A mathematical model is proposed for steady two-dimensional flow around a submerged screen. The general problem analysed is the flow in a parallel-sided channel partially spanned by a screen, and the fluid is considered to be inviscid except at the screen, where the flow has the required pressure drop. The model is constructed by first replacing the screen with a distribution of sources and then manipulating the stream function for this flow so that the mass and momentum balances across the screen are satisfied. Consequently the model predicts a flow field which is realistic except for the expected discontinuity in velocity between the wake and external flow. In general, the governing equations must be solved numerically, but for the important case of a plane screen oriented normal or roughly normal to the approaching flow, an approximate analytical solution is possible. The accuracy of the model was ascertained by conducting wind-tunnel tests on screens of various solidities and orientations, and comparing the measured downstream velocity distributions with those predicted by the numerical and analytical solutions of the model. Overall, the theoretical results agree well with the experimental data, showing that the model is valid for screens of low and high solidity, in fact, for pressure drop coefficients up to 10. Comparisons with the work of others show that the proposed model is also accurate for the special cases of a screen submerged in an infinite flow field and of a screen spanning the full width of the channel.

A laser-Doppler velocimeter suitable for the measurement of mean and fluctuating flow velocities in water is described. Results of a study using this system in an axisymmetric turbulent jet of water and dilute polymer solutions are given. The laser-Doppler technique is better suited for such measurements than either Pitot tubes or heat-transfer gauges because the Doppler velocity measurements are independent of the physical properties of the fluid. Previous velocity measurements in polymer jets have suffered from the effects of the additives upon the sensors.

Turbulent round jets with Reynolds numbers between 5000 and 50000 were studied. For a jet issuing from a convergent nozzle the additives were found to have no effect upon the mean axial velocity or turbulence intensity at any point in the jet. However, for a jet issuing from a long length of circular pipe, the additives reduced the centre-line velocity and increased the turbulence level in the early part of the jet. Thus the principal effect of a polymer additive upon the jet appears to result from its effect upon the initial conditions.

In a horizontal convecting layer of fluid, several distinct transitions occur at certain distinct Rayleigh numbers R, for a given Prandtl number Pr. The regime diagram has been extended to include the Prandtl-number range

2·5 × 10−2 [les ] Pr [les ] 0·85 × 104.

In particular it is found that distinct changes in the slope of the heat-flux curve occur even for Pr = 2·5 × 10−2. The flow is steady up to R = Rt = 2·4 × 103. For R > Rt, the period of oscillation is compared with the theoretical values of Busse. For Pr = 0·71 decreases as well as increases in the slope of the heat-flux curve are observed.

For R just greater than Rc, the preferred orientation of rolls in various side-wall geometries is investigated. For high Prandtl number, the effect of curvature of the roll axis, forced by curved side walls, upon the second transition at RII is investigated. It is found that curvature, as well as previously discussed effects, leads to a lowering of RII. These results, along with the observed hysteresis, support the view that there are metastable states attainable by finite amplitude instability. Finally the nature of the time-dependent flow at large R and high Prandtl number is investigated in a Hele-Shaw cell. It is shown unequivocally that the observed periodicity at a fixed point is due to hot or cold plumes moving past the point.

Detailed measurements of the flow over a compression corner were taken using hot-wire probes. The experiments were performed in supersonic flow (M = 2·64) under adiabatic wall conditions. The incompressible analogues of the boundary-layer profiles were obtained and their integral characteristics and shape factors correlated. Comparison with the self-similar profiles used by Lees & Reeves (1964) to describe interaction problems showed some similarities between the shape factors, but the measured negative shears in the separated bubble proved to be much less.

The transition to turbulence in the convective flow of air between horizontal plates in a circular convection chamber has been investigated. Measurements of the heat flux and the instantaneous spatial temperature field were made simultaneously for a range of Rayleigh number Ra between 3 × 103 and 5 × 104. Ra could be varied by changing the vertical separation or the temperature difference between the plates. The temperature field was measured with either a horizontal or a vertical array of resistance wires mounted so that the flow field could be traversed at velocities much greater than flow velocities characteristic of thermal convection. Slope transitions in the heat flux were found at Ra = 9600 and Ra = 26000. Many measurements of the instantaneous horizontal distribution of temperature for Ra > RaT2 indicate a growth in amplitude of fluctuations with non-dimensional cyclical wavenumbers of 0·4 and greater. The probability of observing these high wavenumber fluctuations also increases as Ra becomes greater than RaT2. The horizontal wavelengths of the different types of temperature fluctuations are compared with the observations of others.

