ALBERT

Description: This note provides some explanation of the fact that, contrary to the requirements of local isotropy, the skewness S of the streamwise temperature derivative ∂θ/∂x1 has been observed to be a non-zero constant of magnitude of about unity in high-Reynolds-number and high-Péclet-number turbulent shear flows. Measurements in slightly heated homogeneous shear flows and in unsheared grid turbulence suggest that S is non-zero only when the mean shear dU1/dx2 and the mean temperature gradient dT/dx2 are both non-zero. The sign of S is given by –sgn (dU1/dx2).sgn (dT/dx2). For fixed dU1/dx2, S is of the form tanh (αdT/dx2), α being a constant, while for fixed dT/dx2, it is of the form S/S* = 1 − β1 exp (− β2τ), where S* is a characteristic value of S, β1 and β2 are positive constants, and τ can be interpreted as a ‘total strain’. The derivative skewness data in other (inhomogeneous) shear flows are also compatible with the latter relation. Predictions from a simplified transport equation for [formula omitted], derived in the light of the present experimental observations, are in reasonable agreement with the measured values of S. A possible physical mechanism maintaining S is discussed. © 1980, Cambridge University Press. All rights reserved.

Description: When pure solvent is separated from a solution of non-zero concentration Cb by a semi-permeable membrane, permeable to solvent (water) but not to solute, water flows osmotically across the membrane towards the solution. Its velocity J is given by J = PΔC, where P is a constant and ΔC is the concentration difference across the membrane. Because the osmotic flow advects solute away from the membrane, ΔC is usually less than Cb, by a factor γ which depends on the thickness of and flow in a concentration boundary layer. In this paper the layer is analysed on the assumption that the stirring motions in the bulk solution, which counter the osmotic advection, can be represented as two-dimensional stagnation-point flow. The steady-state results are compared with those of the standard physiological model in which the layer has a given thickness δ and the osmotic advection is countered only by diffusion. It turns out that the standard theory, although mechanistically inadequate, accurately predicts the value of γ over a wide range of values of the governing parameter β = PCbδ/D (where D is the solute diffusivity) if δ is given by where ν is the kinematic viscosity of the fluid and α is the stirring parameter. The final approach to the steady state is also analysed, and it is shown to be achieved in a time scale (D/ν)1/3/αk′ where k′ is a dimensionless number whose dependence on β is computed. Moreover, if β exceeds a certain critical value (≈ 10), the approach to the steady state is not monotonic but takes the form of a damped oscillation (in practice, however, β is unlikely to rise significantly above 1). The theory is extended to the case where the solute concentration is non-zero on both sides of the membrane and in that case it is shown that J is bounded as β → ∞. © 1980, Cambridge University Press. All rights reserved.

Description: A rational asymptotic theory describing the perturbed flow in a turbulent boundary layer encountering a small two-dimensional hump is presented. The theory is valid in the limit of very high Reynolds number in the case of an aerodynamically smooth surface, or in the limit of small drag coefficient in the case of a rough surface. The method of matched asymptotic expansions is used to obtain a multiple-structured flow, along the general lines of earlier laminar studies. The leading-order velocity perturbations are shown to be precisely the inviscid, irrotational, potential flow solutions over most of the domain. The Reynolds stresses are found to vary across a thin layer adjacent to the surface, and display a singular behaviour near the surface which needs to be resolved by an even thinner wall layer. The Reynolds stress perturbations are calculated by means of a second-order closure model, which is shown to be the minimum level of sophistication capable of describing these variations. The perturbation force on the hump is also calculated, and its order of magnitude is shown to depend on the level of turbulence closure; a cruder turbulence model gives rise to spuriously large forces. © 1980, Cambridge University Press. All rights reserved.

Description: Steady potential flow around a two-dimensional bubble with surface tension, either free or attached to a wall, is considered. The results also apply to a liquid drop. The flow and the bubble shape are determined as functions of the contact angle β and the dimensionless pressure ratio γ = (pb − ps)/½ρU2. Here pb is the pressure in the bubble, ps = p∞ + ½ρU2 is the stagnation pressure, p∞ is the pressure at infinity, ρ is the fluid density and U is the velocity at infinity. The surface tension σ determines the dimensions of the bubble, which are proportional to 2σ/ρU2. As γ tends to ∞, the bubble surface tends to a circle or circular arc, and as γ decreases the bubble elongates in the direction normal to the flow. When γ reaches a certain value γ0(β), opposite sides of the bubble touch each other. The problem is formulated as an integrodifferential equation for the bubble surface. This equation is discretized and solved numerically by Newton's method. Bubble profiles, the bubble area, the surface energy and the kinetic energy are presented for various values of β and γ. In addition a perturbation solution is given for γ large when the bubble is nearly a circular arc, and a slender-body approximation is presented for β ∼ ½π and γ ∼ γ0(β), when the bubble is slender. © 1980, Cambridge University Press. All rights reserved.

Description: The evolution of the shape of a slender inviscid drop in an axisymmetric straining motion is studied at low Reynolds numbers. It is found that the shape equation can be solved by polynommals with time-dependent coefficients. A global stability result can be used to show simply that only one possible equilibrium is stable. It is further shown that if the slender drop starts with a long-wavelength waist then it cannot go to this stable equilibrium and must either extend indefinitely or burst. In the class of trinomial shapes, it is shown that the drop either bursts or goes to the stable equilibrium, depending on whether or not the initial shape has a waist. © 1980, Cambridge University Press. All rights reserved.

