed compressible flows with shocks and turbulence. It shows LAD is faster but prone to instability in strong discontinuities, while C-GBR is more robust and accurate albeit more computationally expensive. A hybrid LAD–GBR framework is proposed and demonstrated, delivering stability in problematic cases and achieving notable speed-ups over C-GBR in challenging tests such as the M2.9 compression ramp. The results indicate that centralised-gradient methods offer reliable shock–turbulence coupling, and the hybrid approach provides a practical path for efficient high-speed flow simulations on modern hardware.

Abstract

The two promising methods for capturing high-speed flows are local artificial diffusivity (LAD) and centralised gradient-based reconstruction (C-GBR), the former being computationally economical and the latter being more robust and stable but expensive. While the LAD approach captures discontinuities by adding artificial fluid transport coefficients, C-GBR employs a wave appropriate discontinuity sensor to obtain cleaner results and utilises the HLLC approximate Riemann solver for computing inviscid fluxes. The efficacy of these schemes is initially demonstrated in single-species 1D and 2D test cases. Moreover, the shock-capturing capability is assessed for 3D supersonic and hypersonic turbulent boundary layers. The accuracy of LAD predictions is comparable to that of C-GBR for the test case of a supersonic turbulent boundary layer. From the stability front, all simulations are found to be stable with the C-GBR scheme, whereas the LAD-based simulations are observed to abruptly diverge for supersonic and hypersonic flows over compression corners with stronger shocks and larger flow separations. From the computational front, the LAD-based schemes are $1.17 - 2.32 \times$ faster than the monotonicity-preserving explicit/implicit C-GBR schemes. A hybrid approach leveraging the strengths of LAD and GBR schemes is proposed as a promising solution for high-speed turbulent flows with strong shock-boundary layer interactions. The efficacy of the hybrid LAD-GBR solver is demonstrated for the compressible triple-point and supersonic compression ramp test cases. For the M2.9, $24^{\circ}$ case, the hybrid solver was stable and achieved a notable $1.67 \times$ speed-up over the C-GBR scheme.

Figures (31)

• Figure 1: Plots of (a) density and (b) velocity profiles for the Sod shock tube problem compared against the exact solution.
• Figure 2: Plots of (a) density and (b) velocity profiles for the Lax problem compared against the exact solution.
• Figure 3: Plots of (a) density and (b) velocity profiles for the Le Blanc problem compared against the exact solution.
• Figure 4: Plots of (a) density profiles and (b) magnified view of the density field ($0 \leq x \leq 2.5$) for the Shu-Osher problem compared against the exact solution chamarthi2023efficient.
• Figure 5: Line contours of density for the 2D shock entropy wave test with (a) LAD-C6, (b) MIG4 and (c) MEG6 schemes; (d) Density variation along the streamwise direction at $y=0$ obtained using various schemes compared against the exact solution chamarthi2021high.