Temporal subharmonic amplitude and phase behaviour in a jet shear layer: wavelet analysis and Hamiltonian formulation

Abstract

Fourier and wavelet transformation techniques are utilized in a complementary manner in order to characterize temporal aspects of the transition of a planar jet shear layer. The subharmonic is found to exhibit an interesting temporal amplitude and phase variation that has not been previously reported. This takes the form of intermittent π-shifts in subharmonic phase between two fixed phase values. These phase jumps are highly correlated with local minima of the subharmonic amplitude. In contrast, the fundamental amplitude and phase show no such behaviour. The temporal phase behaviour of the subharmonic has the effect of intermittently disrupting the phase lock with the fundamental. A dynamical systems model is developed which is based on a classic vortex representation of the shear layer. The Hamiltonian formulation of the problem is shown to provide remarkable agreement with the experimental results. All the essential aspects of the temporal amplitude and phase behaviour of the subharmonic are reproduced by the model including amplitude-dependent effects. The model is also shown to provide a dynamical systems based explanation for time-averaged amplitude and phase behaviour observed in these as well as earlier experiments. The results of experiments involving both bimodal forcing at fundamental and subharmonic frequencies with prescribed initial effective phase angle as well as experiments involving only fundamental excitation over an amplitude range extending two orders of magnitude are presented. The temporal subharmonic amplitude and phase behaviour is observed in bimodal forcing experiments in those regions of the flow characterized by subharmonic mode suppression and vortex tearing events (even if the forcing amplitudes are quite large). In addition, temporal subharmonic amplitude and phase behaviour is the rule in experiments involving low-amplitude forcing of the fundamental and the natural development of the subharmonic.

Wake interaction between two side-by-side square cylinders in channel flow

“…(For further details on the continuous wavelet transform see, for example, [8,12].) By considering the complex values of the wavelet coefficient G(a, s) at the scale corresponding to the shedding frequency f s , the instantaneous phase difference of lift (or drag) between the two cylinders can be obtained as…”

Experimental studies of vortices shed from cylinders with a step-change in diameter

“…As γ is also scaled in the process, each scale corresponds to a 'local' frequency in the time history at a particular time. When the wavelet transform is applied to a discrete time series with a sampling time t, the actual local frequency corresponding to a given scale can be calculated as (Abry 1997;Gordeyev & Thomas 1999)…”

Field Reconstruction From Single Scale Continuous Wavelet Coefficients

“…As a data compaction tool, this approach has been superseded by orthogonal wavelet thresholding. Thus unpromising for data storage and transmission or for computer simulation, continuous wavelets nonetheless present advantages for frequency resolution in the analysis of data, 6 , for the definition of partition functions and other applications based on lines of modulus maxima, 1 , for the connection to structure functions, 12 , and for formal manipulations of partial differential equations, 13 . This paper presents a new tool relevant to the latter.…”