Stratified Shear Flow Research Articles

Characteristics of Shear Stratified Flows in the Conditions of the Sea of Japan Shelf Based on in-situ Measurements in 2022

Characteristics of Shear Stratified Flows in the Conditions of the Sea of Japan Shelf Based on in-situ Measurements in 2022

Regenerative Orr mechanism yielding large non-modal perturbation energy growth in a viscosity stratified plane shear flow

Transiently growing non-modal perturbations can play a crucial role in the transition of plane shear flows in modally stable regimes. In terms of the extent of transient amplification, three-dimensional perturbations are typically more prominent due to the lift-up effect. In contrast, two-dimensional (2D) spanwise-independent perturbations are often considered less important as they typically undergo modest levels of transient growth and are short-lived. The Orr mechanism is key to the amplification of energy for 2D perturbations. In this work, we discuss 2D non-modal perturbations of three-layer viscosity stratified flows with the mean shear rates of the outer layers being equal. Strikingly, a novel regenerative Orr mechanism is found that allows for significant amount of energy amplification despite the 2D nature of the perturbations. Moreover, these perturbations survive for considerably long times. The perturbation structure shows symmetry about the middle layer and evolves such that the Orr mechanism can repeatedly occur in a regenerative manner resulting in the perturbation energy evolving in a markedly non-monotonic fashion. When these same perturbations are introduced in a uniform plane shear flow, the corresponding non-modal transient amplifications are shown to be much smaller.

Energy Budget Characterisation of the Optimal Disturbance in Stratified Shear Flow

Stratified Taylor–Couette flow (STCF) undergoes transient growth. Recent studies have shown that there exists transient amplification in the linear regime of counter-rotating STCF. The kinetic budget of the optimal transient perturbation is analysed numerically to simulate the interaction of the shear production (SP), buoyancy flux (BP), and other energy components that contributes to the total optimal transient kinetic energy. These contributions affect the total energy by influencing the perturbation to extract kinetic energy (KE) from the mean flow. The decay of the amplification factor resulted from the positive amplification of both BP and SP, while the growth is attributed to the negative and positive amplification of BP and SP, respectively. The optimal SP is positively amplified, implying that there is the possibility of constant linear growth. These findings agree with the linear growth rate for increasing values of Grashof number.

Long-wave instabilities of sloping stratified exchange flows

We investigate the linear instability of two-layer stratified shear flows in a sloping two-dimensional channel, subject to non-zero longitudinal gravitational forces. We reveal three previously unknown instabilities, distinct from the well-known Kelvin–Helmholtz instability and Holmboe wave instability, in that they have longer wavelengths (of the order of 10 to $10^3$ shear-layer depths) and often slower growth rates. Importantly, they can grow in background flows with gradient Richardson number $\gg 1$ , which offers a new mechanism to sustain turbulence and mixing in strongly stratified flows. These instabilities are shown to be generic and relatively insensitive to Reynolds number, Prandtl number, base flow profile and boundary conditions. The nonlinear evolution of these instabilities is investigated through a forced direct numerical simulation, in which the background momentum and density are sustained. The growth of long unstable waves in background flows initially stable to short wave causes a decrease in the local gradient Richardson number. This leads to local nonlinear processes that result in small-scale overturns resembling Kelvin–Helmholtz billows. Our results establish a new energy exchange pathway, where the mean kinetic energy of a strongly stratified flow is extracted by primary unstable long waves and secondary short waves, and subsequently dissipated into internal energy.

