The stratified inclined duct: what we have learned about stratified turbulence in a maintained shear flow

The potential for differential turbulent transport of oceanic temperature (T) and salinity (𝒮) is explored using three-dimensional direct numerical simulations of decaying stratified turbulence. The simulations employ a realistic molecular diffusion coefficient for T, and one for a “salt” scalar S that is 10 times smaller. Initially, a uniformly stratified medium is disturbed by a turbulent burst whose initial energy is assigned a range of values. In each instance, transports of T integrated over the subsequent decay of the burst exceed those of S. The more energetic cases occupy parameter ranges similar to, and exhibit spectral characteristics that are essentially indistinguishable from, those of direct observations of turbulence in the stratified ocean interior. In these cases, the turbulent diffusivity of T exceeds that of S by 6%–22%. These simulations underestimate the degree of differential diffusion between T and salinity 𝒮 (which has a molecular diffusivity 100 times less than T); thus at the Reynolds numbers attained by the simulations these results constitute lower bounds for differential diffusion associated with sporadic turbulence in the ocean. The simulation results are consistent with previous laboratory and two-dimensional numerical experiments and suggest that the assumption of equal turbulent diffusivities for T and 𝒮, commonly used in circulation modeling and in interpreting oceanic mixing measurements, should be reconsidered.

We consider numerically the transition to turbulence and associated mixing in stratified shear flows with initial velocity distribution $\overline{U}(z,0)\,\boldsymbol{e}_{x}=U_{0}\,\boldsymbol{e}_{x}\tanh (z/d)$ and initial density distribution $\overline{\unicode[STIX]{x1D70C}}(z,0)=\unicode[STIX]{x1D70C}_{0}[1-\tanh (z/\unicode[STIX]{x1D6FF})]$ away from a hydrostatic reference state $\unicode[STIX]{x1D70C}_{r}\gg \unicode[STIX]{x1D70C}_{0}$. When the ratio $R=d/\unicode[STIX]{x1D6FF}$ of the characteristic length scales over which the velocity and density vary is equal to one, this flow is primarily susceptible to the classic well-known Kelvin–Helmholtz instability (KHI). This instability, which typically manifests at finite amplitude as an array of elliptical vortices, strongly ‘overturns’ the density interface of strong initial gradient, which nevertheless is the location of minimum initial gradient Richardson number $Ri_{g}(0)=Ri_{b}=g\unicode[STIX]{x1D70C}_{0}d/\unicode[STIX]{x1D70C}_{r}U_{0}^{2}$, where $Ri_{g}(z)=-([g/\unicode[STIX]{x1D70C}_{r}]\,\text{d}\overline{\unicode[STIX]{x1D70C}}/\text{d}z)/(\text{d}\overline{U}/\text{d}z)^{2}$ and $Ri_{b}$ is a bulk Richardson number. As is well known, at sufficiently high Reynolds numbers ($Re$), the primary KHI induces a vigorous but inherently transient burst of turbulence and associated irreversible mixing localised in the vicinity of the density interface, leading to a relatively well-mixed region bounded by stronger density gradients above and below. We explore the qualitatively different behaviour that arises when $R\gg 1$, and so the density interface is sharp, with $Ri_{g}(z)$ now being maximum at the density interface $Ri_{g}(0)=RRi_{b}$. This flow is primarily susceptible to Holmboe wave instability (HWI) (Holmboe, Geophys. Publ., vol. 24, 1962, pp. 67–113), which manifests at finite amplitude in this symmetric flow as counter-propagating trains of elliptical vortices above and below the density interface, thus perturbing the interface so as to exhibit characteristically cusped interfacial waves which thereby ‘scour’ the density interface. Unlike previous lower-$Re$ experimental and numerical studies, when $Re$ is sufficiently high the primary HWI becomes increasingly more three-dimensional due to the emergence of shear-aligned secondary convective instabilities. As $Re$ increases, (i) the growth rate of secondary instabilities appears to saturate and (ii) the perturbation kinetic energy exhibits a $k^{-5/3}$ spectrum for sufficiently large length scales that are influenced by anisotropic buoyancy effects. Therefore, at sufficiently high $Re$, vigorous turbulence is triggered that also significantly ‘scours’ the primary density interface, leading to substantial irreversible mixing and vertical transport of mass above and below the (robust) primary density interface. Furthermore, HWI produces a markedly more long-lived turbulence event compared to KHI at a similarly high $Re$. Despite their vastly different mechanics (i.e. ‘overturning’ versus ‘scouring’) and localisation, the mixing induced by KHI and HWI is comparable in both absolute terms and relative efficiency. Our results establish that, provided the flow Reynolds number is sufficiently high, shear layers with sharp density interfaces and associated locally high values of the gradient Richardson number may still be sites of substantial and efficient irreversible mixing.

