A systematic study is made of the effect of latitude on the linear, normal mode stability characteristics of the laminar barotropic Ekman layer. The outcome depends upon the direction of the geostrophic flow (in the case of flows modelling the atmospheric Ekman layer) or, alternatively, upon the direction of the applied stress (in the case of flows modelling the oceanic Ekman layer). The minimum critical Reynolds number Rc is a function of latitude. For the atmospheric Ekman layer Rc = 30.8 for all latitudes less than 26.2° and increases monotonically with latitude to 54.2. At a latitude of 45° N, Rc is 33.9 and arises for a geostrophic wind directed towards a compass heading 252° (clockwise from north), corresponding to rolls with axes pointing due west and having wavenumber k (with unit of length taken to be the Ekman layer depth) of 0.594. The minimum Rc for the oceanic boundary layer is 11.6 for latitudes less than 81.1°, and increases with latitude to 11.8. At 45° N latitude, the critical condition arises for a surface-current compass heading of 345.2°, roll axis of 351° and a wavenumber k = 0.33. The results for Rc are all symmetric about the equator, with roll axes and associated basic flow directions rotated by 180°. As the Reynolds number R increases, the effects of the perturbation Coriolis acceleration on the instability diminish, as has been previously shown, and the error caused by neglect of the horizontal component of angular velocity therefore decreases. The high Reynolds number limit is systematically explored. It is shown that the lower branch of the neutral curve is not inviscid as R → ∞; rather kR → constant. The upper branch is inviscid in the limit R → ∞, and corresponds to a regular or singular neutral mode depending on whether the angle ε between the outer geostrophic flow and the roll axis is greater or less than 15.93°. ‘Inflectional’ modes, thought to be relevant by some investigators, do not exist for ε < 15.93°. Lastly, the most unstable inviscid mode corresponding to zero phase speed, a condition to which certain well-known experiments are sensitive, occurs at ε = 11.8° with wavenumber k = 0.6. This is in good agreement with published experimental data.

This experimental study explores and quantifies mixing characteristics associated with a gaseous round jet injected perpendicularly into cross-flow for a range of flow and injection conditions. The study utilizes acetone planar laser-induced fluorescence imaging to determine mixing metrics in both centreplane and cross-sectional planes of the jet, for a range of jet-to-cross-flow momentum flux ratios ($2\leqslant J\leqslant 41$), density ratios ($0.35\leqslant S\leqslant 1.0$) and injector configurations (flush nozzle, flush pipe and elevated nozzle), all at a fixed jet Reynolds number of 1900. For the majority of conditions explored, there is a direct correspondence between the nature of the jet’s upstream shear layer instabilities and structure, as documented in detail in Getsinger et al. (J. Fluid Mech., vol. 760, 2014, pp. 342–367), and the jet’s mixing characteristics, consistent with diffusion-dominated processes, but with a few notable exceptions. When quantified as a function of distance along the jet trajectory, mixing metrics for jets in cross-flow with an absolutely unstable upstream shear layer and relatively symmetric counter-rotating vortex pair cross-sectional structure tend to show better local molecular mixing than for jets with convectively unstable upstream shear layers and generally asymmetric cross-sectional structures. Yet the spatial evolution of mixing with downstream distance can be greater for a few specific convectively unstable conditions, apparently associated with the initiation and nature of shear layer rollup as a trigger for improved mixing. A notable exception to these trends concerns conditions where the equidensity jet in cross-flow has an upstream shear layer that is already absolutely unstable, and the jet density is then reduced in comparison with that of the cross-flow. Here, density ratios below unity tend to mix less well than for equidensity conditions, demonstrated to result from differences in the nature of higher-density cross-flow entrainment into lower-density shear layer vortices.

Potential flows ${\bm u} = {\bm\nabla} \phi$ are solutions of the Navier–Stokes equations for viscous incompressible fluids for which the vorticity is identically zero. The viscous term $\mu \nabla^2 {\bm u} = \mu{\bm \nabla}\nabla^2\phi$ vanishes, but the viscous contribution to the stress in an incompressible fluid (Stokes 1850) does not vanish in general. Here, we show how the viscosity of a viscous fluid in potential flow away from the boundary layers enters Prandtl's boundary layer equations. Potential flow equations for viscous compressible fluids are derived for sound waves which perturb the Navier–Stokes equations linearized on a state of rest. These linearized equations support a potential flow with the novel features that the Bernoulli equation and the potential as well as the stress depend on the viscosity. The effect of viscosity is to produce decay in time of spatially periodic waves or decay and growth in space of time-periodic waves.

