Instability of mixing layers: Momentum and thermal transport in the continuum breakdown regime

We examine the momentum and thermal transport in the continuum breakdown regime of a mixing layer flow, which exhibits Kelvin-Helmholtz instability under ideal continuum conditions. The Grad 13 moment model is used as it provides an adequate description of the flow physics (second-order accurate in Knudsen number) in the transition regime. Analytical solutions are developed under breakdown conditions for two-dimensional, compressible, parallel shear flows. It is shown that the deviation of viscous stress and heat flux from the Navier-Stokes-Fourier system follows two different scaling regimes depending upon the Mach number. At low Mach numbers, the departure of all stress and heat-flux components depends only upon the Knudsen number. At high Mach number, the scaling of shear stress and transverse heat flux depends on the product of the Knudsen and Mach numbers. The normal stresses depend individually on the Knudsen and Mach number. The scaling results are verified against numerical simulations of compressible mixing layers performed using the unified gas kinetic scheme for various degrees of rarefaction.

Continuum breakdown in compressible mixing layers

Gas-kinetic simulations of rarefied and compressible mixing layers are performed to characterize continuum breakdown and the effect on the Kelvin-Helmholtz instability. The unified gas-kinetic scheme (UGKS) is used to perform the simulations at different Mach and Knudsen numbers. The UGKS stress tensor and heat-flux vector fields are compared against those given by the Navier-Stokes-Fourier constitutive equations. The most significant difference is seen in the shear stress and transverse heat flux. The study demonstrates the existence of two distinct continuum breakdown regimes, one at low and the other at high convective Mach numbers. Overall, at low convective Mach numbers, the deviation from continuum stress and heat flux appears to scale exclusively with the micro-macro length scale ratio given by the Knudsen number. On the other hand, at high convective Mach numbers, the deviation depends on the global micro-macro timescale ratio given by the product of Mach and Knudsen numbers. We further demonstrate that, unlike shear stresses and transverse heat flux, the deviations in normal stresses and the streamwise heat flux depend separately on Knudsen and Mach numbers. A local parameter called the gradient Knudsen number is proposed to characterize the rarefaction effects on the local momentum and thermal transport. Noncontinuum aspects of gas-kinetic stress-tensor and heat-flux behavior that Grad's 13-moment equation model reasonably captures are identified.

Breakdown parameter for kinetic modeling of multiscale gas flows

Multiscale methods built purely on the kinetic theory of gases provide information about the molecular velocity distribution function. It is therefore both important and feasible to establish new breakdown parameters for assessing the appropriateness of a fluid description at the continuum level by utilizing kinetic information rather than macroscopic flow quantities alone. We propose a new kinetic criterion to indirectly assess the errors introduced by a continuum-level description of the gas flow. The analysis, which includes numerical demonstrations, focuses on the validity of the Navier-Stokes-Fourier equations and corresponding kinetic models and reveals that the new criterion can consistently indicate the validity of continuum-level modeling in both low-speed and high-speed flows at different Knudsen numbers.

Multiscale lattice Boltzmann approach to modeling gas flows

For multiscale gas flows, the kinetic-continuum hybrid method is usually used to balance the computational accuracy and efficiency. However, the kinetic-continuum coupling is not straightforward since the coupled methods are based on different theoretical frameworks. In particular, it is not easy to recover the nonequilibrium information required by the kinetic method, which is lost by the continuum model at the coupling interface. Therefore, we present a multiscale lattice Boltzmann (LB) method that deploys high-order LB models in highly rarefied flow regions and low-order ones in less rarefied regions. Since this multiscale approach is based on the same theoretical framework, the coupling precess becomes simple. The nonequilibrium information will not be lost at the interface as low-order LB models can also retain this information. The simulation results confirm that the present method can achieve modeling accuracy with reduced computational cost.