A Shifted Boundary Method for Thermal Flows

TL;DR

The paper addresses the challenge of accurately simulating coupled thermal and incompressible flows in geometrically complex domains where boundary-fitted meshes are costly. It proposes an Octree-based Shifted Boundary Method (Octree-SBM) combined with a linear semi-implicit Navier–Stokes solver and RB-VMS for the energy equation, operating on a surrogate domain with a mapping to enforce Dirichlet and Neumann conditions on the true boundary. The approach is validated across 2D and 3D cases spanning natural, forced, and mixed convection, demonstrating accurate flux calculations (Nu) and boundary enforcement, along with strong parallel scalability on large HPC systems. The results indicate that Octree-SBM provides a robust, efficient framework for multiphysics simulations on complex geometries, enabling rapid meshing and accurate boundary treatment without boundary-fitted grids, with potential extensions to FSI, AMR, and higher-order discretizations.

Abstract

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.

Figures (31)

• Figure 1: The surrogate domain, its boundary, and the distance vector $\boldsymbol{d}$.
• Figure 1: Mesh convergence results (spatial convergence).
• Figure 2: Diagram illustrating the block iteration technique used to perform multiphysics simulations of thermal incompressible flow (NSHT).
• Figure 2: Illustration of the variation in timestep during the MMS testing. The timestep increases linearly from $\frac{k}{4}$ to $k$, followed by a linear decrease from $k$ to $\frac{k}{4}$, with $k = 0.157$. This pattern is shown in the figure, providing a graphical representation of the time step variation throughout the simulation.
• Figure 3: Schematic graph of various NSHT simulations performed in the paper.

Theorems & Definitions (4)

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