This paper considers the mixing of a turbulent radial wall jet with a secondary fluid introduced into the impingement area of the wall jet so as to form a steady axisymmetric state. Similarity of the concentration profiles perpendicular to the wall has been assumed and, by solving the momentum and mass flow equations, the concentration distribution through the layer and a similarity exponent giving the variation of concentration along the wall have been determined.

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.

A new approximate theory is proposed for treating the flow past smoothly contoured two-dimensional bluff bodies in the intermediate Reynolds number range O(1) < Re < 0(102), where the displacement effect of the thick viscous layer near the surface of the body is large and a steady laminar wake is present. The theory is based on a new pressure hypothesis which enables one to take account of the displacement interaction and centrifugal effects in thick viscous layers using conventional first-order boundary-layer equations. The basic question asked is how the wall pressure gradient in ordinary boundary -layer theory must be modified if the pressure gradient along the displacement surface using the Prandtl pressure hypothesis is to be equal to the pressure gradient along this surface using a higher-order approximation to the Navier-Stokes equation in which centrifugal forces are considered. The inclusion of the normal pressure field with displacement interaction is shown to be equivalent to stretching the streamwise body co-ordinate in first-order boundary-layer theory such that the streamwise pressure gradient as a function of distance along the original and displacement body surfaces are equal.

While the new theory is of a non-rigorous nature, it yields results for the location of separation and detailed surface pressure and vorticity distribution which are in remarkably good agreement with the large body of available numerical Navier-Stokes solutions. A novel feature of the new boundary-value problem is the development of a simple but accurate approximate method for determining the inviscid flow past an arbitrary two-dimensional displacement body with its wake.

A flat plate x = 0, |y| < L is initially at rest in an electrically conducting, inviscid, incompressible fluid permeated by a uniform magnetic field (B0, 0, 0). The plate is impulsively accelerated to a small velocity (— U, 0, 0) which is then kept constant. It is assumed that LV/λ 1 and U/V [Lt ] 1, where V is the Alfven velocity, and λ is the magnetic diffusivity.

Four stages in the development of the flow are distinguished, the last three being:

1. (ii) L [Lt ] Vt [Lt ] (λ)½. During this stage the initial potential flow is being disturbed by propagation of electric current and vorticity from the plate. The initial discontinuity on the plate has only propagated a small distance away compared to L, but a large distance compared to the length scale of diffusion (λt)½. Exact solutions to the flow are found in the neighbourhood of y = — L and y = 0 (x = 0).

2. (iii) Vt [Lt ] L [Lt ] (λt)½. The asymptotic behaviour of the electric current and vorticity on the plate are determined showing that a column of fluid of length Vt moves with the plate, aligned to the magnetic field. The transverse diffusion of the current sheets bounding the column accelerates the fluid in layers of thickness O(λt)½.

3. (iv) (λt)½ [Lt ] L. Unlike stages (ii) and (iii), where the motion is dominated by the propagation of vorticity and electric current as Alfven waves from the plate, diffusive mechanisms completely dominate the motion. ‘Slug’ flow is maintained. The nature of the flow, including the structure of the layers bounding the column are determined for |x| [Lt ] (λt)½.

It is shown that the solution of the problem of scattering of long waves by a finite barrier in a rotating system may be obtained directly from the solution of the electromagnetic scattering problem for a finite strip. It is shown that the effect of rotation is to produce on both sides of the barrier, but away from the ends, a wave which is propagated without attenuation parallel to the barrier.

The structure of shear layers due to bottom topography in a rotating stratified fluid is obtained under the restriction σS [Lt ] E½, where σS = ναgΔT/KΩ2L is a measure of the stratification and E = ν/Ω2L is the Ekman number. The layers are found to be similar to the side-wall layers discussed by Barcilon & Pedlosky (19673) if σS [Gt ] E½ and are Stewartson layers if $\sigma S \ll E^{\frac{2}{3}}$. Some comments are made on the possibility of Taylor column formation in a stratified fluid.

