The motion behind the head of a gravity current advancing over a no-slip horizontal surface is a complex three-dimensional flow. There is intense mixing between the current and its surroundings and the foremost part of the head is raised above the surface. Experimental results are obtained from (i) an apparatus in which the head is brought to rest by using an opposing flow and a moving floor and (ii) a modified lock exchange flow. The dimensionless velocity of advance, rate of mixing between the two fluids and the depth of the mixed layer left behind the head and above the following gravity current are determined for an extended range of the dimensionless gravity current depth. The mixing between the two fluids is the result of gravitational and shear instabilities at the gravity current head. A semi-empirical analysis is presented to describe the results. The influence of Reynolds number is discussed and comparison with a documented atmospheric flow is presented.

In 1965 Sullivan made many measurements of concentration in dye plumes in the surface layer of Lake Huron with the primary purpose of estimating the distance–neighbour function (Sullivan 1965, 1971). This paper presents the results of a recent analysis of the concentration fluctuations in these experiments for, despite their great practical and theoretical importance, there are very few published reports of such measurements from natural environments. One reason for this apparent neglect has undoubtedly been the anticipated high noise level, and the present results confirm this expectation. The experimental analysis uses the framework of relative diffusion since this has great advantages compared with that of absolute diffusion. Despite the noise, the results are consistent, to the degree of spatial resolution attained, with the self-similar structure anticipated for relative, but not absolute, diffusion. Further interesting features of the results are that changes in the form of the statistical properties across the plume indicate an unexpectedly strong influence of the central regions, and that certain statistical properties have much less noisy profiles than that of the mean square fluctuations. The influence of molecular diffusion is shown to be strong. Interpretation of the results is based partly on the extension of the theory recently developed by Chatwin & Sullivan (1979a) for a cloud, although the limited spatial resolution attained did not allow direct critical examination of this work.

This paper discusses the influence of turbulence convection on the formation of acoustic momentum and thermal boundary layers over a rigid surface in the presence of a low Mach number wall-turbulence shear flow. Equations which determine the modified boundary-layer profiles are obtained from a consideration of the relaxation of coherent perturbations in the Reynolds stress. These equations can be solved analytically for a wide range of conditions which are investigated in detail. The theory is applied to the problem of sound propagation in fully-developed turbulent pipe flow, and at low Mach numbers good agreement is obtained between predicted acoustic attenuation rates and experimental results available in the literature.

The swimming of a micro-organism by the propagation of helical waves on a long slender flagellum is analysed. The model developed by Higdon (1979) is used to study the motion of an organism with a spherical cell body (radius A) propelled by a cylindrical flagellum (radius a, length L).

The average swimming speed and power consumption are calculated for helical waves (amplitude α, wavenumber k). A wide range of parameter values is considered to determine the optimal swimming motion. The optimal helical wave has ak ≈ 1, corresponding to a pitch angle of 45°. The optimum number of waves on the flagellum increases as the flagellar length L/A increases, such that the optimum wavelength decreases as L/A increases. The efficiency is relatively insensitive to the flagellar radius a/A. The optimum flagellar length is L/A ≈ 10.

The results are compared to calculations using two different forms of resistance coefficients. Gray-Hancock coefficients overestimate the swimming speed by approximately 20% and underestimate the power consumption by 50%. The coefficients suggested by Lighthill (1976) overestimate the swimming speed for large cell bodies (L/A < 15) by 20% and underestimate for small cell bodies (L/A > 15) by 10%. The Lighthill coefficients underestimate the power consumption up to 50% for L/A < 10, and overestimate up to 25% for L/A > 10. Overall, the Lighthill coefficients are superior to the Gray-Hancock coefficients in modelling swimming by helical waves.

