The problem of the resonant response of a gas contained in a two-dimensional rectangular cavity to periodic (sinusoidal) velocity excitations at the walls of the cavity is investigated. It is found that in some neighbourhood of each resonant frequency there are discontinuous pressure disturbances (shock waves). The present theory is an extension of Chester's theory on resonances in closed tubes.

We present a phenomenological model of intermittency called the β-model and related to the Novikov-Stewart (1964) model. The key assumption is that in scales ∼ l02−n only a fraction βn of the total space has an appreciable excitation. The model, the idea of which owes much to Kraichnan (1972, 1974), is dynamical in the sense that we work entirely with inertial-range quantities such as velocity amplitudes, eddy turn-over times and energy transfer. This gives more physical insight than the traditional approach based on probabilistic models of the dissipation.

The β-model leads in an elementary way to the concept of the self-similarity dimension D, a special case of Mandelbrot's (1974, 1976) ‘fractal dimension’. For three-dimensional turbulence, the correction B to the $\frac{5}{3}$ exponent of the energy spectrum is equal to $\frac{1}{3}(3 - D)$ and is related to the exponent μ of the dissipation correlation function by $B = \frac{1}{3}\mu $ (0.17 for the currently accepted value). This is a borderline case of the Mandelbrot inequality $B \leqslant \frac{1}{3}\mu $. It is shown in the appendix that this inequality may be derived from the Navier-Stokes equation under the strong, but plausible, assumption that the inertial-range scaling laws for second- and fourth-order moments have the same viscous cut-off.

The predictions of the β-model for the spectrum and for higher-order statistics are in agreement with recent conjectures based on analogies with critical phenomena (Nelkin 1975) but generally diasgree with the 1962 Kolmogorov lognormal model. However, the sixth-order structure function 〈δv6(l)〉 and the dissipation correlation function 〈ε(r)ε(r + 1)〉 are related by \[ \langle \delta v^6(l)\rangle /l^2\sim\langle \epsilon({\bf r})\epsilon({\bf r}+{\bf l})\rangle \] in both models. We conjecture that this relation is model independent.

Finally, some possible directions for further numerical and experimental work on intermittency are indicated.

A sphere was subjected to a simple harmonic motion in an otherwise undisturbed liquid. Records of the resistance of the liquid to the motion for various amplitudes and frequencies were obtained. The resistance was first represented by an equation consisting of three terms with empirical coefficients: the steady-motion drag, a term due to the ‘added mass’ and a term due to the history of the motion. It was found that the data could be correlated only with a large degree of scatter by this type of equation. Subsequently an attempt was made to represent the resistance by means of a single term, with an empirical coefficient C. It was found that C correlated well with the acceleration number Vd/V2 and the Reynolds number Vd/v, where V, V and d are the acceleration, velocity and diameter of the sphere respectively and v is the kinematic viscosity of the liquid. C increased with Vd/V2 and decreased in the limit to the steady-motion drag coefficient Cd when Vd/V2 became very small. The range of the Reynolds number in the experiments was 102 < Vd/v < 104 and the range of the acceleration number was 0 ≤ Vd/V2 ≤ 10·5.

Steady flow of an incompressible homogeneous fluid over shallow topography in a rotating annulus is considered. The flow is driven by a differentially rotating lid. The recycling nature of the system means that prescribed upstream conditions are not available to close the problem. Consideration of the balance of transport across streamlines for the geostrophic flow leads to a general circulation condition; namely $\Gamma(\psi) = \frac{1}{2}\Gamma_T(\psi)$, where ϕ is the stream function for the geostrophic flow, Λ is the circulation around a streamline and ΛT is the circulation around the same path calculated using the prescribed upper surface velocity. Using this condition, stream functions for the linear viscous and nonlinear quasi-inviscid flow can be found. Solutions for these two limits, and linearized perturbation solutions for the transition regime between them, are presented for flow over ridges in the annulus.

This paper supplements an earlier study (Singh 1974) of the steady case of ‘Entry flow in a curved pipe’. Here we consider an entrance profile of the form \[ w = Q(t)/(1+\delta r\cos\alpha), \] which is physiologically more relevant for blood being pumped from the left ventricle into the ascending aorta. A boundary-layer analysis is applied to determine the effects of curvature and an adverse pressure gradient (associated with the primary flow) on the wall shear. The study shows how the negative wall shear and backflow near the wall develop during the decelerating phase of the cycle as the boundary layer grows. The analysis shows how the increasing effect of the secondary flow due to curvature draws off slower moving fluid azimuthally from the outer bend to the inner bend; this induces a cross-flow of faster moving fluid from the inner bend to the outer bend which results in a thinning of the boundary layer at the outer bend and a thickening at the inner bend. This implies an increased wall shear at the outer bend compared with that at the inner bend as the flow develops further downstream; this is in contrast with the initial stages of the motion near the entrance where the higher wall shear occurs at the inner bend owing to the external flow and to geometric factors. The analysis shows that the displacement effect of the boundary layer on the core is not significant because the boundary layer remains thin, about one-tenth of the tube diameter.

