Most of the nearly zonal, multiple, alternating jets observed in the oceans are latent, that is, their amplitudes are weak relative to the ambient mesoscale eddies. Yet, relatively strong jets are often observed in dynamical simulations. To explore mechanisms controlling the degree of latency, we analyse solutions of an idealized, eddy-resolving and flat-bottom quasigeostrophic model, in which dynamically generated mesoscale eddies maintain and interact with a set of multiple zonal jets. We find that the degree of the latency is controlled primarily by the bottom friction: the larger the friction parameter, the more latent are the jets; and the degree of the latency is substantial for a realistic range of the oceanic bottom friction coefficient. This result not only provides a plausible explanation for the latency of the oceanic jets, but it may also be relevant to the prominent atmospheric multiple jets observed on giant gas planets, such as Jupiter. We hypothesize that these jets can be so strong because of the relative absence of the bottom friction. The mechanism controlling the latency in our solutions is understood in terms of the changes induced in the linear eigenmodes of the time–mean flow by varying the bottom friction coefficient; these changes, in turn, affect and modify the jets. Effects of large Reynolds numbers on the eddies, jets, and the latency are also discussed.

Two-dimensional thermal convection of air between horizontal plates of length much greater than their separation distance is studied numerically by solution constraining motions to lie in a single vertical plane. Rayleigh numbers from 105 to 107 are employed. Steady rolls with wavelength twice the plate-separation were obtained in both cases. As the experimental two-dimensional constraint is relaxed, short-period turbulent fluctuations in temperature develop, the rolls or cells become only quasi-steady, and their wavelength increases. For the three-dimensional case, very large width-to-height ratios and averaging periods are found necessary before the temperature variance in time approaches the variance in the horizontal.

Convection of an internally heated fluid, confined between top and bottom plates of equal temperature, is studied by direct numerical simulation in two and three dimensions. The unstably stratified upper region drives convection that penetrates into the stably stratified lower region. The fraction of produced heat escaping across the bottom plate, which is one half without convection, initially decreases as convection strengthens. Entering the turbulent regime, this decrease reverses in two dimensions but continues monotonically in three dimensions. The mean fluid temperature, which grows proportionally to the heating rate ($H$) without convection, grows proportionally to $H^{4/5}$ when convection is strong in both two and three dimensions. The ratio of the heating rate to the fluid temperature is likened to the Nusselt number of Rayleigh–Bénard convection. Simulations are reported for Prandtl numbers between 0.1 and 10 and for Rayleigh numbers (defined in terms of the heating rate) up to $5\times 10^{10}$.

The transient force exerted by a low-speed liquid droplet impinging onto a flat rigid surface is investigated experimentally. The measurements employ a high-sensitivity piezo-electric sensor, along with a high-speed camera, and cover four decades in droplet Reynolds number and greater than two decades in Weber number. Across these ranges, the peak of individual force profiles span from 3 mN to over 1300 mN. Once normalised, the force–time profiles support the existence of an inertially dominated self-similar regime. Within this regime, previous numerical and theoretical studies predict a $\sqrt{t}$ dependence of impact normal force during the initial pre-peak rise. While our measurements confirm this finding, they also indicate that, after the peak force the profiles exhibit an exponential decay. This long-time decay law suggests treatment of the momentum transport from the droplet using a lumped model. An observed linear dependence between the force and force decay rate supports this approach. The reason for the efficacy of treating this system via a lumped model apparently connects to the physics right at the surface that limit the rate of momentum transport from the droplet to the surface. This is explored by estimating the momentum transfer by solely using the deforming droplet shape, but under the condition of negligible momentum gradients within the droplet. The short- and long-time solutions are combined and the resulting model equation is shown to accurately cover the entire force–time profile.

