Buoyancy Reynolds Number Research Articles

Direct numerical simulations of non-helical and helical stratified turbulent flows

ABSTRACTDirect numerical simulations of weakly and strongly stratified decaying homogenous turbulence are conducted for two transition-to-turbulence cases with different initial conditions: the non-helical Taylor–Green vortex (TGV) and the helical Arnold–Beltrami–Childress (ABC) configurations. These simulations are carried out using a pseudo-spectral formulation with a 5123 grid resolution at varying Froude numbers, . First, an effort is made to classify the various regimes represented by and 0.08, or alternatively, buoyancy Reynolds number () values from to for non-helical and helical stably stratified turbulence. Second, various quantities including enstrophy, dissipation, vertical shearing, coherent vortical structures, density fields, energy spectra and fluxes in the different directions, and helicity are analysed to understand the effects of stratification on the development of turbulence and the turbulent structure in these different regimes. Consistent with previous studies by Riley and deBruynKops (Phys. Fluids, 2003) and Rorai et al. (Phys. Rev. E, 2013), both the TGV and ABC simulation results showed that stratification slowed down the development of turbulence. At the lowest (0.16 and 0.08) tested here, for which , the fluid motions in TGV were inhibited significantly by the strong stratification so as to eliminate the energy cascade regime and in turn, the inertial subrange. This led to an increased dissipation at the large scales in these cases as also observed in Brethouwer et al. (J. Fluid Mech., 2007). Spectral fluxes also showed an inverse energy cascade for very high stratification. Due to the nature of the ABC's initial configuration, it was observed that for the same values, its energy cascade was less inhibited in comparison to TGV and always forward as opposed to inverse for the very low- TGV cases, a behaviour that was also reflected in the different evolutions between the two flows. So compared to TGV, the large scales were more energetic in ABC for the same . In addition, horizontal velocity and density fluctuation fields of ABC simulation results, both showed the density layering effect from increased stratification.

Short-wave vortex instability in stratified flow

In this paper we investigate a new instability of the Lamb–Chaplygin dipole in a stratified fluid. Through numerical linear stability analysis, a secondary peak in the growth rate emerges at vertical scales about an order of magnitude smaller than the buoyancy scale Lb=U/N where U is the characteristic velocity and N is the Brunt–Väisälä frequency. This new instability exhibits a growth rate that is similar to, and even exceeds, that of the zigzag instability, which has the characteristic length of the buoyancy scale. This instability is investigated for a wide range of Reynolds numbers, Re=2000–20000, and horizontal Froude numbers, Fh=0.05–0.2, where Fh=U/NR, Re=UR/ν, R is the radius of the dipole, and ν is the kinematic viscosity. A range of vertical scales is explored from above the buoyancy scale to the viscous damping scale. Additionally, evidence is presented that the length scale and growth rate of this new instability are partially determined by the buoyancy Reynolds number, Reb=Fh2Re.

Diapycnal diffusivity, turbulent Prandtl number and mixing efficiency in Boussinesq stratified turbulence

