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When an acoustic wave propagates through a viscous fluid, it progressively transfers momentum to the fluid through viscous dissipation, which results in the formation of a steady vortical flow called acoustic streaming. Although spawned by viscous effects, the magnitude of the streaming does not depend on the viscosity in most simple geometries. However, viscosity has a profound influence on the acoustic streaming as demonstrated by Riaud et al. (J. Fluid Mech., vol. 821, 2017, pp. 384–420) in their study of sessile mm-sized water–glycerol droplets placed on a piezoelectric substrate with a 20-MHz ultrasound surface acoustic wave propagating along its surface. A detailed experimental and numerical analysis reveals that streaming dynamics is driven by a few ultrasound ray caustics inside the droplet.

Direct numerical simulation of flow past a sphere in a stratified fluid is carried out at a subcritical Reynolds number of 3700 and $Fr=U_{\infty }/ND=1,2$ and 3 to understand the dynamics of moderately stratified flows with $Fr=O(1)$ . Here, $U_{\infty }$ is the free stream velocity, $N$ is the background buoyancy frequency and $D$ is the sphere diameter. The unstratified flow past the sphere consists of a separated shear layer that transitions to turbulence, a recirculation zone and a wake with a mean centreline deficit velocity, $U_{0}$ , that decreases with downstream distance as a power law. With increasing stratification, the separated shear layer plunges inward vertically and its roll up is inhibited, the recirculation zone is shortened and the mean wake decays at a slower rate of $U_{0}\propto (x_{1}/D)^{-0.25}$ in the non-equilibrium (NEQ) region. The transition from the near wake where $U_{0}$ has a decay rate similar to the unstratified case to the NEQ regime occurs as an oscillatory modulation by a steady lee wave pattern with a period of $t=2\unicode[STIX]{x03C0}/N$ that leads to accelerated $U_{0}$ between $Nt=\unicode[STIX]{x03C0}$ and approximately $Nt=2\unicode[STIX]{x03C0}$ . Far downstream, the wake is dominated by coherent horizontal motions. The acceleration of $U_{0}$ by the lee wave and the lower turbulence production in the NEQ regime, thereby less loss to turbulence, prolongs the lifetime of the wake relative to its unstratified counterpart. The intensity, temporal spectra and structure of turbulent fluctuations in the wake are assessed. Buoyancy induces significant anisotropy among the velocity components and between their vertical and horizontal profiles. Consequently, the near wake ( $x_{1}/D<10$ ) exhibits significant differences in turbulence profiles relative to its unstratified counterpart. Spectra of vertical velocity show a discrete peak in the near wake that is maintained further downstream. The turbulent kinetic energy (TKE) balance is computed and contributions from pressure transport and buoyancy are found to become increasingly important as stratification increases. The findings of this investigation will be helpful in designing accurate initial conditions for the temporally evolving model of stratified wakes.

This paper presents the characteristics of the different stages in the evolution of the wake of a circular cylinder rolling without slipping along a wall at constant speed, acquired through numerical stability analysis and two- and three-dimensional numerical simulations. Reynolds numbers between 30 and 300 are considered. Of importance in this study is the transition to three-dimensionality from the underlying two-dimensional periodic flow and, in particular, the way that the associated transitions influence the fluid forces exerted on the cylinder and the development and the structure of the wake. It is found that the steady two-dimensional flow becomes unstable to three-dimensional perturbations at $Re_{c,3D}=37$ , and that the transition to unsteady two-dimensional flow – or periodic vortex shedding – occurs at $Re_{c,2D}=88$ , thus validating and refining the results of Stewart et al. (J. Fluid Mech. vol. 648, 2010, pp. 225–256). The main focus here is on Reynolds numbers beyond the transition to unsteady flow at $Re_{c,2D}=88$ . From impulsive start up, the wake almost immediately undergoes transition to a periodic two-dimensional wake state, which, in turn, is three-dimensionally unstable. Thus, the previous three-dimensional stability analysis based on the two-dimensional steady flow provides only an element of the full story. Floquet analysis based on the periodic two-dimensional flow was undertaken and new three-dimensional instability modes were revealed. The results suggest that an impulsively started cylinder rolling along a surface at constant velocity for $Re\gtrsim 90$ will result in the rapid development of a periodic two-dimensional wake that will be maintained for a considerable time prior to the wake undergoing three-dimensional transition. Of interest, the mean lift and drag coefficients obtained from full three-dimensional simulations match predictions from two-dimensional simulations to within a few per cent.