Understanding how turbulence enhances irreversible scalar mixing in density-stratified fluids is a central problem in geophysical fluid dynamics. While isotropic overturning regions are commonly the focus of mixing analyses, we here investigate whether significant mixing may arise in anisotropic statically stable regions of the flow. Focusing on a single forced direct numerical simulation of stratified turbulence, we analyse spatial correlations between the vertical density gradient $\partial \rho /\partial z$ and the dissipation rates of kinetic energy $\epsilon$ and scalar variance $\chi$ , the latter quantifying scalar mixing. The domain is characterized by relatively well-mixed density layers separated by sharp stable interfaces that are correlated with high vertical shear. While static instability is most prevalent within the mixed layers, much of the scalar mixing is localized to the intervening interfaces, a phenomenon not apparent if considering local static instability or $\epsilon$ alone. While the majority of the domain is characterized by the canonical flux coefficient $\varGamma \equiv \chi /\epsilon =0.2$ , often assumed in ocean mixing parametrizations, extreme values of $\chi$ within the statically stable interfaces, associated with elevated $\varGamma$ , strongly skew the bulk statistics. Our findings suggest that current parametrizations of turbulent mixing may be biased by undersampling, such that the most common, but not necessarily the most significant, mixing events are overweighted. Having focused here on a single simulation of stratified turbulence, it is hoped that our results motivate a broader investigation into the role played by stable density interfaces in mixing, across a wider range of parameters and forcing schemes representative of ocean turbulence.

We consider turbulence driven by a large-scale horizontal shear in Kolmogorov flow (i.e. with sinusoidal body forcing) and a background linear stable stratification with buoyancy frequency $N_{B}^{2}$ imposed in the third, vertical direction in a fluid with kinematic viscosity $\unicode[STIX]{x1D708}$. This flow is known to be organised into layers by nonlinear unstable steady states, which incline the background shear in the vertical and can be demonstrated to be the finite-amplitude saturation of a sequence of instabilities, originally from the laminar state. Here, we investigate the next order of motions in this system, i.e. the time-dependent mechanisms by which the density field is irreversibly mixed. This investigation is achieved using ‘recurrent flow analysis’. We identify (unstable) periodic orbits, which are embedded in the turbulent attractor, and use these orbits as proxies for the chaotic flow. We find that the time average of an appropriate measure of the ‘mixing efficiency’ of the flow $\mathscr{E}=\unicode[STIX]{x1D712}/(\unicode[STIX]{x1D712}+{\mathcal{D}})$ (where ${\mathcal{D}}$ is the volume-averaged kinetic energy dissipation rate and $\unicode[STIX]{x1D712}$ is the volume-averaged density variance dissipation rate) varies non-monotonically with the time-averaged buoyancy Reynolds numbers $\overline{Re}_{B}=\overline{{\mathcal{D}}}/(\unicode[STIX]{x1D708}N_{B}^{2})$, and is bounded above by $1/6$, consistently with the classical model of Osborn (J. Phys. Oceanogr., vol. 10 (1), 1980, pp. 83–89). There are qualitatively different physical properties between the unstable orbits that have lower irreversible mixing efficiency at low $\overline{Re}_{B}\sim O(1)$ and those with nearly optimal $\mathscr{E}\lesssim 1/6$ at intermediate $\overline{Re}_{B}\sim 10$. The weaker orbits, inevitably embedded in more strongly stratified flow, are characterised by straining or ‘scouring’ motions, while the more efficient orbits have clear overturning dynamics in more weakly stratified, and apparently shear-unstable flow.

