Vertical Froude Number Research Articles

Mixing efficiency in large-eddy simulations of stratified turbulence

The irreversible mixing efficiency is studied using large-eddy simulations (LES) of stratified turbulence, where three different subgrid-scale (SGS) parameterizations are employed. For comparison, direct numerical simulations (DNS) and hyperviscosity simulations are also performed. In the regime of stratified turbulence where$Fr_{v}\sim 1$, the irreversible mixing efficiency$\unicode[STIX]{x1D6FE}_{i}$in LES scales like$1/(1+2Pr_{t})$, where$Fr_{v}$and$Pr_{t}$are the vertical Froude number and turbulent Prandtl number, respectively. Assuming a unit scaling coefficient and$Pr_{t}=1$,$\unicode[STIX]{x1D6FE}_{i}$goes to a constant value$1/3$, in agreement with DNS results. In addition, our results show that the irreversible mixing efficiency in LES, while consistent with this prediction, depends on SGS parameterizations and the grid spacing$\unicode[STIX]{x1D6E5}$. Overall, the LES approach can reproduce mixing efficiency results similar to those from the DNS approach if$\unicode[STIX]{x1D6E5}\lesssim L_{o}$, where$L_{o}$is the Ozmidov scale. In this situation, the computational costs of numerical simulations are significantly reduced because LES runs require much smaller computational resources in comparison with expensive DNS runs.

An unambiguous definition of the Froude number for lee waves in the deep ocean

There is a long-standing debate in the literature of stratified flows over topography concerning the correct dimensionless number to refer to as a Froude number. Common definitions using external quantities of the flow include $U/(ND)$, $U/(Nh_{0})$, and $Uk/N$, where $U$ and $N$ are, respectively, scales for the background velocity and buoyancy frequency, $D$ is the depth, and $h_{0}$ and $k^{-1}$ are, respectively, height and width scales of the topography. It is also possible to define an internal Froude number $Fr_{\unicode[STIX]{x1D6FF}}=u_{0}/\sqrt{g^{\prime }\unicode[STIX]{x1D6FF}}$, where $u_{0}$, $g^{\prime }$, and $\unicode[STIX]{x1D6FF}$ are, respectively, the characteristic velocity, reduced gravity, and vertical length scale of the perturbation above the topography. For the case of hydrostatic lee waves in a deep ocean, both $U/(ND)$ and $Uk/N$ are insignificantly small, rendering the dimensionless number $Nh_{0}/U$ the only relevant dynamical parameter. However, although it appears to be an inverse Froude number, such an interpretation is incorrect. By non-dimensionalizing the stratified Euler equations describing the flow of an infinitely deep fluid over topography, we show that $Nh_{0}/U$ is in fact the square of the internal Froude number because it can identically be written in terms of the inner variables, $Fr_{\unicode[STIX]{x1D6FF}}^{2}=Nh_{0}/U=u_{0}^{2}/(g^{\prime }\unicode[STIX]{x1D6FF})$. Our scaling also identifies $Nh_{0}/U$ as the ratio of the vertical velocity scale within the lee wave to the group velocity of the lee wave, which we term the vertical Froude number, $Fr_{vert}=Nh_{0}/U=w_{0}/c_{g}$. To encapsulate such behaviour, we suggest referring to $Nh_{0}/U$ as the lee-wave Froude number, $Fr_{lee}$.

Analogies and differences between the stability of an isolated pancake vortex and a columnar vortex in stratified fluid

