Validation of a high-order finite difference compressible solver

TL;DR

This work develops a high-order compressible solver based on Lele's sixth-order compact finite differences, integrated with third-order Runge–Kutta time stepping, Padé-type filters, and localized artificial diffusivity to balance low dissipation with robustness near shocks. The method is parallelized via domain decomposition and MPI to enable scalable simulations on CPU/GPU clusters and is validated against five canonical cases, including shock-dominated and turbulence-dominated regimes. Results show accurate shock capturing, clear resolution of vortical structures, and good agreement with exact solutions and reference DNS/experimental data for both mean and second-order statistics. The study provides a reproducible, scalable baseline for high-fidelity simulations of compressible turbulent flows with shock interactions, with potential for broader application to curved geometries and complex walls.

Abstract

The verification and validation of a high-order compressible in-house solver based on a compact finite difference scheme are presented. Validation is performed using five canonical cases: the one-dimensional Sod shock tube problem, two-dimensional shock-shear layer interaction, compressible channel flow, compressible turbulent boundary layer, and shock-turbulent boundary layer interaction. Comparisons against exact solutions and reference direct numerical simulation data demonstrate accurate shock capturing, resolution of vortical structures, and good agreement for first and second order statistics.

Figures (10)

• Figure 1: Density profiles for the one-dimensional Sod shock-tube problem at $t^*=0.2$, comparing numerical solutions obtained with different grid resolutions (N=100, 200, 400, and 600 ) against the exact Riemann solution and 5th order WENO results by Jiang and ShuJiang_1996_JCP.
• Figure 2: Instantaneous density contour at $t^*=120$ by (a) 5th order WENO scheme by Wang et al.Wang_2021_CMAME, (b) 5th order WENO-Z scheme by Peng et al.Peng_2019_CaF and (c) Present study based on 6th order compact scheme. (d) Density distribution at $y^*=0$.
• Figure 3: Instantaneous streamwise velocity contour at the cross section.
• Figure 4: Comparison of compressible channel flow with other DNS data Morinishi_2004_JFMModesti_2016_IJHF: (a) Mean velocity, (b) Van-Driest transformed mean velocity, (c) normalized mean temperature and (d) root mean square (rms) velocity fluctuations.
• Figure 5: Iso-surface of Q criterion colored by density of current simulation.