The energy theory of hydrodynamic stability is applied to the viscous incompressible flow of a fluid contained between two concentric spheres which rotate about a common axis with prescribed angular velocities. The critical Reynolds number is calculated for various radius and angular velocity ratios such that it is certain the basic laminar motion is stable to any disturbances. The stability problem is solved by means of a toroidal–poloidal representation of the disturbance flow and numerical integration of the resulting eigenvalue problem.

An axisymmetric flow of a rotating fluid into a point sink was studied experimentally. The type of motion is mainly controlled by the value of the Rossby number R, a ratio of inertial and Coriolis forces. Experimental investigation shows that at a sufficiently large value of R the fluid motion resembles potential flow. However, as R falls below a critical number the withdrawal of the fluid starts to be selective. The flow field then divides into two regions; namely, a central flowing core and an almost stagnant region surrounding it.

It is observed that at a Rossby number below the critical value the flow field, induced by a sudden start of discharge at the sink, experiences several distinct stages during the course of each run. At the initial moment the flow exhibits a feature of potential flow. During the second stage, it develops into a state of selective withdrawal with an inviscid profile of a flowing core, which is the main interest of the present study. In the third stage, due to the unavoidable influence of the free or solid surface at the upstream, flow undergoes another change. The flowing core becomes a fast-spinning jet, in which the viscous force becomes important.

It is also found that once selective withdrawal begins, the angular velocity and flow rate of the flowing core differ substantially from the basic rotation and the actual discharge at the sink. During this second stage of flow development, the flowing core tends to adjust itself such that the intrinsic Rossby number R′(= Wmax/2Ωcδc) based on the properties of the flowing core, virtually remains constant for all values of R below the critical value. This constant value of R′ is found to be about 0·36. The critical value of R which marks the beginning of the selective withdrawal is found to be in the neighbourhood of 0·26.

Galerkin (spectral) methods for numerical simulation of incompressible flows within simple boundaries are shown to possess many advantages over existing finite-difference methods. In this paper, the accuracy of Galerkin approximations obtained from truncated Fourier expansions is explored. Accuracy of simulation is tested empirically using a simple scalar-convection test problem and the Taylor–Green vortex-decay problem. It is demonstrated empirically that the Galerkin (Fourier) equations involving Np degrees of freedom, where p is the number of space dimensions, give simulations at least as accurate as finite-difference simulations involving (2N)p degrees of freedom. The theoretical basis for the improved accuracy of the Galerkin (Fourier) method is explained. In particular, the nature of aliasing errors is examined in detail. It is shown that ‘aliasing’ errors need not be errors at all, but that aliasing should be avoided in flow simulations. An eigenvalue analysis of schemes for simulation of passive scalar convection supplies the mathematical basis for the improved accuracy of the Galerkin (Fourier) method. A comparison is made of the computational efficiency of Galerkin and finite-difference simulations, and a survey is given of those problems where Galerkin methods are likely to be applied most usefully. We conclude that numerical simulation of many of the flows of current interest is done most efficiently and accurately using the spectral methods advocated here.

A general theoretical relationship between the (three-dimensional) vcLacity field and density field is established for geostrophic flow of a non-homogeneous fluid. The practical value of the relationship – which reduces to the celebrated two-dimensional theorem due to Proudman and Taylor when the fluid is homogeneous – is illustrated by means of three examples of flows in rapidly rotating fluids, namely (i) baroclinic waves in laboratory systems and in the atmosphere, (ii) the gyroscopic ‘steering’ processes in Jupiter's atmosphere that are implied by the ‘Taylor column’ theory of the Great Red Spot, and (iii) certain striking properties of ocean currents revealed by recent observations in the Atlantic and Mediterranean.

In problems where a viscous fluid flows past a porous solid it has frequently been assumed that the tangential component of surface velocity is zero. When the porous solid has an open structure with large pores the external surface stress may produce a tangential flow below the surface. Recently, Beavers & Joseph (1967) have assumed that the surface velocity UB depends on the mean tangential stress $[\mu(d\overline{u}/dy)]_{y=0}$ in the fluid outside the porous solid through the relation \[ \left[\mu\frac{d\overline{u}}{dy}\right]_{y=0} = \frac{\mu\alpha}{k^{\frac{1}{2}}}(U_B-Q), \] where Q is the volume flow rate per unit cross-section within the porous material due to the pressure gradient, k is the Darcy constant and α is a constant which depends only on the nature of the porosity. An artificial mathematical model of a porous medium is proposed for which the flow can be calculated both inside and outside the surface. This conceptual model was materialized and the experimental results agree with the calculations. The calculated values of α so found are not quite independent of the external means of producing the external tangential stress.

