A discrepancy exists between the predictions of analytical solutions of approximate Navier–Stokes equations and numerical finite-difference solutions of the full Navier–Stokes equations regarding the development of laminar flow at the entrance to cylindrical pipes for Newtonian fluids. Starting from a uniform velocity profile at the entrance to the pipe, analytical solutions of approximate Navier–Stokes equations predict the velocity profile to have a maximum at the centre of the pipe at all times. In contrast, numerical finite-difference solutions of the full Navier–Stokes equations have suggested that the location of the velocity maximum moves from the wall towards the centre of the pipe at a short distance from the entrance, after which it remains at the centre of the pipe. This study presents the first experimental evidence of the moving velocity maximum from the wall towards the centre of the pipe. The initial uniform velocity profile was achieved by flowing the fluid through a monolith composed of narrow parallel channels and the flow development was investigated using magnetic resonance velocimetry. The experimentally observed variation of the position and size of the velocity maximum with the Reynolds number and the distance from the entrance to the pipe is shown to be in good agreement with the predictions of numerical finite-difference solutions of the full Navier–Stokes equations.

This theoretical investigation of steady fluid flow through a rigid three-dimensional branching geometry is motivated by applications to haemodynamics in the brain especially, while the flow through a tube with a blockage or through a collapsed tube provides another motivation with a biomedical background. Three-dimensional motion without symmetry is addressed through one mother vessel to two or several daughters. A comparatively long axial length scale of the geometry leads to a longitudinal vortex system providing a slender-flow model for the complete mother-and-daughters flow response. Computational studies and subsequent analysis, along with comparisons, are presented. The relative flow rate varies in terms of an effective Reynolds number dependence, allowing a wide range of flow rates to be examined theoretically; also any rigid cross-sectional shape and ratio of cross-sectional area expansion or contraction from the mother vessel to the daughters can be accommodated in principle in both the computations and the analysis. Swirl production with substantial crossflows is found. The analysis shows that close to any carina (the ridge separating daughter vessels) or carinas at a branch junction either forward or reversed motion can be observed locally at the saddle point even though the bulk of the motion is driven forward into the daughters. The local forward or reversed motion is controlled, however, by global properties of the geometry and incident conditions, a feature which applies to any of the flow rates examined.

A new amplitude equation is derived for high-frequency acoustic waves propagating through an incompressible vortical flow using multi-time-scale asymptotic analysis. The reduced model is derived without an explicit spatial-scale separation ansatz between the wave and vortical fields. As a consequence, the model is seen to capture very well the features of the wave field in the regime where the spatial scales of the wave and vortical fields are comparable, a regime for which an optimal reduced model does not seem to be available.

The dynamical behaviour of a system of parallel line vortices in an inviscid fluid is studied numerically. The initial configuration of the system is assumed to be such that the points of intersection of the line vortices with a plane normal to the vorticity form a regular polygon. The numerical experiments show that the vortex polygon is rearranged due to non-linear interactions among the line vortices in such a way as to produce a more or less uniform distribution of vortices inside the fluid with an approximately constant mean separation. The average angular velocity of the rotation of the vortex lines about the instantaneous centroid of the vortex system remains approximately constant. These results agree with the conjecture of Raja Gopal (1964). The results may prove to be of some value in a macroscopic model of liquid helium based on hydrodynamical principles.

We investigate in detail the dynamics of bouncing of a fluid-immersed solid sphere onto a textured wall at moderate Reynolds and Stokes numbers. Using high-frequency interferometric measurements, the dynamics of the sphere is resolved in time and space, before, during and after collision with the wall. The critical Stokes number for bouncing is shown to be significantly influenced by the geometry of the texture, i.e. the surface fraction and the height of the micro-pillars. A modified Hertz model is developed to take into account the influence of this texture geometry on the collision dynamics. The predicted scaling for the collision time and penetration depth of the sphere into the textured wall is found to be in good agreement with the experimental measurements.

We present the first solution to a boundary motion planning problem for the Navier–Stokes equations, linearized around the parabolic equilibrium in a three-dimensional channel flow. The pressure and skin friction at one wall are chosen as the reference outputs as they are the most readily measurable ‘wall-restricted’ quantities in experimental fluid dynamics and also because they play a special role as performance metrics in aerodynamics. The reference velocity input is applied at the opposite wall. We find the exact (method independent) solution to the motion planning problem using the PDE (partial differential equation) backstepping theory. The motion planning solution results in open-loop controls, which produce the reference output trajectories only under special initial conditions for the flow velocity field. To achieve convergence to the reference trajectory from other (nearby) initial conditions, we design a feedback controller. We also present a detailed examination of the closed-form solutions for gains and the behaviour of the motion planning solution as the wavenumbers grow or the Reynolds number grows. Numerical results are shown for the motion planning problem.