When subjected to a steady cross-flow, the deformation of flexible blades is known to result in the alleviation of the internal stresses in comparison to rigid structures. In the field of biomechanics, the flow-induced deformations of flexible structures leading to stress reduction have been often referred to as ‘reconfiguration’ in order to highlight the alleged benefits of such an adaptive process. In this paper, we investigate the reconfiguration of thin elastic blades and the resulting internal stresses when the flow about the blade is oscillatory. Our approach, based on numerical simulations using reduced order fluid force models, is validated by experimental observations. Through a systematic investigation of the response of the structure, we identify four kinematic regimes depending on the excursion of the fluid particles relative to the dimensions of the blade and on the frequency of the flow oscillations relative to the characteristic frequency of the blade. When the flow amplitude is smaller than the structural width, fluid inertia dominates over drag and the fluid–structure coupling triggers resonances that may cause a magnification of the internal stresses. But the small magnitude of the fluid load in this regime is unlikely to cause any severe damage in practice. Otherwise, when drag is the dominant load, flexibility always permits a reduction of the internal stresses. As in the static case, dynamic reconfiguration results in the concentration of the stresses within a small bending length whose scaling depends on the kinematic regime. The magnitude of the stresses does not depend on the actual length of the structure anymore, which suggests the absence of mechanical limitations to the axial growth of wave-swept plants. However, the risk of resonances originating from the inertial load when the blade width compares with the flow excursion favours elongated shapes that best accommodate the oscillatory fluid loadings.

Time-averaged two-point measurements of the fluctuating wall pressure and instantaneous measurements of the full fluctuating wall pressure field were performed in incompressible turbulent impinging jets with nozzle-to-plate spacings of 2 and 4 diameters, Reynolds numbers of 23 300, and Mach numbers of 0.03. An azimuthal Fourier decomposition of the fluctuating wall pressure revealed that the fluctuations in the stagnation region were dominated by azimuthal mode 1 to 3, and the contributions from these modes were larger for the jet with $H/D=4$. Azimuthal modes 0 and 1 made significant contributions to the pressure fluctuations in the wall jet region of both jets, indicating that the large-scale ring structures formed in the jet shear-layer play a prominent role in this region. The contributions from modes 0 and 1 in the wall jet were smaller for the jet with $H/D=4$ indicating that the ring structures play a less prominent role in the wall jet region as the nozzle-to-plate distance increased. A wavelet analysis of the transient fluctuations indicated that azimuthal mode 1 had dominant high-frequency and low-frequency components, while azimuthal mode 0 had only the higher-frequency oscillations. The higher-frequency components of azimuthal modes 0 and 1 occurred in both the stagnation and the wall jet regions and are attributed to the asymmetric evolution of the ring structures in the jet. The low-frequency oscillations were primarily associated with azimuthal mode 1 and were evident only in the stagnation region. These oscillations became more prominent as the nozzle-to-plate distance was increased.

A general numerical method using the boundary integral equation technique of Pozrikidis (1994) for Stokes flow in an axisymmetric domain is used to obtain the first solutions to the Brinkman equation for the motion of a particle in the presence of planar confining boundaries. The method is first applied to study the perpendicular and parallel motion of a sphere in a fibre-filled medium bounded by either a solid wall or a planar free surface which remains undeformed. By accurately evaluating the singular integrals arising from the discretization of the resulting integral equation, one can efficiently and accurately treat flow problems with high α defined by rs/K1/2p in which rs is the radius of the sphere and Kp is the Darcy permeability. Convergence and accuracy of the new technique are tested by comparing results for the drag with the solutions of Kim & Russell (1985a) for the motion of two spheres perpendicular to their line of centres in a Brinkman medium. Numerical results for the drag and torque exerted on the particle moving either perpendicular or parallel to a confining planar boundary are presented for ε[ges ]0.1, in which εrs is the gap between the particle and the boundary. When the gap width is much smaller than rs, a local analysis using stretched variables for motion of a sphere indicates that the leading singular term for both drag and torque is independent of α provided that α = O(1). These results are of interest in modelling the penetration of the endothelial surface glycocalyx by microvilli on rolling neutrophils and the motion of colloidal gold and latex particles when they are attached to membrane receptors and observed in nanovid (video enhanced) microscopy. The method is then applied to investigate the motion of a sphere translating in a channel. The drag and torque exerted on the sphere are obtained for various values of α, the channel height H and particle position b. These numerical results are used to describe the diffusion of a spherical solute molecule in a parallel walled channel filled with a periodic array of cylindrical fibres and to assess the accuracy of a simple multiplicative formula proposed in Weinbaum et al. (1992) for diffusion of a solute in the interendothelial cleft.

