A Tensor Calculus Formulation of the Lumley Decomposition applied to the Turbulent Axi-symmetric Jet Far-field

The current work studies the far-field of the turbulent round jet at a Reynolds number of 20 000. A formalism based on tensor calculus is introduced for the study of flows for which equilibrium similarity holds, motivated by the complexity of the coordinate system that the far-field region is represented in.

The tensor-based approach ensures that the invariance of the field is preserved and allows for a formulation of the Lumley Decomposition (LD) in tensor form, which is valid in any welldefined coordinate system. This expression of the LD reveals that the inner product space must be equipped with a specific weight function in order to conclude that the modes in the streamwise direction are Fourier modes.

It is furthermore shown that the use of the so-called similarity coordinates introduced by Ewing et al. 2007 fails to yield a Hermitian form of the LD matrix operator due to the non-orthogonality of this coordinate system. The orthogonal stretched spherical coordinates (SSC) are therefore proposed as an alternative, since they ensure the Hermitian property of the LD operator and allow the LD to be performed on the turbulent jet with a amplitude decaying stretched Fourier decomposition along the streamwise coordinate.

The energy equation is spanned by the LD eigenfunctions coupling the eigenfunctions to the various energy transport mechanisms for a solenoidal fluid in curvilinear coordinates. This allows for a reconstruction of the energy production term leading to energy production and transport analysis. The reconstructions of the component energy spectra in the −5/3-range reveal the varying spectral fluxes for each individual modal component across the jet.

This indicates the presence of a more complex underlying energy transfer mechanism across modes than what is usually assumed by the Richardson cascade. The reconstruction of the energy production is then performed in order to quantify the hypothesis stated in Wänström 2009 that multiple modes have the ability to obtain energy directly from the mean flow.

It is demonstrated that every mode must participate in the energy production in some way, for flows where the LD operator is self-adjoint. Furthermore, each mode’s production of energy is shown to vary over wave and mode number across the span of the flow. In particular, it is demonstrated that the relative energy production of modes is substantial even in the −5/3-range.

This means that modes in this region obtain a substantial part of their energy directly from the mean flow.