Description: Conditions are found for the appearance of non-uniform progressive waves of permanent form from a long-wave modulation of a finite-amplitude Stokes wave on deep water. The waveheight at which the modulated waves can occur is a very slowly decreasing function of the modulation wavelength for values up to 150 times the original wavelength. Some qualitative remarks are made about the problem of determining the stability of the new waves. © 1980, Cambridge University Press. All rights reserved.

Description: Based on the parabolic approximation, a refraction—diffraction model for linear water waves is developed. With the assumption that the water depth (refraction index) is slowly varying, the model equation describes the forward-scattered wavefield. Two examples are considered in particular: (i) wave diffraction by a long thin barrier on a uniform slope, and (ii) wave convergence over a semicircular step shoal. For the former problem, a similarity solution in terms of Fresnel integrals is obtained for the wavefield in the neighbourhood of the shadow boundary. For the latter problem, the resulting Schrödinger equation is solved numerically. The wavefield near the caustics as well as in the shadow region is obtained and compared with experimental data. © 1980, Cambridge University Press. All rights reserved.

Description: The coherent structure dynamics in the near field of a circular jet has been experimentally explored by inducing ‘stable’ vortex pairing through controlled excitation (see Zaman & Hussain 1980) and applying phase-averaging techniques. Hot-wire measurements were made in a 7·62 cm air jet with laminar exit boundary layer at the Reynolds number ReD = 3·2 × 104, excited at the Strouhal number StD = 0·85. At a particular phase during the pairing process, spatial distributions of the phase-average longitudinal and lateral velocity perturbations (〈u)〉, 〈v〉), vorticity, streamlines, the coherent and background Reynolds stresses and turbulence intensities have been educed. These data have been obtained for four different locations occupied by the vortices at the same phase (preceding, during, and following the pairing event), in the region 0 〈 x/D 〈 5. Spatial distributions of these measures at four successive phases during the pairing process are also educed in an attempt to further understand the vortex-pairing dynamics. The flow physics is discussed on the basis of measurements over the physical extent of the vortical structures, phase-locked to specific phases of the pairing event and thus do not involve use of the Taylor hypothesis. The computed pseudostream functions at particular phases are compared with the corresponding streamlines drawn by the method of isoclines. Transition of the vortices is examined on the basis of vorticity diffusion, the superimposed random fluctuation field intensities and Reynolds stress and phase-locked circumferential correlation measurements. The peak vorticity drops rapidly owing to transition and interaction of the vortices during pairing but, farther downstream, the decay can be attributed to destruction of the coherent vorticity by the background turbulence Reynolds stress, especially at the locations of the latter's ‘saddle points’. Controlled excitation enhances the initial circumferential coherence of the vortical structures, but is ineffective in delaying turbulent breakdown near the end of the potential core; the breakdown appears to occur through evolution of the circumferential lobe structures. The coherent structure Reynolds stress is found to be much larger than the background turbulence Reynolds stress for 0 〈 x/D ≲ 3, but these two are comparable near the end of the jet potential core. The zone average of the coherent structure Reynolds stress over the cross-section of the merging vortex pair is much larger than that over a single vortical structure either before or after the completion of pairing. During the pairing process, such average correlations are found to be the largest at an early phase of the process while entrainment, turbulent breakdown as well as rapid diffusion of vorticity occur at a later phase. The regions of alternate positive and negative coherent Reynolds stresses associated with the structures and their interactions help explain ‘negative production’. © 1980, Cambridge University Press. All rights reserved.

Description: 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 [formula omitted]), while the experimental results (extracted from instantaneous flow rate-pressure drop measurements) extend up to [formula omitted]. 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. © 1980, Cambridge University Press. All rights reserved.

Description: This paper reports on an extensive experimental study of the flows due to under-expanded axisymmetric jets impinging on flat plates. The range of plate locations extends to a point where the jet is just subsonic but the main emphasis is on the behaviour in the first shock cell. Plate inclinations from 90° to 30° were investigated by means of comprehensive surface pressure measurements and shadowgraph pictures. Wherever possible, the main features of the results have been reconstructed using inviscid analyses of the wave interactions. The flows are shown to be extremely complex due to the local structure of the free jet and, particularly, due to interactions between shock waves in the free jet and those created by the plate. In the near field, these interactions tend to be the controlling factors but at larger distances from the nozzle, mixing effects become increasingly important. The maximum pressure on the plate when it is inclined can be very much larger than when the plate is perpendicular, owing to the possibility of high pressure recoveries through multiple shock systems. Correlations are presented for some of the main features on perpendicular plates and it is shown that the integrated pressure loads for both normal and inclined plates can be predicted well by a simple momentum balance. © 1980, Cambridge University Press. All rights reserved.

Description: When a pipe, connected to reservoirs at both ends, is subjected to an oscillating compression and expansion, fluid is alternately squeezed into and out of the reservoirs, causing a considerable amount of dissipation. This and two other related geometries (involving a symmetrical channel, and a pair of circular disks) are analysed for their flow patterns and energy dissipation, as a function of frequency. It is found that, under certain circumstances, the impedance due to transverse flow can greatly exceed the acoustical impedance (due to longitudinal flow). © 1980, Cambridge University Press. All rights reserved.

Description: 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. © 1980, Cambridge University Press. All rights reserved.

Description: 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. © 1980, Cambridge University Press. All rights reserved.