In order to understand the dynamics of pancake shaped vortices in stably stratified fluids, we perform a linear stability analysis of an axisymmetric vortex with Gaussian angular velocity in both the radial and axial directions with an aspect ratio of ${\it\alpha}$. The results are compared to those for a columnar vortex (${\it\alpha}=\infty$) in order to identify the instabilities. Centrifugal instability occurs when $\mathscr{R}>c(m)$ where $\mathscr{R}=ReF_{h}^{2}$ is the buoyancy Reynolds number, $F_{h}$ the Froude number, $Re$ the Reynolds number and $c(m)$ a constant which differs for the three unstable azimuthal wavenumbers $m=0,1,2$. The maximum growth rate depends mostly on $\mathscr{R}$ and is almost independent of the aspect ratio ${\it\alpha}$. For sufficiently large buoyancy Reynolds number, the axisymmetric mode is the most unstable centrifugal mode whereas for moderate $\mathscr{R}$, the mode $m=1$ is the most unstable. Shear instability for $m=2$ develops only when $F_{h}\leqslant 0.5{\it\alpha}$. By considering the characteristics of shear instability for a columnar vortex with the same parameters, this condition is shown to be such that the vortex is taller than the minimum wavelength of shear instability in the columnar case. For larger Froude number $F_{h}\geqslant 1.5{\it\alpha}$, the isopycnals overturn and gravitational instability can operate. Just below this threshold, the azimuthal wavenumbers $m=1,2,3$ are unstable to baroclinic instability. A simple model shows that baroclinic instability develops only above a critical vertical Froude number $F_{h}/{\it\alpha}$ because of confinement effects.

Dynamics of stratified turbulence decaying from a high buoyancy Reynolds number

We present direct numerical simulations (DNS) of unforced stratified turbulence with the objective of testing the strongly stratified turbulence theory. According to this theory the characteristic vertical scale of the turbulence is given by $\ell _{v}\sim u_{h}/N$, where $u_{h}$ is the horizontal velocity scale and $N$ the Brunt–Väisälä frequency. Combined with the hypothesis of the energy dissipation rate scaling as ${\it\epsilon}\sim u_{h}^{3}/\ell _{h}$, this theory predicts inertial range scalings for the horizontal spectrum of horizontal kinetic energy and of potential energy, according to $E(k_{h})\propto k_{h}^{-5/3}$. We begin by presenting a scaling analysis of the horizontal vorticity equation from which we recover the result regarding the vertical scale, $\ell _{v}\sim u_{h}/N$, highlighting in the process the important dynamical role of large-scale vertical shear of horizontal velocity. We then present the results from decaying DNS, which show a good agreement with aspects of the theory. In particular, the vertical Froude number is found to reach a constant plateau in time, of the form $Fr_{v}=u_{h}/(N\ell _{v})=C$ with $C=O(1)$ in all the runs. The derivation of the dissipation scaling ${\it\epsilon}\sim u_{h}^{3}/\ell _{h}$ at low Reynolds number in the context of decaying stratified turbulence highlights that the same scaling holds at high $\mathscr{R}=ReFr_{h}^{2}\gg 1$ as well as at low $\mathscr{R}\ll 1$, which is known (see Brethouwer et al., J. Fluid Mech., vol. 585, 2007, pp. 343–368) but not sufficiently emphasized in recent literature. We find evidence in our DNS of the dissipation scaling holding at $\mathscr{R}=O(1)$, which we interpret as being in the viscous regime. We also find ${\it\epsilon}_{k}\sim u_{h}^{3}/\ell _{h}$ and ${\it\epsilon}_{p}\sim u_{h}^{3}/\ell _{h}$ (with ${\it\epsilon}={\it\epsilon}_{k}+{\it\epsilon}_{p}$), in our high-resolution run at earlier times corresponding to $\mathscr{R}=O(10)$, which is in the transition between the strongly stratified and the viscous regimes. The horizontal spectrum of horizontal kinetic energy collapses in time using the scaling $E_{h}(k_{h})=C_{1}{\it\epsilon}_{k}^{2/3}k_{h}^{-5/3}$ and the horizontal potential energy spectrum is well described by $E_{p}(k_{h})=C_{2}{\it\epsilon}_{p}{\it\epsilon}_{k}^{-1/3}k_{h}^{-5/3}$. The presence of an inertial range in the horizontal direction is confirmed by the constancy of the energy flux spectrum over narrow ranges of $k_{h}$. However, the vertical energy spectrum is found to differ significantly from the expected $E_{h}(k_{v})\sim N^{2}k_{v}^{-3}$ scaling, showing that $Fr_{v}$ is not of order unity on a scale-by-scale basis, thus providing motivation for further investigation of the vertical structure of stratified turbulence.