We propose a theory of a steady circular hydraulic jump based on the shallow-water model obtained from the depth-averaged Navier–Stokes equations. The flow structure both upstream and downstream of the jump is determined by considering the flow over a plate of finite radius. The radius of the jump is found using the far-field conditions together with the jump conditions that include the effects of surface tension. We show that a steady circular hydraulic jump does not exist if the surface tension is above a certain critical value. The solution of the problem provides a basis for the hydrodynamic stability analysis of the hydraulic jump. An analogy between the hydraulic jump and a detonation wave is pointed out.

The Markovian random coupling (MRC) model is a modified form of the stochastic model of the Navier-Stokes equations introduced by Kraichnan (1958, 1961). Instead of constant random coupling coefficients, white-noise time dependence is assumed for the MRC model. Like the Kraichnan model, the MRC model preserves many structural properties of the original Navier-Stokes equations and should be useful for investigating qualitative features of turbulent flows, in particular in the limit of vanishing viscosity. The closure problem is solved exactly for the MRC model by a technique which, contrary to the original Kraichnan derivation, is not based on diagrammatic expansions. A closed equation is obtained for the functional probability distribution of the velocity field which is a special case of Edwards’ (1964) Fokker-Planck equation; this equation is an exact consequence of the stochastic model whereas Edwards’ equation constitutes only the first step in a formal expansion based directly on the Navier-Stokes equations. From the functional equation an exact master equation is derived for simultaneous second-order moments which happens to be essentially a Markovianized version of the single-time quasi-normal approximation characterized by a constant triad-interaction time.

The explicit form of the MRC master equation is given for the Burgers equation and for two- and three-dimensional homogeneous isotropic turbulence.

Experiments were conducted to test the linear theory of internal gravity waves produced in a stably stratified liquid by the steady vertical motion of a sphere. The measurements of the phase configuration in a medium whose density increased linearly with depth were made by means of a Toepler-schlieren system. The agreement between observation and prediction was found to be good.

This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variable-coefficient Korteweg–de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adiabatic evolution of isolated solitary waves and periodic wave trains in the presence of variable topography and bottom friction, modelled by the Chezy law, but also, importantly, to study the effects of these factors on the propagation of undular bores, which are essentially unsteady in the system under consideration. In particular, it is shown that the combined action of variable topography and bottom friction generally imposes certain global restrictions on the undular bore propagation so that the evolution of the leading solitary wave can be substantially different from that of an isolated solitary wave with the same initial amplitude. This non-local effect is due to nonlinear wave interactions within the undular bore and can lead to an additional solitary wave amplitude growth, which cannot be predicted in the framework of the traditional adiabatic approach to the propagation of solitary waves in slowly varying media.

A nonlinear analysis of the interaction between a water wave and a floating ice cover in river channels is presented. The one-dimensional weakly nonlinear equation for shallow water wave propagation in a uniform channel with a floating ice cover is derived. The ice cover is assumed to be a thin uniform elastic plate. The weakly nonlinear equation is a fifth-order KdV equation. Analytical solutions of the nonlinear periodic wave equation are obtained. These solutions show that the shape, wavelength and celerity of the nonlinear waves depend on the wave amplitude. The wave celerity is slightly smaller than the open water wave celerity. The wavelength decreases as the wave amplitude increases. Based on these solutions the fracture of the ice cover is analysed. The spacing between transverse cracks varies form 50 m to a few hundred metres with the corresponding wave amplitude varying from 0.2 to 0.8 m, depending on the thickness and strength of the cover. These results agree well with limited field observations.

A simple two-phase model for steady fully developed flows of particles and water over erodible inclined beds is developed for situations in which the water and particles have the same depth. The rheology of the particles is based on recent numerical simulations and physical experiments, the rheology of the fluid is based on an eddy viscosity, and the interaction between the particles and the fluid is through drag and buoyancy. Numerical solutions of the resulting differential equations and boundary conditions provide velocity profiles of the fluid and particles, the concentration profile of the particles, and the depth of the flow at a given angle of inclination of the bed. Simple approximations permit analytical expressions for the flow velocities and the depth of flow to be obtained that agree with the numerical solutions and those measured in experiments.

