The adjustment under gravity of a fluid containing a horizontal density gradient is described.’ The fluid is initially at rest and the resulting motion is calculated as the flow accelerates, driven by the baroclinic density field. Two forms of the initial density distribution are considered. In the first the initial horizontal gradient is constant. A purely horizontal motion develops as the isopycnals rotate towards the horizontal. The vertical density gradient increases continually with time but the horizontal density gradient remains unchanged. The horizontal velocity has a uniform vertical shear, and the gradient Richardson number is constant in space and decreases monotonically with time to ½. The second density distribution consists of a piecewise constant gradient with a jump in the gradient along a vertical isopycnal. The density is continuous. In this case frontogenesis is predicted to occur on the isopycnal between the two constant-density-gradient regions, and the timescale for the formation of a front is determined. Laboratory experiments are reported which confirm the results of these calculations. In addition, lock exchange experiments have been carried out in which the horizontal mean gradient is represented by a series of step density differences separated by vertical gates.

Viscous oscillatory flow past particles, governed by the unsteady Stokes equation, is considered. The problem is addressed in its general form for arbitrary flows and particle shapes using the boundary-integral method. It is shown that the leading-order correction to the force exerted on a particle in unsteady flow may be inferred directly from the drag in steady translational motion. For axisymmetric flow, a numerical procedure for solving the boundary-integral equation is developed, and is applied to study streaming oscillatory flow past spheroids, dumbbells, and biconcave disks. The effect of the particle geometry on the structure of the flow is studied by comparing the streamline pattern associated with these particles to that for the sphere. The results reveal the existence of travelling stagnation points on the surface of non-spherical particles, and the formation of unsteady viscous eddies in the interior of the flow. These eddies grow during the decelerating flow period, and shrink during the accelerating flow period. For particles with concave boundaries, unsteady free eddies may originate from an expansion of wall eddies that reside within the concave regions.

The Marginal Ice Zone includes wide areas covered by dispersed ice floes in which wave conditions are significantly affected by the ice. When the wind blows from the solid ice pack, towards the open sea, growing waves are scattered by the floes, and their spectral characteristics modified. To further understand this problem, a model for the evolution of wind waves in a sparse field of ice floes has been developed. The sea state is described by a two-dimensional discrete spectrum. Time-limited wave growth is obtained by numerical integration of the energy balance equation using the exact nonlinear transfer integral. Wave scattering by a single floe is represented in terms of far-field expressions of the diffracted and forced potentials obtained numerically by the Green function method. The combined effect of a homogeneous field of floes on the wave spectrum is expressed in terms of the Foldy–Twersky integral equations under the assumption of single scattering. The results show a strong dependence of the spectrum amplitude and directional properties on the ratio of the ice floe diameter to the wavelength. For a certain range of this parameter, the ice cover appears to be very effective in dispersing the energy; the wave spectrum rapidly tends to isotropy, a tendency which prevents the normal growth of wave energy and the decrease in peak frequency. Therefore, in the Marginal Ice Zone, the ability of an offshore wind to generate a significant wave field is severely limited.

We develop a simple model in which longitudinal, compressible, unsteady heat transfer between heater and gas is computed in the small-Mach-number limit. This calculation is used to determine the transfer function of the heater, which plays an important role in the stability limits of the thermoacoustic instability of the Rijke tube. The transfer function is determined analytically in the limit of small expansion parameter γ, and numerically for γ of order unity. In the case ρμ/cp = constant, an analytical solution can be found.

We present results of numerical simulations of stably stratified, randomly forced turbulence. The selection of forcing and damping are designed to give insight into the question of whether cascade of energy to large scales is possible for strongly stratified three-dimensional turbulence in a manner similar to two-dimensional turbulence. We consider narrow-band wavenumber forcing, whose angular distribution ranges from two-dimensional to three-dimensional isotropic. Our principal results are as follows; for two-dimensional forcing, and for sufficiently small Froude number, the statistically steady state is characterized by a weakly inverse-cascading horizontal-velocity variance field. The vertical variability of the horizontal-velocity field is pronounced, but seems to approach a limit independent of the Brunt–Väisälä frequency N, as N → ∞. If, on the other hand, the Froude number exceeds a critical value, the vertical variability is weak, and the statistics of the scales larger than the forcing scale is near that predicted by inviscid equipartitioning. For all forcing functions considered the vertical motion and temperature field (w, T), centred at smaller scales, are more three-dimensionally isotropic, with no large-scale organization. At large N, (small Froude number) the w-field scales as 1/N, with horizontal motion field nearly independent of N. Furthermore, at large N and for horizontal forcing, the horizontal motion field is consistent with the condition that a substantial fraction of the total dissipation is attributable to an effective drag acting upon all horizontal scales of motion, which in turn flattens the slope of the energy spectrum in the inverse-cascade range, and increases it in the enstrophy-cascade range.

