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DSMC SIMULATION OF TRANSITION AND TURBULENT FLOW IN A LID-DRIVEN CAVITY AT HIGH MACH NUMBER
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The flow in a 2D and 3D lid-driven cavity, with two opposite walls moving in opposite directions with equal velocity, simulated using the direct simulation Monte Carlo (DSMC) method, has been used as a test bed for examining different aspects of transition and turbulence at high Mach M = (U_w/\sqrt(k_B T_w /m)) and Reynolds numbers Re = (ρ_av U_w L_y/μ_w). Here, L_y is the smallest box dimension, U_w and T_w are the wall velocity and temperature, ρ_av is the volume-averaged gas density, and μ_w is the gas viscosity at the temperature corresponding to the wall temperature.
The transition is found to be highly sub-critical in 2D even at high Mach number, with well-separated lower and upper critical Reynolds numbers for the transition from turbulent-laminar and laminar-turbulent transitions. The transition Reynolds number increases faster than linearly with Mach number. This implies that the Knudsen number at transition (ratio of mean-free-path and system size, 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 0.025, indicating that transition is a continuum phenomenon even at high Mach numbers. The transition to turbulence is sub-critical in 3D as well. The transition Reynolds number does increase with Mach number, and the Knudsen number increases monotonically with Mach number over the parameter range studied here. There appear to be two linear regimes in the variation of the transition Reynolds number with Mach number, one with higher slope at lower Mach number, and the other with lower slope at higher Mach number.
In the present analysis of high Mach turbulent flows using DSMC method, wall slip in the temperature and the velocities are found to be significant. Slip occurs because the temperature/velocity of the molecules incident on the wall could be very different from that of the the wall, even though the temperature/velocity of the reflected molecules is equal to that of the wall. There is slip even in the mean velocity as well as the intensity of the turbulent velocity fluctuations tangential to the wall. Due to the non-zero variation in the tangential velocities, it is found that the wall-normal fluctuating velocity increases proportional to the distance from the wall, in contrast to an incompressible flow with no-slip boundary conditions where the velocity increases as the square of the distance from the wall. We find that the amplitudes of the tangential fluctuating velocities increase close to the one-third power of the distance from the wall.
In a compressible turbulent flow, we examine the seemingly contradictory result that the ratio of the mean free path (\lambda) and Kolmogorov scale (\eta) increases proportional as (M/Re^(1/4)), and it increases asymptotically with Mach number in the high Mach number limit. The simulation show that the ratio does decrease as Re^(−1/4), but it does not increase linearly with Mach number. The resolution suggested by our simulation is that even though the Mach number based on the wall velocity and temperature is large, the local Mach number based on the local dissipation velocity in regions of high shear decreases due to an increase in temperature. Due to this, the ratio of the mean free path and Kolmogorov scale appears to taper off in the high Mach number limit.
An important finding is that the ratio of the mean free path and Kolmogorov scale shows very little variation across the domain, even though the mean free path and Kolmogorov scale individually show larger variations. The ratio of the mean free path and Kolmogorov scale was also shown to be equal to the local Mach number based on the local dissipation velocity, and can also be interpreted as the square root of the ratio of the strain rate and collision frequency. All of these quantities are remarkably invariant across the domain, indicating a coupling between the local temperature and the dissipation rate in a high Mach number turbulent flow.
Author(s):
Dr. Sahadev Pradhan
Department of Chemical Engineering, Indian Institute of Science, Bangalore-560 012
India
Prof. Viswanathan Kumaran
Department of Chemical Engineering, Indian Institute of Science, Bangalore-560 012
India