Validation of a high-order finite difference compressible solver

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.

💡 Research Summary

The paper presents the verification and validation of an in‑house compressible flow solver that employs a sixth‑order compact finite‑difference scheme originally proposed by Lele (1992). Spatial derivatives are obtained by solving tridiagonal systems, which are parallelized using a Thomas‑algorithm‑based approach to achieve scalability on both CPU and GPU clusters. Temporal integration is performed with a third‑order explicit Runge‑Kutta method, supplemented by high‑order non‑dispersive filters and a localized artificial diffusivity model (Kawamura‑Lele) to suppress spurious high‑frequency oscillations while preserving physical dissipation.

Validation is carried out through five canonical benchmark cases that together span a wide range of compressible flow phenomena:

1. One‑dimensional Sod shock‑tube – Simulations on 200, 400 and 600 uniform cells are compared against the exact Riemann solution and a fifth‑order WENO scheme. The compact scheme captures the shock, contact discontinuity and expansion fan without noticeable overshoots, and demonstrates clear grid convergence as resolution increases.

2. Two‑dimensional shock–shear‑layer interaction – A 500 × 100 grid resolves an oblique shock impinging on a developing mixing layer. Density contours show that the compact scheme reproduces vertical vortical structures more sharply than fifth‑order WENO‑Z on the same mesh, and matches results obtained on finer meshes with other authors.

3. Compressible channel flow DNS – Conducted at bulk Mach 1.5 and bulk Reynolds number 6000 (≈ Reτ 218). Two grid resolutions (120 × 180 × 120 and 256 × 200 × 230) are examined. Mean streamwise velocity, Van‑Driest‑transformed velocity, temperature profile, and rms velocity fluctuations all agree closely with reference DNS data from Morrison et al. (2014) and Modesti & Pirozzoli (2016), confirming that the near‑wall resolution (Δy⁺≈0.5) and compressibility treatment are adequate.

4. Compressible turbulent boundary layer – Simulated at free‑stream Mach 2.25 and momentum‑thickness Reynolds number Reθ≈2100. A turbulent tripping body force is used to generate fully developed turbulence on a 1300 × 250 × 301 grid (Δx⁺≈16, Δy⁺≈0.5, Δz⁺≈6). Van‑Driest‑scaled mean velocity and density‑scaled Reynolds normal stresses match DNS results from Poggi et al. (2015), Pirozzoli et al. (2011) and incompressible data from Schlatter & Örlü (2010). Notably, the peak streamwise Reynolds stress is captured accurately despite relatively coarse streamwise and spanwise spacing.

5. Shock‑turbulent boundary‑layer interaction on a 24° compression ramp – A DNS with 2865 × 298 × 420 points reproduces the experimental conditions of Bookey et al. (2015) (M∞≈2.9, Reθ≈2400). The inflow turbulent boundary layer is generated via the same tripping method, and statistical comparisons of mean profiles and fluctuations show excellent agreement with both the experiment and prior DNS studies.

Overall, the results demonstrate that the sixth‑order compact finite‑difference approach provides spectral‑like resolution for turbulent structures while retaining robust shock‑capturing capability through modest artificial diffusion. The parallel tridiagonal solver mitigates the traditional computational overhead of compact schemes, enabling efficient large‑scale three‑dimensional compressible DNS. Consequently, the presented solver constitutes a powerful tool for high‑fidelity simulations of complex compressible flows where shock–turbulence interactions are critical.