The joint effects of weak nonlinearity and weak linear damping on the dominant, antisymmetric gravity wave (excited by torsional oscillations of a plane wavemaker about a vertical axis) near its cutoff frequency in a rectangular channel are investigated, following Barnard, Mahony & Pritchard (1977). The evolution equations for the envelope of this mode are derived from the variational formulation previously developed for the parametrically excited cross-wave problem (Miles & Becker 1988). They are equivalent to those of Barnard et al., after correcting their damping and self-interaction terms, and, after appropriate normalization, differ from the cross-wave evolution equations only in the boundary condition at the wavemaker. Analytical approximations and the results of numerical integration for stationary envelopes (as observed in the experiments of Barnard et al.) are presented. The present results are somewhat closer to the observations of Barnard et al. than are their calculations, but the differences between the two calculations are not qualitatively significant.

The propagation of nonlinear hydromagnetic waves in a highly conducting, self-gravitating fluid in a cylindrical geometry, subject to the convective forces produced by a radial temperature gradient, is treated in a Boussinesq approximation. Exact wave solutions of the nonlinear magnetohydrodynamic equations (in the Boussinesq approximation) in the presence of convective forces are obtained for the case when the physical quantities are independent of the axial coordinate or the azimuthal angle in the cylindrical coordinates. The solutions represent waves propagating in the azimuthal or axial direction under the influence of the helical magnetic and velocity fields and the convective forces. The solutions may be applicable to the hydromagnetic waves in the Earth's fluid core and the solar convection zone.

The paper is a first attempt at theoretical investigation of the experimentally observed enigmatic phenomenon of surface manifestations of bottom topography reproducing well the image of the relief despite several kilometres of ocean depth. Both satellite observations and direct measurements have been repeatedly reported in the last two decades. We suggest a possible mechanism for these manifestations in a large scale range (of the order 5 × 101–103 km), based on the hydrodynamic theory of quasi-geostrophic stratified flow over topography on a β-plane.

The classical theories of quasi-geostrophic flow over topography on a β-plane do not include vertical shear, and it is well-known that the disturbance caused by topography cannot reach the surface of a stratified ocean unless the stratification or current velocity is unrealistic. The new element changing the situation qualitatively is the taking into account of the influence of near-bottom and near-surface boundary layers, where flow velocities, velocity gradients and stratification can significantly exceed the corresponding values for the flow in the main body. The asymptotic solution derived shows the considerable increase of the normal mode amplitude towards the boundaries. Thus, this specific distortion of the eigenmode structure results in effective forcing of the modes by topography and, on the other hand, leads to pronounced disturbances in the fields of near-surface characteristics. The mechanism effectiveness is demonstrated by the fact that the surface disturbance amplitude normally significantly exceeds the corresponding value for the barotropic current equal to the maximum of the shear flow. A remarkable feature of the solution is that the Green's function is strongly localized in the horizontal plane for a wide range of relevant parameters, thus leading to the close resemblance of surface patterns and bottom relief. To get a better understanding of the quantitative characteristics of the mechanism, the dependence of the effect on the parameters of an N-layer model was studied in detail. The amplification of the surface manifestations due to the presence of boundary layers can reach several orders of magnitude and thus make the manifestations easily observable. The surface temperature anomalies due to topography were estimated and found to be observable under favourable conditions.

Some exact solutions of the steady M.H.D. equations for an inviscid infinitely conducting fluid are discussed, and a generalization of the Prendergast model for an idealized magnetic star is obtained, in which there is a finite azimuthal swirl velocity of the fluid. The flow in all examples is bounded by a sphere and expressions are found for the distribution of energy between the fluid motion and magnetic field.

The effect of tube elasticity on the stability of Poiseuille flow to infinitesimal axisymmetric disturbances is investigated. The disturbance equations for the fluid are solved numerically while those for the arbitrarily thick tube are solved analytically in terms of Bessel functions of complex argument. It is shown that an elastic tube can cause instability of Poiseuille flow, unlike a rigid tube, in which the flow is always stable. Neutral curves are presented for various values of the tube parameters. It is found that the critical Reynolds number varies almost as the square root of the Young's modulus of the tube material while the critical dimensionless frequency is almost invariant, being about 1·1 for the cases studied.

