We consider the nonlinear optimisation of the mixing of a passive scalar, initially arranged in two layers, in a two-dimensional plane Poiseuille flow at finite Reynolds and Péclet numbers, below the ...linear instability threshold. We use a nonlinear-adjoint-looping approach to identify optimal perturbations leading to maximum time-averaged energy as well as maximum mixing in a freely evolving flow, measured through the minimisation of either the passive scalar variance or the so-called mix-norm, as defined by Mathew, Mezić & Petzold (Physica D, vol. 211, 2005, pp. 23–46). We show that energy optimisation appears to lead to very weak mixing of the scalar field whereas the optimal mixing initial perturbations, despite being less energetic, are able to homogenise the scalar field very effectively. For sufficiently long time horizons, minimising the mix-norm identifies optimal initial perturbations which are very similar to those which minimise scalar variance, demonstrating that minimisation of the mix-norm is an excellent proxy for effective mixing in this finite-Péclet-number bounded flow. By analysing the time evolution from initial perturbations of several optimal mixing solutions, we demonstrate that our optimisation method can identify the dominant underlying mixing mechanism, which appears to be classical Taylor dispersion, i.e. shear-augmented diffusion. The optimal mixing proceeds in three stages. First, the optimal mixing perturbation, energised through transient amplitude growth, transports the scalar field across the channel width. In a second stage, the mean flow shear acts to disperse the scalar distribution leading to enhanced diffusion. In a final third stage, linear relaxation diffusion is observed. We also demonstrate the usefulness of the developed variational framework in a more realistic control case: mixing optimisation by prescribed streamwise velocity boundary conditions.
Understanding how turbulence leads to the enhanced irreversible transport of heat and other scalars such as salt and pollutants in density-stratified fluids is a fundamental and central problem in ...geophysical and environmental fluid dynamics. This review discusses recent research activity directed at improving community understanding, modeling, and parameterization of the subtle interplay between energy conversion pathways, instabilities, turbulence, external forcing, and irreversible mixing in density-stratified fluids. The conceptual significance of various length scales is highlighted, and in particular, the importance is stressed of overturning or scouring in the formation and maintenance of layered stratifications, i.e., robust density distributions with relatively deep and well-mixed regions separated by relatively thin interfaces of substantially enhanced density gradient.
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 (
$Re=U_{m}h/\unicodeSTIX{x1D708}=500$
and
$Re=3000$
, where
$U_{m}$
is the maximum flow speed of the unperturbed flow,
$h$
is the channel half-depth and
$\unicodeSTIX{x1D708}$
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.
We compare the properties of the turbulence induced by the breakdown of Kelvin–Helmholtz instability (KHI) at high Reynolds number in two classes of stratified shear flows where the background ...density profile is given by either a linear function or a hyperbolic tangent function, at different values of the minimum initial gradient Richardson number ${{Ri}}_0$. Considering global and local measures of mixing defined in terms of either the irreversible mixing rate $\mathscr {M}$ associated with the time evolution of the background potential energy, or an appropriately defined density variance dissipation rate $\chi$, we find that the proliferation of secondary instabilities strongly affects the efficiency of mixing early in the flow evolution, and also that these secondary instabilities are highly sensitive to flow perturbations that are added at the point of maximal (two-dimensional) billow amplitude. Nevertheless, mixing efficiency does not appear to depend strongly on the far field density structure, a feature supported by the evolution of local horizontally averaged values of the buoyancy Reynolds number ${Re}_b$ and gradient Richardson number ${Ri}_g$. We investigate the applicability of various proposed scaling laws for flux coefficients $\varGamma$ in terms of characteristic length scales, in particular discussing the relevance of the overturning ‘Thorpe scale’ in stratified turbulent flows. Finally, we compare a variety of empirical model parameterizations used to compute diapycnal diffusivity in an oceanographic context, arguing that for transient flows such as KHI-induced turbulence, simple models that relate the ‘age’ of a turbulent event to its mixing efficiency can produce reasonably robust mixing estimates.