The effect of free stream coherent structures in the asymptotic suction boundary layer on the initiation of Görtler vortices is considered from both the ‘imperfect’ bifurcation and receptivity viewpoints. Firstly a weakly nonlinear and a full numerical approach are used to describe Görtler vortices in the asymptotic suction boundary layer in the absence of forcing from the free stream. It is found that interactions between different spanwise harmonics occur and lead to multiple secondary bifurcations in the fully nonlinear regime. Furthermore it is shown that centrifugal instabilities of the asymptotic suction boundary layer behave quite differently than their counterparts in either fully developed flows such as Couette flow or growing boundary layers. A significant result is that the most dangerous disturbance is found to bifurcate subcritically from the unperturbed state. Within the weakly nonlinear regime the receptivity of Görtler vortices to the free stream exact coherent structures discovered by Deguchi & Hall (J. Fluid Mech., vol. 752, 2014, pp. 602–625; J. Fluid Mech., vol. 778, 2015, pp. 451–484) is considered. The presence of free stream structures results in a resonant excitation of Görtler vortices in the main boundary layer. This leads to imperfect bifurcations reminiscent of those found by Daniels (Proc. R. Soc. Lond. A, vol. 358, 1977, pp. 173–197) and Hall & Walton (Proc. R. Soc. Lond. A, vol. 358, 1977, pp. 199–221; J. Fluid Mech., vol. 90, 1979, pp. 377–395) in the context of transition to finite amplitude Bénard convection in a bounded region. In order to understand the receptivity problem for the given flow the spatial initial value problem for this interaction is also considered when the free stream structure begins at a fixed position along the wall. Remarkably, it will be shown that free stream structures are incredibly efficient generators of Görtler vortices; indeed the induced vortices are found to be larger than the free stream structure which provokes them! The relationship between the imperfect bifurcation approach and receptivity theory is described.

We present a fully predictive model for the impact of a smooth, convex and perfectly hydrophobic solid onto the free surface of an incompressible fluid bath of infinite depth in a regime where surface tension is important. During impact, we impose natural kinematic constraints along the portion of the fluid interface that is pressed by the solid. This provides a mechanism for the generation of linear surface waves and simultaneously yields the pressure applied on the impacting masses. The model compares remarkably well with data of the impact of spheres and bouncing droplet experiments, and is completely free of any of impact parametrisation.

The stability of two-layer Couette flow is investigated under variations in viscosity ratio, thickness ratio, interfacial tension and density ratio. The effects of the base flow on eigenvalue spectra are explained. A new type of interfacial mode is discovered at low viscosity ratio with properties that are different from Yih’s original interfacial mode (Yih, J. Fluid Mech., vol. 27, 1967, pp. 337–352). No unstable Tollmien–Schlichting waves were found over the range of parameters considered in this work. The results for thin films with different thicknesses can be collapsed onto a single curve if the Reynolds number and wavenumber are suitably defined based on the parameters of the thin layer. Interfacial tension always has a stabilizing effect, but the effects of density ratio cannot be so easily generalized. Neutral stability curves for water–alkane and water–air systems are presented as an initial step towards better understanding the effects of flow stability on the longevity and performance of liquid-infused surfaces and superhydrophobic surfaces.

