The steady and quasi-steady motion achieved in a rotating stratified sphere of fluid is studied in the context of a linearized Boussinesq model. In certain parameter ranges an explicit expression is obtained for the flow field as a functional of the surface stress. The non-singular interior solution is used to examine the behaviour of the boundary layer close to the equator. The results agree with previous conclusions about the behaviour of a rotating stratified fluid in simpler geometries. Viewing the problem as a simple model for the interior core of the sun, this work indicates a solar spin-down time that is within the lifetime of the sun.

The growth rates and phase speeds of gravity-capillary wind waves are investigated through numerical solution of a linear, viscous, coupled, shear-flow perturbation model. Numerical results are obtained by transforming the boundary-value problem of a perturbed mean laminar shear flow into a matrix-eigenvalue problem using standard finite-difference methods.

Detailed calculations are performed for a basic state composed of a logarithmic-linear mean flow profile in the air and a linear-logarithmic mean flow profile in the water. We exclude turbulent Reynolds stresses. Calculated growth rates show excellent agreement with corresponding experimental growth rates. This implies that the initial growth of gravity-capillary wind waves is almost certainly due to the instability of the coupled laminar shear flow in the air and water.

The investigation also demonstrates that the shear flow in the water cannot be ignored in wave growth studies, since the usual 3-4%, highly sheared, wind-induced surface drift produces a significant increase in the growth of wind-generated gravity-capillary waves.

Boundary-layer damping of one-dimensional gravity waves of slowly varying amplitude a(t), characteristic wavenumber k, and characteristic frequency ω in water of depth d and kinematic viscosity v is calculated for a [Lt ] d, d [Lt ] 1/k and (2ν/ω)½ [Lt ] d. General results are given for the temporal evolution of the power spectral density determined by either a Fourier-integral (spatially aperiodic) or Fourier-series (spatially periodic) representation of the wave. Solitary and cnoidal waves are considered as examples. Keulegan's (1948) inverse-fourth-power decay for the solitary wave is recovered, and the numerical parameter therein is evaluated by reduction to a Riemann zeta function. A universal decay curve is obtained for the Stokes-scaled amplitude S = a/k2d3 of the cnoidal wave as a function of the boundary-layer-scaled time (νω)½t/d; the result is both more flexible and more compact than that obtained by Isaacson (1976). The decay is within 5% of that for a solitary wave (inverse fourth power) for S > 2 or that for an infinitesimal wave (exponential) for S < 2. An analytical approximation with a maximum error of less than 1% is obtained by joining an asymptotic approximation for S > 1 to the exponential approximation for S < 1.

The reflexion of an internal gravity wave with stream function \[ \psi = \exp\{i(kx + lz - \omega t)\} \] from a corrugated surface of the form z = a cos px is investigated using an extension of Rayleigh's method for an inviscid non-diffusive fluid. It is found that the method converges for many wall slopes ap and incoming-wave phase propagation incidence angles θ = tan−1l/k but that the appropriate series solution is only asymptotic in other cases. The accuracy of the calculations is assured by requiring that the solution satisfy the boundary condition at the wall using a least-squares error minimization technique. The accuracy is then verified through the conservation of energy flux. It is found that, as the surface slope is increased for constant θ, less energy appears in specular reflexions and more is either back-scattered or redistributed into other forward-scattered modes. As the horizontal internal wavelength is decreased to become comparable to the corrugation wavelength of the wall, substantially less energy appears in specular reflexion, but of the order of 95% of the incoming energy is specularly reflected for 30° < θ < 60° when the two wavelengths are equal. In contrast to this, it is found that the general level of the ratio of back-scattered to forward-scattered energy is reduced by O(10−3) to O(10−2) as the incident horizontal internal wavelength becomes smaller than the corrugation wavelength. The results are compared with the linear theory of Baines (1971); agreement is good for forward-scattered energy and excellent for the back-scattered flux when p ≥ k.

We consider a kinetic theory model of a gas, whose molecular velocities are restricted to a set of fourteen given vectors. For this model we study the Couette flow problem, the boundary conditions on the walls being the conditions of pure diffuse reflexion. The kinetic equations can be integrated by quadrature under the assumption that the walls have opposite velocities and equal temperatures. The presence on the walls of tangential velocities leads to the consequence that the velocity slip coefficient does not in general vanish when the Knudsen number goes to zero.

Considering the same problem again after the suppression of tangential velocities, we obtain formulae for the velocity and temperature slip coefficients which generalize results of Broadwell (1964b), and which agree qualitatively with experiments.

The stability of the Stewartson layer in a rotating incompressible fluid is investigated within the framework of a linear theory. The boundary-layer structure of the shear layer is correctly taken into account and the effect of viscous dissipation on the disturbance is included in the governing equations. The growth rate ωi of the disturbance is given as a function of the unified parameter mRo/(γ½), where m, an integer, is the azimuthal component of the wavenumber vector, γ the radius of the layer, Ro the Rossby number and E the Ekman number. Instability occurs when m Ro/(γ½) > 9·5. The time evolution of a growing disturbance is given schematically. Comparison of our results with the experiments by Hide & Titman shows good agreement.

This paper reports an experimental investigation into the impingement of underexpanded axisymmetric jets on each of three wedges arranged symmetrically in the jets. Three jets were used; two were produced by a convergent-divergent nozzle of exit Mach number 2.2 which was operated at underexpansion ratios of 1·2 and 2, while the third jet was produced by a convergent nozzle operated at an underexpansion ratio of 4. The base widths of the wedges equalled the exit diameter of the convergent-divergent nozzle, their apex angles were 90°, 60° and 45° and they were situated within or just downstream of the first cell of each jet.

