Canonical Stochastic Realization of Turbulent Sound Sources via Forced Linear Advection-Diffusion-Dissipation Equation

Abstract

Stochastic sound sources derived from Reynolds Averaged Navier-Stokes (RANS) solution are recognized in Computational Aeroacoustics as one possible way to efficiently predict broadband sound. In this paper a stochastically forced linear advection-diffusion-dissipation equation is introduced. The model provides spectra and anisotropic two-point correlations that otherwise have to be incorporated in datum stochastic methods as additional model assumptions. The output are fluctuating velocity components, from which vortex sound sources derive. The forcing is white (delta-correlated) in time and possess a finite correlation length scale in space. The well-posedness is demonstrated in the paper. A solenoidal forcing term is shown to realize the correlation tensor of homogeneous isotropic turbulence together with a longitudinal turbulence spectrum that exhibits a plateau for lower frequencies followed by a characteristic power law roll-off and final cut-off. Exponent of decay and cut-off are adjustable. The stochastic partial differential equation involves a diffusion parameter, a time-scale, a re-distribution tensor, and a forcing variance. Transport equations for Reynolds stresses and turbulence kinetic energy derive from it that have the canonical form of major RANS transport equations. In particular, all parameters needed can be assigned to corresponding RANS parameters so that an accurate reproduction of RANS one-point statistics becomes feasible. For the generation of two-point statistics the hypothesis from turbulence modeling is adopted that the present model calibrated for homogeneous isotropic turbulence is also applicable for more general flows.