Comparison of Higher-Order Numerical Schemes and Several Filtering Methods Applied to Navier-Stokes Equations with Applications to Computational Aeroacoustics (CAA)

Abstract

In Computational Aeroacoustics (CAA), fluid-acoustic coupling methods for the computation of sound have been widely used by researchers for the last five decades. In the first part of the coupling procedure, the fully unsteady incompressible or compressible flow equations for the near-field of the unsteady flow are solved by using a Computational Fluid Dynamics (CFD) technique, i.e., direct numerical simulation (DNS), Large Eddy Simulation (LES) or unsteady Reynolds averaged Navier-Stokes equations (RANS) [1]; results of these simulations are then used to calculate sound sources using the acoustic analogy or by solving a set of acoustic perturbation equations (APE) leading to the solution of the acoustic field. In this paper, a coupling method in which the near-field of the unsteady flow is simulated by a fme-mesh-small-timestep-LES-alike numerical method applied in two-dimension, and the acoustic propagation of a particular frequency in a medium, where the effect of the flow motion may be neglected, is resolved by Helmholtz equation [2] to predict noise distribution inside a car compartment due to the aerodynamic flow over an open sunroof generating the frequency of interests.

In essence filtering of the unsteady compressible Navier-Stokes equations by certain filters is expected to lead to the same numerical results of the above LES-alike method. The aim of this paper is to examine the similarities between the effect of filters and the combined effect of high-order schemes and discretisation meshes, in which the latter has been employed in the coupling procedure described above. Numerical tests are carried out on a non-linear elliptic boundary value problem and an unsteady parabolic initial boundary value problem, all with solutions exhibiting highly oscillatory behaviour across the spatial domain. Numerical solutions are obtained by using high-order differencing schemes. These results are compared with several filtered solutions. The study provides a systematic way of classifying the effective filter being used in LES [3] in terms of the mesh size, the number of grid points, and the order of the numerical scheme.