A horizontal channel flow of two immiscible fluid layers with different densities, viscosities and thicknesses, subject to vertical gravitational forces and with an insoluble surfactant monolayer present at the interface, is investigated. The base Couette flow is driven by the uniform horizontal motion of the channel walls. Linear and nonlinear stages of the (inertialess) surfactant and gravity dependent long-wave instability are studied using the lubrication approximation, which leads to a system of coupled nonlinear evolution equations for the interface and surfactant disturbances. The (inertialess) instability is a combined result of the surfactant action characterized by the Marangoni number $Ma$ and the gravitational effect corresponding to the Bond number $Bo$ that ranges from $-\infty$ to $\infty$ . The other parameters are the top-to-bottom thickness ratio $n$ , which is restricted to $n\geqslant 1$ by a reference frame choice, the top-to-bottom viscosity ratio $m$ and the base shear rate $s$ . The linear stability is determined by an eigenvalue problem for the normal modes, where the complex eigenvalues (determining growth rates and phase velocities) and eigenfunctions (the amplitudes of disturbances of the interface, surfactant, velocities and pressures) are found analytically by using the smallness of the wavenumber. For each wavenumber, there are two active normal modes, called the surfactant and the robust modes. The robust mode is unstable when $Bo/Ma$ falls below a certain value dependent on $m$ and $n$ . The surfactant branch has instability for $m<1$ , and any $Bo$ , although the range of unstable wavenumbers decreases as the stabilizing effect of gravity represented by $Bo$ increases. Thus, for certain parametric ranges, even arbitrarily strong gravity cannot completely stabilize the flow. The correlations of vorticity-thickness phase differences with instability, present when gravitational effects are neglected, are found to break down when gravity is important. The physical mechanisms of instability for the two modes are explained with vorticity playing no role in them. This is in marked contrast to the dynamical role of vorticity in the mechanism of the well-known Yih instability due to effects of inertia, and is contrary to some earlier literature. Unlike the semi-infinite case that we previously studied, a small-amplitude saturation of the surfactant instability is possible in the absence of gravity. For certain $(m,n)$ -ranges, the interface deflection is governed by a decoupled Kuramoto–Sivashinsky equation, which provides a source term for a linear convection–diffusion equation governing the surfactant concentration. When the diffusion term is negligible, this surfactant equation has an analytic solution which is consistent with the full numerics. Just like the interface, the surfactant wave is chaotic, but the ratio of the two waves turns out to be constant.

An analytical model is developed for the prediction of noise radiated by an aerofoil with leading-edge serration in a subsonic turbulent stream. The model makes use of Fourier expansion and Schwarzschild techniques in order to solve a set of coupled differential equations iteratively and express the far-field sound power spectral density in terms of the statistics of incoming turbulent upwash velocity. The model has shown that the primary noise-reduction mechanism is due to the destructive interference of the scattered pressure induced by the leading-edge serrations. It has also shown that in order to achieve significant sound reduction, the serration must satisfy two geometrical criteria related to the serration sharpness and hydrodynamic properties of the turbulence. A parametric study has been carried out and it is shown that serrations can reduce the overall sound pressure level at most radiation angles, particularly at small aft angles. The sound directivity results have also shown that the use of leading-edge serration does not significantly change the dipolar pattern of the far-field noise at low frequencies, but it changes the cardioid directivity pattern associated with radiation from straight-edge scattering at high frequencies to a tilted dipolar pattern.

The steady laminar flow of a moderately inertial wall jet is examined theoretically near the exit of a channel. The free-surface jet emerges asymmetrically from the channel as it adheres to an infinite (upper) wall subject to a pressure gradient. The problem is solved using the method of matched asymptotic expansions. The small parameter involved in the expansions is the inverse cubic power of the Reynolds number. The flow field is obtained by matching the inviscid rotational core flow separately with the free-surface and the two wall layers. The upstream influence is examined as well as the break in the symmetry between the two wall layers. The wall jet exhibits a contraction near the channel exit that is independent of inertia, and eventually expands for any Reynolds number. Unlike the flow of a wall jet emerging into the same ambient fluid, the free-surface jet experiences a limited weakening in shear stress along the infinite wall, suggesting the possibility of separation for a jet with relatively low inertia. Significant shearing and elongation ensue at the exit, accompanied by flattening of the velocity profile near the upper wall.

