The emergence of localized vortex–wave interaction states in plane Couette flow

Abstract

The recently understood relationship between high Reynolds number vortex-wave interaction theory and computationally-generated self-sustaining processes provides a possible route to an understanding of some of the underlying structures of fully turbulent flows. Here vortex-wave interaction theory, which we now refer to as VWI, is used in the long streamwise wavelength limit to continue the development found at order one wavelengths by Hall and Sherwin (2010). The asymptotic description given reduces the Navier-Stokes equations to the so-called boundary-region equations for which we find equilibrium states describing the change in the VWI as the wavelength of the wave increases from O(h) to O(Rh) where R is the Reynolds number and 2h is the depth of the channel. The reduced equations do not include the streamwise pressure gradient of the perturbation or the effect of streamwise diffusion of the wave-vortex states. The solutions we calculate have an asymptotic error proportional to R −2 when compared to the full Navier-Stokes equations. The results found correspond to the minimum drag configuration for VWI states and might therefore be of relevance to the control of turbulent flows. The key feature of the new states discussed here is the thickening of the critical layer structure associated with the wave part of the flow to completely fill the channel so that the roll part of the flow is driven throughout the flow rather than as in Hall and Sherwin (2010) as a stress discontinuity across the critical layer. We identify a critical streamwise wavenumber scaling which when approached causes the flow to localise and take on similarities with computationally-generated or experimentally-observed turbulent spots. In effect the identification of this critical wavenumber for a given value of the assumed high Reynolds number fixes a minimum box length necessary for the emergence of localized structures. Whereas nonlinear equilibrium states of the Navier-Stokes equations are thought to form a backbone on which turbulent flows hang, our results suggest that the localized states found here might play a related role for turbulent spots.

A doubly localized equilibrium solution of plane Couette flow

“…Avila et al (2013) found a streamwise-localized relative periodic orbit of pipe flow that closely resembles the transient turbulent puffs of Hof et al (2006). Deguchi et al (2013) and Gibson & Brand (2014) independently found spanwise-localized forms of the periodic EQ7/HVS solution of Itano & Generalis (2009);Gibson et al (2009). Gibson & Brand (2014) also presented a number of spanwise-localized and wall-normal-localized traveling waves of channel flow.…”

Localized vortex/Tollmien–Schlichting wave interaction states in plane Poiseuille flow

“…As a result we will clearly be able to witness the localization phenomenon seen in the solutions. We shall also reveal that the nature of the localization in the vortex/TS wave interaction state is quite different from that seen in Deguchi, Hall & Walton (2013).…”

“…In this sense our use of the word 'localization' is different from that used in recent numerical investigations of fully isolated vortex structures, e.g. Schneider, Gibson & Burke (2010), Deguchi, Hall & Walton (2013) and Gibson & Brand (2014). Similar computations are now presented for larger values of β.…”

“…The interaction is therefore governed by the so-called boundary-region equations in which the roll, streak and wave now evolve over a single O(Re) streamwise lengthscale. Such an asymptotic problem was recently solved for plane Couette flow by Deguchi, Hall & Walton (2013) where the solution was found to strongly localize in the spanwise direction while also exhibiting streamwise localization as the amplitude of the motion was increased. Although we have not investigated the solution of the corresponding boundary-region problem for plane Poiseuille flow, we would expect to observe similar localization for this flow configuration.…”

“…For the former Rayleigh-type interaction, Hall & Sherwin (2010) have verified that the numerical solutions of the governing interaction equations for plane Couette flow predict excellently the asymptotic behaviour of full Navier-Stokes equilibrium solutions, generated using a Newton-Rhaphson approach by Wang, Gibson & Waleffe (2007). Subsequently Deguchi, Hall & Walton (2013) considered the long-wavelength development of vortex/Rayleigh-wave interaction states in plane Couette flow and uncovered a family of localized solutions. In phase space the plane Couette flow equilibrium solutions studied by Nagata (1990) and Wang et al (2007) have been found to be associated directly with the stable structure on the hyper surface that separates the regions of laminar and turbulent attraction: a number of plane Poiseuille flow solutions, for example those seen in Itano & Toh (2001), are likely to be of a similar type.…”