(435j) Binary Gas Mixture in a High Speed Channel

Title: Binary gas mixture in a high speed channel

Author : Dr. Sahadev Pradhan

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

ABSTRACT:

The viscous, compressible flow in a 2D wall-bounded channel, with bottom wall moving in positive x direction, simulated using the direct simulation Monte Carlo (DSMC) method [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], has been used as a test bed for examining different aspects of flow phenomenon and separation performance of a binary gas mixture at Mach number Ma = U_w / √(γ k_B T_w /m) in the range 0.1 < Ma < 30, and Knudsen number Kn = (1/(√2 π d^2 n_d H) in the range 0.1 < Kn < 10. Here, H is the channel width, U_w is the wall velocity, T_w is the wall temperature, m and d are the molecular mass and molecular diameter, n_d is the number density, and k_B is the Boltzmann constant. The generalized analytical model is formulated which includes the fifth order differential equation for the boundary layer at the channel wall in terms of master potential (χ), which is derived from the equations of motion in a rectangular (x−y) plane. The starting point of the analytical model is the Navier-Stokes, mass, momentum and energy conservation equations in the rectangular (x−y) coordinate, where x and y are the streamwise and wall-normal directions. The linearization approximation is used, where the equations of motion are truncated at linear order in the velocity and pressure disturbances to the base flow, which is an isothermal compressible Couette flow. Additional assumptions in the analytical model include high aspect ratio (length of the channel L is large compared to the height H), constant temperature in the base state (isothermal condition), and low Reynolds number (laminar flow). In this limit, the gas flow is restricted to a boundary layer of thickness (Re^{−1/2} H)at the wall of the channel. The solutions of the generalized analytical model in a high-speed channel are compared with direct simulation Monte Carlo (DSMC) simulations. The comparison reveals that the boundary conditions in the simulations and analysis have to be compared with care ((Pradhan & Kumaran, J. Fluid Mech., vol. 686, 2011, pp. 109-159); (Kumaran & Pradhan, J. Fluid Mech., vol. 753, 2014, pp. 307- 359)). The commonly used ’diffuse reflection’ boundary conditions at solid walls in DSMC simulations result in a non-zero slip velocity as well as a ’temperature slip’ (gas temperature at the wall is different from wall temperature). These have to be incorporated in the analysis in order to make quantitative predictions. When these precautions are taken,there is excellent agreement between analysis and simulations, to within 10%.

KEY WORDS: High speed channel flow, DSMC Simulations, rarefied gas flow.

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.