Recent measurements of the skewness of the derivative of the temperature fluctuation θ, implying the breakdown of local isotropy even in high Reynolds number shear flows, are examined. Using the temperature signal in a slightly heated axisymmetric jet, a detailed quantitative analysis is made of the suggestion that the observed presence of a well-defined large-scale pattern of the temperature signal in these flows is responsible for this breakdown. A selective ensemble averaging technique is used for separating this pattern from fluctuations superposed on it. The technique is extended to extract the large-scale patterns in simultaneously measured axial (u), radial (v) velocity fluctuations, and the products uv, uθ and vθ, so that it is possible to separate contributions of these patterns from those of the superposed fluctuations to several important turbulent quantities. The mean shape of the patterns, their degree of anisotropy and correlation, and their contribution to turbulence intensities and Reynolds shear stress are obtained. Probability densities and spectra of these quasi-homogeneous superposed fluctuations are also obtained. Results show that the fluctuations are consistent with local isotropy and make the dominant contribution to the turbulence intensities, that the large-scale patterns are responsible for the observed skewness values of the derivative of v, and that the fluctuations may be responsible for a significant part of the turbulent momentum and heat transport, especially in the region of the jet where the turbulent energy production is substantial.

The method of characteristics is applied to the nonlinear equations describing two-phase flow in a fluidized bed. The method shows how a small disturbance changes with time and distance and can, eventually, produce a flow discontinuity similar to a shock wave in gases. The parameters entering the analysis are the amplitude of the initial disturbance, the wavelength of the original disturbance, the particulate pressure function, the particulate size, the uniform fluidization voidage, the uniform fluidization velocity, the fluid viscosity, the particulate density, and the fluid density. A parametric study shows that the following factors delay shock formation: a decrease in particulate size, an increase in bed density, an increase in fluid viscosity, and a decrease in particulate density. Experimental data on bubble formation in gas-particulate fluidized beds show that these same factors delay bubble formation. It is concluded, therefore, that the shock front and the bubble front are one and the same thing.

To gain insight into the orbital motion in waves on the point of breaking, we first study the trajectories of particles in some ideal irrotational flows, including Stokes’ 120° corner-flow, the motion in an almost-highest wave, in periodic deep-water waves of maximum height, and in steep, solitary waves.

In Stokes’ corner-flow the particles move as though under the action of a constant force directed away from the crest. The orbits are expressible in terms of an elliptic integral. The trajectory has a loop or not according as q [sqcup ] c where q is the particle speed at the summit of each trajectory, in a reference frame moving with speed c. When q = c, the trajectory has a cusp. For particles near the free surface there is a sharp vertical gradient of the horizontal displacement.

The trajectories of particles in almost-highest waves are generally similar to those in the Stokes corner-flow, except that the sharp drift gradient at the free surface is now absent.

In deep-water irrotational waves of maximum steepness, it is shown that the surface particles advance at a mean speed U equal to 0·274c, where c is the phase-speed. In solitary waves of maximum amplitude, a particle at the surface advances a total distance 4·23 times the depth h during the passage of each wave. The initial angle α which the trajectory makes with the horizontal is close to 60°.

The orbits of subsurface particles are calculated using the ‘hexagon’ approximation for deep-water waves. Near the free surface the drift has the appearance of a thin forwards jet, arising mainly from the flow near the wave crest. The vertical gradient is so sharp, however, that at a mean depth of only 0.01L below the surface (where L is the wavelength) the forwards drift is reduced to less than half its surface value. Under the action of viscosity and turbulence, this sharp gradient will be modified. Nevertheless the orbital motion may contribute appreciably to the observed ‘winddrift current’.

Implications for the drift motions of buoys and other floating bodies are also discussed.

