We generalize a recent method for computing optimal 2D convection cooling flows in a horizontal layer to a wide range of geometries, including those relevant for technological applications. We write the problem in a conformal pair of coordinates which are the pure conduction temperature and its harmonic conjugate. We find optimal flows for cooling a cylinder in an annular domain, a hot plate embedded in a cold surface, and a channel with a hot interior and cold exterior. With a constraint of fixed kinetic energy, the optimal flows are all essentially the same in the conformal coordinates. In the physical coordinates, they consist of vortices ranging in size from the length of the hot surface to a small cutoff length at the interface of the hot and cold surfaces. With the constraint of fixed enstrophy (or fixed rate of viscous dissipation), a geometry-dependent metric factor appears in the equations. The conformal coordinates are useful here because they map the problems to a rectangular domain, facilitating numerical solutions. With a small enstrophy budget, the optimal flows are dominated by vortices that have the same size as the flow domain.

To investigate turbulent properties and the developing mechanisms of the tidally induced bottom boundary layer in the linearly stratified ocean, numerical experiments have been executed with a non-hydrostatic three-dimensional model in the rotating frame, changing the temporal Rossby number Rot = |σ/f|, i.e. the ratio of the tidal frequency σ to the Coriolis parameter f. After the flow transitions to turbulence, the entire water column can be characterized by three layers: the mixed layer where density is homogenized and the flow is turbulent (z < zm); the stratified layer where the initial stratification remains and the flow is laminar (z > zt); and the interfacial layer between them where the flow is turbulent but the stratification remains (zm < z < zt). Turbulence is scaled by the frictional velocity uτ and the mixed-layer thickness zm (uτ and uτ/N where N is the buoyancy frequency) in the mixed (interfacial) layer, and has similarity. The mixed layer is thickened by the process where light water of the upper stratified layer is mixed with the lower unstratified layer water through the interfacial layer. As Rot approaches unity, i.e. near the critical latitude, the mixed layer develops more rapidly according to the following mechanism. As becomes Rot closer to unity, the current shear in the interfacial layer is intensified, since the difference of velocity becomes larger between the lower turbulent mixed and upper laminar stratified layers, and this leads to thickening of the interfacial layer. As a result, density deviation of the water entrained from above becomes larger, and this causes more rapid development of the mixed layer. In terms of the energy conversion from the eddy kinetic energy (EKE) to the potential energy (PE), the efficiency factor β which is the ratio of the conversion rate from EKE to PE to that from the tidal shear to EKE increased from 0.25% for Rot = 0.5 to 3.5% for Rot = 1.05 on average. When the time is normalized by the period required for the mixed layer to be thickened to the unstratified turbulent boundary layer δ = uτ/|f+σ|, the mixed layer development occurred in a similar manner in all cases. This similarity suggests the possibility of universal formulation for the turbulent tidal mixing under stratification.

An efficient method of constructing inviscid Batchelor-model flows is developed. The method is based on an analytic continuation of the potential part of the flow into the closed-streamline vortex region. Numerical solutions are presented for Batchelor-model flows past airfoils with cavities. With the airfoil and dividing streamline shape, the eddy vorticity, and the jump in the Bernoulli constant across the eddy boundary given, the program calculates the corresponding cavity shape and the entire flow.

Coupled nonlinear equations are derived for the amplitudes of two small-amplitude resonantly interacting gravity waves in water of non-uniform depth. Such resonance is possible only for wavelengths long compared to the depth. It is shown that the same equations are obtained from the exact Euler equations, from the nonlinear shallow water theory, and from the Boussinesq equations.

The diffraction of a gravity wave of length λ that is obliquely incident upon, and the radiation from, a periodic row of vertical circular ducts of radii a and horizontal spacing b with mouths at a distance h below the free surface of a deep ocean are determined through integral-equation and variational formulations. Numerical results for the reflection coefficient, the pressure-amplification factor (the ratio of the complex amplitude of the wave-induced pressure in the depths of the duct to that of the incident wave at the level of the mouths), and the radiation impedance (the real and imaginary parts of which are measures of the radiation damping and the virtual mass or stiffness of the fluid external to the ducts) are presented as functions of a/λ with a/b and a/h as parameters for the special case of normal incidence with λ > b (which implies that the crests of the scattered waves are parallel to the midplane of the ducts). These results, which complement those of Simon (1981) for a single duct, are of practical interest for wave-power absorption.

The linear stability of a basic flow of two homogeneous inviscid incompressible fluids under the action of gravity is treated mathematically. In the basic state, one fluid is at rest below a horizontal plane z = 0; and the other flows above in the x direction, its speed varying slowly with the lateral co-ordinate y. The eigenvalue problem for normal modes is derived; its equation is a partial differential one, the co-ordinates y and z not being separable. The problem is solved approximately by taking the modes locally as if the basic velocity were independent of y, though the lateral wavenumber is allowed to vary slowly with y. This leads to an ordinary differential equation in y which is solved by the JWKB method. Detailed calculations are made for a parabolic profile, representing the blowing of air over water in a wide channel, and for other profiles.