Results of direct numerical simulations of stably stratified, freely evolving, homogeneous turbulence are presented. An examination of initial data designed to give insight into laboratory flows suggests that the numerical simulations have a satisfactory degree of realism, insofar as statistical parameters such as total energy and length scales are concerned. The motion is decomposed into a stratified turbulence (vortical) component and a wave component. For initial-value problems similar to laboratory studies of stratified flows, the vortical component decays at a rate virtually identical to that of the non-buoyant case up to t = 6N−1 (N is the Brunt-Väisälä frequency). The decay rate decreases after this time, suggesting a kind of turbulence ‘collapse’. The temperature structure that emerges clearly shows the development of the collapse stage of the flow, which is also diagnosed by the behaviour of parameters such as the Thorpe scale.

We next examine the very small-Froude-number regime in order to understand possible universal aspects of the flow. An examination of various initial conditions with different proportions of stratified and wave components indicates a lack of universality. For initial data containing only vortical motion (motions derived from the vertical vorticity field), the vortical field tends to dominate, in subsequent evolution, at strong stratification. However, contrary to two-dimensional turbulence, the flow is more strongly dissipative than two-dimensional flows due to the frictional effect associated with layering. Other quantities examined are frequency spectra, and the probability distribution for vertical shear. The frequency spectra exhibit some features in common with spectra extracted from oceanographic data.

An ageostrophic version of Phillips’ model is studied. All instabilities found are systematically interpreted in terms of resonance of wave components. The instability occurs if there is a pair of wave components which propagate in the opposite direction to the basic flow and these wave components have almost the same Doppler-shifted frequency. A new instability, identified as a resonance between the Kelvin wave and the Rossby waves, is found at Froude number F ≈ 0.7. The Rossby waves are almost completely in geostrophic balance while the ageostrophic Kelvin wave is the same as in a one-layer system. Doppler shifting matches frequencies which would otherwise be very different. This instability is presumably the mechanism of the frontal instability observed by Griffiths & Linden (1982) in a laboratory experiment. Ageostrophic, baroclinic instability with non-zero phase speed is also observed in the numerical calculation. This instability is caused by resonance between different geostrophic modes.

A method of using local wave properties to provide a detailed description of breakings in a random wave field is developed. These properties, derived through the Hilbert transform, include the angular frequency, phase velocity, and surface-velocity components. The breaking characteristics are presented, including the probability of breaking, its time- and lengthscales, its intensity, and the phase of its inception. The time- and lengthscales, of breaking events were found to be linearly proportional to the corresponding scales of underlying waves, and to indicate that the breaking region is geometrically similar. Consistent results were obtained from temporal and spatial measurements. Finally, on the basis of these results we have evaluated geometric and kinematic criteria for identifying breaking waves.

When a long horizontal cylinder filled with fluid is differentially heated at the end walls at high Rayleigh number, A, the axial flow in the midsection consists of boundary layers at the top and bottom of the cylinder flowing in reverse directions, and the temperature is stably and linearly distributed in the vertical. Both the temperature is stably and linearly distributed in the vertical. Both the temperature and the flows are almost independent of the axial dimension. The adiabatic boundary condition on the cylinder requires temperature corrections which can induce cross-section boundary layers on the cylindrical wall and vertical internal boundary layers in the middle. Both types of boundary layers are O(A−¼) in width. Matching different boundary layers at the poles is achieved through additional A−⅙ and A−⅛ layers. The maximum boundary-layer velocity is calculated to be almost one-quarter of the axial velocity observed in experiments for A = 108.

The relative viscosities of suspensions of randomly oriented rods in a Newtonian fluid were measured using falling-ball rheometry. The rods were monodisperse and sufficiently large to render colloidal and Brownian forces negligible. Steel and brass ball bearings were dropped along the centreline of cylindrical columns containing the suspensions. The terminal velocities of the falling balls were measured and used to determine the average viscosities of the suspensions. The suspensions behaved as Newtonian fluids in that they were characterized by a constant viscosity. They exhibited a linear relative viscosity-volume fraction relationship for volume fractions less than 0.125, and, for volume fractions between 0.125 and 0.2315, the specific viscosity increased with the cube of the volume fraction. The relative viscosity was found to be independent of falling-ball size for a ratio of falling ball to fibre length greater than 0.3. It was found to be independent of the diameter of the containing cylindrical column for a ratio of column diameter to fibre length greater than 3.2. The value determined for the intrinsic viscosity is in good agreement with theoretical predictions for suspensions of randomly oriented rods.