We have used the technique of laser-Doppler velocimetry to study the transition to turbulence in a fluid contained between concentric cylinders with the inner cylinder rotating. The experiment was designed to test recent proposals for the number and types of dynamical regimes exhibited by a flow before it becomes turbulent. For different Reynolds numbers the radial component of the local velocity was recorded as a function of time in a computer, and the records were then Fourier-transformed to obtain velocity power spectra. The first two instabilities in the flow, to time-independent Taylor vortex flow and then to time-dependent wavy vortex flow, are well known, but the present experiment provides the first quantitative information on the subsequent regimes that precede turbulent flow. Beyond the onset of wavy vortex flow the velocity spectra contain a single sharp frequency component and its harmonics; the flow is strictly periodic. As the Reynolds number is increased, a previously unobserved second sharp frequency component appears at R/Rc = 10·1, where Rc is the critical Reynolds number for the Taylor instability. The two frequencies appear to be irrationally related; hence this is a quasi-periodic flow. A chaotic element appears in the flow at R/Rc ≃ 12, where a weak broadband component is observed in addition to the sharp components; this flow can be described as weakly turbulent. As R is increased further, the component that appeared at R/Rc= 10·1 disappears at R/Rc = 19·3, and the remaining sharp component disappears at R/Rc = 21·9, leaving a spectrum with only the broad component and a background continuum. The observance of only two discrete frequencies and then chaotic flow is contrary to Landau's picture of an infinite sequence of instabilities, each adding a new frequency to the motion. However, recent studies of nonlinear models with a few degrees of freedom show a behaviour similar in most respects to that observed.

Weis-Fogh discovered a remarkable new principle of aerodynamic lift. Hovering wasps exploit the principle and fly with an aerodynamic performance superior in some respects to anything previously known. In this paper we address the question of whether the Weis-Fogh effect can be exploited in turbomachinery. We think the answer is yes.

Normal turbomachinery design is based on the analysis of isolated cascades of blades with steady entry and exit flows. The interactions between adjacent cascades and nonuniformities of the flow are usually regarded as problems which have to be minimized. Unsteadiness gives rise to noise. In this paper we take the opposite view and examine a novel type of turbomachinery stage that depends on the interaction between rotor and stator for its normal operation. The stage exploits the Weis-Fogh principle and has the unusual property that when started from rest it generates a pressure rise without shedding any vorticity into the fluid. We argue that there may be a performance advantage for stages of this new type.

Experiments were done to check the validity of the theoretical model and these are described. The results seem to show that under certain circumstances a strong rotor-stator interaction can result in an improved stage performance, and we suggest that this improvement may be due to the Weis-Fogh effect.

The effect of a steady deterministic drift on the dispersion of particles suspended in a stationary homogeneous turbulent field is examined using previously reported closed equations of the mean-squared particle displacement. The existence of negative particle velocity correlations is demonstrated. The dependence of the particle diffusivity and kinetic energy on the inertia and drift factors is evaluated for a model turbulence spectrum. Asymptotic formulae are obtained for the case of dominant deterministic motion. The theory is applied to analysis of the experimental data of Snyder & Lumley (1971), and good and consistent agreement with these asymptotic formulae was obtained after accounting for the quasi-stationarity of particle velocity correlation functions in decaying turbulence.

A new integro-differential equation is derived for steady free-surface waves. Numerical solutions of this equation for periodic gravity-capillary waves on a fluid of infinite depth are presented. For the two limiting cases of gravity waves and capillary waves, our results are in excellent agreement with previous calculations. For gravity-capillary waves, detailed calculations are performed near the wave-number at which the classical second-order perturbation solution breaks down. Our calculations yield two solutions in this region, which in the limit of small amplitudes agree with the results obtained by Wilton in 1915; one solution has the small amplitude behaviour of a gravity wave and the other that of a capillary wave, but the numerical results show that at large amplitudes both waves have the characteristics of capillary waves. The calculations also show that the wavenumber range in which two solutions exist increases with increasing wave height.

A method is outlined by which high-order solutions are obtained for steadily progressing shallow water waves. It is shown that a suitable expansion parameter for these cnoidal wave solutions is the dimensionless wave height divided by the parameter m of the cn functions: this explicitly shows the limitation of the theory to waves in relatively shallow water. The corresponding deep water limitation for Stokes waves is analysed and a modified expansion parameter suggested.

Cnoidal wave solutions to fifth order are given so that a steady wave problem with known water depth, wave height and wave period or length may be solved to give expressions for the wave profile and fluid velocities, as well as integral quantities such as wave power and radiation stress. These series solutions seem to exhibit asymptotic behaviour such that there is no gain in including terms beyond fifth order. Results from the present theory are compared with exact numerical results and with experiment. It is concluded that the fifth-order cnoidal theory should be used in preference to fifth-order Stokes wave theory for wavelengths greater than eight times the water depth, when it gives quite accurate results.

