This paper reports a comprehensive set of hot-wire measurements of a round buoyant plume which was generated by forcing a jet of hot air vertically up into a quiescent environment. The boundary conditions of the experiment were measured, and are documented in the present paper in an attempt to sort out the contradictory mean flow results from the earlier studies. The ambient temperature was monitored to ensure that the facility was not stratified and that the experiment was conducted in a neutral environment. The axisymmetry of the flow was checked by using a planar array of sixteen thermocouples and the mean temperature measurements from these are used to supplement the hot-wire measurements. The source flow conditions were measured to ascertain the rate at which the buoyancy was added to the flow. The measurements conserve buoyancy within 10%. The results are used to determine balances of the mean energy and momentum differential equations. In the mean energy equation it is found that the vertical advection of energy is primarily balanced by the radial turbulent transport. In the mean momentum equation the vertical advection of momentum and the buoyancy force balance the radial turbulent transport. The buoyancy force is the second largest term in this balance and is responsible for the wider (and higher) velocity profiles in plumes as compared to jets. Budgets of the temperature variance and turbulent kinetic energy are also determined in which thermal and mechanical dissipation rates are obtained as the closing terms. Similarities and differences between the two balances are discussed. It is found that even though the direct effect of buoyancy in turbulence, as evidenced by the buoyancy production term, is substantial, most of the turbulence is produced by shear. This is in contrast to the mean velocity field where the effect of the buoyancy force is quite strong. Therefore, it is concluded that in a buoyant plume the primary effect of buoyancy on turbulence is indirect, and enters through the mean velocity field (giving larger shear production).

The three-dimensional unsteady boundary layer induced by a vortex filament moving outside a circular cylinder is considered. In the present paper, we focus attention on the situation where the inviscid flow is fully three-dimensional but is symmetric with respect to the top centreline of the cylinder. The motion of the vortex toward the cylinder leads to separation of the boundary layer; in the present work a large unsteady adverse pressure gradient develops as well. Results for the three-dimensional streamlines, the vorticity distribution, and the velocity component normal to the cylinder indicate the presence of a region of unsteady three-dimensional secondary flow structure of rather complex shape located deep within the boundary layer. Within this three-dimensional secondary flow the fluid is progressively squeezed into a narrow region under the main vortex and it is expected that a local three-dimensional jet will develop sending boundary-layer fluid out into the main stream. It is pointed out that such three-dimensional eruptive behaviour has been observed in experiments. The results indicate the development of a three-dimensional singularity in the boundary-layer equations.

Velocity data were obtained in the turbulent wake of a circular cylinder with an orthogonal array of sixteen X-wires, eight in the (x, y)-plane and eight in the (x, z)-plane. By applying the phase-plane technique to these data, three types of critical points (where the velocity is zero and the streamline slope is indeterminate) were identified. Of these, foci and saddle points occurred most frequently, although a significant number of nodes was also found. Flow topology and properties associated with these points were obtained in each plane. Saddle-point regions associated with spanwise vortices provide the dominant contribution to the Reynolds shear stress and larger contributions to the normal stresses than focal regions. The topology was found to be in close agreement with that obtained from other methods of detecting features of the organized motion. The inter-relationship between critical points simultaneously identified in the two planes can provide some insight into the three-dimensionality of the organized motion. Foci in the (x, z)-plane correspond, with relatively high probability and almost negligible streamwise separation, to saddle points in the (x, y)-plane and are interpreted in terms of ribs aligned with the diverging separatrix between consecutive spanwise vortex rolls. Foci in the (x, z)-plane which correspond, with relatively weak probability, to foci in the (x, y)-plane seem consistent with a distortion of the vortex rolls in the (y, z)-plane.