Mixing of the Connecticut River Plume During Ambient Flood Tides: Spatial Heterogeneity and Contributions of Bottom‐Generated and Interfacial Mixing

AbstractThe Connecticut River plume is influenced by energetic ambient tides in the Long Island Sound receiving waters. The objectives of this modeling study are (a) characterizing the spatial heterogeneity of turbulent buoyancy fluxes, (b) partitioning turbulent buoyancy fluxes into bottom‐generated and interfacial shear contributions, and (c) quantifying contributions to plume‐integrated mixing within the tidal plume. The plume formed during ambient flood tides under low river discharge, spring tides, and no winds is analyzed. Turbulent buoyancy fluxes (B) and depth‐integrated B through the plume (Bd) are characterized by pronounced spatial heterogeneity. Strong mixing (Bd ∼ 10−5‐10−4 m3/s3) occurs near the mouth, in the nearfield plume turning region, over shoals, and nearshore shallow areas. Low to moderate mixing (Bd ∼ 10−8‐10−6 m3/s3) occupies half the plume. Buoyancy fluxes are first partitioned based on the depth of the shear stress minimum between plume‐generated and bottom‐generated shear maxima. Four other tested partitioning methods are based on open channel flow and stratified shear flow parameterizations. Interfacial and bottom‐generated shear contribute to different areas of intense and moderate mixing. All methods indicate a significant plume mixing role for bottom‐generated mixing, but interfacial mixing is a bigger contributor. Plume‐integrated total and interfacial mixing peak at max ambient flood and the timing of peak bottom‐generated mixing varies among partitioning methods. Two‐thirds of the mixing occurs in concentrated intense mixing areas. A parameter space with the ambient tidal Froude number and plume thickness to depth ratio as axes indicates many tidally modulated plumes are moderately to dominantly influenced by bottom‐generated tidal mixing.

Symmetrization and asymptotic stability in non-homogeneous fluids around stratified shear flows

Symmetrization and asymptotic stability in non-homogeneous fluids around stratified shear flows

Settling versus mixing in stratified shear flows

We study the coupled settling, deformation and mixing dynamics of a dense blob of fluid falling in an axially (vertically) linearly stratified Taylor–Couette cell (operated in a laminar stable regime). This configuration allows the independent analysis of stretching dynamics, driven by radial (horizontal) velocity variations, and settling dynamics, driven by buoyancy forces associated with vertical density variations. As the blob settles, it is stretched in the horizontal plane and forms an elongated lamella. Through the competing effects of transverse compression of the lamella due to this shear-induced stretching and broadening due to diffusion, the lamella irreversibly mixes with ambient fluid, thus progressively adjusting its own density towards that of the ambient fluid. Eventually, the lamella settling stops at a final equilibrium position that depends on the ambient vertical density gradient and the rate at which it has been deformed by the horizontal shear. We show how this final position is determined by stretching-enhanced diffusion, i.e. mixing. We demonstrate that a theoretical mixing model compares favourably with experiments with various Froude numbers (quantifying the relative strength of the horizontal shear and the vertical stratification) and construct a new criterion for the energetic ‘efficiency’ of this mixing process that explicitly captures its inherently diffusive character.

A novel (2+1)-dimensional nonlinear evolution equation for weakly stratified free-surface boundary layers

To get an insight into the dynamics of the oceanic surface boundary layer we develop an asymptotic model of the nonlinear dynamics of linearly decaying three-dimensional long-wave perturbations in weakly stratified boundary-layer flows. Although in nature the free-surface boundary layers in the ocean are often weakly stratified due to solar radiation and air entrainment caused by wave breaking, weak stratification has been invariably ignored. Here, we consider an idealized hydrodynamic model, where finite-amplitude three-dimensional perturbations propagate in a horizontally uniform unidirectional weakly stratified shear flow confined to a boundary layer adjacent to the water surface. Perturbations satisfy the no-stress boundary condition at the surface. They are assumed to be long compared with the boundary-layer thickness. Such perturbations have not been studied even in a linear setting. By exploiting the assumed smallness of nonlinearity, wavenumber, viscosity and the Richardson number, on applying triple-deck asymptotic scheme and multiple-scale expansion, we derive in the distinguished limit a novel essentially two-dimensional nonlinear evolution equation, which is the main result of the work. The equation represents a generalization of the two-dimensional Benjamin–Ono equation modified by the explicit account of viscous effects and new dispersion due to weak stratification. It describes perturbation dependence on horizontal coordinates and time, while its vertical structure, to leading order, is given by an explicit analytical solution of the linear boundary value problem. It shows the principal importance of weak stratification for three-dimensional perturbations.