In order that it be correctly characterized, irreversible turbulent mixing in stratified fluids must distinguish between adiabatic ‘stirring’ and diabatic ‘mixing’. Such a distinction has been formalized through the definition of a diapycnal diffusivity, $K_{{\it\rho}}$ (Winters & D’Asaro, J. Fluid Mech., vol. 317, 1996, pp. 179–193) and an appropriate mixing efficiency, $\mathscr{E}$ (Caulfield & Peltier, J. Fluid Mech., vol. 413, 2000, pp. 1–47). Equivalent attention has not been paid to the definitions of a corresponding momentum diffusivity $K_{m}$ and hence an appropriately defined turbulent Prandtl number $\mathit{Pr}_{t}=K_{m}/K_{{\it\rho}}$. In this paper, the diascalar framework of Winters & D’Asaro (1996) is first reformulated to obtain an ‘Osborn-like’ formula in which the correct definition of irreversible mixing efficiency $\mathscr{E}$ is shown to replace the flux Richardson number which Osborn (J. Phys. Oceanogr., vol. 10, 1980, pp. 83–89) assumed to characterize this efficiency. We advocate the use of this revised representation for diapycnal diffusivity since the proposed reformulation effectively removes the simplifying assumptions on which the original Osborn formula was based. We similarly propose correspondingly reasonable definitions for $K_{m}$ and $\mathit{Pr}_{t}$ by eliminating the reversible component of the momentum production term. To explore implications of the reformulations for both diapycnal and momentum diffusivity we employ an extensive series of direct numerical simulations (DNS) to investigate the properties of the shear-induced density-stratified turbulence that is engendered through the breaking of a freely evolving Kelvin–Helmholtz wave. The DNS results based on the proposed reformulation of $K_{{\it\rho}}$ are compared with available estimations due to the mixing length model, as well as both the Osborn–Cox and the Osborn models. Estimates based upon the Osborn–Cox formulation are shown to provide the closest approximation to the diapycnal diffusivity delivered by the exact representation. Through compilation of the complete set of DNS results we explore the characteristic dependence of $K_{{\it\rho}}$ on the buoyancy Reynolds number $\mathit{Re}_{b}$ as originally investigated by Shih et al. (J. Fluid Mech., vol. 525, 2005, pp. 193–214) in their idealized study of homogeneous stratified and sheared turbulence, and show that the validity of their results is only further reinforced through analysis of the turbulence produced in the more geophysically relevant Kelvin–Helmholtz wave life-cycle ansatz. In contrast to the results described by Shih et al. (2005) however, we show that, besides $\mathit{Re}_{b}$, a vertically averaged measure of the gradient Richardson number $\mathit{Ri}_{b}$ may equivalently characterize the turbulent mixing at high $\mathit{Re}_{b}$. Based on the dominant driving processes involved in irreversible mixing, we categorize the intermediate (i.e. $\mathit{Re}_{b}=O(10^{1}{-}10^{2})$) and high (i.e. $\mathit{Re}_{b}>O(10^{2})$) range of $\mathit{Re}_{b}$ as ‘buoyancy-dominated’ and ‘shear-dominated’ mixing regimes, which together define a transition value of $\mathit{Ri}_{b}\sim 0.2$. Mixing efficiency varies non-monotonically with both $\mathit{Re}_{b}$ and $\mathit{Ri}_{b}$, with its maximum (on the order of 0.2–0.3) occurring in the ‘buoyancy-dominated’ regime. Unlike $K_{{\it\rho}}$ which is very sensitive to the correct choice of $\mathscr{E}$ (i.e. $K_{{\it\rho}}\propto \mathscr{E}/(1-\mathscr{E})$), we show that $K_{m}$ is almost insensitive to the choice of $\mathscr{E}$ (i.e. $K_{m}\propto 1/(1-\mathscr{E})$) so long as $\mathscr{E}$ is not close to unity, which implies $K_{m}\approx \mathit{Ri}_{b}\mathit{Re}_{b}$ for the entire range of $\mathit{Re}_{b}$. The turbulent Prandtl number is consequently shown to decrease monotonically with $\mathit{Re}_{b}$ and may be (to first order) simply approximated by $\mathit{Re}_{b}$ itself. Assuming $\mathit{Pr}_{t}=1$, or $\mathit{Pr}_{t}=10$ (as is common in large-scale numerical models of the ocean general circulation), is also suggested to be a questionable assumption.

Classical scaling and intermittency in strongly stratified Boussinesq turbulence

Classical scaling arguments of Kolmogorov, Oboukhov and Corrsin (KOC) are evaluated for turbulence strongly influenced by stable stratification. The simulations are of forced homogeneous stratified turbulence resolved on up to$8192\times 8192\times 4096$grid points with buoyancy Reynolds numbers of$\mathit{Re}_{b}=13$, 48 and 220. A simulation of isotropic homogeneous turbulence with a mean scalar gradient resolved on$8192^{3}$grid points is used as a benchmark. The Prandtl number is unity. The stratified flows exhibit KOC scaling only for second-order statistics when$\mathit{Re}_{b}=220$; the$4/5$law is not observed. At lower$\mathit{Re}_{b}$, the$-5/3$slope in the spectra occurs at wavenumbers where the bottleneck effect occurs in unstratified cases, and KOC scaling is not observed in any of the structure functions. For the probability density functions (p.d.f.s) of the scalar and kinetic energy dissipation rates, the lognormal model works as well for the stratified cases with$\mathit{Re}_{b}=48$and 220 as it does for the unstratified case. For lower$\mathit{Re}_{b}$, the dominance of the vertical derivatives results in the p.d.f.s of the dissipation rates tending towards bimodal. The p.d.f.s of the dissipation rates locally averaged over spheres with radius in the inertial range tend towards bimodal regardless of$\mathit{Re}_{b}$. There is no broad scaling range, but the intermittency exponents at length scales near the Taylor length are in the range of$0.25\pm 0.05$and$0.35\pm 0.1$for the velocity and scalar respectively.