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
$Re_{b}\equiv \unicodeSTIX{x1D700}/(\unicodeSTIX{x1D708}N^{2})$
, where
$\unicodeSTIX{x1D700}$
is the dissipation,
$\unicodeSTIX{x1D708}$
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
$L^{+}\equiv L/\unicodeSTIX{x1D6FF}_{\unicodeSTIX{x1D708}}$
, where
$\unicodeSTIX{x1D6FF}_{\unicodeSTIX{x1D708}}$
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
$Pr_{t}\equiv \unicodeSTIX{x1D708}_{t}/\unicodeSTIX{x1D705}_{t}$
of order unity, where
$\unicodeSTIX{x1D708}_{t}$
and
$\unicodeSTIX{x1D705}_{t}$
are the turbulent viscosity and diffusivity respectively, which is consistent with Reynolds analogy. The turbulent Froude number
$Fr_{h}\equiv \unicodeSTIX{x1D700}/(NU^{\prime 2})$
, 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
$\unicodeSTIX{x1D705}_{t}/\unicodeSTIX{x1D708}\sim \unicodeSTIX{x1D708}_{t}/\unicodeSTIX{x1D708}\sim Re_{b}Ri_{g}/(1-Ri_{g})$
. 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.
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
...$\unicodeSTIX{x1D708}$
and density diffusivity
$\unicodeSTIX{x1D705}$
. The initial diffusive error function density distribution varies continuously so that
$\unicodeSTIX{x1D70C}\in \bar{\unicodeSTIX{x1D70C}}-\unicodeSTIX{x1D70C}_{0}/2,\bar{\unicodeSTIX{x1D70C}}+\unicodeSTIX{x1D70C}_{0}/2$
. A constant pressure gradient is imposed in a plane two-dimensional channel of depth
$2h$
. We consider flows with a finite Péclet number
$Pe=U_{m}h/\unicodeSTIX{x1D705}=500$
and Prandtl number
$Pr=\unicodeSTIX{x1D708}/\unicodeSTIX{x1D705}=1$
, and a range of bulk Richardson numbers
$Ri_{b}=g\unicodeSTIX{x1D70C}_{0}h/(\bar{\unicodeSTIX{x1D70C}}U^{2})\in 0,1$
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.
We present a new, simple and physically motivated parameterization, based on the ratio of Thorpe and Ozmidov scales, for the irreversible turbulent flux coefficient $\varGamma _{\mathcal {M}}= ...{\mathcal {M}}/\epsilon$, i.e. the ratio of the irreversible rate ${\mathcal {M}}$ at which the background potential energy increases in a stratified flow due to macroscopic motions to the dissipation rate of turbulent kinetic energy $\epsilon$. Our parameterization covers all three key phases (crucially, in time) of a shear-induced stratified turbulence life cycle: the initial, ‘hot’ growing phase, the intermediate energetically forced phase and the final ‘cold’ fossilization decaying phase. Covering all three phases allows us to highlight the importance of the intermediate one, to which we refer as the ‘Goldilocks’ phase due to its apparently optimal (and so neither too hot nor too cold, but just right) balance, in which energy transfer from background shear to the turbulent mixing is most efficient. The value of $\varGamma _{\mathcal {M}}$ is close to 1/3 during this phase, which we demonstrate appears to be related to an adjustment towards a critical or marginal Richardson number for sustained turbulence ${\sim }0.2\text {--}0.25$. Importantly, although buoyancy effects are still significant at leading order for the turbulent dynamics during this intermediate phase, the marginal balance in the flow ensures that the turbulent mixing of the (density) scalar is nevertheless effectively ‘locked’ to the turbulent mixing of momentum. We present supporting evidence for our parameterization through comparison with six oceanographic datasets that span various turbulence generation regimes and a wide range of geographical location and depth. Using these observations, we highlight the significance of parameterizing an inherently variable flux coefficient for capturing the turbulent flux associated with rare energetic, yet fundamentally shear-driven (and so not strongly stratified) overturns that make a disproportionate contribution to the total mixing. We also highlight the importance of representation of young turbulent patches in the parameterization for connecting the small scale physics to larger scale applications of mixing such as ocean circulation and tracer budgets. Shear-induced turbulence is therefore central to irreversible mixing in the world's oceans, apparently even close to the seafloor, and it is critically important to appreciate the inherent time dependence and evolution of mixing events: history matters to mixing.
