The influence of hydrodynamic dispersion on thermal convection in porous media is studied theoretically. The fluid-saturated porous layer is homogeneous, isotropic and bounded by two infinite horizontal planes kept at constant temperatures. The supercritical, steady two-dimensional motion, the heat transport and the stability of the motion are investigated. The dispersion effects depend strongly on the Rayleigh number and on the ratio of grain diameter to layer depth. The present results provide new and closer approximations to experimental data of the heat transport.

European Mechanics Colloquium number 119 was held at Imperial College on 16–18 July 1979, when the subject of vortex shedding from bodies in unidirectional flow and oscillatory flow, was discussed. A wide range of experimental work was presented including low-Reynolds-number flows around circular cylinders, the influence of disturbances on bluff body flow, the measurement of fluctuating forces and the influence of oscillations of the stream. About a third of the 33 papers presented concentrated on theoretical aspects and the majority of these were concerned with the ‘method of discrete vortices’.

In arc welding electromagnetic forces are thought to be the major cause of motion in the weldpool, and it has recently been found by experimentalists that the application to the workpiece of a uniform magnetic field normal to the weldpool surface appears to stabilize the welding process. In this paper we investigate motion in a hemisphere due to a stationary distributed current source with a superimposed uniform magnetic field applied parallel to the axis of symmetry. When inertial effects are ignored and a simple source-sink model of the current source is introduced, we obtain an analytic solution for the fluid flow valid for low currents and low applied fields. A numerical scheme is then developed to solve the full nonlinear-flow problem and results are obtained for the source-sink model of the current source and for a more realistic numerically constructed distributed current source. For values of the externally applied magnetic field and the current flowing through the hemisphere typical of those occurring in practical welding situations we discover that the flow in an axial section through the pool is radially outwards on the free surface. This is in the opposite direction to that generally predicted by the theory when there is no superimposed magnetic field.

A numerical method is described which is suitable for solving the equations governing the steady motion of a viscous fluid through a slightly curved tube of circular cross-section but which is also applicable to the solution of any problem governed by the steady two-dimensional Navier–Stokes equations in the plane polar co-ordinate system. The governing equations are approximated by a scheme which yields finite-difference equations which are of second-order accuracy with respect to the grid sizes but which have associated matrices which are diagonally dominant. This makes them generally more amenable to solution by iterative techniques than the approximations obtained using standard central differences, while preserving the same order of accuracy.

The main object of the investigation is to obtain numerical results for the problem of steady flow through a curved tube which corroborate previous numerical work on this problem in view of a recent paper (Van Dyke 1978) which tends to cast doubt on the accuracy of previous calculations at moderately high values of the Dean number; this is the appropriate Reynolds-number parameter in this problem. The present calculations tend to verify the accuracy of previous results for Dean numbers up to 5000, beyond which it is difficult to obtain accurate results. Calculated properties of the flow are compared with those obtained in previous numerical work, with the predictions of boundary-layer theory for large Dean numbers and with the predictions of Van Dyke (1978).

The nonlinear evolution and breakdown of laminar flow in the boundary layer on a flat plate is examined with the aim of making a closer comparison of theory and experiment than has been attempted previously. The importance of three-dimensionality is emphasized. It is concluded that many features of the nonlinear instability are consistent with existing linear and weakly nonlinear theories even as breakdown is approached. The development of the secondary instability, or ‘spike’, is also considered and suggestions for an improved theory of its growth are made.

Decaying turbulence in neutral and stratified fluids has been studied experimentally for relatively high mesh Reynolds numbers and long time-histories. The neutral case indicates an initial period decay law, q2∝ t−1, through non-dimensional time \[ W_gt/M \simeq 800 \] which is considerably longer than previous measurements at the same mesh Reynolds number (Re = 48260). The stratified experiment resulted in a decay rate virtually identical to that of the neutral case through Wgt/M = 275. However the decay rate sharply decreased after this time when the field of turbulence was replaced by internal gravity waves. A critical Richardson number marks the transition from the turbulence to an internal gravity wave domain.

Calculations of the flow field under laminar conditions in a helical semicircular duct have been made by numerically solving the Navier–Stokes equations. With the flat wall of the duct being the outer wall, the solution of the momentum equations for Dean numbers below 105 gave, for the secondary flow, twin counter-rotating vortices of Taylor–Goertler type. However, above a Dean number of Dn = 105, two solutions were possible. One solution was similar to that obtained for Dn < 105. The other solution revealed four vortices for the secondary flow. For Dn > 105, convergence to either flow pattern depended on the initial guess used in the numerical solution. Flow visualization confirmed the possibility of the presence of both types of secondary flow patterns.