Evidence for Bolgiano-Obukhov scaling in rotating stratified turbulence using high-resolution direct numerical simulations

We report results on rotating stratified turbulence in the absence of forcing and with large-scale isotropic initial conditions using direct numerical simulations computed on grids of up to 40963 points. The Reynolds and Froude numbers are, respectively, equal to Re = 5.4 × 104 and Fr = 0.0242. The ratio of the Brunt-Väisälä to the inertial wave frequency, N/f, is taken to be equal to 4.95, a choice appropriate to model the dynamics of the southern abyssal ocean at mid latitudes. This gives a global buoyancy Reynolds number RB = ReFr2 ≈ 32, a value sufficient for some isotropy to be recovered in the small scales beyond the Ozmidov scale, but still moderate enough that the intermediate scales where waves are prevalent are well resolved. We concentrate on the large-scale dynamics, for which we find a spectrum compatible with the Bolgiano-Obukhov scaling. This scaling is also found for geostrophically balanced initial conditions on a run at a lower resolution and hence lower RB ≈ 4. Furthermore, we confirm that the Froude number based on a typical vertical length scale is of order unity, with strong gradients in the vertical. Two characteristic scales emerge from this computation and are identified from sharp variations in the spectral distribution of either total energy or helicity. A spectral break is also observed at a scale at which the partition of energy between the kinetic and potential modes changes abruptly, and beyond which a Kolmogorov-like spectrum recovers. Large slanted layers are ubiquitous in the flow, in the velocity and temperature fields, with local overturning events indicated by small local Richardson numbers and strong localized vortex tangles . Finally, a small large-scale enhancement of energy directly attributable to the effect of rotation is also observed.

Stratified turbulence forced with columnar dipoles: numerical study

This paper builds upon the investigation of Augier et al. (Phys. Fluids, vol. 26 (4), 2014) in which a strongly stratified turbulent-like flow was forced by 12 generators of vertical columnar dipoles. In experiments, measurements start to provide evidence of the existence of a strongly stratified inertial range that has been predicted for large turbulent buoyancy Reynolds numbers $\mathscr{R}_{t}={\it\varepsilon}_{\!K}/({\it\nu}N^{2})$, where ${\it\varepsilon}_{\!K}$ is the mean dissipation rate of kinetic energy, ${\it\nu}$ the viscosity and $N$ the Brunt–Väisälä frequency. However, because of experimental constraints, the buoyancy Reynolds number could not be increased to sufficiently large values so that the inertial strongly stratified turbulent range is only incipient. In order to extend the experimental results toward higher buoyancy Reynolds number, we have performed numerical simulations of forced stratified flows. To reproduce the experimental vortex generators, columnar dipoles are periodically produced in spatial space using impulsive horizontal body force at the peripheries of the computational domain. For moderate buoyancy Reynolds number, these numerical simulations are able to reproduce the results obtained in the experiments, validating this particular forcing. For higher buoyancy Reynolds number, the simulations show that the flow becomes turbulent as observed in Brethouwer et al. (J. Fluid Mech., vol. 585, 2007, pp. 343–368). However, the statistically stationary flow is horizontally inhomogeneous because the dipoles are destabilized quite rapidly after their generation. In order to produce horizontally homogeneous turbulence, high-resolution simulations at high buoyancy Reynolds number have been carried out with a slightly modified forcing in which dipoles are forced at random locations in the computational domain. The unidimensional horizontal spectra of kinetic and potential energies scale like $C_{1}{\it\varepsilon}_{\!K}^{2/3}k_{h}^{-5/3}$ and $C_{2}{\it\varepsilon}_{\!K}^{2/3}k_{h}^{-5/3}({\it\varepsilon}_{\!P}/{\it\varepsilon}_{\!K})$, respectively, with $C_{1}=C_{2}\simeq 0.5$ as obtained by Lindborg (J. Fluid Mech., vol. 550, 2006, pp. 207–242). However, there is a depletion in the horizontal kinetic energy spectrum for scales between the integral length scale and the buoyancy length scale and an anomalous energy excess around the buoyancy length scale probably due to direct transfers from large horizontal scale to small scales resulting from the shear and gravitational instabilities. The horizontal buoyancy flux co-spectrum increases abruptly at the buoyancy scale corroborating the presence of overturnings. Remarkably, the vertical kinetic energy spectrum exhibits a transition at the Ozmidov length scale from a steep spectrum scaling like $N^{2}k_{z}^{-3}$ at large scales to a spectrum scaling like $C_{K}{\it\varepsilon}_{\!K}^{2/3}k_{z}^{-5/3}$, with $C_{K}=1$, the classical Kolmogorov constant.