The formation and pinch-off of non-axisymmetric vortex rings is considered experimentally. Vortex rings are generated using a non-circular piston–cylinder arrangement, and the resulting velocity fields are measured using digital particle image velocimetry. Three different nozzle geometries are considered: an elliptical nozzle with an aspect ratio of two, an elliptical nozzle with an aspect ratio of four and an oval nozzle constructed from tangent circular arcs. The formation of vortices from the three nozzles is analysed by means of the vorticity and circulation, as well as by investigation of the Lagrangian coherent structures in the flow. The results indicate that, in all three nozzles, the maximum circulation the vortex can attain is determined by the equivalent diameter of the nozzle: the diameter of a circular nozzle of identical cross-sectional area. In addition, the time at which the vortex rings pinch off is found to be constant along the nozzle contours, and independent of relative variations in the local curvature. A formation number for this class of vortex rings is defined based on the equivalent diameter of the nozzle, and the formation number for vortex rings of the three geometries considered is found to lie in the range of 3–4. The implications of the relative shape and local curvature independence of the formation number on the study and modelling of naturally occurring vortex rings such as those that appear in biological flows is discussed.

Flow-induced vibration (FIV) of an elastically mounted circular cylinder with an attached splitter plate in uniform flow is studied numerically via a stabilized space–time finite element method. The Reynolds number based on the cylinder diameter $D$ and the free-stream speed is restricted to 150. The ratio of the density of the body to that of the fluid, for the major part of the study, is 10. Two different reduced speeds are defined to quantify the compliance of the elastic support and flexibility of the splitter plate, respectively: $U_{s}^{\ast }$ based on the natural frequency of the spring–mass system and $U_{p}^{\ast }$ based on the fundamental natural frequency of the plate. Flow past a stationary cylinder ($U_{s}^{\ast }=0$) with a flexible splitter plate of length $3.5D$ is studied at different values of $U_{p}^{\ast }$. The vibration response of the plate exhibits lock-in with various eigenmodes of the plate in different ranges of $U_{p}^{\ast }$. The onsets of these lock-in regions are abrupt and hysteretic. The elastically mounted cylinder, without the splitter plate, undergoes large-amplitude vortex-induced vibration (VIV) for $4<U_{s}^{\ast }<7$. These large-amplitude oscillations are a consequence of synchronization, wherein the vortex shedding frequency locks in to the cylinder oscillation frequency. A rigid splitter plate attached to the cylinder reduces significantly the peak amplitude during VIV. Increasing the length of the plate from $1.5D$ to $3.5D$ only marginally affects the peak amplitude. It, however, leads to a wider range of lock-in. Unlike the case of an isolated cylinder, the lock-in and desynchronization regimes are not well demarcated in the presence of the splitter plate. Further, galloping is observed beyond a critical value of $U_{s}^{\ast }$; the amplitude of vibration increases with an increase in $U_{s}^{\ast }$ while the vibration frequency is relatively low and remains nearly constant. Increase in plate length delays, in terms of $U_{s}^{\ast }$, the onset of galloping. It is also found that the flexibility of the plate affects the maximum oscillation amplitude in the VIV regime. It also dictates the presence/absence of galloping. The system behaves similar to an isolated cylinder for a very flexible plate. The response is devoid of galloping, but relatively large amplitude of oscillation is observed during lock-in. The behaviour of the cylinder with a stiff plate is similar to that with the rigid one. The galloping instability sets in when the flexibility of the plate is less than a certain value ($U_{p}^{\ast }<4.7$, approximately for $U_{s}^{\ast }=22$). The VIV and galloping are separated by a range of $U_{s}^{\ast }$ in which the flow is either steady, for longer plates, or exhibits very weak vortex shedding. In the VIV regime, the plate tip and cylinder vibrate in phase for low $U_{p}^{\ast }$; their motion is out of phase for larger $U_{p}^{\ast }$. The change in phase is also associated with change in the frequency of vibration. At low $U_{p}^{\ast }$, the frequency of vibration is close to the first natural frequency of the system, while at high $U_{p}^{\ast }$ it becomes closer to the second natural frequency. The vibration amplitude of the cylinder is close to maximum in the VIV regime for $U_{s}^{\ast }=6$. Computations for various $U_{p}^{\ast }$, for $U_{s}^{\ast }=6$ and $22$, are utilized to determine optimal flexibility that leads to minimal FIV. The effect of the length of the flexible splitter plate, mass ratio and damping ratio is studied. A strategy is proposed to utilize the computations from various combinations of $U_{s}^{\ast }$ and $U_{p}^{\ast }$ to choose the appropriate flexibility of the attached splitter plate to minimize FIV.

