Exact coherent structures in stably stratified plane Couette flow

Olvera, D.; Kerswell, Rich R.

doi:10.1017/jfm.2017.447

Cited by 14 publications

(30 citation statements)

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“…To minimise its memory footprint, our direct solver code restricts solutions to a symmetry subspace, thereby reducing the size of the basis set needed to represent the states at a given spatial resolution: see Appendix A for details. Following the previous work (Deguchi 2017;Olvera & Kerswell 2017), the symmetries imposed (in addition to L x -and L z -periodicity) are: 'shift-and-reflect' S, rotation about the z-axis Ω and spanwise reflection Z, defined by…”

Section: Methodsmentioning

confidence: 99%

“…, 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.…”

Section: Low Prandtl Number (Pr 1)mentioning

confidence: 98%

“…2, since states demand increasingly high wall-normal and (to a lesser extent) spanwise resolution as Pr increases. Accurately converging states in situations where structures become spatially localised is an important future challenge (see also Olvera & Kerswell 2017). In the following section we obtain higher Prandtl numbers by moving to a regime where we need only solve for the density field (see Appendix A.)…”

Section: Low Prandtl Number (Pr 1)mentioning

confidence: 99%

“…Drawing motivation from the above examples and the recent interest in stratified ECS, our plan here is to complement the stratified plane Couette flow study of Olvera & Kerswell (2017), by probing the – space at fixed Reynolds number. We are particularly interested here in the as-yet unexplored limits and , which are relevant for understanding astrophysical and geophysical flows respectively.…”

Section: Introductionmentioning

confidence: 99%

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

2019

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We examine how known unstable equilibria of the Navier-Stokes equations in plane Couette flow adapt to the presence of an imposed stable density difference between the two boundaries for varying values of the Prandtl number Pr, the ratio of viscosity to density diffusivity, and fixed moderate Reynolds number, Re = 400. In the two asymptotic limits Pr → 0 and Pr → ∞, it is found that such solutions exist at arbitrarily high bulk stratification but for different physical reasons. In the Pr → 0 limit, density variations away from a constant stable density gradient become vanishingly small as diffusion of density dominates over advection, allowing equilibria to exist for bulk Richardson number Ri b O(Re −2 Pr −1 ). Alternatively, at high Prandtl numbers, density becomes homogenised in the interior by the dominant advection which creates strongly stable stratified boundary layers that recede into the wall as Pr → ∞. In this scenario, the density stratification and the flow essentially decouple, thereby mitigating the effect of increasing Ri b . An asymptotic analysis is presented in the passive scalar regime Ri b O(Re −2 ), which reveals O(Pr −1/3 )-thick stratified boundary layers with O(Pr −2/9 )wide eruptions, giving rise to density fingers of O(Pr −1/9 ) length and O(Pr −4/9 ) width that invade an otherwise homogeneous interior. Finally, increasing Re to 10 5 in this regime reveals that interior stably stratified density layers can form away from the boundaries, separating well-mixed regions.

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Section: Methodsmentioning

confidence: 99%

Section: Low Prandtl Number (Pr 1)mentioning

confidence: 98%

Section: Low Prandtl Number (Pr 1)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

2019

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“…At the present stage it is not known where the regime boundaries lie in the Re-Ri plane, and the main aim of the current investigation is to be able to identify the intermittency boundary. On the other hand, tracing the relaminarisation boundary is beyond the scope of the present study, although there have been some recent investigations of this issue focusing on the identification of minimal seeds (Eaves & Caulfield 2015) and edge states (Olvera & Kerswell 2014) in stably stratified PCF. The two boundaries are distinct and well separated at low Re.…”

Section: Introductionmentioning

confidence: 99%

The intermittency boundary in stratified plane Couette flow

2015

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We study stratified turbulence in plane Couette flow using direct numerical simulations. Two external dimensionless parameters control the dynamics, the Reynolds number Re = Uh/ν and the bulk Richardson number Ri = gα V Th/U 2 , where U and T are half the velocity and temperature difference between the two walls respectively, h is the half channel depth, ν is the kinematic viscosity and gα V is the buoyancy parameter. We focus on spatio-temporal intermittency due to stratification and we explore the boundary between fully developed turbulence and intermittent flow in the Re-Ri plane. The structures populating the intermittent flow regime show coexistence between laminar and turbulent patches, and we demonstrate that there are qualitative differences between the previously studied low-Re low-Ri intermittent regime and the high-Re high-Ri intermittent regime. At low-Re low-Ri, turbulent regions span the entire gap, whereas at high-Re high-Ri, turbulence is confined vertically with complex dynamics arising from interacting turbulent layers. Consistent with a previous investigation of Flores & Riley (Boundary-Layer Meteorol., vol. 129 (2), 2010, pp. 241-259), we present evidence suggesting that intermittency in the asymptotic regime of high-Re Couette flows appears for L + < 200, where L + = Lu τ /ν, with L being the Monin-Obukhov length scale, L = u 3 τ /C κ q w , q w the wall heat flux, C κ the von Kármán constant and u τ = √ τ w /ρ 0 the friction velocity determined from the wall shear stress τ w , where ρ 0 is the constant background density. We also consider the mixing as quantified by various versions of the flux Richardson number Ri f , defined as the ratio of the conversion rate from kinetic to potential energy to the turbulent kinetic energy injection rate due to shear. We investigate how laminar and turbulent regions separately contribute to the overall mixing. Remarkably, we find that although fluctuations are greatly suppressed in the laminar regions, Ri f does not change significantly compared with its value in turbulent regions. As we observe a tight coupling between the mean temperature and velocity fields, we demonstrate that both Monin-Obukhov self-similarity theory (Monin & Obukhov, Contrib. Geophys. Inst. Acad. Sci. USSR, vol. 151, 1954, pp. 163-187) and the explicit algebraic model of Lazeroms et al. (J. Fluid Mech., vol. 723, 2013, pp. 91-125) predict the mean profiles well. We thus use these models to trace out the boundary between fully developed turbulence and intermittency in the Re-Ri plane.

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Disruption of states by a stable stratification

Eaves

Caulfield

2015

J. Fluid Mech.

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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 ).

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Exact coherent structures in stably stratified plane Couette flow

Cited by 14 publications

References 58 publications

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

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

The intermittency boundary in stratified plane Couette flow

Disruption of states by a stable stratification

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