In all cases in which potential flows satisfy the Navier–Stokes equations, which includes all potential flows of incompressible fluids as well as potential flows in the acoustic approximation derived here, it is neither necessary nor useful to put the viscosity to zero.

In constant area back pressured ducts, shock trains exhibit inherent unsteadiness where the shock system fluctuates about its time-averaged position despite constant bulk inflow and outflow conditions. In this work, the underlying causes of inherent unsteadiness are identified and the flow dynamics of the system is studied for a shock train in a Mach 2.0 ducted flow that is mechanically back pressured. High-speed schlieren movies and pressure measurements are collected to quantify the shock system fluctuations. Cross-spectral analysis of this data is used to identify specific perturbations, i.e. the fluid phenomena that impact the shock train motion. Key information about each perturbation is also obtained, including where it originates, what direction it travels and how it impacts each shock. Oil flow visualization and particle image velocimetry are then used to gain insight into the physical structure of perturbations and the flow phenomena that generate them. The results identify a complex, frequency-dependent dynamical system that is influenced by (i) upstream propagating acoustic waves that emanate from separation bubbles, (ii) vortices that shed from separation bubbles and convect downstream and (iii) upstream propagating acoustic waves generated in the diffuser. With this information, a scaling argument for the shock train inherent unsteadiness is presented.

Uniform flow in curved, wide, erodible-bed channels is formulated on the basis of the conservation of flux of moment-of-momentum, to obtain relations for the vertical distributions of radial-plane velocity and radial shear stress. The expression for the radial stress exerted on the bed is utilized in a force-equilibrium analysis of the moving bed layer to obtain relations for the average transverse slope of the bed and for the radial bed profile. The reduction of primary bed shear stress due to the net radial transport of streamwise momentum toward the outer (concave) bank is then calculated, by introducing the derived expressions for the velocity components into the momentum equation for the primary-flow direction. It is found that the stress reductions in deep, narrow channels can exceed 50%. Bed profiles and velocity distributions measured in natural and laboratory streams are found to be in good conformity with those calculated from this moment-based theory.

The interaction of an array of voids collapsing after passage of a stress wave is studied as a model problem relevant to porous materials, for example, to energy localization leading to hotspot formation in energetic materials. Dynamic experiments are designed to illuminate the hydrodynamic processes of collapsing void interactions for eventual input into device-scale initiation models. We examine a stress wave loading representative of accidental mechanical insult, for which the wave passage length scale is comparable with the void and inter-void length scales. A single void, two-void linear array, and a four-void staggered array are studied. Diagnostic techniques include high-speed imaging of cylindrical void collapse and the first particle image velocimetry measurements in the surrounding material. Voids exhibit an asymmetrical collapse process, with the formation of a high-speed internal jet. Volume and diameter versus time data for single void collapse under stress wave loading are compared with literature results for single voids under shock-wave loading. The internal volume history does not fall on a straight line and is in agreement with simulations, but in contrast to existing linear experimental data fits. The velocity field induced in the surrounding material is measured to quantify a region of influence at selected stages of single void collapse. In the case of multiple voids, the stress wave diffracts in response to the presence of the upstream void, affecting the loading condition on the downstream voids. Both collapse-inhibiting (shielding) and collapse-triggering effects are observed.

The problem studied here is that of the attenuation of internal waves through turbulent mixing in a weakly and exponentially stratified fluid. The equations are linearized and it is assumed that the action of turbulence can be parametrically represented by eddy mixing coefficients and that the influence of bottom friction is restricted to a thin bottom boundary layer. The simple case where there is no rotation and only one component to the stratification is first examined in detail, and the modifications caused by introducing rotation and a second component are subsequently investigated. Subject quantitatively to the choice made for the eddy coefficients, but qualitatively not strongly dependent on that choice, the following conclusions are drawn: (i) very short internal waves (length < 100 m) are strongly damped in basins of all depths; (ii) long internal waves or seiches in shallow seas (depth ≃ 100 m) will not last more than a few cycles as free oscillations; (iii) the attenuation rate for long internal tides is small enough that these should be observable very far from the coasts, but large enough to exclude the possibility of oceanic standing wave systems; (iv) for very long internal waves the damping is predominantly due to the effect of bottom friction, and the attenuation rate becomes almost independent of the actual form of the stratification present in the fluid.

