A scattering configuration consists of a primary acoustic source S and a scattering body B, and we examine the effect of convection of S and B, at a uniform subsonic Mach number M, on the scattered field, for various types of source and scatterer. It is shown that if B is a compact rigid body scattering the near field of the source S it is not equivalent to a convected point dipole, but rather its pressure field is augmented by the quadrupole convection factor (1 − M cosθ)−3. If, on the other hand, the compact body B scatters the distant field of S, convective effects introduce an O(M) monopole field which does not vanish in the sideline directions. A number of problems are examined in which B is a rigid half-plane, and there it is shown that the effect of convection is to augment the pressure by $(1 - M \cos \theta)^{-\frac{3}{2}}$ in the case of diffraction of the field of a distant source S, and by $(1 - M \cos \theta)^{-\frac{5}{2}}$ for scattering of the near field of S. Effects associated with the multipole order of S are discussed, as are those arising from the satisfaction of a Kutta condition at the trailing edge of the half-plane, and the application of these results to current problems in aerodynamic sound is mentioned.

This paper studies several geometric aspects of the Poisson and Gaussian random fields approximating Burgers k−2 and Kolmogorov $k^{-\frac{5}{3}}$ homogeneous turbulence. In particular, simulated sample scalar iso-surfaces (e.g. surfaces of constant temperature or concentration) are exhibited, and their relative degrees of wiggliness are shown to be best characterized by saying that the corresponding fractal dimensions are respectively equal to 3−½ and $3-\frac{1}{3}$.

The hydromagnetic flow of a thermally stratified fluid confined between two rotating parallel plates is studied. The flow is assumed to be linear, steady and axially symmetric. The flow is driven both mechanically and thermally and general thermal boundary conditions are applied. Attention is focused upon the mechanism controlling the interior fluid (diffusion, Ekman pumping or hydro-magnetic forces) and upon the conditions under which laminated flow (∂v/∂z ≠ 0) may occur. It is found that the occurrence of laminated flow is very sensitive to the thermal boundary conditions and is suppressed by hydromagnetic effects. For mixed boundary conditions, hydromagnetic forces control the interior and laminated flow is suppressed if α [ges ] O(1), where α2 represents the ratio of hydro-magnetic to Coriolis forces. For a constant heat flux, this occurs for a much weaker magnetic field: if α [ges ] O(E¼). For a restricted range of the parameters, a new boundary layer, called the thermomagnetic layer, in which Coriolis, thermal and hydromagnetic forces balance may occur.

The properties of edge waves confined by the interaction of buoyancy and Coriolis forces to the vicinity of a rigid plane boundary in a rotating, stratified, electrically conducting fluid pervaded by a magnetic field are established in some simple cases. The background shear is taken to be zero, the basic Alfvén velocity V and Brunt–Väisälä frequency N are assumed uniform, and all dissipative effects are taken to be vanishingly small. It is shown that waves trapped against the bounding wall can occur only if V is parallel to the wall. When the basic rotation vector Ω is also parallel to the wall, the hydromagnetic edge waves have a higher frequency and smaller spatial extent perpendicular to the wall than their non-hydromagnetic counterparts, but more complex behaviour is found when Ω possesses a component normal to the wall. There are conditions under which edge waves may exist even when the basic density stratification is top-heavy (i.e. when N2 < 0).

Exact solutions in closed form have been found using the singularity method for various quadratic flows of an unbounded incompressible viscous fluid at low Reynolds numbers past a prolate spheroid with an arbitrary orientation with respect to the fluid. The quadratic flows considered here include unidirectional paraboloidal flows, with either an elliptic or a hyperbolic velocity distribution, and stagnation-like quadratic flows as typical representations. The motion of a force-free spheroidal particle in a paraboloidal flow has been determined. It is shown that the spheroid rotates about three principal axes with angular velocities governed by a set of Jeffery orbital equations with the rate of shear evaluated at the centre of the spheroid. These angular velocities depend on the minor-to-major axis ratio of the spheroid and its instantaneous orientation, but are independent of its actual size. The spheroid also translates at a variable speed, depending on its orientation relative to the surrounding fluid, along a straight path parallel to the main flow direction without any side drift or migration. This ‘jerk’ motion obeys a trajectory equation which is size dependent.

