When a bluff body is placed in a crossflow, the total temperature in its wake can become substantially less than the incoming one, as manifested by the fact that the recovery factor R on its rearmost surface takes negative values at high subsonic flow: this is the phenomenon referred to here as the Eckert-Weise effect. Although a vortex street has been a suspected cause, the issue of whether this is so, and what the mechanism is, has remained unsettled. In this experimental and theoretical investigation, we first examine the cause of the Eckert-Weise effect by enhancing the vortex shedding through acoustic synchronization: resonance between the vortex shedding and transversely standing acoustic waves in a wind tunnel. At the lowest synchronization, where a ringing sound emanates from the wind tunnel, R at the rearmost section of the cylinder is found to become negative even at a Mach number of 0.2; the base pressure (Cpb) takes dips correspondingly, indicative of the intensification of the vortex street. At this lowest acoustic resonance, the decrease of R and Cpb, uniform along the span, agrees with the expectation based on the spanwise uniformity of the lowest standing wave. At the next acoustic resonance where the standing wave now varies along the span, the corresponding dips in R and Cpb, non-uniform along the span, reveals an interesting ‘strip-theory’-like behaviour of the vortex intensities in the vortex street. These results correlating the change in R with Cpb confirm that the Eckert-Weise effect is indeed caused by the vortex shedding, the mechanism of which is examined theoretically in the latter half of the paper.

A simple theoretical argument, bolstered by a full numerical simulation, shows that the time-varying static pressure field due to the vortex movement separates the instantaneous total temperature into hot and cold spots located around vortices; once time-averaged, however, the total temperature distribution conceals the presence of hot spots and takes the guise of a colder wake, the Eckert-Weise effect. Therefore the correct explanation of the Eckert-Weise effect, a time-averaged phenomenon, emerges only out of, and only as a superposition of, instantaneous total temperature separation around vortices. Such a separation is not confined to the outside of vortex cores; every vortex in its entirety becomes thermally separated. Nor is it limited to the far downstream equilibrium configuration of the Kármán vortex street but applies to the important near-wake vortices, and to any three-dimensional vortical structure as well. For low subsonic flows in particular, this dynamical explanation also leads to a similar separation of total pressure; these features may thus be potentially exploited as a general marker to identify and quantify vortices.

The evolution of weakly-nonlinear two-layer flow over topography is considered. The governing equations are formulated to consider the effects of quadratic and cubic nonlinearity in the transcritical regime of the internal mode. In the absence of cubic nonlinearity an inhomogeneous Korteweg-de Vries equation describes the interfacial displacement. Numerical solutions of this equation exhibit undular bores or sequences of Boussinesq solitary waves upstream in a transcritical regime. For sufficiently large supercritical Froude numbers, a locally steady flow is attained over the topography. In that regime in which both quadratic and cubic nonlinearity are comparable, the evolution of the interface is described by an inhomogeneous extended Kortewegde Vries (EKdV) equation. This equation displays undular bores upstream in a subcritical regime, but monotonic bores in a transcritical regime. The monotonic bores are solitary wave solutions of the corresponding homogeneous EKdV equation. Again, locally steady flow is attained for sufficiently large supercritical Froude numbers. The predictions of the numerical solutions are compared with laboratory experiments which show good agreement with the solutions of the forced EKdV equation for some range of parameters. It is shown that a recent result of Miles (1986), which predicts an unsteady transcritical regime for single-layer flows, may readily be extended to two-layer flows (described by the forced KdV equation) and is in agreement with the results presented here.

Numerical experiments exploiting the symmetry of the homogeneous EKdV equation show that solitary waves of fixed amplitude but arbitrary length may be generated in systems described by the inhomogeneous EKdV equation.

A boundary-layer analysis is presented for the two-dimensional nonlinear convection of an infinite-Prandtl-number fluid in a rectangular enclosure, in the limit of large Rayleigh numbers. Particular emphasis is given to the analysis of the periodic boundary layers, and on the removal of the singularities that appear near the corners of the cell. It is argued that this later step is necessary to ensure the correctness of the boundary-layer assumptions. Numerical values are obtained for the heat transfer and stress characteristics of the flow.

