Transient growth in strongly stratified shear layers

Kaminski, Alexis; Caulfield, C. P.; Taylor, John R.

doi:10.1017/jfm.2014.552

Cited by 30 publications

(46 citation statements)

References 23 publications

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“…Optimal perturbation analysis of shear flows renders the entire problem to identifying the initial condition that maximizes an objective functional, perturbation kinetic energy being traditionally chosen. A density stratification suggests that the total perturbation energy, the sum of kinetic and potential energies, is the relevant measure [21,32]. We mention in passing that, in situations which do not allow for an evident measure of potential energy, other measures of optimality will need to be defined.…”

Section: Discussionmentioning

confidence: 99%

Analytical solutions for algebraic growth of disturbances in a stably stratified shear flow

et al. 2015

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We investigate analytically the short-time response of disturbances in a density-varying Couette flow without viscous and diffusive effects. The complete inviscid problem is also solved as an initial value problem with a density perturbation. We show that the kinetic energy of the disturbances grows algebraically at early times, contrary to the wellknown algebraic decay at time tending to infinity. This growth can persist for arbitrarily long times in response to sharp enough initial perturbations. The simplest in our three-stage study is a model problem forced by a buoyancy perturbation in the absence of background stratification. A linear growth with time is obtained in the vertical velocity component. This model provides an analogy between the transient mechanism of kinetic energy growth in a twodimensional density-varying flow and the lift-up mechanism of the three-dimensional constant density flow. Next we consider weak stable background stratification. Interestingly, the lowest order solution here is the same as that of the model flow. Our final study shows that a strong background stratification results in a sub-linear growth with time of the perturbation. A framework is thus presented where two-dimensional streamwise disturbances can lead to large transient amplification, unlike in constant density flow where three dimensions are required.

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Section: Discussionmentioning

confidence: 99%

Analytical solutions for algebraic growth of disturbances in a stably stratified shear flow

et al. 2015

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“…We now consider the case with a uniform background stratification, so that U = tanh z but B = z. This is also a commonly studied problem (Drazin 1958;Churilov & Shukhman 1987;Kaminski et al 2014) as again, linear stability analysis can be performed analytically. As before, Ri m = Ri b for this flow.…”

Section: Uniform Stratification: the Drazin Modelmentioning

confidence: 99%

Kelvin–Helmholtz billows above Richardson number

2019

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We study the dynamical system of a forced stratified mixing layer at finite Reynolds number Re, and Prandtl number P r = 1. We consider a hyperbolic tangent background velocity profile in the two cases of hyperbolic tangent and uniform background buoyancy stratifications. The system is forced in such a way that these background profiles are a steady solution of the governing equations. As is well-known, if the minimum gradient Richardson number of the flow, Ri m , is less than a certain critical value Ri c , the flow is linearly unstable to Kelvin-Helmholtz instability in both cases. Using Newton-Krylov iteration, we find steady, two-dimensional, finite amplitude elliptical vortex structures, i.e. 'Kelvin-Helmholtz billows', existing above Ri c . Bifurcation diagrams are produced using branch continuation, and we explore how these diagrams change with varying Re. In particular, when Re is sufficiently high we find that finite amplitude Kelvin-Helmholtz billows exist at Ri m > 1/4, where the flow is linearly stable by the Miles-Howard theorem. For the uniform background stratification, we give a simple explanation of the dynamical system, showing the dynamics can be understood on a two-dimensional manifold embedded in state space, and demonstrate the cases in which the system is bistable. In the case of a hyperbolic tangent stratification, we also describe a new, slowgrowing, linear instability of the background profiles at finite Re, which complicates the dynamics. † Email address for correspondence: jpp39@cam.ac.uk

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“…More recent work has shown that it is possible for disturbances to grow in stratified flows even if the Miles-Howard stability criterion is met and all normal-mode perturbations are stable (Farrell & Ioannou 1993b;Kaminski et al 2014). Owing to the nonnormality of the Navier-Stokes equations, it is possible for perturbations to grow at finite times and then subsequently decay in these ostensibly 'stable' flows (Trefethen et al 1993;Farrell & Ioannou 1993a).…”

Section: Introductionmentioning

confidence: 99%

“…In fact, though the maximum perturbation growth decreased with increasing stratification, no particular significance was associated with the critical value of Ri = 1/4. More recently, as reported in Kaminski et al (2014), we computed the linear optimal perturbations of a flow with uniform background stratification and hyperbolic tangent shear. We showed that while unstable normal-mode perturbations dominated at long target times, at shorter target times the perturbations leading to maximum growth were not necessarily the normal modes predicted by the Taylor-Goldstein equation.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear evolution of linear optimal perturbations of strongly stratified shear layers

2017

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The Miles–Howard theorem states that a necessary condition for normal-mode instability in parallel, inviscid, steady stratified shear flows is that the minimum gradient Richardson number, $Ri_{g,min}$, is less than $1/4$ somewhere in the flow. However, the non-normality of the Navier–Stokes and buoyancy equations may allow for substantial perturbation energy growth at finite times. We calculate numerically the linear optimal perturbations which maximize the perturbation energy gain for a stably stratified shear layer consisting of a hyperbolic tangent velocity distribution with characteristic velocity $U_{0}^{\ast }$ and a uniform stratification with constant buoyancy frequency $N_{0}^{\ast }$. We vary the bulk Richardson number $Ri_{b}=N_{0}^{\ast 2}h^{\ast 2}/U_{0}^{\ast 2}$ (corresponding to $Ri_{g,min}$) between 0.20 and 0.50 and the Reynolds numbers $\mathit{Re}=U_{0}^{\ast }h^{\ast }/\unicode[STIX]{x1D708}^{\ast }$ between 1000 and 8000, with the Prandtl number held fixed at $\mathit{Pr}=1$. We find the transient growth of non-normal perturbations may be sufficient to trigger strongly nonlinear effects and breakdown into small-scale structures, thereby leading to enhanced dissipation and non-trivial modification of the background flow even in flows where $Ri_{g,min}>1/4$. We show that the effects of nonlinearity are more significant for flows with higher $\mathit{Re}$, lower $Ri_{b}$ and higher initial perturbation amplitude $E_{0}$. Enhanced kinetic energy dissipation is observed for higher-$Re$ and lower-$Ri_{b}$ flows, and the mixing efficiency, quantified here by $\unicode[STIX]{x1D700}_{p}/(\unicode[STIX]{x1D700}_{p}+\unicode[STIX]{x1D700}_{k})$ where $\unicode[STIX]{x1D700}_{p}$ is the dissipation rate of density variance and $\unicode[STIX]{x1D700}_{k}$ is the dissipation rate of kinetic energy, is found to be approximately 0.35 for the most strongly nonlinear cases.

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Transient growth in strongly stratified shear layers

Cited by 30 publications

References 23 publications

Analytical solutions for algebraic growth of disturbances in a stably stratified shear flow

Analytical solutions for algebraic growth of disturbances in a stably stratified shear flow

Kelvin–Helmholtz billows above Richardson number

Nonlinear evolution of linear optimal perturbations of strongly stratified shear layers

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