Our 1975 paper reported the results of experiments on shock reflexion in a wind tunnel and a shock tube; further results are presented here. For strong shocks it is shown that transition to Mach reflexion takes place continuously at the shock wave incidence angle ω0 corresponding to the normal shock point ω0 = ωN, unless the downstream boundaries form a throat. In this event transition can be promoted anywhere within the range ω0 [les ] ωN, and it is even possible to suppress regular reflexion altogether! However when ω0 < ωN the transition is discontinuous and accompanied by hysteresis. Again for strong shocks evidence is presented which suggests that the famous persistence of regular reflexion beyond the ωN point ω0 > ωN is spurious. For weak shocks the transition condition is not known but it is found that even for regular reflexion a marked discrepancy between theory and experiment develops as the shocks become progressively weaker. Also when weak shocks diffract over single concave corners there is a somewhat surprising discontinuity in the regular reflexion range. It seems that none of these phenomena can be adequately explained by real gas effects such as viscosity and variation of specific heats.

A finite-difference procedure, is employed to predict the turbulent flow and heat transfer in horizontal, inclined and vertical pipes when influenced by buoyancy. The flow is treated as parabolic; and the turbulence model used involves the solution of two differential equations, one for the kinetic energy of turbulence and the other for its dissipation rate. Results are presented for the velocity and temperature fields, and the associated flow-resistance and heat-transfer coefficients. The predictions for horizontal and vertical pipes have been compared with the available experimental results, and the agreement obtained is good.

This paper applies the test field model developed by Kraichnan to the study of an isotropic, passive scalar contaminant convected by decaying isotropic turbulence. Test field model predictions of scalar and velocity dissipation spectra at large Reynolds and Péclet numbers are shown to be in excellent agreement with atmospheric data, after intrinsic scale constants in the model are adjusted to give valid inertial range coefficients. Theoretical values for the inertial range coefficients are obtained for large and small Prandtl numbers. Simulation results for velocity and scalar energy, dissipation and transfer spectra and second- and third-order velocity, scalar and velocity–scalar correlations at moderate Reynolds and Péclet numbers are shown to agree moderately well with heated grid turbulence data. Simulation results are presented for the normalized decay rates of the scalar and velocity dissipation rates and for the ratio of the velocity to scalar decay time scales; these quantities are employed in second-order modelling. In the self-similar decay mode the simulations yield unity levels of the normalized decay rates and of the ratio of decay time scales over the moderate range of Reynolds and Prandtl numbers investigated. These results are compared with data from heated grid turbulence experiments and are discussed in the light of asymptotic decay of concomitant scalar and velocity fields.

The development in space and time of a plane initial disturbance to a spatially uniform exploding atmosphere is analysed on the assumption that the disturbance amplitude is comparable in magnitude with the inverse (dimensionless) activation energy of the explosion reaction. Particular attention is focused on the shock-fitting problem, which has features that distinguish it from its inert-atmosphere counterpart.

Using the positive half of a sine wave to typify an isolated compression perturbation, it is found that the amplifying effect of the ambient reaction leads to very rapid shock wave development, which depends significantly on the spatial extent of the disturbance. The latter also influences the question of whether local explosion (local explosion is recognized here as a logarithmically unbounded growth of the disturbance amplitude; in other words as a local breakdown of the present approximations) occurs at the shock wave or some distance behind it. The subsequent evolution of these two states will no doubt be significantly different, but the answer to this speculation must await extension of the present theory to encompass the rapid events that ensue near the local explosion regions.

Results from a series of pipe-flow experiments using a range of water-soluble drag-reducing polymers are presented. Degradation has been investigated by means of multiple passes of the solutions through a pipe. A theory predicting drag reduction in pipe flow has been devised which agrees with the experimental results. Changes in polymer molecular weight due to degradation are taken into account. The analysis is then applied to a turbulent boundary layer with polymer injection.