Disruption of states by a stable stratification

Abstract

We identify 'minimal seeds' for turbulence, i.e. initial conditions of the smallest possible total perturbation energy density E c that trigger turbulence from the laminar state, in stratified plane Couette flow, the flow between two horizontal plates of separation 2H, moving with relative velocity 2 U, across which a constant density difference 2 ρ from a reference density ρ r is maintained. To find minimal seeds, we use the 'direct-adjoint-looping' (DAL) method for finding nonlinear optimal perturbations that optimise the time-averaged total dissipation of energy in the flow. These minimal seeds are located adjacent to the edge manifold, the manifold in state space that separates trajectories which transition to turbulence from those which eventually decay to the laminar state. The edge manifold is also the stable manifold of the system's 'edge state'. Therefore, the trajectories from the minimal seed initial conditions spend a large amount of time in the vicinity of some states: the edge state; another state contained within the edge manifold; or even in dynamically slowly varying regions of the edge manifold, allowing us to investigate the effects of a stable stratification on any coherent structures associated with such states. In unstratified plane Couette flow, these coherent structures are manifestations of the self-sustaining process (SSP) deduced on physical grounds by Waleffe (Phys. Fluids, vol. 9, 1997, pp. 883-900), or equivalently finite Reynolds number solutions of the vortex-wave interaction (VWI) asymptotic equations initially derived mathematically by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641-666). The stratified coherent states we identify at moderate Reynolds number display an altered form from their unstratified counterparts for bulk Richardson numbers Ri B = g ρH/(ρ r U 2 ) = O(Re −1 ), and exhibit chaotic motion for larger Ri B . We demonstrate that at hith Reynolds number the suppression of vertical motions by stratification strongly disrupts input from the waves to the roll velocity structures, thus preventing the waves from reinforcing the viscously decaying roll structures adequately, when Ri B = O(Re −2 ).

Exact coherent structures in stably stratified plane Couette flow

“…In the critical layer, stratification needs to be Figure 20 (left) confirms this scaling as Ri b increases from zero which is just a statement that the Rayleigh number, Ra := −Ri b Re 2 P r, is O(1). This scaling was proposed by Eaves & Caulfield (2015) by heuristic arguments and has also been independently found recently by Deguchi (2017).…”

Stably stratified exact coherent structures in shear flow: the effect of Prandtl number

“…, equivalent to the Rayleigh number Ra −Ri b PrRe 2 being O(1), was first observed numerically in the Pr = 1 case by Eaves & Caulfield (2015) and then in the analysis of Deguchi (2017) and Olvera & Kerswell (2017). There, at least for Pr = O(1), once the stratification strength increases beyond this scaling, new dynamic balances emerge [e.g.…”

Nonlinear Nonmodal Stability Theory

“…Pringle et al (2012) argue for why energy growth after time T might be the most natural objective functional for this procedure, but actually any objective functional which takes on heightened values in the basin of attraction of the turbulent state will do (e.g. total dissipation over the trajectory -Monokrousos et al 2011, Duguet et al 2013and Eaves & Caulfield 2015. The key point is that the optimization procedure should seek out any initial condition on the energy hypersurface which is in the turbulent basin of attraction if such states maximise the objective functional.…”