Experimental stabilisation of 2D vortex patterns using time-dependent forcing

Experimental results of the effect of time-periodic and “chirped” (electro-magnetic) forcing on vortex patterns in shallow-water-layer flows are presented. Analogously to vibrational control, the use of a time-periodic forcing results in stabilisation of otherwise unstable vortex patterns. Chirped frequency forcing yields self-organising patterns that are different from those in stationary and periodically forced experiments. The results are shown to be consistent with theoretical analysis of 2D Taylor-Green vortices, i.e. unstable analytical solutions of the 2D NavierStokes equation. These results imply that, compared to the more often analysed stationary forced flows, time-varying forcing can stabilise different vortex patterns in shallow-water-layer flows. Copyright c © EPLA, 2013 Introduction. – Shallow fluid flows behave quite differently from fully three-dimensional (3D) flows. Contrary to the small length scales observed in 3D turbulence, these quasi–two-dimensional (Q2D) flows typically self-organise into large vortices [1–4]. Self-organisation is, for example, seen in geophysical flows (both in the atmosphere and in the oceans), in soap films, and in electromagnetically forced shallow-water-layer experimental setups. If 3D effects are completely negligible, these flows can be modelled by the two-dimensional (2D) Navier-Stokes equation. The Q2D behaviour of cellular flows (square arrays of vortices in a bounded domain) has been studied extensively. These flows are experimentally realised in conductive shallow fluid layers using electromagnetic forcing [5–7]. Contrary to the self-organisation in decaying Q2D flows, stationary forced flows attain a stationary state that is organised for moderate forcing amplitudes [8]. However, if the forcing amplitude is increased, the flow typically undergoes a bifurcation, consistent with (a)E-mail: [email protected] theoretical analysis [9], and becomes time-periodic and eventually chaotic in time and spatially disorganised. For certain cases, Q2D cellular flows can be described analytically as 2D Taylor-Green vortices [10]. These square vortex arrays are a family of exact solutions of the 2D Navier-Stokes equation. Their stability depends on several factors like boundary conditions, the value of the Reynolds number and magnitude and type of the forcing. Stability analysis for decaying and stationary forced 2D Taylor-Green vortices have revealed that the vortices remain stable for small Reynolds numbers [11–14]. For decaying Taylor-Green vortex arrays at a higher Reynolds number, the self-organisation into a domain filling vortex can be explained using variational techniques [15,16]. In the present paper, we report on an experimental study of the influence of different types of forcing on the stability and self-organisation of Q2D cellular flows. In particular the effect of time-varying forcing is discussed. We show that time-periodic forcing can stabilise a cellular flow that is similar to a 2D Taylor-Green vortex pattern, but different from the Taylor-Green vortex resulting from