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

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