Transition to stably stratified states in open channel flow with radiative surface heating

AbstractDirect numerical simulations (DNS) of turbulent stratified flow in an open channel with an internal heat source following the Beer–Lambert law from the surface are used to investigate the transition from neutral to strongly stable flow. Our buoyancy bulk parameter is defined through the ratio of the domain height ${\it\delta}$ to $\mathscr{L}$, a bulk Obukhov length scale for the flow. We cover the range ${\it\lambda}={\it\delta}/\mathscr{L}=0{-}2.0$, from neutral conditions to the onset of the stable regime, with the Reynolds number range $Re_{{\it\tau}}=200{-}800$, at a Prandtl number of 0.71. The result is a boundary layer flow where the effects of stratification are weak in the wall region but progressively stronger in the outer layer up to the free surface. At ${\it\lambda}\simeq 1$ the turbulent kinetic energy (TKE) budget is in local equilibrium over a region extending from the near-wall region to a free-surface affected region a distance $l_{{\it\nu}}$ from the surface, with $l_{{\it\nu}}/{\it\delta}\sim Re^{-1/2}$. In this equilibrium region the flow can be characterised by the flux Richardson number $R_{f}$ and the local Obukhov length scale ${\it\Lambda}$. At higher ${\it\lambda}$ local mixing limit conditions are observed over an extended region. At ${\it\lambda}=2$ the flux Richardson number approaches critical limit values of $R_{f,c}\simeq 0.18$ and gradient Richardson number $Ri_{c}\simeq 0.2$. At high ${\it\lambda}$, we obtain a flow field where buoyancy interacts with the smallest scales of motion and the turbulent shear stress and buoyancy flux are suppressed to molecular levels. We find that this regime can be identified in terms of the parameter $Re_{\mathscr{L},c}=\mathscr{L}u_{{\it\tau}}/{\it\nu}\lesssim 200{-}400$ (where $u_{{\it\tau}}$ is the friction velocity and ${\it\nu}$ the kinematic viscosity), which is related to the $L_{\ast }$ parameter of Flores and Riley (Boundary-Layer Meteorol., vol. 139 (2), 2011, pp. 241–259) and buoyancy Reynolds number $\mathscr{R}$. With energetic equilibrium attained, the local buoyancy Reynolds number, $Re_{{\it\Lambda}}={\it\Lambda}\langle u^{\prime }w^{\prime }\rangle ^{1/2}/{\it\nu}$, is directly related to the separation of the Ozmidov ($l_{O}$) and Kolmogorov (${\it\eta}$) length scales in the outer boundary layer by $Re_{{\it\Lambda}}\simeq \mathscr{R}\equiv (l_{O}/{\it\eta})^{4/3}$. The inner wall region has the behaviour $\mathscr{R}\sim Re_{\mathscr{L}}Re_{{\it\tau}}$, in contrast to stratified boundary layer flows where the buoyancy flux is non-zero at the wall and $\mathscr{R}\sim Re_{\mathscr{L}}$.