ABSTRACTDetailed measurements of flow velocity and its turbulent fluctuation were obtained over fixed, two‐dimensional dunes in a laboratory channel. Laser Doppler anemometry was used to measure the downstream and vertical components of velocity at more than 1800 points over one dune wavelength. The density of the sampling grid allowed construction of a unique set of contour maps for all mean flow and turbulence parameters, which are assessed using higher moment measures and quadrant analysis. These flow field maps illustrate that: (1) the time‐averaged downstream and vertical velocities agree well with previous studies of quasi‐equilibrium flow over fixed and mobile bedforms and show a remarkable symmetry from crest to crest; (2) the maximum root‐mean‐square (RMS) of the downstream velocity values occur at and just downstream of flow reattachment and within the flow separation cell; (3) the maximum vertical RMS values occur within and above the zone of flow separation along the shear layer and this zone advects and diffuses downstream, extending almost to the next crest; (4) positive downstream skewness values occur within the separation cell, whereas positive vertical skewness values are restricted to the shear layer; (5) the highest Reynolds stresses are located within the zone of flow separation and along the shear layer; (6) high‐magnitude, high‐frequency quadrant‐2 events (‘ejections’) are concentrated along the shear layer (Kelvin‐Helmholtz instabilities) and dominate the contribution to the local Reynolds stress; and (7) high‐magnitude, high‐frequency quadrant‐4 events occur bounding the separation zone, near reattachment and close to the dune crest, and are significant contributors to the local Reynolds stress at each location. These data demonstrate that the turbulence structure associated with dunes is controlled intrinsically by the formation, magnitude and downstream extent of the flow separation zone and resultant shear layer. Furthermore, the origin of dune‐related macroturbulence lies in the dynamics of the shear layer rather than classical turbulent boundary layer bursting. The fluid dynamic distinction between dunes and ripples is reasoned to be linked to the velocity differential across the shear layer and hence the magnitude of the Kelvin‐Helmholtz instabilities, which are both greater for dunes than ripples. These instabilities control the local flow and turbulence structure and dictate the modes of sediment entrainment and their transport rates.

Vortical structures and turbulent bursts behind magnetic obstacles in transitional flow regimes

Abstract. Shear flow in a stable stratification provides a waveguide for internal gravity waves. In the inviscid approximation, internal gravity waves are known to be unstable below a threshold in Richardson number. However, in a viscous fluid, at low enough Reynolds number, this threshold recedes to Ri = 0. Nevertheless, even the slightest viscosity strongly damps internal gravity waves when the Richardson number is small (shear forces dominate buoyant forces). In this paper we address the dynamics that approximately govern wave propagation when the Richardson number is small and the fluid is viscous. When Ri << 1, to a first approximation, the transport equations for thermal energy and momentum decouple. Thus, a large amplitude temperature wave then has little effect on the fluid velocity. Under such conditions in the atmosphere, a small amplitude "turbulent burst" is observed, transporting momentum rapidly and seemingly randomly. A regular perturbation scheme from a base state of a passing temperature wave and no velocity disturbance is developed here. Small thermal energy convection-momentum transport coupling is taken into account. The elements of forcing, wave dispersion, (turbulent) dissipation under strong shearing, and weak nonlinearity lead to this dynamical equation for the amplitude A of the turbulent burst in velocity: Aξ = λ1A + λ2Aξξ + λ3Aξξξ + λ4AAξ + b(ξ) where ξ is the coordinate of the rest frame of the passing temperature wave whose horizontal profile is b(ξ). The parameters λi are constants that depend on the Reynolds number. The above dynamical system is know to have limit cycle and chaotic attrators when forcing is sinusoidal and wave attenuation negligible.

