Lumley Decomposition of the Turbulent Round Jet Far-field. Part 1 -- Kinematics

TL;DR

This paper develops a tensor formulation of Lumley Decomposition for curvilinear coordinates and applies it to analyze the far-field of a turbulent jet, revealing self-similar Fourier modes with specific spectral characteristics.

Contribution

It introduces a tensor-based Lumley Decomposition in curvilinear coordinates and identifies optimal eigenfunctions as stretched amplitude-decaying Fourier modes in turbulent jet analysis.

Findings

Eigenfunctions are stretched Fourier modes with linearly increasing wavelength.

Energy spectra exhibit -5/3 and -7/3 slopes consistent with turbulence theory.

Method can be extended to other self-similar turbulent flows.

Abstract

The current work presents a tensor formulation of the Lumley Decomposition (LD), introduced in its original form by Lumley (1967b), allowing decompositions of turbulent flow fields in curvilinear coordinates. The LD in his form is shown to enable semi-analytical decompositions of self-similar turbulent flows in general coordinate systems. The decomposition is applied to the far-field region of the fully developed turbulent axi-symmetric jet, which is expressed in stretched spherical coordinates in order to exploit the self-similar nature of the flow while ensuring the self-adjointness of the LD integral. From the LD integral it is deduced that the optimal eigenfunctions in the streamwise direction are stretched amplitude-decaying Fourier modes (SADFM). The SADFM are obtained from the LD integral upon the introduction of a streamwise-decaying weight function in the vector space definition. The wavelength of the Fourier modes is linearly increasing in the streamwise direction with an amplitude which decays with the -3/2 power of distance from the virtual origin. The streamwise evolution of the SADFM re-sembles reversed wave shoaling known from surface waves. The energy- and cross-spectra obtained from these SADFM exhibit a -5/3- and a -7/3-slope region, respectively, as would be expected for regular Fourier modes in homogeneous and constant shear flows. The approach introduced in this work can be extended to other flows which admit to equilibrium similarity, such that a Fourier-based decomposition along inhomogeneous flow directions can be performed.