A new method of generating turbulence in a stratified fluid is presented. The flow is forced by a symmetric array of sources and sinks placed around the perimeter of a tank containing stratified fluid. The sources and sinks are located in a horizontal plane and the flow from the sources is directed horizontally, so that fluid is withdrawn from and re-injected at its neutral density level with some horizontal momentum. The sources and sinks are arranged so that no net impulse or angular momentum is imparted to the flow. Measurements of the mixing produced by the turbulence are made using a conductivity probe to record the vertical density profile. The flow field is measured by tracking small neutrally buoyant particles which are placed within the fluid. The tracking of the particles and analysis of the flow fields are done automatically using DigImage, a recently developed suite of particle tracking software. The characteristics of the flow are found to depend on the forcing parameter F = V/Nd, where V is the mean velocity of the flow through the source orifices, d is the diameter of the sources and sinks and N is the buoyancy frequency of the stratification. At large F three-dimensional turbulence is produced within a mixed layer centred on the level of the sources and sinks. A comparison of mixing rates measured in this and more conventional experiments is made, and it is concluded that in terms of the local turbulence parameters the entrainment rates are similar. At low F, no significant mixing occurs and the flow is approximately two-dimensional with very small vertical velocities. Under these circumstances a qualitative change in the characteristics of the flow occurs after the experiment has been running for some hours. It is observed that the scale of the motion increases until there is an accumulation of the energy at the largest scale that can be accommodated within the tank. The structure of this large-scale circulation is analysed and it is found that a form of vorticity expulsion from the interior of the circulation has occurred. These results are compared with numerical simulations of two-dimensional turbulence, and some measurements of turbulent decay are discussed.

The main goal of this paper is to clarify the spatial instability of a piecewise linear free shear flow. We do this by obtaining numerical solutions to the Orr–Sommerfeld equation at high Reynolds numbers. The velocity profile chosen is very much like a piecewise linear one, with the exception that the corners have been rounded so that the entire profile is infinitely differentiable. We find that the (viscous) spatial instability of this modified profile is virtually identical to the inviscid spatial instability of the piecewise linear profile and agrees qualitatively with the inviscid results for the tanh profile when the shear layers are convectively unstable. The unphysical features, previously identified for the piecewise linear velocity profile, arise only when the flow is absolutely unstable. In a nutshell, we see nothing wrong with the inviscid spatial instability of piecewise linear shear flows.

The topology of vortex lines and surfaces is examined in incompressible viscous axisymmetric flows with swirl. We argue that the evolving topology of the vorticity field must be examined in terms of axisymmetric vortex surfaces rather than lines, because only the surfaces enjoy structural stability. The meridional cross-sections of these surfaces are the orbits of a dynamical system with the azimuthal circulation being a Hamiltonian H and with time as a bifurcation parameter μ. The dependence of H on μ is governed by the Navier–Stokes equations; their numerical solutions provide H. The level curves of H establish a time history for the motion of vortex surfaces, so that the circulation they contain remains constant. Equivalently, there exists a virtual velocity field in which the motion of the vortex surfaces is frozen almost everywhere; the exceptions occur at critical points in the phase portrait where the virtual velocity is singular. The separatrices emerging from saddle points partition the phase portrait into islands; each island corresponds to a structurally stable vortex structure. By using the flux of the meridional vorticity field, we obtain a precise definition of reconnection: the transfer of flux between islands. Local analysis near critical points shows that the virtual velocity (because of its singular behaviour) performs ‘cut-and-connect’ of vortex surfaces with the correct rate of circulation transfer - thereby validating the long-standing viscous ‘cut-and-connect’ scenario which implicitly assumes that vortex surfaces (and vortex lines) can be followed over a short period of time in a viscous fluid. Bifurcations in the phase portrait represent (contrary to reconnection) changes in the topology of the vorticity field, where islands spontaneously appear or disappear. Often such topology changes are catastrophic, because islands emerge or perish with finite circulation. These and other phenomena are illustrated by direct numerical simulations of vortex rings at a Reynolds number of 800.