Description: Decaying turbulence in neutral and stratified fluids has been studied experimentally for relatively high mesh Reynolds numbers and long time-histories. The neutral case indicates an initial period decay law, q2∝ t−1, through non-dimensional time which is considerably longer than previous measurements at the same mesh Reynolds number (Re = 48260). The stratified experiment resulted in a decay rate virtually identical to that of the neutral case through Wgt/M = 275. However the decay rate sharply decreased after this time when the field of turbulence was replaced by internal gravity waves. A critical Richardson number marks the transition from the turbulence to an internal gravity wave domain. © 1980, Cambridge University Press. All rights reserved.

Description: A generalized eddy-viscosity function νT, is introduced in order to express the Reynolds stress in an incompressible dusty gas as a linear combination of the Kronecker and rate-of-strain tensors. On the basis of Saffman's dusty-gas model a transport equation for the eddy viscosity is derived from the general turbulence energy equations, thereby introducing two additional functions, the specific turbulence kinetic energy E1, and a scale variable s. In order to determine the three variables modified Prandtl–Wieghardt relation among them is accepted and a transport equation for s is postulated in the same manner as in the clean-gas turbulence transport model (firstly proposed by Harlow & Nakayama 1967) but with the inclusion of an additional term accounting for the dust particles stabilizing action. We are considering values of loading (mass ratio of particles) of order of unity, with particle/gas density ratios of order of 103 and volume concentrations of the order of 10−3, so that particle–particle interactions are neglected. Supposing that the particles nearly follow the gas motion, following well at large scales and poorly at small, an application of the theory to problem of numerical calculations of the dusty-gas parameters such as mean velocity profile of turbulent pipe flow is given. © 1980, Cambridge University Press. All rights reserved.

Description: Measurements of the thermal and velocity structure of the near-surface mixing layer of a freshwater lake in moderate wind conditions from fixed or mobile arrays of sensors reveal large-scale coherent structures consisting of narrow fronts across which both the temperature and the horizontal component of the current increase. These fronts are generally transverse to the wind direction and are inclined to the vertical, and appear to be similar to fronts, reported as temperature ‘ramps’, in the near-surface atmospheric boundary layer. The time derivatives of the temperature are skewed in a sense consistent with observations in laboratory and atmospheric boundary layers, and of a magnitude consistent with measurements in the latter. Evidence is presented to show that bubbles generated by breaking waves are carried down in the large-scale pattern of flow associated with the fronts in the mixing layer. The presence of a Langmuir circulation associated with wind rows has not been established in these experiments. The relevance of the observations to the ocean mixing layer is discussed. © 1980, Cambridge University Press. All rights reserved.

Description: The convective instability of a layer of fluid heated from below is studied on the assumption that the flux of heat through the boundaries is unaffected by the motion in the layer. It is shown that when the heat flux is above the critical value for the onset of convection, motion takes place on a horizontal scale much greater than the layer depth. Following Childress & Spiegel (1980) the disparity of scales is exploited in an expansion scheme that results in a nonlinear evolution equation for the leading-order temperature perturbation. This equation which does not depend on the vertical co-ordinate, is solved analytically where possible and numerically where necessary; most attention is concentrated on solutions representing two-dimensional rolls. It is found that for any given heat flux a continuum of steady solutions is possible for all wave numbers smaller than a given cut off. Stability analysis reveals, however, that each mode is unstable to one of longer wavelength than itself, so that any long box will eventually contain a single roll, even though the most rapidly growing mode on linear theory has much shorter wavelength. © 1980, Cambridge University Press. All rights reserved.

Description: A model is presented of the two-dimensional boundary-layer and interior flow in a rectangular box resulting from the application of a quadratic temperature variation on its lower surface. The other walls are insulating. It is shown that a similarity form exists for the narrow, thermocline-like layer near the lower surface, and that this can satisfy all known consistency conditions with the interior, together with either laminar or turbulent side-wall regions. The interior temperature and Nusselt number are shown to be insensitive to Prandtl number, and to be primarily functions of the horizontal Rayleigh number RaL. Specifically, the interior temperature, relative to the coldest applied value, is 60% of the total applied temperature range. The Nusselt number is predicted to vary as [formula omitted] for a box with unit aspect ratio. The dynamics of the side-wall region, and the details of the imposed temperature variation appear to be unimportant in determining the overall buoyancy exchange. The solutions are compared with numerical and observational results, and generally good agreement is found. © 1980, Cambridge University Press. All rights reserved.

Description: Energy stability theory has been formulated for two-dimensional buoyancy–thermocapillary convection in a layer with a free surface. The theory yields a critical Rayleigh number RE for which R 〈 RE is a sufficient condition for stability of the layer. RE emerges from the variational formulation as an eigenvalue of a nonlinear system of Euler–Lagrange equations. For the case of small capillary number (large mean surface tension) explicit values are obtained for RE. The analogous linear-theory results for this case are obtained in terms of a critical Rayleigh number RL. These are compared. It is found that the existence of the deformable interface can lead to a stabilization relative to the case of a planar interface. This result is explained in physical terms. The energy theory is then generalized to include general flow problems having three-dimensional disturbances, non-Newtonian bulk fluids and general interfacial mechanics such as surface viscosity and elasticity. © 1980, Cambridge University Press. All rights reserved.

