Testing linear marginal stability in stratified shear layers

Howland, Christopher J.; Taylor, John R.; Caulfield, C. P.

doi:10.1017/jfm.2018.79

Cited by 16 publications

(20 citation statements)

References 31 publications

(30 reference statements)

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“…This apparent inconsistency between our results and the theoretical analysis of Howland et al. (2018) gives a clue to the mechanism underlying the ‘relaxation’ process that we have suggested occurs in the initial stages of the evolution of our flow. An important point of difference between conditions in the near-surface region of our equilibrium state flow and those in the analysis of Howland et al.…”

Section: Vertical Profilescontrasting

confidence: 99%

“…In the near-surface region the mean velocity and temperature profiles (see figure 6) contain an inflection and hence are similar to the canonical conditions under which Kelvin–Helmholtz and Holmboe instabilities form. Howland, Taylor & Caulfield (2018) investigated marginal stability associated with the formation of K–H waves from laminar initial conditions and showed the marginal stability limit to be . In fact Kaminski, Caulfield & Taylor (2017) have shown that Kelvin–Helmholtz-like billows can form for up to 0.4 in the presence of perturbations that are sufficiently large and that have the optimal structure for amplification.…”

Section: Vertical Profilesmentioning

confidence: 99%

“…2017; Howland et al. 2018, for example). These studies typically use mathematically prescribed initial mean vertical velocity and temperature profiles, and apply artificial perturbations in order to catalyse the formation of the K–H instability.…”

Section: Vertical Profilesmentioning

confidence: 99%

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Evolution of thermally stratified turbulent open channel flow after removal of the heat source

et al. 2019

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Evolution of thermally stratified open channel flow after removal of a volumetric heat source is investigated using direct numerical simulation. The heat source models radiative heating from above and varies with height due to progressive absorption. After removal of the heat source the initial stable stratification breaks down and the channel approaches a fully mixed isothermal state. The initial state consists of three distinct regions: a near-wall region where stratification plays only a minor role, a central region where stratification has a significant effect on flow dynamics and a near-surface region where buoyancy effects dominate. We find that a state of local energetic equilibrium observed in the central region of the channel in the initial state persists until the late stages of the destratification process. In this region local turbulence parameters such as eddy diffusivity $k_{h}$ and flux Richardson number $R_{f}$ are found to be functions only of the Prandtl number $Pr$ and a mixed parameter ${\mathcal{Q}}$, which is equal to the ratio of the local buoyancy Reynolds number $Re_{b}$ and the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$. Close to the top and bottom boundaries turbulence is also affected by $Re_{\unicode[STIX]{x1D70F}}$ and vertical position $z$. In the initial heated equilibrium state the laminar surface layer is stabilised by the heat source, which acts as a potential energy sink. Removal of the heat source allows Kelvin–Helmholtz-like shear instabilities to form that lead to a rapid transition to turbulence and significantly enhance the mixing process. The destratifying flow is found to be governed by bulk parameters $Re_{\unicode[STIX]{x1D70F}}$, $Pr$ and the friction Richardson number $Ri_{\unicode[STIX]{x1D70F}}$. The overall destratification rate ${\mathcal{D}}$ is found to be a function of $Ri_{\unicode[STIX]{x1D70F}}$ and $Pr$.

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Section: Vertical Profilescontrasting

confidence: 99%

Section: Vertical Profilesmentioning

confidence: 99%

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Evolution of thermally stratified turbulent open channel flow after removal of the heat source

et al. 2019

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“…Figure 5 shows the vorticity structure of the steady states at two different values of Ri b . In the case of the Hopf bifurcation, billow-like structures are clearly seen, bearing a strong resemblance to the saturated, unsteady billows found by Howland et al (2018). Increasing Ri b along the upper branch to the saddle-node bifurcation, these structures remain but become significantly less pronounced.…”

Section: Uniform Stratification: the Drazin Modelmentioning

confidence: 69%

Kelvin–Helmholtz billows above Richardson number

2019

Self Cite

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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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“…Miles (1961) and Howard (1961) . Numerous studies have since extended this analysis and shown that, as long as Ri is not too large, shear flows are unstable when there is a viscous-diffusive fluid (Howland et al, 2018), when there is preexisting ambient turbulence in the fluid (Kaminski & Smyth, 2019), or when the shear is unsteady (Radko, 2019…”

Section: Shear Mixingmentioning

confidence: 99%

Roles of Shear and Convection in Driving Mixing in the Ocean

Ivey

Bluteau

Gayen

et al. 2021

Geophysical Research Letters

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Using field, numerical, and laboratory studies, we consider the roles of both shear and convection in driving mixing in the interior of the density‐stratified ocean. Shear mixing dominates when the Richardson number Ri < 0.25, convective mixing dominates when Ri > 1.0, and in the intermediate regime when 0.25 < Ri < 1.0 both shear and convection can contribute to mixing. For pure shear mixing the mixing efficiency Rif approaches 0.5, while for pure convective mixing the mixing efficiency Rif approaches 0.75. The diapycnal diffusivities for the two mechanisms are given by very different expressions. Despite these complexities, a simple mixing length model using the mean flow shear S provides robust estimates of diffusivity across the range 0 < Ri < 2. To account for the roles of both shear and convection over this range of Ri, we also formulate a modified version of the empirical KPP model for parameterizing ocean mixing in numerical models.

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Testing linear marginal stability in stratified shear layers

Cited by 16 publications

References 31 publications

Evolution of thermally stratified turbulent open channel flow after removal of the heat source

Evolution of thermally stratified turbulent open channel flow after removal of the heat source

Kelvin–Helmholtz billows above Richardson number

Roles of Shear and Convection in Driving Mixing in the Ocean

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