Efficient computation of unsteady turbulent flows using an implicit solver and an adaptive time-stepping algorithm

The Courant–Friedrichs–Lewy (CFL) condition restricts the time-step size in explicit time-stepping methods, resulting in substantial overall simulation times for unsteady problems. Implicit methods are stable for larger CFL numbers (10-1000), thus reducing the computational time needed to achieve a steady-state solution. However, larger time-step sizes and approximations made in the LUSGS solver destroy the temporal accuracy of the solution, necessitating the application of dual time-stepping (DTS) for unsteady flows. DTS methods employ a pseudo-time-loop inside each physical time-step calculation and are even more computationally expensive. In this work, we show that an adaptive time-stepping (ATS) algorithm based on the variable time-step backward differentiation formula can be used to compute unsteady flows in a fraction of the time taken by the DTS method. This adaptive algorithm which employs a local error control strategy to calculate time-step sizes is implemented in an in-house parallel implicit unstructured grid solver and is also employed to obtain rapid convergence in false transient simulations. For transient flow solutions, the enhanced computational efficiency is demonstrated through Stokes second problem and 3D turbulent flow over a circular cylinder at Reynolds number 5000. The ATS algorithm is based on the block LUSGS iterative solver and is shown to be at least five times faster compared to the DTS method in terms of wall clock time.