Kelvin–Helmholtz Instability “Tube” and “Knot” Dynamics. Part I: Expanding Observational Evidence of Occurrence and Environmental Influences

Abstract Multiple recent observations in the mesosphere have revealed large-scale Kelvin–Helmholtz instabilities (KHI) exhibiting diverse spatial features and temporal evolutions. The first event reported by Hecht et al. exhibited multiple features resembling those seen to arise in early laboratory shear-flow studies described as “tube” and “knot” (T&K) dynamics by Thorpe. The potential importance of T&K dynamics in the atmosphere, and in the oceans and other stratified and sheared fluids, is due to their accelerated turbulence transitions and elevated energy dissipation rates relative to KHI turbulence transitions occurring in their absence. Motivated by these studies, we survey recent observational evidence of multiscale Kelvin–Helmholtz instabilities throughout the atmosphere, many features of which closely resemble T&K dynamics observed in the laboratory and idealized initial modeling. These efforts will guide further modeling assessing the potential importance of these T&K dynamics in turbulence generation, energy dissipation, and mixing throughout the atmosphere and other fluids. We expect these dynamics to have implications for parameterizing mixing and transport in stratified shear flows in the atmosphere and oceans that have not been considered to date. Companion papers describe results of a multiscale gravity wave direct numerical simulation (DNS) that serendipitously exhibits a number of KHI T&K events and an idealized multiscale DNS of KHI T&K dynamics without gravity wave influences. Significance Statement Kelvin–Helmholtz instabilities (KHI) occur throughout the atmosphere and induce turbulence and mixing that need to be represented in weather prediction and other models of the atmosphere and oceans. This paper documents recent atmospheric evidence for widespread, more intense, features of KHI dynamics that arise where KH billows are initially discontinuous, misaligned, or varying along their axes. These features initiate strong local vortex interactions described as “tubes” and “knots” in early laboratory experiments, suggested by, but not recognized in, earlier atmospheric and oceanic profiling, and only recently confirmed in newer, high-resolution atmospheric imaging and idealized modeling to date.

On the Bounded Region for the Stratified Shear Flows in \(\beta\)-plane

On the Bounded Region for the Stratified Shear Flows in \(\beta\)-plane

Stratified inclined duct: direct numerical simulations

The stratified inclined duct (SID) experiment consists of a zero-net-volume exchange flow in a long tilted rectangular duct, which allows the study of realistic stratified shear flows with sustained internal forcing. We present the first three-dimensional direct numerical simulations (DNS) of SID to explore the transitions between increasingly turbulent flow regimes first described by Meyer & Linden (J. Fluid Mech., vol. 753, 2014, pp. 242–253). We develop a numerical set-up that faithfully reproduces the experiments and sustains the flow for arbitrarily long times at minimal computational cost. We recover the four qualitative flow regimes found experimentally in the same regions of parameter space: laminar flow, waves, intermittent turbulence and fully developed turbulence. We find good qualitative and quantitative agreement between DNS and experiments and highlight the added value of DNS to complement experimental diagnostics and increase our understanding of the transition to turbulence, both temporally (laminar/turbulent cycles) and parametrically (as the tilt angle of the duct and the Reynolds number are increased). These results demonstrate that numerical studies of SID – and deeper integration between simulations and experiments – have the potential to lead to a better understanding of stratified turbulence.

