An adjoint-based approach for finding invariant solutions of Navier–Stokes equations

Abstract

We consider the incompressible Navier-Stokes equations with periodic boundary conditions and time-independent forcing. For this type of flow, we derive adjoint equations whose trajectories converge asymptotically to the equilibrium and traveling wave solutions of the Navier-Stokes equations. Using the adjoint equations, arbitrary initial conditions evolve to the vicinity of a (relative) equilibrium at which point a few Newton-type iterations yield the desired (relative) equilibrium solution. We apply this adjoint-based method to a chaotic two-dimensional Kolmogorov flow. A convergence rate of 100% is observed, leading to the discovery of 21 new steady state and traveling wave solutions at Reynolds number Re = 40. Some of the new invariant solutions have spatially localized structures that were previously believed to only exist on domains with large aspect ratios. We show that one of the newly found steady state solutions underpins the temporal intermittencies, i.e., high energy dissipation episodes of the flow. More precisely, it is shown that each intermittent episode of a generic turbulent trajectory corresponds to its close passage to this equilibrium solution.

Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems

“…A ubiquitous feature of turbulent fluid flow is the intermittent bursts observed in the time series of their measured quantities such as energy dissipation [34,35]. Even at moderate Reynolds numbers, the dimension of the turbulent attractors are high.…”

Are extreme dissipation events predictable in turbulent fluid flows?

“…We first present the joint and conditional statistics for the energy E, dissipation rate Z and the indicator λ. In this first step, we do not include any time shifts, thus setting t 0 = ∆t = 0 so that X t = X t in equation (14). Figure 13 shows the joint and conditional PDFs of the indicator versus energy E and the energy dissipation Z at Re = 2200.…”

“…This is clearly the case in figure 12, however, we show that this is true most of the time during the longterm simulations. Figure 14 shows the conditional PDFs pẼ |λ and pZ |λ where the future maximaẼ andZ are computed with t 0 = t e and ∆t = 10t e (see equation (14)). The extreme value threshold E e (respectively, Z e ) are set as the mean of energy (respectively, dissipation) plus one standard deviation.…”

Extreme Events: Mechanisms and Prediction

“…Because of the simplicity of the forcing F and the boundary conditions, the Kolmogorov flow (i.e. the Navier-Stokes equations driven by the Kolmogorov forcing) has been studied extensively both by numerical and analytical methods [142,143,144,145,146,113]. Similar variants of the Kolmogorov flow have also been investigated experimentally [147,148,149,150].…”