For incompressible three-dimensional (two-dimensional) turbulence of finite energy, bounds are obtained on energy (enstrophy) flux. To estimate the nonlinear terms, we use a decomposition of the Fourier space into shells of exponentially increasing radii and the property of boundedness in position space of square-integrable functions with Fourier transforms of compact support. In the limit of zero viscosity, it is shown that the three-dimensional (two-dimensional) energy (enstrophy) inertial range, if it exists, cannot have an energy spectrum steeper than $k^{-\frac{8}{3}} (k^{-4})$. Similar results are obtained for the advection of a passive scalar. The connexion with the problem of homogeneous turbulence and intermittency is briefly discussed.

The flow around a cylinder and a sphere rotating freely in a simple shear was studied experimentally for moderate values of the shear Reynolds number Re. For a freely rotating cylinder, the data were found to be consistent with the results obtained numerically by Kossack & Acrivos (1974), at least for Reynolds numbers up to about 10. Rates of rotation of a freely suspended sphere were also obtained over the same range of Reynolds numbers and showed that, with increasing Re, the dimensionless angular velocity does not decrease as fast for a sphere as it does for a cylinder. In both cases, photographs of the streamline patterns around the objects were consistent with this behaviour. Furthermore, it was found in each case that the asymptotic solutions for Re [Lt ] 1 derived by Robertson & Acrivos (1970) for a cylinder and by Lin, Peery & Schowalter (1970) for a sphere are not valid for Reynolds numbers greater than about 0.1, and that the flow remains steady only up to values of Re of about 6.

A numerical method incorporating some of the ideas underlying the wake source model of Parkinson & Jandali (1970) is presented for calculating the incompressible potential flow external to a bluff body and its wake. The effect of the wake is modelled by placing sources on the rear of the wetted surface of the body. Unlike Parkinson & Jandali's method, however, the body shapes that can be treated are not limited by the restrictions imposed by the use of conformal transformation. In the present method the wetted surface of the body is represented by a distribution of discrete vortices. Good agreement has been found between the pressure distributions predicted by the numerical method and the analytic expressions of Parkinson & Jandali for a ‘two-dimensional’ circular cylinder and flat plate. A flat plate at incidence and other asymmetric two-dimensional flows have also been treated. The method has been extended to axisymmetric bluff bodies and the results show good agreement with measured pressure distributions on a circular disk and a sphere.

The inviscid instability of heterogeneous swirling flows with radius-dependent density is investigated and secular relations for the instability growth rates for several different flow configurations are obtained from explicit solutions of the governing equations. It is found, in agreement with a sufficiency condition for the stability of such flows obtained earlier, that they are stable to both axisymmetric and non-axisymmetric infinitesimal modes whenever the density is a monotonic increasing function of radius and at the same time the radial variations in both the angular and axial velocity components remain small. The instability mechanisms present in these flows are both of centrifugal and of shear origin, the classical Rayleigh–Synge criterion being a condition for centrifugal stability. It is shown, via several counter examples, that the Rayleigh–Synge criterion for the stability of swirling flows is generally neither a necessary nor a sufficient condition when non-axisymmetric disturbances are considered or large shears exist in the flow. Very stable flows occur when the angular and axial velocity components have no radial variation and simultaneously the density increases with radius as is the case in a typical centrifuge.

The initial flow field of an incompressible, viscous fluid around a circular cylinder, set impulsively to move normal to its axis, is studied in detail. The nonlinear vorticity equation is solved by the method of matched asymptotic expansions. Analytic solutions for the stream function in terms of exponential and error functions for the inner flow field, and of circular functions for the outer, are obtained to the third order, from which a uniformly valid composite solution is found. Also presented are the vorticity, pressure, separation point and drag. These quantities agree with the numerical computations of Collins & Dennis. Extended solutions developed by Padé approximants indicate that higher than third-order approximations will yield only minor improvements.

