Boundary conditions in the Transitional Flow interface are kinetically motivated and assume the diffuse reflection or emission of molecules from surfaces. Molecules can be emitted from surfaces with a drifting or nondrifting velocity distribution, which enables a range of boundary conditions to be formulated. In the Navier–Stokes limit, the emission of molecules from a surface with a drifting Maxwellian enables pressure and velocity boundary conditions to be formulated. In the molecular flow limit a range of boundary conditions based on the diffuse emission of molecules from stationary walls are available. The Wall boundary condition itself is valid across the full range of Knudsen numbers. When constructing models the validity of the boundary conditions must be carefully considered.

The diffuse reflection boundary condition was first derived for the lattice Boltzmann method (which is used by the Transitional Flow interface) by Ansumali and Karlin (Ref. 3). COMSOL Multiphysics uses a slightly modified version of this boundary condition, for consistency with the Free Molecular Flow interface. In Ref. 3 a detailed formal derivation of the boundary condition is given, based on general kinetic theory arguments. Here, the boundary condition is derived using the assumption of Knudsen’s law for the emission of molecules from the boundary.

To account for the motion of the wall (at velocity uw), consider the distribution functions in a moving frame, such that the wall is stationary. In this case the continuum velocity distribution function, f(ξ ,x,t), can be written as g(χ ,x,t), where χ = ξ – uw. Knudsen’s law states that the emitted flux of molecules, J(χ ,x,t), is related to the equilibrium distribution by (Ref. 5)

where α is a normalization constant, and geq(χ ,0,ρ) is given by Equation 3-1. α is determined by setting the total outgoing flux, J, to a fixed value, giving:

For a wall boundary condition, the total outgoing flux is usually equated to the total incoming flux, G, given by

The symbol χ' is used to represent velocity vectors in the incoming half of velocity space. The fluxes J(χ ,x ,t) and G(χ' ,x ,t) can be written in terms of the distribution function by considering that the rate of molecules arriving at the surface is related to the normal velocity component of the molecules traveling at velocity χ, as well as the number density of these molecules:

Assuming J = G and substituting Equations 3-5, 3-6, and 3-7 into Equation 3-4 gives the result

The asterisk indicates that the variable g* refers to only the outgoing velocity distribution function. Considering the form of geq(χ ,0 ,ρ) from Equation 3-1 it is clear that the equilibrium function is completely symmetric about the origin, so the following result applies:

Equation 3-8 is therefore equivalent to the result derived in Ref. 3 (note that the COMSOL Multiphysics convention of n represents the outward normal of the domain, whilst in Ref. 3 n is defined as the inward normal).

Considering Equation 3-1 again, it is clear that geq(χ ,0 ,ρ) = feq(ξ ,u, ρ) so Equation 3-5 can be transformed back to the stationary reference frame in the following manner:

Similarly Equation 3-6 becomes

The validity of the discretization form was proved by Ansumali and Karlin (Ref. 3).

Equation 3-9 and Equation 3-10 form the basis of all the boundary conditions used in the Transitional Flow interface. For wall boundary conditions at all Knudsen numbers the relation J = G is used to define the boundary condition. In the molecular flow limit J is usually defined on the basis of kinetic theory expressions for the effusive molecular flux, with uw = 0. In the Navier–Stokes limit a fluid velocity boundary condition can be implemented by setting uw = ufluid and J = G (ufluid is the user-defined fluid velocity). The pressure boundary condition in the Navier–Stokes limit is more complex, and is described next.

Specifying a pressure in the fluid in the Navier–Stokes limit is in general difficult with a kinetic equation. In the molecular flow limit the kinetic theory expressions for the flux due to molecular effusion from a reservoir at constant pressure can be employed. In this case the flux enters the domain with a stationary Maxwellian. In the Navier–Stokes limit the incoming flux is harder to specify because the molecules entering the domain do so with a Maxwellian that is drifting at the velocity of the fluid. Pressure boundary conditions based on solutions of the Boltzmann equation within Grad’s 13 moment system have been developed (Ref. 2 and Ref. 6), but these are complex. A simpler approach is used in the Transitional Flow interface, which produces good results when compared with Navier–Stokes fluid flow benchmarks.

In Equation 3-9 there are two unknowns: the emitted flux and the velocity of the Maxwellian. To make the incoming molecules enter with a velocity distribution determined by the fluid flow in the Navier–Stokes limit, uw = u is set, where u is the local fluid velocity (a function of both the outgoing and incoming distribution functions; note that at this stage the outgoing distribution functions are not determined). The amount of gas entering the fluid flow domain is not straightforward to determine, but it is possible to adjust the flux entering the domain continuously to ensure that the pressure remains constant. In COMSOL Multiphysics this is done by means of a weak form equation for the emitted flux, defined on the boundary. This approach ensures that the pressure is fixed on the boundary whilst allowing the velocity to be determined by the overall fluid flow. This is the desired behavior for a pressure boundary condition in the Navier–Stokes limit. Note that for converged solutions it is observed that the computed emitted flux is equal to the incoming flux, which justifies the use of the moving wall boundary condition in this manner.