A Shifted Boundary Method for Thermal Flows

This paper presents an incomplete Octree mesh implementation of the Shifted Boundary Method (Octree-SBM) for multiphysics simulations of coupled flow and heat transfer. Specifically, a semi-implicit formulation of the thermal Navier-Stokes equations is used to accelerate the simulations while maintaining accuracy. The SBM enables precise enforcement of field and derivative boundary conditions on cut (intercepted) elements, allowing for accurate flux calculations near complex geometries, when using non-boundary fitted meshes. Both Dirichlet and Neumann boundary conditions are implemented within the SBM framework, with results demonstrating that the SBM ensures precise enforcement of Neumann boundary conditions on Octree-based meshes. We illustrate this approach by simulating flows across different regimes, spanning several orders of magnitude in both the Rayleigh number ($Ra \sim 10^3$--$10^9$) and the Reynolds number ($Re \sim 10^0$--$10^4$), and covering the laminar, transitional, and turbulent flow regimes. Coupled thermal-flow phenomena and their statistics across all these regimes are accurately captured without any additional numerical treatments, beyond a Residual-based Variational Multiscale formulation (RB-VMS). This approach offers a reliable and efficient solution for complex geometries, boundary conditions and flow regimes in computational multiphysics simulations.