An undisturbed geostrophic density current flows along a vertical wall (the coast) with the free streamline (the front) located at a distance L from the wall which is comparable to the Rossby radius of deformation. Finite amplitude perturbations with downstream wavelengths much larger than L are discussed, and it is shown that the slope of the front in the horizontal plane increases with time. Some perturbations tend to ‘break’ seaward by developing large transverse velocities away from the coast. The temporal evolution of some perturbations is such as to completely ‘block’ the upstream flow, but the subsequent behaviour is beyond the scope of the theory. We also discuss the propagation of the nose of the intrusion when a density current debouches from a coastal source and then flows along the coastal boundary.

The diffraction of a steady Stokes wave train by a thin wedge with vertical walls is studied when the incident wave is directed along the wedge axis (grazing incidence). Parabolic approximation applied recently by Mei & Tuck (1980) to linear diffraction is extended to this nonlinear case. Significant effects of nonlinearity are found numerically, in particular the sharp forward bending of wave crests near the wedge. The computed features are found to corroborate the existing experiments only qualitatively; the controlling factors in the latter being not completely understood. An analytical model of stationary shock is proposed to approximate the numerical results of Mach stems.

Steady rotating flow over topography in a periodic channel is examined, with emphasis on the interaction of waves, topography and mean flow. A simple quasi-linear theory is presented that features an implicit equation relating the net zonal flow to the forcing and topography. A good description of the dynamics is obtained, even when resonant Rossby waves appear. Multiple solutions for given external parameters are predicted in some cases, and confirmed by comparison with a fully nonlinear numerical model.

The nonlinear results also indicate that the zonally averaged shear can be important when topographic effects or Rossby numbers are large. With this factor taken into account the theory gives good agreement with the fully nonlinear model, as long as eddy–eddy interactions are minor.

The theory is relevant to the dynamics of planetary waves in the atmosphere, and may also be applied to some oceanic problems.

Lake & Yuen (1978) have suggested that in very steep wind waves the modulation-frequency of the wave amplitude may correspond to the frequency of the fastest-growing subharmonic instability of a uniform train of waves whose amplitude equals the mean wave amplitude $\overline{a}$. The approximate theory of Benjamin & Feir (1967) gives this frequency as $(\overline{a}k)f_d$, where κ is the wavenumber and fd the frequency of the unperturbed waves. This expression applies strictly only to very small values of the wave steepness $\overline{a}k$.

More recently (Longuet-Higgins 1978) the present author calculated accurately all the normal-mode instabilities of steep gravity waves on deep water. In this note these calculations are used to determine the frequency of the fastest-growing sub-harmonic instabilities precisely. When compared with the experimental data of Lake & Huen, these frequencies show even closer agreement.

The subject is the small-scale structure of a magnetic field in a turbulent conducting fluid, ‘small scale’ meaning lengths much smaller than the characteristic dissipative length of the turbulence. Philip Saffman developed an approximation to describe this structure and its evolution in time. Its usefulness invites a closer examination of the approximation itself and an attempt to place sharper limits on the numerical parameters that appear in the approximate correlation functions, topics to which the present paper is addressed.

A Lagrangian approach is taken, wherein one makes a Fourier decomposition of the magnetic field in a neighbourhood that follows a fluid element. If one construes the viscous-convective range narrowly, by ignoring magnetic dissipation entirely, then results for a magnetic field in two dimensions are consistent with Saffman's approximation, but in three dimensions no steady state could be found. Thus, in three dimensions, turbulent amplification seems to be more effective than Saffman's approximation implies. The cause seems to be a matter of geometry, not of correlation times or relative time scales.

Strictly-outward spectral transfer is a characteristic of Saffman's approximation, and this may be an accurate description only when dissipation suppresses the contributions from inwardly directed spectral transfer. In the spectral region where dominance passes from convection to dissipation, one can generate expressions for the parameters that arise in Saffman's approximation. Their numerical evaluation by computer simulation may enable one to sharpen the limits that Saffman had already set for those parameters.

The Craik–Leibovich (CL) equations for Langmuir circulations are shown to be an Eulerian approximation to an exact theory of the generalized Lagrangian mean (GLM) due to Andrews and McIntyre. Derivation of the CL equations using the GLM formalism is decisively simpler than the original method. The CL theory is then compared to other wave–current interaction theories of Langmuir circulations, notably those of Garrett and of Moen.

A linear non-homogeneous analysis is presented for the standing waves produced on the hollow core of an irrotational vortex by an arbitrary obstacle on the wall of the tube containing the vortex. The group-velocity criterion based upon Kelvin's corresponding dispersion relation predicts whether a certain asymptotic wave pattern appears upstream or downstream of the obstacle. The analysis leads to amplitude singularities for the standing waves at certain critical radii of the core. The particularly interesting case of a counter-helix for which the wave energy is propagating upstream appears for a first-mode angular disturbance. For this situation it seems to be possible that the helix ends in a hydraulic jump and is continued by a counter-helix downstream, as the core size gradually diminishes due to the deceleration of the flow caused by viscous effects (not included in the analysis). The capillary-wave pattern produced by surface tension is also considered. A brief outline for the analogous wave problem is given for the case where the fluid rotates like a rigid body.