Experimental study of stratified turbulence forced with columnar dipoles

We present a novel experimental setup aimed at producing a forced strongly stratified turbulent flow. The flow is forced by an arena of 12 vortex pair generators in a large tank. The continuous interactions of the randomly produced vortex pairs give rise to a statistically stationary disordered flow in contrast to previous experiments where the stratified turbulence is decaying. The buoyancy frequency N is set to its highest value N = 1.7 rad/s using salt as stratifying agent so that the horizontal Froude number Fh = Ω/N is low, while the buoyancy Reynolds number \documentclass[12pt]{minimal}\begin{document}$\mathcal {R} = Re {F_h}^2$\end{document}R=ReFh2, where Re = Ωa2/ν is the classical Reynolds number, is as high as possible given the experimental constraints (Ω is the maximum angular velocity of the vortices, a their radius and ν the viscosity). PIV measurements show that the flow is not homogeneous in the horizontal plane and is organised into horizontal layers along the vertical. When \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}$\end{document}R is increased, we observe a progressive evolution from the viscosity dominated regime with smooth layers to a regime with small scales superimposed on the layers and for which the vertical Froude number is of order one. The latter regime resembles the strongly stratified turbulent regime with a downscale cascade that has been predicted for large \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}$\end{document}R. However, horizontal second order structure functions do not exhibit a clear inertial range for the largest \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}$\end{document}R achieved \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}=310$\end{document}R=310. In addition, the corresponding turbulent buoyancy Reynolds number \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}_t=P/(\nu N^2)$\end{document}Rt=P/(νN2) based on an estimation of the injection rate of energy P is only of order unity \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}_t \simeq 0.4$\end{document}Rt≃0.4 indicating that only the edge of the strongly stratified turbulent regime has been reached. However, these results suggest that sufficiently large turbulent buoyancy Reynolds numbers, \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}_t \simeq 10$\end{document}Rt≃10, could be achieved experimentally by scaling up five times this novel set-up.

Sensitivity of stratified turbulence to the buoyancy Reynolds number

AbstractIn this article we present direct numerical simulations of stratified flow at resolutions of up to $204{8}^{2} \times 513$, to explore scalings for the dynamics of stably stratified turbulence. Recent work suggests that for strong enough stratification, the vertical integral scale of the turbulence adjusts to yield a vertical Froude number, ${F}_{v} $, of order unity at high enough Reynolds number, whilst the horizontal Froude number, ${F}_{h} $, decreases as stratification is increased. Our numerical simulations are consistent with predictions by Lindborg (J. Fluid Mech., vol. 550, 2006, pp, 207–242), and with numerical simulations at lower resolution, in that the horizontal kinetic energy spectrum follows a Kolmogorov spectrum (after replacing the wavenumber with the horizontal wavenumber) and that the horizontal potential energy spectrum similarly follows the Corrsin–Obukhov spectrum for a passive scalar. Most importantly, we build upon these previous results by thoroughly exploring the dependence of the horizontal spectrum of horizontal kinetic energy on both the stratification and the relative size of the vertical dissipation terms, as quantified by the buoyancy Reynolds number. Our most important result is that variations in the power-law exponent scale entirely with the buoyancy Reynolds number and not with the stratification itself, lending considerable support to the Lindborg (2006) hypothesis that horizontal spectra are independent of stratification at large Reynolds numbers. We further demonstrate that even at the large numerical resolution of this study, the spectrum and hence the dynamics are affected by the buoyancy Reynolds number unless it is larger than $O(10)$, indicating that extreme care must be taken when assessing claims made from previous numerical simulations of stratified flow at low or moderate resolution and extrapolating the results to geophysical or astrophysical Reynolds numbers.

An explanation for salinity- and SPM-induced vertical countergradient buoyancy fluxes

Measurements of turbulent fluctuations of velocity, salinity, and suspended particulate matter (SPM) are presented. The data show persistent countergradient buoyancy fluxes. These countergradient fluxes are controlled by the ratio of vertical turbulent kinetic energy (VKE) and available potential energy (APE) terms in the buoyancy flux equation. The onset of countergradient fluxes is found to approximately coincide with larger APE than VKE. It is shown here that the ratio of VKE to APE can be written as the square of a vertical Froude number. This number signifies the onset of the dynamical significance of buoyancy in the transport of mass. That is when motions driven by buoyancy begin to actively determine the vertical turbulent transport of mass. Spectral and quadrant analyses show that the occurrence of countergradient fluxes coincides with a change in the relative importance of turbulent energetic structures and buoyancy-driven motions in the transport of mass. Furthermore, these analyses show that with increasing salinity-induced Richardson number (Ri), countergradient contributions expand to the larger scales of motions and the relative importance of outward and inward interactions increases. At the smaller scales, at moderate Ri, the countergradient buoyancy fluxes are physically associated with an asymmetry in transport of fluid parcels by energetic turbulent motions. At the large scales, at large Ri, the countergradient buoyancy fluxes are physically associated with convective motions induced by buoyancy of incompletely dispersed fluid parcels which have been transported by energetic motions in the past. Moreover, these convective motions induce restratification and enhanced settling of SPM. The latter is generally the result of salinity-induced convective motions, but SPM-induced buoyancy is also found to play a role.