The stability characteristics of compressible spanwise-periodic open-cavity flows are investigated with direct numerical simulation (DNS) and biglobal stability analysis for rectangular cavities with aspect ratios of $L/D=2$ and 6. This study examines the behaviour of instabilities with respect to stable and unstable steady states in the laminar regime for subsonic as well as transonic conditions where compressibility plays an important role. It is observed that an increase in Mach number destabilizes the flow in the subsonic regime and stabilizes the flow in the transonic regime. Biglobal stability analysis for spanwise-periodic flows over rectangular cavities with large aspect ratio is closely examined in this study due to its importance in aerodynamic applications. Moreover, biglobal stability analysis is conducted to extract two-dimensional (2-D) and 3-D eigenmodes for prescribed spanwise wavelengths $\unicode[STIX]{x1D706}/D$ about the 2-D steady state. The properties of 2-D eigenmodes agree well with those observed in the 2-D nonlinear simulations. In the analysis of 3-D eigenmodes, it is found that an increase of Mach number stabilizes dominant 3-D eigenmodes. For a short cavity with $L/D=2$ , the 3-D eigenmodes primarily stem from centrifugal instabilities. For a long cavity with $L/D=6$ , other types of eigenmodes appear whose structures extend from the aft-region to the mid-region of the cavity, in addition to the centrifugal instability mode located in the rear part of the cavity. A selected number of 3-D DNS are performed at $M_{\infty }=0.6$ for cavities with $L/D=2$ and 6. For $L/D=2$ , the properties of 3-D structures present in the 3-D nonlinear flow correspond closely to those obtained from linear stability analysis. However, for $L/D=6$ , the 3-D eigenmodes cannot be clearly observed in the 3-D DNS due to the strong nonlinearity that develops over the length of the cavity. In addition, it is noted that three-dimensionality in the flow helps alleviate violent oscillations for the long cavity. The analysis performed in this paper can provide valuable insights for designing effective flow control strategies to suppress undesirable aerodynamic and pressure fluctuations in compressible open-cavity flows.

The structure of the incompressible steady three-dimensional flow in a two-sided anti-symmetrically lid-driven cavity is investigated for an aspect ratio $\unicode[STIX]{x1D6E4}=1.7$ and spanwise-periodic boundary conditions. Flow fields are computed by solving the Navier–Stokes equations with a fully spectral method on $128^{3}$ grid points utilizing second-order asymptotic solutions near the singular corners. The supercritical flow arises in the form of steady rectangular convection cells within which the flow is point symmetric with respect to the cell centre. Global streamline chaos occupying the whole domain is found immediately above the threshold to three-dimensional flow. Beyond a certain Reynolds number the chaotic sea recedes from the interior, giving way to regular islands. The regular Kolmogorov–Arnold–Moser tori grow with increasing Reynolds number before they shrink again to eventually vanish completely. The global chaos at onset is traced back to the existence of one hyperbolic and two elliptic periodic lines in the basic flow. The singular points of the three-dimensional flow which emerge from the periodic lines quickly change such that, for a wide range of supercritical Reynolds number, each periodic convection cell houses a double spiralling-in saddle focus in its centre, a spiralling-out saddle focus on each of the two cell boundaries and two types of saddle limit cycle on the walls. A representative analysis for $\mathit{Re}=500$ shows chaotic streamlines to be due to chaotic tangling of the two-dimensional stable manifold of the central spiralling-in saddle focus and the two-dimensional unstable manifold of the central wall limit cycle. Embedded Kolmogorov–Arnold–Moser tori and the associated closed streamlines are computed for several supercritical Reynolds numbers owing to their importance for particle transport.

