Optimizing energy growth as a tool for finding exact coherent structures

Significance 

Optimization provides a powerful tool for extracting information from the Navier-Stokes equations. For the shear flow transition problem, optimizing over all the possible infinitesimal disturbances in a bid to find the one that maximizes the subsequent energy growth after, say some preselected time period T, has proven invaluable in exposing the generic energy amplification mechanism present. Unfortunately, such an approach says nothing about the behaviour of finite-amplitude disturbances which ultimately determine transition from a linearly stable reference state. This can, however, be remedied by allowing the competing disturbances seeking to maximize the energy growth to all have the same initial finite energy E0 and incorporating the fully nonlinear Navier-Stokes equations as a constraint. The downside to this is that the number of parameters over which the results must be interpreted double (from T to E0 & T) and a fully nonlinear, nonconvex optimization problem has to be solved about which little is known.

Presently the solution technique is iterative and results so far have revealed several interesting new insights into the transition problem. In particular, if the target time T is taken as large, and the optimal growth found as a function of E0 , then a rapid change in the optimal growth is found to occur at some critical value of E0.  This corresponds to the minimal energy for a disturbance to be outside the basin of attraction of the reference state. The key idea is that the optimization procedure naturally selects such disturbances if they exist since their energy remains finite for large T as opposed to that for other disturbances which ultimately decay.

Recently, a research paper published in the journal, Physical Review Fluids, by Dr. Daniel Olvera from Bristol University and Professor Rich Kerswell (Cambridge University) have extended this technique to demonstrate how nonlinear optimization can also be used to find  unstable states nearby to the reference state. Using the context of stably-stratified plane Couette flow, they successfully recovered a stratified version of the well known (unstable) spanwise-localised `snake-like’ flow state which co-exists with the (stable) constant-shear basic state at low Reynolds numbers (see Figure for a convergence to a global state – known as Nagata’s solution – as well). They then find that the snake solution is connected, via a global 3D state (Nagata’s solution), to the 2D rolls of the Rayleigh-Bénard problem where now the stratification is unstable. In doing so, a particularly simple delocalization process was found where the spanwise tails of the snake gradually reduce their spatial decay rate until this vanishes at the global linear instability threshold whereupon the state is then global. This would seem a very generic phenomenon where a localized state moves from a region of subcriticality to one of supercriticality (or vice versa). The authors also probe unstratified plane Couette flow at very low Reynolds numbers to look for further states and examine a transitional bursting phenomenon caused by the stable stratification.

The Olvera and Kerswell study demonstrates that an optimization technique in which the energy growth of a finite-amplitude disturbance to a reference solution is maximized can be used to generate flow fields which are subsequently convergeable via a Newton GMRES algorithm to another nearby solution of the Navier-Stokes equations. This represents an important new use of  optimization in fluid mechanics with possible application to other  dynamical systems.

 

Optimizing energy growth as a tool for finding exact coherent structures

Figure legend: Convergence of Nagata solution from evolution of optimal perturbation NLOP2 at Re = 130. Top, initial condition of NLOP2 (t = 0) for target time T = 40 at E = 6.0×10-3. Middle, evolution in time of NLOP2 at t = 16. This state is used as initial guess for the Newton-GMRES method. Bottom, converged Nagata solution. Left column, contours of yz cross-sections of streamwise perturbation velocity (arrows indicate velocity field in plane). Plots use 8 contour levels set by the extremes of state at t = 16 (arrows rescaled as well). Right column, isosurfaces of +60% of maximum streamwise perturbation velocity.

About the author

Dr Daniel Olvera is a research associate at the Research Centre in Flow Measurement and Fluid Mechanics at Coventry University.  He began working in astrophysics and numerical methods at the Institute of Astronomy (UNAM, Mexico). He moved to England to specialized in scientific computation and high performance computing. He joined the Fluids and Materials  group of the School of Mathematics at the University of Bristol, working on turbulence and optimization techniques in stratified flows with Prof. Rich Kerswell.

He completed his PhD in 2017 and later  the joined Dr. Chris Pringle in the research project ‘Transition to Turbulence in Complex fluids’.

About the author

Rich Kerswell is currently the G.I. Taylor Professor of Fluid Mechanics in the Department of Applied Mathematics & Theoretical Physics (DAMTP), Cambridge University, England

Reference

D. Olvera and R. R. Kerswell. Optimizing energy growth as a tool for finding exact coherent structures. Physical Review Fluids 2, 083902 (2017)

 

Go To Physical Review Fluids

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