Large eddy simulation of stably stratified open channel flow

Large eddy simulation has been used to study flow in an open channel with stable stratification imposed at the free surface by a constant heat flux and an adiabatic bottom wall. This leads to a stable pycnocline overlying a well-mixed turbulent region near the bottom wall. The results are contrasted with studies in which the bottom heat flux is nonzero, a difference analogous to that between oceanic and atmospheric boundary layers. Increasing the friction Richardson number, a measure of the relative importance of the imposed surface stratification with respect to wall-generated turbulence, leads to a stronger, thicker pycnocline which eventually limits the impact of wall-generated turbulence on the free surface. Increasing stratification also leads to an increase in the pressure-driven mean streamwise velocity and a concomitant decrease in the skin friction coefficient, which is, however, smaller than in the previous channel flow studies where the bottom buoyancy flux was nonzero. It is found that the turbulence in any given region of the flow can be classified into three regimes (unstratified, buoyancy-affected, and buoyancy-dominated) based on the magnitude of the Ozmidov length scale relative to a vertical length characterizing the large scales of turbulence and to the Kolmogorov scale. Since stratification does not strongly influence the near-wall turbulent production in the present configuration, the behavior of the buoyancy flux, turbulent Prandtl number, and mixing efficiency is qualitatively different from that seen in stratified shear layers and in channel flow with fixed temperature walls, and, furthermore, collapse of quantities as a function of gradient Richardson number is not observed. The vertical Froude number is a better measure of stratified turbulence in the upper portion of the channel where buoyancy, by providing a potential energy barrier, primarily affects the transport of turbulent patches generated at the bottom wall. The characteristics of free-surface turbulence including the kinetic energy budget and pressure-strain correlations are examined and found to depend strongly on the surface stratification.

Effect of the Schmidt number on the diffusion of axisymmetric pancake vortices in a stratified fluid

An asymptotic analysis of the equations for quasi-two-dimensional flow in stratified fluids is conducted, leading to a model for the diffusion of pancake-like vortices in cyclostrophic balance. This analysis permits one to derive formally the model for the diffusion of an axisymmetric monopole proposed by Beckers et al. [J. Fluid Mech. 433, 1 (2001)], and to extend their results. The appropriate parameter for the perturbation analysis is identified as the square of the vertical Froude number Fv=U/LvN, where U is the horizontal velocity scale, N is the Brunt–Väisälä frequency, and Lv the vertical length scale. The physical mechanisms involved in the vortex decay are examined under the light of the asymptotic analysis results. In particular we discuss the effects of the Schmidt number, Sc, which measures the balance between the diffusion of momentum and the diffusion of the stratifying agent. Remarkably, the vertical transport due to the slow cyclostrophic adjustment is shown to slowdown the velocity decay when Sc is larger than unity whereas it accelerates it when Sc is smaller than unity.

An investigation of stably stratified turbulent channel flow using large-eddy simulation