The flow induced by the combined torsional and transverse oscillations of a sphere with amplitude ratio $\unicode[STIX]{x1D6FC}$ and phase difference $\unicode[STIX]{x1D6FD}$ in a concentric spherical container is examined. Analytical solutions of the leading-order flow field and shear stress profiles have been obtained. Steady streaming flows are also analysed not only for the case of unrestricted Womersley number $|M|$ , but also in the low-frequency $(|M|\ll 1)$ and high-frequency ( $|M|\gg 1$ ) limits. At high frequency, the flow field has been divided into three regions: two boundary layers and the outer region. The streaming flow field is determined for the limiting case of the streaming Reynolds number $R_{s}\ll 1$ . The results are compared with those of single torsional or transverse oscillation, and found to match very well. The amplitude ratio $\unicode[STIX]{x1D6FC}$ and phase difference $\unicode[STIX]{x1D6FD}$ , in determining the streaming, are also discussed. The number and direction of steady streaming recirculation on the $r$ – $\unicode[STIX]{x1D703}$ plane depend on value of the amplitude ratio $\unicode[STIX]{x1D6FC}$ . The phase difference $\unicode[STIX]{x1D6FD}$ plays a dominant role in the azimuthal velocity $u_{1\unicode[STIX]{x1D719}}^{(s)}$ of steady streaming. When $\unicode[STIX]{x1D6FD}$ is approximately $(2n+1)\unicode[STIX]{x03C0}/2$ , $u_{1\unicode[STIX]{x1D719}}^{(s)}$ vanishes under low-frequency oscillation, while steady streaming has a recirculation on the $r$ – $\unicode[STIX]{x1D719}$ plane under higher-frequency oscillation.

The distinctive pitching of hinged splitters in the trailing edge of elliptic cylinders was experimentally studied at various angles of attack ( $AoA$ ) of the cylinder, Reynolds numbers, splitter lengths, aspect ratios ( $AR$ ) of the cylinder and freestream turbulence levels. High-resolution telemetry and hotwire anemometry were used to characterize and gain insight on the dynamics of splitters and wake flow. Results show that the motions of the splitters contain various dominating modes, e.g. $f_{p}$ and $f_{v}$ , which are induced by the mean flow and wake dynamics. High background turbulence dampens the coherence of the regular vortex shedding leading to negligible $f_{v}$ . For a sufficiently long splitter, namely twice the semimajor axis of the cylinder, dual vortex shedding mode exists close to the leading and trailing edges of the splitter. In general, the splitters oscillate around an equilibrium position nearly parallel to the mean direction of the flow; however, a skewed equilibrium is also possible with a strong recirculation region. This is the case with cylinders of low $AR$ and high $AoA$ , where higher lift and drag occurs. Flow measurements at various transverse locations within the wake of the cylinder–splitter system indicate that the signature of the low-frequency splitter pitching is shifted in the wake in the cases with non-zero $AoA$ of the cylinder. Although the splitter pitching exhibits two dominant vortex shedding modes in various configurations, only the higher frequency is transmitted to the wake.

This paper performs a linear stability analysis to investigate the stability of plane Poiseuille–Couette flow in a fluid layer overlying a porous medium saturated with the same fluid. The effect of superimposed Couette flow on the associated Poiseuille flow in such a two-layer system is explored carefully. The result shows that the presence of Couette flow may destabilize the Poiseuille flow at small depth ratio $\hat{d}$ , defined by the ratio of the depth of the fluid layer to the depth of the porous layer, and induce a tri-modal structure to the neutral curves. At moderate $\hat{d}$ , the Couette component generally produces a stabilization effect on the flow. When the velocity of the upper moving wall is large enough, a bi-modal behaviour of the neutral curves appears and a shift of instability mode occurs from the long-wave fluid-layer mode to the porous-layer mode with higher wavenumber. These stability characteristics are remarkably different from those of the plane Poiseuille–Couette flow in a single fluid layer in that the flow becomes absolutely stable when the wall velocity is over 70 % of the maximum velocity of the Poiseuille component of flow. The stability of pure Couette flow in such a two-layer system is also studied. It is found that the flow is still absolutely stable with respect to infinitesimal disturbances, which is the same as the stability characteristic of a single-layer plane Couette flow.