For many years there has been debate regarding why shock wave reflection off a solid surface has allowed regular reflection to persist beyond the incidence angles where it becomes theoretically impossible. Theory predicts that at some limiting angle the reflection point will move away from the wall and Mach reflection will occur. Previous studies have suggested that the paradox could be due to the presence of the growing viscous boundary layer immediately behind the point of reflection, and some numerical studies support this view. This paper takes the approach of establishing an experimental facility in which the theoretical assumptions regarding the surface of reflection are met, i.e. that the reflecting surface is perfectly smooth, perfectly rigid, and adiabatic. This is done by constructing a bifurcated shock tube facility in which a shock wave is split into two plane waves that are then allowed to reflect off each other at the trailing edge of wedge. The plane of symmetry between the waves then acts as the perfect reflection surface.

Through a careful set of visualization experiments, and the use of multivariate analysis to take account of the uncertainty in shock Mach number, triple-point trajectory angle, and slightly different shock wave arrival times at the trailing edge, the current work shows that the transition from one type of reflection to the other does indeed occur at the theoretical value. Conventional tests of reflection off a solid wall show significantly different transition results. Furthermore, it is also shown that the thermal boundary layer plays an important role in this regard. It is thus confirmed that viscous and thermal effects are the reason for the paradox. Reasons are also suggested for the counter-intuitive behaviour of the reflected shock wave angle.

The method of characteristics is used to calculate numerical solutions for the one-dimensional motion of a plane shock-wave through a stationary gas which contains a region of non-uniform density. These solutions are compared with those given by the Chisnell-Whitham approach which ignores the effects on the shock-wave of the disturbances which are generated in the flow behind it, and also with the asymptotic solution given by the simple theory which regards the nonuniform region as a contact-surface discontinuity. It is concluded that the results of the simplified theories must be applied with caution.

Oscillatory forcing of a porous medium may have a dramatic effect on the mean flow rate produced by a steady applied pressure gradient. The oscillatory forcing may excite nonlinear inertial effects leading to either enhancement or retardation of the mean flow. Here, in Part 1, we consider the effects of non-zero inertial forces on steady flows in porous media, and investigate the changes in the flow character arising from changes in both the strength of the inertial terms and the geometry of the medium. The steady-state Navier–Stokes equations are solved via a Galerkin finite element method to determine the velocity fields for simple two-dimensional models of porous media. Two geometric models are considered based on constricted channels and periodic arrays of circular cylinders. For both geometries, we observe solution multiplicity yielding both symmetric and asymmetric flow patterns. For the cylinder arrays, we demonstrate that inertial effects lead to anisotropy in the effective permeability, with the direction of minimum resistance dependent on the solid volume fraction. We identify nonlinear flow phenomena which might be exploited by oscillatory forcing to yield a net increase in the mean flow rate. In Part 2, we take up the subject of unsteady flows governed by the full time-dependent Navier–Stokes equations.

Using a novel experimental set-up, we examine the stability of an axisymmetric dense current of fluid on a sloping bottom in a rotating frame of reference. In a cylindrical tank on a rotating table, saline fluid is injected through an annular mesh encircling a cone of 1/15 slope. The resulting ring of fluid becomes unstable to periodic sinusoidal waves, whose characteristics are examined as a function of the ambient fluid depth, the rotation rate, and the salinity and depth of the dense current, the latter being related to the injection time. For experiments with relatively low-salinity currents, we find the phase speed of the instability is proportional to the rotation rate, $\Omega$, rather than being proportional to the Nof speed, which varies as $1/\Omega$. Likewise, the number of waves observed is found to be inconsistent with existing theories for baroclinic instability of a dense current on a sloping bottom with a stationary ambient.

Surface tracers reveal that significant currents are induced in the ambient by the injection of the saline fluid and we propose that it is the coupling between the surface and bottom flow that ultimately controls the observed dynamics. In particular, we propose that barotropic instability of the surface flow controls the behaviour of the low-salinity current and, using potential vorticity conservation, we predict the phase speed of the instability should indeed be proportional to $\Omega$. The experimental results are also compared with related work in which eddies and Ekman layers evolve from dense fluid injected from a localized source.