A gas in a tube, one end of which is open, is driven by a periodic applied velocity or pressure at or near a resonant frequency. Damping is introduced into the system by radiation of energy through the open end. It is shown that shocks are possible at an open end and that there is a critical level of damping which ensures a continuous gas response for all frequencies. At the critical level the amplitude of the response is O(ε1/3), where ε is the amplitude of the input, and it is bounded by the amplitude predicted by linear theory. There is agreement with the qualitative experimental results available.

The period of oscillations for the Bénard problem in a two-component system taking into account thermal diffusion is given. Schmidt-Milverton plots are presented for water-methanol and water-isopropanol systems. Anomalous heating curves are observed. Thermocouple responses are given for some heating powers and show oscillations in the temperature field. A qualitative agreement exists with the predicted values of the period given by the theory.

An accurate method is described for integrating the Navier-Stokes equations numerically for the time-dependent flow past an impulsively started circular cylinder. Results of integrations over the range of Reynolds numbers, based on the diameter of the cylinder, from 5 to ∞ are presented and compared with previous numerical, theoretical and experimental results. In particular, the growth of the length of the separated wake behind the cylinder has been calculated for R = 40, 100 and 200 and is found to be in very good agreement with the results of recent experimental measurements. The calculated pressure distribution over the surface of the cylinder for R = 500 is also found to be in reasonable agreement with experimental measurements for the case R = 560.

For Reynolds numbers up to 100 the equations were integrated until most of the features of the flow showed a close approximation to steady-state conditions. The results obtained are in good agreement with previous calculations of the steady flow past a circular cylinder. For R > 100 the integrations were continued until the implicit method of integration broke down by reason of its failure to converge. A secondary vortex appeared on the surface of the cylinder in the case R = 500, but for higher Reynolds numbers, including the case R = ∞, the procedure broke down before the appearance of a secondary vortex. In all cases the flow was assumed to remain symmetrical.

The spin-up of a thermally stratified Boussinesq fluid in a circular cylinder with insulated side walls is analysed under the conditions of strong stratification (Brunt-Väisälä frequency N [Gt ] rotation frequency Ω) and small Prandtl number. An earlier paper (Sakurai, Clark & Clark 1971) showed that complete spin-up is achieved in the Eddington-Sweet time. The present work considers in detail the spin-up transients corresponding to shorter time scales.

The analysis reveals a complicated system of merging and bifurcating horizontal layers in the interior flow. Following the spin-up of the cylindrical container, a rotational shear layer, of the kind discovered by Holton (1965), forms near each horizontal boundary. At the same time, a thermal boundary layer begins diffusing outward from each boundary. When the thermal layer reaches the shear layer, the two merge and form a higher order layer, which diffuses at a rate proportional to t¼. At a later time, the layer splits into a steady layer and another diffusing layer, this time following a t½ law. One important conclusion from the analysis is that the lifetime of the rotational shear layer is not great: it is of order (Ω/N)2 (R2/χ), where R is the radius of the cylinder and χ is the thermal diffusivity.

The problem of computing the angular velocity from the poorly converging series is dealt with in some detail, and graphs are given of representative values. The results show that, for spin-up, there are appreciable adverse gradients of angular momentum near the side wall, and thus there is some question about the stability of the spin-up configuration.

Finally, a discussion is given of continuous spin-up of the container and the results are applied qualitatively to the solar spin-down problem. The principal conclusion is that the Ekman time scale is unimportant in the solar case.

In 1964, Yeh & Cummins demonstrated that coherent light sources could be used for the measurement of steady fluid velocities by observing the Doppler shift in the frequency of light scattered from small particles moving with the fluid. Since 1964 many investigators have attempted to extend this technique to the measurement of turbulent velocity fluctuations.

A fundamental limitation on this type of velocimeter is the Doppler ambiguity introduced by the finite transit time of particles through the scattering volume, turbulent velocity fluctuations across the scattering volume, mean velocity gradients and electronic noise. A unified account of the effect of the Doppler ambiguity on the measurement of the instantaneous velocities is presented and results are interpreted using the power spectrum. The influence of the ambiguity on the measurement of other statistical quantities is also examined.

Limitations on the spatial and temporal resolution imposed by the finite sampling volume are examined using the power spectrum and criteria for optimization of the response are proposed.

