Coherent structures in stably stratified plane Couette flow - eScholarship

Coherent structures in stably stratified plane Couette flow D. Olvera * & R. R. Kerswell School of Mathematics, University of Bristol, Bristol, UK. * do12542@bristol.ac.uk Abstract A large body of recent work in wall-bounded, unstratified shear flow has been dedicated to finding Exact Coherent Structures (ECS) (e.g. see the review Kawahara et al. (2012)). These are fully non-linear, invariably unstable, exact solutions which help understand how turbulence arises. The growing consensus is that the existence of these solutions is a necessary (but not sufficient) condition for turbulent dynamics to be possible in parameter space. Plane Couette flow - a fluid sheared between two parallel, differentially moving plates - is an exemplar of this in that the flow is linearly stable for all Reynolds numbers Re yet turbulence can occur at a finite Re beyond that where ECS disconnected from the basic sheared state start to exist. In this presentation we add extra physics in the form of stable stratification (gravity perpendicular to the plates) to plane Couette flow to explore this hypothesis further and to investigate transition in a stably stratified shear flow. Introduction Mathematically, adding stratification introduces two further control parameters to the plane Couette flow problem, the bulk Richardson number Ri b and the Prandtl number P r. Fixing P r (here to 1), means that the boundary between laminar flow and turbulent transition is no longer essentially a point on the Reynolds number line but now a line in the 2D Ri-Re plane which plausibly extends to infinite Re for some Ri b = Ri b (Re) bounded away from 0. Using a small domain for our computations (2π long and π wide where the plate separation is 2), we characterise this laminar-turbulent boundary at least for low Re and identify important ECS via edge tracking which appear to organise the transition process. The latter is demonstrated by using a new approach based on nonlinear optimal growth (Kerswell et al., 2014). This technique has recently been used to explore how the ‘minimal seed’ - the initial condition of minimum energy that can reach the turbulent state - is modified by stratification (Eaves and Caulfield, 2015). Here, we show how the approach can more generally used beyond the laminar-turbulent boundary to illustrate bursting events associated with the presence of ECS. The Boussinesq approximation, ρ = ρ 0 + ∆ρ, where ∆ρ << ρ 0 , is employed to obtain the nondimensional equations of the stratified plane Couette flow (pCf). Using the following scales, u = U u ∗ , t = h ∗ t , U y = h y ∗ , p = ρ 0 U 2 p ∗ − ρ 0 g y where u = (u, v, w) is the velocity field, 2U is the velocity differential across the plates (separated by a distance 2h, see figure 1), κ is the thermal diffusivity, ρ 0 is the reference density, p is the pressure and ν is the kinematic viscosity. The nondimensional governing VIII th Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016