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The dynamics of stratified horizontal shear flows at low Péclet number
- Laura Cope, P. Garaud, C. P. Caulfield
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- Journal:
- Journal of Fluid Mechanics / Volume 903 / 25 November 2020
- Published online by Cambridge University Press:
- 17 September 2020, A1
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We consider the dynamics of a vertically stratified, horizontally forced Kolmogorov flow. Motivated by astrophysical systems where the Prandtl number is often asymptotically small, our focus is the little-studied limit of high Reynolds number but low Péclet number (which is defined to be the product of the Reynolds number and the Prandtl number). Through a linear stability analysis, we demonstrate that the stability of two-dimensional modes to infinitesimal perturbations is independent of the stratification, whilst three-dimensional modes are always unstable in the limit of strong stratification and strong thermal diffusion. The subsequent nonlinear evolution and transition to turbulence are studied numerically using direct numerical simulations. For sufficiently large Reynolds numbers, four distinct dynamical regimes naturally emerge, depending upon the strength of the background stratification. By considering dominant balances in the governing equations, we derive scaling laws for each regime which explain the numerical data.
Mixing in forced stratified turbulence and its dependence on large-scale forcing
- Christopher J. Howland, John R. Taylor, C. P. Caulfield
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- Journal:
- Journal of Fluid Mechanics / Volume 898 / 10 September 2020
- Published online by Cambridge University Press:
- 25 June 2020, A7
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We study direct numerical simulations of turbulence arising from the interaction of an initial background shear, a linear background stratification and an external body force. In each simulation the turbulence produced is spatially intermittent, with dissipation rates varying over orders of magnitude in the vertical. We focus analysis on the statistically quasi-steady states achieved by applying large-scale body forcing to the domain, and compare flows forced by internal gravity waves with those forced by vertically uniform vortical modes. By considering the turbulent energy budgets for each simulation, we find that the injection of potential energy from the wave forcing permits a reversal in the sign of the mean buoyancy flux. This change in the sign of the buoyancy flux is associated with large, convective density overturnings, which in turn lead to more efficient mixing in the wave-forced simulations. The inhomogeneous dissipation in each simulation allows us to investigate localised correlations between the kinetic and potential energy dissipation rates. These correlations lead us to the conclusion that an appropriate definition of an instantaneous mixing efficiency, $\unicode[STIX]{x1D702}(t):=\unicode[STIX]{x1D712}/(\unicode[STIX]{x1D712}+\unicode[STIX]{x1D700})$ (where $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D712}$ are the volume-averaged turbulent viscous dissipation rate and fluctuation density variance destruction rate respectively) in the wave-forced cases is independent of an appropriately defined local turbulent Froude number, consistent with scalings proposed for low Froude number stratified turbulence.
The viscous Holmboe instability for smooth shear and density profiles
- J. P. Parker, C. P. Caulfield, R. R. Kerswell
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- Journal:
- Journal of Fluid Mechanics / Volume 896 / 10 August 2020
- Published online by Cambridge University Press:
- 01 June 2020, A14
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The Holmboe wave instability is one of the classic examples of a stratified shear instability, usually explained as the result of a resonance between a gravity wave and a vorticity wave. Historically, it has been studied by linear stability analyses at infinite Reynolds number, $Re$, and by direct numerical simulations at relatively low $Re$ in the regions known to be unstable from the inviscid linear stability results. In this paper, we perform linear stability analyses of the classical ‘Hazel model’ of a stratified shear layer (where the background velocity and density distributions are assumed to take the functional form of hyperbolic tangents with different characteristic vertical scales) over a range of different parameters at finite $Re$, finding new unstable regions of parameter space. In particular, we find instability when the Richardson number is everywhere greater than $1/4$, where the flow would be stable at infinite $Re$ by the Miles–Howard theorem. We find unstable modes with no critical layer, and show that, despite the necessity of viscosity for the new instability, the growth rate relative to diffusion of the background profile is maximised at large $Re$. We use these results to shed new light on the wave-resonance and over-reflection interpretations of stratified shear instability. We argue for a definition of Holmboe instability as being characterised by propagating vortices above or below the shear layer, as opposed to any reference to sharp density interfaces.
Kelvin–Helmholtz billows above Richardson number $1/4$
- J. P. Parker, C. P. Caulfield, R. R. Kerswell
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- Journal:
- Journal of Fluid Mechanics / Volume 879 / 25 November 2019
- Published online by Cambridge University Press:
- 23 September 2019, R1
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We study the dynamical system of a two-dimensional, forced, stratified mixing layer at finite Reynolds number $Re$, and Prandtl number $Pr=1$. We consider a hyperbolic tangent background velocity profile in the two cases of hyperbolic tangent and uniform background buoyancy stratifications, in a domain of fixed, finite width and height. 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 when $Ri_{m}>1/4$ for the background flow, which 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, slow-growing, linear instability of the background profiles at finite $Re$, which complicates the dynamics.
