In this paper we examine the nonlinear interaction of a forced internal gravity wave in a stratified fluid with its critical level. The representative Richardson number J is taken to be large and the undisturbed state consists of a hyperbolic-tangent velocity profile and an almost constant density gradient. It is assumed that at large values of a non-dimensional time t the flow outside the critical layer is steady, consisting of the mean shear together with a disturbance periodic in x that corresponds to the single harmonic of the incident wave of small amplitude ε. The requirements of a match across the critical layer lead to a reflected wave and a transmitted wave both of whose amplitudes are O(εe−νπ) when 1 [Lt ] t [Lt ] ε−2/3, where ν = (J − ¼)½. For ν [Gt ] 1 the layer therefore acts as a wave absorber, and the purpose of this investigation is to ascertain whether this property persists on an even longer time scale. At times t = O(ε−2/3) the layer has thickness O(ε2/3) and the first few terms of an expansion in powers of ε2/3t show that higher harmonics are forced on the outer flow, and the reflexion and transmission coefficients develop with time. The leading-order correction to these coefficients is calculated explicitly; that to the transmission coefficient is again exponentially small in ν though that to the reflexion coefficient is O(ν−1). The reflexion coefficient is therefore increasing and the critical layer begins to restore wave energy to the outer flow. Owing to the complexity of the calculation higher-order corrections are not obtained here, but the results presented are in agreement with predictions of earlier workers that the layer acts as an absorber and a reflector but not as a transmitter.

Burgers’ equation, a one-dimensional analogue of the Navier–Stokes equation, has been solved numerically in full detail at high (equivalent) Reynolds numbers. These fine-mesh solutions have been used to study the dynamics of the Burgers’ equation analogue of three-dimensional turbulence and in particular the drain of energy from the large to the small structures.

The equation has also been solved on a coarse mesh, using various forms of subgrid model. The solutions so derived have been compared with filtered solutions of the same problem on a fine mesh. In this way it has been possible to test directly the performance of subgrid models at high Reynolds numbers, a test which cannot be made on the Navier–Stokes equations themselves.

With proper choice of the parameters, the performance of the subgrid models is very satisfactory.

The shape of the liquid/gas interface of a weld pool is determined by the balance of gravitational and surface tension forces. Equilibrium profiles and stability criteria are derived for vertical and horizontal situations.

A theoretical model has been developed to investigate the stability of a disturbance in an incompressible turbulent shear flow dominated by turbulence scales that are small with respect to the cross-stream dimension of the flow. The approach utilizes the ‘phase average’ concept to derive the differential equations governing the mechanics of a potential flow disturbance. Turbulence closure is effected at second order. The result is an Orr–Sommerfeld-type equation with complications introduced by the turbulence model. Integration of the linear eigenvalue problem for a wake disturbance leads to the result that the critical eddy-viscosity-based Reynolds number is markedly increased by decreasing the turbulence scale. The viscoelastic behaviour of background turbulence, further complicated by the effects of mean shear, appears to provide stabilization and is discussed in some detail.

To study the mixing of a passive scalar in nearly isotropic turbulence, experiments have been made in isotropically mixed thermal fields with thermal mesh size Mθ (a) equal to the momentum mesh size M, (b) larger than M (obtained by heating only alternate rods of the turbulence generating grid), and (c) smaller than M. This last condition was achieved by inserting a fine heating screen with Mθ < M, at locations downstream of the turbulence grid. The heating screen was designed to produce negligible statistical change in the velocity field a short distance downstream. In all the heated grid experiments, for a given initial configuration of the thermal field, the intensity of temperature fluctuations θ normalized by the mean temperature rise ΔT, and the decay rate of $\overline{\theta^2} $ were both independent of the temperature of the grid. The principal effect of having Mθ > M was an increase in the relative intensity of temperature fluctuations compared with the Mθ = M case, and a marginal increase in their decay rate; contrary to expectation, the ratio R of temperature to velocity integral scales in the region of approximate homogeneity did not differ from that corresponding to Mθ = M. In heated screen experiments, the relative decay rate was independent of Mθ/M and ΔT. For the three locations of the heating screen used in these experiments, the decay rate was also independent of the relative distance xs of the heating screen from the turbulence generating grid; however, larger xs was associated with larger relative intensity of fluctuations. To a first approximation, the ratio R approached unity according to the empirical relation R = 1 − A exp [− αxθ/(UT0)], where xθ is downstream distance measured from the heating screen, and T0 is a characteristic turbulence decay time scale at x0 = 0. It was also verified that the skewness of the streamwise temperature derivative is approximately zero sufficiently downstream of the heating screen. Where the present study overlaps with previous measurements, an extensive comparison reveals several points of agreement as well as departure.