An operational laser-Doppler velocimeter is described and measurements of spectra in both laminar and turbulent flow are presented. The experimental results are seen to be in excellent agreement with theoretical predictions.

The method of energy is used to discuss the stability of time-dependent diffusive temperature profiles in fluid layers subject to impulsive changes in surface temperature.

Bounds for the ratio of disturbance energy production to dissipation are found to be parametric functions of time because the basic temperature develops through diffusion. This time dependence leads to the demarcation of regions of stability in a Rayleigh number-time plane and the interpretation of these regions is given. Numerical results are presented for the cases of impulsive heating and cooling of initialty isothermal fluid layers. New global stability results which give the Rayleigh number below which the diffusive solution to the Boussinesq equations is unique are reported for these cases.

An analysis of the experimentally determined resonant amplitude and frequencies of oscillation of a gyroscope whose rotor contains liquid-filled cylindrical cavities reveals a surprising small amplitude instability region that is not predicted by the Stewartson instability criterion for a liquid-filled spinning top. Along with the experimental results and speculation concerning the cause of the observed phenomenon, there are presented some practical implications of the observations of use to designers of spinning vehicles containing liquid.

Overall heat transfer and mean temperature distribution measurements have been made of turbulent thermal convection in horizontal water layers heated from below. The Nusselt number is found to be proportional to Ra0·278 in the range 2·76 × 105 < Ra < 1·05 × 108. Eight discrete heat flux transitions are found in this Rayleigh number range. An interferometric method is used to measure the mean temperature distribution for Rayleigh numbers between 3·11 × 105 and 1·86 × 107. Direct visual and photographic observations of the fluctuating interferogram patterns show that the main heat transfer mechanism is the release of thermals from the boundary layers. For relatively low Rayleigh numbers (up to 5 × 105) many of the thermals reach the opposite surface and coalesce to form large masses of relatively warm fluid near the cold surface and masses of cold fluid near the warm surface, resulting in a temperature-gradient reversal. With increasing Rayleigh numbers, fewer and fewer thermals reach the opposite bounding surface and the thermals show persistent horizontal movements near the bounding surfaces. The central region of the layer becomes an isothermal core. The mean temperature distributions for the high Rayleigh number range are found to follow a Z−2 power law over a considerable range, where Z is the distance from the bounding surface. A very limited agreement with the theoretically predicted Z−1 power law is also found.

A two-dimensional unsteady transonic flow of a perfect gas with constant specific heats is considered, solutions being found in the form of perturbations from a uniform, sonic, isentropic flow. Longitudinal viscous stress terms are retained so that shock waves can be included. The case where the characteristic time of a temporal flow disturbance is large compared with the time taken by a sonic disturbance to traverse the transonic regime is studied. A similarity solution involving an arbitrary function of time is employed, such that the channel walls are in general not stationary. Solutions are presented for thick (shock fills transonic region) and thin (shock tends to a discontinuity) shock waves for both decelerating and accelerating channel flows. For the thin-shock case, both numerical and asymptotic solutions are given. Flow pictures illustrating variations in shock position and structure as well as velocity distributions are shown for exponentially decreasing and for harmonic temporal flow disturbances.

A body is started impulsively from rest and moves in a curved path in a stably stratified fluid. The phase configuration of the internal waves which are generated is studied using small amplitude wave theory. The theory is compared with experiment for a few special cases which include a horizontal cylinder moving in a circular path in a vertical plane, oscillating through a large amplitude in a horizontal plane, and moving with constant velocity past stationary plates. Theory and experiment show reasonable agreement except where the waves produced by the wake dominate the flow.

A Poiseuille-flow problem in a cylindrical capillary in the whole range of Knudsen numbers with incomplete tangential momentum accommodation of molecules incident on the wall has been worked out. The linear non-homogeneous integral equation for the macroscopic gas velocity flow has been solved by the Bubnov-Galerkin method. For a limited range of Knudsen numbers, generally known results have been obtained.

An experimental investigation of the rare gases helium, neon and argon in the range of Knudsen numbers 103−10−3 has been made on packets consisting of 10 and 100 glass capillaries with molten walls. Comparison of theoretical and experimental data enables us to define both slip constants and tangential momentum accommodation coefficients. In the free-molecule flow regime the accommodation coefficients are 0·935, 0·929 and 0·975 for helium, neon and argon, respectively. In the viscous slip-flow regime these coefficients are equal to 0·895, 0·865 and 0·919, respectively. This difference in the tangential momentum accommodation coefficients is, most probably, due to the variable density of adsorbed molecules coating the capillary wall. Gas viscosity coefficients which coincide with those of Kestin within 0.5% have also been calculated. Argon was used as the calibrating gas.