An International Advanced Course on the Thermomechanics of Magnetic Fluids was held from 3 to 7 October 1977 at the International Centre for Mechanical Sciences (CISM), Udine, Italy, under the direction of B. Berkovsky. The course was a joint CISM/UNESCO undertaking and its general objectives were to assess knowledge in the field, to identify specific problem areas and promising directions for future research and applications, and to assist in the establishment of active relations between national research groups. It was the first conference to bring together the diverse and dispersed community of researchers in the field; there were 34 participants from 16 countries and 19 papers were included in the programme.

Nonlinear thermosolutal convection is investigated using the mean-field approximation (Herring 1963; Busse 1970). The boundary-layer method for rigid boundaries is used by assuming a large Rayleigh number R for different ranges of the diffusivity ratio τ and the solute Rayleigh number RS. The heat and solute fluxes F and FS are determined for the values of the wavenumbers αn which optimize F. The αN mode (where N denotes the total number of modes) is shown to have a solute-layer thickness of order $\tau^{\frac{1}{3}}\delta _{tN}$ (δtN denoting the temperature-layer thickness in the αN mode), and it is also proved that it is only for this mode that the solute concentration affects the boundary-layer structure. Solutions are possible if K = FSRS/FR < 1. For K ≥ 1, the stabilizing effect of solute is unimportant and there can be infinitely many modes. However, as K → 1 −, N, αn, F and FS decrease rapidly and the maximizing convection is suppressed entirely by the solute concentration. A simple interpretation of the model for the diffusive system leads also to the results for the salt-finger system.

Large-scale atmospheric flow shares certain attributes with two-dimensional turbulence. In this paper, we study the effect of spherical geometry on two-dimensional turbulence.

Energy transfer is multi-component in spherical geometry in contrast to energy transfer among triads of wave vectors in Cartesian geometry. It follows that energy transfer is more local in spherical than in Cartesian geometry. Enstrophy transfer to higher wavenumbers in spherical geometry is less than enstrophy transfer to higher wavenumbers in Cartesian geometry. Since both energy and enstrophy are inviscid constants of motion, the back transfer of energy is also less in spherical than in Cartesian geometry. Therefore, with a finite viscosity, enstrophy decays more slowly in spherical geometry than in Cartesian geometry. Here these conjectures are tested numerically by spectral methods. The numerical results agree well with the conjectures.

The response of a Hill's spherical vortex to an irrotational axisymmetric small perturbation is examined on the assumption that viscous diffusion of vorticity is negligible. The problem of determining the response is first reduced to a system of differential equations for the evolution of the Legendre coefficients of the disturbance stream function. This system is then solved approximately and it is shown that, if the initial disturbance is such as to make the vortex prolate spheroidal, the vortex detrains a fraction ⅗ε of its original volume, where ε is the fractional extension of the axis of symmetry in the imposed distortion. The detrained fluid forms a thin spike growing from the rear stagnation point. If the vortex is initially oblate, irrotational fluid is entrained at the rear stagnation point to the interior of the vortex.

The linear hydrodynamic stability problem for plane Poiseuille flow of a dilute suspension of rigid fibres is solved numerically. The constitutive equation given by Batchelor (1970a, b, 1971) is used to model the rheological properties of the suspension. The resulting eigenvalue problem is shown to be singular. The appropriate contour in the complex plane is determined by considering an initial-value problem. It is shown that, for a fixed, but not too large, inclination of the wave front to the mean flow, the fibres cause the critical Reynolds number to increase monotonically with the product of the volume fraction of the fibres and the square of their aspect ratio. The stabilizing influence of the fibres seems to vanish for large wave inclination angles.