For the special hydrodynamic situation of unbounded homogeneous shear layers, turbulence closure models of Mellor‐Yamada type (MY) and k‐ε type are put into a single canonical form. For this situation we show that conventional versions of MY and various k‐ε versions lack a proper steady state, and are unable to simulate the most basic properties of stratified shear flows exemplified in, for example, the Rohr et al. [1988] experiments: exponential growth at sufficiently low gradient Richardson number (Rg), exponential decay at sufficiently large Rg, and a steady state in between. Proper choice of one special model parameter readily solves the problems. In the fairly general case of structural equilibrium (state of exponential evolution) in weakly to moderately stratified turbulence (Rg ≲ 0.25), the ratio between the Thorpe scale (or Ellison scale) and the Ozmidov scale varies like the gradient Richardson number (Rg) to the power 3/4, and the ratio of the Thorpe scale to the buoyancy scale varies like Rg1/2. Length scales predicted by our current model are consistent with laboratory measurements of Rohr et al. [1988], with large‐eddy numerical simulations of Schumann and Gerz [1995], and with microstructure measurements from the 1987 Tropic Heat Experiment in the equatorial Pacific by Peters et al. [1995].

The geometry of the experiment proposed as a benchmark is a glass channel of rectangular cross section that contains two immiscible fluid of different densities. The channel is suddenly tilted at a small angle from the horizontal and a Kelvin-Helmholtz instability develops under certain conditions. It is suggested to compare the numerical simulation with the experimental values of the most unstable wave number, the starting time and the growth rate of the instability, and the wave speed.

The entrapped wedge is a specific type of stratified flow problem that is found in estuaries when a barrage is used to control water levels. To avoid problems with water quality, it is essential that any wedge be removed. The sensible mechanism by which this should occur is via the momentum of the freshwater flow. To investigate this, a laboratory investigation of the erosion of an entrapped saline wedge caused by a buoyant overflow has been undertaken. The shear flow has been found to induce mixing at the interface between the two fluids, and a mixed layer results. The characteristics of the layer have been found to change with time with two different mechanisms being identified. The behavior of the layer depends on the test conditions, in that certain combinations of velocity, depth, and density difference yield different mechanisms of erosion. The temporal behavior of the interfacial layer with various freshwater velocities and depths and density differences are presented.

This chapter presents and discusses aero-optical distortions due to typical turbulent flows, like shear layers, boundary layers, and flows around commonly used side-mounted turrets. First, it introduces useful scaling laws, which allows re-scaling of aero-optical distortions to different flow regimes and speeds. Physical mechanisms and semi-empirical models of aero-optical distortions for both shear layers and boundary layers are presented and discussed. Effect of the heated/cooled wall on the aero-optical distortions of turbulent boundary layers is discussed. Instantaneous far-field intensity drop-outs, caused by boundary layers are extensively discussed. Aero-optical distortions caused by the flow around turrets at subsonic and transonic speeds, including data collected in flight, are presented. Finally, aero-optical effects of the local shock formed on top of the turret at high transonic speeds are analyzed.