Mixing and entrainment are suppressed in inclined gravity currents
- Maarten van Reeuwijk, Markus Holzner, C. P. Caulfield
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- Journal:
- Journal of Fluid Mechanics / Volume 873 / 25 August 2019
- Published online by Cambridge University Press:
- 28 June 2019, pp. 786-815
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We explore the dynamics of inclined temporal gravity currents using direct numerical simulation, and find that the current creates an environment in which the flux Richardson number $\mathit{Ri}_{f}$, gradient Richardson number $\mathit{Ri}_{g}$ and turbulent flux coefficient $\unicode[STIX]{x1D6E4}$ are constant across a large portion of the depth. Changing the slope angle $\unicode[STIX]{x1D6FC}$ modifies these mixing parameters, and the flow approaches a maximum Richardson number $\mathit{Ri}_{max}\approx 0.15$ as $\unicode[STIX]{x1D6FC}\rightarrow 0$ at which the entrainment coefficient $E\rightarrow 0$. The turbulent Prandtl number remains $O(1)$ for all slope angles, demonstrating that $E\rightarrow 0$ is not caused by a switch-off of the turbulent buoyancy flux as conjectured by Ellison (J. Fluid Mech., vol. 2, 1957, pp. 456–466). Instead, $E\rightarrow 0$ occurs as the result of the turbulence intensity going to zero as $\unicode[STIX]{x1D6FC}\rightarrow 0$, due to the flow requiring larger and larger shear to maintain the same level of turbulence. We develop an approximate model valid for small $\unicode[STIX]{x1D6FC}$ which is able to predict accurately $\mathit{Ri}_{f}$, $\mathit{Ri}_{g}$ and $\unicode[STIX]{x1D6E4}$ as a function of $\unicode[STIX]{x1D6FC}$ and their maximum attainable values. The model predicts an entrainment law of the form $E=0.31(\mathit{Ri}_{max}-\mathit{Ri})$, which is in good agreement with the simulation data. The simulations and model presented here contribute to a growing body of evidence that an approach to a marginally or critically stable, relatively weakly stratified equilibrium for stratified shear flows may well be a generic property of turbulent stratified flows.
Layer formation and relaminarisation in plane Couette flow with spanwise stratification
- Dan Lucas, C. P. Caulfield, Rich R. Kerswell
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- Journal:
- Journal of Fluid Mechanics / Volume 868 / 10 June 2019
- Published online by Cambridge University Press:
- 10 April 2019, pp. 97-118
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In this paper we investigate the effect of stable stratification on plane Couette flow when gravity is oriented in the spanwise direction. When the flow is turbulent we observe near-wall layering and associated new mean flows in the form of large-scale spanwise-flattened streamwise rolls. The layers exhibit the expected buoyancy scaling $l_{z}\sim U/N$ where $U$ is a typical horizontal velocity scale and $N$ the buoyancy frequency. We associate the new coherent structures with a stratified modification of the well-known large-scale secondary circulation in plane Couette flow. We find that the possibility of the transition to sustained turbulence is controlled by the relative size of this buoyancy scale to the spanwise spacing of the streaks. In parts of parameter space transition can also be initiated by a newly discovered linear instability in this system (Facchini et al., J. Fluid Mech., vol. 853, 2018, pp. 205–234). When wall turbulence can be sustained the linear instability opens up new routes in phase space for infinitesimal disturbances to initiate turbulence. When the buoyancy scale suppresses turbulence the linear instability leads to more ordered nonlinear states, with the possibility for intermittent bursts of secondary shear instability.
Self-organized criticality of turbulence in strongly stratified mixing layers
- Hesam Salehipour, W. R. Peltier, C. P. Caulfield
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- Journal:
- Journal of Fluid Mechanics / Volume 856 / 10 December 2018
- Published online by Cambridge University Press:
- 02 October 2018, pp. 228-256
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Motivated by the importance of stratified shear flows in geophysical and environmental circumstances, we characterize their energetics, mixing and spectral behaviour through a series of direct numerical simulations of turbulence generated by Holmboe wave instability (HWI) under various initial conditions. We focus on circumstances where the stratification is sufficiently ‘strong’ so that HWI is the dominant primary instability of the flow. Our numerical findings demonstrate the emergence of self-organized criticality (SOC) that is manifest as an adjustment of an appropriately defined gradient Richardson number, $Ri_{g}$, associated with the horizontally averaged mean flow, in such a way that it is continuously attracted towards a critical value of $Ri_{g}\sim 1/4$. This self-organization occurs through a continuously reinforced localization of the ‘scouring’ motions (i.e. ‘avalanches’) that are characteristic of the turbulence induced by the breakdown of Holmboe wave instabilities and are developed on the upper and lower flanks of the sharply localized density interface, embedded within a much more diffuse shear layer. These localized ‘avalanches’ are also found to exhibit the expected scale-invariant characteristics. From an energetics perspective, the emergence of SOC is expressed in the form of a long-lived turbulent flow that remains in a ‘quasi-equilibrium’ state for an extended period of time. Most importantly, the irreversible mixing that results from such self-organized behaviour appears to be characterized generically by a universal cumulative turbulent flux coefficient of $\unicode[STIX]{x1D6E4}_{c}\sim 0.2$ only for turbulent flows engendered by Holmboe wave instability. The existence of this self-organized critical state corroborates the original physical arguments associated with self-regulation of stratified turbulent flows as involving a ‘kind of equilibrium’ as described by Turner (1973, Buoyancy Effects in Fluids, Cambridge University Press).
Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Péclet and Richardson numbers
- F. Marcotte, C. P. Caulfield
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- Journal:
- Journal of Fluid Mechanics / Volume 853 / 25 October 2018
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- 23 August 2018, pp. 359-385
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We consider the nonlinear optimisation of irreversible mixing induced by an initial finite amplitude perturbation of a statically stable density-stratified fluid with kinematic viscosity $\unicode[STIX]{x1D708}$ and density diffusivity $\unicode[STIX]{x1D705}$. The initial diffusive error function density distribution varies continuously so that $\unicode[STIX]{x1D70C}\in [\bar{\unicode[STIX]{x1D70C}}-\unicode[STIX]{x1D70C}_{0}/2,\bar{\unicode[STIX]{x1D70C}}+\unicode[STIX]{x1D70C}_{0}/2]$. A constant pressure gradient is imposed in a plane two-dimensional channel of depth $2h$. We consider flows with a finite Péclet number $Pe=U_{m}h/\unicode[STIX]{x1D705}=500$ and Prandtl number $Pr=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}=1$, and a range of bulk Richardson numbers $Ri_{b}=g\unicode[STIX]{x1D70C}_{0}h/(\bar{\unicode[STIX]{x1D70C}}U^{2})\in [0,1]$ where $U_{m}$ is the maximum flow speed of the laminar parallel flow, and $g$ is the gravitational acceleration. We use the constrained variational direct-adjoint-looping (DAL) method to solve two optimisation problems, extending the optimal mixing results of Foures et al. (J. Fluid Mech., vol. 748, 2014, pp. 241–277) to stratified flows, where the irreversible mixing of the active scalar density leads to a conversion of kinetic energy into potential energy. We identify initial perturbations of fixed finite kinetic energy which maximise the time-averaged perturbation kinetic energy developed over a finite time interval, and initial perturbations that minimise the value (at a target time, chosen to be $T=10$) of a ‘mix-norm’ as first introduced by Mathew et al. (Physica D, vol. 211, 2005, pp. 23–46), further discussed by Thiffeault (Nonlinearity, vol. 25, 2012, pp. 1–44) and shown by Foures et al. (2014) to be a computationally efficient and robust proxy for identifying perturbations that minimise the long-time variance of a scalar distribution. We demonstrate, for all bulk Richardson numbers considered, that the time-averaged kinetic-energy maximising perturbations are significantly suboptimal at mixing compared to the mix-norm minimising perturbations, and also that minimising the mix-norm remains (for density-stratified flows) a good proxy for identifying perturbations which minimise the variance at long times. Although increasing stratification reduces the mixing in general, mix-norm minimising optimal perturbations can still trigger substantial mixing for $Ri_{b}\lesssim 0.3$. By considering the time evolution of the kinetic energy and potential energy reservoirs, we find that such perturbations lead to a flow which, through Taylor dispersion, very effectively converts perturbation kinetic energy into ‘available potential energy’, which in turn leads rapidly and irreversibly to thorough and efficient mixing, with little energy returned to the kinetic-energy reservoirs.
Optimal mixing in three-dimensional plane Poiseuille flow at high Péclet number
- L. Vermach, C. P. Caulfield
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- Journal of Fluid Mechanics / Volume 850 / 10 September 2018
- Published online by Cambridge University Press:
- 10 July 2018, pp. 875-923
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We consider a passive zero-mean scalar field organised into two layers of different concentrations in a three-dimensional plane channel flow subjected to a constant along-stream pressure gradient. We employ a nonlinear direct-adjoint-looping method to identify the optimal initial perturbation of the velocity field with given initial energy which yields ‘maximal’ mixing by a target time horizon, where maximal mixing is defined here as the minimisation of the spatially integrated variance of the concentration field. We verify in three-dimensional flows the conjecture by Foures et al. (J. Fluid Mech., vol. 748, 2014, pp. 241–277) that the initial perturbation which maximises the time-averaged energy gain of the flow leads to relatively weak mixing, and is qualitatively different from the optimal initial ‘mixing’ perturbation which exploits classical Taylor dispersion. We carry out the analysis for two different Reynolds numbers ($Re=U_{m}h/\unicode[STIX]{x1D708}=500$ and $Re=3000$, where $U_{m}$ is the maximum flow speed of the unperturbed flow, $h$ is the channel half-depth and $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid) demonstrating that this key finding is robust with respect to the transition to turbulence. We also identify the initial perturbations that minimise, at chosen target times, the ‘mix-norm’ of the concentration field, i.e. a Sobolev norm of negative index in the class introduced by Mathew et al. (Physica D, vol. 211, 2005, pp. 23–46). We show that the ‘true’ variance-based mixing strategy can be successfully and practicably approximated by the mix-norm minimisation since we find that the mix-norm-optimal initial perturbations are far less sensitive to changes in the target time horizon than their optimal variance-minimising counterparts.