This work introduces a similarity solution to the problem of a viscous, incompressible and rotational fluid in a right-cylindrical chamber with uniformly porous walls and a non-circular cross-section. The attendant idealization may be used to model the non-reactive internal flow field of a solid rocket motor with a star-shaped grain configuration. By mapping the radial domain to a circular pipe flow, the Navier–Stokes equations are converted to a fourth-order differential equation that is reminiscent of Berman’s classic expression. Then assuming a small radial deviation from a fixed chamber radius, asymptotic expansions of the three-component velocity and pressure fields are systematically pursued to the second order in the radial deviation amplitude. This enables us to derive a set of ordinary differential relations that can be readily solved for the mean flow variables. In the process of characterizing the ensuing flow motion, the axial, radial and tangential velocities are compared and shown to agree favourably with the simulation results of a finite-volume Navier–Stokes solver at different cross-flow Reynolds numbers, deviation amplitudes and circular wavenumbers.

Flat plate turbulent boundary layers under zero pressure gradient are simulated using synthetic turbulence generated by the fast random particle–mesh method. The stochastic realisation is based on time-averaged turbulence statistics derived from Reynolds-averaged Navier–Stokes simulation of flat plate turbulent boundary layers at Reynolds numbers $\mathit{Re}_{\unicode[STIX]{x1D70F}}=2513$ and $\mathit{Re}_{\unicode[STIX]{x1D70F}}=4357$ . To determine fluctuating pressure, a Poisson equation is solved with an unsteady right-hand side source term derived from the synthetic turbulence realisation. The Poisson equation is solved via fast Fourier transform using Hockney’s method. Due to its efficiency, the applied procedure enables us to study, for high Reynolds number flow, the effect of variations of the modelled turbulence characteristics on the resulting wall pressure spectrum. The contributions to wall pressure fluctuations from the mean-shear turbulence interaction term and the turbulence–turbulence interaction term are studied separately. The results show that both contributions have the same order of magnitude. Simulated one-point spectra and two-point cross-correlations of wall pressure fluctuations are analysed in detail. Convective features of the fluctuating pressure field are well determined. Good agreement for the characteristics of the wall pressure fluctuations is found between the present results and databases from other investigators.

Thinning and rupture of a thin film of a power-law fluid on a solid substrate under the balance between destabilizing van der Waals pressure and stabilizing capillary pressure is analysed. In a power-law fluid, viscosity is not constant but is proportional to the deformation rate raised to the $n-1$ power, where $0 is the power-law exponent ( $n=1$ for a Newtonian fluid). In the first part of the paper, use is made of the slenderness of the film and the lubrication approximation is applied to the equations of motion to derive a spatially one-dimensional nonlinear evolution equation for film thickness. The variation with time remaining until rupture of the film thickness, the lateral length scale, fluid velocity and viscosity is determined analytically and confirmed by numerical simulations for both line rupture and point rupture. The self-similarity of the numerically computed film profiles in the vicinity of the location where the film thickness is a minimum is demonstrated by rescaling of the transient profiles with the scales deduced from theory. It is then shown that, in contrast to films of Newtonian fluids undergoing rupture for which inertia is always negligible, inertia can become important during thinning of films of power-law fluids in certain situations. The critical conditions for which inertia becomes important and the lubrication approximation is no longer valid are determined analytically. In the second part of the paper, thinning and rupture of thin films of power-law fluids in situations when inertia is important are simulated by solving numerically the spatially two-dimensional, transient Cauchy momentum and continuity equations. It is shown that as such films continue to thin, a change of scaling occurs from a regime in which van der Waals, capillary and viscous forces are important to one where the dominant balance of forces is between van der Waals, capillary and inertial forces while viscous force is negligible.