Numerical solutions of finite-difference approximations to the Navier–Stokes equations have been obtained for the axisymmetric motion of a Boussinesq liquid in a rigidly bounded rotating annulus. For most of the cases studied, a temperature difference is maintained between the top and bottom surfaces such that essentially a basic stable density stratification is imposed on the fluid. The side walls are thermally insulated and the motion is driven by a differential rotation of the top surface. Approximate steady-state solutions are obtained for various values of the Rossby number ε and the stratification parameter S = N2/Ω2, where N is the Brunt–Väisälä frequency and Ω is the rotational frequency. The changes in the flow field with the variation of these parameters is studied. Particular attention is given to an investigation of the meridional, or up welling, circulation and its dependence on the stratification parameter. The effects on the flow of different boundary conditions, such as an applied stress driving, specified temperature at the side walls and an applied heat flux at the top and bottom surfaces, are also investigated.

The dynamics of a near-surface vortex are examined in a two-layer setting on the beta-plane. Initially, the vortex is radially symmetric and localized in the upper layer. Two non-dimensional parameters govern its evolution and translation: the ratio δ of the thickness of the vortex to the total depth of the fluid, and the non-dimensional beta-effect number α = βL/f (f and β are the Coriolis parameter and its meridional gradient respectively, L is the radius of the vortex). We assume, as suggested by oceanic observations, that α < δ < 1: A simple set of asymptotic equations is derived, which describes the beta-induced translation of the vortex and a dipolar perturbation developing on and under the vortex (in both layers).

This set was solved numerically for oceanic lenses, and the following features were observed: (i) The meridional (southward) component of the translation speed of the lens rapidly ‘overtakes’ the zonal (westward) component. The former grows approximately linearly, whereas the latter oscillates about the Nof (1981) value (i.e. about the speed of translation of a vortex in a one-layer reduce-gravity fluid). (ii) Vortices of the same shape, but different radii and amplitudes, follow the same trajectory. The amplitude and radius affect only the absolute value, but not the direction, of the translation speed. (iii) In the lower layer below the vortex, a ‘region’ is generated where the velocity of the fluid is growing linearly with time. The velocity field in the region becomes more and more homogeneous (and equal to the translation speed of the vortex).

Experiments are described in which a body was towed at large Richardson numbers through a uniformly stratified fluid, with particular attention being given to the parameter range in which diffusive effects are likely to play an important role. When the Richardson number is only moderately large (i.e. less than about 106) it appears that a linear viscous diffusive theory fails to model the experimental observations, but at larger Richardson numbers it seems that such a theory is a strong candidate for modelling the physical situation. At Richardson numbers of about 103 a non-diffusive viscous model (see the theories of Graebel 1969; Janowitz 1971) appears to give a fairly good description of the experimental results, except for some discrepancies between the theories and the observed flows in the wake downstream from the body. However, the conditions under which the experiments were carried out were not completely favourable for the application of these theories.

A brief survey and new interpretations of some of the work on the rotatingfluid counterparts of the present experiments are also given. In view of the theoretical similarities between the two situations we have tried to assess and compare (where possible) the results of experimental investigations in each field and we feel that the observations in the rotating-fluid experiments are in good qualitative agreement with the present results. No measurements of the drag acting on the body were made in the present experiments but this has been done carefully in the rotating-fluid experiments, and the results of those measurements are in poor agreement with the theoretical predictions. Because of the close similarities between the axisymmetric model for viscous rotating fluids (cf. Moore & Saffman 1969) and the two-dimensional model for diffusive stratified fluids (Freund & Meyer 1972), an explanation for the discrepancy should be sought before the linear viscous diffusive model can be applied with any confidence to the flow of stratified fluids at very large Richardson numbers.

We examine the flows induced by horizontally modulated, vertically confined (or guided), internal wavepackets in a stratified, Boussinesq fluid. The wavepacket induces both an Eulerian flow and a Stokes drift, which together determine the Lagrangian transport of passive tracers. We derive equations describing the wave-induced flows in arbitrary stable stratification and consider four special cases: a two-layer fluid, symmetric and asymmetric piecewise constant (‘top-hat’) stratification and, more representative of the ocean, exponential stratification. In a two-layer fluid, the Stokes drift is positive everywhere with the peak value at the interface, whereas the Eulerian flow is negative and uniform with depth for long groups. Combined, the net depth-integrated Lagrangian transport is zero. If one layer is shallower than the other, the wave-averaged interface displaces into that layer making the Eulerian flow in that layer more negative and the Eulerian flow in the opposite layer more positive so that the depth-integrated Eulerian transports are offset by the same amount in each layer. By contrast, in continuous stratification the depth-integrated transport due to the Stokes drift and Eulerian flow are each zero, but the Eulerian flow is singular if the horizontal phase speed of the induced flow equals the group velocity of the wavepacket, giving rise to a single resonance in uniform stratification (McIntyre, J. Fluid Mech., vol. 60, 1973, pp. 801–811). In top-hat stratification, this single resonance disappears, being replaced by multiple resonances occurring when the horizontal group velocity of the wavepacket matches the horizontal phase speed of higher-order modes. Furthermore, if the stratification is not vertically symmetric, then the Eulerian induced flow varies as the inverse squared horizontal wavenumber for shallow waves, the same as for the asymmetric two-layer case. This ‘infrared catastrophe’ also occurs in the case of exponential stratification suggesting significant backward near-surface transport by the Eulerian induced flow for modulated oceanic internal modes. Numerical simulations are performed confirming these theoretical predictions.