In part 1 (Woods 1979) generalized constitutive relations for the fluid stress p and the heat flux q were established by mean-free-path arguments. These relations are expected to hold over a wide range of Knudsen numbers K, but in the earlier paper, which presented the general theory, they were successfully tested only to O(K2), the order to which the Burnett constitutive equations are valid. It remains to verify the general theory at much higher K values, and to this end we have applied it to the propagation of forced sound waves in a rarefied monatomic gas.

Theories for such waves (based on the linearized Boltzmann equation) are available, and there are also experimental results for the speed and attenuation of waves up to K values of about 10. Our theory is in good agreement with experiment in the range 0 < K < 1. For K > 1 we obtain close agreement with experimental values of the wave speed, but as wave damping in this range is largely due to the unavoidable proximity of the wave transmitter – an effect not included in our calculations – we have found smaller values of the attenuation than obtaining in the experiments.

The classical series expansion procedure of Michell is used to calculate some new highest-wave solutions. These solutions are shown to correspond to the types of gravity waves studied recently by Chen & Saffman (1980). Results are presented for wave profiles, phase speeds, and kinetic and potential energies.

The mechanism of the creation of secondary vortices behind an impulsively started circular cylinder is analysed in this paper by a higher order of accuracy numerical method. This is a combination of second-order and fourth-order compact finite difference schemes to resolve complete unsteady Navier–Stokes equations. The fourth-order compact scheme is used to calculate the Poisson equation of the stream function and the second-order alternating direction implicit scheme to resolve the vorticity transport equation.

In particular, the growth of primary and secondary vortices with time is analysed for Reynolds numbers equal to 300, 550 and 1000. A single secondary vortex first appears at a Reynolds number equal to 300 on the surface of the cylinder. At R = 550, this creation is found numerically at dimensionless time t about 2·85, and this single secondary vortex is transformed into a pair of secondary vortices at t about 5. For R = 1000, two single vortices can be observed at t about 2·5, one near the separation point and another more important, easily identified in flow structure. These secondary vortices are transformed into a pair of secondary vortices at t about 4·5.

A numerical analysis of the influence of the grid systems and the time step is also given. All numerical results presented here are compared with experimental visualizations. The comparison is found satisfactory.

Fully developed periodic flow (with non-zero mean) of a Newtonian fluid in a rigid curved tube has been investigated both numerically and experimentally. Results are reported for the mean friction factor, the amplitude ratio and phase angle between flow rate and pressure drop, the axial velocity profile, and the wall shear stress distribution. The numerical results (obtained by a finite difference method) are restricted to rather slow flows (mean Dean number $\tilde{Dn} < 100 $), while the experimental results (extracted from instantaneous flow rate-pressure drop measurements) extend up to $\tilde{Dn} \sim 300 $. A ‘resonant’ interaction between the axial and secondary flows at intermediate frequencies appears to be a characteristic feature of periodic flow in a curved tube.

In this note we confirm the time dependence of the decay of disturbances in a stratified linear shear flow found by Eliassen, Høiland & Riis (1953) and by Booker & Bretherton (1967). The result differs from those of Case (1960b) and of Chimonas (1979) which are incorrect.

This paper discusses experiments and a theoretical model for the convective circulation driven by a surface buoyancy flux in a horizontal layer of fluid. The layer is closed at one end and, at the other end, the buoyancy has a fixed value over a given depth. Such circulation occurs in side arms of cooling lakes used for waste-heat disposal from power generation. Some geophysical circulations, such as in the Red Sea, are also of the above type.

The experiments were done in a 35 ft long flume using heat transfer between the heated water and the atmosphere to generate the surface buoyancy flux. The observed circulation was characterized by two distinct layers flowing in opposite directions and separated by a density interface. For upper-layer depths less than about half the total depth at the open end, the downflow was observed to be concentrated near the closed end. Circulation flowrates and vertical temperature profiles were measured.

