Using the Fourier components of the velocity potential at the surface in an iterative perturbation theory, the author derives expressions for the modal frequency shifts and modal decay rates in a spectrum of in viscid gravity–capillary surface waves. The expressions differ from those calculated by the perturbation methods of Hasselmann (1962) and Valenzuela & Laing (1972), which are based on expanding the velocity potential about the equilibrium fluid level z = 0. This difference is shown to be due to the failure of the z = 0 perturbation method to converge rapidly enough to produce lowest-order corrections that are smaller than unperturbed quantities. A physical explanation for this failure is given.

The thermal structure in the boundary layer and its relation to the heat flux from the cooling and evaporating surface of a deep tank of water are investigated. When a deep layer of water in contact with still air above loses heat to the air, the cooled water in a region just under the surface converges along lines and then plunges down in sheets. These sheets of falling water dissipate as they move into the body of the water, which is in turbulent motion. The vertical profiles of the horizontally averaged temperature and its standard deviation agree fairly closely with theoretical profiles based on time averages of the solution to the heat diffusion equation. The differences between observed and thus predicted profile shapes are consistent with the expected effects of the falling cold thermals and the warm return flow, which are neglected in the theories. The profiles of the standard deviation have large values up to the interface and lie between predictions based on boundary conditions of constant surface temperature and constant heat flux, in keeping with the experimental conditions.

The relation between the net heat flux and the temperature difference across the boundary layer is given in non-dimensional form by N = 0[sdot ]156R0[sdot ]33, which is in good agreement with the asymptotic similarity prediction N [vprop ] R1/3 but lower than theoretical calculations of the upper bound of N vs. R.

The effect of Brownian motion of particles in a statistically homogeneous suspension is to tend to make uniform the joint probability density functions for the relative positions of particles, in opposition to the tendency of a deforming motion of the suspension to make some particle configurations more common. This smoothing process of Brownian motion can be represented by the action of coupled or interactive steady ‘thermodynamic’ forces on the particles, which have two effects relevant to the bulk stress in the suspension. Firstly, the system of thermodynamic forces on particles makes a direct contribution to the bulk stress; and, secondly, thermodynamic forces change the statistical properties of the relative positions of particles and so affect the bulk stress indirectly. These two effects are analysed for a suspension of rigid spherical particles. In the case of a dilute suspension both the direct and indirect contributions to the bulk stress due to Brownian motion are of order ø2, where ø([Lt ] 1) is the volume fraction of the particles, and an explicit expression for this leading approximation is constructed in terms of hydrodynamic interactions between pairs of particles. The differential equation representing the effects of the bulk deforming motion and the Brownian motion on the probability density of the separation vector of particle pairs in a dilute suspension is also investigated, and is solved numerically for the case of relatively strong Brownian motion. The suspension has approximately isotropic structure in this case, regardless of the nature of the bulk flow, and the effective viscosity representing the stress system to order ϕ2 is found to be \[ \mu^{*} = \mu(1+2.5\phi + 6.2\phi^2). \] The value of the coefficient of ø2 for steady pure straining motion in the case of weak Brownian motion is known to be 7[sdot ]6, which indicates a small degree of ‘strain thickening’ in the ø2-term.

Calculated values of the three velocity components and measured values of the longitudinal component are reported for the flow of water in a 90° bend of 40 x 40mm cross-section; the bend had a mean radius of 92mm and was located downstream of a 1[sdot ]8m and upstream of a 1[sdot ]2m straight section. The experiments were carried out at a Reynolds number, based on the hydraulic diameter and bulk velocity, of 790 (corresponding to a Dean number of 368). Flow visualization was used to identify qualitatively the characteristics of the flow and laser-Doppler anemometry to quantify the velocity field. The results confirm and quantify that the location of maximum velocity moves from the centre of the duct towards the outer wall and, in the 90° plane, is located around 85% of the duct width from the inner wall. Secondary velocities up to 65% of the bulk longitudinal velocity were calculated and small regions of recirculation, close to the outer corners of the duct and in the upstream region, were also observed.

The calculated results were obtained by solving the Navier–Stokes equations in cylindrical co-ordinates. They are shown to exhibit the same trends as the experiments and to be in reasonable quantitative agreement even though the number of node points used to discretize the flow for the finite-difference solution of the differential equations was limited by available computer time and storage. The region of recirculation observed experimentally is confirmed by the calculations. The magnitude of the various terms in the equations is examined to determine the extent to which the details of the flow can be represented by reduced forms of the Navier–Stokes equations. The implications of the use of so-called ‘partially parabolic’ equations and of potential- and rotational-flow analysis of an ideal fluid are quantified.