The dynamics of rotating fluids was, in the main, developed by methods special to the field, using the equationsof motion of a fluid inarotatingframe of reference. It is, nevertheless, possible to derive all the leading results from the classical principles of fluid dynamics in non-rotating frames; specifically, from the rules governing rate of change of vorticity. Although writers on the subject have adopted this approach increasingly often in recent years, the author believes that a broad survey of the field, deriving results from those classical rules concerning vorticity, has not previously been given and may be of some interest to fluid dynamicists in general.

The present survey was read to the IUTAM Symposium on Rotating Fluid Systems at La Jolla, California, on 28 March 1966. It states briefly (§2) the rules governing rate of change of vorticity, and then applies them, first, to problems of steady relative motion of rotating fluids; in particular, of the atmosphere (§3), of rotating fluids in the laboratory (§4), and of the oceans (§5). Waves and wavy movements are then studied, first (§6) for systems with constant Coriolis parameter, including inertial waves, surface waves, ‘long waves’ and internal waves, and, secondly (§7)) for systems with variable Coriolis parameter, including Rossby-Haurwitz waves with and without the influence of tidal effects, as well as problems of barotropic and baroclinic instability. Vorticity principles are used as the sole theoretical tool throughout the survey.

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

Geophysical granular flows, such as avalanches, debris flows, lahars and pyroclastic flows, are always strongly influenced by the basal topography that they flow over. In particular, localised bumps or obstacles can generate rapid changes in the flow thickness and velocity, or shock waves, which dissipate significant amounts of energy. Understanding how a granular material is affected by the underlying topography is therefore crucial for hazard mitigation purposes, for example to improve the design of deflecting or catching dams for snow avalanches. Moreover, the interactions with solid boundaries can also have important applications in industrial processes. In this paper, small-scale experiments are performed to investigate the flow of a granular avalanche over a two-dimensional smooth symmetrical bump. The experiments show that, depending on the initial conditions, two different steady-state regimes can be observed: either the formation of a detached jet downstream of the bump, or a shock upstream of it. The transition between the two cases can be controlled by adding varying amounts of erodible particles in front of the obstacle. A depth-averaged terrain-following avalanche theory that is formulated in curvilinear coordinates is used to model the system. The results show good agreement with the experiments for both regimes. For the case of a shock, time-dependent numerical simulations of the full system show the evolution to the equilibrium state, as well as the deposition of particles upstream of the bump when the inflow ceases. The terrain-following theory is compared to a standard depth-averaged avalanche model in an aligned Cartesian coordinate system. For this very sensitive problem, it is shown that the steady-shock regime is captured significantly better by the terrain-following avalanche model, and that the standard theory is unable to predict the take-off point of the jet. To retain the practical simplicity of using Cartesian coordinates, but have the improved predictive power of the terrain-following model, a coordinate mapping is used to transform the terrain-following equations from curvilinear to Cartesian coordinates. The terrain-following model, in Cartesian coordinates, makes identical predictions to the original curvilinear formulation, but is much simpler to implement.

Motions of a single vortex filament in a background flow are studied by numerical simulation of a set of model equations. The model, which in essence is due to Hama, treats the self-interaction of the filament through the so-called ‘localized-induction approximation’ (LIA). Interaction with the prescribed background field is treated by simply advecting the filament appropriately. We are particularly interested in elucidating the evolution of sinuous vortices such as the ‘wiggle’ seen by Breidenthal in the transition to three-dimensionality in the mixing layer. The model studied embodies two of the simplest ingredients that must enter into any dynamical explanation: induction and advection. For finite-amplitude phenomena we make contact with the theory of solitons on strong vortices developed by Betchov and Hasimoto. In a shear, solitons cannot exist, but solitary waves can, and their interactions with the shear are found to be key ingredients for an understanding of the behaviour of the vortex filament. When sheared, a soliton seems to act as a ‘nucleation site’ for the generation of a family of waves. Computed sequences are shown that display a remarkable morphological similarity to flow-visualization studies. The present application of fully nonlinear dynamics to a model presents an attractive alternative to the extrapolations from linearized stability theory applied to the full equations that have so far constituted the theoretical basis for understanding the experimental results.