Description: Linear stability analysis is applied to the problem of a density-stratified fluid contained in an inclined slot being subjected to a lateral temperature gradient. Stability equations are solved using the Galerkin technique with 12 terms in the truncated expansion series. Within the range of θ considered, |θ| 〈 75°, critical instability was found to be of the stationary type. Results of critical thermal Rayleigh numbers and wavenumbers at all inclination angles are in good agreement with the experimental results obtained earlier (Paliwal & Chen 1980). Contrary to intuition, these results show that the system is more stable when the lower wall is heated. This is shown to be the result of the increased vertical solute gradient in the steady state prior to the onset of instabilities when the heating is from below. © 1980, Cambridge University Press. All rights reserved.

Description: Results from idealized ocean models indicate that equatorially trapped baroclinic waves incident on an eastern boundary may be partially transmitted north and south along the coast as boundary-trapped internal Kelvin waves. The offshore scale of the coastal internal Kelvin waves is the internal Rossby radius of deformation δR, which decreases as the Coriolis parameter f increases. The effect of the presence of a continental slope of width Ls, along a north–south oriented coastline, on the poleward propagation of coastal trapped internal Kelvin waves is studied in a two-layer β-plane model. The waves propagate from regions near the equator where δR 〉 Ls to mid-latitudes where δR 〈 Ls. It is assumed that f varies slowly on the alongshore scale of the waves L, that L ≫ Ls, and that either the topographic slope is weak or that the upper-layer depth is small compared to the lower-layer depth. All of the coastal trapped waves present in the model are non-dispersive. For most values of f, the cross-shelf eigen-functions consist of the internal Kelvin wave and an infinite set of continental shelf waves whose vertical structure depends on δR/Ls. For δR/Ls ≫ 1, the shelf waves are bottom trapped while for δR/Ls ≪ 1 they are barotropic. The wave speeds Cn of the shelf waves vary linearly with f, whereas the wave speed c0 of the internal Kelvin wave is independent of f. As f increases through critical values fCn, where Cn approaches C0, the phase speeds and the eigenfunctions vary so that the eigenfunctions represent a different type of wave on either side of fCn. In the slowly-varying approximation, the alongshore energy flux in each eigenfunction is a constant. It follows that an internal Kelvin wave which has a wavelength short enough that the slowly-varying approximation remains valid and which propagates poleward from the equatorial region where f 〈 fC1 will transform into a shelf wave, at values of f near fC1, and will continue propagation poleward in that form. As a result, coastal trapped baroclinic disturbances may be able to propagate efficiently from the equatorial region to mid-latitudes where they may take the form of barotropic shelf waves. © 1980, Cambridge University Press. All rights reserved.

Description: It is shown that all parallel inviscid shear flows of constant density are unstable to a wide class of initial infinitesimal three-dimensional disturbances in the sense that, according to linear theory, the kinetic energy of the disturbance will grow at least as fast as linearly in time. This can occur even when the disturbance velocities are bounded, because the streamwise length of the disturbed region grows linearly with time. This finding may have implications for the observed tendency of turbulent shear flows to develop a longitudinal streaky structure. © 1980, Cambridge University Press. All rights reserved.

Description: We consider the onset of Rayleigh–Bénard convection from random fluctuations arising within a fluid. In the specific case in which the fluctuations are thermodynamically determined, we reduce the problem to a random initial value problem for the Fourier modes. For the case of weak nonlinear convection, it is possible to truncate the number of modes and this truncated set is solved both by a Monte Carlo technique and by moment methods for various Rayleigh numbers. We find three stages in the evolution of ordered convection from random fluctuations which correspond to time intervals in which the fluctuations and the nonlinearity have different degrees of importance. It is shown that no simple moment truncation method will succeed and that the time for onset of convection is a mean over a distribution of times for which members of an ensemble exhibit appreciable convective transport. © 1980, Cambridge University Press. All rights reserved.

Description: The thermodynamics of irreversible processes is normally limited to processes that can be adequately described by linear constitutive relations, like those of Fourier and Newton in a simple gas. In this paper we use thermodynamic arguments to derive the (nonlinear) Burnett equations for a monatomic gas, thus avoiding the complicated kinetic theory by which the equations were discovered and which somewhat obscures the origin of the various terms in the equations. Expressions are given for the entropy, its flux and its production rate correct to second-order in Knudsen number. The theory involves five phenomenological parameters, and as there are eleven coefficients in the second-order terms of Burnett's equations, we are able to deduce several necessary constraints between these coefficients. Compact forms for the equations are found that clarify their physical significance. The general method we have developed is applicable to media other than simple gases. In a final section we use our theory of Burnett's equations to draw some general conclusions concerning the second law of thermodynamics. It is shown that the Clausius-Duhem inequality holds only for the linear theory of constitutive relations; and that axiomatic generalizations of the inequality to nonlinear processes – common in continuum mechanics–fail because the vital distinction between reversible and irreversible processes is not made. © 1980, Cambridge University Press. All rights reserved.