The waves generated in a two-dimensional fluid domain of infinite lateral extent and uniform depth by a deformation of the bounding solid boundary are investigated both theoretically and experimentally. An integral solution is developed for an arbitrary bed displacement (in space and time) on the basis of a linear approximation of the complete (nonlinear) description of wave motion. Experimental and theoretical results are presented for two specific deformations of the bed; the spatial variation of each bed displacement consists of a block section of the bed moving vertically either up or down while the time-displacement history of the block section is varied. The presentation of results is divided into two sections based on two regions of the fluid domain: a generation region in which the bed deformation occurs and a downstream region where the bed position remains stationary for all time. The applicability of the linear approximation in the generation region is investigated both theoretically and experimentally; results are presented which enable certain gross features of the primary wave leaving this region to be determined when the magnitudes of parameters which characterize the bed displacement are known. The results indicate that the primary restriction on the applicability of the linear theory during the bed deformation is that the total amplitude of the bed displacement must remain small compared with the uniform water depth; even this restriction can be relaxed for one type of bed motion.

Wave behaviour in the downstream region of the fluid domain is discussed with emphasis on the gradual growth of nonlinear effects relative to frequency dispersion duringpropagationand the subsequent breakdown of the linear theory. A method is presented for finding the wave behaviour in the far field of the downstream region, where the effects of nonlinearities and frequency dispersion have become about equal. This method is based on the use of a model equation in the far field (which includes both linear and nonlinear effects in an approximate manner) first used by Peregrine (1966) and morerecently advocated by Ben jamin, Bona & Mahony (1972) as a preferable model to the more commonly used equation of Korteweg & de Vries (1895). An input-output approach is illustrated for the numerical solution of this equation where the input is computed from the linear theory in its region of applicability. Computations are presented and compared with experiment for the case of a positive bed displacement where the net volume of the generated wave is finite and positive; the results demonstrate the evolution of a train of solitary waves (solitons) ordered by amplitude followed by a dispersive train of oscillatory waves. The case of a negative bed displacement in which the net wave volume is finite and negative (and the initial wave is negative almost everywhere) is also investigated; the results suggest that only a dispersive train of waves evolves (no solitons) for this case.

Approximate equations are derived for the slow variation in amplitude of free oscillations at the inertial frequency in a slightly stratified ocean. In an ocean of constant depth with horizontally uniform stratification the evolution equation for waves of small vertical mode number can be reduced to a Schrödinger equation, with the increment in Coriolis parameter playing the role of the potential. An exact solution of the Schrödinger equation is presented which demonstrates the transience of inertial oscillations.

A method due to Chester is applied to the theoretical study of resonant wave motion of a radiatively active gas. The inviscid non-conducting grey gas is confined between two infinite parallel walls, and a one-dimensional wave motion is driven by a sinusoidally varying input of black-body radiation from one of the walls. For sufficiently weak driving radiation, the motion can be described by the general solution of the classical wave equation (with the functional form of the solution still undetermined) plus a particular solution due to the driving radiation. When the driving is done at or near a resonant frequency, however, nonlinearities and perturbations in spontaneous emission from the gas must be taken into account before application of the boundary conditions. Such application then leads to a nonlinear integral equation governing the undetermined function in the general solution of the wave equation. This equation is solved numerically by the method of parametric differentiation.

In a frequency range around resonance and for a sufficiently weak relative level of spontaneous emission, the nonlinearities give rise to shock waves (numbering N at the Nth resonant frequency) that are repeatedly reflected at the walls. The perturbations in spontaneous emission give rise to damping, however, and for sufficiently high levels of emission the shock waves disappear. Specific results for various values of optical thickness and various relative levels of spontaneous emission are presented at frequencies in a range around the first resonance.

An analytic expression is found for an infinite subset of the coefficients of the perturbation expansion. They are the coefficients of the terms most rapidly varying at each order, which are also the first terms in the expansion of each Fourier coefficient. The sum of these terms gives a nonlinear approximation to the solution. At greatest height this approximation has a profile with a 90° corner.