Fundamentals of Turbulent Flows

We discuss properties of numerical methods that are essential for high-fidelity (LES, DNS) simulations of turbulent flows. In choosing a numerical method, one must be cognizant of the broadband nature of the solution spectra and the resolution of turbulent structures. These requirements are substantially different than those in the RANS approach, where the solutions are smooth and agnostic to turbulent structures. We focus on spatial discretization of the governing equations in canonical flows where Fourier analysis is helpful in revealing the effect of discretization on the solution spectra. In high-fidelity numerical simulations of turbulent flows, it is necessary that conservation properties inherent in the governing equations, such as kinetic energy conservation in the inviscid limit, are also satisfied discretely. An important benefit of adhering to conservation principles is the prevention of nonlinear numerical instabilities that may manifest after long-time integration of the governing equations. We end by discussing the appropriate choice of domain size, grid resolution, and boundary conditions in the context of canonical flows with uniform Cartesian mesh spacing.