Coherent Structures from Spatio-Temporal Frequency ResponsesAt first view, turbulent flows look 'statistical’, i.e. random and unorganized. Depending on the flow conditions and geometry, there are however multitudes of organized flow structures buried in this randomness. They are commonly referred to as coherent structures. One effective technique to dig them out of the random mess is to look for peaks in the spatio-temporal power spectral densities of flow velocity or vorticity fields. This kind of 'data mining’ is done on the results of Direct Numerical Simulations (DNS) data of the full Navier-Stokes equations. If on the other hand one deals with only the Linearized Navier-Stokes (LNS) equations (linearized around some mean or laminar flow condition, not to be confused with just the Stokes equations), then one can obtain the power spectral densities -in a 'simulation-free’ manner- from computations of the spatio-temporal frequency response. In this setting, coherent structures in turbulent flow correspond to the most resonant structures when the flow is excited by random body forces. Input-output analysis of the linearized Navier-Stokes equations
In the above setting, the LNS around laminar Poiseuille flow with body forces as inputs are shown. This is an input-output view of the flow dynamics with the 'disturbance’ body force field Component-wise view of the spatio-temporal frequency response
The 3x3 matrix of spatio-temporal frequency responses from each of the polarized body forces ( ![]() Note that if we integrate in The upper-right two components (from The most 'resonant’ flow structures
Isosurface plots of the most resonant flow structures according to the frequency response above. Note the greatly compressed streamwise axis scale. These structures are very long, alternating high and low streamwise velocity 'streaks’, interwoven with counter-rotating streamwise vortices. Comparison with DNS data
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