The development of a turbulent streamwise vortex core in the wake of a half delta wing has been examined using high-resolution DPIV. The objective of this work was to gain understanding of the transport processes at work a short distance downstream of the wing trailing edge as the wake vortex developed. Experiments were conducted in the Rutgers Free Surface Water Tunnel using an in-house DPIV system. A turbulent streamwise vortex was generated by a half delta wing, with 44 cm chord length and 60° sweep angle, mounted at 30° angle of attack. Reynolds number based on chord length was 65 000. Laser sheets oriented perpendicular to the flow direction were positioned 1, 3.5, and 7 chord lengths downstream of the wing trailing edge. Instantaneous vortex centres were identified in order to track vortex meandering as well as for better quantification of turbulence levels in the vortex core. Mean and fluctuating turbulence terms in the mean streamwise vorticity transport equation along with turbulent kinetic energy dissipation and production were evaluated relative to an inertial reference frame as well as relative to a vortex-centred frame. The results of this analysis highlight the importance of this near-wake region to the downstream evolution of the trailing vortices. There is a high degree of dissipation as well as streamwise vorticity convection in the very near wake which decreases rapidly with increasing distance from the trailing edge.

This paper addresses the dam-break problem for particle suspensions, that is, the flow of a finite volume of suspension released suddenly down an inclined flume. We were concerned with concentrated suspensions made up of neutrally buoyant non-colloidal particles within a Newtonian fluid. Experiments were conducted over wide ranges of slope, concentration and mass. The major contributions of our experimental study are the simultaneous measurement of local flow properties far from the sidewalls (velocity profile and, with lower accuracy, particle concentration) and macroscopic features (front position, flow depth profile). To that end, the refractive index of the fluid was adapted to closely match that of the particles, enabling data acquisition up to particle volume fractions of 60 %. Particle migration resulted in the blunting of the velocity profile, in contrast to the parabolic profile observed in homogeneous Newtonian fluids. The experimental results were compared with predictions from lubrication theory and particle migration theory. For solids fractions as large as 45 %, the flow behaviour did not differ much from that of a homogeneous Newtonian fluid. More specifically, we observed that the velocity profiles were closely approximated by a parabolic form and there was little evidence of particle migration throughout the depth. For particle concentrations in the 52–56 % range, the flow depth and front position were fairly well predicted by lubrication theory, but taking a closer look at the velocity profiles revealed that particle migration had noticeable effects on the shape of the velocity profile (blunting), but had little impact on its strength, which explained why lubrication theory performed well. Particle migration theories (such as the shear-induced diffusion model) successfully captured the slow evolution of the velocity profiles. For particle concentrations in excess of 56 %, the macroscopic flow features were grossly predicted by lubrication theory (to within 20 % for the flow depth, 50 % for the front position). The flows seemed to reach a steady state, i.e. the shape of the velocity profile showed little time dependence.

Three-dimensional convection in a Boussinesq fluid confined between horizontal rigid boundaries is studied in a series of numerical experiments. Convection in air, whose Prandtl number Pr = 0·71, is systematically investigated, together with another model for Pr = 1. Convection with a steadily changing mean temperature is also considered. Two-dimensional rolls over the Rayleigh number range 4500 [les ] Ra [les ] 24000 and three-dimensional flow patterns over the range 26000 [les ] Ra ≤ 32000 are shown to be stable in air when the mean temperature of the layer is constant ($\partial \overline{T}/\partial t = \eta = 0$). Discrete changes in the slope of the heat-flux curve are shown to exist in the ranges \[ 7000\leqslant Ra\leqslant 8000,\quad 12000\leqslant Ra\leqslant 14000\quad{\rm and}\quad 24000\leqslant Ra\leqslant 26000 \] in air. Only the last discrete transition in the heat flux is asSociated with a significant transition in the flow pattern. Two-dimensional rolls with a horizontally asymmetric distribution of upward and downward motions over the range 4500 [les ] Ra [les ] 8000, and three-dimensional flow patterns over the range 10 000 [les ] Ra [les ] 20 000 are shown to be stable when the mean temperature varies with time. The circulation in a three-dimensional cell depends on the sign of the mean temperature change: downward motions occupy the centre of the cell when $\partial\overline{T}/\partial t > 0$, and upward motions when $\partial\overline{T}/\partial t < 0 $. Motions start to be time dependent for Ra > 20000. Transitions in the planform are asSociated with discrete changes in the slope of the heat-flux curve. Transitions in both the heat flux and flow pattern depend quantitatively on the Prandtl number.