Description: An electrically conducting Boussinesq fluid is confined between two rigid horizontal planes. The fluid is permeated by a strong uniform horizontal magnetic field and the entire system rotates rapidly about a vertical axis. By heating the fluid from below and cooling it from above the system becomes unstable to small perturbations when the adverse temperature gradient becomes sufficiently large. Attention is restricted to small values of the Ekman number E and the ratio q of the thermal and magnetic diffusivities (see (1.2) and (1.3) below). In this parameter range marginal convection is steady and its character depends on the relative sizes of the Coriolis and Lorentz forces as measured by the parameter λ (see (1.1) below). When λ ≥ 2/3½, motion consists of a single roll, whose axis is perpendicular to the applied magnetic field. On the other hand, when λ 〈 2/3½, two distinct rolls are possible: the axis of each roll lies oblique but with equal angle to the applied magnetic field. Only the latter case is discussed here. Once the Rayleigh number R exceeds its critical value Rc only one of the two sets of single rolls remains stable, while its squared amplitude increases linearly with R – Rc. For certain values of the parameters λ and τ (see (1.6) below) a second critical value may be isolated at which the system becomes unstable to unidirectional geostrophic flow perturbations aligned with the applied magnetic field. The instability sets in as either a steady or oscillatory shear flow dependent on the values taken by λ and τ. In both cases, after the secondary instability sets in, the roll amplitude remains largely insensitive to further increase in the Rayleigh number with the consequence that the geostrophic flow is stabilized. The amplitude of the shear, on the other hand, increases with R, adjusting its magnitude to ensure stability of the convection rolls. © 1980, Cambridge University Press. All rights reserved.

Description: A rivulet is a narrow stream of liquid located on a solid surface and sharing a curved interface with the surrounding gas. Capillary instabilities are investigated by a linearized stability theory. The formulation is for small, static rivulets whose contact (common or three-phase) lines (i) are fixed, (ii) move but have fixed contact angles or (iii) move but have contact angles smooth functions of contact-line speeds. The linearized stability equations are converted to a disturbance kinetic-energy balance showing that the disturbance response exactly satisfies a damped linear harmonic-oscillator equation. The ‘damping coefficient’ contains the bulk viscous dissipation, the effect of slip along the solid and all dynamic effects that arise in contact-line condition (iii). The ‘spring constant’, whose sign determines stability or instability in the system, incorporates the interfacial area changes and is identical in cases (ii) and (iii). Thus, for small disturbances changes in contact angle with contact-line speed constitute a purely dissipative process. All the above results are independent of slip model at the liquid–solid interface as long as a certain integral inequality holds. Finally, sufficient conditions for stability are obtained in all cases (i), (ii) and (iii). © 1980, Cambridge University Press. All rights reserved.

Description: The development of a three-dimensional water boundary layer along a heated longitudinal horizontal cylinder is studied by a finite-difference method. The secondary flow is induced in an otherwise axially symmetric laminar boundary layer by the buoyancy force. The development of the boundary layer is studied under two heating conditions: constant wall heat flux and constant wall temperature. In general, close to the leading edge, the magnitude of the secondary flow is small and the boundary-layer flow is forced-convection dominant. The secondary flow grows downstream, and the interaction of the free and forced convection becomes important. The flow becomes free-convection dominant further downstream. The temperature-dependent viscosity of water has the effect of thinning the heated boundary layer. The buoyancy effect and the variable viscosity effect enhance each other over the lower part of the cylinder and compete with each other over the upper part of the cylinder. The numerical results compare with the forced convection dominant asymptotic solution and indicate that the asymptotic solution is only valid when x 〈 0·1a/ε½. Since the boundary layer is thin compared with the radius of the cylinder, the transverse curvature effect is small and can be neglected. Therefore, the solution can be applied to the entrance region of heated straight pipes as the zeroth-order boundary-layer flow. © 1980, Cambridge University Press. All rights reserved.

Description: It is shown that for a small sphere freely suspended in a linear shear flow at small Reynolds numbers, the Nusselt number N is given by [formula omitted], where P is the Péclet number. For any given type of shear flow, the numerical value of the constant α can be obtained from a general expression derived by Batchelor (1979). The corresponding result for a particle of arbitrary shape is N/N0 = {1 − αN0P½ + O(P3/2)}−1, where N0 is the Nusselt number for pure conduction. © 1980, Cambridge University Press. All rights reserved.

Description: The problem of acoustic radiation generated by instability waves of a compressible plane turbulent shear layer is solved. The solution provided is valid up to the acoustic far-field region. It represents a significant improvement over the solution obtained by classical hydrodynamic-stability theory which is essentially a local solution with the acoustic radiation suppressed. The basic instability-wave solution which is valid in the shear layer and the near-field region is constructed in terms of an asymptotic expansion using the method of multiple scales. This solution accounts for the effects of the slightly divergent mean flow. It is shown that the multiple-scales asymptotic expansion is not uniformly valid far from the shear layer. Continuation of this solution into the entire upper half-plane is described. The extended solution enables the near-and far-field pressure fluctuations associated with the instability wave to be determined. Numerical results show that the directivity pattern of acoustic radiation into the stationary medium peaks at 20 degrees to the axis of the shear layer in the downstream direction for supersonic flows. This agrees qualitatively with the observed noise-directivity patterns of supersonic jets. © 1980, Cambridge University Press. All rights reserved.

Description: Harris, Graham & Corrsin (1977) have measured the properties of the quasi-homo-geneous turbulence field induced by a large mean shear, and in analysing their measurements they neglect the diagonal components of the mean-field contribution to the pressure–strain correlation. Their measurements are re-analysed using the recommendations of Gibson & Launder (1978) for modelling this correlation, the imbalance between production and dissipation being allowed for by the algebraic modelling technique of Rodi (1976): the experimental data give strong support to Rodi's basic assumption. The data suggest that the only substantial defect in the Gibson–Launder model is its failure to predict the anisotropy measured in the 23 plane normal to the mean flow. The longitudinal predictions are good, and those for the shear (12) component are much improved when measured values of the anisotropy are substituted into the calculation. This analysis does not suggest any clear need for a nonlinear representation of the pressure–strain correlation. However, the most general linear representation of the mean-field term is even more complex than the analysis of Launder, Reece & Rodi (1975) would suggest: their model is disproved by an example. Attempts to deduce the dissipation directly from the experiments, rather than by energy balance, are not very successful. © 1980, Cambridge University Press. All rights reserved.

