On the Use of Higher-Order Projection Methods for Incompressible Turbulent Flow

An important issue in the development of higher-order methods for incompressible and low Mach number flows is how they perform when the flow is turbulent. A useful diagnostic of a method for turbulent flow is the minimum resolution that is required to adequately resolve the turbulent energy cascade at a given Reynolds number. In this paper, we present careful numerical experiments to assess the utility of higher-order numerical methods based on this metric. We first introduce a numerical method for the incompressible Navier-Stokes equations that is fourth-order accurate in space and time for smooth flows. The method is based on an auxiliary variable formulation and combines fourth-order finite volume differencing with a semi-implicit spectral deferred correction temporal integration scheme. We also introduce, for comparison purposes, second-order versions of both the spatial and temporal discretizations. We demonstrate that for smooth problems, each of the methods exhibits the expected order of convergence. We next examine the behavior of these schemes on prototypical turbulent flows; in particular, we consider homogeneous isotropic turbulence in which long wavelength forcing is used to maintain the overall level of turbulent intensity. We provide comparisons of the fourth-order method with the comparable second-order method as well as with a second-order semi-implicit projection method based on a shock-capturing discretization. The results demonstrate that, for a given Reynolds number, the fourth-order scheme leads to dramatic reduction in the required resolution relative to either of the second-order schemes. In addition, the resolution requirements appear to be reasonably well predicted by scaling relationships based on dimensional analysis, providing a characterization of resolution requirements as a function of Reynolds number.