Conditions are investigated under which a body lying at rest or rocking on a solid horizontal surface can be removed from the surface by hydrodynamic forces or instead continues rocking. The investigation is motivated by recent observations on Martian dust movement as well as other small- and large-scale applications. The nonlinear theory of fluid–body interaction here has unsteady motion of an inviscid fluid interacting with a moving thin body. Various shapes of body are addressed together with a range of initial conditions. The relevant parameter space is found to be subtle as evolution and shape play substantial roles coupled with scaled mass and gravity effects. Lift-off of the body from the surface generally cannot occur without fluid flow but it can occur either immediately or within a finite time once the fluid flow starts up: parameters for this are found and comparisons are made with Martian observations.

We examine the integral properties of freely decaying homogeneous magnetohydrodynamic (MHD) turbulence subject to an imposed magnetic field B0 at low-magnetic Reynolds number. We confirm that, like conventional isotropic turbulence, the fully developed state possesses a Loitsyansky-like integral invariant, in this case I// = − ∫ r⊥2 〈 u⊥·u′⊥〉 dr, where 〈u(x) ·u(x + rc)〉 = 〈u·u′〉 is the usual two-point velocity correlation and the subscript ⊥ indicates components perpendicular to the imposed field. The conservation of I// for fully developed turbulence places a fundamental restriction on the way in which the integral scales can develop, i.e. it implies u⊥2 ℓ⊥4 ℓ// ≈ constant where u⊥, ℓ⊥ and ℓ// are integral scales. This constraint can be used to estimate the evolution of u⊥(t; B0), ℓ⊥(t; B0) and ℓ//(t; B0), and these theoretical decay laws are shown to be in good agreement with numerical simulations.

We examine two-dimensional motion of a stably stratified fluid containing two solutes with different molecular diffusivities in an inclined slot. The two solutes have continuous opposing gradients with the slower-diffusing one more dense at the bottom. It is found that, in the steady state, there exists a slow upward flow along the slope driven by the slight buoyancy difference near the wall, not unlike the solution found by Phillips (1970) for a single solute. The magnitude of the flow is less than that in Phillips’ solution by a factor of approximately (1-λ)/(1-λτ), where λ is the ratio of the density gradient and τ−1 is the ratio of the diffusivity of the faster-diffusing solute to that of the slower-diffusing one. For the time-dependent flow resulting from switching on the diffusivities at t = 0, there may be a dramatic reversal of the flow near the walls depending on the relative magnitude of λ and τ. If λ is somewhat greater than τ, the initial flow is downward, along the slope, reaching a maximum magnitude about one order of magnitude greater than the steady-state value. Then the ‘reverse’ flow gradually diminishes and approaches the steady state rather slowly. For λ [gsim ] τ, the approach to the steady state is monotonic; there is no ‘reverse’ flow near the wall. The existence of the downward flow, which was observed by Turner & Chen (1974), may lead to double-diffusive instabilities which eventually result in horizontal convecting layers.

A second-order nonlinear frequency-domain model extending the linear complementary mild-slope equation (CMSE) is presented. The nonlinear model uses the same streamfunction formulation as the CMSE. This allows the vertical profile assumption to accurately satisfy the kinematic bottom boundary condition in the case of nonlinear triad interactions as well as for the linear refraction–diffraction part. The result is a model with higher accuracy of wave–bottom interactions including wave–wave interaction. The model's validity is confirmed by comparison with accurate numerical models, laboratory experiments over submerged obstacles and analytical perturbation solutions for class III Bragg resonance.

Shore protection by small seabed bars was once considered possible because seafloor undulations strongly reflect surface waves of twice the wavelength by the so-called Bragg resonance mechanism. The idea, however, proved ‘unreliable’ when it was realized that a patch of longshore seabed bars adjacent to a reflective shore could result in larger waves at the shoreline than for the case of a flat seabed. Here we propose to revamp the Bragg resonance mechanism as a means of coastal protection by considering oblique seabed bars that divert, rather than reflect, shore-normal incident waves to the shore-parallel direction. We show, via multiple-scale analysis supported by direct numerical simulations, that the creation of a large protected wake near the shoreline requires a bi-chromatic patch to deflect the incident waves to the shore-parallel direction. With two superposed sets of oblique seabed bars, the incident wave energy becomes efficiently deflected far to the sides, leaving a wake of decreased wave activity downstream of the patch. We demonstrate that the shore protection efficiency provided by this novel arrangement is not affected by reflection of leaked waves at the shoreline, and that it is relatively robust against small frequency detuning.

