On the All-Speed Roe-type Scheme for Large Eddy Simulation of Homogeneous Decaying Turbulence

As the representative of the shock-capturing scheme, the Roe scheme fails to LES because important turbulent characteristics cannot be reproduced such as the famous k-5/3 spectral law owing to large numerical dissipation. In this paper, the Roe scheme is divided into five parts: , , , , and , which means basic upwind dissipation, pressure-difference-driven and velocity-difference-driven modification of the interface fluxes and pressure, respectively. Then, the role of each part on LES is investigated by homogeneous decaying turbulence. The results show that the parts , , and have little effect on LES. It is important especially for because it is necessary for computation stability. The large numerical dissipation is due to and , and each of them has much larger dissipation than SGS dissipation. According to these understanding, an improved all-speed LES-Roe scheme is proposed, which can give enough good LES results for even coarse grid resolution with usually adopted reconstruction.

💡 Research Summary

The paper investigates why the classical Roe shock‑capturing scheme, widely used for compressible flow computations, performs poorly in large‑eddy simulation (LES) of homogeneous decaying turbulence (HDT). The authors first decompose the Roe scheme’s numerical dissipation into five distinct components: (1) ξ, the basic upwind dissipation that guarantees stability but introduces excessive artificial viscosity; (2) pUΔ, a pressure‑difference‑driven modification of the interface velocity, which has negligible physical dissipation but is essential for suppressing the checkerboard pressure‑velocity decoupling; (3) uUΔ, a velocity‑difference‑driven flux modification that contributes almost no dissipation; (4) ppΔ, a pressure‑difference‑driven pressure correction with a small negative dissipation; and (5) upΔ, a velocity‑difference‑driven pressure correction that is responsible for the well‑known non‑physical behaviour problem (pressure scaling with Mach number) and adds a large amount of artificial dissipation comparable to ξ.

To assess the impact of each component on LES, nine test cases are constructed on three grid resolutions (32³, 64³, 128³). The cases range from a pure central‑difference scheme with the Smagorinsky SGS model (Cen‑SMA) to the full Roe scheme, and to variants where only one of the five components is active. The simulations employ a third‑order MUSCL reconstruction, a four‑stage Runge‑Kutta time integrator, and a standard finite‑volume framework.

Results show that the central‑difference scheme without any dissipation (Case 2) and the variants containing only pUΔ, uUΔ, or ppΔ diverge because they lack any stabilising dissipation. The ξ‑only case (Case 4) and the upΔ‑only case (Case 8) both produce spectra that are far steeper than the theoretical k⁻⁵⁄³ law; the former because ξ is overly dissipative, the latter because upΔ mimics a large SGS term while also causing non‑physical pressure scaling. The full Roe scheme (Case 3) suffers from the same problem, yielding a high‑wavenumber slope close to –5, confirming that the combined effect of ξ and upΔ overwhelms the physical SGS dissipation.

Crucially, the pUΔ component, while not needed for reproducing the k⁻⁵⁄³ inertial range, is indispensable for numerical stability: removing it leads to checkerboard pressure oscillations and eventual blow‑up. The uUΔ and ppΔ components have negligible influence on both stability and spectral accuracy.

Guided by these observations, the authors propose an improved all‑speed LES‑Roe scheme. The key modifications are: (i) reduce the coefficient of ξ to 0.5, thereby halving the basic upwind dissipation; (ii) completely eliminate the upΔ term to remove the non‑physical pressure scaling and excess artificial viscosity; (iii) retain pUΔ to keep the checkerboard‑suppressing mechanism; (iv) keep uUΔ and ppΔ unchanged because they have little effect. The resulting formulation is expressed in equations (29) and (30) of the paper.

Testing the new scheme (denoted 0.5 ξ) demonstrates that, even on the coarse 32³ grid, the kinetic‑energy spectrum closely follows the k⁻⁵⁄³ law over a substantially wider wavenumber range than the original Roe scheme, and is comparable to the reference Cen‑SMA results. Visualisations of vorticity iso‑surfaces confirm that small‑scale vortex tubes are preserved, unlike the original Roe case where they are almost completely damped. Statistical measures such as the resolved skewness tensor also fall within the same range as the reference LES, indicating that the modified scheme reproduces key turbulence statistics without the need for an explicit SGS model.

In summary, the paper provides a systematic decomposition of the Roe scheme’s numerical dissipation, identifies the components responsible for excessive artificial viscosity and non‑physical pressure behaviour, and constructs a lightweight, all‑speed Roe variant that satisfies LES requirements (numerical dissipation ≪ physical SGS dissipation) while retaining stability on low‑Mach, low‑resolution grids. This work opens a practical pathway for using shock‑capturing schemes in LES of mixed‑Mach-number flows, such as those encountered in aerospace, propulsion, and atmospheric applications, without resorting to high‑order reconstructions or complex preconditioning.