An investigation based on the elongated-body theory of Lighthill (1960, 1970) and Wu (1971) is made of the carangiform swimming motions of fish, a mode of propulsion in which the amplitude of body undulation becomes significant only in the posterior half, or even third, of the length of the fish and the anterior part is relatively inflexible. For typical slender fish performing undulatory swimming motions, three hydro-dynamic features are taken into account: (a) the resistance to the lateral undulatory motions as well as longitudinal (tangential) frictional resistance, (b) the forces of interaction with the water associated with its inertial (virtual-mass) response to the lateral motions of the fish and (c) the reaction forces due to the vortex sheets shed from sharp trailing edges to the rear of the section of maximum span (depth), where a great variety of fins are generally found.

The active tail oscillations give rise to oscillatory side forces, to which the remainder of the body responds passively. These passive yawing motions are studied to find their amplitude, the yawing axis and any associated energy dissipation. The contribution of each of the above three forces is examined and the effects of the oscillation frequency, a slenderness parameter δ of the body, and the shape of the transverse cross-sections are considered.

The present theory is further applied to predict the turning movement when the fish changes direction, the way in which the above forces act during this process being investigated. A small perturbation analysis relative to a uniform rectilinear motion not only reveals whether or not the motion is dynamically stable, but also leads to a full description of the motion that follows a given initial perturbation. For finite perturbations a numerical method is adopted to find the time development of both the direction of motion relative to the direction of the initial uniform motion and the ratio of the kinetic energy of the body to the initial kinetic energy for different magnitudes I0 of the initial impulsive perturbation in the transverse direction. The final angle of turn is obtained in terms of I0 for different values of δ. The loss of kinetic energy during the turn divided by the initial kinetic energy is found to be very small for a small angle of turn.

Detailed analyses are made for rigid fish models in which the distribution of depth (or span) along the body length is quadratic with a maximum at the centre. Agreement of the present analyses with observations is fairly good at least qualitatively and some quantitative estimates are made of the side forces, both oscillatory and impulsive, exerted on the body by the tail. It is found that the vortex sheets shed from trailing edges increase the energy loss due to the yawing oscillations, in addition to the resistive dissipation, but contribute to the directional stability of rectilinear motion.

This paper deals with a two-dimensional turbulent boundary layer disturbed by a circular cylinder. The cylinder was placed inside or outside the boundary layer with its axis parallel to the wall and normal to the flow direction. The mean velocity, wall shear stress, longitudinal turbulent intensity, autocorrelations and turbulent length scale were measured and here the relaxation features of the disturbed boundary layer are discussed. The measurements were made for a ratio of the cylinder diameter d to the undisturbed boundary-layer thickness δ0 equal to 0·30 and for three values of the ratio of the height h of the cylinder axis to δ0 equal to 0·222, 0·556 and 1·24.

The results show that the near-wall region of the disturbed boundary layer recovers much more quickly than the outer region and that in the case h/δ0 = 0·222 the recovery is faster than in other cases, as reported by Clauser (1956). Moreover, it is found that the fluctuating velocity field recovers more slowly than the mean velocity field, and that the characteristics of the turbulence in the outer region are still close to those in the wake of an isolated cylinder at the last measurement station, although the mean velocity profile has almost completely returned to its natural shape.

In the paper we consider harbour oscillations excited by wind-generated gravity waves. The analysis is based on the fact that waves propagate along rays (wave orthogonals). In this way the elliptic boundary-value problem is turned into an initial-value problem along each ray. When a ray strikes the boundary (the harbour walls), reflected rays are produced in accordance with the law of reflexion. When a ray strikes an edge point of the boundary (e.g. the tip of a breakwater) diffracted rays are produced and emitted in all directions into the harbour. Algorithms for the tracing of incident, multiply reflected and singly diffracted rays as well as the computation of the field on each ray are presented. Attenuation mechanisms (e.g. partial reflexion), which limit the number of rays needed to compute the field, are included. Numerical examples for a rectangular and an actual harbour are given. A comparison between the results obtained by ray methods and finite difference methods is included.

Von Kármán's momentum-balance method is adopted to investigate the earthquake forces on a rigid dam with an inclined upstream face of constant slope. The distribution of the hydrodynamic pressure along the upstream face is determined. It is found that the maximum hydrodynamic pressure occurs at the base of the dam for any inclination angle between 0 and 90°. Explicit analytical formulae for evaluating the total horizontal, vertical and normal loads are presented and a useful approximate rule for practical engineers is given.

An inconsistency in a previous stability analysis of a cylindrical vortex sheet is resolved. It is found that disturbances with azimuthal wavenumbers m = 1 and 2 are stable, whereas previously the stability of these modes was uncertain. It has been hypothesized that the multiple-vortex phenomenon is initiated by such an instability (Ward 1972). Since double vortices (m = 2) arise, we are led to consider a central downdraft surrounded by a uniform updraft; for a non-zero vertical wavenumber γ, the modes m = 1 and 2 are destabilized. Our theory is supported by the observation that double vortices tend to form as intertwining spirals (i.e. they have vertical structure).

