(458j) Transition and Turbulence in a Wall-Bounded Channel Flow At High Mach Number :

Title : Transition and turbulence in a wall-bounded channel flow at high Mach number

Author(s) :  S. Pradhan  and  V. Kumaran

 Affiliation : Department of Chemical Engineering, Indian Institute of Science, Bangalore-560 012, India

Abstract :

The turbulence in the viscous, compressible flow in a wall-bounded channel is studied at high Mach M = U_m / sqrt(γk_BT_w/m), and Reynolds numbers Re = (ρ_mU_mH)/μ_w using direct simulation Monte Carlo (DSMC) simulations. Here, H is the channel half-width, U_m is the mean velocity, ρ_m is the mean density, T_w is the wall temperature, m is the molecular mass, μ_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 simulations, though its variation with Mach number is qualitatively similar.

 In the turbulent channel flow,  there is slip in both the mean and  fluctuating velocities at the wall;  there is also significant temperature slip. These have to be incorporated in continuum DNS models to get accurate results. The variation of the ratio of the mean free path and Kolmogorov scale show the same power law behavior as those in a lid-driven cavity.  In the inertial sub-layer,  the Van-Driest transformed velocity is found to accurately follow the logarithmic law of the wall.

We have also find 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 intermolecular distance is smaller than the cell size, 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, are found to be in good agreement with simulations, for two different molecular models, the hard-sphere and the variable hard-sphere.