A solution to the dispersion of small particles suspended in a turbulent fluid is presented, based on the approximation proposed by Phythian for the dispersion of fluid points in an incompressible random fluid. Motion is considered in a frame moving with the mean velocity of the fluid, the forces acting on the particle being taken as gravity and a fluid drag assumed linear in the particle velocity relative to that of the fluid. The probability distribution of the fluid velocity field in this frame is taken as Gaussian, homogeneous, isotropic, stationary and of zero mean. It is shown that, in the absence of gravity, the long-time particle diffusion coefficient is in general greater than that of the fluid, approaching with increasing particle relaxation time a value consistent with the particle being in an Eulerian frame of reference. The effect of gravity is consistent with Yudine's effect of crossing trajectories, reducing unequally the particle diffusion in directions normal to and parallel to the direction of the gravitational field. To characterize the effect of flow and gravity on particle diffusion it has been found useful to use a Froude number defined in terms of the turbulent intensity rather than the mean velocity. Depending upon the value of this number, it is found that the particle integral time scale may initially decrease with increasing particle relaxation time though it eventually rises and approaches the particle relaxation time. It is finally shown how this analysis may be extended to include the extra forces generated by the fluid and particle accelerations.

In this paper we examine how the unsteady flow field radiated from an oscillating body is altered from the result of acoustic theory as the direct consequence of disturbances propagating through the non-uniform flow produced by the presence of the body. Taking the specific example of an oscillating airfoil placed in supersonic flow and having the contour of a parabolic arc, we derive a closed-form representation for the unsteady flow field in terms of the confluent hypergeometric function. The analytical expression reveals explicitly that, though the body shape has a negligible effect in the near field, it inextricably affects the unsteady flow at a large distance, both in its amplitude and phase, and substantially modifies the results of acoustic theory. In addition, we display the relation of this solution to the ‘fundamental solution’ and the other salient physical features connected with disturbances propagating through non-uniform flow. The present results recover Whitham's rule in the limit of zero frequency of oscillation and also include, as another special case, the unsteady solution for a wedge obtained by Carrier and Van Dyke.

By the use of high-speed photography, instabilities occurring in high Reynolds number water jets discharging into air have been made visible. These instabilities include the axisymmetric mode accompanying the transition from laminar to turbulent flow at the nozzle exit, spray formation as a culmination of the axisymmetric disturbances, and, further downstream, helical disturbances which result in the entire jet assuming a helical form. The final disruption of the jet is due to amplification of the helical waves. It is further shown that the amplification of the helical disturbances is due in part to aerodynamic form drag, since jets discharging into surrounding air moving at the same speed as the jet remain relatively stable, compared with the case when the jet is discharged into stagnant air.

Using a normal and tangential co-ordinate approach, a perturbation theory is developed for wind-forced linear and nonlinear Kelvin waves propagating along an irregular coastline. The theory is valid for coastline curvatures which are non-dimensionally small, the curvature being non-dimensionalized with respect to the reciprocal of the boundary-layer trapping scale, i.e. the reciprocal of the radius of deformation. According to linear theory, the main effect of a coastline of small curvature is to cause a phase-speed change in the wave (from – c to – c(1 – ½k(s)), where k(s) is the non-dimensional curvature a distance s along the coast from the origin) and to make the offshore Ekman transport change more rapidly along the coast, the latter effect implying a more ‘wavelike’ ocean or lake response. Two discernible nonlinear effects were found to be an increase (decrease) in the linear-solution longshore gradients in regions of positive (negative) isopycnal displacement and a tendency for increased (decreased) isopyncal displacement at capes (bays).

High-frequency fluctuations in temperature and velocity were measured at a height of 2 m above a harvested, nearly level field of rye grass. Conditions were both stably and unstably stratified. Reynolds numbers ranged from 370000 to 740000. Measurements of velocity were made with a hot-wire anemometer and measurements of temperature with a platinum resistance element which had a diameter of 0[sdot ]5 μm and a length of 1 mm. Thirteen runs ranging in length from 78 to 238 s were analysed.

Spectra of velocity fluctuations are consistent with previously reported universal forms. Spectra of temperature, however, exhibit an increase in slope with increasing wavenumber as the maximum in the one-dimensional dissipation spectrum is approached. The peak of the one-dimensional dissipation spectrum for temperature fluctuations occurs at a higher wavenumber than that of simultaneous spectra of the dissipation of velocity fluctuations. It is suggested that the change in slope of the temperature spectra and the dissimilarity between temperature and velocity spectra may be due to spatial dissimilarity in the dissipation of temperature and velocity fluctuations. The temperature spectra are compared with a theoretical prediction for fluids with large Prandtl number, due to Batchelor (1959). Even though air has a Prandtl number of 0[sdot ]7, the observations are in qualitative agreement with predictions of the theory. The non-dimensional wavenumber at which the increase in slope occurs is about 0[sdot ]02, in good agreement with observations in the ocean reported by Grant et al. (1968).

