A wall-function approach to incorporating Knudsen-layer effects in gas micro flow simulations

For gas flows in microfluidic configurations, the Knudsen layer close to the wall can comprise a substantial part of the entire flowfield and has a major effect on quantities such as the mass flow rate through micro devices. The Knudsen layer itself is characterized by a highly nonlinear relationship between the viscous stress and the strain rate of the gas, so even if the Navier-Stokes equations can be used to describe the core gas flow they are certainly inappropriate for the Knudsen layer itself. In this paper we propose a “wall-function” model for the stress/strain rate relations in the Knudsen layer. The constitutive structure of the Knudsen layer has been derived from results from kinetic theory for isothermal shear flow over a planar surface. We investigate the ability of this simplified model to predict Knudsen-layer effects in a variety of configurations. We further propose a semi-empirical Knudsen-number correction to this wall function, based on high-accuracy DSMC results, to extend the predictive capabilities of the model to greater degrees of rarefaction. THE KNUDSEN LAYER A Knudsen layer (or kinetic boundary layer) always exists at the interface between a solid boundary and a moving gas. In Figure 1, the solid line schematically shows the structure of the Knudsen layer within a shear flow bounded by a planar wall. In this region the gas is far from a state of local thermodynamic equilibrium; consequently the Navier-Stokes equations and their associated no-slip boundary condition are inapplicable. For many flow situations the scale of the Knudsen layer (at most a few mean free paths in thickness) is negligible in comparison to the macroscopic length scale of interest. However, when the Knudsen number of the flow is greater than about 0.01 the Knudsen layer starts to impact on the entire flow field. Since at relatively low Knudsen numbers (0.01<Kn<0.1) the Navier-Stokes equations outside of the Knudsen layer may still be at least approximately valid, slip boundary conditions are often employed to incorporate these Knudsen-layer effects within a conventional fluid dynamic solution. Fundamentally, the slip boundary condition attempts to predict the actual velocity slip that occurs at the gassurface interface: uslip in Figure 1. The famous slip conditions proposed by Maxwell [1] are of this type, as are the slip conditions based on adsorption chemistry proposed by Myong [2]. However, within a Navier-Stokes formulation for the core flow, boundary conditions of this type fail to capture completely the macroscopic impact of the Knudsen layer. The dashed and dotted line in Figure 1 demonstrates why, for a simple shear flow, Navier-Stokes simulations with ordinary slip boundary conditions are inaccurate outside of the Knudsen layer. As an alternative, more sophisticated boundary conditions, such as those due to Kogan [3] and Cercignani [4], attempt to compensate for the structure of the Knudsen layer by adding a “fictitious” velocity slip to the boundary condition: uslip in Figure 1. The additional slip ensures that, at least outside the Knudsen layer, a Navier-Stokes solution can be accurate. The dashed line in Figure 1 demonstrates this principle. (Note: by comparison to kinetic theory and DSMC data it can be shown that Maxwell’s boundary condition significantly overestimates the amount of actual velocity slip. Consequently, there is a large amount of fictitious slip incorporated within Maxwell’s formulation, and this fortuitously improves its predictive capabilities.)