The impulsive linear spin-down of a stably stratified Boussinesq fluid in a circular cylinder is analyzed under the assumption that the Prandtl number P = ν/κ is small (ν is the kinematic viscosity, κ the thermal diffusivity). The nature of the spin-down process depends on the ordering of P with respect to the Ekman number E. For P < E½, the spin-down is similar to that of an unstratified fluid. For P > E½ the process is similar to that for a stratified fluid with P = O(1). The distinctive case P = O(E½) is analyzed in detail. For that case it is shown that for N [Gt ] Ω, the asymptotic state of rigid rotation is reached in a time of the order of (N/Ω)2τ, where N is the Brunt-Väisälä frequency, Ω the angular velocity and τ the thermal diffusion time for the cylinder. We calculate spin-down times for parameter values corresponding to the solar interior. For an angular velocity as large as that suggested by Dicke (5·74 × 10−5 sec −1) the spin-down time is less than the age of the sun. For an angular velocity comparable to the surface value (2middot;87 × 10−6 sec −1), the spin-down time is greater than the age of the sun. These results suggest that a uniformly and rapidly rotating solar interior is not possible, but we cannot rule out a state of non-uniform rotation producing an oblateness as large as that measured by Dicke & Goldenberg (1967).

This paper considers the linear inviscid reflexion of internal/inertial waves from smooth bumpy surfaces where a characteristic (or ray) is tangent to the surface at some point. There are two principal cases. When a characteristic associated with the incident wave is tangent to the surface we have diffraction; when the tangential characteristic is associated with a reflected wave we have split reflexion, a phenomenon which has no counterpart in classical non-dispersive wave theory. In both these cases the problem of determining the wave field may be reduced to a set of coupled integral equations with two unknown functions. These equations are solved for the simplest topography for each case, and the properties of the wave fields for more general topographies are discussed. For both split reflexion and diffraction, the fluid velocity has an inverse-square-root singularity on the tangential characteristic, and the energy density has a corresponding logarithmic singularity. The diffracted wave field penetrates into the shadow region a distance which is of the order of the incident wavelength. Possibilities for instability and mixing are discussed.

Davey, Di Prima & Stuart's (1968) double amplitude expansion for disturbances in flow between concentric cylinders is formulated in matrix notation. The stability of the secondary equilibrium (Taylor-vortex) flow is calculated using fifth-order terms in amplitude, and using the full equations rather than the small-gap approximation. Qualitative confirmation is found of instabilities to the Taylor-vortex flow to non-a.xisymmetric disturbances at about 10 % above the first critical Taylor number.

The present paper contains an analysis of the model of a porous material proposed in part 1, and carries out calculations which allow comparison between theory and the experiments described therein. The relevant boundary conditions to be applied at an interface between a fluid and such a material are considered.

This paper describes some laboratory and numerical experiments made on the longitudinal dispersion in an open channel flow. Particular attention has been paid to the initial stages of the process.

Physical arguments suggest that the streamwise dispersion of a line of marked fluid elements across a two-dimensional turbulent shear flow occurs in three distinct stages. These stages are identified by a change in the form of the distribution of marked fluid elements in the flow direction. The skewed distribution of the first stage is readily identified by a constant value (approximately 1·1) for the ratio of the peak velocity (V1) of the distribution to the mean-flow velocity $\overline{U}$; experiments using dyed fluid, made at this stage of the process, have revealed six identifiable features of the suggested distribution. The distributions suggested for the second and the third stage are consistent with the experimental findings of Elder (1959) for the second stage and Taylor (1954) for the third stage.

An attempt has been made to simulate the process numerically using a Markovian model. The results of the simulation confirm features suggested by physical arguments and are in agreement with the open channel experiments.

The Lagrangian autocorrelation function is found to be related to the Lagrangian velocity-history of marked fluid released from extreme positions on the flow cross-section. The correlation function, as expressed in terms of the velocity-history function provided by the numerical simulation, is \begin{eqnarray*} && R(t^{\prime}) = \exp (-bt^{\prime})\int_0^{1}U^{+2}dy^{\prime};\\ && t^{\prime} = tu_{*}/d,\quad U^{+} = \frac{U(y^{\prime})-\overline{U}}{u_{*}}, \end{eqnarray*} where u* is the friction velocity and U(y′) is the temporal mean velocity at a (non-dimensionalized) distance y′ from the flow boundary. In an open channel flow at a Reynolds number (based on friction velocity and channel depth) of 500, the numerical simulation provides the value of b = 0·536.

The results of an experiment, in which the three-dimensional motion of small neutrally buoyant spheres was recorded in many small discrete time intervals, corroborate the theoretical suggestions and simulation results.

The modified Oseen method is extended to provide a description of the boundary layers which accompany certain swirling flows over a rigid boundary in a rotating container. By comparison with known results it is shown that a refined procedure has errors of the order of 1% when the inviscid flow is a rigid body rotation; it is anticipated that, for the more interesting flows, the error is of the order of 30%.

A theory is presented which describes the propagation of large amplitude tsunamis across a basin of variable depth in the limit when this depth is varying slowly on a scale deiined by the wavelength. In part 1 only the off-shore behaviour is considered; in part 2 some features of the final run up are described.