The structure and origin of confined Holmboe waves
- Adrien Lefauve, J. L. Partridge, Qi Zhou, S. B. Dalziel, C. P. Caulfield, P. F. Linden
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- Journal of Fluid Mechanics / Volume 848 / 10 August 2018
- Published online by Cambridge University Press:
- 05 June 2018, pp. 508-544
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Finite-amplitude manifestations of stratified shear flow instabilities and their spatio-temporal coherent structures are believed to play an important role in turbulent geophysical flows. Such shear flows commonly have layers separated by sharp density interfaces, and are therefore susceptible to the so-called Holmboe instability, and its finite-amplitude manifestation, the Holmboe wave. In this paper, we describe and elucidate the origin of an apparently previously unreported long-lived coherent structure in a sustained stratified shear flow generated in the laboratory by exchange flow through an inclined square duct connecting two reservoirs filled with fluids of different densities. Using a novel measurement technique allowing for time-resolved, near-instantaneous measurements of the three-component velocity and density fields simultaneously over a three-dimensional volume, we describe the three-dimensional geometry and spatio-temporal dynamics of this structure. We identify it as a finite-amplitude, nonlinear, asymmetric confined Holmboe wave (CHW), and highlight the importance of its spanwise (lateral) confinement by the duct boundaries. We pay particular attention to the spanwise vorticity, which exhibits a travelling, near-periodic structure of sheared, distorted, prolate spheroids with a wide ‘body’ and a narrower ‘head’. Using temporal linear stability analysis on the two-dimensional streamwise-averaged experimental flow, we solve for three-dimensional perturbations having two-dimensional, cross-sectionally confined eigenfunctions and a streamwise normal mode. We show that the dispersion relation and the three-dimensional spatial structure of the fastest-growing confined Holmboe instability are in good agreement with those of the observed confined Holmboe wave. We also compare those results with a classical linear analysis of two-dimensional perturbations (i.e. with no spanwise dependence) on a one-dimensional base flow. We conclude that the lateral confinement is an important ingredient of the confined Holmboe instability, which gives rise to the CHW, with implications for many inherently confined geophysical flows such as in valleys, estuaries, straits or deep ocean trenches. Our results suggest that the CHW is an example of an experimentally observed, inherently nonlinear, robust, long-lived coherent structure which has developed from a linear instability. We conjecture that the CHW is a promising candidate for a class of exact coherent states underpinning the dynamics of more disordered, yet continually forced stratified shear flows.
Horizontal locomotion of a vertically flapping oblate spheroid
- Jian Deng, C. P. Caulfield
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- Journal of Fluid Mechanics / Volume 840 / 10 April 2018
- Published online by Cambridge University Press:
- 15 February 2018, pp. 688-708
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We consider the self-induced motions of three-dimensional oblate spheroids of density $\unicode[STIX]{x1D70C}_{s}$ with varying aspect ratios $AR=b/c\leqslant 1$, where $b$ and $c$ are the spheroids’ centre-pole radius and centre-equator radius, respectively. Vertical motion is imposed on the spheroids such that $y_{s}(t)=A\sin (2\unicode[STIX]{x03C0}ft)$ in a fluid of density $\unicode[STIX]{x1D70C}$ and kinematic viscosity $\unicode[STIX]{x1D708}$. As in strictly two-dimensional flows, above a critical value $Re_{C}$ of the flapping Reynolds number $Re_{A}=2Afc/\unicode[STIX]{x1D708}$, the spheroid ultimately propels itself horizontally as a result of fluid–body interactions. For $Re_{A}$ sufficiently above $Re_{C}$, the spheroid rapidly settles into a terminal state of constant, unidirectional velocity, consistent with the prediction of Deng et al. (Phys. Rev. E, vol. 94, 2016, 033107) that, at sufficiently high $Re_{A}$, such oscillating spheroids manifest $m=1$ asymmetric flow, with characteristic vortical structures conducive to providing unidirectional thrust if the spheroid is free to move horizontally. The speed $U$ of propagation increases linearly with the flapping frequency, resulting in a constant Strouhal number $St(AR)=2Af/U$, characterising the locomotive performance of the oblate spheroid, somewhat larger