Two-point similarity in the round jet revisited

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. Certain symmetries have previously been believed to provide the rationale to transform the basis functions into Fourier counterparts (from the approach of equilibrium similarity theory) based on convergence rate arguments. This would naturally be highly desirable both from a computational and an analytical perspective. Promising symmetries have been identified in the past by earlier researchers - such as a two-point correlation coeffcient which is separable into a product of a function depending on the absolute coordinates and a displacement invariant function. The present work, however, shows that the twopoint 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 ow at hand and the ideal case of homogeneous turbulence.