A large-eddy simulation of laminar transonic buffet on an airfoil at a Mach number $M=0.735$, an angle of attack $\unicode[STIX]{x1D6FC}=4^{\circ }$, a Reynolds number $Re_{c}=3\times 10^{6}$ has been carried out. The boundary layer is laminar up to the shock foot and laminar/turbulent transition occurs in the separation bubble at the shock foot. Contrary to the turbulent case for which wall pressure spectra are characterised by well-marked peaks at low frequencies ($St=f\cdot c/U_{\infty }\simeq 0.06{-}0.07$, where $St$ is the Strouhal number, $f$ the shock oscillation frequency, $c$ the chord length and $U_{\infty }$ the free-stream velocity), in the laminar case, there are also well-marked peaks but at a much higher frequency ($St=1.2$). The shock oscillation amplitude is also lower: 6 % of chord and limited to the shock foot area in the laminar case instead of 20 % with a whole shock oscillation and intermittent boundary layer separation and reattachment in the turbulent case. The analysis of the phase-averaged fields allowed linking of the frequency of the laminar transonic buffet to a separation bubble breathing phenomenon associated with a vortex shedding mechanism. These vortices are convected at $U_{c}/U_{\infty }\simeq 0.4$ (where $U_{c}$ is the convection velocity). The main finding of the present paper is that the higher frequency of the shock oscillation in the laminar regime is due to a different mechanism than in the turbulent one: laminar transonic buffet is due to a separation bubble breathing phenomenon occurring at the shock foot.

Jets arising from rivers, streams and tidal flows entering still waters differ from most experimental studies of jets both in aspect ratio and in the presence of a solid bottom boundary and an upper free surface. Despite these differences, the applicability of experimental jet studies to these systems remains largely untested by either field or realistically scaled experimental studies. Here we present experimental results for a wall-bounded plane jet scaled to jets created by flow discharging into floodplain lakes. A characteristic feature of both our prototype and experimental jets is the presence of large-scale meandering turbulent structures that span the width of the jets. In our experimental jets, we observe self-similarity in the distribution of mean streamwise velocities by a distance of six channel widths downstream of the jet outlet. After a distance of nine channel widths the velocity decay and the spreading rates largely agree with prior experimental results for plane jets. The magnitudes and distributions of the cross-stream velocity and lateral shear stresses approach self-preserving conditions in the upper half of the flow, but decrease in magnitude, and deviate from self-preserving distributions with proximity to the bed. The presence of the meandering structure has little influence on the mean structure of the jet, but dominates the jet turbulence. A comparison of turbulence analysed at time scales both greater than and less than the period of the meandering structure indicates that these structures increase turbulence intensities by 3–5 times, and produce lateral shear stresses and momentum diffusivities that are one and two orders of magnitude greater, respectively, than turbulence generated by bed friction alone.

Three-dimensional temporal instabilities, leading to spray formation of a round liquid jet segment with an outer, coaxial high-density gas flow, are studied with Navier–Stokes and level-set computations. These computations predict the liquid surface shape showing the smaller structures on the conical wave crests, i.e. lobes, holes, bridges and ligaments, which are the precursors to droplet and spray formations. These structures and their time scales affect droplet size and velocity distributions as well as spray cone angles. The gas-to-liquid density ratio, liquid Reynolds number ($Re$) and liquid Weber number ($We$) range between 0.02–0.9, 320–16 000 and 2000–230 000, respectively, which cover three distinct physical domains. (1) At higher $Re$ and $We$, ligaments and then drops develop following hole and liquid bridge formations. (2) At higher gas densities throughout the $Re$ range, several holes merge forming two bridges per lobe before breaking to form ligaments; this hole merging is explained by slower development of hairpin vortices and lobe shape. (3) In cases where both gas density and $Re$ or $We$ are lower, the well-ordered lobes are replaced by more irregular, smaller-scale corrugations along the conical wave crest edge; ligaments form differently by stretching from the lobes before holes form. Thicker ligaments and larger droplets form in the low $Re$, low gas density range. The surface wave dynamics, vortex dynamics and their interactions are explained. Understandings of liquid stream break up and concurrent smaller structure formation are built upon an examination of both translation and rotation of the fluid. In all cases, hole formation is correlated with hairpin and helical vortices; fluid motion through a perforation in the thin sheet near the wave crest corresponds to these vortices. The hole formation process is dominated by inertial forces rather than capillary action, which differs from mechanisms suggested previously for other configurations. Circulation due to streamwise vorticity increases while the lobes thin and holes form. For larger surface tension, cavities in the jet core rather than perforations in a sheet occur. The more rapid radial extension of the two-phase mixture with increasing gas density is explained by greater circulation in the ring (i.e. wave crest) region. Experimental descriptions of the smaller structures are available only at lower $Re$ and lower density, agreeing with the computations. Computed scales of bridges, ligaments, early droplets and emerging spray radii agree qualitatively with experimental evidence through the high $Re$ and $We$ domains.

