# Volume penalization for inhomogeneous Neumann boundary conditions modeling scalar flux in complicated geometry

## TL;DR

This paper introduces a volume penalization method for inhomogeneous Neumann boundary conditions, enabling modeling of scalar flux in complex geometries within simple Cartesian domains, and validates it through numerical experiments.

## Contribution

The paper generalizes existing flux-based volume penalization to inhomogeneous Neumann conditions, facilitating easier simulation of scalar flux in complex geometries.

## Key findings

- The method accurately models scalar flux in complex geometries.
- Convergence analysis shows reliable numerical behavior.
- Application to convection demonstrates practical effectiveness.

## Abstract

We develop a volume penalization method for inhomogeneous Neumann boundary conditions, generalizing the flux-based volume penalization method for homogeneous Neumann boundary condition proposed by Kadoch et al. [J. Comput. Phys. 231 (2012) 4365]. The generalized method allows us to model scalar flux through walls in geometries of complex shape using simple, e.g. Cartesian, domains for solving the governing equations. We examine the properties of the method, by considering a one-dimensional Poisson equation with different Neumann boundary conditions. The penalized Laplace operator is discretized by second order central finite-differences and interpolation. The discretization and penalization errors are thus assessed for several test problems. Convergence properties of the discretized operator and the solution of the penalized equation are analyzed. The generalized method is then applied to an advection-diffusion equation coupled with the Navier-Stokes equations in an annular domain which is immersed in a square domain. The application is verified by numerical simulation of steady free convection in a concentric annulus heated through the inner cylinder surface using an extended square domain.