Implicit Velocity Correction Schemes for Scale-Resolving Simulations of Incompressible Flow: Stability, Accuracy, and Performance

Abstract

Scale-resolving simulations of high Reynolds number incompressible flows are often limited by the Courant-Friedrichs-Lewy (CFL) stability restriction imposed by explicit time-stepping schemes, resulting in small time step sizes and long time-to-solution. In this work, we systematically compare two implicit formulations of the velocity correction scheme -- a linear-implicit approach and a sub-stepping (or semi-Lagrangian) method -- against a standard semi-implicit formulation within a high-order spectral/hp element framework. The schemes are assessed in terms of stability limits, temporal accuracy, and computational performance for implicit large-eddy simulation of the Imperial Front Wing benchmark, a complex high Reynolds number geometry with curved surfaces that imposes strict CFL constraints. Both implicit schemes extend the stability limit by up to two orders of magnitude in time step size. While increasing the cost per time step, they reduce the overall time-to-solution by up to a factor of eleven. Accuracy analysis shows that time step sizes up to twenty times larger than the explicit limit have only minor impact on resolving laminar-turbulent transition and key flow statistics. The results quantify the trade-off between stability, accuracy, and computational cost for implicit velocity correction schemes on complex geometries and provide guidance for selecting time integration strategies in large-scale scale-resolving simulations.