Description: An alternative to Hunt's (1973) extension of classical rapid distortion theory is used to calculate the turbulence downstream of a rapid contraction. This problem was originally studied by Batchelor & Proudman (1954) and Ribner & Tucker (1953), but their analyses were restricted to flows in which the characteristic turbulence scales were small compared to the spatial scales of the mean flow (usually the characteristic dimension of the apparatus). We now consider the case where the turbulence scale can have the same magnitude as the mean-flow spatial scale. Relatively simple formulae are obtained by calculating the turbulence only in the downstream region where the mean flow is no longer affected by the potential field of the contraction. The results are then further simplified by assuming that the contraction is large and expanding in inverse powers of the contraction ratio. The calculations show that effects of finite turbulence scale can be quite significant. We also obtain some important new results for small-scale turbulence by expanding the solutions in inverse powers of the turbulence spatial scale. © 1980, Cambridge University Press. All rights reserved.

Description: A series of LDA measurements and visual observations of confined turbulent vortex flow are described. The experiments were performed with water as the fluid medium in a vortex tube of length-to-diameter ratio L/D = 3.8 for a range of exit diameters De between De/D = 1 and 0.18. The experiments reveal a remarkable change in the vortex structure as De is reduced: from a thick core with an axial-velocity defect in the centre, and even reversed flow, to a thin annular jet-like core with a peak axial velocity more than an order of magnitude greater than the average value and again a central velocity deficit. The corresponding swirl profiles are not remarkable and are well-represented under all conditions by the solution of Burgers (1948), albeit with a velocity maximum which is strongly dependent upon De. © 1980, Cambridge University Press. All rights reserved.

Description: A theoretical and experimental investigation of oscillatory flow in curved pipes is presented. The equations for fully developed laminar flow are found to depend on an amplitude parameter G and a frequency parameter α. Initially the Navier–Stokes equations are expanded in terms of G and the resulting linearized equations are solved numerically using finite Hankel integral transforms. A further expansion is used for the case α → 0 and closed-form solutions are presented. Laser-Doppler anemometry has been used to obtain velocity information in oscillatory air flow in small-diameter curved glass tubes. Using this technique, low-Reynolds-number experiments were carried out and comparison between theory and experiments is presented. For α ≤ 1 the velocity distributions found are essentially those for steady flow, while for α ≥ 11·0 the results are not at variance with earlier work. It is for the transition stage between these two regimes that unexpected behaviour is recorded, but a satisfactory explanation is found in terms of general trends within the flow. © 1980, Cambridge University Press. All rights reserved.

Description: The fluid mechanics of self-propelling, slender uniflagellar micro-organisms is examined theoretically. The mathematical analysis of these motions is based upon the Stokes equations, and the body is represented by a continuous distribution of stokeslets and doublets of undetermined strength. Since the body is self-propelling, additional constraints on the total force and moment upon it are applied. A system of singular integral and auxiliary equations, in which the propulsive velocity and viscous force per unit length are the unknowns, is derived. The vector integral equation is decomposed into near- and far-field contributions, and the solution is determined by a straightforward iterative procedure. The flagella considered are of constant radius and are restricted to planar undulations. The analysis is applied to a small amplitude wave form of infinite length, and a third-order analytic solution is obtained. By means of numerical computation, the method is extended to large amplitude wave forms of both infinite and finite length. The validity and accuracy of the solution method, the effect of local curvature, and an approximate model for an attached cell body-proper are evaluated in light of alternative theories. The solution method is systematically applied to a variety of wave-form shapes representative of actual flagella. For a sinusoidal wave form, the variations in propulsive velocity, power output and propulsive efficiency are examined as functions of the number of wavelengths on the flagellum, the amplitude and the flagellar radius. Wave forms of variable amplitude and variable wavelength are also considered. Among the significant results are the effect of the cell body on pitching, the significant differences between constant frequency and constant phase-speed undulations for variable wavelength wave forms, and comparisons with other pertinent theories. © 1980, Cambridge University Press. All rights reserved.

Description: The lateral extent of the large-scale features of a high-Reynolds-number, turbulent mixing layer are studied by means of an array of twelve hot wires placed across the span of the wind tunnel. Correlation measurements show that the large-scale structure rapidly approaches an asymptotic state, in which the lateral correlation length becomes proportional to the local mixing-layer thickness. Visualizations of the instantaneous hot-wire outputs show the large structure to extend across the wind tunnel, but to contain spanwise irregularities. It is argued that these spanwise irregularities are produced by the pairing interactions between adjacent vortices. © 1980, Cambridge University Press. All rights reserved.

Description: In an ocean of uniform depth the propagation of small-amplitude plane waves is impeded by two vertical semi-infinite perfectly reflecting barriers extending from the bottom of the sea to above the free surface. The two screens do not lie in general in the same plane, and they are separated by a gap through which wave energy is transmitted from the open sea into the sheltered region. A transmission coefficient is established for small gap widths relative to the wavelength and the agreement with existing theoretical results for special cases is found to be very good. © 1980, Cambridge University Press. All rights reserved.