For the two runs for which the stratification was stable, the normalized spectra of the temperature derivative fall on average slightly below the mean of the spectra of the remaining runs in the range in which the slope is approximately one-third. Hence the Reynolds number may not have always been sufficiently high to satisfy completely the conditions for an inertial subrange.

Universal inertial-subrange constants were directly evaluated from one-dimensional dissipation spectra and found to be 0[sdot ]54 and 1[sdot ]00 for velocity and temperature, respectively. The constant for velocity is consistent with previously reported values, while the value for temperature differs from some of the previous direct estimates but is only 20% greater than the mean of the indirect estimates. This discrepancy may be explained by the neglect in the indirect estimates of the divergence terms in the conservation equation for the variance of temperature fluctuations. There is weak evidence that the one-dimensional constant, and hence the temperature spectra, may depend upon the turbulence Reynolds number, which varied from 1200 to 4300 in the observations reported.

Systematic renormalized perturbation expansions for turbulence and turbulent convection are constructed which are invariant at each order under random Galilean transformations. Two types of expansion are developed whose lowest truncations give, respectively, the Lagrangian-history direct-interaction approximation and the abridged Lagrangian-history direct-interaction approximation. These approximations previously were derived as heuristic modifications of the Eulerian direct-interaction approximation (Kraichnan 1965). The techniques used involve reversion of primitive perturbation expansions for the generalized velocity field u(x, t/s), defined as the velocity measured at time s in the fluid element which passes through x at time t. The new expansions are illustrated by application to a random linear oscillator, to passive-scalar convection by a random velocity and to the Lagrangian velocity covariance. The lowest term of the expansion for the passive scalar gives Taylor's (1921) exact result for dispersion of fluid elements, and higher terms describe the deviations of the particle-displacement distribution from Gaussian form. In all the applications the assumed underlying statistics are more general than Gaussian statistics, which appear as a special case.

In this paper we formulate the aerodynamic sound problem for a relaxing medium in a rather general way, independent of the details of the relaxation process. The medium is characterized by an appropriate relaxation time τ0 and by a frozen (af0) and an equilibrium (ae0) sound speed. The equation describing aerodynamic sound in such a medium is the familiar one describing acoustic waves in a non-equilibrium medium but subjected to aerodynamic sound sources expressed in terms of a frozen and an equilibrium form of the Lighthill stress tensor. The far-field result for both compact and non-compact sources in the frequency range ω [Gt ] τ0−1 can be expressed as the ratio of far-field densities for the relaxing and non-relaxing propagation medium: \[ \frac{\rho}{\rho_L} = \exp\left[-\left(1-\frac{a^2_{e_0}}{a^2_{f_0}}\right)\frac{x}{2a_{f0}\tau_0}\right], \] where x is the observation distance and the subscript L stands for ‘Lighthill’. The result for the main radiated aerodynamic sound, which comes from sources in the range ω [Lt ] τ0−1, essentially propagates in a manner described by the lower-order equilibrium waves, the diffusive effects from the higher-order waves being small, and the result for compact sources is a restatement of Lighthill's result in terms of the equilibrium propagation speed with the source region identically in equilibrium. For non-compact sources the propagation is still given by ae0 but the source region is now understood to encompass relaxation effects, the details of which are left unspecified.

An exact Eulerian formulation of the problem of diffusion of passive scalar and vector fields by a turbulent velocity field is obtained. It is shown that, in the short autocorrelation time limit, the diffusion equation is exact for any turbulence. For non-zero autocorrelation times the form of the first few correction terms to the diffusion equation is found. As a result of these corrections the diffusion of scalar, divergence-free and curl-free vector fields will be different. The calculations use the Kubo–Van Kampen–Terwiel technique and are carried out for zero ordinary diffusivity and for homogeneous, stationary, isotropic, incompressible, helical turbulence.

Linearized theory is used to study the unsteady flow in a supersonic cascade with in-passage shock waves. We use the Wiener–Hopf technique to obtain a closed-form analytical solution for the supersonic region. To obtain a solution for the rotational flow in the subsonic region we must solve an infinite set of linear algebraic equations. The analysis shows that it is possible to correlate quantitatively the oscillatory shock motion with the Kutta condition at the trailing edges of the blades. This feature allows us to account for the effect of shock motion on the stability of the cascade.

