Two-point similarity in the round jet revisited

TL;DR

This paper revisits the two-point correlation in axi-symmetric jets, revealing that the correlation tensor's properties challenge the assumption that Fourier modes are optimal basis functions, due to flow inhomogeneity.

Contribution

It demonstrates that the correlation tensor is not displacement invariant under common coordinate transformations, questioning the applicability of Fourier-based spectral methods for this flow.

Findings

Correlation tensor multiplied by Jacobian is not displacement invariant.

Fourier modes are not optimal basis functions for this flow.

Inability to achieve displacement invariance relates to flow inhomogeneity.

Abstract

The similarity of the two-point correlation tensor along the streamwise direction in the axi-symmetric jet far-field is analyzed, herein its utility in spectral theory. A separable two-point correlation coefficient has been the basis for the argument that the energy-optimized basis functions along the streamwise direction are Fourier modes (from the approach of equilibrium similarity theory). This would naturally be highly desirable both from a computational and an analytical perspective. The present work, however, shows that the two-point correlation tensor multiplied by the Jacobian is not displacement invariant even in logarithmically stretched coordinates. This result directly impacts the motivation for a Fourier-based representation of the correlation function in spectral space in relation to the Proper Orthogonal Decomposition (POD) of the field. It is demonstrated that a displacement invariant form of the kernel is impossible to achieve using the suggested coordinate transformations from earlier works. This inability is shown to be related to the fundamental differences between the turbulent flow at hand and the ideal case of homogeneous turbulence.