than the equivalent $St$ for two-dimensional spheroids, demonstrating that the three-dimensional flow is less efficient at driving locomotion. $St$ decreases with increasing aspect ratio for both two-dimensional and three-dimensional flows, although the relative disparity (and hence relative inefficiency of three-dimensional motion) decreases. For flows with $Re_{A}\gtrsim Re_{C}$, we observe two distinct types of inherently three-dimensional motion for different aspect ratios. The first, associated with a disk of aspect ratio $AR=0.1$ at $Re_{A}=45$, consists of a ‘stair-step’ trajectory. This trajectory can be understood through consideration of relatively high azimuthal wavenumber instabilities of interacting vortex rings, characterised by in-phase vortical structures above and below an oscillating spheroid, recently calculated using Floquet analysis by Deng et al. (Phys. Rev. E, vol. 94, 2016, 033107). Such ‘in-phase’ instabilities arise in a relatively narrow band of $Re_{A}\gtrsim Re_{C}$, which band shifts to higher Reynolds number as the aspect ratio increases. (Indeed, for horizontally fixed spheroids with aspect ratio $AR=0.2$, Floquet analysis actually predicts stability at $Re_{A}=45$.) For such a spheroid ($AR=0.2$, $Re_{A}=45$, with sufficiently small mass ratio $m_{s}/m_{f}=\unicode[STIX]{x1D70C}_{s}V_{s}/(\unicode[STIX]{x1D70C}V_{s})$, where $V_{s}$ is the volume of the spheroid) which is free to move horizontally, the second type of three-dimensional motion is observed, initially taking the form of a ‘snaking’ trajectory with long quasi-periodic sweeping oscillations before locking into an approximately elliptical ‘orbit’, apparently manifesting a three-dimensional generalisation of the $QP_{H}$ quasi-periodic symmetry breaking discussed for sufficiently high aspect ratio two-dimensional elliptical foils in Deng & Caulfield (J. Fluid Mech., vol. 787, 2016, pp. 16–49).
Testing linear marginal stability in stratified shear layers
- C. J. Howland, J. R. Taylor, C. P. Caulfield
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- Journal of Fluid Mechanics / Volume 839 / 25 March 2018
- Published online by Cambridge University Press:
- 02 February 2018, R4
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We use two-dimensional direct numerical simulations of Boussinesq stratified shear layers to investigate the influence of the minimum gradient Richardson number $Ri_{m}$ on the early time evolution of Kelvin–Helmholtz instability to its saturated ‘billow’ state. Even when the diffusion of the background velocity and density distributions is counterbalanced by artificial body forces to maintain the initial profiles, in the limit as $Ri_{m}\rightarrow 1/4$, the perturbation growth rate tends to zero and the saturated perturbation energy becomes small. These results imply, at least for such canonical inflectional stratified shear flows, that ‘marginally unstable’ flows with $Ri_{m}$ only slightly less than 1/4 are highly unlikely to become ‘turbulent’, in the specific sense of being associated with significantly enhanced dissipation, irreversible mixing and non-trivial modification of the background distributions without additional externally imposed forcing.
Layer formation in horizontally forced stratified turbulence: connecting exact coherent structures to linear instabilities
- Dan Lucas, C. P. Caulfield, Rich R. Kerswell
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- Journal:
- Journal of Fluid Mechanics / Volume 832 / 10 December 2017
- Published online by Cambridge University Press:
- 26 October 2017, pp. 409-437
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We consider turbulence in a stratified ‘Kolmogorov’ flow, driven by horizontal shear in the form of sinusoidal body forcing in the presence of an imposed background linear stable stratification in the third direction. This flow configuration allows the controlled investigation of the formation of coherent structures, which here organise the flow into horizontal layers by inclining the background shear as the strength of the stratification is increased. By numerically converging exact steady states from direct numerical simulations of chaotic flow, we show, for the first time, a robust connection between linear theory predicting instabilities from infinitesimal perturbations to the robust finite-amplitude nonlinear layered state observed in the turbulence. We investigate how the observed vertical length scales are related to the primary linear instabilities and compare to previously considered examples of shear instability leading to layer formation in other horizontally sheared flows.
