Characterizing deviation from equilibrium in direct simulation Monte Carlo simulations

Abstract

A fundamental and yet computationally feasible parameter based on the characteristic function of the velocity distribution function (VDF) is proposed for determining the deviation from near-equilibrium conditions in rarefied flow simulations using the direct simulation Monte Carlo (DSMC) method. The proposed parameter utilizes the one-to-one correspondence between the VDF and its characteristic function (or Fourier transform), thereby correlating the deviation of the VDF (from a Chapman-Enskog VDF) with the deviation of the characteristic function (also from that of a Chapman-Enskog VDF). The results are first presented for an unsteady Bobylev solution for approach to equilibrium in 0-D, free-molecular Fourier-Couette flow problem and the Mott-Smith solution for the shock wave all of which have analytical solutions for the VDF, thereby confirming that the proposed parameter indeed captures the deviation from near-equilibrium conditions accurately. The utility of the proposed parameter is then demonstrated using two benchmark problems—Couette flow (over a range of Knudsen numbers) and structure of a normal shock (for upstream Mach numbers of 1.5, 3, and 5)—solved using the DSMC method. While the current work only presents results for benchmark one-dimensional DSMC simulations, the approach can be extended easily to rarefied flows in higher dimensions. Therefore, the proposed parameter has the potential to be used for understanding the nature of VDF and its deviation from near-equilibrium conditions at all locations in a flow field without the need for explicitly sampling the VDF.