We consider a pair of like-signed, initially elliptical vortices with uniform vorticity distribution embedded in an incompressible, inviscid fluid occupying a two-dimensional, infinite domain. We characterize this finite-time, aperiodic, dynamical system in terms of its fixed points and separatrices, which divide the flow into inner core, inner recirculation, outer recirculation regions and outer flow. We numerically simulate the time evolution of the vortex pair using a contour dynamics algorithm. The rotational and co-rotational motion of the vortices perturb the separatrices, which undergo to deformations, yielding a tangle whose complexity increases as the amplitude of the perturbation increases. We analyse the dynamics of the tangle and explain the transport of fluid between different regions. We use two diagnostics to quantify stirring: stretching of the interface and the mix-norm. These two diagnostics characterize stirring in contradicting ways and present different sensitivity to the parameters considered. We find that stretching is dominated by the chaotic advection induced within the inner core and inner recirculation regions, whereas the mix-norm is dominated by the laminar transport induced within the outer recirculation regions. For pairs of vortices of small aspect ratio, stretching is piecewise linear and the mix-norm does not decrease monotonically. We show that these two effects are strongly coupled and synchronized with the rotational motion of the vortices. Since the nominal domain is unbounded, we quantify stirring on three concentric, circular domains. One domain nearly encloses the outer separatrices of the vortex pair, one is smaller and one larger than the first one. We show that the mix-norm is very sensitive to the size of the domain, while stretching is not. To quantify the sensitivity of stirring to the geometry of the initial concentration field, we consider, as an initial scalar field, two concentrations delimited by a straight-line interface of adjustable orientation. We show that the interface passing through the centroids of the vortices is the one most efficiently stretched, while the initial concentration field with an orthogonal interface is the most efficiently stirred. Finally, we investigate the effects of the angular impulse on the stirring performance of the vortex pair. Stretching is very sensitive to the angular impulse, while the mix-norm is not. We show that there is a value of the angular impulse which maximizes stretching and argue that this is due to two competing mechanisms.

The time evolution of localized disturbances in an elliptical flow confined in an elliptical cylinder is studied by direct numerical simulation (DNS). The base flow is subject to the elliptic instability. The unstable growth of localized disturbances predicted by the short-wavelength stability analysis is captured. The time evolution can be divided into four stages: linear, weakly nonlinear, nonlinear and turbulent. In the linear stage a single wavepacket grows exponentially without changing its shape. The exponential growth is accompanied by large oscillations which have time period half that of the fluid particles in the elliptical flow. An averaged wavepacket, which is a train of bending waves that has a finite spatial extent, also grows exponentially, while the oscillations of the growth rate are small. The averaged growth rate increases as the kinematic viscosity decreases; the inviscid limit is close to the value predicted by the short-wavelength stability analysis. In the weakly nonlinear stage the energy stops growing. The vortical structure of the initial disturbances is deformed into wavy patterns. The energy spectrum loses the peak at the initial wavenumber, developing a broad spectrum, and the flow goes into the next stage. In the nonlinear stage weak vorticity is scattered in the whole domain although strong vorticity is still localized. The probability density functions (p.d.f.) of a velocity component and its longitudinal derivative are similar to those of isotropic turbulence; however, the energy spectrum does not have an inertial range showing the Kolmogorov spectrum. Finally in the turbulent stage fine-scale structures appear in the vorticity field. The p.d.f. of the longitudinal derivative of velocity shows the strong intermittency known for isotropic turbulence. The energy spectrum attains an inertial range showing the Kolmogorov spectrum. The turbulence is not symmetric because of rotation and strain; the component of vorticity in the compressing direction is smaller than the other two components. The energy of the mean flow as well as the total energy decreases. The ratio of the lost energy to the initial energy of the mean flow is large in the core region.

Expressions are derived for a three-dimensional polar in the pressure ratio, deflexion angle space to represent conditions downstream of a detonation wave with given upstream conditions. An analysis of reflected waves is undertaken and the representation of three wave confluences in a three-dimensional hodograph space studied. These techniques are applied to experimental results available for spin detonation in gaseous mixtures of acetylene-oxygen-argon and carbon monoxide-oxygen. Assumptions made in earlier papers concerning local extinction of combustion at the confluence point are not necessary in this picture.