Simulation of the viscous flow past a cylinder with a discontinuous Galerkin spectral element method on a hierarchical Cartesian mesh with adaptive mesh refinement (AMR). The polynomial degree is N = 3 and the mesh uses five refinement levels. The cylinder boundary is imposed with an immersed boundary method (IBM) based on volume penalization [1], which adds a source term to the compressible Navier-Stokes equations,
$$
\partial_t \mathbf{u} + \nabla \cdot \vec{\mathbf{f}} (\mathbf{u},\nabla \mathbf{u}) = \underbrace{\beta (\mathbf{u}_s – \mathbf{u})}_{\text{IBM source}},
$$
where $\mathbf{u}$ is the vector of conservative quantities, $\vec{\mathbf{f}}$ is the Navier-Stokes flux, $\beta$ is the penalty parameter, and $\mathbf{u}_s$ is an imposed state. For this simulation, $\mathbf{u}_s$ mimics an isothermal no-slip wall boundary condition.

The simulation was obtained with the open-source code FLUXO (https://github.com/project-fluxo/fluxo).

References
[1] Kou, J., Joshi, S., Hurtado-de-Mendoza, A., Puri, K., Hirsch, C., & Ferrer, E. (2022). Immersed boundary method for high-order flux reconstruction based on volume penalization. Journal of Computational Physics, 448, 110721.