Flows induced by centrifugal buoyancy occur in rotating systems in which the centrifugal force is large when compared to other body forces and are of interest for geophysicists and also in engineering problems involving rapid rotation and unstable temperature gradients. In this numerical study we analyse the onset of centrifugal buoyancy in a rotating cylindrical cavity bounded by two plane, insulated disks, adopting a geometrical configuration relevant to fundamental studies of buoyancy-induced flows occurring in gas turbine’s internal air systems. Using linear stability analysis, we obtain critical values of the centrifugal Rayleigh number $Ra$ and corresponding critical azimuthal wavenumbers for the onset of convection for different radius ratios. Using direct numerical simulation, we integrate the solutions starting from a motionless state to which small sinusoidal perturbations are added, and show that nonlinear triadic interactions occur before energy saturation takes place. At the lowest Rayleigh number considered, the final state is a limit-cycle oscillation affected by the presence of the disks, having a spectrum dominated by a certain mode and its harmonics. We show that, for this case, the limit-cycle oscillations only develop when no-slip end walls are present. For the largest $Ra$ considered chaotic motion occurs, but the critical wavenumber obtained from the linear analysis eventually becomes the most energetic even in the turbulent regime.

The flow over an open cavity is an example of supercritical Hopf bifurcation leading to periodic limit-cycle oscillations. One of its distinctive features is the existence of strong higher harmonics, which results in the time-averaged mean flow being strongly linearly unstable. For this class of flows, a simplified formalism capable of unravelling how exactly the instability grows and saturates is lacking. This study builds on previous work by Mantič-Lugo et al. (Phys. Rev. Lett., vol. 113, 2014, 084501) to fill in the gap using a parametrized approximation of an instantaneous, phase-averaged mean flow, coupled in a quasi-static manner to multiple linear harmonic disturbances interacting nonlinearly with one another and feeding back on the mean flow via their Reynolds stresses. This provides a self-consistent modelling of the mean flow–fluctuation interaction, in the sense that all perturbation structures are those whose Reynolds stresses force the mean flow in such a way that the mean flow generates exactly the aforementioned perturbations. The first harmonic is sought as the superposition of two components, a linear component generated by the instability and aligned along the leading eigenmode of the mean flow, and a nonlinear orthogonal component generated by the higher harmonics, which progressively distorts the linear growth rate and eigenfrequency of the eigenmode. Saturation occurs when the growth rate of the first harmonic is zero, at which point the stabilizing effect of the second harmonic balances exactly the linear instability of the eigenmode. The model does not require any input from numerical or experimental data, and accurately predicts the transient development and the saturation of the instability, as established from comparison to time and phase averages of direct numerical simulation data.

Turbulent mixing plays a major role in enabling the large-scale ocean circulation. The accuracy of mixing rates estimated from observations depends on our understanding of basic fluid mechanical processes underlying the nature of turbulence in a stratified fluid. Several of the key assumptions made in conventional mixing parameterizations have been increasingly scrutinized in recent years, primarily on the basis of adequately high resolution numerical simulations. We add to this evidence by compiling results from a suite of numerical simulations of the turbulence generated through stratified shear instability processes. We study the inherently intermittent and time-dependent nature of wave-induced turbulent life cycles and more specifically the tight coupling between inherently anisotropic scales upon which small-scale isotropic turbulence grows. The anisotropic scales stir and stretch fluid filaments enhancing irreversible diffusive mixing at smaller scales. We show that the characteristics of turbulent mixing depend on the relative time evolution of the Ozmidov length scale $L_{O}$ compared to the so-called Thorpe overturning scale $L_{T}$ which represents the scale containing available potential energy upon which turbulence feeds and grows. We find that when $L_{T}\sim L_{O}$ , the mixing is most active and efficient since stirring by the largest overturns becomes ‘optimal’ in the sense that it is not suppressed by ambient stratification. We argue that the high mixing efficiency associated with this phase, along with observations of $L_{O}/L_{T}\sim 1$ in oceanic turbulent patches, together point to the potential for systematically underestimating mixing in the ocean if the role of overturns is neglected. This neglect, arising through the assumption of a clear separation of scales between the background mean flow and small-scale quasi-isotropic turbulence, leads to the exclusion of an highly efficient mixing phase from conventional parameterizations of the vertical transport of density. Such an exclusion may well be significant if the mechanism of shear-induced turbulence is assumed to be representative of at least some turbulent events in the ocean. While our results are based upon simulations of shear instability, we show that they are potentially more generic by making direct comparisons with $L_{T}-L_{O}$ data from ocean and lake observations which represent a much wider range of turbulence-inducing physical processes.