Analytical reconstruction of isotropic turbulence spectra based on the Gaussian transform

TL;DR

This paper introduces an analytical method to reconstruct realistic isotropic turbulence spectra, like von Kármán, by superposing weighted Gaussian spectra, enhancing turbulence simulation accuracy.

Contribution

It derives analytical weighting functions for arbitrary isotropic spectra and proposes a discretisation method for practical turbulence spectrum synthesis.

Findings

Successfully reconstructs von Kármán spectrum using the proposed method.

Demonstrates the effectiveness of the approach with turbulence spectrum synthesis.

Provides analytical tools for improved turbulence modeling in aeroacoustic simulations.

Abstract

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.