The properties of turbulence subgrid-scale stresses are studied using experimental data in the far field of a round jet, at a Reynolds number of Rλ ≈ 310. Measurements are performed using two-dimensional particle displacement velocimetry. Three elements of the subgrid-scale stress tensor are calculated using planar filtering of the data. Using a priori testing, eddy-viscosity closures are shown to display very little correlation with the real stresses, in accord with earlier findings based on direct numerical simulations at lower Reynolds numbers. Detailed analysis of subgrid energy fluxes and of the velocity field decomposed into logarithmic bands leads to a new similarity subgrid-scale model. It is based on the ‘resolved stress’ tensor Lij, which is obtained by filtering products of resolved velocities at a scale equal to twice the grid scale. The correlation coefficient of this model with the real stress is shown to be substantially higher than that of the eddy-viscosity closures. It is shown that mixed models display similar levels of correlation. During the a priori test, care is taken to only employ resolved data in a fashion that is consistent with the information that would be available during large-eddy simulation. The influence of the filter shape on the correlation is documented in detail, and the model is compared to the original similarity model of Bardina et al. (1980). A relationship between Lij and a nonlinear subgrid-scale model is established. In order to control the amount of kinetic energy backscatter, which could potentially lead to numerical instability, an ad hoc weighting function that depends on the alignment between Lij and the strain-rate tensor, is introduced. A ‘dynamic’ version of the model is shown, based on the data, to allow a self-consistent determination of the coefficient. In addition, all tensor elements of the model are shown to display the correct scaling with normal distance near a solid boundary.

Dyakov (1954) and Kontorovich (1957) formulated the conditions for corrugation instability of shock waves as well as for spontaneous emission of sound and entropy-vortex waves from them. For the first time since then, it is shown here that physical circumstances do exist under which shocks in gases spontaneously emit sound waves. Such circumstances are provided by strong ionizing shocks. In order to see that, the coefficient of reflection of an acoustic wave from a shock is derived as a function of the wave's frequency and the ionization degree. Spontaneous emission of sound occurs when the reflection coefficient becomes infinitely large. It is shown that the relevant frequency range for the occurrence of spontaneous emission is that for which the electrons are not in local thermodynamic equilibrium with the heavy particles. The special properties of acoustic perturbations behind the ionizing shock are considered for this frequency range and the sound velocity in a partially ionized gas is derived. In addition, the condition for spontaneous emission of sound is modified in order to take into account the difference between the electrons and heavy-particle perturbed temperatures. It is shown, by numerical calculations, that the criterion for spontaneous emission is satisfied behind ionizing shocks in argon. In particular, for an initial pressure of 5 Torr, the threshold for the occurrence of the spontaneous emission is found to be M1 = 15. This critical value of the shock Mach number, as well as other calculated physical features, agree very well with those obtained experimentally by Glass & Liu (1978) who observed the occurrence of instability behind shocks in argon.

A simple formula for ‘wave damping’ is derived, exact within the context of the proposed theory, namely: potential flow correct to second order in the wave amplitude and to leading order in U/c, where U is the drift velocity and c the wave celerity. The analysis is restricted to a two-dimensional problem although the extension to three dimensions seems possible.

Dynamic simulations of the pressure-driven flow in a channel of a non-Brownian suspension at zero Reynolds number were conducted using Stokesian Dynamics. The simulations are for a monolayer of identical particles as a function of the dimensionless channel width and the bulk particle concentration. Starting from a homogeneous dispersion, the particles gradually migrate towards the centre of the channel, resulting in an homogeneous concentration profile and a blunting of the particle velocity profile. The time for achieving steady state scales as (H/a)3a/〈u〉, where H is the channel width, a the radii of the particles, and 〈u〉 the average suspension velocity in the channel. The concentration and velocity profiles determined from the simulations are in qualitative agreement with experiment.

A model for suspension flow has been proposed in which macroscopic mass, momentum and energy balances are constructed and solved simultaneously. It is shown that the requirement that the suspension pressure be constant in directions perpendicular to the mean motion leads to particle migration and concentration variations in inhomogeneous flow. The concept of the suspension ‘temperature’ – a measure of the particle velocity fluctuations – is introduced in order to provide a nonlocal description of suspension behaviour. The results of this model for channel flow are in good agreement with the simulations.

A body moves at uniform speed in an unbounded inviscid fluid. Initially, the body is infinitely far upstream of an infinite plane of marked fluid; later, the body moves through and distorts the plane and, finally, the body is infinitely far downstream of the marked plane. Darwin (1953) suggested that the volume between the initial and final positions of the surface of marked fluid (the drift volume) is equal to the volume of fluid associated with the ‘added-mass’ of the body.