Stratified shear flow: experiments in an inclined duct

AbstractWe present results of experiments on stratified shear flow in an inclined duct. The duct connects two reservoirs of fluid with different densities, and contains a counterflow with a dense layer flowing beneath a less dense layer moving in the opposite direction. We identify four flow states in this experiment, depending on the fractional density differences, characterised by the dimensionless Atwood number, and the angle of inclination $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\theta $, which is defined to be positive (negative) when the along-duct component of gravity reinforces (opposes) the buoyancy-induced pressure differences across the ends of the duct. For sufficiently negative angles and small fractional density differences, the flow is observed to be laminar ($\mathsf{L}$ state), with an undisturbed density interface separating the two layers. For positive angles and/or high fractional density differences, three other states are observed. For small angles of inclination, the flow is wave-dominated and exhibits Holmboe modes ($\mathsf{H}$ state) on the interface, with characteristic cusp-like wave breaking. At the highest positive angles and density differences, there is a turbulent ($\mathsf{T}$ state) high-dissipation interfacial region typically containing Kelvin–Helmholtz (KH)-like structures sheared in the direction of the mean shear and connecting both layers. For intermediate angles and density differences, an intermittent state ($\mathsf{I}$ state) is found, which exhibits a rich range of spatio-temporal behaviour and an interfacial region that contains features of KH-like structures and of the other two lower-dissipation states: thin interfaces and Holmboe-like structures. We map the state diagram of these flows in the Atwood number–$\theta $ plane and examine the force balances that determine each of these states. We find that the $\mathsf{L}$ and $\mathsf{H}$ states are hydraulically controlled at the ends of the duct and the flow is determined by the pressure difference associated with the density difference between the reservoirs. As the inclination increases, the along-slope component of the buoyancy force becomes more significant and the $\mathsf{I}$ and $\mathsf{T}$ states are associated with increasing dissipation within the duct. We replot the state space in the Grashof number–$\theta $ phase plane and find the transition to the $\mathsf{T}$ state is governed by a critical Grashof number. We find that the corresponding buoyancy Reynolds number of the transition to the $\mathsf{T}$ state is of the order of 100, and that this state is also found to be hydraulically controlled at the ends of the duct. In this state the dissipation balances the force associated with the along-slope component of buoyancy and the counterflow has a critical composite Froude number.

The quest for an unambiguous parameterization of mixing efficiency in stably stratified geophysical flows

AbstractEstimates of turbulent mixing in geophysical settings typically depend on the efficiency at which shear‐driven turbulence mixes density across isopycnals. To date, however, no unifying parameterization of diapycnal mixing efficiency exists due to the variability of natural flows and also due to certain ambiguities that arise from descriptions based on a single parameter. Here we highlight important ambiguities of some common single‐parameter schemes in the context of a multiparameter framework that considers the independent effects of shear, buoyancy, and viscosity. Parameterizations based on the gradient Richardson number (Ri), the turbulent Froude number (FrT), and the buoyancy Reynolds number (Reb) are considered. The diagnostic ability of these parameters is examined using published data from both direct numerical simulations and field observations.

Entrainment and mixing in a laboratory model of oceanic overflow

AbstractWe present experimental measurements of a wall-bounded gravity current, motivated by characterizing natural gravity currents such as oceanic overflows. We use particle image velocimetry and planar laser-induced fluorescence to simultaneously measure the velocity and density fields as they evolve downstream of the initial injection from a turbulent channel flow onto a plane inclined at$10^\circ $with respect to horizontal. The turbulence level of the input flow is controlled by injecting velocity fluctuations upstream of the output nozzle. The initial Reynolds number based on the Taylor microscale of the flow,$R_{\lambda }$, is varied between 40 and 120, and the effects of the initial turbulence level are assessed. The bulk Richardson number$\mathit{Ri}$for the flow is${\sim }$0.3 whereas the gradient Richardson number$\mathit{Ri}_g$varies between 0.04 and 0.25, indicating that shear dominates the stabilizing effect of stratification. Kelvin–Helmholtz instability results in vigorous vertical transport of mass and momentum. We present baseline characterization of standard turbulence quantities and calculate, in several different ways, the fluid entrainment coefficient$E$, a quantity of considerable interest in mixing parameterization for ocean circulation models. We also determine the properties of mixing as represented by the flux Richardson number$\mathit{Ri}_f$as a function of$\mathit{Ri}_g$and diapycnal mixing parameter$K_{\rho }$versus the buoyancy Reynolds number$\mathit{Re}_b$. We find reasonable agreement with results from natural flows.

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.

Potential enstrophy in stratified turbulence

AbstractDirect numerical simulations are used to investigate potential enstrophy in stratified turbulence with small Froude numbers, large Reynolds numbers, and buoyancy Reynolds numbers ($R{e}_{b} $) both smaller and larger than unity. We investigate the conditions under which the potential enstrophy, which is a quartic quantity in the flow variables, can be approximated by its quadratic terms, as is often done in geophysical fluid dynamics. We show that at large scales, the quadratic fraction of the potential enstrophy is determined by $R{e}_{b} $. The quadratic part dominates for small $R{e}_{b} $, i.e. in the viscously coupled regime of stratified turbulence, but not when $R{e}_{b} \gtrsim 1$. The breakdown of the quadratic approximation is consistent with the development of Kelvin–Helmholtz instabilities, which are frequently observed to grow on the layerwise structure of stratified turbulence when $R{e}_{b} $ is not too small.

