Dynamics of channel flows:
Discovering coherent structures in transitional and turbulent flows through impulse and frequency response analysis of the linearized Navier-Stokes equations

It's stil not widely recognized that the Linearized Navier-Stokes (LNS) equations contain a lot of information about transition and turbulence dynamics. This appears to be especially true in the case of wall-bounded shear flows, for which satisfactory transition scenarios and turbulence models have been elusive. In the past 15 yrs a new analysis of the LNS equations has emerged, with substantially different consequences than classical hyrdodynamic stability analysis based on normal modes. A recent summary of this approach appears in the book Stability and Transition in Shear Flows by Schmid & Henningson.

One way to carry out this analysis is from an "input-output" point of view. This means introducing inputs (e.g. body force fields) and considering various velocity and vorticity fields as outputs. As such, we have studied the spatio-temporal frequency and impulse responses of the LNS. This reveals an amazing richness of behavior of the LNS equations; something which is not at all obvious from normal mode analysis. Coherent flow structures and transition mechanisms that are sometimes balmed on "nonlinear mechanisms" show up in these linear models. This should not be as surprising as it sounds. It is another example of how the widely spread treminology of "nonlinear mechanisms" and "nonlinear phenomena" leads to confusion (see "Myths, Misconceptions ...." in the talks page).

Each of the pictures below represent commonly observed phenomena of transition or turbulence that can related to either the frequency or impulse response of the LNS.


Click on each of the pictures below for a more detailed description

Coherent structures from frequency response

 TS waves vs. streamwise vortices & streaks in supercritical channel flows
Impulse excitation of channel flow

 Impulse excitation movie
Component-wise input-output analysis


See also the web page of Mihailo Jovanovic for more on this.



Related Publications
  1. M. Jovanovic and B. Bamieh. Componentwise Energy Amplification in Channel Flows. J. Fluid Mechanics, 534:145-183, 2005.


  2. B. Bamieh and M. Dahleh. Energy Amplification in Channel Flows with Stochastic Excitation. Physics of Fluids, 13(11):3258-3269, 2001.


  3. M. Jovanovic and B. Bamieh. Unstable modes versus non-normal modes in supercritical channel flows. In Proc. of The American Control Conference, Boston, MA, pages 2245-2250, 2004.


  4. M. Jovanovic and B. Bamieh. Modeling flow statistics using the linearized Navier-Stokes equations. In Proc. 40'th IEEE Conference on Decision and Control, Orlando, FL, 2001.


  5. M. Jovanovic and B. Bamieh. The Spatio-Temporal Impulse Response of the Linearized Navier-Stokes Equations. In Proc. American Control Conference, 2001.





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