Wind Dependencies of Deep Cycle Turbulence in the Equatorial Cold Tongues

Abstract Several years of moored turbulence measurements from χpods at three sites in the equatorial cold tongues of Atlantic and Pacific Oceans yield new insights into proxy estimates of turbulence that specifically target the cold tongues. They also reveal previously unknown wind dependencies of diurnally varying turbulence in the near-critical stratified shear layers beneath the mixed layer and above the core of the Equatorial Undercurrent that we have come to understand as deep cycle (DC) turbulence. Isolated by the mixed layer above, the DC layer is only indirectly linked to surface forcing. Yet, it varies diurnally in concert with daily changes in heating/cooling. Diurnal composites computed from 10-min averaged data at fixed χpod depths show that transitions from daytime to nighttime mixing regimes are increasingly delayed with weakening wind stress τ. These transitions are also delayed with respect to depth such that they follow a descent rate of roughly 6 m h−1, independent of τ. We hypothesize that this wind-dependent delay is a direct result of wind-dependent diurnal warm layer deepening, which acts as the trigger to DC layer instability by bringing shear from the surface downward but at rates much slower than 6 m h−1. This delay in initiation of DC layer instability contributes to a reduction in daily averaged values of turbulence dissipation. Both the absence of descending turbulence in the sheared DC layer prior to arrival of the diurnal warm layer shear and the magnitude of the subsequent descent rate after arrival are roughly predicted by laboratory experiments on entrainment in stratified shear flows. Significance Statement Only recently have long time series measurements of ocean turbulence been available anywhere. Important sites for these measurements are the equatorial cold tongues where the nature of upper-ocean turbulence differs from that in most of the world’s oceans and where heat uptake from the atmosphere is concentrated. Critical to heat transported downward from the mixed layer is the diurnally varying deep cycle of turbulence below the mixed layer and above the core of the Equatorial Undercurrent. Even though this layer does not directly contact the surface, here we show the influence of the surface winds on both the magnitude of the deep cycle turbulence and the timing of its descent into the depths below.

Restratification in Late Winter Lakes Induced by Cabbeling

AbstractThe under‐ice fluid dynamics during late winter in many freshwater settings remain poorly understood. One example is how cabbeling, the generation of dense water by mixing water masses on different sides of the temperature of maximum density (Tmd), affects the vertical transport. Using high resolution numerical simulations of the development of a stratified parallel shear flow, we show that when the temperature stratification passes through Tmd, cabbeling modifies the three‐dimensional aspects of the instability with the net effect of forming an emergent, stable stratification after a brief period of strong mixing. This stratification effectively separates the quiescent upper layer from the turbulent one below, thereby limiting transport and mixing between them. We propose a simple model for the vertical fluid flux during the mixing regime, and discuss a potential mechanism responsible for the emergent stratification as well as its sensitivity to the bulk Richardson number.

Thermo-hydraulic characteristics of thermal interface induced by natural convection in water tank with internal heat source

Large-scale water tank or pool with internal heat source is common structures in solar energy and nuclear power. In this paper, a visible thermal interface is captured in the experiments in a square water tank with a horizontal heating rod, which is an interesting phenomenon hardly reported before. The direct shadowgraphy method is employed to record gray graphs of the forming process and flow regimes of the thermal interface. Corresponding transient numerical simulations with the large eddy simulation (LES) method are conducted to display detailed thermo-hydraulic characteristics of the thermal interface. Based on the temperature distribution and flow field, a natural convection circulation driven by buoyancy is formed above a stagnant zone, which is regarded as a two-layer stratified shear flow. The horizontal shear flow above the stagnant zone induces the visible thermal interface. Because the heat transfer mode switches from convection to conduction when crossing the interface, the sharp temperature gradient and density variation occur at the interface. The effect of the heat flux of the heating rod is investigated, and three different flow regimes of the thermal interface are observed under different heat loads. The stability of the flow regimes is analyzed by the gradient Richardson number, which is the ratio of the buoyancy to the turbulent shear. At low heat load, the thermal interface is destabilized because the turbulent shear overcomes the density barrier. As the heat loading increases, the buoyancy increasing rate exceeds the shear increasing rate, and the stronger potential barrier dominates the flow structure, which is beneficial to the interface stability.

