A scale-aware dispersion-relation-preserving finite difference scheme for computational aeroacoustics

The numerical schemes for computational aeroacoustics (CAA) should have minimal dispersion and proper dissipation in order to accurately capture the amplitude and phase of waves. In this paper, we propose a scale-aware dispersion-relation-preserving (SA-DRP) finite difference scheme based on an improved scale sensor and a new dispersion control strategy. The scale sensor quantifies the local length scale of the solution in the form of the effective scaled wavenumber. The new feature of this scale sensor is the accurate prediction of the wavenumber for a pure sine wave. The new dispersion control strategy determines the dispersion parameter of the scheme in terms of the scale sensor. In contrast to the traditional dispersion-relation-preserving (DRP) scheme that minimizes the integral dispersion error, the new strategy directly solves the dispersion parameter by requiring the numerical dispersion relation to be equal to the exact one. As a result, precise dispersion relation can be realized within a very broad wavenumber range. The approximate dispersion relation analysis shows that the SA-DRP scheme maintains an accurate dispersion relation up to the scaled wavenumber k = 2.5. Moreover, the overshoot in the dispersion relation of the DRP scheme is not presented in that of the SA-DRP scheme. To suppress nonphysical oscillations, we also add proper dissipation that is adjusted automatically according to the effective scaled wavenumber. Several CAA benchmark test cases are presented to demonstrate the higher resolution and higher efficiency achieved by the proposed scheme compared with the conventional spectrally optimized schemes.