Analytical reconstruction of isotropic turbulence spectra based on the Gaussian transform

The Random Particle Mesh (RPM) method used to simulate turbulence-induced broadband noise in several aeroacoustic applications is extended to realise isotropic turbulence spectra. With this method turbulent fluctuations are synthesised by filtering white noise with a Gaussian filter kernel that in turn gives a Gaussian spectrum. The Gaussian function is smooth and its derivatives and integrals are again Gaussian functions. The Gaussian filter is efficient and finds wide-spread applications in stochastic signal processing. However in many applications Gaussian spectra do not correspond to real turbulence spectra. Thus in turbo-machines the von K\'arm\'an, Liepmann, and modified von K\'arm\'an spectra are more realistic model spectra. In this note we analytically derive weighting functions to realise arbitrary isotropic solenoidal spectra using a superposition of weighted Gaussian spectra of different length scales. The analytic weighting functions for the von K\'arm\'an, the Liepmann, and the modified von K\'arm\'an spectra are derived subsequently. Finally a method is proposed to discretise the problem using a limited number of Gaussian spectra. The effectivity of this approach is demonstrated by realising a von K\'arm\'an velocity spectrum using the RPM method.