Lower branch equilibria in Couette flow: the emergence of canonical states for arbitrary shear flows

Abstract

We consider the development of nonlinear three-dimensional vortex-wave interaction equilibria of laminar plane Couette flow for a range of spanwise wavenumbers. The results are computed using a hybrid approach that captures the required asymptotic structure while at the same time providing a direct link with full numerical calculations of equilibrium states. Each equilibrium state consists of a streak flow, a roll flow and a wave propagating on the streak. Direct numerical simulations at finite Reynolds numbers using initial conditions constructed from these parts confirm that the scheme generates equilibrium solutions of the Navier-Stokes equations. Consideration of the form of the vortex-wave interaction equations in the highspanwise-wavenumber limit predicts that for small wavelengths the equilibria take on a self-similar structure confined near the centre of the flow. These states feel no influence from the walls, and lead to a class of canonical states relevant to arbitrary shear flows. These predictions are supported by an analysis of computational results at increasing values of the spanwise wavenumber. For the wave part of these new canonical states, it is shown that the mass-specific kinetic energy density per unit wavenumber scales with the 5/3 power of the wavenumber.

Exact coherent structures in stably stratified plane Couette flow

“…The rescaling forū andρ comes from the fact that the streaks andρ only now extend over a cross-stream distance of O( ) of the underlying applied shear and density fields. A similar scaling was discussed in Blackburn et al (2013) for unstratified flows when considering the large spanwise wavenumber limit of the SSP/VWI process. Here, the rescaling is driven by increasing Ri b but the overall effect is the same except there is no localisation in the streamwise direction.…”

“…oceanic flows are almost always stably stratified - Thorpe (2007)). While there has been previous work on unstably-stratified shear flow -Rayleigh-Benard convection with imposed shear (Clever et al 1977;Clever & Busse 1992), internally-heated shear flow (Generalis & Nagata 2003) or natural convection with imposed shear Hall (2012) -the only work in computing ECS for stable stratification is that of Busse (1992,2000) who established that Nagata's (1990) now famous first solution in pCf could be continued back to Rayleigh-Benard convection with shear. This lack of attention might well be because stable stratification is perceived as a universally stabilizing influence although this is now appreciated as an oversimplification (Howard & Maslowe 1973;Huppert 1973;Davey & Reid 1977) or because introducing stratification increases the dimension of parameter space from a very manageable 1 (Re, the Reynolds number) to a more daunting 3 (Re, Ri b and P r where the bulk Richardson and Prandtl numbers are defined below).…”

Vertically localised equilibrium solutions in large-eddy simulations of homogeneous shear flow

“…Continuation in R S reveals that, as this branch extends towards higher R S its solutions concentrate in a relatively thin critical layer for R S > 70. This is a common feature of lower-branch solutions in wall-bounded flows at high Reynolds numbers (Wang et al 2007;Viswanath 2009;Hall & Sherwin 2010;Deguchi et al 2013;Blackburn et al 2013). Figure 3 shows isosurfaces of |ω x | = 0.6|ω x | max , and of u = 0, representing the geometry of the vortical structures and of the velocity streak in the upper-and lower-branch solutions, showing that they are localised around y = 0.…”

“…Such critical layer-type solutions are described by an asymptotic theory called as vortex-wave interaction (VWI) (Hall & Smith 1991;Hall & Sherwin 2010) and their instability has the edge-mode (Deguchi & Hall 2016). It is shown that vertically-localised equilibrium states can be embedded in any shear flow at high Reynolds number (Blackburn et al 2013;Deguchi & Hall 2014a;Deguchi 2015).…”

Exact coherent structures at extreme Reynolds number

“…By explicitly recognizing the emergence of multiple spatial and temporal scales, simplified partial differential equations (PDEs) are derived that enable a reduction in computational complexity. For example, Beaume (2012) and Blackburn, Hall & Sherwin (2013) show that the computation of certain ECS can be reduced to the coupled solution of two 2-D problems: a nonlinear problem for the x-averaged flow at unit effective Reynolds number; and a quasi-linear problem for inviscid wavy instabilities riding on the streamwise-averaged flow. In fact, a time-dependent version of these reduced PDEs is equivalent (for large Re) to the 'restricted nonlinear model' of turbulence in wall-bounded parallel shear flows recently proposed by Thomas et al (2015).…”