We re-examine Darwin's (1953) concept of drift and, as an illustration, we study flow around a sphere. Two lengthscales are introduced: ρmax, the radius of a circular plane of marked particles; and x0, the initial separation of the sphere and plane. Numerical solutions and asymptotic expansions are derived for the horizontal Lagrangian displacement of fluid elements. These calculations show that depending on its initial position, the Lagrangian displacement of a fluid element can be either positive – a Lagrangian drift – or negative – a Lagrangian reflux. By contrast, previous investigators have found only a positive horizontal Lagrangian displacement, because they only considered the case of infinite x0. For finite x0, the volume between the initial and final positions of the plane of marked fluid is defined to be the ‘partial drift volume’, which is calculated using a combination of the numerical solutions and the asymptotic expansions. Our analysis shows that in the limit corresponding to Darwin's study, namely that both x0 and ρmax become infinite, the partial drift volume is not well-defined: the ordering of the limit processes is important. This explains the difficulties Darwin and others noted in trying to prove his proposition as a mathematical theorem and indicates practical, as well as theoretical, criteria that must be satisfied for Darwin's result to hold.

We generalize our results for a sphere by re-considering the general expressions for Lagrangian displacement and partial drift volume. It is shown that there are two contributions to the partial drift volume. The first contribution arises from a reflux of fluid and is related to the momentum of the flow; this part is spread over a large area. It is well-known that evaluating the momentum of an unbounded fluid is problematic since the integrals do not converge; it is this first term which prevented Darwin from proving his proposition as a theorem. The second contribution to the partial drift volume is related to the kinetic energy of the flow caused by the body: this part is Darwin's concept of drift and is localized near the centreline. Expressions for partial drift volume are generalized for flow around arbitrary-shaped two- and three-dimensional bodies. The partial drift volume is shown to depend on the solid angles the body subtends with the initial and final positions of the plane of marked fluid. This result explains why the proof of Darwin's proposition depends on the ratio ρmax/x0.

An example of drift due to a sphere travelling at the centre of a square channel is used to illustrate the differences between drift in bounded and unbounded flows.

A boundary integral method is presented for analysing particle motion in a rotating fluid for flows where the Taylor number ${\cal T}$ is arbitrary and the Reynolds number is small. The method determines the surface traction and drag on a particle, and also the velocity field at any location in the fluid.

Numerical results show that the dimensionless drag on a spherical particle translating along the rotation axis of an unbounded fluid is determined by the empirical formula $D/6\pi = 1+(4/7){\cal T}^{1/2}+(8/9\pi){\cal T}$, which incorporates known results for the low and high Taylor number limits. Streamline portraits show that a critical Taylor number ${\cal T}_c\ap 50$ exists at which the character of the flow changes. For ${\cal T} < {\cal T}_c$ the flow field appears as a perturbation of a Stokes flow with a superimposed swirling motion. For ${\cal T} > {\cal T}_c$ the flow field develops two detached recirculating regions of trapped fluid located fore and aft of the particle. The recirculating regions grow in size and move farther from the particle with increasing Taylor number. This recirculation functions to deflect fluid away from the translating particle, thereby generating a columnar flow structure. The flow between the recirculating regions and the particle has a plug-like velocity profile, moving slightly slower than the particle and undergoing a uniform swirling motion. The flow in this region is matched to the particle velocity in a thin Ekman layer adjacent to the particle surface.

A further study examines the translation of spheroidal particles. For large Taylor numbers, the drag is determined by the equatorial radius; details of the body shape are less important.

An investigation is presented that analyses the energy flows that are connected to the dynamical behaviour of coherent structures in a transitional flat-plate boundary layer. Based on a mathematical description of the three-dimensional coherent structures of this flow as provided by the Karhunen–Loève procedure, energy equations for the coherent structures are derived by Galerkin projection of the Navier–Stokes equations in vorticity transport formulation onto the corresponding basis of eigenfunctions. In a first step, the time-averaged energy balance – showing the energy flows that support the different coherent structures and thus maintain the fluctuations of the velocity field – is considered. In a second step, the instantaneous power budget is investigated for the particularly interesting case of a coherent structure providing a prime contribution to the characteristic spike events of the transitional boundary layer. As this structure shows a strong variation in energy, the question about which mechanisms cause these variations is addressed. Our results show that the occurrence of a spike must be attributed to an autonomous event and cannot be interpreted as just an epiphenomenon of the passage of a Λ-vortex.