Close interactions between pairs of two-dimensional vortices of like sign were investigated in experiments with barotropic vortices and baroclinic vortices. The vortices were generated by sources or sinks in a rotating fluid which, respectively, was homogeneous or contained a two-layer density stratification. For two identical anticyclonic, unstratified vortices there was a critical separation distance beyond which the vortices coalesced to form a single larger anticyclone. The critical distance d*, scaled by the radius R of a core having non-zero relative vorticity, was d*/R = 3.3 ± 0.2. This value is in agreement with results of previous numerical simulations for finite-area vortices in non-rotating flows. The effects on vortex structure of Ekman pumping due to the presence of a rigid boundary caused cyclonic vortices to coalesee from larger distances. Baroclinic vortices in a two-layer stratification were also found to coalesce despite a potential-energy barrier. However, the critical separation distance depended on the internal Rossby radius. When the Rossby radius was large compared with the core radius, vortices coalesced from distances much greater than the critical distance for barotropic vortices. Coalescing of two vortices of equal size and strength led to two symmetric entwined spirals of water, while close interaction of unequal vortices caused the weaker vortex to be wrapped around the outer edge of the stronger. Implications of these results are discussed for ocean eddies and intense atmospheric cyclones.

Reflection of a shallow-water soliton at a plane beach is investigated numerically based on the edge-layer theory developed in Part 1 of this series. The offshore behaviour in the shallow-water region is first obtained by solving the Boussinesq equation under the ‘reduced’ boundary condition. The spatial and temporal variations of the surface elevation are displayed for two typical values of the inclination angle of the beach. Using these solutions, the nearshore behaviour is then evaluated to obtain the surface elevation and the velocity distribution in the edge layer. Both offshore and nearshore behaviours furnish a full knowledge of the reflection problem of a shallow-water soliton. To check the applicability of the edge-layer theory, a ‘computational experiment’ is carried out based on the boundary-element method, in which the Laplace equation is solved numerically under the full nonlinear boundary conditions at the free surface without introducing the edge-layer concept. Both results show a fairly good agreement for the overall reflection behaviour of a shallow-water soliton except for the surging movement at the shoreline.

A model problem is analysed to study the microscopic flow near the surface of porous media. In the idealized system, we consider two-dimensional media consisting of infinite and semi-infinite periodic lattices of cylindrical inclusions. In Part 1, results for axial flow were presented. In this work results for transverse flow are presented and discussed in the context of macroscopic approaches such as slip coefficients and Brinkman's equation.

We consider the evolution of an isolated elliptical vortex in a weakly dissipative fluid. It is shown computationally that a spatially smooth vortex relaxes inviscidly towards axisymmetry on a circulation timescale as the result of filament generation. Heuristically, we derive a simple geometrical formula relating the rate of change of the aspect ratio of a particular vorticity contour to its orientation relative to the streamlines (where the orientation is defined through second-order moments). Computational evidence obtained with diagnostic algorithms validates the formula. By considering streamlines in a corotating frame and applying the new formula, we obtain a detailed kinematic understanding of the vortex's decay to its final state through a primary and a secondary breaking. The circulation transported into the filaments although a small fraction of the total, breaks the symmetry and is the chief cause of axisymmetrization.

It is shown that the Rayleigh-Taylor instability of an accelerating incompressible, inviscid fluid layer is the result of pressure gradients, not gravitational acceleration. As in the classical Rayleigh-Taylor instability of a semi-infinite layer, finite fluid layers form long thin spikes whose structure is essentially independent of the initial thickness of the layer. A pressure maximum develops above the spike that effectively uncouples the flow in the spike from the rest of the fluid. Interspersed between the spikes are rising bubbles. The bubble motion is seriously affected by the thickness of the layer. For thin layers, the bubbles accelerate upwards exponentially in time and the layer thins so rapidly that it may disrupt at finite times.

The classical point-vortex model for a Kármán vortex street is linearly stable only for an isolated case. This property has been shown numerically to hold for other, more complicated, models of the same flow. It is shown here that it is a consequence of the Hamiltonian structure of the model, related to the codimension of the set of matrices with a particular Jordan block structure in the space of Hamiltonian matrices, and that it can be expected to hold generically for any two-dimensional inviscid array of vortices that has back-to-fore symmetry, and that is ‘close enough’ to the point-vortex model.