We report the first experimental study of Lagrangian acceleration in turbulent Rayleigh–Bénard convection, using particle tracking velocimetry. A method has been developed to quantitatively evaluate and eliminate the uncertainties induced by temperature and refraction index fluctuations caused by the thermal plumes. It is found that the acceleration p.d.f. exhibits a stretched exponential form and that the probability for large magnitude of acceleration in the lateral direction is higher than those in the vertical directions, which can be attributed to the vortical motion of the thermal plumes. The local acceleration variance was obtained for various values of the three control parameters: the Rayleigh number (), the Prandtl number ( and 6.1) and the system size . These were then compared with the theoretically predicted dependence on these parameters for buoyancy-dominated turbulent flows and for homogeneous and isotropic turbulence, respectively. It is found that in the central region is dominated by contributions from the turbulent background rather than from the buoyancy force, and the Heisenberg–Yaglom relation holds in this region. From this, we obtain the first experimental results of the constant of the acceleration variance in the micro-scale Reynolds number range , which fills a gap in this constant in the lower end from the experimental side, and provides possible constraints for its high behaviour if a certain fitting function is attempted. In addition, acceleration correlation functions were obtained for different . It is found that the zero crossing time of acceleration correlation functions is at ( is the Kolmogorov time scale) over the range of spanned in our experiments, which is the same as the simulation results in isotropic turbulence, and the exponential decay time , which is larger than found experimentally for other types of turbulent flows with larger .

Stuart (1960) has developed a theory of the stability of plane Poiseuille flow to periodic disturbances of finite amplitude which, in the neighbourhood of the neutral curve, leads to an equation of the Landau (1944) type for the amplitude A of the disturbance: \[ d|A|^2/dt = k_1|A|^2 - k_2|A|^4. \] If k2 is positive in the supercritical region (R > RC) where k1 is positive, then, according to Stuart, there is a possibility of the existence of periodic solutions of finite amplitude which asymptotically approach a constant value of (k1/k2)½. We have evaluated the coefficient k2 and found that there indeed exists a zone in the (α, R)-plane where it is positive. This is the zone inside the dashed curve shown in figure 1, with the region of instability predicted by the linear theory included inside the ‘neutral curve’. Stuart's theory and Eckhaus's generalization thereof could apply in the overlapping zone just above the lower branch of the neutral curve.

We consider the free-convective flow from a heated sphere, in the Boussinesq approximation, at high Grashof number. The characteristics of the boundary layer close to the surface of the sphere are evaluated numerically, and the eruption of the fluid from the boundary layer into the plume above the sphere is discussed.

The formation of segregation patterns in initially homogeneous, fluidized, binary mixtures of particles has been studied. The adjustment of the bed depends on the proportions of fine and coarse particles in the mixture and the gas flow rate relative to the minimum fluidization velocities of the two components. The particles are immobile until the gas flow rate is sufficiently large to fluidize the mixture of particles. When the gas flow rate exceeds this critical value, alternating vertical bands of coarse and fine particles form. At a second critical gas velocity this pattern breaks down and the more familiar pattern of a mixed horizontal band on top of a layer of coarse particles forms. A phase diagram, constructed from experimental observations, shows the conditions for which each of these regimes exists. Its structure is explained in terms of the fluidization and consequent mobility of the mixture components. When horizontal bands are present, the thickness of the lower layer of coarse particles decreases with increasing gas flow rate depending on the proportion of fine particles in the bed. This, and its development, can be understood by analogy with the sedimentation of particles through a turbulent fluid. The experiments imply that the efficiency of mixing by the bubbles in the fluidized bed is very much less than that expected from gas bubbles in a liquid.

In a previous paper (Booker 1976) we found experimentally that the convective heat transport in a fluid with temperature-dependent viscosity decreased significantly as the ratio of the viscosities at the top and bottom boundaries increased. In this note, we show that this decrease in heat transport can be entirely accounted for by an increase in the critical Rayleigh number with variable viscosity.