Photographic observations of hollow-core vortices in water flow are presented which confirm the qualitative predictions of the analysis, both for the response to an axisymmetric area contraction and also to a 90° bend at the downstream end of the vortex tube.

A diagnostic visualization study of turbulence in stagnation flow around a circular cylinder was carried out to gain physical insight into the coherent structure of turbulence in flow around a bluff body advanced by the vorticity-amplification theory. The visualization was conducted at a cylinder-diameter Reynolds number of 8 × 103 utilizing titanium dioxide white smoke for an approaching flow containing turbulence at scales larger than the neutral wavelength of the stagnation flow. Analyses of the flow events focused on tracing out the temporal and spatial evolution of a cross-vortex tube outlined by the entrained smoke filaments from its emergence near the stagnation zone through its penetration into the cylinder boundary layer.

The selective stretching of cross-vortex tubes, their streamwise tilting, the emergence of an organized turbulent flow pattern near the stagnation zone, the interaction of the amplified vorticity with the body laminar boundary layer and the growth of a turbulent boundary layer were revealed by the visualization. In particular, the visualization indicated that the cross-vortex tubes conveyed by the diverging stagnation flow constitute a coherent substructure within the overall turbulent flow that is triggered to its fullest manifestation by the stretching mechanism.

The flow near the mouth of an open tube is examined, experimentally and theoretically, under conditions in which resonant acoustic waves are excited in the tube at the other end. If the edge of the tube is round, separation does not occur at high Strouhal numbers, which enables us to verify theoretical predictions for dissipation in the boundary layer and for acoustic radiation. Observation with the aid of schlieren pictures shows that in the case of a sharp edge vortices are formed during inflow. The vortices are shed from the pipe during outflow. Based on these observations a mathematical model is developed for the generation and shedding of vorticity. The main result of the analysis is a boundary condition for the pressure in the wave, to be applied near the mouth. The pressure amplitudes in the acoustic wave measured under resonance are compared with theoretical predictions made with the aid of the boundary condition obtained in the paper.

The extensional viscosity of a dilute suspension of spherical particles (rigid spheres, viscous drops or gas bubbles) is computed for the case when the Reynolds number of the microscale disturbance motion R is not restricted to be small, as in the classical analysis of Einstein and Taylor. However, the present theory is restricted to steady axisymmetric pure straining flow (uniaxial extension). The rate of energy dissipation is expressed using the Bobyleff-Forsythe formula and then conditionally convergent integrals are removed explicitly. The problem is thereby reduced to a determination of the flow around a particle, subject to pure straining at infinity, followed (for rigid particles) by an evaluation of the volume integral of the vorticity squared. In the case of fluid particles, further integrals over the volume and surface of the particle are required. In the present paper, results are obtained numerically for 1 [les ] R [les ] 1000 for a rigid sphere, for a drop whose viscosity is equal to the viscosity of the ambient fluid, and for an inviscid drop (gas bubble). For the last case, limiting results are also obtained for R → ∞ using Levich's approach.

All of these results show a strain-thickening behaviour which increases with the viscosity of the particle. The possibility of experimental verification of the results, which is complicated by the inapplicability of the approximation of material frame-indifference in this case, is discussed.

A comprehensive programme of research is being undertaken on short-crested waves produced by obliquely reflecting waves from a rigid vertical wall. This has included a new wave theory to third-order approximation. The second-order Eulerian water-particle velocities throughout the bottom boundary layer are now investigated. From this the resulting mass transport is considered to the first approximation. The vertical velocity component has a non-zero value within and just beyond the boundary layer. The limiting two-dimensional cases of progressive and standing waves are obtained and compared with published results. Comparison is also made with available experimental data. Graphs of some analytical solutions are presented.

The stability of natural convection in a vertical annular enclosure has been studied by the linear theory. It was found that for all Prandtl numbers the instability sets in as a wave travelling upward. For low Prandtl numbers, the larger the curvature the more stable the flow; the reverse is true for high Prandtl numbers. The theoretical predictions of the mode of instability were verified for air. A multicellular flow pattern was observed to drift upward with the predicted wave speed. The measured wavelength of the cells is in good agreement with the linear analysis.

This paper considers the nature of stationary axisymmetric convection in a shallow rotating annulus or cylinder heated from below. The horizontal and vertical walls of the container are assumed to be perfect conductors and insulators respectively. The critical regime is considered in which the circulations due to the centrifugal force and the vertical gravitational instability are of equal significance. Among the effects of the centrifugal acceleration on the gravitational instability are a shift in the value of the Rayleigh number at which instability sets in, a modification in the distribution and amplitude of convection cells in the container and, most significantly, a smooth transition to finite amplitude convection as the Rayleigh number is increased.