A numerical method is developed for nonlinear three-dimensional but axisymmetric free-surface problems using a mixed Eulerian-Lagrangian scheme under the assumption of potential flow. Taking advantage of axisymmetry, Rankine ring sources are used in a Green's theorem boundary-integral formulation to solve the field equation; and the free surface is then updated in time following Lagrangian points. A special treatment of the free surface and body intersection points is generalized to this case which avoids the difficulties associated with the singularity there. To allow for long-time simulations, the nonlinear computational domain is matched to a transient linear wavefield outside. When the matching boundary is placed at a suitable distance (depending on wave amplitude), numerical simulations can, in principle, be continued indefinitely in time. Based on a simple stability argument, a regriding algorithm similar to that of Fink & Soh (1974) for vortex sheets is generalized to free-surface flows, which removes the instabilities experienced by earlier investigators and eliminates the need for artificial smoothing. The resulting scheme is very robust and stable.

For illustration, three computational examples are presented: (i) the growth and collapse of a vapour cavity near the free surface; (ii) the heaving of a floating vertical cylinder starting from rest; and (iii) the heaving of an inverted vertical cone. For the cavity problem, there is excellent agreement with available experiments. For the wave-body interaction calculations, we are able to obtain and analyse steady-state (limit-cycle) results for the force and flow field in the vicinity of the body.

We extend a recent linearized theory on Bragg scattering of surface waves by periodic sandbars to include second-order effects of the free surface and of the bars. New experiments are performed to verify the existence of the cutoff detuning frequency, the dispersive nature of the first-order wave envelope, and the radiation of second-order long waves. Measured transient and quasi-steady responses to incident wave packets and uniform wavetrains are compared with corresponding theoretical results. For quasi-steady incident waves of relatively small steepness it is found necessary to improve the theory to the second order in bar slope, in order that the calculated short-wave envelopes agree with those measured over the bars.

The two-dimensional flow of a thin film down a vertical or tilted plane wall into an infinite pool is studied in the Stokes approximation, the principal aim being to determine the shape of the fluid surface. Results are obtained for fluids with or without surface tension. Earlier results by Ruschak, that the surface tension gives rise to thickness variation of the film, are confirmed. For small or vanishing surface tension a dip of the pool surface is found to exist close to the wall. The case of a wall moving downwards is also considered.

A simple two-equation model is derived which has the properties that the total contaminant exposure, the mean time of arrival, the temporal spread, and the skewness, are asymptotically correct at large distances downstream of a discharge. The role of changes in the breadth of a river upon the dispersion process is investigated by a means of an illustrative example. This reveals cubic dependence upon the breadth, and hence the great importance of wide reaches of rivers as regards contaminant dispersion.

The reflection of a train of two-dimensional finite-amplitude internal waves propagating at an angle β to the horizontal in an inviscid fluid of constant buoyancy frequency and incident on a uniform slope of inclination α is examined, specifically when β > α. Expressions for the stream function and density perturbation are derived to third order by a standard iterative process. Exact solutions of the equations of motion are chosen for the incident and reflected first-order waves. Whilst these individually generate no harmonics, their interaction does force additional components. In addition to the singularity at α = β when the reflected wave propagates in a direction parallel to the slope, singularities occur for values of α and β at which the incident-wave and reflected-wave components are in resonance; strong nonlinearity is found at adjacent values of α and β. When the waves are travelling in a vertical plane normal to the slope, resonance is possible at second order only for α < 8.4° and β < 30°. At third order the incident wave is itself modified by interaction with reflected components. Third-order resonances are only possible for α < 11.8° and, at a given α, the width of the β-domain in which nonlinearities connected to these resonances is important is much less than at second order. The effect of nonlinearity is to reduce the steepness of the incident wave at which the vertical density gradient in the wave field first becomes zero, and to promote local regions of low static stability remote from the slope. The importance of nonlinearity in the boundary reflection of oceanic internal waves is discussed.

In an Appendix some results of an experimental study of internal waves are described. The boundary layer on the slope is found to have a three-dimensional structure.

This paper is primarily concerned with Mach-number effects on the vortex shedding behind a square cylinder (side length D = 20 mm) in a Reynolds-number range of 0.696 × 105 < Re < 4.137 × 105, and a Mach-number range of 0.1522 < M < 0.9049.

Regular periodic vortex shedding is present, irrespective of the appearance of shock waves around a square cylinder. The shape of the vortices is, however, deformed by the shock waves, and each vortex centre becomes non-uniform while the vortex passes through the gap between the upper and lower shock waves. Weak shock waves around the square cylinder do not alter the Strouhal number, but strong shock waves weaken the vortex shedding and increase the Strouhal number suddenly. Acoustic waves have been recorded by the Mach-Zehnder interferometer when the Mach number is close to the critical value. The acoustic waves are generated most strongly at the instant when each vortex hits the foot of the shock waves formed above and below the vortex formation region.