A sequence of two- and three-dimensional simulations are conducted for the double-diffusive convection (DDC) flows in the diffusive regime subjected to an imposed shear. For a wide range of control ...parameters, and for sufficiently strong perturbation of the conductive initial state, staircase-like structures spontaneously develop, with relatively well-mixed layers separated by sharp interfaces of enhanced scalar gradient. Such staircases appear to be robust even in the presence of strong shear over very long times, with early-time coarsening of the observed layers. For the same set of control parameters, different asymptotic layered states, with markedly different vertical scalar fluxes, can arise for different initial perturbation structures. The imposed shear significantly spatio-temporally modifies the vertical transport of the various scalars. The flux ratio $\gamma ^*$ (i.e. the ratio between the density fluxes due to the total salt flux and the total heat flux) is found, at steady state, to be essentially equal to the square root of the ratio of the salt diffusivity to the thermal diffusivity, consistent with the physical model proposed by Linden & Shirtcliffe (J. Fluid Mech., vol. 87, 1978, pp. 417–432) and the variational arguments presented by Stern (J. Fluid Mech., vol. 114, 1982, pp. 105–121) for unsheared double-diffusive convection.
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, $\unicodeSTIX{x1D702}(t):=\unicodeSTIX{x1D712}/(\unicodeSTIX{x1D712}+\unicodeSTIX{x1D700})$ (where $\unicodeSTIX{x1D700}$ and $\unicodeSTIX{x1D712}$ 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.
We consider numerically the transition to turbulence and associated mixing in stratified shear flows with initial velocity distribution
...$\overline{U}(z,0)\,\boldsymbol{e}_{x}=U_{0}\,\boldsymbol{e}_{x}\tanh (z/d)$
and initial density distribution
$\overline{\unicodeSTIX{x1D70C}}(z,0)=\unicodeSTIX{x1D70C}_{0}1-\tanh (z/\unicodeSTIX{x1D6FF})$
away from a hydrostatic reference state
$\unicodeSTIX{x1D70C}_{r}\gg \unicodeSTIX{x1D70C}_{0}$
. When the ratio
$R=d/\unicodeSTIX{x1D6FF}$
of the characteristic length scales over which the velocity and density vary is equal to one, this flow is primarily susceptible to the classic well-known Kelvin–Helmholtz instability (KHI). This instability, which typically manifests at finite amplitude as an array of elliptical vortices, strongly ‘overturns’ the density interface of strong initial gradient, which nevertheless is the location of minimum initial gradient Richardson number
$Ri_{g}(0)=Ri_{b}=g\unicodeSTIX{x1D70C}_{0}d/\unicodeSTIX{x1D70C}_{r}U_{0}^{2}$
, where
$Ri_{g}(z)=-(g/\unicodeSTIX{x1D70C}_{r}\,\text{d}\overline{\unicodeSTIX{x1D70C}}/\text{d}z)/(\text{d}\overline{U}/\text{d}z)^{2}$
and
$Ri_{b}$
is a bulk Richardson number. As is well known, at sufficiently high Reynolds numbers (
$Re$
), the primary KHI induces a vigorous but inherently transient burst of turbulence and associated irreversible mixing localised in the vicinity of the density interface, leading to a relatively well-mixed region bounded by stronger density gradients above and below. We explore the qualitatively different behaviour that arises when
$R\gg 1$
, and so the density interface is sharp, with
$Ri_{g}(z)$
now being maximum at the density interface
$Ri_{g}(0)=RRi_{b}$
. This flow is primarily susceptible to Holmboe wave instability (HWI) (Holmboe, Geophys. Publ., vol. 24, 1962, pp. 67–113), which manifests at finite amplitude in this symmetric flow as counter-propagating trains of elliptical vortices above and below the density interface, thus perturbing the interface so as to exhibit characteristically cusped interfacial waves which thereby ‘scour’ the density interface. Unlike previous lower-
$Re$
experimental and numerical studies, when
$Re$
is sufficiently high the primary HWI becomes increasingly more three-dimensional due to the emergence of shear-aligned secondary convective instabilities. As
$Re$
increases, (i) the growth rate of secondary instabilities appears to saturate and (ii) the perturbation kinetic energy exhibits a
$k^{-5/3}$
spectrum for sufficiently large length scales that are influenced by anisotropic buoyancy effects. Therefore, at sufficiently high
$Re$
, vigorous turbulence is triggered that also significantly ‘scours’ the primary density interface, leading to substantial irreversible mixing and vertical transport of mass above and below the (robust) primary density interface. Furthermore, HWI produces a markedly more long-lived turbulence event compared to KHI at a similarly high
$Re$
. Despite their vastly different mechanics (i.e. ‘overturning’ versus ‘scouring’) and localisation, the mixing induced by KHI and HWI is comparable in both absolute terms and relative efficiency. Our results establish that, provided the flow Reynolds number is sufficiently high, shear layers with sharp density interfaces and associated locally high values of the gradient Richardson number may still be sites of substantial and efficient irreversible mixing.
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