Turbulent disruption of density staircases in stratified shear flows

The formation of step-like ‘density staircase’ distributions induced by stratification and turbulence has been widely studied, and can be explained by the ‘instability’ of a sufficiently strongly stably stratified turbulent flow due to the decrease of the turbulent density flux with increasing stratification via the ‘Phillips mechanism’ (Phillips, Deep Sea Research and Oceanographic Abstracts, vol. 19, 1972, pp. 79–81. Elsevier). However, such density staircases are not often observed in ocean interiors, except in regions where double-diffusion processes are important, leading to thermohaline staircases. Using reduced-order models for the evolution of velocity and density gradients, we analyse staircase formation in stratified and sheared turbulent flows. Under the assumption of inertial scaling $\epsilon \sim U^3/L$ for the kinetic energy dissipation rate $\epsilon$ , where $U$ and $L$ are characteristic velocity and length scales, we determine ranges of bulk Richardson numbers ${Ri}_{b}$ and turbulent Prandtl numbers ${Pr}_{T}$ for which staircases can potentially form and show that the Phillips mechanism only survives in the limit of sufficiently small turbulent Prandtl numbers. For relevant oceanic parameters, a range of turbulent Prandtl numbers above which the system is not prone to staircases is found to be ${Pr}_T \simeq 0.5\unicode{x2013}0.8$ . Since several studies indicate that the turbulent Prandtl number in stably stratified turbulence and in ocean interiors is usually above this threshold, this result supports the empirical observation that staircases are not favoured in ocean interiors in the presence of relatively homogeneous and sustained turbulence. We also show that our analysis is robust to other scalings for $\epsilon$ (such as the more strongly stratified scaling $\epsilon \sim U^{2}N_{c}$ , where $N_{c}$ is a characteristic value of the buoyancy frequency), supporting our results in both shear-dominated and buoyancy-dominated turbulent regimes as well as in weakly and strongly stratified regimes.

Regime transitions in stratified shear flows: the link between horizontal and inclined ducts

We present the analytical solution for the two-dimensional velocity and density fields within an approximation for laminar stratified inclined duct (SID) flows where diffusion dominates over inertia in the along-channel momentum equation but is negligible in the density transport equation. We refer to this approximation as the hydrostatic/gravitational/viscous in momentum and advective in density (HGV-A) approximation due to the leading balances in the governing equations. The analytical solution is valid for laminar flows in a two-layer configuration in the limit of long ducts. The non-dimensional volume flux within the HGV-A approximation is given by $Fr^* ={{Re}}_g/(AK)$ , which is a control parameter with ${{Re}}_g$ the gravitational Reynolds number, $A$ the aspect ratio of the duct and $K$ a geometrical parameter that depends on the tilt of the duct and is obtained from the analytical solution. This analytical solution was validated against results from laboratory experiments, and allows us to gain new insight into the dynamics and properties of SID flows. Most importantly, constant values of $Fr^*$ describe, in both horizontal and inclined ducts, the transitions between increasingly turbulent flow regimes: from laminar flow, to interfacial waves, to intermittent turbulence and sustained turbulence.

Nonlinear dynamics of unstably stratified two-layer shear flow in a horizontal channel

The Rayleigh–Taylor instability at the interface of two sheared fluid layers in a horizontal channel is investigated in the absence of inertia. The dynamics of the flow is described by a nonlinear lubrication equation which is solved numerically for adverse density stratifications. The early-time dynamics features a number of finger-like protrusions of different heights at the interface. The fingers travel at different speeds leading to a sequence of merging events after which the interface eventually settles to a near-saturated state, comprising only one finger that includes most of the lower fluid. For sufficiently large density stratifications, the final state spans the height of the channel and includes two thin fluid films at each wall, both of which undergo chaotic dynamics, but finite-time touch-down/touch-up is shown to be precluded by the shear flow. An asymptotic analysis in the large-Bond-number limit (intense density stratification) reveals the finer structure of the final state including Landau–Levich-type connection regions. The asymptotic solutions are compared with numerical results of the lubrication model as well as direct numerical simulations, and excellent agreement is observed between the three in terms of interfacial structure, wave speed and film thicknesses.