From the present work and that of Okajima (1982), it is suggested that the Strouhal number of alternating vortices shed from a square cylinder can be estimated to be about 0.13 in the Reynolds-number range between 102 and 3.4 × 105.

We analyse the steady streaming generated in an infinite elliptical tube containing a viscous, incompressible fluid when the boundary oscillates in such a way that the area and ellipticity of the cross-section vary with time but remain independent of the longitudinal coordinate. The parameters α−1 = (ν/Ωa02)½ and ε = U0/a0Ω, where ν is the kinematic viscosity, Ω is the oscillation frequency, a0 is the undisturbed semi-major axis and U0 is a typical wall velocity, are taken to be small, so that the Stokes layer is thin and the interaction which leads to the steady streaming can be analysed as a small perturbation. Coupled axial and transverse velocities, both oscillatory and steady, are generated. A complication is the need to specify the tangential as well as the normal velocity component on the tube wall, which requires an assumption concerning its elastic properties. We have considered two cases: (i) constant major axis, in which all boundary points move parallel to the minor axis, and (ii) an inextensible wall. The three-dimensional steady streaming in the core of the tube is computed only in the limit of small steady-streaming Reynolds number, Rs = ε2α2.

The early three-dimensional stages of transition in the Blasius boundary layer are studied by numerical solution of the Navier-Stokes equations. A finite-amplitude two-dimensional wave and low-amplitude three-dimensional random disturbances are introduced. Rapid amplification of the three-dimensional components is observed and leads to transition. For intermediate amplitudes of the two-dimensional wave the breakdown is of subharmonic type, and the dominant spanwise wavenumber increases with the amplitude. For high amplitudes the energy of the fundamental mode is comparable to the energy of the subharmonic mode, but never dominates it; the breakdown is of mixed type. Visualizations, energy histories, and spectra are presented. The sensitivity of the results to various physical and numerical parameters is studied. The agreement with experimental and theoretical results is discussed.

The Rayleigh-Janzen expansion method is extended to plane and steady flows which contain one or more point vortices interacting with a smooth or sharp-edged obstacle. A uniformly valid approximate solution of the compressible-flow equations is deduced by applying a perturbation method and by using matched asymptotic expansions to solve the resulting singular perturbation problem. The method yields compressibility corrections for the vortex positions and for the velocities. Results are presented for the flow past a circle and a pair of symmetric vortices (Föppl's flow). They show that the compressibility effects are substantial and are consistent with experimental data.

The turbulent diffusion process is investigated for a continuous point source of a non-buoyant plume in grid-generated water turbulence. Two kinds of biplanar grids with a mesh length of 10 mm and 20 mm were used. The mesh Reynolds numbers were 1480 and 2970, respectively. The mean and fluctuating concentration fields of aqueous dye solution were measured by the light absorption method. Experimental results for both grids were compared.

For both grids, the mean concentration radial profiles proved to have a similar Gaussian shape, and the mean concentration on the plume axis obeys the hyperbolic decay law well. These mean concentration profiles and their decay show an excellent agreement with the results deduced from the similarity analysis for the mean concentration field.

Radial profiles of the fluctuation r.m.s. value and relative intensity (i.e. the ratio of the r.m.s. value to the mean concentration) were found also to be nearly similar, and the centreline r.m.s. value decays downstream as a hyperbola. The relative intensity on the centreline tends to increase slightly downstream. All experimental results obtained were much less scattered and more reliable than those reported earlier.

The similarity for the concentration fluctuation intensity has been analysed using a thin-layer approximation. Also, an approximate analysis of the fluctuating concentration field is given by replacing the fluctuating concentration signal by a randomly spaced sequence of rectangular waves with various heights and widths.

The wave packets located at the wingtips of turbulent spots in plane Poiseuille flow have been investigated by hot-film anemometry. The streamwise velocity disturbances associated with the waves were found to be antisymmetric with respect to the channel centreline. The amplitude of the waves had a maximum close to the wall that was about 4% of the centreline velocity. The modified velocity field outside the spot was measured and linear stability analysis of the measured velocity profiles showed that the flow field was less stable than the undisturbed flow. The phase velocity and amplitude distribution of the waves were in reasonable agreement with the theory, which together with the symmetry properties indicate that the wave packet consisted of the locally least stable Tollmien-Schlichting mode.