Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow

Abstract

We consider long-time simulations of two-dimensional turbulence body forced by $\sin 4y\hat {\boldsymbol{x}} $ on the torus $(x, y)\in \mathop{[0, 2\mathrm{\pi} ] }\nolimits ^{2} $ with the purpose of extracting simple invariant sets or ‘exact recurrent flows’ embedded in this turbulence. Each recurrent flow represents a sustained closed cycle of dynamical processes which underpins the turbulence. These are used to reconstruct the turbulence statistics using periodic orbit theory. The approach is found to be reasonably successful at a low value of the forcing where the flow is close to but not fully in its asymptotic (strongly) turbulent regime. Here, a total of 50 recurrent flows are found with the majority buried in the part of phase space most populated by the turbulence giving rise to a good reproduction of the energy and dissipation p.d.f. However, at higher forcing amplitudes now in the asymptotic turbulent regime, the generated turbulence data set proves insufficiently long to yield enough recurrent flows to make viable predictions. Despite this, the general approach seems promising providing enough simulation data is available since it is open to extensive automation and naturally generates dynamically important exact solutions for the flow.

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

“…This flow (with identical geometry and parameters as the ones chosen here) was studied thoroughly by Chandler & Kerswell (2013), providing a basis for comparison with our results.…”

“…The NGh iterations are explained in detail by Chandler & Kerswell (2013). Our implementation and choice of parameters are identical to theirs with the exception that we use the velocity formulations of the Kolmogorov flow, while Chandler & Kerswell (2013) use the vorticity formulation. We chose to use the velocity formulation since the adjoint equations (3.3) and (3.9) are derived for the general three-dimensional flow, and hence are in the velocity-pressure form.…”

“…Recently, Chandler & Kerswell (2013) and Lucas & Kerswell (2014) carried out the most exhaustive search for invariant solutions of a chaotic two-dimensional Kolmogorov flow using Newton-GMRES-hook iterations. Here, we apply our adjoint-based method to the same Kolmogorov flow and discover several new steady state and traveling wave solutions.…”

Exact coherent structures in stably stratified plane Couette flow

“…As a consequence, properties of the ECS have something to say about how transition is triggered (Viswanath & Cvitanović 2009;Duguet et al 2010;Pringle et al 2012), the subsequent transitional process (Itano & Toh 2001;Skufca et al 2006;Schneider et al 2007;Mellibovsky et al 2009) and features of the turbulent state itself (e.g. Kawahara & Kida (2001); Hof et al (2004); Viswanath (2007); Gibson et al (2008) ;Chandler & Kerswell (2013) ;Willis et al (2013); Lucas & Kerswell (2015)). …”

Exact coherent structures in an asymptotically reduced description of parallel shear flows

“…In addition, much insight into the complex dynamics exhibited by transitional flows has come from viewing the flow in terms of a temporal sequence of transitions between weakly unstable coherent structures (Auerbach et al 1987, Christiansen et al 1997, Halcrow et al 2009. This notion has met with great success, but depends on our ability to identify a large number of ECS in these flows and the connections between them (Duguet et al 2008, Chandler and Kerswell 2013, Lucas and Kerswell 2014b. In this paper we propose a systematic but general procedure that leads to a simplified but selfconsistent description of the required ECS.…”