Unlike the theory for a completely supersonic flow, the present study predicts the occurrence of supersonic bending flutter. It therefore provides a possible explanation for the bending flutter that has recently been detected in aircraft-engine compressors at higher blade loadings.

A train of surface gravity waves of wavelength λ in a channel of depth H is incident on a horizontal plate of length l that is submerged to a depth c. Under the assumption that both λ and l are large compared with H, the method of matched asymptotic expansions is used to show that, to first order, the reflexion coefficient R and the transmission coefficient T are given by \[ R = \chi \left\{\frac{\sigma l}{(gH)^{\frac{1}{2}}}\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}-2\bigg(\frac{c}{H}\bigg)^{\frac{1}{2}}\bigg(1-\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)\right\} \] and \[ T =\chi\left\{2i\left[\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}+\frac{\sigma l}{b}\bigg(\frac{c}{g}\bigg)^{\frac{1}{2}}\right]\right\} \] where \begin{eqnarray*} \chi &=& 1\left/ \left\{2\bigg(\frac{c}{H}\bigg)^{\frac{1}{2}}\bigg(1-\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)+\frac{\sigma l}{b}\bigg(\frac{H}{g}\bigg)^{\frac{1}{2}}\bigg(1+\frac{c}{H}\bigg)\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}\right.\right.\\ &&\left. +2i\bigg(\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}+\frac{\sigma l}{b}\bigg(\frac{c}{g}\bigg)^{\frac{1}{2}}\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)\right\}, \end{eqnarray*} σ is the angular frequency and g the acceleration due to gravity.

Measurements of the overall heat flux in steady convection have been made in a horizontal layer of dilute aqueous electrolyte. The layer is bounded below by a rigid zero-heat-flux surface and above by a rigid isothermal surface. Joule heating by an alternating current passing horizontally through the layer provides a uniformly distributed volumetric energy source. The Nusselt number at the upper surface is found to be proportional to Ra0[sdot ]227 in the range 1[sdot ]4 ≤ Ra/Rac ≤ 1[sdot ]6 × 109, which covers the laminar, transitional and turbulent flow regimes. Eight discrete transitions in the heat flux are found in this Rayleigh number range. Extrapolation of the heat-transfer correlation to the conduction value of the Nusselt number yields a critical Rayleigh number which is within -6[sdot ]7% of the value given by linearized stability theory. Measurements have been made of the time scales of developing convection after a sudden start of volumetric heating and of decaying convection when volumetric heating is suddenly stopped. In both cases, the steady-state temperature difference across the layer appears to be the controlling physical parameter, with both processes exhibiting the same time scale for a given steady-state temperature difference, or [mid ]ΔRa[mid ]. For step changes in Ra such that [mid ]ΔRa[mid ] > 100Rac, the time scales for both processes can be represented by Fo [vprop ] [mid ]ΔRa[mid ]m, where Fo is the Fourier number of the layer. Temperature profiles of developing convection exhibit a temperature excess in the upper 15–20 % of the layer in the early stages of flow development for Rayleigh numbers corresponding to turbulent convection. This excess disappears when the average core temperature becomes large enough to permit eddy transport and mixing processes near the upper surface. The steady-state limits in the transient experiments yield heat-transfer data in agreement with the results of the steady-state experiments.

A numerical proceure is introduced to solve the one-dimensional equations of gasdynamics for a cylindrically or spherically symmetric flow. The method consists of a judicious combination of Glimm's method and operator splitting. The method is applied to the problem of a converging cylindrical shock.

The boundary-value problem for irrotational surface waves is derived from a variational integral I with the Lagrangian density [Lscr ] = Ξ ηt - [Hscr ] where Ξ (X, t) is the value of the velocity potential at the free surface, y = η(x, t), and [Hscr ] is the energy density in x space. [Hscr ] then is expressed as a functional of Ξ and η, qua canonical variables, by solving a reduced boundary-value problem for the potential, after which the requirement that I be stationary with respect to independent variations of Ξ and η yields a pair of evolution equations for Ξ and η. The Fourier expansions Ξ = pn(t)ϕn*(x) and η = qn(t)ϕn(x), where {ϕn} is an orthogonal set of basis functions, reduce I to Hamilton's action integral, in which the complex amplitudes pn and qn appear as canonically conjugate co-ordinates, and yield canonical equations for pn and qn that are the spectral transforms of the evolution equations for Ξ and η. The evolution equations are reduced (asymptotically) to partial differential equations for pn and qn by expanding [Hscr ] in powers of α = a/d and β = (d/l)2, where a and l are scales of amplitude and wavelength. Explicit third approximations are developed for β = O(α).