A gas-kinetic scheme for the simulation of turbulent flows

TL;DR

This paper introduces a gas-kinetic numerical scheme for simulating turbulent flows, demonstrating improved accuracy over traditional Navier-Stokes methods, especially in regions with high turbulent Knudsen numbers such as shock layers.

Contribution

The paper presents a physically consistent gas-kinetic scheme for turbulent flow simulation that outperforms Navier-Stokes-based methods without relying on eddy viscosity assumptions.

Findings

Gas-kinetic scheme performs better in shock-boundary layer interactions.

It handles high turbulent Knudsen numbers more effectively.

Outperforms Navier-Stokes in turbulent regimes with complex flow features.

Abstract

Numerical schemes derived from gas-kinetic theory can be applied to simulations in the hydrodynamics limit, in laminar and also turbulent regimes. In the latter case, the underlying Boltzmann equation describes a distribution of eddies, in line with the concept of eddy viscosity developed by Lord Kelvin and Osborne Reynolds at the end of the nineteenth century. These schemes are physically more consistent than schemes derived from the Navier-Stokes equations, which invariably assume infinite collisions between gas particles (or interactions between eddies) in the calculation of advective fluxes. In fact, in continuum regime too, the local Knudsen number can exceed the value 0.001 in shock layers, where gas-kinetic schemes outperform Navier-Stokes schemes, as is well known. Simulation of turbulent flows benefit from the application of gas-kinetic schemes, as the turbulent Knudsen number (the ratio between the eddies' mean free path and the mean flow scale) can locally reach values well in excess of 0.001, not only in shock layers. This study has investigated a few cases of shock - boundary layer interaction comparing a gas-kinetic scheme and a Navier-Stokes one, both with a standard k-\omega turbulence model. Whereas the results obtained from the Navier-Stokes scheme are affected by the limitations of eddy viscosity two-equation models, the gas-kinetic scheme has performed much better without making any further assumption on the turbulent structures.