The Rayleigh–Taylor instability of an inclined buoyant cylinder of one very viscous fluid rising through another is examined through linear stability analysis, numerical simulation and experiment. The stability analysis represents linear eigenmodes of a given axial wavenumber as a Fourier series in the azimuthal direction, allowing the use of separable solutions to the Stokes equations in cylindrical polar coordinates. The most unstable wavenumber k∗ is long-wave if both the inclination angle α and the viscosity ratio λ (internal/external) are small; for this case, k∗ ∝ max{α, (λ ln λ−1)1/2} and thus a small angle in experiments can have a significant effect for λ ≪ 1. As α increases, the maximum growth rate decreases and the upward propagation rate of disturbances increases; all disturbances propagate without growth if the cylinder is sufficiently close to vertical, estimated as α ≳ 70°. Results from the linear stability analysis agree with numerical calculations for λ = 1 and experimental observations. A point-force numerical method is used to calculate the development of instability into a chain of individual plumes via a complex three-dimensional flow. Towed-source experiments show that nonlinear interactions between neighbouring plumes are important for α ≳ 20° and that disturbances can propagate out of the system without significant growth for α ≳ 40°.

The hydromagnetic flow of a thermally stratified fluid confined between two rotating parallel plates is studied. The flow is assumed to be linear, steady and axially symmetric. The flow is driven both mechanically and thermally and general thermal boundary conditions are applied. Attention is focused upon the mechanism controlling the interior fluid (diffusion, Ekman pumping or hydro-magnetic forces) and upon the conditions under which laminated flow (∂v/∂z ≠ 0) may occur. It is found that the occurrence of laminated flow is very sensitive to the thermal boundary conditions and is suppressed by hydromagnetic effects. For mixed boundary conditions, hydromagnetic forces control the interior and laminated flow is suppressed if α [ges ] O(1), where α2 represents the ratio of hydro-magnetic to Coriolis forces. For a constant heat flux, this occurs for a much weaker magnetic field: if α [ges ] O(E¼). For a restricted range of the parameters, a new boundary layer, called the thermomagnetic layer, in which Coriolis, thermal and hydromagnetic forces balance may occur.

We consider the linear evolution of a localized vortex with Gaussian potential vorticity that is superposed on a horizontal Couette flow in a rapidly rotating strongly stratified fluid. The Rossby number, defined as the ratio of the shear of the Couette flow to the Coriolis frequency, is assumed small. Our focus is on the inertia–gravity waves that are generated spontaneously during the evolution of the vortex. These are exponentially small in the Rossby number and hence are neglected in balanced models such as the quasi-geostrophic model and its higher-order generalizations. We develop an exponential-asymptotic approach, based on an expansion in sheared modes, to give an analytic description of the three-dimensional structure of the inertia–gravity waves emitted by the vortex. This provides an explicit example of the spontaneous radiation of inertia–gravity waves by localized balanced motion in the small-Rossby-number regime.

The inertia–gravity waves are emitted as a burst of four wavepackets propagating downstream of the vortex. The approach employed reduces the computation of inertia–gravity-wave fields to a single quadrature, carried out numerically, for each spatial location and each time. This makes it possible to unambiguously define an initial state that is entirely free of inertia–gravity waves, and circumvents the difficulties generally associated with the separation between balanced motion and inertia–gravity waves.

The flow field around pairs of small particles moving and rotating in a shear flow close to a wall at low but finite Reynolds number (Re) is computed as a function of time by means of the lattice-Boltzmann technique. The total force and torque acting on each particle is computed at each time step and the position of the particles is updated. By considering the lift force and the disturbances induced by the particles, the trajectories of the pair of particles are explained as a function of the distances from the wall and the Reynolds number. It is shown that when particles are positioned in a particular form, they collide forming strings. In particular, we are interested in particle-bridge formation in shear flows, and two collided particles (a string) can be considered as a nucleus of a particle bridge.