The theoretical model uses a ‘two-layer’ approach. The mass, momentum, and buoyancy conservation equations are integrated vertically on each side of the interface. Mass and buoyancy transfer across the interface are neglected. The interfacial shear stress is proportional to the square of the difference of the average layer velocities. For small layer densimetric Froude numbers, the free surface is shown to be approximately horizontal and the problem reduces to one ordinary differential equation for the thickness of the upper layer. General solutions of this interface equation are presented for horizontal and sloping bottoms. Different configurations are possible depending on the nature of the singular points which occur in the phase plan.

For the convective circulation, the interface is shown to go through a singular point. This condition leads to a simple analytical solution for the circulation flowrate in the case of constant surface buoyancy flux and horizontal bottom. This solution compares well with the experimental data and with measurements on the Red Sea circulation.

We study the initial-value problem posed by the small-amplitude (linearized) free oscillations of free drops, gas bubbles, and drops in a host liquid when viscous effects cannot be neglected. It is found that the motion consists of modulated damped oscillations, with the damping parameter and frequency approaching only asymptotically the results of the normal-mode analysis. The connexion with the normal-mode method is demonstrated explicitly and the experimental relevance of our results is discussed.

Measurements of velocity fluctuations in the wake of a thin non-lifting aerofoil are presented: in one set of experiments the flow was symmetrical, while in the other the upper surface of the aerofoil was roughened to increase the surface shear stress. Measurements were confined to the near wake, where the disturbed region lies within the inner layers of the original boundary layers; thus the boundary-layer thickness is not a relevant length scale. The practical relevance of the experiment is to the prediction of flow over aerofoils, where only the initial region of the wake significantly affects the aerofoil pressure distribution. Temperature-conditioned sampling techniques were used, one boundary layer at a time being heated so that fluid from each boundary layer could be traced within the wake. In contrast to the behaviour of merging shear layers in ducts and jets, the wake interaction involves significant fine-scale mixing; the results reveal a three-layer structure, with a fine-scale inner wake of mixed fluid separating two layers in which structural changes are confined to the region of time sharing, or internal intermittency, between mixed and unmixed fluid. The implications of the results for calculation methods are discussed.

An experimental investigation on the IHSM 4 × 8 cm Shock Tube has confirmed the hypothesis of Ben-Dor (1978) and Ben-Dor & Glass (1979) that in truly non-stationary flows the transitions from regular to Mach reflexion (RR → MR) and from Mach to regular reflexion (MR → RR) are different. Consequently it is shown that a hysteresis loop exists in the ${\rm RR}\rightleftarrows {\rm MR}$ transition phenomenon.

An experimental examination of the fundamental properties of free turbulent interactions in a mixing flow has been conducted. A turbulent, two-dimensional, incompressible, uniform-density mixing layer was produced by using a parallel bar grid of different bar spacings but the same space to bar diameter ratio M/D = 2. In this unique mixing-flow geometry, two initially parallel flowing streams of air were formed in such a way as to possess the same mean structure but different turbulent kinetic energy. The mixing phenomenon was driven solely by the turbulent kinetic-energy self-diffusion process.

The data showed the development of the turbulent properties without the interference caused by turbulence generated by a mean shear flow. A simple diffusion model was constructed by assuming an analogy between the unsteady gradient diffusion process and the transport of turbulent kinetic energy by the fluctuations. An empirical error function least squares curve fit, a diffusional power law and diffusion coefficients were derived. The empirical coefficients of self-diffusion were found to be 10.4 and 68.6 cm2 s−1 for the longitudinal and transverse turbulent energy components, or about a hundred and five hundred times faster than molecular viscous shear diffusion, respectively. This self-diffusion mechanism by microscale size eddies may account for some of the error found by other investigators in balancing the measured terms of the turbulent kinetic-energy equation.

The incompressible flow in a labyrinth seal is computed using the ‘κ−ε’ turbulence model with a pressure-velocity computer code in order to explain leakage phenomena against the mean pressure gradient. The flow is axisymmetric between a rotating shaft and an enclosing cylinder at rest. The main stream in circumferential direction induces a secondary mean flow vortex pattern inside annular cavities on the surface of the shaft. The domain of interest is one such cavity of an enlarged model of a labyrinth seal, where the finite difference result of a computer program is compared with measurements obtained by a back-scattering laser-Doppler anemometer at a cavity Reynolds number of ∼ 3 × 104 and a Taylor number of ∼ 1·2 × 104. The turbulent kinetic energy and the turbulence dissipation rate are verified experimentally for a comparison with the result of the turbulence model.