The stability of forward and rearward slug flows due to the motion of a body through a perfectly conducting liquid with an embedded magnetic field aligned to the direction of motion is studied. It is found that, for sub-Alfvénic flows, the forward and rearward slug flows predicted by Stewartson, Leibovich and Ludford are stable. For super-Alfvénic flows, if at all meaningful, both the forward and the rearward slug flows are stable also.

Spin-up in a rectangular container with a free surface is investigated numerically and experimentally. In the formulation of two-dimensional numerical computation, we use a potential-like function in addition to the stream function to deal with the first-order Ekman pumping model. It is shown that our numerical results are in good agreement with those obtained by the experiment when either the leading-order or first-order pumping model is used. On the other hand, when no pumping effect is considered the numerical results show, except in the initial development, a considerable discrepancy from those of the experiment. Our attention in this study is focused on clarifying the physical mechanism of cyclonic vortex merging. At low Reynolds numbers and/or liquid depths the Ekman pumping damps the vortical flows fast, resulting in non-merging. At moderate Reynolds numbers, it enhances merging because the cyclonic vortices expand due to the Ekman pumping. We discuss the influence of various parameters, including Reynolds number, Rossby number, and dimensionless liquid depth, on the evolution of the vortical flows.

The linear stability of the impulsively started flow through a pipe of circular cross-section is studied at high Reynolds number R. A crucial non-dimensional time of O(R7/9) is identified at which the disturbance acquires internal flow characteristics. It is shown that even if the disturbance amplitude at this time is as small as O(R−22/27) the subsequent evolution of the perturbation is nonlinear, although it can still be followed analytically using a multiple-scales approach. The amplitude and wave speed of the nonlinear disturbance are calculated as functions of time and we show that as t → ∞, the disturbance evolves into the long-wave limit of the neutral mode structure found by Smith & Bodonyi in the fully developed Hagen–Poiseuille flow, into which our basic flow ultimately evolves. It is proposed that the critical amplitude found here forms a stability boundary between the decay of linear disturbances and ‘bypass’ transition, in which the fully developed state is never attained.

A theory is developed for the speed and structure of steady-state non-dissipative gravity currents in rotating channels. The theory is an extension of that of Benjamin (1968) for non-rotating gravity currents, and in a similar way makes use of the steady-state and perfect-fluid (incompressible, inviscid and immiscible) approximations, and supposes the existence of a hydrostatic ‘control point’ in the current some distance away from the nose. The model allows for fully non-hydrostatic and ageostrophic motion in a control volume V ahead of the control point, with the solution being determined by the requirements, consistent with the perfect-fluid approximation, of energy and momentum conservation in V, as expressed by Bernoulli's theorem and a generalized flow-force balance. The governing parameter in the problem, which expresses the strength of the background rotation, is the ratio W = B/R, where B is the channel width and R = (g′H)1/2/f is the internal Rossby radius of deformation based on the total depth of the ambient fluid H. Analytic solutions are determined for the particular case of zero front-relative flow within the gravity current. For each value of W there is a unique non-dissipative two-layer solution, and a non-dissipative one-layer solution which is specified by the value of the wall-depth h0. In the two-layer case, the non-dimensional propagation speed c = cf(g′H)−1/2 increases smoothly from the non-rotating value of 0.5 as W increases, asymptoting to unity for W → ∞. The gravity current separates from the left-hand wall of the channel at W = 0.67 and thereafter has decreasing width. The depth of the current at the right-hand wall, h0, increases, reaching the full depth at W = 1.90, after which point the interface outcrops on both the upper and lower boundaries, with the distance over which the interface slopes being 0.881R. In the one-layer case, the wall-depth based propagation speed Froude number c0 = cf(g′h0)−1/2 = 21/2, as in the non-rotating one-layer case. The current separates from the left-hand wall of the channel at W0 ≡ B/R0 = 2−1/2, and thereafter has width 2−1/2R0, where R0 = (g′h0)1/2/f is the wall-depth based deformation radius.

We develop asymptotic solutions for passive suspended sediment transport under a flood surge on a uniform slope. Our solutions provide predictions of the net scour under a surge, and simple estimates of the conditions under which it may ‘bulk up’ into a mud or debris flow, as well as illustrating their sensitivity to sediment entrainment rates.

An upper bound on the rate of vertical heat transport is established in three dimensions for stress-free velocity boundary conditions on horizontally periodic plates. A variation of the background method is implemented that allows negative values of the quadratic form to yield ‘small’ ($O\left(1/ \mathit{Pr}\right)$) corrections to the subsequent bound. For large (but finite) Prandtl numbers this bound is an improvement over the ‘ultimate’ $R{a}^{1/ 2} $ scaling and, in the limit of infinite $Pr$, agrees with the bound of $R{a}^{5/ 12} $ recently derived in that limit for stress-free boundaries.