Analytical reconstruction of isotropic turbulence spectra based on the Gaussian transform

• Realisation of isotropic turbulence spectra by analytically superposing Gaussian spectra.
• Analytical derivation of the weighting function for von Karman and Liepmann spectra.
• Discretisation and truncation for a small number of Gaussian spectra.
• Validation on the example of a turbulence velocity von Karman spectrum.

The field of application of the Random Particle Mesh (RPM) method used to simulate turbulence-induced broadband noise in several aeroacoustic applications is improved 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 filter is efficient and finds wide-spread applications in stochastic signal processing. However Gaussian spectra do not correspond to real turbulence spectra. Thus in turbo-machines the von Kármán, Liepmann, and modified von Kármán 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 Kármán, the Liepmann, and the modified von Kármán 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 Kármán velocity spectrum using the RPM method.