Transition and Turbulence in a Wall-Bounded Channel Flow at High Mach Number
Authors: Pradhan Dr. Sahadev 
Published: February 2020

The turbulence in the viscous, compressible flow in a 3D wall-bounded channel, simulated using the direct simulation Monte Carlo (DSMC) method, has been used as a test bed for examining different aspects of transition and turbulence ([1, 2, 3]) at high Mach Ma = U_m / \sqrt(gamma k_B T_w /m), and Reynolds numbers Re = (rho_m U_m H)/mu_w. Here, H is the channel half-width, U_m is the mean velocity, rho_m is the mean density, T_w is the wall temperature, m is the molecular mass, mu_w is the molecular viscosity based on the temperature at the isothermal wall, and k_B is the Boltzmann constant.
The laminar-turbulent transition is accompanied by a discontinuous change in the friction factor even at high Mach number. The transition Reynolds number increases faster than linearly with Mach number, and the Knudsen number at transition (also proportional to the ratio of Mach and Reynolds numbers) passes through a maximum as the Mach number is increased. This maximum value is small, less than 0.009, indicating that transition is a continuum phenomenon even at high Mach numbers. The transition Reynolds number predicted by the linear stability analysis is significantly higher than that observed in the simulations, though its variation with Mach number is qualitatively similar.
In a high Mach turbulent channel flow wall slip in the temperature and the velocities ([2]) are found to be significant. Interestingly, we find that the slip in the streamwise fluctuating velocity is higher than that in the mean velocity at high Mach number. We find that the amplitudes of the tangential fluctuating velocities increase between (0.2 - 0.4) power of the distance from the wall.
In a compressible turbulent channel flow, we examine the result that the ratio of the mean free path and Kolmogorov scale increases proportional as (Ma/Re^{1/4}), and it increases asymptotically with Mach number in the high Mach number limit. The simulation show that the ratio (mean free path to Kolmogorov scale) does decrease as (Re^{-1/4}), but it does not increase linearly with Mach number. This is due to the decrease in the local Mach number within the channel, due to the increase in the temperature by viscous heating.
We have also found that the smallest length scale for the velocity gradients is comparable to, or smaller than, the mean free path. Though this appears unusual, it should be noted that the smallest length for the gradients is the distance between molecules, and not the mean free path. In our simulations, the inter-molecular distance turns out to be much smaller than the mean free path. In fact, the inter-molecular distance is smaller than the cell size (we have 200 simulated molecules per cell), whereas the Kolmogorov scale and the mean free path are larger than the cell size.
Even though the distance between molecules is the smallest length scale for gradients, the mean free path is the length scale for molecular transport, since the kinematic viscosity and thermal conductivity are proportional to the product of the mean free path and the fluctuating velocity. The present results suggest that the smallest scale for transport could be much larger than the smallest scale for gradients, thereby suggesting non-local transport in high Mach number turbulent flows at the smallest scales.
A modification of the linear velocity profile in the viscous sub-layer near the wall, which takes into account temperature and density variations, is derived. The power law variation of the velocity and temperature is predicted under the assumption that the increase in temperature across the viscous sub-layer is larger than the wall temperature. It is found that the scaling laws do depend on the molecular model, through the dependence of viscosity and thermal conductivity on the temperature. The predicted power law, is found to be in good agreement with simulations, for two different molecular models, the hard-sphere and the variable hard-sphere.

References:

[1] Bird, G.A. 1994 Molecular gas dynamics and the direct simulation of gas flows. Clarendon Press, Oxford.
[2] Pradhan, S. & Kumaran, V. 2011 The generalized Onsager model for the secondary flow in a high-speed rotating cylinder. J. Fluid Mech. 686, 109.
[3] Spina, E.F. & Smits, A.J. & Robinson, S. K. 1994 The physics of supersonic turbulent boundary layers. Ann. Rev. Fluid Mech. 26, 287.