The stability is investigated of the swirling flow between two concentric cylinders in the presence of stable axial linear density stratification, for flows not satisfying the well-known Rayleigh criterion for inviscid centrifugal instability, d(Vr)2/dr < 0. We show by a linear stability analysis that a sufficient condition for non-axisymmetric instability is, in fact, d(V/r)2/dr < 0, which implies a far wider range of instability than previously identified. The most unstable modes are radially smooth and occur for a narrow range of vertical wavenumbers. The growth rate is nearly independent of the stratification when the latter is strong, but it is proportional to it when it is weak, implying stability for an unstratified flow. The instability depends strongly on a non-dimensional parameter, S, which represents the ratio between the strain rate and twice the angular velocity of the flow. The instabilities occur for anti-cyclonic flow (S < 0). The optimal growth rate of the fastest-growing mode, which is non-oscillatory in time, decays exponentially fast as S (which can also be considered a Rossby number) tends to 0. The mechanism of the instability is an arrest and phase-locking of Kelvin waves along the boundaries by the mean shear flow. Additionally, we identify a family of (probably infinitely many) unstable modes with more oscillatory radial structure and slower growth rates than the primary instability. We determine numerically that the instabilities persist for finite viscosity, and the unstable modes remain similar to the inviscid modes outside boundary layers along the cylinder walls. Furthermore, the nonlinear dynamics of the anti-cyclonic flow are dominated by the linear instability for a substantial range of Reynolds numbers.

A temporal, inviscid, linear stability analysis of a liquid jet and the co-flowing gas stream surrounding the jet has been performed. The basic liquid and gas velocity profiles have been computed self-consistently by solving numerically the appropriate set of coupled Navier–Stokes equations reduced using the slenderness approximation. The analysis in the case of a uniform liquid velocity profile recovers the classical Rayleigh and Weber non-viscous results as limiting cases for well-developed and very thin gas boundary layers respectively, but the consideration of realistic liquid velocity profiles brings to light new families of modes which are essential to explain atomization experiments at large enough Weber numbers, and which do not appear in the classical stability analyses of non-viscous parallel streams. In fact, in atomization experiments with Weber numbers around 20, we observe a change in the breakup pattern from axisymmetric to helicoidal modes which are predicted and explained by our theory as having an hydrodynamic origin related to the structure of the liquid-jet basic velocity profile. This work has been motivated by the recent discovery by Gañán-Calvo (1998) of a new atomization technique based on the acceleration to large velocities of coaxial liquid and gas jets by means of a favourable pressure gradient and which are of emerging interest in microfluidic applications (high-quality atomization, micro-fibre production, biomedical applications, etc.).

Turbulent flow in a rectangular channel is investigated to determine the scale and pattern of the eddies that contribute most to the total turbulent kinetic energy and the Reynolds shear stress. Instantaneous, two-dimensional particle image velocimeter measurements in the streamwise-wall-normal plane at Reynolds numbers Reh = 5378 and 29 935 are used to form two-point spatial correlation functions, from which the proper orthogonal modes are determined. Large-scale motions – having length scales of the order of the channel width and represented by a small set of low-order eigenmodes – contain a large fraction of the kinetic energy of the streamwise velocity component and a small fraction of the kinetic energy of the wall-normal velocities. Surprisingly, the set of large-scale modes that contains half of the total turbulent kinetic energy in the channel, also contains two-thirds to three-quarters of the total Reynolds shear stress in the outer region. Thus, it is the large-scale motions, rather than the main turbulent motions, that dominate turbulent transport in all parts of the channel except the buffer layer. Samples of the large-scale structures associated with the dominant eigenfunctions are found by projecting individual realizations onto the dominant modes. In the streamwise wall-normal plane their patterns often consist of an inclined region of second quadrant vectors separated from an upstream region of fourth quadrant vectors by a stagnation point/shear layer. The inclined Q4/shear layer/Q2 region of the largest motions extends beyond the centreline of the channel and lies under a region of fluid that rotates about the spanwise direction. This pattern is very similar to the signature of a hairpin vortex. Reynolds number similarity of the large structures is demonstrated, approximately, by comparing the two-dimensional correlation coefficients and the eigenvalues of the different modes at the two Reynolds numbers.