Irreversible mixing by unstable periodic orbits in buoyancy dominated stratified turbulence
- Dan Lucas, C. P. Caulfield
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- Journal of Fluid Mechanics / Volume 832 / 10 December 2017
- Published online by Cambridge University Press:
- 26 October 2017, R1
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We consider turbulence driven by a large-scale horizontal shear in Kolmogorov flow (i.e. with sinusoidal body forcing) and a background linear stable stratification with buoyancy frequency $N_{B}^{2}$ imposed in the third, vertical direction in a fluid with kinematic viscosity $\unicode[STIX]{x1D708}$. This flow is known to be organised into layers by nonlinear unstable steady states, which incline the background shear in the vertical and can be demonstrated to be the finite-amplitude saturation of a sequence of instabilities, originally from the laminar state. Here, we investigate the next order of motions in this system, i.e. the time-dependent mechanisms by which the density field is irreversibly mixed. This investigation is achieved using ‘recurrent flow analysis’. We identify (unstable) periodic orbits, which are embedded in the turbulent attractor, and use these orbits as proxies for the chaotic flow. We find that the time average of an appropriate measure of the ‘mixing efficiency’ of the flow $\mathscr{E}=\unicode[STIX]{x1D712}/(\unicode[STIX]{x1D712}+{\mathcal{D}})$ (where ${\mathcal{D}}$ is the volume-averaged kinetic energy dissipation rate and $\unicode[STIX]{x1D712}$ is the volume-averaged density variance dissipation rate) varies non-monotonically with the time-averaged buoyancy Reynolds numbers $\overline{Re}_{B}=\overline{{\mathcal{D}}}/(\unicode[STIX]{x1D708}N_{B}^{2})$, and is bounded above by $1/6$, consistently with the classical model of Osborn (J. Phys. Oceanogr., vol. 10 (1), 1980, pp. 83–89). There are qualitatively different physical properties between the unstable orbits that have lower irreversible mixing efficiency at low $\overline{Re}_{B}\sim O(1)$ and those with nearly optimal $\mathscr{E}\lesssim 1/6$ at intermediate $\overline{Re}_{B}\sim 10$. The weaker orbits, inevitably embedded in more strongly stratified flow, are characterised by straining or ‘scouring’ motions, while the more efficient orbits have clear overturning dynamics in more weakly stratified, and apparently shear-unstable flow.
Role of overturns in optimal mixing in stratified mixing layers
- A. Mashayek, C. P. Caulfield, W. R. Peltier
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- Journal of Fluid Mechanics / Volume 826 / 10 September 2017
- Published online by Cambridge University Press:
- 08 August 2017, pp. 522-552
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Turbulent mixing plays a major role in enabling the large-scale ocean circulation. The accuracy of mixing rates estimated from observations depends on our understanding of basic fluid mechanical processes underlying the nature of turbulence in a stratified fluid. Several of the key assumptions made in conventional mixing parameterizations have been increasingly scrutinized in recent years, primarily on the basis of adequately high resolution numerical simulations. We add to this evidence by compiling results from a suite of numerical simulations of the turbulence generated through stratified shear instability processes. We study the inherently intermittent and time-dependent nature of wave-induced turbulent life cycles and more specifically the tight coupling between inherently anisotropic scales upon which small-scale isotropic turbulence grows. The anisotropic scales stir and stretch fluid filaments enhancing irreversible diffusive mixing at smaller scales. We show that the characteristics of turbulent mixing depend on the relative time evolution of the Ozmidov length scale $L_{O}$ compared to the so-called Thorpe overturning scale $L_{T}$ which represents the scale containing available potential energy upon which turbulence feeds and grows. We find that when $L_{T}\sim L_{O}$, the mixing is most active and efficient since stirring by the largest overturns becomes ‘optimal’ in the sense that it is not suppressed by ambient stratification. We argue that the high mixing efficiency associated with this phase, along with observations of $L_{O}/L_{T}\sim 1$ in oceanic turbulent patches, together point to the potential for systematically underestimating mixing in the ocean if the role of overturns is neglected. This neglect, arising through the assumption of a clear separation of scales between the background mean flow and small-scale quasi-isotropic turbulence, leads to the exclusion of an highly efficient mixing phase from conventional parameterizations of the vertical transport of density. Such an exclusion may well be significant if the mechanism of shear-induced turbulence is assumed to be representative of at least some turbulent events in the ocean. While our results are based upon simulations of shear instability, we show that they are potentially more generic by making direct comparisons with $L_{T}-L_{O}$ data from ocean and lake observations which represent a much wider range of turbulence-inducing physical processes.
Nonlinear evolution of linear optimal perturbations of strongly stratified shear layers
- A. K. Kaminski, C. P. Caulfield, J. R. Taylor
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- Journal:
- Journal of Fluid Mechanics / Volume 825 / 25 August 2017
- Published online by Cambridge University Press:
- 20 July 2017, pp. 213-244
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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.
