Stretched Amplitude Decaying Fourier Modes in the Jet Far-Field

A framework is presented for decomposing the self-similar region of the round jet along the streamwise direction using an analytical series expansion deduced from the four dimensional (space-time) Proper Orthogonal Decomposition
(POD) integral. A specific inner product weight function is introduced in order to deduce the Stretched Amplitude Decaying Fourier Modes. The inner product weight function, maps the POD kernel into a displacement invariant kernel. The SADFM are demonstrated to be orthogonal with respect to the weighted inner product and therefore constituting a basis for the weighted Hilbert space, L ² w (Ω,C³⁾ over the domain Ω⊂R⁴. The SADFM allow for an Fast Fourier Transform (FFT) algorithm to be used along the streamwise inhomogeneous direction reducing the computational workload for the four-dimensional POD of the jet far-field. Applying a Galerkin projection to the governing equations using the SADFM provides additional insight into the far-field dynamics. This may be useful in relation to the study of the interaction of various terms in the turbulence kinetic energy transport equation, thereby extending the analytical tools that can be used in the study of the turbulence in the self-similar region of the round jet.