A novel technique for injecting buoyancy (heat) into a liquid is described and demonstrated. When buoyancy was injected for a short time a laminar vortex ring formed. Its vertical displacement was found to be only approximately proportional to the square root of time (measured from an apparent initial time). Approximate geometrical similarity was also observed although the Reynolds number decreased from 28 to about 14.

Secular instability of unsteady flow of a horizontal layer of viscoelastic liquid set in motion by simple harmonic motion of the lower boundary in its own plane is investigated. Using an extension of Floquet's theory for ordinary differential equations, the stability of long waves is studied by a regular perturbation method. The elastic parameter of the fluid is found to be destabilizing and stabilizing in different ranges of frequency.

This paper investigates the three-dimensional laminar boundary layer over a blunt body (a prolate spheroid) at low incidence and with reversed flow. Results reflecting the general characteristics of such a problem are presented. More significant are the features relating to the circumferential flow reversal. Some of these features confirm our early hypotheses concerning the existence of a reversed region ahead of separation and the role of the zero-cfθ line in the general context of separation in three dimensions. Other features are unexpected, including the distribution of cfμ and the shape of the separation line. Here cfθ and cfμ denote, respectively, the circumferential and meridional components of the skin friction.

The paper studies convection in a horizontal layer of fluid rotating about a vertical axis. The flows at large Rayleigh number R, with a single horizontal wave-number, are investigated using the mean-field approximation of Herring (1963). The flow that maximizes the heat flux is the same as that which gives an upper bound to the heat flux in the limit of infinite Prandtl number as calculated by the methods of Howard (1963) and Chan (1971, 1974).

Rotation is not significant until the Taylor number Ta exceeds O(R). For $O(R) \ll Ta \ll O[(R\log R)^{\frac{4}{3}}]$, it can increase the rate of heat transfer, a phenomenon noted experimentally by Rossby (1969). It does so because an Ekman layer is formed outside the thermal boundary layer, causing a thinning of the thermal layer. The maximum value of the Nusselt number N is approximately $0.177R^{\frac{1}{5}} Ta^{\frac{1}{10}}[\log Ta]^{\frac{1}{5}}$. As the Taylor number increases further into the region $O[(R\log R)^{\frac{4}{3}}] \ll Ta \ll O(R^{\frac{3}{2}})$, the maximum value of N drops sharply, and becomes approximately $0.029 R^{\frac{3}{2}} Ta^{-1} \log(R^{\frac{3}{2}}/Ta)$. Hence, N now decreases with a further increase of Ta and eventually becomes O(1) as $Ta\rightarrow O(R^{\frac{3}{2}})$ and the layer becomes stable.

A numerical solution of the complete Navier–Stokes equations of motion by means of an implicit finite-difference method is presented for the following developing-flow problem: a piston forced with constant speed through an infinitely long tube of circular cross-section. The transition of the velocity profile of an incompressible isothermal Newtonian fluid from the plug-flow profile in front of the piston to the parabolic profile of developed flow is analysed. Streamlines, vorticity distributions, velocity profiles, the excess pressure drop and the entrance length are given for Reynolds numbers from 0 to 800.

An attempt is made to explain theoretically two curious phenomena involving the motion of the liquid in a spinning, gyrating, liquid-filled gyroscope. One of the phenomena is the periodic breakdown of the free-surface wave form of the spinning liquid in the gyroscope when it gyrates at angles larger than about 1°. The other is the resonant amplitude growth rate of the liquid-filled gyroscope at these angles, for then the small angle stability theory of Stewartson (1959) fails to make the correct predictions.