Detailed front-face pressure distributions were obtained for 15 different configurations. Shadowgraph photographs were taken of these and other cases. The results show a variety of possible flow patterns. The major factors determining which flow pattern occurs are the combination of the centre-line Mach number and wedge apex angle and the jet shock strength and position. It was found that observed shock intersections can be reconstructed by means of shock polars, but that a four-shock confluence will not always occur when it is possible according to the shock polars. In one case there is evidence of the existence of a β triple-shock confluence. An interesting and unusual flow involving a stagnation bubble was obtained when the wedges were situated downstream of the free-jet Mach disk.

The study of natural convection in a saturated porous medium bounded by two concentric, horizontal, isothermal cylinders reveals different types of evolution according to the experimental conditions and the geometrical configuration of the model. At small Rayleigh numbers the state of the system corresponds to a regime of pseudo-conduction. The isotherms are coaxial with the cylinders. At larger Rayleigh numbers a regime of steady two-dimensional convection sets in between the two cylinders. Finally, for Rayleigh numbers above the critical Rayleigh number Ra*c the phenomena become three-dimensional and fluctuating. The appearance of these different regimes depends, moreover, on the geometry considered and, in particular, on two numbers: R, the ratio of the radii of the cylinders, and A, the ratio of the length of the cylinders to the radius of the inner one. In order to approach these experimental observations and to obtain realistic theoretical models, several methods of solving the equations have been used.

The perturbation method yields information about the thermal field and the heat transfer between the cylinders under conditions close to the equilibrium state.

A numerical two-dimensional model enables us to extend the range of investigation and to represent properly the phenomena when steady convection appreciably modifies the temperature distribution and the velocities within the porous layer.

Neither of these models allows account to be taken of the instabilities observed experimentally above a critical Rayleigh number Ra*c. For this reason, a study of stability has been carried out using a Galerkin method based on equations corresponding to an initial state of steady convection. The results obtained show the importance of three-dimensional effects for the onset of fluctuating convection. The critical transition Rayleigh number Ra*c is thus determined in terms of the ratio of the radii R by solving an eigenvalue problem.

A numerical three-dimensional model based on the method of finite elements has thus been developed in order to point out the different types of evolution with time. Steady two-dimensional convection and fluctuating three-dimensional convection have been actually found by calculation. The solution of the system of equations by the method of finite elements is briefly described.

The experimental and theoretical results are then compared and a general physical interpretation is given.

The problem of uniform flow past a flat plate whose surface has a constant velocity λU opposite in direction to that of the mainstream is considered for large values of the Reynolds number R. In a previous communication (Klemp & Acrivos 1972) it was shown that, if the region of reverse flow which is established next to the plate as a consequence of its motion is O(R−1/2) in thickness, the appropriate laminar boundary-layer equations have a solution provided λ ≤ 0·3541. Here the analysis is extended to the range λ > 0·3541, which cannot be treated using a conventional boundary-layer approach. Specifically, it is found that for λ > 0·3541 the flow consists of three overlapping domains: (a) the external uniform flow; (b) a conventional boundary layer with reverse flow for xs < x < 1, where xs, refers to the point of detachment of the ψ = 0 streamline and x = 1 is the trailing edge of the plate; and (c) an inviscid collision region in the neighbourhood of xs, having dimensions O(R−1/2) in both the streamwise and the normal direction, within which the reverse moving stream collides with the uniform flow, turns around and then proceeds downstream. It is established furthermore that xs = 0 for 0 ≤ λ ≤ 1 and that xs < 0 for λ > 1.

Also, detailed streamline patterns were obtained numerically for various λ's in the range of 0 ≤ λ ≤ 2 using a novel computational scheme which was found to be more efficient than that previously reported. Interestingly enough, the drag first decreased with λ, reached a minimum at λ = 0.3541, and then increased monotonically until, at λ = 2, it was found to have attained essentially the value predicted from the asymptotic λ → ∞ similarity solution available in the literature. Thus it is felt that the present numerical results plus the two similarity solutions for λ = 0 and for λ → ∞ fully describe the high-R steady flow for all non-negative values of λ.

The properties of a turbulent boundary layer were investigated as they relate to the form drag on a two-dimensional fence. Detailed measurements were performed at zero pressure gradient of velocity profiles along smooth, rough and transitional flat plates. Upon comparison with other published data, these measurements resulted in simple formulae for the displacement thickness and the local shear coefficient and in a modification to the universal velocity defect law for equilibrium boundary layers.

With these boundary layers, experiments were performed to determine the drag on a two-dimensional fence. These data were analysed along with data from previous investigations. It was found that after suitable blockage corrections all form-drag coefficients for two-dimensional fences collapsed on a single curve if they were calculated with the shear velocity as the reference velocity and plotted against the ratio of the fence height to the characteristic roughness parameter of the approaching flow.

We consider in this paper the hydromagnetic flow of a viscous, incompressible, electrically conducting fluid induced by the differential rotation of an insulating boundary. A uniform magnetic field parallel to the axis of rotation is applied at a large distance from the interface. The method of matched asymptotic expansions is employed to study the hydromagnetic coupling between different regions of fluid flow and the interior of the insulating boundary.