A volume penalization method for solving conjugate scalar transport with interfacial jump conditions

TL;DR

This paper advances a volume penalization approach for solving conjugate scalar transport with interfacial jump conditions on complex geometries. By introducing a divergence-form source term and an equivalent scalar representation, it yields a unified governing equation capable of handling both Neumann boundary conditions and jumps in the scalar and its flux across fluid–solid interfaces. Verification against analytical solutions and body-fitted mesh results demonstrates first-order accuracy and sub-3% deviations, while mitigating non-physical diffusion in the solid region. The method's demonstrated accuracy and compatibility with OpenFOAM highlight its potential for robust, geometry-agnostic simulations of multiphase transport in engineering applications.

Abstract

Conjugate scalar transport with interfacial jump conditions on complex interfacial geometries is common in thermal and chemical processes, while its accurate and efficient simulations are still quite challenging. In the present study, a novel treatment of a two-phase interface in the volume penalization method, a kind of immersed boundary method, for solving conjugate scalar transport with general interfacial boundary conditions is developed. We first propose an interfacial treatment for solving an advection-diffusion equation with a Neumann boundary condition, and then extend it to general conjugate scalar transport with both interfacial flux and scalar jumps. A one-dimensional diffusion problem is solved to verify the present scheme and demonstrate the advantage of the present scheme in improving accuracy and unifying the governing equations in the two phases with an additional source term representing the local jump condition of the interfacial scalar flux. Then, the present scheme is further applied to fluid-solid coupled scalar diffusion and advection-diffusion problems with the scalar and its flux jumps across the interface. The simulation results of the present scheme generally show good agreement with reference results obtained by body-fitted mesh simulations with average relative deviations less than 3.0%.

Figures (18)

• Figure 1: Schematic of conjugate scalar transport with interfacial jump conditions.
• Figure 2: Profiles of the level-set function $\phi_{0}$, the phase indicator $\phi$, and its gradient $\frac{d\phi}{d\phi_0}$ in the vicinity of the fluid-solid interface.
• Figure 3: (a) Schematic and (b) profile of the distribution of the scalar flux around the interface.
• Figure 4: Schematic of the control volume over the interfacial region. Here, $\psi$ and $\theta$ denote the azimuthal and the polar directions with the corresponding radius of curvature $\mathit{R_\psi}$ and $\mathit{R_\theta}$, respectively, in a three-dimensional polar coordinate system with the origin $\mathit{O}$.
• Figure 5: Schematic of the one-dimensional diffusion problem.