On the application of higher-order Backward Difference (BDF) methods for computing turbulent flows

Abstract

The Backward Differentiation Formulae (BDF) of orders one to six are implemented in an in-house three-dimensional finite-volume (FVM) compressible flow code that operates on unstructured meshes. The lower-upper symmetric Gauss-Seidel (LUSGS) implicit relaxation technique based on the splitting of convective Jacobians is used to obtain a variable-order stable implementation in the solver. The temporal order of accuracy of the implemented schemes is verified using the Method of Manufactured Solutions (MoMS). Practical engineering flow problems are simulated to investigate the operational stability of BDF methods so that these can be used as a cheaper alternative to multi-stage methods in RANS, hybrid RANS-LES, and LES implementations. Steady-state flow simulations of supersonic flow over a flat plate and high Reynolds number flow over a NACA-0012 airfoil show that algorithm with BDF schemes of orders 1-5 is stable at high CFL numbers (700-1000) and produces a converged solution. Stokes' second problem and a high-resolution delayed detached eddy simulation (DDES) of flow over a 3D circular cylinder using the BDF1-BDF6 methods demonstrate the use of these higher-order temporal schemes in unsteady laminar and turbulent flows. It is concluded that BDF methods of orders 3-5 can be practically employed to achieve higher levels of temporal accuracy in flow simulations when the level of spatial accuracy is also high (ENO/WENO schemes, spectral methods, etc.) in hybrid RANS-LES, LES or DNS. Even without higher-order spatial accuracy, the same BDF schemes can be used in Adaptive Time-stepping (ATS) methods to obtain prescribed temporal accuracy efficiently by algorithmically switching between the different schemes.