Linear- and Non-Linear Perturbation Equations with Relaxation Source Terms for Forced Eddy Simulation of Aeroacoustic Sound Generation

Abstract

Turbulence related sound is generated by the dynamics of fluctuating vorticity. For example, trailing edge noise is caused by vorticity traveling past the trailing edge. To excite fluctuating vorticity by forcing the linearized Euler equations (LEE) with right-hand side source terms, one peculiar problem is observable: while the rise of vorticity levels by external sources poses no problem, to properly lower them, the right-hand side terms must act as a sink, being exactly in anti-phase to the vorticity levels as present in the LEE solution. However, the accurate prediction of vorticity in terms of phase cannot be guaranteed, especially for approximately modeled sources e.g. using stochastic methods. Thus in general there will be a mismatch between actual induced and intended levels of vorticity. In this paper a new class of relaxation source terms is introduced that enables the proper excitation of vorticity levels in linear and non-linear perturbation equations and as such enables an accurate control over the vorticity magnitudes. The source can be formulated to act selectively in wave-number space, i.e. without directly affecting the dynamics of resolved low wave-number vorticity components whereas the resolved high wave-number part is piloted by the fluctuating vorticity imposed as a reference solution. The reformulation of the Navier-Stokes equations in primitive variables and non-linear perturbation form is presented. Direct noise computation of sound radiated from a vortex shedding cylinder in laminar cross flow verify their implementation. The relaxation source term without forcing is applied to the unstable jet problem of the 4th CAA Workshop on Benchmark Problems. The forcing of frozen and decaying stochastic turbulence in conjunction with the relaxation source term is studied. First results for high-lift noise prediction with forced eddy simulation are presented.