Vorticity-Based Polynomial Adaptation for Unsteady Flows

Aerospace applications, including but not limited to the aerodynamics of slats and flaps, the aeroacoustics of complete landing gear configurations, and the performance prediction of air brakes, spoilers, fans, and turbines, benefit from high-fidelity scale resolving approaches such as direct numerical simulation and large eddy simulation. These approaches are shown to be accurate in complex unsteady flow regimes where current industry-standard Reynolds-averaged Navier-Stokes methods often fail. For example, recent research has demonstrated that direct numerical simulation can be utilized for analysis of complete jet engine low-pressure turbine cascades at their designed operating condition. High-order unstructured spatial discretizations such as flux reconstruction are particularly appealing for large eddy simulation of unsteady turbulent flows as they are capable of resolving the flow more efficiently; however, their computational cost remains relatively high, preventing their industrial use. Therefore, given the inherent computational cost of high-order methods, it is crucial to deploy strategies to minimize the overall computational cost of large eddy simulation while attaining comparable accuracy. A practical approach is to limit the use of high-order elements to regions where higher resolution is required for an accurate discrete approximation of the solution, reducing the total number of degrees of freedom. This can be achieved by locally increasing or decreasing the solution polynomial degree to adjust resolution and order of accuracy to achieve an accurate and cost-efficient simulation, a practice called polynomial adaptation or p-adaptation. Large eddy simulation can also benefit from the computational power of state-of-the-art hardware architectures by taking advantage of parallel computing, which can achieve a significant speed-up by decomposing the domain and solving the problem concurrently. Parallel computing also provides a larger memory capacity and higher bandwidth, resulting in better performance. However, combining polynomial adaptation with a massively parallel computation can cause overhead because p-adaptation tends to change the degree of solution polynomials locally, leaving the computational domain unbalanced. To maintain parallel efficiency, dynamic load balancing techniques have been exploited. The current work introduces a novel dynamically load-balanced polynomial adaptation method for simulations of unsteady flows in the vicinity of complex geometries using the flux reconstruction scheme. This approach is applied to both two-dimensional and three-dimensional studies, while for the latter, we make use of implicit large eddy simulation, which relies on the dissipation error of the numerical scheme to act as a subgrid-scale model to dissipate the kinetic energy from the smallest turbulent scales in the flow. We verify the utility of our dynamically load-balanced adaptation approach when applied to the arbitrary Lagrangian-Eulerian form of the compressible Navier–Stokes equations for a range of applications on moving and deforming domains. Specifically, we verify the arbitrary Lagrangian-Eulerian and dynamic load balancing implementation by performing simulations of an Euler vortex, and then illustrate the accuracy and efficiency of the adaptation routine by performing simulations of flow over an oscillating circular cylinder with two different flow settings, dynamic stall of a 2D NACA 0012 airfoil undergoing heaving and pitching motions, and flow over a vertical axis wind turbine composed of two NACA 0012 airfoils. We further illustrate the accuracy and efficiency of the routine when applied to transitional and turbulent flows by performing simulations of a shallow dynamic stall of a 3D SD 7003 airfoil undergoing heaving and pitching motions and transitional flow over a 3D circular cylinder. Results demonstrate that the dynamically load-balanced adaptation algorithm can track regions of interest, such as vortices and boundary layers, and yields a significant speed-up when applied to parallel simulations. We also demonstrate the scalability of the algorithm and the capability of dynamic load balancing technique to distribute uniform computational load among processors