Investigation of the linear stability problem for rapidly rotating convection on an f-plane has revealed the existence of two distinct scales in the vertical structure of the critical eigenfunctions: a small length scale whose vertical wavenumber kz is comparable with the large horizontal wavenumber k⊥ selected at onset, and a large-scale modulation which forms an envelope on the order of the layer depth d. The small-scale structure in the vertical results from a geostrophic balance imposed by the Taylor–Proudman constraint. This primary balance forces rotational alignment and confines fluid motions to planes perpendicular to the rotation axis. For convective transport in the vertical this constraint must be relaxed. This is achieved by molecular dissipation which allows weak upward (downward) spiralling of hot (cold) fluid elements across the Taylor–Proudman planes and results in a large-scale vertical modulation of the Taylor columns.

In the limit of fast rotation (i.e. large Taylor number) a multiple-scales analysis leads to the determination of a critical Rayleigh number as a function of wavenumber, roll orientation and the tilt angle of the f-plane. The corresponding critical eigenfunction represents the core solution; matching to passive Ekman boundary layers is required for a complete solution satisfying boundary conditions.

An extension of this analysis, introduced by Bassom & Zhang (1994), is used to describe strongly nonlinear two-dimensional convection, characterized by significant departures of the mean thermal field from its conduction profile. The analysis requires the solution of a nonlinear eigenvalue problem for the Nusselt number (for steady convection) and the Nusselt number and oscillation frequency (for the overstable problem). The solutions of this problem are used to calculate horizontal and vertical heat fluxes, as well as Reynolds stresses, as functions of both the latitude and roll orientation in the horizontal, and these are used to calculate self-consistently north–south and east–west mean flows. These analytical predictions are in good agreement with the results of three-dimensional simulations reported by Hathaway & Somerville (1983).

A numerical investigation of transonic and low-supersonic flows of dense gases of the Bethe–Zel'dovich–Thompson (BZT) type is presented. BZT gases exhibit, in a range of thermodynamic conditions close to the liquid/vapour coexistence curve, negative values of the fundamental derivative of gasdynamics. This can lead, in the transonic and supersonic regime, to non-classical gasdynamic behaviours, such as rarefaction shock waves, mixed shock/fan waves and shock splitting. In the present work, inviscid and viscous flows of a BZT fluid past an airfoil are investigated using accurate thermo-physical models for gases close to saturation conditions and a third-order centred numerical method. The influence of the upstream kinematic and thermodynamic conditions on the flow patterns and the airfoil aerodynamic performance is analysed, and possible advantages deriving from the use of a non-conventional working fluid are pointed out.

We analyse and improve a recently-proposed two-phase flow model for the statistical evolution of two-fluid mixing. A hyperbolic equation for the volume fraction, whose characteristic speed is the average interface velocity v*, plays a central role. We propose a new model for v* in terms of the volume fraction and fluid velocities, which can be interpreted as a constitutive law for two-fluid mixing. In the incompressible limit, the two-phase equations admit a self-similar solution for an arbitrary scaling of lengths. We show that the constitutive law for v* can be expressed directly in terms of the volume fraction, and thus it is an experimentally measurable quantity. For incompressible Rayleigh–Taylor mixing, we examine the self-similar solution based on a simple zero-parameter model for v*. It is shown that the present approach gives improved agreement with experimental data for the growth rate of a Rayleigh–Taylor mixing layer.

Closure of the two-phase flow model requires boundary conditions for the surfaces that separate the two-phase and single-phase regions, i.e. the edges of the mixing layer. We propose boundary conditions for Rayleigh–Taylor mixing based on the inertial, drag, and buoyant forces on the furthest penetrating structures which define these edges. Our analysis indicates that the compatibility of the boundary conditions with the two-phase flow model is an important consideration. The closure assumptions introduced here and their consequences in relation to experimental data are compared to the work of others.