An efficient treatment of heat-flux boundary conditions in GSIS for rarefied gas flows

TL;DR

The paper tackles the problem of efficiently enforcing heat-flux boundary conditions in rarefied gas simulations within the GSIS framework. It introduces a consistent macroscopic boundary-flux update that splits the boundary flux into outgoing and incoming parts and uses Maxwellian-based increments together with closed-form half-space moments to update wall parameters $(T_w,\rho_w)$ so that impermeability and energy constraints are satisfied in the macroscopic stage. This approach preserves the asymptotic-preserving and fast-convergence properties of GSIS across Knudsen regimes while significantly reducing wall-parameter iterations. Three challenging test cases—the hypersonic flow through a 3D nozzle, hypersonic flow around an adiabatic 2D cylinder, and steady heat transfer in a 2D annulus with an inner heat-flux wall—show good agreement with DSMC and substantial efficiency gains, e.g., order-of-magnitude reductions in iterations and substantial wall-clock savings compared with conventional CIS. The method provides a practical and robust route to accurate simulations of heat-flux boundary conditions in rarefied gas dynamics and can be extended to more complex gas models and surface interactions.

Abstract

Heat-flux boundary conditions are challenging to implement efficiently in rarefied gas flow simulations because the wall-reflected gas temperature and density must be determined dynamically during the computation. This paper aims to tackle this problem within the general synthetic iterative scheme (GSIS), where the Boltzmann kinetic equation is solved deterministically in an outer loop and macroscopic synthetic equations are solved in an inner loop. To avoid kinetic-macroscopic boundary-flux mismatch and the resulting convergence bottlenecks, for the macroscopic boundary flux at every inner iteration, the incident increment is estimated using a Maxwellian distribution, and then the reflected contribution is obtained by boundary conditions consistent with those in the kinetic solver. In addition to retaining the fast-converging and asymptotic-preserving properties of GSIS, the proposed method significantly reduces the iterations required to determine the wall-reflected gas parameters. Numerical simulations of rarefied gas flows in and around a 3D nozzle, a 2D adiabatic cylinder, and a 2D annular heat-transfer configuration show good agreement with the direct simulation Monte Carlo method, while achieving substantial efficiency gains over conventional iterative schemes.

Figures (5)

• Figure 1: Schematic for the GSIS algorithm: the kinetic solver computes an intermediate distribution $f^{n+1/2}$ using Eq. \ref{['eq:cis']} and the associated macroscopic quantities, followed by $M$ macroscopic inner iterations to solve Eq. \ref{['eq:macro_update_mono']}, where $m$ denotes the $m$-th inner macroscopic iteration, and a final VDF update as per Eq. \ref{['eq:updatef_mono']} to obtain $f^{n+1}$.
• Figure 2: 3D computational mesh and nozzle geometry. Top: overall computational domain and hexahedral mesh with $335{,}740$ cells. Bottom: nozzle contour in the $x$--$z$ plane following the axisymmetric 2D geometry reported in jin2024nozzle. Only the geometric contour is adopted, while the present flow conditions and numerical setup are newly defined.
• Figure 3: Comparisons of $C_p$ and $C_h$ along the nozzle inner wall as functions of $x/L_0$ between DSMC and GSIS at $\mathrm{Kn}=0.5276$ (top) and $\mathrm{Kn}=0.2638$ (bottom) with $\mathrm{Ma}=5$.
• Figure 4: Comparisons of velocity and temperature between DSMC (contours) and GSIS (black lines) with $\text{Ma} = 5$. In each figure, the top and bottom half spatial regions show the results of $\text{Kn} = 0.1$ and 0.01, respectively.
• Figure 5: For the annular configuration with $\text{Kn}=0.01$, comparisons between the CIS and the GSIS method are presented for the (a) temperature, (b) heat flux, and (c) the convergence history of the net heat flux at the outer isothermal wall. Note that the first five steps in the GSIS method evolve the initial field using the CIS update.