Formulations for scalar boundedness in simulations of turbulent compressible multi-component flows using high-order finite-difference methods

TL;DR

This paper introduces high-order finite-difference formulations that preserve scalar boundedness in turbulent compressible multi-component flow simulations without predefined bounds, balancing accuracy and numerical dissipation.

Contribution

It proposes adaptive formulations combining non-dissipative and dissipative fluxes, improving scalar boundedness without sacrificing high-order accuracy.

Findings

The monotonicity-preserving limiter scheme outperforms others in boundedness and accuracy.

The schemes effectively handle sharp scalar gradients in under-resolved turbulent flows.

Adaptive switching between flux formulations maintains low numerical dissipation.

Abstract

Preserving scalar boundedness is important for numerical schemes used in turbulent compressible multi-component flow simulations to prevent unphysical results and unstable simulations. However, ensuring scalar boundedness for high-order, low-dissipation numerical schemes poses challenges in highly under-resolved conditions due to inherent dispersion errors that generate spurious oscillations. Numerical dissipation is needed to mitigate these oscillations, but excessive dissipation negatively affects resolution. In this work, we propose formulations for high-order finite-difference schemes to preserve scalar boundedness without predefined bounds, while maintaining high accuracy and low numerical dissipation. The proposed formulations augment a non-dissipative numerical flux of a high-order central-difference scheme with an explicit dissipative numerical flux that adaptively switches between high-order and low-order formulations. Building on a deliberate choice of the non-dissipative flux, we construct two schemes using Jameson's artificial viscosity method and a monotonicity-preserving limiter as the dissipative flux. We examine the schemes in one-dimensional scalar advection problems and a three-dimensional temporal turbulent mixing-layer case involving sharp scalar gradients and under-resolved conditions, evaluating their accuracy, boundedness of species mass fractions, and numerical diffusivity. The scheme with the monotonicity-preserving limiter demonstrates superior performance.