(222j) Dsmc Simulations of Leading Edge Flat-Plate Boundary Layer Flows at High Mach Number

The flow over a 2D leading-edge flat plate is studied at Mach number Ma = (U_inf / â??{k_B T_inf / m}) in the range 3 < Ma < 10, and at Reynolds number number Re = (L_T U_inf ρ_inf )/ μ_inf  equal to 10⁴ using two-dimensional (2D) direct simulation Monte Carlo (DSMC) simulations to understand the flow phenomena of the leading-edge flat plate boundary layer at high Mach number [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Here, L_T is the characteristic dimension, U_inf and T_inf  are the free stream velocity and temperature, ρ_inf is the free stream density, m is the molecular mass, μ_inf is the molecular viscosity based on the free stream temperature T_inf, and k_Bis the Boltzmann constant.

The variation of streamwise velocity, temperature, number-density, and mean free path along the wall normal direction away from the plate surface is studied. The qualitative nature of the streamwise velocity at high Mach number is similar to those in the incompressible limit (parabolic profile). However, there are important differences. The amplitudes of the streamwise velocity increase as the Mach number increases and turned into a more flatter profile near the wall. There is significant velocity and temperature slip ((Pradhan and Kumaran, J. Fluid Mech-2011); (Kumaran and Pradhan, J. Fluid Mech-2014)) at the surface of the plate, and the slip increases as the Mach number is increased. It is interesting to note that for the highest Mach numbers considered here, the streamwise velocity at the wall exceeds the sound speed, and the flow is supersonic throughout the flow domain. The subsonic region near the wall, expected when a no-slip boundary condition is applied, is not present when there is wall slip at sufficiently high Mach number.

In a compressible leading-edge flat plate boundary layer flows we determine the mean free path at different streamwise direction (x = 0.2, and 0.8m) away from the plate surface and found significant differences due to the variation in the streamwise location. This is due to the variation in the local temperature across the boundary layer by the viscous heating. The leading edge shock wave is evidently captured in the present DSMC simulations. An important finding is that the wall heating (by increasing the wall temperature in the simulations) increases the local mean free path at the wall. However away from the wall the mean free path profiles remains unaffected by the wall heating.

Key words:Compressible boundary layer, leading-edge flat plate, DSMC simulations.

References:

[1] Pradhan, S. & Kumaran, V. 2011 The generalized Onsager model for the secondary flow in a high-speed rotating cylinder. J. Fluid Mech. 686, 109.

[2] Kumaran, V. & Pradhan, S. 2014 The generalized Onsager model for a binary gas mixture. J. Fluid Mech. 753, 307.

[3] Bird, G. A. 1994 Molecular gas dynamics and the direct simulation of gas flows. Clarendon Press, Oxford.

[4] Bird, G. A. 1963 Approach to translational equilibrium in a rigid sphere gas. Physics of fluids 6, 1518.

[5] Cercignani, C. 2000 Rarefied gas dynamics. From basic concepts to actual calculations. Cambridge University Press.

[6] Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.

[7] Gaviglio, J. 1987 Reynolds analogies and experimental study of heat transfer in the supersonic boundary layer. Intl J. Heat Mass Transfer 30, 911.

[8] Thompson, K. W. 1987 Time dependent boundary conditions for hyperbolic systems. J. Comput. Phys. 68, 1, 24.

[9] Garcia, A. L. & Wagner, W. 2000 Time step truncation error in direct simulation Monte Carlo. Phys. Fluids. 12 , 2621.

[10] Vincenti, W. G. and Kruger, C. H. 1967 Introduction to Physical Gas Dynumics. Wiley, New York.