The natural frequencies and damping ratios for surface waves in a circular cylinder are calculated on the assumptions of a fixed contact line, Stokes boundary layers, and either a clean or a fully contaminated surface. These theoretical predictions are compared with the measurements for the first six modes in a brimfull, sharp-edged cylinder of radius 2.77 cm and depth 3.80 cm. The differences between the predicted and observed frequencies were less than 0.5% for all but the fundamental axisymmetric mode with a clean surface. The difference between the predicted and observed damping ratio for the dominant mode with a clean surface was 20%; this difference was significantly larger for the higher modes with a clean surface and for all of the modes with a contaminated surface.

This paper is concerned with the generation of short gravity waves and their radiation from the outer edge of the turbulent boundary layer and wake of a ship. They arise primarily near the ship's stern. The wave spectrum in the direction of wavenumber vector at an angle (90° – δ) to the ship's track is: \[\Phi_{\delta}(\omega) = \Psi\left(\frac{UT_d}{2l},\frac{U\sin\delta}{c_g},\frac{R}{UT_d}\right)\frac{1}{k_0R}\frac{2l\omega^2}{g^2}\gamma\left(0,\frac{\pi}{l};0,\omega \right),\] where Ψ is dimensionless and a function of three dimensionless parameters. γ is the spectrum of the oscillating motion at the boundary, U the ship speed, Td the decay timescale of the oscillating motion, 2l the lengthscale of the eddies, and R the distance away from the boundary along the wavenumber vector. Generally, Φδ has large values near δ = 0 and small values at large δ; it behaves as 1/R at distances not far from the ship, then may vary slower than 1/R at intermediate distances, and finally behaves as 1/R again at distances far from the ship. These are consistent with the pattern found in SAR images of narrow V-like ship wakes. The method developed here is also applicable to various problems of surface wave generation by turbulence in water.

An instability mechanism that can amplify wind-forced inertial oscillations in the upper ocean is investigated. This forced instability happens because of the phase relationship between the mixed-layer depth and the surface current. It allows the inertial oscillations propagating against the wind to extract energy from it and amplify. The key ingredients for the instability to work are (a) a non-zero mean wind stress, (b) a spatial variability of the oscillations in the direction of the wind stress. The amplification is demonstrated using a simple shallow-water model in a few situations: the dispersion of a localized disturbance with steady and time-varying wind forcing, generation of inertial waves at a coast, and spatial variability induced by mesoscale eddies. Estimates of the growth rate are provided for both dissipative and non-dissipative cases.

Steady states of a translationally-symmetric cylindrical bubble protruding from a slot in a solid wall into a liquid undergoing a simple shear flow are investigated. Deformations of and the flow past the bubble are determined by solving the nonlinear free-boundary problem comprised of the two-dimensional Navier–Stokes system by the Galerkin/finite element method. Under conditions of creeping flow, the results of finite element computations are shown to agree well with asymptotic results. When the Reynolds number Re is finite, flow separates from the free surface and a recirculating eddy forms behind the bubble. The length of the separated eddy measured in the flow direction increases with Re, whereas its width is confined to within the region that lies between the supporting solid surface and the separation point at the free surface. By tracking solution branches in parameter space with an arc-length continuation method, curves of bubble deformation versus Reynolds number are found to exhibit turning points when Re reaches a critical value Rec. Therefore, along a family of bubble shapes, solutions do not exist when Re > Rec. The locations of turning points and the structure of flow fields are found to be governed virtually by a single parameter, We = Ca Re, where We and Ca are Weber and capillary numbers. Two markedly different modes of bubble deformation are identified at finite Re. One is dominant when Re is small and is tantamount to a plain skewing or tilting of the bubble in the downstream direction; the other becomes more pronounced when Re is large and corresponds to a pure upward stretching of the bubble tip.

We report the results of an experimental study of flow in a Taylor–Couette system where the usual circular outer cylinder is replaced by one with a square cross-section. The objective is to determine the validity of low-dimensional dynamical systems as a descriptive framework for flows in a domain without the special continuous symmetry of the original problem. We focus on a restricted version of the flow, where the steady flow consists of a single cell, thereby minimizing the multiplicity of solutions. The steady-state bifurcation structure is found to be qualitatively unchanged from that of the standard system. A complex but self-consistent bifurcation structure is uncovered for time-dependent flows, culminating in observations of dynamics similar to those of the finite-dimensional Sil’nikov mechanism. Such behaviour has been observed in the standard system with continuous azimuthal symmetry. The present results extend the range of closed-flow problems where there is an apparent connection between the infinite-dimensional Navier-Stokes equations and finite-dimensional dynamical systems.