Diapycnal mixing in layered stratified plane Couette flow quantified in a tracer-based coordinate
- Qi Zhou, J. R. Taylor, C. P. Caulfield, P. F. Linden
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- Journal:
- Journal of Fluid Mechanics / Volume 823 / 25 July 2017
- Published online by Cambridge University Press:
- 15 June 2017, pp. 198-229
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The mixing properties of statically stable density interfaces subject to imposed vertical shear are studied using direct numerical simulations of stratified plane Couette flow. The simulations are designed to investigate possible self-maintaining mechanisms of sharp density interfaces motivated by Phillips’ argument (Deep-Sea Res., vol. 19, 1972, pp. 79–81) by which layers and interfaces can spontaneously form due to vertical variations of diapycnal flux. At the start of each simulation, a sharp density interface with the same initial thickness is introduced at the midplane between two flat, horizontal walls counter-moving at velocities $\pm U_{w}$. Particular attention is paid to the effects of varying Prandtl number $\mathit{Pr}\equiv \unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}$, where $\unicode[STIX]{x1D708}$ and $\unicode[STIX]{x1D705}$ are the molecular kinematic viscosity and diffusivity respectively, over two orders of magnitude from 0.7, 7 and 70. Varying $\mathit{Pr}$ enables the system to access a considerable range of characteristic turbulent Péclet numbers $\mathit{Pe}_{\ast }\equiv {\mathcal{U}}_{\ast }{\mathcal{L}}_{\ast }/\unicode[STIX]{x1D705}$, where ${\mathcal{U}}_{\ast }$ and ${\mathcal{L}}_{\ast }$ are characteristic velocity and length scales, respectively, of the motion which acts to ‘scour’ the density interface. The dynamics of the interface varies with the stability of the interface which is characterised by a bulk Richardson number $\mathit{Ri}\,\equiv \,b_{0}h/U_{w}^{2}$, where $b_{0}$ is half the initial buoyancy difference across the interface and $h$ is the half-height of the channel. Shear-induced turbulence occurs at small $\mathit{Ri}$, whereas internal waves propagating on the interface dominate at large $\mathit{Ri}$. For a highly stable (i.e. large $\mathit{Ri}$) interface at sufficiently large $\mathit{Pe}_{\ast }$, the complex interfacial dynamics allows the interface to remain sharp. This ‘self-sharpening’ is due to the combined effects of the ‘scouring’ induced by the turbulence external to the interface and comparatively weak molecular diffusion across the core region of the interface. The effective diapycnal diffusivity and irreversible buoyancy flux are quantified in the tracer-based reference coordinate proposed by Winters & D’Asaro (J. Fluid Mech., vol. 317, 1996, pp. 179–193) and Nakamura (J. Atmos. Sci., vol. 53, 1996, pp. 1524–1537), which enables a detailed investigation of the self-sharpening process by analysing the local budget of buoyancy gradient in the reference coordinate. We further discuss the dependence of the effective diffusivity and overall mixing efficiency on the characteristic parameters of the flow, such as the buoyancy Reynolds number and the local gradient Richardson number, and highlight the possible role of the molecular properties of fluids on diapycnal mixing.
Self-similar mixing in stratified plane Couette flow for varying Prandtl number
- Qi Zhou, John R. Taylor, C. P. Caulfield
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- Journal of Fluid Mechanics / Volume 820 / 10 June 2017
- Published online by Cambridge University Press:
- 04 May 2017, pp. 86-120
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We investigate fully developed turbulence in stratified plane Couette flows using direct numerical simulations similar to those reported by Deusebio et al. (J. Fluid Mech., vol. 781, 2015, pp. 298–329) expanding the range of Prandtl number $Pr$ examined by two orders of magnitude from 0.7 up to 70. Significant effects of $Pr$ on the heat and momentum fluxes across the channel gap and on the mean temperature and velocity profile are observed. These effects can be described through a mixing length model coupling Monin–Obukhov (M–O) similarity theory and van Driest damping functions. We then employ M–O theory to formulate similarity scalings for various flow diagnostics for the stratified turbulence in the gap interior. The midchannel gap gradient Richardson number $Ri_{g}$ is determined by the length scale ratio $h/L$, where $h$ is the half-channel gap depth and $L$ is the Obukhov length scale. As $h/L$ approaches very large values, $Ri_{g}$ asymptotes to a maximum characteristic value of approximately 0.2. The buoyancy Reynolds number $Re_{b}\equiv \unicode[STIX]{x1D700}/(\unicode[STIX]{x1D708}N^{2})$, where $\unicode[STIX]{x1D700}$ is the dissipation, $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $N$ is the buoyancy frequency defined in terms of the local mean density gradient, scales linearly with the length scale ratio $L^{+}\equiv L/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$, where $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ is the near-wall viscous scale. The flux Richardson number $Ri_{f}\equiv -B/P$, where $B$ is the buoyancy flux and $P$ is the shear production, is found to be proportional to $Ri_{g}$. This then leads to a turbulent Prandtl number $Pr_{t}\equiv \unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D705}_{t}$ of order unity, where $\unicode[STIX]{x1D708}_{t}$ and $\unicode[STIX]{x1D705}_{t}$ are the turbulent viscosity and diffusivity respectively, which is consistent with Reynolds analogy. The turbulent Froude number $Fr_{h}\equiv \unicode[STIX]{x1D700}/(NU^{\prime 2})$, where $U^{\prime }$ is a turbulent horizontal velocity scale, is found to vary like $Ri_{g}^{-1/2}$. All these scalings are consistent with our numerical data and appear to be independent of $Pr$. The classical Osborn model based on turbulent kinetic energy balance in statistically stationary stratified sheared turbulence (Osborn, J. Phys. Oceanogr., vol. 10, 1980, pp. 83–89), together with M–O scalings, results in a parameterization of $\unicode[STIX]{x1D705}_{t}/\unicode[STIX]{x1D708}\sim \unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D708}\sim Re_{b}Ri_{g}/(1-Ri_{g})$. With this parameterization validated through direct numerical simulation data, we provide physical interpretations of these results in the context of M–O similarity theory. These results are also discussed and rationalized with respect to other parameterizations in the literature. This paper demonstrates the role of M–O similarity in setting the mixing efficiency of equilibrated constant-flux layers, and the effects of Prandtl number on mixing in wall-bounded stratified turbulent flows.
