Tree-Based Adaptive Mesh Refinement on Line-Based Unstructured Grids for Airfoil Flows

For unstructured grids, a solution algorithm using Hamiltonian paths is computationally efficient because it utilizes hidden one-dimensional structures in the grids to perform finite difference reconstruction and line-implicit time integration along each of the inherent Hamiltonian paths to improve the accuracy of the solutions. However, a point-continuous pure quadrilateral grid is a prerequisite for identifying these hidden Hamiltonian line structures, and this requirement limits the capability of the algorithm to introduce local mesh refinement because the grid refinement process generates hanging nodes in the quadrilateral grid. In this paper, a tree-based grid data structure is introduced to enable multilevel local mesh refinement/coarsening to improve the accuracy of a solution algorithm utilizing Hamiltonian paths. Solution-based adaptive mesh refinement (AMR) is applied using Hessian-based criteria to identify areas where mesh refinement/coarsening is required. The finite difference reconstructions and line-implicit time integration algorithms are modified to include multilevel ghost cells to accommodate hanging nodes introduced by the refinement/coarsening process. The proposed AMR technique is evaluated using various two-dimensional viscous flow simulations for three airfoils: NACA 0012, Eppler 387, and CRM-HL. Both steady and unsteady flow simulations are performed at a variety of flow conditions to assess the capability of the AMR algorithm to capture various flow phenomena including laminar-turbulent flow transition point, flow separation bubble, boundary layer profile, aerodynamic load hysteresis, and shock wave formation/dissolution. The AMR results demonstrate significant improvements in solution accuracy with minimal increment in computational cost compared to the corresponding non-AMR simulations at the same initial grid resolution level. In particular, the proposed AMR algorithm shows approximately 10 times speed-up and reduced number of computational cells over comparable non-AMR simulations at the same highest grid resolution level with similar solution accuracy.