Spectral analysis of the transition to turbulence from a dipole in stratified fluid

AbstractWe investigate the spectral properties of the turbulence generated during the nonlinear evolution of a Lamb–Chaplygin dipole in a stratified fluid for a high Reynolds number $Re= 28\hspace{0.167em} 000$ and a wide range of horizontal Froude number ${F}_{h} \in [0. 0225~0. 135] $ and buoyancy Reynolds number $\mathscr{R}= Re{{F}_{h} }^{2} \in [14~510] $. The numerical simulations use a weak hyperviscosity and are therefore almost direct numerical simulations (DNS). After the nonlinear development of the zigzag instability, both shear and gravitational instabilities develop and lead to a transition to small scales. A spectral analysis shows that this transition is dominated by two kinds of transfer: first, the shear instability induces a direct non-local transfer toward horizontal wavelengths of the order of the buoyancy scale ${L}_{b} = U/ N$, where $U$ is the characteristic horizontal velocity of the dipole and $N$ the Brunt–Väisälä frequency; second, the destabilization of the Kelvin–Helmholtz billows and the gravitational instability lead to small-scale weakly stratified turbulence. The horizontal spectrum of kinetic energy exhibits a ${{\varepsilon }_{K} }^{2/ 3} { k}_{h}^{\ensuremath{-} 5/ 3} $ power law (where ${k}_{h} $ is the horizontal wavenumber and ${\varepsilon }_{K} $ is the dissipation rate of kinetic energy) from ${k}_{b} = 2\lrm{\pi} / {L}_{b} $ to the dissipative scales, with an energy deficit between the integral scale and ${k}_{b} $ and an excess around ${k}_{b} $. The vertical spectrum of kinetic energy can be expressed as $E({k}_{z} )= {C}_{N} {N}^{2} { k}_{z}^{\ensuremath{-} 3} + C{{\varepsilon }_{K} }^{2/ 3} { k}_{z}^{\ensuremath{-} 5/ 3} $ where ${C}_{N} $ and $C$ are two constants of order unity and ${k}_{z} $ is the vertical wavenumber. It is therefore very steep near the buoyancy scale with an ${N}^{2} { k}_{z}^{\ensuremath{-} 3} $ shape and approaches the ${{\varepsilon }_{K} }^{2/ 3} { k}_{z}^{\ensuremath{-} 5/ 3} $ spectrum for ${k}_{z} \gt {k}_{o} $, ${k}_{o} $ being the Ozmidov wavenumber, which is the cross-over between the two scaling laws. A decomposition of the vertical spectra depending on the horizontal wavenumber value shows that the ${N}^{2} { k}_{z}^{\ensuremath{-} 3} $ spectrum is associated with large horizontal scales $\vert {\mathbi{k}}_{h} \vert \lt {k}_{b} $ and the ${{\varepsilon }_{K} }^{2/ 3} { k}_{z}^{\ensuremath{-} 5/ 3} $ spectrum with the scales $\vert {\mathbi{k}}_{h} \vert \gt {k}_{b} $.

Kinetic energy dynamics in forced, homogeneous, and axisymmetric stably stratified turbulence

Direct numerical simulations with up to 4096×4096×2048 grid points are used to study the dynamics of kinetic energy in forced, homogeneous, and axisymmetric stably stratified flow with unit Prandtl number. No mean shear or internal waves are introduced into the flows, forcing represents persistent large horizontal motions with a small amount of noise, and turbulence develops in response. Three different stratification levels result in buoyancy Reynolds numbers ranging from 11 to 230, and multiple cross-checks for consistency in various dimensionless ratios show that the flows are consistent with theory for strong stratification. The balance equations for the horizontal and vertical contributions to kinetic energy are examined in terms of two-dimensional spectra. Downscale cascades of horizontal energy are evident at both the horizontal and the vertical, as is upscale transfer of energy at large horizontal scales but small vertical scales. Vertical energy also cascades down scale in both directions, so the dissipation rate of vertical energy relative to that of horizontal energy is the same as for unstratified isotropic turbulence. The horizontal and vertical length scales responsible for dissipation rate and the relative importance of dissipation rate compared with the energy transfer rate depend on the strength of the stratification. Both three-dimensional turbulence at scales smaller than the Ozmidov length scale and stratified turbulence at scales between the Ozmidov and the buoyancy scale produce power law scaling of in one-dimensional spectra. A sharper power law is observed at scales larger than the buoyancy scale. The second-order longitudinal structure functions, while exhibiting power law over the same scaling range, do not corroborate the Kolmogorov–Obukhov–Corrsin scaling.