Description: Periodic wave trains propagating over water which varies in depth in the direction of wave propagation are studied by using accurate solutions for wave trains in constant depth of water. The accurate solutions are (i) Cokelet's (1977) extension of Stokes' approximation and, for the longer waves, (ii) a solution for trains of solitary waves using the solitary-wave solution of Longuet-Higgins & Fenton (1974). A representative selection of results is shown in diagrams. A feature which arises from the use of these accurate solutions is that near the highest wave two solutions are possible for a given incoming wave. Although the solutions cannot describe waves that break, it is shown that as depth is decreased a point is reached beyond which no solution can be found. This is taken to indicate the region in which waves break. The limitations of the theory are discussed and analysed. Comparisons with experimental measurements of Hansen & Svendsen (1979) are included. © 1980, Cambridge University Press. All rights reserved.

Description: The role of large vortex structures in the evolution of a two-dimensional shear layer is studied numerically. The motion of up to 4096 vortices is followed on a 256 × 256 grid using the cloud-in-cell algorithm. The scaling predictions of self-preservation theory are confirmed for low-order velocity correlations, although the existence of vortex structures produces large fluctuations even in a simulation of this size. The simple picture of the shear layer as a line of vortex blobs, that merge pairwise thus thickening the layer, is not seen. On the contrary, the layer seems to thicken by the scattering of vortex structures of roughly fixed size about the midline. The size of the vortex structures does not scale with the layer thickness. A study of the entrainment of a passive marker shows that flow visualization experiments may have overestimated the size of the vortex structures. It appears that the finite area vortices have time to equilibrate between mergings, and the consequences of applying equilibrium statistical mechanics to their internal structure are explored. A simple model is presented which demonstrates how the size and separation of vortex structures may lock into a fixed ratio. This is precisely the type of mechanism that is needed to produce simple scaling in a flow that has initially several distinct length scales. A number of consistency checks on the numerical results are performed. In particular, the evolution of the same vortex configuration on two grids of different size is compared. This test showed that, although errors on subgrid scales do propagate to small wavenumbers, the dominant wavenumber of vorticity cascades back ahead of the peak in the error spectrum. © 1980, Cambridge University Press. All rights reserved.

Description: A study has been made of the flow past thin, inflated lenticular aerofoils at zero angle of incidence. The configuration is an idealization of the flow past long low inflated buildings when the effects of the earth's boundary layer are neglected. It is shown theoretically, and confirmed experimentally, that the non-dimensional tension coefficient in the inflated membranes comprising the aerofoil is a unique function of the internal pressure coefficient divided by the square root of the excess-length ratio. This applies for excess length ratios up to at least 0·05 which corresponds to a thickness-to-chord ratio, wind off, of 0·28. The aerofoil is found to become unstable at negative internal pressures corresponding to tension coefficients of approximately 0·4. The theory has also been used to predict the shape of the aerofoil and the pressure distribution over it. With increasing wind speed at positive internal pressures, the membrane theoretically changes shape from a circular arc to a form involving inflexion points and in all cases the contours are symmetrical about the half-chord point. The non-dimensional membrane slope at the leading and trailing edges agrees fairly well with the experimental values. However, the measured pressure distributions show some fore-and-aft asymmetry and the maximum thickness is slightly downwind of the half-chord point. Nevertheless the comparison between theory and experiment is satisfactory for small values of the excess-length ratio. © 1980, Cambridge University Press. All rights reserved.

Description: The mixing produced by dropping a horizontal grid through a sharp density interface is examined experimentally. The fraction of the available kinetic energy used to mix the fluid, the flux Richardson number Rf, is measured as a function of the overall Richardson number Ri0. It is found that Rf increases from zero as Ri0 does, reaches a maximum, and then decreases with further increase in Ri0. The final interface thickness is found to be a decreasing function of Ri0, and the equivalent vertical diffusion coefficient is calculated. Qualitative observations of the flow show that the final stages of the decay of motion in the interface are characterized by pancake-shaped modes with large horizontal and small vertical scales. © 1980, Cambridge University Press. All rights reserved.

Description: 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. © 1980, Cambridge University Press. All rights reserved.

Description: 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. © 1980, Cambridge University Press. All rights reserved.

Description: 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. © 1980, Cambridge University Press. All rights reserved.

Description: This paper presents a study of steady gravity currents entering a two-layer system, with the current travelling either along the boundary to form a boundary current, or between the two different layers to form an intrusion. It is shown that, at the front of an intrusion, the streamlines meet at angles of 120° at a stagnation point. For an energy-conserving current the volume inflow rate to the current, the velocity of propagation and the downstream depths are determined. In contrast to the pioneering study of Benjamin (1968), it is found that the depth of the current is not always uniquely determined and it is necessary to use some principle additional to the conservation relationships to determine which solution occurs. An appropriate principle is obtained by considering dissipative currents. In general, if the volume inflow rate to a current is prescribed, the current loses energy in order to maintain a momentum balance. We thus suggest the criterion that the energy dissipation is a maximum for a fixed volume inflow rate. It is postulated that the energy which is lost will go to form a stationary wave train behind the current. A nonlinear calculation is carried out to determine the amplitude and wavelength of these waves for intrusions. Such waves have been observed on intrusions in laboratory experiments and the results of the calculation are found to agree well with the experiments. Similar waves have not been observed on boundary currents because the resulting waves have too much energy and break. © 1980, Cambridge University Press. All rights reserved.

