Stretched Amplitude Decaying Fourier Modes in the Jet Far-Field

Abstract

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, $L^2_w(\Omega ,\mathbb {C}^3)$, over the domain $\Omega \subset \mathbb {R}^4$. 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.