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The similarity of the two-point correlation tensor along the streamwisedirection in the axi-symmetric jet far-field is analyzed, herein its utility inspectral theory. A separable two-point correlation coefficient has been thebasis for the argument that the energy-optimized basis functions along thestreamwise direction are Fourier modes (from the approach of equilibriumsimilarity theory). This would naturally be highly desirable both from acomputational and an analytical perspective. The present work, however, showsthat the two-point correlation tensor multiplied by the Jacobian is notdisplacement invariant even in logarithmically stretched coordinates. Thisresult directly impacts the motivation for a Fourier-based representation ofthe correlation function in spectral space in relation to the Proper OrthogonalDecomposition (POD) of the field. It is demonstrated that a displacementinvariant form of the kernel is impossible to achieve using the suggestedcoordinate transformations from earlier works. This inability is shown to berelated to the fundamental differences between the turbulent flow at hand andthe ideal case of homogeneous turbulence.

Two-point similarity in the round jet revisited