Boundary-forced stratified turbulence is studied in the prototypical case of turbulent channel flow subject to stable stratification. The large-eddy simulation approach is used with a mixed subgrid model that involves a dynamic eddy viscosity component and a scale-similarity component. After an initial transient, the flow reaches a new balanced state corresponding to active wall-bounded turbulence with reduced vertical transport which, for the cases in our study with moderate-to-large levels of stratification, coexists with internal wave activity in the core of the channel. A systematic reduction of turbulence levels, density fluctuations and associated vertical transport with increasing stratification is observed. Countergradient buoyancy flux is observed in the outer region for sufficiently high stratification.Mixing of the density field in stratified channel flow results from turbulent events generated near the boundaries that couple with the outer, more stable flow. The vertical density structure is thus of interest for analogous boundary-forced mixing situations in geophysical flows. It is found that, with increasing stratification, the mean density profile becomes sharper in the central region between the two turbulent layers at the upper and lower walls, similar to observations in field measurements as well as laboratory experiments with analogous density-mixing situations.Channel flow is strongly inhomogeneous with alternative choices for the Richardson number. In spite of these complications, the gradient Richardson number, Rig, appears to be the important local determinant of buoyancy effects. All simulated cases show that correlation coefficients associated with vertical transport collapse from their nominal unstratified values over a narrow range, 0.15 < Rig < 0.25. The vertical turbulent Froude number, Frw, has an O(1) value across most of the channel. It is remarkable that stratified channel flow, with such a large variation of overall density difference (factor of 26) between cases, shows a relatively universal behaviour of correlation coefficients and vertical Froude number when plotted as a function of Rig. The visualizations show wavy motion in the core region where the gradient Richardson number, Rig, is large and low-speed streaks in the near-wall region, typical of unstratified channel flow, where Rig is small. It appears from the visualizations that, with increasing stratification, the region with wavy motion progressively encroaches into the zone with active turbulence; the location of Rig ≃ 0.2 roughly corresponds to the boundary between the two zones.

Theoretical analysis of the zigzag instability of a vertical columnar vortex pair in a strongly stratified fluid

A general theoretical account is proposed for the zigzag instability of a vertical columnar vortex pair recently discovered in a strongly stratified experiment.The linear inviscid stability of the Lamb–Chaplygin vortex pair is analysed by a multiple-scale expansion analysis for small horizontal Froude number (Fh = U/LhN, where U is the magnitude of the horizontal velocity, Lh the horizontal lengthscale and N the Brunt–Väisälä frequency) and small vertical Froude number (Fv = U/LvN, where Lv is the vertical lengthscale) using the scaling of the equations of motion introduced by Riley, Metcalfe & Weissman (1981). In the limit Fv = 0, these equations reduce to two-dimensional Euler equations for the horizontal velocity with undetermined vertical dependence. Thus, at leading order, neutral modes of the flow are associated, among others, to translational and rotational invariances in each horizontal plane. To each broken invariance is related a phase variable that may vary freely along the vertical. Conservation of mass and potential vorticity impose at higher order the evolution equations governing the phase variables that we derive for Fh [Lt ] 1 and Fv [Lt ] 1 in the spirit of phase dynamics techniques established for periodic patterns. In agreement with the experimental observations, this asymptotic analysis shows the existence of an instability consisting of a vertically modulated rotation and a translation of the columnar vortex pair perpendicular to the travelling direction. The dispersion relation as well as the spatial eigenmode of the zigzag instability are determined. The analysis predicts that the most amplified vertical wavelength should scale as U/N and the maximum growth rate as U/Lh.Our main finding is thus that the typical thickness of the ensuing layers will be such that Fv = O(1) and not Fv [Lt ] 1 as assumed by Riley et al. (1981) and Lilly (1983). This implies that such strongly stratified flows are not described by two- dimensional horizontal equations. These results may help to understand the layering commonly observed in stratified turbulence and the fundamental differences with strictly two-dimensional turbulence.

The mechanics of running: How does stiffness couple with speed?

A mathematical model for terrestrial running is presented, based on a leg with the properties of a simple spring. Experimental force-platform evidence is reviewed justifying the formulation of the model. The governing differential equations are given in dimensionless form to make the results representative of animals of all body sizes. The dimensionless input parameters are: U, a horizontal Froude number based on forward speed and leg length; V, a vertical Froude number based on vertical landing velocity and leg length, and K LEG, a dimensionless stiffness for the leg-spring. Results show that at high forward speed, K LEG is a nearly linear function of both U and V, while the effective vertical stiffness is a quadratic function of U. For each U, V pair, the simulation shows that the vertical force at mid-step may be minimized by the choice of a particular step length. A particularly useful specification of the theory occurs when both K LEG and V are assumed fixed. When K LEG = 15 and V = 0.18, the model makes predictions of relative stride length S and initial leg angle θ 0 that are in good agreement with experimental data obtained from the literature.