A self-induced free-surface oscillation termed ‘self-induced sloshing’ was observed in a rectangular tank with a submerged and horizontally injected water jet. Self-induced sloshing is excited by the flow itself without any external force. Its behaviour was examined by experiment. The dominant frequency was found to be close to the first or second eigenvalue of fluid in a tank. The conditions of sloshing excitation were obtained for four tank geometries. They were called the ‘sloshing condition’, and defined in terms of inlet velocity and water level. Sloshing conditions were found to be strongly dependent on inlet velocity and tank geometry. A two-dimensional numerical simulation code was developed to simulate self-induced sloshing. The code was based on the boundary-fitted coordinate (BFC) method with height function. The numerical results were qualitatively verified by the experimental results, and were found to correlate well in terms of flow pattern, free-surface shape and sloshing conditions. In this study, sloshing growth was evaluated quantitatively using the simulation results. Oscillation energy supplied for the sloshing motion during a sloshing period (Econ) was calculated from simulation results. Sloshing growth was found to be strongly related to the sign and magnitude of Econ. The distribution of Econ showed that jet flow had a strong correlation with the sloshing growth. It was clarified that sloshing growth was primarily dependent on the spatial phase state of jet fluctuation. A governing parameter of self-induced sloshing, the modified Strouhal number Sts, was proposed on the basis of numerical evaluations of oscillation energy. The value of Sts suggests that one or two large vortices generated by jet fluctuations exist between the inlet and outlet during a sloshing period. When Sts is approximately either 1 (first stage) or 2 (second stage), self-induced sloshing occurs consistently in all experimental cases. The dependence of sloshing on inlet velocity, water level and tank geometry was revealed using Sts. For several tank geometries, a sloshing mode shift or jet mode (stage) transition was found to occur due to changes in inlet jet velocity. The combination of sloshing mode and jet stage can determine the state of the self-induced sloshing. As a result of this study, we propose a new excitation mechanism of self-induced sloshing, represented by a simple feedback loop closed by sloshing motion and jet fluctuation. The overall physical oscillation mechanism of self-induced sloshing was clarified using this feedback loop.

A new implementation of the conventional Stokesian Dynamics (SD) algorithm, called accelerated Stokesian Dynamics (ASD), is presented. The equations governing the motion of N particles suspended in a viscous fluid at low particle Reynolds number are solved accurately and efficiently, including all hydrodynamic interactions, but with a significantly lower computational cost of O(N ln N). The main differences from the conventional SD method lie in the calculation of the many-body long-range interactions, where the Ewald-summed wave-space contribution is calculated as a Fourier transform sum and in the iterative inversion of the now sparse resistance matrix. The new method is applied to problems in the rheology of both structured and random suspensions, and accurate results are obtained with much larger numbers of particles. With access to larger N, the high-frequency dynamic viscosities and short-time self-diffusivities of random suspensions for volume fractions above the freezing point are now studied. The ASD method opens up an entire new class of suspension problems that can be investigated, including particles of non-spherical shape and a distribution of sizes, and the method can readily be extended to other low-Reynolds-number-flow problems.

The interaction of two conical shock waves, one converging and straight and the other diverging and curvilinear, in an axisymmetric flow was investigated both experimentally and numerically. A double-loop hysteresis was discovered in the course of the experimental investigation. The double-loop hysteresis consisted of a major one, associated with the interaction between the boundary layer and the wave configuration, and a minor one, associated with the dual-solution phenomenon, which is known to be non-viscous-dependent. The minor hysteresis loop was found to be an internal hysteresis loop of the major one. As expected the numerical Euler calculations failed to detect the viscous-dependent major hysteresis loop but did succeed in obtaining the non-viscous-dependent minor (internal) hysteresis loop. In addition, multiple hysteresis loops, associated with the interaction between the shock wave configuration and the edge of the curvilinear mobile cone were also observed. The non-viscous minor hysteresis loop involved different overall shock wave reflection configurations, and the other hysteresis loops involved the same shock wave reflection configuration but different flow patterns.

Thin wires are attached on the outer surface and parallel to the axis of a smooth circular cylinder in a steady cross-stream, modelling the effect of protrusions and attachments. The impact of the wires on wake properties, and vortex-induced loads and vibration are studied at Reynolds numbers up to 4.6 × 104, with 3.0 × 104 as a focus point. For a stationary cylinder, wires cause significant reductions in drag and lift coefficients, as well as an increase in the Strouhal number to a value around 0.25–0.27. For a cylinder forced to oscillate harmonically, the main observed wire effects are: (a) an earlier onset of frequency lock-in, when compared with the smooth cylinder case; (b) at moderate amplitude/cylinder diameter (A/D) ratios (0.2 and 0.5), changes in the phase of wake velocity and of lift with respect to motion are translated to higher forcing frequencies, and (c) at A/D = 1.0, no excitation region exists; the lift force is always dissipative.