The analysis exploits the experimental fact that the axis of rotation of liquid in the rotor of a spinning gyrating gyroscope does not remain coincident with the axis of rotation of the rotor when the gyroscope gyrates at amplitudes greater than the above-mentioned 1°. It is shown that this lack of coincidence generates Rossby waves and modifies the inertial wave frequencies that would ordinarily occur in a right circular cylinder. There is no nonlinear interaction between these Rossby and inertial waves; hence the free-surface breakdown remains unexplained. However, the modification of the inertial wave frequencies does seem to account for the curious amplitude growth rate.

The problem of the pattern of motion realized in a convectively unstable system with spherical symmetry can be considered without reference to the physical details of the system. Since the solution of the linear problem is degenerate because of the spherical homogeneity, the nonlinear terms must be taken into account in order to remove the degeneracy. The solvability condition leads to the selection of patterns distinguished by their symmetries among spherical harmonics of even order. It is shown that the corresponding convective motions set in as subcritical finite amplitude instabilities.

Experimental investigations of natural convection in a porous layer placed between two horizontal and isothermal plane surfaces have revealed a new type of convection as the Rayleigh number Ra* increases: fluctuating convection. A numerical study carried out on a two-dimensional model in order to simulate this phenomenon shows that, apart from the influence of the Rayleigh number, the aspect ratio A (length/height) of a vortical cell is the most important parameter for the occurrence of this type of convection. These quasi-periodic fluctuations induce important variations in the temperature field and in the streamlines. The total heat transport, as defined by the Nusselt number Nu*, varies within limits which may be separated by 80% of the mean value. Using the Galerkin method it is possible to deduce the conditions for the onset of convection from a state of pure conduction and also to define the critical conditions for the development of fluctuating convection from another perturbed state. A physical interpretation of the results is given for each type of convection. The results seem to agree with the experimental and numerical results obtained by different authors.

The properties of reflexion, refraction and absorption of a gravity wave incident upon a shear layer are investigated. It is shown that one must expect these properties to be very different depending upon the parameters (such as the Richardson number Ri, the wavelength normalized by the length scale of the shear and the ratio of the flow speed to the phase speed of the wave) characterizing the interaction of a gravity wave with a shear layer. In particular, it is shown that for all Richardson numbers there is a discontinuity in the net wave-action flux across the critical level, i.e. at a height where the flow speed matches the horizontal phase speed of the wave. When Ri > ¼, this is accompanied by absorption of part of the energy of the incident wave into the mean flow. In addition it is shown that the phenomenon of wave amplification (over-reflexion) can arise provided that the ultimate shear flow speed exceeds the horizontal phase speed of the wave and Ri is less than a certain critical value Ric ≃ 0·1129, in which case the reflected wave extracts energy from the streaming motion. It is also pointed out that wave amplification can lead to instability if the boundary conditions are altered in such a way that the system can behave like an ‘amplifier’.

Measurements of velocity fluctuations u (axial) and v (radial) and temperature fluctuations θ averaged over the turbulent zone have been made in a turbulent heated jet with a co-flowing stream and compared with the conventionally averaged results. The zone-averaged mean temperature and temperature fluctuation intensity appear to be nearly homogeneously distributed in the outer, intermittent region of the jet. This homogeneity does not apply to the u and v fluctuations. The flatness factor of the temperature within the turbulent part of the flow is remarkably constant throughout the intermittent region. Although the skewness of the turbulent θ fluctuations is non-zero, it is smaller than the skewness of the turbulent u and v fluctuations. The average $\overline{\theta v}$ of the heat flux over the turbulent zone increases in the intermittent region whereas the zone-averaged momentum flux $\overline{uv}$ and zone-averaged heat flux $\overline{\theta u}$ continuously decrease. This leads to the Prandtl number of the turbulent fluid being smaller than the conventional Prandtl number in the outer part of the flow. A budget for the square of the temperature fluctuations within the turbulent part of the flow indicates a constant distribution of temperature dissipation. The transport of heat by the large-scale structure of the flow is briefly discussed in the light of available experimental information on other turbulent shear flows.