Numerical study of vertical dispersion by stratified turbulence

Numerical simulations are carried out to investigate vertical fluid particle dispersion in uniformly stratified stationary turbulent flows. The results are compared with the analysis of Lindborg & Brethouwer (J. Fluid Mech., vol. 614, 2008, pp. 303–314), who derived long- and short-time relations for the mean square vertical displacement 〈δz〉 of fluid particles. Several direct numerical simulations (DNSs) with different degrees of stratification and different buoyancy Reynolds numbers are carried out to test the long-time relation 〈δz2〉 = 2ϵPt/N2. Here, ϵP is the mean dissipation of turbulent potential energy; N is the Brunt–Väisälä frequency; and t is time. The DNSs show good agreement with this relation, with a weak dependence on the buoyancy Reynolds number. Simulations with hyperviscosity are carried out to test the relation 〈δz2〉 = (1+πCPL)2ϵPt/N2, which should be valid for shorter time scales in the range N−1 ≪ t ≪ T, where T is the turbulent eddy turnover time. The results of the hyperviscosity simulations come closer to this prediction with CPL about 3 with increasing stratification. However, even in the simulation with the strongest stratification the growth of 〈δz2〉 is somewhat slower than linear in this regime. Based on the simulation results it is argued that the time scale determining the evolution of 〈δz2〉 is the eddy turnover time, T, rather than the buoyancy time scale N−1, as suggested in previous studies. The simulation results are also consistent with the prediction of Lindborg & Brethouwer (2008) that the nearly flat plateau of 〈δz2〉 observed at t ~ T should scale as 4EP/N2, where EP is the mean turbulent potential energy.

Mixed Convection Heat Transfer from a Cylinder in Power-Law Fluids: Effect of Aiding Buoyancy

Mixed convection heat transfer from an isothermally heated horizontal cylinder immersed in incompressible power-law fluids is considered here in the steady flow regime when both the imposed flow and the buoyancy induced motion are in the same direction, i.e., the so-called buoyancy aiding configuration. Within the framework of the Boussinesq approximation, the suitable forms of the momentum and thermal energy equations for the power-law fluid model have been solved numerically using the finite volume based FLUENT (version 6.3) solver for the following ranges of conditions: the buoyancy parameter (Richardson number, 0 ≤ Ri ≤ 2), power-law index (0.2 ≤ n ≤ 1.8), Reynolds number (1 ≤ Re ≤ 40), and Prandtl number (1 ≤ Pr ≤ 100). In particular, the effects of these dimensionless parameters on the detailed local and global kinematics of flow and heat transfer characteristics such as streamline, vorticity, and pressure profiles, individual and total drag coefficients, and local and average Nusselt numbers have b...

Turbulence in coastal fronts near the mouths of Block Island and Long Island Sounds