The flow-induced response of a flexibly mounted cylinder with attached wires is significantly altered as well, even far away from lock-in. Parameterizing the response using nominal reduced velocity Vrn = U/fnD, we found that frequency lock-in occurs and lift phase angles change through 180° at Vrn [thkap ] 4.9; anemometry in the wake confirms that a mode transition accompanies this premature lock-in. A plateau of constant response is established in the range Vrn = 5.1–6.0, reducing the peak amplitude moderately, and then vibrations are drastically reduced or eliminated above Vrn = 6.0. The vortex-induced vibration response of the cylinder with wires is extremely sensitive to angular bias near the critical value of Vrn = 6.0, and moderately so in the regime of suppressed vibration.

When a drop moves in a uniform vertical temperature gradient under the combined action of gravity and thermocapillarity at small values of the thermal Péclet number, it is shown that inclusion of inertia is crucial in the development of an asymptotic solution for the temperature field. If inertia is completely ignored, use of the method of matched asymptotic expansions, employing the Péclet number (known as the Marangoni number) as the small parameter, leads to singular behaviour of the outer temperature field. The origin of this behaviour can be traced to the interaction of the slowly decaying Stokeslet, arising from the gravitational contribution to the motion of the drop, with the temperature gradient field far from the drop. When inertia is included, and the method of matched asymptotic expansions is used, employing the Reynolds number as a small parameter, the singular behaviour of the temperature field is eliminated. A result is obtained for the migration velocity of the drop that is correct to O(Re2 log Re).

Theory and lattice-Boltzmann simulations are used to examine the effects of fluid inertia, at small Reynolds numbers, on flows in simple cubic, face-centred cubic and random arrays of spheres. The drag force on the spheres, and hence the permeability of the arrays, is determined at small but finite Reynolds numbers, at solid volume fractions up to the close-packed limits of the arrays. For small solid volume fraction, the simulations are compared to theory, showing that the first inertial contribution to the drag force, when scaled with the Stokes drag force on a single sphere in an unbounded fluid, is proportional to the square of the Reynolds number. The simulations show that this scaling persists at solid volume fractions up to the close-packed limits of the arrays, and that the first inertial contribution to the drag force relative to the Stokes-flow drag force decreases with increasing solid volume fraction. The temporal evolution of the spatially averaged velocity and the drag force is examined when the fluid is accelerated from rest by a constant average pressure gradient toward a steady Stokes flow. Theory for the short- and long-time behaviour is in good agreement with simulations, showing that the unsteady force is dominated by quasi-steady drag and added-mass forces. The short- and long-time added-mass coefficients are obtained from potential-flow and quasi-steady viscous-flow approximations, respectively.

Lattice-Boltzmann simulations are used to examine the effects of fluid inertia, at moderate Reynolds numbers, on flows in simple cubic, face-centred cubic and random arrays of spheres. The drag force on the spheres, and hence the permeability of the arrays, is calculated as a function of the Reynolds number at solid volume fractions up to the close-packed limits of the arrays. At Reynolds numbers up to O(102), the non-dimensional drag force has a more complex dependence on the Reynolds number and the solid volume fraction than suggested by the well-known Ergun correlation, particularly at solid volume fractions smaller than those that can be achieved in physical experiments. However, good agreement is found between the simulations and Ergun's correlation at solid volume fractions approaching the close-packed limit. For ordered arrays, the drag force is further complicated by its dependence on the direction of the flow relative to the axes of the arrays, even though in the absence of fluid inertia the permeability is isotropic. Visualizations of the flows are used to help interpret the numerical results. For random arrays, the transition to unsteady flow and the effect of moderate Reynolds numbers on hydrodynamic dispersion are discussed.

The anisotropy of small-scale temperature fluctuations in shear flows is analysed by making measurements in high-Reynolds-number atmospheric surface layers. A spherical harmonics representation of the moments of scalar increments is proposed, such that the isotropic part corresponds to the index j = 0 and increasing degrees of anisotropy correspond to increasing j. The parity and angular dependence of the odd moments of the scalar increments show that the moments cannot contain any isotropic part (j = 0), but can be satisfactorily represented by the lowest-order anisotropic term corresponding to j = 1. Thus, the skewnesses of scalar increments (and derivatives) are inherently anisotropic quantities, and are not suitable indicators of the tendency towards isotropy.

Observed atmospheric and oceanic internal wave spectra, when analysed in an Eulerian frame of reference, exhibit a large-wavenumber ‘tail’. In one-dimensional vertical-wavenumber (k3) spectra, it is typically proportional to |k3|−3.

In 1989, K. R. Allen and R. I. Joseph showed that a large-wavenumber tail was to be anticipated as a consequence of Eulerian nonlinearity, and they derived relations for the coefficients of both horizontal and vertical spectra of the form |k|−3. The coefficients were obtained only for the wave-induced vertical-displacement spectra, and only for an input spectrum having a certain ‘canonical’ frequency variation derived on other grounds.

