A new method for isolating turbulent states in transitional stratified plane Couette flow

We present a new adaptive control strategy to isolate and stabilize turbulent states in transitional, stably stratified plane Couette flow in which the gravitational acceleration (non-dimensionalized as the bulk Richardson number$Ri$) is adjusted in time to maintain the turbulent kinetic energy (TKE) of the flow. We demonstrate that applying this method at various stages of decaying stratified turbulence halts the decay process and allows a succession of intermediate turbulent states of decreasing energy to be isolated and stabilized. Once the energy of the initial flow becomes small enough, we identify a single minimal turbulent spot, and lower-energy states decay to laminar flow. Interestingly, the turbulent states which emerge from this process have very similar time-averaged$Ri$, but TKE levels different by an order of magnitude. The more energetic states consist of several turbulent spots, each qualitatively similar to the minimal turbulent spot. This suggests that the minimal turbulent spot may well be the lowest-energy turbulent state which forms a basic building block of stratified plane Couette flow. The fact that a minimal spot of turbulence can be stabilized, so that it neither decays nor grows, opens up exciting opportunities for further study of spatiotemporally intermittent stratified turbulence.The EPSRC grant EP/K034529/1 entitled ‘Mathematical Underpinnings of Stratified Turbulence’ is gratefully acknowledged for supporting the research presented here.This is the author accepted manuscript. The final version is available from Cambridge University Press via https://doi.org/10.1017/jfm.2016.62

Ethical challenges in researching and telling the stories of recently deceased people

This paper explores ethical challenges encountered when conducting research about, and telling, the stories of individuals who had died before the research began. Cases were explored where individuals who lived alone had died alone at home and where their bodies had been undiscovered for an extended period. The ethical review process had not had anything significant to say about the deceased ‘participants’. As social researchers we considered whether it was ethical to involve deceased people in research when they had no opportunity to decline, and we were concerned about how to report such research. The idea that the dead can be harmed did not help our decision-making processes, but the notion of the dead having limited human rights conferred upon them was useful and aided us in clarifying how to conduct our research and disseminate our findings.Peer reviewe

Testing the assumptions underlying ocean mixing methodologies using direct numerical simulations

AbstractDirect numerical simulations of stratified turbulence are used to test several fundamental assumptions involved in the Osborn, Osborn–Cox, and Thorpe methods commonly used to estimate the turbulent diffusivity from field measurements. The forced simulations in an idealized triply periodic computational domain exhibit characteristic features of stratified turbulence including intermittency and layer formation. When calculated using the volume-averaged dissipation rates from the simulations, the vertical diffusivities inferred from the Osborn and Osborn–Cox methods are within 40% of the value diagnosed using the volume-averaged buoyancy flux for all cases, while the Thorpe-scale method performs similarly well in the simulation with a relatively large buoyancy Reynolds number (Reb ≃ 240) but significantly overestimates the vertical diffusivity in simulations with Reb &lt; 60. The methods are also tested using a limited number of vertical profiles randomly selected from the computational volume. The Osborn, Osborn–Cox, and Thorpe-scale methods converge to their respective estimates based on volume-averaged statistics faster than the vertical diffusivity calculated directly from the buoyancy flux, which is contaminated with reversible contributions from internal waves. When applied to a small number of vertical profiles, several assumptions underlying the Osborn and Osborn–Cox methods are not well supported by the simulation data. However, the vertical diffusivity inferred from these methods compares reasonably well to the exact value from the simulations and outperforms the assumptions underlying these methods in terms of the relative error. Motivated by a recent theoretical development, it is speculated that the Osborn method might provide a reasonable approximation to the diffusivity associated with the irreversible buoyancy flux.</jats:p

Robust identification of dynamically distinct regions in stratified turbulence

We present a new robust method for identifying three dynamically distinct regions in a stratified turbulent flow, which we characterise as quiescent flow, intermittent layers and turbulent patches. The method uses the cumulative filtered distribution function of the local density gradient to identify each region. We apply it to data from direct numerical simulations of homogeneous stratified turbulence, with unity Prandtl number, resolved on up to $8192\times 8192\times 4096$ grid points. In addition to classifying regions consistently with contour plots of potential enstrophy, our method identifies quiescent regions as regions where \unicode[STIX]{x1D716}/\unicode[STIX]{x1D708}N^{2}\sim O(1), layers as regions where \unicode[STIX]{x1D716}/\unicode[STIX]{x1D708}N^{2}\sim O(10) and patches as regions where \unicode[STIX]{x1D716}/\unicode[STIX]{x1D708}N^{2}\sim O(100). Here, \unicode[STIX]{x1D716} is the dissipation rate of turbulence kinetic energy, \unicode[STIX]{x1D708} is the kinematic viscosity and $N$ is the (overall) buoyancy frequency. By far the highest local dissipation and mixing rates, and the majority of dissipation and mixing, occur in patch regions, even when patch regions occupy only 5 % of the flow volume. We conjecture that treating stratified turbulence as an instantaneous assemblage of these different regions in varying proportions may explain some of the apparently highly scattered flow dynamics and statistics previously reported in the literature.The research activities of G.D.P. and S.dB.K. were funded by the US Office of Naval Research via grant N00014-15-1-2248. Additional support to G.D.P. and S.dB.K. was provided from the UK Engineering and Physical Sciences Research Council grant EP/K034529/1 entitled ‘Mathematical Underpinnings of Stratified Turbulence’, which also funds the research activity of J.R.T. and C.P.C. H.S. gratefully acknowledges the award of a Crighton Fellowship at the Department of Applied Mathematics & Theoretical Physics, University of Cambridge. High-performance computing resources were provided through the US Department of Defense High Performance Computing Modernization Program by the Army Engineer Research and Development Center and the Army Research Laboratory under Frontier Project FP-CFD-FY14-007.This is the author accepted manuscript. The final version is available from Cambridge University Press via https://doi.org/10.1017/jfm.2016.61