Turbulent Transport in a Stratified Shear Flow

Within the framework of the theory of unsteady turbulent flows in a stratified fluid, a new parameterization of the turbulent Prandtl number is proposed. The parameterization is included in the k-ε-closure and used within the three-dimensional model of thermohydrodynamics of an enclosed water body where density distribution includes pycnocline. This allows us to describe turbulence in a stratified shear flow without the restrictions associated with the gradient Richardson number and justify the choice of closure constants. Numerical experiments, where the downward penetration of turbulence was considered, confirm the advantage of the developed approach in describing the effects neglected in the classical closures.

Effects of stabilizing and destabilizing thermal gradients on reversed shear-stratified flows: Combined Kelvin–Helmholtz Rayleigh–Taylor instability

Numerical investigation of the coupled Kelvin–Helmholtz Rayleigh–Taylor instability (KHRTI) is presented here by solving the compressible Navier–Stokes equations for two air streams differentially heated in two halves of a three-dimensional (3D) box. Here, we explore the role of a stabilizing and destabilizing thermal gradient and that of reversing the direction of the air streams considered for Atwood numbers of ±0.1567 and dimensionless tangential shear of ΔU=0.68 and 4.1. The onset of the KHRTI and development of the turbulent mixing layer are explored via time-resolved and instantaneous distributions of temperature and vorticity. Early stages of the KHRTI with reversed air streams follows a Kelvin–Helmholtz (KH) mechanism, with Rayleigh–Taylor (RT) dynamics becoming important at later times. This leads to an earlier development of the turbulent mixing layer. The KHRTI with stabilizing or destabilizing thermal gradients shows a dominance of the buoyancy-driven mechanism, right from the onset. The transition from laminar to turbulent mixing layer involves the creation of coherent structures of spikes, bubbles, and KH whirls for the destabilizing, stabilizing thermal gradient, and reversed shear cases, respectively. The spectra of the turbulent signals reveal a −5/3 scaling when the shear-driven mechanism is prevalent in the flow and −11/5 scaling when the buoyancy-driven effects become prominent. The compressible enstrophy budget of the KHRTI shows that the onset process is dominated by vortex stretching or compressibility effects, followed by a sharp rise in baroclinic torque contribution once the buoyancy effects become relevant.

How does turbulence mix a stratified fluid?

A principal topic of interest and importance in stably stratified flows is how turbulence irreversibly mixes the ambient density field. Because the density field is coupled dynamically to the velocity field through the stable stratification, this mixing affects the overall flow dynamics, and its accurate parameterisation has become a ‘grand challenge’ in environmental fluid mechanics (Dauxois et al., Phys. Rev. Fluids, vol. 6, issue 2, 2021, 020501). In order to better understand the detailed kinematics of mixing in a stably stratified fluid, Jiang et al. (2022) perform experiments using a unique laboratory facility, capable of generating controllable stratified shear flows, and providing almost instantaneous density and three-component velocity measurements on a high-resolution, three-dimensional grid. Using three-dimensional data sets from the experiments, they employ the rortex–shear decomposition to identify the morphology of instantaneous rortices in the flow fields, leading to the interpretation of the motion of the rortices and ultimately to how the rortices cause irreversible mixing of the density field. This marks one of the first studies where, in a laboratory setting, full use has been made of somewhat high-resolution, three-dimensional near-instantaneous measurements; it demonstrates what can be accomplished in the laboratory, setting a new standard for future experiments.