Nonlinear Nonmodal Stability Theory

Abstract

This review discusses a recently developed optimization technique for analyzing the nonlinear stability of a flow state. It is based on a nonlinear extension of nonmodal analysis and, in its simplest form, consists of finding the disturbance to the flow state of a given amplitude that experiences the largest energy growth at a certain time later. When coupled with a search over the disturbance amplitude, this can reveal the disturbance of least amplitude—called the minimal seed—for transition to another state such as turbulence. The approach bridges the theoretical gap between (linear) nonmodal theory and the (nonlinear) dynamical systems approach to fluid flows by allowing one to explore phase space at a finite distance from the reference flow state. Various ongoing and potential applications of the technique are discussed.

Stabilisation and drag reduction of pipe flows by flattening the base profile

“…As suggested by Kerswell (2018) and explained using a simple example in appendix B of Kerswell et al (2014), the minimal seed couples together (via the nonlinear effects) the Orr, oblique-wave and lift-up mechanisms, which occur on different time scales and are uncoupled in the linearised dynamics. By ensuring that the energy of the preceding phase feeds into the following, the minimal seed is thus able to produce a much larger overall growth than any possible in the linearised problem.…”

“…However, these methods are impractical at finding the smallest possible solution capable of just kicking the system away from the laminar state, as they require a large number of simulations/experiments. Recent developments have been achieved using variational methods to construct fully nonlinear optimisation problems that seek the minimal seed (Pringle & Kerswell 2010;Pringle et al 2012;Cherubini et al 2012;Duguet et al 2013;Cherubini & Palma 2014); see Kerswell (2018) for a review. From a dynamical-systems point of view, the minimal seed represents the closest (in a chosen norm) point of approach of the laminar-turbulent boundary, or 'edge', to the basic state in phase space, as shown in figure 1.…”

Feedback Control for Transition Suppression in Direct Numerical Simulations of Channel Flow

“…The optimal disturbance is usually computed based on the linearized operator, 23 though nonlinear notions can also be determined. 24 In the present work, we focus on the initial optimal disturbance computed based on the linear model. The optimal disturbance is calculated for the uncontrolled and controlled systems, respectively.…”

Finite Amplitude Stability of Internal Steady Flows of the Giesekus Viscoelastic Rate-Type Fluid

“…This happens even though the "energy budget" formulae paradoxically stem from various simplifications of the original system of nonlinear governing equations. Third, some methods developed for the detailed nonlinear stability analysis of flows of the standard incompressible Navier-Stokes fluid rely on an optimisation technique that allows one to find the perturbation of least amplitude necessary for transition from the base steady state to another state, see Kerswell et al (2014), Olvera and Kerswell (2017) and Kerswell (2018). In the Navier-Stokes case the objective functional used in the optimisation procedure is tantamount to the "kinetic energy" of the perturbation, ∫ Ω 1 2 ṽ 2 dv.…”

“…In the case of Giesekus fluid, the counterpart of the kinetic energy functional is the functional (6.1). Consequently, if the optimisation technique such as that presented in Kerswell (2018) is to be generalised to the case of Giesekus fluid, then the suitable objective functional might be the functional (6.1). Fourth, the Lyapunov functional has been designed using thermodynamical arguments.…”