The present work builds on that of Allen & Joseph. It is more general in some respects, more limited in others. It provides a more transparent form of analysis, it treats a broad class of wave variables, and it does so for input (Lagrangian) spectra that can be chosen by the user, free from any constraint to canonical or other restricted forms. It provides relations whereby the full Eulerian spectrum may be determined numerically, once the input spectrum has been chosen, and it provides analytic forms applicable at large wavenumbers for horizontally isotropic spectra. The derived one-dimensional vertical-wavenumber spectra are discussed in relation to observations.

Certain shortcomings in the development, both as given by Allen & Joseph and as found here, are identified and discussed.

Most reservoirs contain stratified fluid and selective withdrawal is used to obtain water with the desired properties. We initially deal with a layered density distribution. The theory for the critical discharge for a single layer and a point sink is reviewed and extended to cover the case where there is gate discharge (a line sink). The theory for the case when the upper layer depth is large and the flow is coming from both layers is reviewed and it is shown that the valve controls the discharge and a virtual control determines the ratio of the discharge in each layer. This virtual control moves further from the valve as the total discharge increases. We determine the position of the virtual control and the criteria for the maximum for two layers when the upper layer is finite and below a stationary layer. Before this maximum, we show that when the discharge is increased above the critical discharge for the single layer, the finite upper layer does not affect the ratio of the flows from each layer until the virtual control reaches that for the maximum discharge. At this stage, the upper layer becomes tangential to the dam face and this condition and the smoothness of the lower interface determine both the total discharge and the ratio of the flow from each layer. Indeed, at this stage, virtual control and the control of the discharge are at the same section.

In a similar way, with a stationary layer above and below two flowing layers, we derive the maximum discharge from the two flowing layers. For this case, the solution is self-similar. This is then extended to a stable stratified continuous density distribution. The experiments of Gariel (1949) and Lawrence & Imberger (1979) suggest that the predictions of the theory are within the experimental errors.

McDonald (1998) has studied the motion of an intense, quasi-geostrophic, equivalent-barotropic, singular vortex near an infinitely long escarpment. The present work considers the remaining cases of the motion of weak and moderate intensity singular vortices near an escarpment. First, the limit that the vortex is weak is studied using linear theory. For times which are short compared to the advective time scale associated with the vortex it is found that topographic waves propagate rapidly away from the vortex and have no leading-order influence on the vortex drift velocity. The vortex propagates parallel to the escarpment in the sense of its image in the escarpment. The mechanism for this motion is identified and is named the pseudoimage of the vortex. Large-time asymptotic results predict that vortices which move in the same direction as the topographic waves radiate non-decaying waves and drift slowly towards the escarpment in response to wave radiation. Vortices which move in the opposite direction to the topographic waves reach a steadily propagating state. Contour dynamics results reinforce the linear theory in the limit that the vortex is weak, and show that the linear theory is less robust for vortices which move counter to the topographic waves. Second, contour dynamics results for a moderate intensity vortex are given. It is shown that dipole formation is a generic feature of the motion of moderate intensity vortices and induces enhanced motion in the direction perpendicular to the escarpment.

This paper examines and compares spectral measurements from a turbulent round jet and a turbulent boundary layer. The conjecture that is examined is that both flows consist of coherent structures immersed in a background of isotropic turbulence. In the case of the jet, a single size of coherent structure is considered, whereas in the boundary layer there are a range of sizes of geometrically similar structures. The conjecture is examined by comparing experimental measurements of spectra for the two flows with the spectra calculated using models based on simple vortex structures. The universality of the small scales is considered by comparing high-wavenumber experimental spectra. It is shown that these simple structural models give a good account of the turbulent flows.

We consider the stability of the steady free-surface thin-film flows over topography examined in detail by Kalliadasis et al. (2000). For flow over a step-down, their computations revealed that the free surface develops a ridge just before the entrance to the step. Such capillary ridges have also been observed in the contact line motion over a planar substrate, and are a key element of the instability of the driven contact line. In this paper we analyse the linear stability of the ridge with respect to disturbances in the spanwise direction. It is shown that the operator of the linearized system has a continuous spectrum for disturbances with wavenumber less than a critical value above which the spectrum is discrete. Unlike the driven contact line problem where an instability grows into well-defined rivulets, our analysis demonstrates that the ridge is surprisingly stable for a wide range of the pertinent parameters. An energy analysis indicates that the strong stability of the capillary ridge is governed by rearrangement of fluid in the flow direction flowing to the net pressure gradient induced by the topography at small wavenumbers and by surface tension at high wavenumbers.