Rapid distortion theory is used to calculate surface pressure fluctuations beneath a turbulent flow incident on a two-dimensional bluff body. These pressures depend on the ratio, L∞/a, of integral scale to body dimension: we give results in the two asymptotic limits L∞/a [Gt ] 1 and L∞/a [Lt ] 1. The large-scale limit is described by ‘quasi-steady’ theory – which we review here; and for the small-scale limit we introduce a ‘quasi-homogeneous’, or ‘slowly-varying’, approximation. The theory is compared with field and laboratory measurements and it is found that most measurements lie between the theoretical asymptotes, following the predicted trends.

A number of general conclusions have been obtained for which there are new physical explanations – and which laboratory and field experiments appear to confirm.

1. The r.m.s. pressure fluctuations, p′, caused by upwind turbulence, decrease in strength with distance from the stagnation point when L∞/a [Lt ] 1; but p′ increases with distance from the stagnation point when L∞/a [Gt ] 1.

2. For a given dimension of an obstacle, a, transverse to the flow field, p′ increases as the dimension, b, parallel to the flow field increases. At the stagnation point of an elliptical cylinder, when L∞/a [Lt ] 1, \[ p^{\prime} = {\textstyle\frac{1}{2}}\rho u^{\prime}_{\infty} U_{\infty}(1+b/a)(L_{\infty}/a)^{\frac{1}{2}} \] where u′∞, U∞ are the upwind r.m.s. turbulent and mean velocities and ρ is the density.

3. The fluctuating pressure has a cross-correlation length in the flow direction a factor of a/L∞ higher when L∞/a [Lt ] 1 than when L∞/a [Gt ] 1. In the axial direction the correlation length is again greater (though this time of the same order in L∞/a) when the incident turbulence is of small scale than when it is of large scale.

The motion of a spherical microcapsule freely suspended in a simple shear flow is studied. The particle consists of a thin elastic spherical membrane enclosing an incompressible Newtonian viscous fluid. The motions of the internal liquid and of the suspending fluid are both described by Stokes equations. On the deformed surface of the membrane, continuity of velocities is imposed together with dynamic equilibrium of viscous and elastic forces. Since this problem is highly nonlinear, a regular perturbation solution is sought in the limiting case where the deviation from sphericity is small. In particular, the nonlinear theory of large deformation of membrane shells is expanded up to second-order terms. The deformation and orientation of the microcapsule are obtained explicitly in terms of the magnitude of the shear rate, the elastic coefficients of the membrane, the ratio of internal to external viscosities. It appears that the very viscous capsules are tilted towards the streamlines, whereas the less viscous particles are oriented at nearly 45° to the streamlines. The tank-treading motion of the membrane around the liquid contents is predicted by the model and appears as the consequence of a solid-body rotation superimposed upon a constant elastic deformation.

In our previous qualitative paper, it was shown that when a vertical ice surface melts into a stable salinity gradient, the melt water spreads out into the interior in a series of nearly horizontal layers. The experiments reported here are aimed at quantifying this effect, which could be of some importance in the application to melting icebergs. Experiments have also been carried out with heated and cooled vertical walls at larger Rayleigh numbers R than those of previous experiments.

The main result is that for most of our experiments there is no significant difference between these three cases when properly scaled. The layer thickness over a wide range of R is described to within the experimental accuracy by \[ h=0.65 [\rho(T_w,S_{\infty}) - \rho(T_{\infty},S_{\infty})]\left/\frac{d\rho}{dz}\right., \] where the term in brackets is the horizontal buoyancy difference evaluated at the mean salinity and dp/dz is the vertical density gradient due to salinity. In the case of ice melting into warm water the effective wall temperature Tw is approximately 0°C, whereas in colder water the freezing point depression must be taken explicitly into account. A detailed examination of the vertically flowing inner melt water layer in both homogeneous and salinity stratified cases has been made. This layer and the melt water which is mixed outwards from it into the turbulent horizontal layers have little effect on the outer flow. At high R and large external salinity, however, mixing can reduce the effective salinity at the inner edge of the horizontal layers, and thus the layer scale. A puzzling feature is the relatively weak dependence of layer scale on local salinity, though the vigour of convection and the rate of melting are greater where the salinity is high.

The direct application of our results to oceanographic situations predicts layer scales under typical summer conditions of order tens of metres in the Antarctic and of order metres in the Arctic. More measurements will be needed, especially close to icebergs, before the application of these ideas to polar regions can be properly evaluated.