The diffusion of a passive scalar (heat) from a continuous line source placed in a uniformly sheared, nearly homogeneous, turbulent shear flow is examined. Measurements indicate that the mean temperature profile is approximately Gaussian near the source but, further downstream, it becomes asymmetric and its peak shifts towards the region of lower velocity. The r.m.s. temperature fluctuation profile is double peaked close to the source, it is single peaked at intermediate distances and it demonstrates a re-emergence of double peaks which grow in relative magnitude far away from the source. In comparison to similar experiments in isotropic turbulence, the centreline mean temperature appears to have a comparable decay rate, whereas the centreline mean-square temperature fluctuations appear to decay at a faster rate. The spread of the plume is faster than that in isotropic turbulence. The measured turbulent heat fluxes and triple temperature-velocity correlations demonstrate self-similar features. The development of temperature integral lengthscales and microscales is comparable to that in other heated, uniformly sheared flows, while the temperature p.d.f. and the temperature-velocity joint p.d.f. are distinctly non-Gaussian, especially away from the centreline. The relative magnitudes of the two measured components of the turbulent diffusivity tensor are in agreement with earlier measurements.

The development of the Reynolds stress field was studied for flows in which an initially two-dimensional boundary layer was skewed sideways by a spanwise pressure gradient ahead of an upstream-facing wedge. Two different wedges were used, providing a variation in the boundary-layer skewing. Measurements of all components of the Reynolds stress tensor and all ten triple products were measured using a rotatable cross-wire anemometer. The results show the expected lag of the shear stress vector behind the strain rate. Comparison of the two present experiments with previous data suggests that the lag can be estimated if the radius of curvature of the free-stream streamline is known. The magnitude of the shear stress vector in the plane of the wall is seen to decrease rapidly as the boundary-layer skewing increases. The amount of decrease is apparently related to the skewing angle between the wall and the free stream. The triple products evolve rapidly and profiles in the three-dimensional boundary layer are considerably different than two-dimensional profiles, leaving little hope for gradient transport models for the Reynolds stresses. The simplified model presented by Rotta (1979) performs reasonably well providing that an appropriate value of the T-parameter is chosen.

The non-isotropic effects of solid-body rotation on homogeneous turbulence are investigated in this paper. A spectral formalism using eigenmodes introduces the spectral Coriolis effects more easily and leads to simpler expressions for the integral quadratic terms which come mostly from classical two-point closures. The analysis is then applied to a specific eddy damped quasi-normal Markovian model, which includes the inertial waves regime in the evaluation of triple correlations. This procedure allows for a departure from isotropy by external rotation effects. When started with rigorously isotropic initial data, the various trends observed on the Reynolds stresses and the integral lengthscales remain in accordance with the results from direct simulations; moreover they reflect a very specific spectral angular distribution. Such an angular dependence allows a drain of spectral energy from the parallel to the normal wave vectors (with respect to the rotation axis), and thus appears consistent with a trend toward two-dimensionality.

A calculation is made of the turbulent transport terms (third moments) that occur in the Reynolds stress equation for buoyant and/or sheared fluids. This calculation is based on neglect of a two-time fourth-order cumulant – a weaker approximation than neglect of the usual single-time fourth-order cumulant. The previously used eddy-damping assumption for single point moments is avoided. This assumption is then examined critically. Comparison is afterward made between the turbulent transport terms derived here and those derived previously by the eddy-damping method, and between the respective derivations. Also the dissipation of third moments is calculated. The calculation is formally limited to mean quantities which vary but slowly in space and time, and to small anisotropy.

In this paper we consider the effect of a model boundary-layer flow on the hydrodynamic and morphological stability of a simple model of the solidification of a binary alloy. We conduct a linear analysis and develop asymptotic solutions for large Schmidt number and large Reynolds number. We also present numerical solutions for data appropriate to a lead–tin alloy. We show that for modes parallel to the free-stream velocity the flow is responsible for the appearance of travelling waves and, for common values of the material parameters, may stabilize the morphological stability of the interface. However the morphological stability of modes perpendicular to the free-stream velocity is unaffected by the presence of the flow. The hydrodynamic stability of the boundary layer is very weakly affected by the presence of the interface, which we attribute to the large Schmidt numbers associated with real crystal growth situations.

Linear stability analyses and nonlinear flow simulations reveal several important features of transverse secondary instabilities of two-dimensional Kelvin–Helmholtz billows and Stuart vortices. Vortex pairing is found to be the most rapidly amplified mode in a continuous spectrum of vortex merging instabilities. In certain not uncommon circumstances it is possible for more than two vortices to amalgamate in a single interaction, demonstrating that the phenomenon that has become known as the pairing resonance in fact has a rather low quality factor. Another form of merging instability in which a vortex is deformed and drained by its neighbours has been revealed by our linear stability analyses of nonlinear shear-layer disturbances. It appears, however, that this vortex draining instability may be important only in unstratified or very weakly stratified flows, since in moderately stratified Kelvin–Helmholtz flow, it is replaced by a highly localized instability which leads to a temporary distortion of the braids. Nonlinear simulations of vortex merging events in moderately stratified, high-Reynolds-number shear layers are compared to the theoretical predictions of our stability analyses. We investigate and quantify the sensitivity of merging events to variations in the initial conditions. The character of the flow after merging instability saturates and the nonlinear aspects of multiple merging events are also considered.