Measurements of turbulence were performed in four frontal locations near the mouths of Block Island Sound (BIS) and Long Island Sound (LIS). These measurements extend from the offshore front associated with BIS and Mid-Atlantic Bight Shelf water, to the onshore fronts near the Montauk Point (MK) headland, and the Connecticut River plume front. The latter feature is closely associated with the major fresh water input to LIS. Turbulent kinetic energy (TKE) dissipation rate, ε, was obtained using shear probes mounted on an autonomous underwater vehicle. Offshore, the BIS estuarine outflow front showed, during spring season and ebb tide, maximum TKE dissipation rate, ε, estimates of order 10 − 5 W/kg, with background values of order 10 − 6 to 10 − 9 W/kg. Edwards et al. [Edwards, C.A., Fake, T.A., and Bogden, P.S., 2004a. Spring–summer frontogenesis at the mouth of Block Island Sound: 1. A numerical investigation into tidal and buoyancy-forced motion. Journal of Geophysical Research 109 (C12021), doi:10.1029/2003JC002132.] model this front as the boundary of a tidally driven, baroclinically adjusted BIS flow around the MK headland eddy. At the entrance to BIS, near MK, two additional fronts are observed, one of which was over sand waves. For the headland site front east of MK, without sand waves, during ebb tide, ε estimates of 10 − 5 to 10 − 6 W/kg were observed. The model shows that this front is at the northern end of an anti-cyclonic headland eddy, and within a region of strong tidal mixing. For the headland site front further northeast over sand waves, maximum ε estimates were of order 10 − 4 W/kg within a background of order 10 − 7 –10 − 6 W/kg. From the model, this front is at the northeastern edge of the anti-cyclonic headland eddy and within the tidal mixing zone. For the Connecticut River plume front, a surface trapped plume, during ebb tide, maximum ε estimates of 10 − 5 W/kg were obtained, within a background of 10 − 6 to 10 − 8 W/kg. Of all four fronts, the river plume front has the largest finescale mean-square shear, S 2 ~ 0.15 s − 2 . All of the frontal locations had local values of the buoyancy Reynolds number indicating strong isotropic turbulence at the dissipation scales. Local values of the Froude number indicated shear instability in all of the fronts.

Turbulence Observations in the Northern Bight of Monterey Bay from a Small AUV

In this manuscript we show that in shallow water an autonomous underwater vehicle (AUV) can be used to quantify the horizontal and vertical structure of turbulence. We present data obtained during the Layered Organization in the Coastal Ocean (LOCO) experiment, which took place in a very shallow region of Monterey Bay at nighttime in the summer of 2006. The AUV employed was the T-REMUS vehicle, which contains a variety of the microscale and finescale sensors as well as supporting “hotel” sensors which monitor its position and performance. The vehicle was run in a 5-degree yoyo mode and, using the Rockland Microstructure Measurement System (RMMS), was able to obtain direct estimates of the turbulent dissipation rate on a vertical scale of 0.5 m. The yoyo sampling scheme allowed the finestructure and microstructure data to be obtained with an average horizontal sampling distance of 150 m. We examine the spatial structure of turbulence over a horizontal range of 2.5 km and throughout most of the water column, from 1 m from the surface to 4 m above the bottom. Eight hours of such data were collected. The experiment took place during a time period of very light wind forcing, little air sea exchange, and weak tidal flow. Nonetheless, strong turbulence, characterized by a turbulent kinetic energy dissipation rate of ε > 10 − 7 W/kg and buoyancy Reynolds number of Re b > 10 3, was observed throughout the experimental region. The turbulence, for the most part, was patchy and typically confined to the surface and bottom boundary layer regions. However, towards the end of the experiment a spatially continuous region of turbulence of at least 2.5 km extent was observed in the upper part of the thermocline. This occurred during the time period when an internal wave train appeared in the experimental area. Evidence is presented which suggests that the internal wave induced vertical strain gradient was responsible for producing this turbulent field.

Slowly oscillating modes in the passive scalar diffusion in stratified turbulence

Scalar fluxes in unsteady stratified turbulence are investigated when vertical mean gradients of both the density (active scalar) and the passive scalar exist. Analytical solutions by the rapid distortion theory show that there is a “slowly” oscillating mode in the passive scalar flux, whose oscillation period is twice as large as that of the density flux. For passive scalars with high Schmidt number Sc>2Pr∕(1+Pr) (Pr: Prandtl number of density), this slow mode with small initial decay becomes dominant, so that temporal oscillation of Kc (turbulent diffusion coefficient of the passive scalar) persists with larger amplitude than the density diffusion coefficient Kρ. On the other hand, at low Schmidt number Sc<2Pr∕(1+Pr)<2, slow mode suffers strong initial decay, so that Kc and Kρ decay at the similar rate. These results illustrate the importance of molecular diffusivity and unsteadiness in weak turbulence which would make significant difference between Kc and Kρ. A criterion for which the linear mechanisms become important is also derived. When the buoyancy Reynolds number Reb (=ϵ∕νN2, ϵ: energy dissipation rate; N: Brunt–Väisälä frequency) is smaller than Re(l0)[=u(l0)l0∕ν], i.e., the Reynolds number defined at the integral scale l0, turbulent energies, and fluxes would be affected by linear processes. This suggests that large value of Reb does not necessarily mean strong nonlinearity, and the differential diffusion previously observed in stratified turbulence for Reb≲103 could be due to linear processes.