The Navier-Stokes equations for a two-layer flow are written in a curvilinear system of co-ordinates in which the height is measured from the interface. A technique for averaging the equations over an ensemble of wave surfaces which are not very different from each other is proposed. Moments which include deviations in the rate of surface displacement and in the slopes are dropped. It is assumed that the averaged equations describe the evolution of a large-scale velocity field. The moments of the turbulent velocity field are parameterized using the isotropic coefficient of turbulent viscosity, which, from dimensional considerations, is expressed in terms of the length scale growing linearly upwards and downwards from the interface and the turbulent kinetic energy. The equation defining the evolution of turbulent energy is derived without allowing for the curvilinearity of the system of co-ordinates. Laboratory experiments in a wind-water tunnel are simulated by integrating the equations numerically. The results are compared with measurements.

The equations for the earthquake forces on a rigid dam with an inclined upstream face of constant slope are solved exactly by two-dimensional potential-flow theory. The distribution of the hydrodynamic pressure along the upstream face and the total horizontal, vertical and normal loads on the dam are computed from the integral solutions. The results obtained from the exact theory are compared with those derived from the momentum-balance method and there is reasonable agreement.

A numerical method has been developed for the calculation of the pressure distribution, forces and moments on a two-dimensional aerofoil undergoing an arbitrary unsteady motion in an inviscid incompressible flow. In a discussion of the appropriate Kutta condition(s) it is argued that two Kutta conditions are required to obtain a satisfactory solution. The method is applied to (i) a sudden change in aerofoil incidence, (ii) an aerofoil oscillating at high frequency and (iii) an aerofoil passing through a sharp-edged gust.

The initial-value problem for a one-dimensional gravity wave of amplitude a and characteristic length l in water of depth d is examined for 0 < a/d [Lt ] d2/l2 [Lt ] 1. A preliminary reduction leads to a Korteweg-de Vries (KdV) equation in which the nonlinear term is O(ε) relative to the linear terms, where ε = 3al2/4d3 [Lt ] 1 is a measure of nonlinearity/dispersion. The linear approximation (ε ↓ 0) is found to be valid if and only if $\epsilon\tau^{\frac{1}{3}}\ll 1 $ where $\tau = \frac{1}{2}(d/l)^2(gd)^{\frac{1}{2}}({\rm time})/l $ is the slow time in the KdV equation. The asymptotic solution of the KdV equation is obtained with the aid of inverse-scattering theory and is found to comprise not only a decaying wave train that is qualitatively similar to that predicted by the linear approximation, but also a soliton of amplitude 3V2/4d3 = O(εa) if V > 0, where V is the cross-sectional area of the initial displacement, or of amplitude = O(ε3a) if V = 0 (there is no soliton if V < 0). This soliton is fully evolved, and dominates the solution, only for $\epsilon \tau^{\frac{1}{3}} \gg 1$ if V > 0 or $\epsilon^2\tau^{\frac{1}{3}}\gg 1 $ if V = 0, but nonlinearity already has significant effects for $\epsilon\tau^{\frac{1}{3}} = O(1)$.

Free-stream turbulence is modelled by a low-intensity array of vortices where the vorticity is distributed continuously throughout the flow. This vorticity approaches and convects along a semi-infinite flat plate and the structure of the free-stream disturbances is altered by the impermeability condition at the plate. The analysis consists of tracking the vorticity as it convects with the uniform mean flow, determining the stream function induced by that vorticity field without imposing the impermeability condition, and finally superimposing an irrotational flow field which effects impermeability. The bisecting of a vortex as it encounters the plate yields a pair of vortices which rotate in the same direction. The combined heights of these vortices are less than the height of the original vortex. Small segments of a vortex which has been cut by the plate, but not through its centre, are completely absorbed by neighbouring vortices. Far from the leading edge in any direction, the disturbance pressure is O(ρq2), where q is the characteristic disturbance speed, and the streamline patterns convect with the mean flow. Near the leading edge, the fluctuating pressure is O(ρqU∞) because of unsteady vortex distortion and the velocity correlations reveal that Taylor's hypothesis is not valid. These correlations are based on averages over time and over all possible orientations of the vortex array. Vortex structures based on iso-vorticity contours are sometimes quite different from the structures based on disturbance streamlines.