Multiple instability of layered stratified plane Couette flow
- T. S. Eaves, C. P. Caulfield
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- Journal:
- Journal of Fluid Mechanics / Volume 813 / 25 February 2017
- Published online by Cambridge University Press:
- 17 January 2017, pp. 250-278
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We present the linear stability properties and nonlinear evolution of two-dimensional plane Couette flow for a statically stable Boussinesq three-layer fluid of total depth $2h$ between two horizontal plates driven at constant velocity $\pm \unicode[STIX]{x0394}U$. Initially the three layers have equal depth $2h/3$ and densities $\unicode[STIX]{x1D70C}_{0}+\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$, $\unicode[STIX]{x1D70C}_{0}$ and $\unicode[STIX]{x1D70C}_{0}-\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$, such that $\unicode[STIX]{x1D70C}_{0}\gg \unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$. At finite Reynolds and Prandtl number, we demonstrate that this flow is susceptible to two distinct primary linear instabilities for sufficiently large bulk Richardson number $Ri_{B}=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}h/(\unicode[STIX]{x1D70C}_{0}\unicode[STIX]{x0394}U^{2})$. For a given bulk Richardson number $Ri_{B}$, the zero phase speed ‘Taylor’ instability is always predicted to have the largest growth rate and to be an inherently two-dimensional instability. An inherently viscous instability, reminiscent of the ‘Holmboe’ instability, is also predicted to have a non-zero growth rate. For flows with Prandtl number $Pr=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}=1$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity, and $\unicode[STIX]{x1D705}$ is the diffusivity of the density distribution, we find that the most unstable Taylor instability, maximized across wavenumber and $Ri_{B}$, has a (linear) growth rate which is a non-monotonic function of Reynolds number $Re=\unicode[STIX]{x0394}Uh/\unicode[STIX]{x1D708}$, with a global maximum at $Re=700$ over 50 % larger than the growth rate as $Re\rightarrow \infty$. In a fully nonlinear evolution of the flows with $Re=700$ and $Pr=1$, the two interfaces between the three density layers diffuse more rapidly than the underlying instabilities can grow from small amplitude. Therefore, we investigate numerically the nonlinear evolution of the flow at $Re=600$ and $Pr=300$, and at $Re=5000$ and $Pr=70$ in two-dimensional domains with streamwise extent equal to two wavelengths of the Taylor instability with the largest growth rate. At both sets of parameter values, the primary Taylor instability undergoes a period of identifiable exponential ‘linear’ growth. However, we demonstrate that, unlike the so-called ‘Kelvin–Helmholtz’ instability that it superficially resembles, the Taylor instability’s finite-amplitude state of an elliptical vortex in the middle layer appears not to saturate into a quasiequilibrium state, but is rapidly destroyed by the background shear. The decay process reveals $Re$-dependent secondary processes. For the $Re=600$ simulation, this decay allows the development to finite amplitude of the co-existing primary ‘viscous Holmboe wave instability’, which has a substantially smaller linear growth rate. For the $Re=5000$ simulation, the Taylor instability decay induces a non-trivial modification of the mean velocity and density distributions, which nonlinearly develops into more classical finite-amplitude Holmboe waves. In both cases, the saturated nonlinear Holmboe waves are robust and long-lived in two-dimensional flow.
Optimal perturbation growth in axisymmetric intrusions
- Bruce R. Sutherland, C. P. Caulfield
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- Journal of Fluid Mechanics / Volume 811 / 25 January 2017
- Published online by Cambridge University Press:
- 14 December 2016, R4
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The cylindrical lock-release laboratory experiments of Sutherland & Nault (J. Fluid Mech., vol. 586, 2007, pp. 109–118) showed that a radially advancing symmetric intrusive gravity current spreads not as an expanding annulus (as is the case for bottom-propagating gravity currents), but rather predominantly along azimuthally periodic radial ‘spokes’. Here, we investigate whether the spokes are associated with azimuthal perturbations that undergo ‘optimal’ growth. We use a nonlinear axisymmetric numerical simulation initialised with the experimental parameters to compute the time-evolving axisymmetric base state of the collapsing lock fluid. Using fields from this rapidly evolving base state together with the linearised perturbation equations and their adjoint, the ‘direct–adjoint looping’ method is employed to identify, as a function of the azimuthal wavenumber $m$, the vertical–radial structure of the set of initial perturbations that exhibit the largest total perturbation energy gain over a target time $T$. Of this set of perturbations, the one that extracts energy fastest, and so is expected to be observed to emerge first from the base flow, has azimuthal wavenumber comparable to the number of spokes observed in the experiment.