Nonequilibrium Thermodynamics and the Optimal Path to Turbulence in Shear Flows

Abstract

We determine the initial condition on the laminar-turbulent boundary closest to the laminar state using nonlinear optimization for plane Couette flow. Resorting to the general evolution criterion of nonequilibrium systems we optimize the route to the statistically steady turbulent state, i.e., the state characterized by the largest entropy production. This is the first time information from the fully turbulent state is included in the optimization procedure. We demonstrate that the optimal initial condition is localized in space for realistic flow domains.

Nonlinear control of unsteady finite-amplitude perturbations in the Blasius boundary-layer flow

“…This quest has brought to the discovery of nonlinear optimal perturbations, which are characterized by a very different structure with respect to the linear optimal ones and largely outgrow them in energy due to nonlinear mechanisms. The nonlinear optimal perturbation of minimal energy capable of bringing the flow to transition (i.e., the minimal seed of turbulent transition), has been recently found for a pipe flow (Pringle & Kerswell (2010); Pringle et al (2012)); a boundary layer flow (Cherubini et al (2010a(Cherubini et al ( , 2011b); and a Couette flow (Monokrousos et al (2011);Rabin et al (2012); Cherubini & De Palma (2012)). For the boundary-layer and the Couette flow, the minimal seed is characterized by a fundamental invariant structure, composed of a localized array of vortices and low-momentum regions of typical length scale, capable of maximizing the energy growth most rapidly.…”

A space–time variational approach to hydrodynamic stability theory

“…The Brezzi-Rappaz-Raviart perturbation bound permits a rigorous quantification of nonlinear hydrodynamic stability by identifying the amplitude condition under which the linear theory is valid. The approach is different from a direct nonlinear analysis of the most sensitive finite-amplitude initial disturbance conducted in Pringle & Kerswell [38] and Monokrousos et al [39]. The latter, unlike our Brezzi-Rappaz-Raviart perturbation bound, on the one hand incorporates fully nonlinear information, but, on the other hand, does not provide a rigorous global bound statement in the presence of multiple local optima.…”

Minimal perturbations approaching the edge of chaos in a Couette flow

“…For instance, some of these minimal seeds show a symmetry in the spanwise direction, such as for the case of the boundary layer flow. Concerning the plane Couette flow, some of these minimal perturbations have been found to reach the steady lower branch solution (Duguet et al 2010, Rabin et al 2012, whereas some others don't, probably due to the larger Reynolds number considered (Monokrousos et al 2011, Duguet et al 2013. Thus, such a procedure has been shown to be able to find the perturbation of minimal energy which brings the flow in the neighborhood of the edge of chaos in a rather large time and then to turbulence.…”

“…However, relying on low-dimensional models or on expansions on a basis of modes, such analyses could not accurately describe the minimal seed for turbulence transition, i.e., the perturbation of minimal energy which is able to induce transition in the considered flow. Very recently, a more thorough attempt has been made to search for the perturbations of minimal energy on the laminarturbulent boundary for the pipe (Pringle & Kerswell 2010, Pringle et al 2012, the boundary layer (Cherubini et al 2010, Cherubini et al 2011) and the pCf (Monokrousos et al 2011, Rabin et al 2012, Cherubini & De Palma 2013, Duguet et al 2013 flows.…”

“…To determine the initial condition of minimal energy leading eventually to the edge state (i.e., the minimal seed of turbulent transition), these authors optimize at a large target time a functional linked to the turbulent dynamics (the perturbation kinetic energy (Pringle & Kerswell 2010, Cherubini et al 2010, Pringle et al 2012, Rabin et al 2012, Cherubini & De Palma 2013, Cherubini et al 2011 or the time-averaged dissipation (Monokrousos et al 2011, Duguet et al 2013), and then bisect the initial energy of such "optimal" perturbation until reaching the laminar-turbulent boundary.…”