Screening, intervention and outcome in autism and other developmental disorders: the role of randomized controlled trials

We draw attention to a number of important considerations in the arguments about screening and outcome of intervention in children with autism and other developmental disorders. Autism screening in itself never provides a final clinical diagnosis, but may well identify developmental deviations indicative of autism—or of other developmental disorders—that should lead to referral for further clinical assessment. Decisions regarding population or clinic screening cannot be allowed to be based on the fact that prospective longitudinal RCT designs over decades could never be performed in complex developmental disorders. We propose an alternative approach. Early screening for autism and other developmental disorders is likely to be of high societal importance and should be promoted and rigorously evaluated

Testing linear marginal stability in stratified shear layers

We use two-dimensional direct numerical simulations of Boussinesq stratified shear layers to investigate the influence of the minimum gradient Richardson number Rim 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 Rim→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 Rim 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

Nonlinear evolution of linear optimal perturbations of strongly stratified shear layers

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 $U_{0}^{*}$ and a uniform stratification with constant buoyancy frequency $N_{0}^{*}$. We vary the bulk Richardson number $Ri_b$=$N_{0}^{*}$$^2$$h^{*2}$/$U_{0}^{*}$$^2$ (corresponding to $Ri_{g,min}$) between 0.20 and 0.50 and the Reynolds numbers $Re$=$U_{0}^{*}$$h^{*}$/$v^{*}$ between 1000 and 8000, with the Prandtl number held fixed at $Pr$=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 $Re$, 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 $\epsilon_p$/($\epsilon_p$+$\epsilon_k$) where $\epsilon_p$ is the dissipation rate of density variance and $\epsilon_k$is the dissipation rate of kinetic energy, is found to be approximately 0.35 for the most strongly nonlinear cases.EPSR

Self-similar mixing in stratified plane Couette flow for varying Prandtl number

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 $\textit{h/L}$, where $\textit{h}$ is the half-channel gap depth and $\textit{L}$ is the Obukhov length scale. As $\textit{h/L}$ approaches very large values, $Ri_g$ asymptotes to a maximum characteristic value of approximately 0.2. The buoyancy Reynolds number $Re_b$ $\equiv$ $\varepsilon$/($\nu$$N^2$), where $\varepsilon$ is the dissipation, $\nu$ is the kinematic viscosity and $N$ is the buoyancy frequency defined in terms of the local mean density gradient, scales linearly with the length scale ratio $L$+ $\equiv$ $L$/$\delta$$_\nu$, where $\delta$$_\nu$ is the near-wall viscous scale. The flux Richardson number $Ri_f$ $\equiv$ -$B$/$P$, where $B$ is the buoyancy flux and $P$ is the shear production, is found to be proportional to $Ri_g$. This then leads to a turbulent Prandtl number $Pr_t$ $\equiv$ $\nu_t$/$\kappa_t$ of order unity, where $\nu_t$ and $\kappa_t$ are the turbulent viscosity and diffusivity respectively, which is consistent with Reynolds analogy. The turbulent Froude number $Fr_h$ $\equiv$ $\varepsilon$/($NU^\prime$$^2$), where $U^\prime$ is a turbulent horizontal velocity scale, is found to vary like $Ri_g$$^{-1/2}$. All these scalings are consistent with our numerical data and appear to be independent of $Pr$. The classical Osborn model based on turbulent kinetic energy balance in statistically stationary stratified sheared turbulence (Osborn, J. Phys. Oceanogr., vol. 10, 1980, pp. 83-89), together with M-O scalings, results in a parameterization of $\kappa_t$/$\nu$ ~ $\nu_t$/$\nu$ ~ $Re_b$$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.The EPSRC Programme grant EP/K034529/1 entitled ‘Mathematical Underpinnings of Stratified Turbulence’ is gratefully acknowledged for supporting the research presented here

Diapycnal mixing in layered stratified plane Couette flow quantified in a tracer-based coordinate

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 ±$U_w$. Particular attention is paid to the effects of varying Prandtl number $Pr$ ≡ ν/κ, where ν and κ are the molecular kinematic viscosity and diffusivity respectively, over two orders of magnitude from 0.7, 7 and 70. Varying $Pr$ enables the system to access a considerable range of characteristic turbulent Péclet numbers $Pe_∗$ ≡ $U_∗$$L_∗$/κ, where $U_∗$ and $L_∗$ are characteristic velocity and length scales, respectively, of the motion which acts to ‘scour’ the density interface. The dynamics of the interface varies with the stability of the interface which is characterised by a bulk Richardson number $Ri$ ≡ $b_0$$h$/$U^2_w$, 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 $Ri$, whereas internal waves propagating on the interface dominate at large $Ri$. For a highly stable (i.e. large $Ri$) interface at sufficiently large $Pe_∗$, 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