The technique used is to regard the wave as a slowly modulated simple wave with a slowly changing Riemann invariant. One of the most significant results is that over distances where the effect of depth variation modulates the amplitude of the wave, but does not disperse it, the variations of the amplitudes of the flow variables, such as maximum surface elevation, can be calculated as functions of the undisturbed depth without knowing how this depth varies in distance and without knowing the wave profile. These variations are fully calculated.

The work continues the investigation on large amplitude acoustic pulses in stratified media described in an earlier paper by Varley & Cumberbatch (1970). It is a generalization of Whitham's work (1953) on the sonic boom.

Expansions are obtained for the large Prandtl number structure of the laminar natural convection boundary layer, together with its linear stability characteristics, for the case of a uniform-heat-flux semi-infinite vertical plate. The primary source of instability is shown to arise from a temperature-coupling effect associated with the inner heated region of the boundary layer. Based upon an empirical correlation between the results of linear stability theory and experimentally determined regimes of the turbulent-transition process, it is shown that the flow can be expected to become turbulent before the outer vorticity region of the laminar boundary layer is fully established. The results are generalized to the isothermal plate case.

An account is given of the dynamic balance and the thermodynamic considerations which underlie the intensification of a tropical disturbance into a mature hurricane. In particular, it is deduced that the e-folding time for the later stages of the intensification process is of the order of 16 h but that the eye formation itself can occur on a much shorter time scale.

An investigation of the mechanism responsible for the self-excited oscillations of a harmonium reed is presented. Experiments show that the amplitude of vibration of the reed grows exponentially, and measurements of the growth rate as a function of the flow past the reed are reported. Flow visualization studies lead to the conclusion that jet or wake instability is not important in exciting the reed vibration. An analysis of the flow around the reed as an unsteady potential flow results in the evaluation of the aerodynamic forces exciting the reed. The analysis shows that the pressure which excites the reed motion is of the order of $\rho U_0\dot{a}$, where U0 is the flow velocity and $\dot{a}$ is the reed velocity.

The variation of the translational temperature, rotational temperature, and density through shock waves in oxygen and nitrogen was studied using classical laws of mechanics and a Monte Carlo scheme. The collision dynamics were calculated using an intermolecular potential by Parker with both a two-dimensional approximation and the full three-dimensional calculations. The rotational velocity frequency distributions were also calculated. The average number of collisions a molecule will experience a t various stages passing through a shock wave were found and plotted with the temperature and density profiles. The nitrogen results were compared with experimental results and good agreement was found. This also provided a method for giving a first approximation to the three-dimensional intermolecular potential.

Waves generated at the stern of a ship must travel through the ship's wake. The effect of the mean flow in the wake refracting the waves is calculated by using a much simplified model. It is found that the waves diverging from the stern of a ship may differ considerably from the bow waves, in qualitative agreement with observation.

Three problems are considered. In the first, a uniform magnetic field is suddenly switched on outside an infinitely long circular cylinder of incompressible con- ducting fluid. As the field diffuses into the fluid, electric currents are generated and hence a Lorentz force field. Under the assumptions of small magnetic Reynolds number and small magnetic Prandtl number, the initial flow (before it has been modified by convection or viscous diffusion of vorticity) is oalculated. In the second problem an initially uniform field is suddenly switched off outside a similar cylinder of fluid. It is shown that the switching off of the field produces a vorticity distribution identical with that produced by the switching on. The third problem considered is that of a circular cylinder of conducting fluid placed in a rapidly alternating magnetic field. It is shown that the alternating field produces vorticity at a constant rate, and a qualitative description of the resulting flow is given.

The statistical initial-value problem for a class of weakly coupled waves whose linear dispersion relation is ω ∞ ± |k| is examined. It is found that in two and higher dimensions a natural asymptotic closure is possible. The redistribution of energy is achieved by means of two mechanisms; the first by a resonance between collinear wave vectors; the second by a local transfer between adjacent rays. The entropy functional is ∫ log n(k) dk and corresponds to particles obeying Bose–Einstein statistics.

A new theory is developed for the wake far downstream of a cylindrical body of height h, placed with its generators perpendicular to the flow on a surface above which there is a boundary layer of thickness δ. If the streamwise (x) velocity in the wakeis (U + u), then assuming (h/δ) is small enough that the velocity profile in the boundary layer may be regarded as U = αy, and assuming |u| [Lt ] U, linear differential equations governing u are derived. It is found that a constant along the wake is \[ I=\frac{3}{2}\int_0^{\infty} yUu\,dy. \] This result can be used to find an order of magnitude estimate for u, because I is related to the forces on the body producing the wake by the approximate formula \[ I \simeq - C_1/\rho, \] where C1 is that component of the couple on the body produced by pressure and viscous stresses in the x direction. For the particular case of a small hump on the boundary of height h and length b, such that h [Lt ] b, the above relation is shown to be exact. The perturbation velocity in the wake is found to have a similarity solution \[ u = [I/(xv)]f(y^3/[xv/\alpha]), \] the physical implications of which are discussed in detail. The relevance of the theory to the problem of transition behind a trip wire is also mentioned.