For an anisotropic topographic feature in a large-scale flow, the orientation of the topography with respect to the flow will affect the vorticity production that results from the topography–flow interaction. This in turn affects the amount of form drag that the ambient flow experiences. Numerical simulations and perturbation theory are used to explore these effects of change in topographic orientation. The flow is modelled as a quasi-geostrophic homogeneous fluid on an f-plane. The topography is taken to be a hill of limited extent, with an elliptical cross-section in the horizontal. It is shown that, as a result of a basic asymmetry of the quasi-geostrophic flow, the strength of the form drag depends not only on the magnitude of the angle that the topographic axis makes with the oncoming stream, but also on the sign of this angle. For sufficiently low topography, it is found that a positive angle of attack leads to a stronger form drag than that for the corresponding negative angle. For strong topography, this relation is reversed, with the negative angle then resulting in the stronger form drag.

Plane and axisymmetric (radial), horizontal laminar jet flows, produced by natural convection on a horizontal finite plate acting as a heat dipole, are considered at large distances from the plate. It is shown that physically acceptable self-similar solutions of the boundary-layer equations, which include buoyancy effects, exist in certain Prandtl-number regimes, i.e. 0.5<Pr[les ]1.470588 for plane, and Pr>1 for axisymmetric flow. In the plane flow case, the eigenvalues of the self-similar solutions are independent of the Prandtl number and can be determined from a momentum balance, whereas in the axisymmetric case the eigenvalues depend on the Prandtl number and are to be determined as part of the solution of the eigenvalue problem. For Prandtl numbers equal to, or smaller than, the lower limiting values of 0.5 and 1 for plane and axisymmetric flow, respectively, the far flow field is a non-buoyant jet, for which self-similar solutions of the boundary-layer equations are also provided. Furthermore it is shown that self-similar solutions of the full Navier–Stokes equations for axisymmetric flow, with the velocity varying as 1/r, exist for arbitrary values of the Prandtl number.

Comparisons with finite-element solutions of the full Navier–Stokes equations show that the self-similar boundary-layer solutions are asymptotically approached as the plate Grashof number tends to infinity, whereas the self-similar solution to the full Navier–Stokes equations is applicable, for a given value of the Prandtl number, only to one particular, finite value of the Grashof number.

In the Appendices second-order boundary-layer solutions are given, and uniformly valid composite expansions are constructed; asymptotic expansions for large values of the lateral coordinate are performed to study the decay of the self-similar boundary-layer flows; and the stability of the jets is investigated using transient numerical solutions of the Navier–Stokes equations.

Three-dimensional numerical integrations of the Navier-Stokes equations have been made for parameters corresponding to some previous laboratory studies of transverse flow past obstacles in a rotating fluid. In the laboratory experiments the character of the flow was found to depend upon the parameter [Sscr ]L = L/DR, where R is the Rossby number U0/ΩL, L is the horizontal scale of the obstacle, D the depth of the fluid, U0 the flow speed and ω the angular rate of rotation. For [Sscr ]L [Gt ] 1 the flows appeared twodimensional and our results confirm the applicability of this assumption in previous asymptotic theories. For [Sscr ]L ∼ 1 a leaning disturbance is produced which can look columnar in character (‘leaning Taylor column’) and our results enable a detailed examination of this structure. To clarify the importance of nonlinear effects in the leaning Taylor column we compare them with the predictions of a linear inertial wave theory. This theory is valid only for small obstacle slopes but provided it includes the effects of viscosity it gives good predictions of the amplitude of the disturbances. The main difference between the viscous linear theory and the Navier-Stokes solution is a flow asymmetry of nonlinear origin. The role of viscosity is important but passive in the sense that it does not alter the flow structure near the obstacle but progressively dissipates the disturbance with increasing distance from the obstacle. This viscous confinement of the disturbance makes the lee wave flow structure look columnar and is important in allowing some laboratory flows to seem unbounded. The results also confirm the conjecture of Mason (1975, 1977) that the large drag forces occurring in these flows are due to inertial wave radiation.