An unusually low-frequency oscillation in the flow over an airfoil is studied experimentally as well as computationally. Wind-tunnel measurements are carried out with two-dimensional airfoil models in the chord Reynolds number (Rc) range of 0.15 × 105−3.0 × 105. During deep stall, at α [gsim ] 18°, the usual ‘bluff-body shedding’ occurs at a Strouhal number, Sts ≈ 0.2. But at the onset of static stall around α = 15°, a low-frequency periodic oscillation is observed, the corresponding Sts being an order of magnitude lower. The phenomenon apparently takes place only with a transitional state of the separating boundary layer. Thus, on the one hand, it is not readily observed with a smooth airfoil in a clean wind tunnel, while on the other, it is easily removed by appropriate ‘acoustic tripping’. Details of the flow field for a typical case are compared with a case of bluff-body shedding. The flow field is different in many ways from the latter case and does not involve a Kármán Vortex street. The origin of the flow fluctuations traces to the upper surface of the airfoil and is associated with a periodic switching between stalled and unstalled states. The mechanism of the frequency selection remains unresolved, but any connection to blower instabilities, acoustic standing waves or structural resonances has been ruled out.

A similar result has been encountered computationally using a two-dimensional Navier–Stokes code. While with the assumption of laminar flow, wake oscillation akin to the bluff-body shedding has been observed previously, the Sts is found to drop to about 0.03 when a ‘turbulent’ boundary layer is assumed. Details of the flow field and unsteady forces, computed for the same conditions as in the experiment, compare reasonably well with the experimental data.

The phenomenon produces intense flow fluctuations imparting much larger unsteady forces to the airfoil than that experienced in bluff-body shedding, and may represent the primary aerodynamics of stall flutter of blades and wings.

The onset of instability is investigated in a triply diffusive fluid layer in which the density depends on three stratifying agencies having different diffusivities. It is found that, in some cases, three critical values of the Rayleigh number are required to specify the linear stability criteria. As in the case of another problem requiring three Rayleigh numbers for the specification of linear stability criteria (the rotating doubly diffusive case studied by Pearlstein 1981), the cause is traceable to the existence of disconnected oscillatory neutral curves. The multivalued nature of the stability boundaries is considerably more interesting and complicated than in the previous case, however, owing to the existence of heart-shaped oscillatory neutral curves. An interesting consequence of the heart shape is the possibility of ‘quasi-periodic bifurcation’ to convection from the motionless state when the twin maxima of the heart-shaped oscillatory neutral curve lie below the minimum of the stationary neutral curve. In this case, there are two distinct disturbances, with (generally) incommensurable values of the frequency and wavenumber, that simultaneously become unstable at the same Rayleigh number. This work complements the earlier efforts of Griffiths (1979a), who found none of the interesting results obtained herein.

Binary interactions between water droplets of nearly equal size in the flow field behind a weak shock wave were studied experimentally. The droplets had diameters of about 270 mm, and the Reynolds numbers, based on this diameter and on the relative velocity between the droplets and the free stream, ranged from about 130 to about 600. In this paper we report only data for Re < 150, corresponding to non-deforming droplets. The droplets in a given pair were aligned so that each pair fell on a plane parallel to the direction of the incoming flow. In this manner, the second droplet in the pair was ‘behind’ the first, at horizontal distances ranging from 1.5 to 11 diameters, and at vertical distances from the dividing streamline ranging from −3 to 6 diameters. We have quantified the interaction in terms of drag force changes on the droplets, and show that the first, or upstream, droplet is not affected by the second, but that the second experiences significant reductions for vertical distances of about one droplet diameter or less. At the smallest horizontal distances, the maximum decrease observed was about 50%, relative to its isolated value. We also show that the drag changes clearly demarcate a wake behind the first droplet. Further, on the basis of these changes, we define a region of influence attached to the first droplet, where the free-stream velocity is significantly reduced. For the droplets used in this study, this region is a slender paraboloid of revolution, having a length of about 15 diameters and a radius of about one diameter.

An effluent outlet within thirty channel breadths upstream of a branching in a river can lead to high shoreline pollution levels along one of the branches, or near the tip of the central region. This paper identifies an optimal splitting of a steady discharge that minimizes the shoreline pollution.