Flows in square channels are common in applications, such as automotive after-treatment systems and heat exchangers. Flows with axial flow entry are well understood, but for oblique flow entry, there is no clarity on the additional pressure loss magnitude or the flow regime. Laminar flow is often assumed, even though flow separation at the channel entrance can cause a transition to turbulence. Here, the impact of oblique flow entry on the flow is investigated using LES (large eddy simulation) and RANS (Reynolds averaged Navier–Stokes) models, and their advantages and limitations are identified. The LES simulations show that the shear layer at the channel entrance produces continuous shedding of eddies that persist downstream even at moderate channel Reynolds numbers (≈ 2000). The LES predicted pressure losses mostly agree with experimental data, and the differences observed are attributed to the difficulty of accurately replicating the experimental geometry. It is shown that both LES and RANS results are sensitive to the rounding of the leading edge (present in experiments). Including edge rounding improves the pressure predictions. RANS simulations mostly agree with experimens, but unlike LES did not predict transitional flow phenomena for sharp leading edge. This study provides insight into the flow structure and sources of pressure losses in square channels, highlights the importance of understanding key flow and geometric features when using LES to predict complex flows involving flow separation and shear layers, and indicates the need to further investigate the complex instability arising from interactions between secondary flow and shear-layer roll-up.

Acoustic metasurfaces (AMs) are artificial structures possessing exotic acoustic properties, such as negative refraction and perfect absorption, etc., in sub-wavelength thickness scales. Most previous studies on AMs were conducted in laboratory conditions with a static medium, limiting their applications in practical conditions, e.g., the aircraft engines where an aerodynamic background flow is present. To fill this gap, we study the effects of a grazing shear flow on the reflection performance of periodic AMs in this work. An AMs design method is proposed to realize effective wavefront manipulation with the consideration of the shear flow. The strategy is to form a linear reflected phase-shifting covering 0 to 2π above the shear layer. The reflection characteristics of two acoustic porous metasurfaces (APMs), i.e., APM1 and APM2 designed for the uniform and the sheared flows, respectively, are numerically investigated. Results show that the presence of the shear layer reduces the effectiveness of APM1 under oblique incident waves due to the considerable refraction effect. In contrast, under the same flow conditions, APM2 realizes desired wave manipulation behaviors, such as anomalous reflection and surface wave conversion. This study provides guidelines for AMs design in flow conditions, which offers potential for noise control in aeroacoustic problems.

Based on the budget of kinetic energy and simple estimates to relate dissipation and temperature or concentration fluctuations to shear, stratification, and the vertical velocity fluctuations, a consistent set of equations is deduced to estimate vertical fluxes of momentum and heat or mass. The estimates are designed for strongly sheared, neutral and stratified flows at high Reynolds numbers under approximately homogeneous conditions. The set is closed by using basically two empirical coefficients together with the turbulent Prandtl number and the growth rate of kinetic energy as a function of the gradient Richardson number. The correlations are tested using data from previous laboratory experiments and numerical simulations.

High-resolution observations of velocity, salinity, and turbulence quantities were collected in a salt wedge estuary to quantify the efficiency of stratified mixing in a high-energy environment. During the ebb tide, a midwater column layer of strong shear and stratification developed, exhibiting near-critical gradient Richardson numbers and turbulent kinetic energy (TKE) dissipation rates greater than 10−4 m2 s−3, based on inertial subrange spectra. Collocated estimates of scalar variance dissipation from microconductivity sensors were used to estimate buoyancy flux and the flux Richardson number Rif. The majority of the samples were outside the boundary layer, based on the ratio of Ozmidov and boundary length scales, and had a mean Rif = 0.23 ± 0.01 (dissipation flux coefficient Γ = 0.30 ± 0.02) and a median gradient Richardson number Rig = 0.25. The boundary-influenced subset of the data had decreased efficiency, with Rif = 0.17 ± 0.02 (Γ = 0.20 ± 0.03) and median Rig = 0.16. The relationship between Rif and Rig was consistent with a turbulent Prandtl number of 1. Acoustic backscatter imagery revealed coherent braids in the mixing layer during the early ebb and a transition to more homogeneous turbulence in the midebb. A temporal trend in efficiency was also visible, with higher efficiency in the early ebb and lower efficiency in the late ebb when the bottom boundary layer had greater influence on the flow. These findings show that mixing efficiency of turbulence in a continuously forced, energetic, free shear layer can be significantly greater than the broadly cited upper bound from Osborn of 0.15–0.17.