We study the linear stability of surface-tension-driven unidirectional shear flow in an unbounded electrically conducting liquid layer heated from the side and subjected to a uniform magnetic field in the plane of the layer. The threshold of convective instability with respect to oblique travelling waves is calculated depending on the strength and orientation of the magnetic field. For longitudinal waves the critical Marangoni number and the corresponding wavelength are found to increase directly with the induction of a sufficiently strong magnetic field. In general, a coplanar magnetic field causes stabilization of all disturbances except those aligned with the field, which are not influenced at all. With increase of the magnetic field this effect results in the alignment of the most unstable disturbance along the magnetic flux lines. The maximal stabilization is ensured by the magnetic field being imposed spanwise to the basic flow. The corresponding critical Marangoni number is found to be almost insensitive to the thermal properties of the bottom. The strength of the magnetic field necessary to attain the maximal stabilization for a thermally well-conducting bottom is considerably lower than that for an insulating bottom. The basic return flow is found to be linearly stable with respect to purely hydrodynamic disturbances. This effect determines the stability of the basic state with respect to transverse hydrothermal waves at Prandtl number Pr<Prc=0.018. For such a small Pr no alignment of the critical perturbation with a spanwise magnetic field is possible, and the critical Marangoni number can be increased almost directly with the strength of the magnetic field without limit.

In two earlier papers (Gallagher & Mercer 1962, 1964) the results for the first four eigenvalues of the problem of the stability of plane Couette flow were given. The first twelve eigenvalues have been calculated by the same method and the results show that the manner in which the eigenvalues join to form complex pairs depends on α, the wavenumber of the disturbance. They also would appear to indicate that mode-crossing always occurs.

The Green's functions are found for a line source of internal waves in a wedge of stratified fluid of constant Brunt–Väisälä frequency, and are used to discuss the diffraction of internal waves by a wedge in all cases when the vertex angle of the wedge of fluid exceeds the acute angle between a characteristic and the horizontal. Robinson's (1970) results are confirmed and extended.

It is found that the diffracted waves are as important as the incident and reflected ones at all points that lie within a quarter-wavelength or so of either characteristic that passes through the apex. Also, in cases when all the reflected waves are inclined forwards, the diffracted waves lead to a positive backscatter of energy. When the vertex angle of the fluid wedge is less than the characteristic angle, the diffraction problem appears to be ill-posed, and, instead, the motion due to a vibrating body in the wedge of fluid is considered.

A general conclusion is that the so-called ray theory for internal waves, in which the incident and reflected waves alone are considered, has similar limitations to the geometrical theory of optics. Both theories involve the assumption that the typical dimensions in the problem are large compared to the wavelength.

The gas-flow pattern that occurs when a bubble grows during its passage through a fluidized bed is investigated by applying Davidson's (1961) model in an unsteady manner. It is found that the form of the cloud perceived by an observer of this unsteady motion is significantly different from that predicted by the steady theory. Growth is shown to lead to a reduction in the penetration of the cloud ahead of the bubble and a reduction in the area of overlap between cloud and particles. The cloud boundary also interaepts the bubble boundary, and the gas entering the bubble to sustain its growth does so through its base. There is good agreement with experimental observation of clouds.

The mean velocity profile near the surface of turbulent flow in a broad open channel is discussed with dimensional arguments, and a new empirical constant m is introduced which is analogous to von Kármán's constant for flow near a rigid boundary. It is shown that, while the velocity profile depends only rather weakly on m, the dependence of the coefficient of apparent longitudinal diffusion is stronger, and measurements of diffusion could, in principle, provide an accurate determination of its value. The new profiles for various values of m are compared with those in current use, and finally the correction for finite Reynolds number is discussed.

We investigate numerically the time evolution of a two-dimensional flow submitted to a spatially periodic shear force. Initially, the flow is at equilibrium, the forcing balancing viscous stresses. At Reynolds numbers slightly above critical, a large-scale, linear instability drives the fluid towards a stable laminar state. At larger Reynolds number turbulence finally develops after several transient states. These transient states are described by measuring the divergence rate of linearized trajectories from the turbulent flow. This rate gives asymptotically a measure of the first Lyapunov exponent of the flow. We find that the first Lyapunov exponent scales as the characteristic frequency of the flow at large scale. We show here data on incompressible, isothermal and perfect gas (subsonic) two-dimensional flows with unit Prandtl number, and Reynolds number around 30.

Prior modal stability analysis (Kojima et al., Phys. Fluids, vol. 27, 1984, pp. 19–32) predicted that a rising or sedimenting droplet in a viscous fluid is stable in the presence of surface tension no matter how small, in contrast to experimental and numerical results. By performing a non-modal stability analysis, we demonstrate the potential for transient growth of the interfacial energy of a rising droplet in the limit of inertialess Stokes equations. The predicted critical capillary numbers for transient growth agree well with those for unstable shape evolution of droplets found in the direct numerical simulations of Koh & Leal (Phys. Fluids, vol. 1, 1989, pp. 1309–1313). Boundary integral simulations are used to delineate the critical amplitude of the most destabilizing perturbations. The critical amplitude is negatively correlated with the linear optimal energy growth, implying that the transient growth is responsible for reducing the necessary perturbation amplitude required to escape the basin of attraction of the spherical solution.