Description: Using automated laser-Doppler methods we have identified four distinct sequences of instabilities leading to turbulent convection at low Prandtl number (2·5–5·0), in fluid layers of small horizontal extent. Contour maps of the structure of the time-averaged velocity field, in conjunction with high-resolution power spectral analysis, demonstrate that several mean flows are stable over a wide range in the Rayleigh number R, and that the sequence of time-dependent instabilities depends on the mean flow. A number of routes to non-periodic motion have been identified by varying the geometrical aspect ratio, Prandtl number, and mean flow. Quasi-periodic motion at two frequencies leads to phase locking or entrainment, as identified by a step in a graph of the ratio of the two frequencies. The onset of non-periodicity in this case is associated with the loss of entrainment as R is increased. Another route to turbulence involves successive subharmonic (or period doubling) bifurcations of a periodic flow. A third route contains a well-defined regime with three generally incommensurate frequencies and no broadband noise. The spectral analysis used to demonstrate the presence of three frequencies has a precision of about one part in 104 to 105. Finally, we observe a process of intermittent non-periodicity first identified by Libchaber & Maurer at lower Prandtl number. In this case the fluid alternates between quasi-periodic and non-periodic states over a finite range in R. Several of these processes are also manifested by rather simple mathematical models, but the complicated dependence on geometrical parameters, Prandtl number, and mean flow structure has not been explained. © 1980, Cambridge University Press. All rights reserved.

Description: We consider the free-convective flow from a heated sphere, in the Boussinesq approximation, at high Grashof number. The characteristics of the boundary layer close to the surface of the sphere are evaluated numerically, and the eruption of the fluid from the boundary layer into the plume above the sphere is discussed. © 1980, Cambridge University Press. All rights reserved.

Description: 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 [formula omitted] 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. © 1980, Cambridge University Press. All rights reserved.

Description: Fully developed oil flow in a smooth circular pipe at a mean Reynolds number of about 2100 was subjected to a nominally sinusoidal flow modulation at frequencies ranging from 0·05−1·75 Hz. It was observed that flow oscillation increased the critical Reynolds number and, under certain conditions, even brought about laminarization of the flow, which would be intermittently turbulent at the mean Reynolds number under quasi-steady (infinitely small oscillation frequency) conditions. The occurrence and extent of laminarization was, however, found to depend on factors such as the intermittency of turbulent puffs in the mean quasi-steady flow, frequency of oscillation, etc. Two series of experiments were performed. In one series, the oscillatory flow was almost completely laminarized. In the other series, the oscillatory flow was fully turbulent. In both the cases, instantaneous velocities in the flow were measured using laser-Doppler anemometry (LDA). The instantaneous velocity was decomposed into time-mean, periodic and random components employing ensemble-averaging techniques. The experiments indicated that the laminarized oscillatory flow behaves very similarly to laminar oscillatory flow at either end of the Strouhal-number range studied. The oscillatory turbulent flow was found to depend on both the Strouhal number and the ratio of the oscillation frequency (f) to some characteristic frequency (ft) of turbulence in the flow. The design of the present experimental facility made it possible to study the flow at f/ft ≈ 1 (‘high’ oscillation frequency), a condition that could not be attained in most previous investigations. Another unique feature of the present experiment was that the viscous sublayer and Stokes layer were both large enough (several millimetres in thickness) to allow detailed measurements to be made in these regions. It was found, that at this high frequency of oscillation, the Reynolds stresses generally remained frozen at an average state during the entire oscillation cycle. The turbulent structure showed significant departures from equilibrium at all times during the oscillation cycle. As a result, there was a net change in the time-mean velocity profile near the wall and a net increase in the time-mean wall shear stress and power loss due to friction. The observation that unsteadiness can indeed affect the mean flow behaviour in a significant way is new and contradicts the view presently held by many researchers (based on their studies at relatively low oscillation frequencies, i.e. f/ft ≪ 1). The data also indicated that the direct interaction between oscillation and the turbulent structure was essentially confined to the Stokes layer. The study suggests that (again contrary to the existing belief) quasi-steady turbulence models may not be adequate to describe unsteady flows when the time scale of unsteadiness is comparable to that of dominant turbulent eddies. © 1980, Cambridge University Press. All rights reserved.

Description: Eulerian direct interaction is used to close Liouville's equation central to the transport of particles in a turbulent fluid where the dominant drag force is derived from the particle and local fluid velocities. The reliability of the equation is then tested by comparison of solutions with those of a computer simulation of particle motion with Stokes drag in a random velocity field. Using an empirical drag law accurate for particle Reynolds numbers up to 500, formulae are derived for the statistical moments central to particle dispersion for a weak drag force operating in a Gaussian isotropic stationary velocity field. These show for instance that the long time particle diffusion coefficient is in general greater than the equivalent value based on Stokes drag. © 1980, Cambridge University Press. All rights reserved.

Description: The design of most aerodynamic surfaces, as for example the helicopter rotor, is based essentially on quasi-steady theories. However, the dynamics of a rotating blade introduce unexpected fluctuations and overshoots of properties like lift, drag, etc. The phenomenon of unsteady stall is intimately connected with the development of an oscillating boundary layer and separation. Experimental investigation of such flows was undertaken by a method of visualization developed especially for the study of laminar or turbulent boundary layers and separation. The method captures the instantaneous two-dimensional flow field, including regions of separated flow, and provides accurate quantitative information. Laser-doppler anemometer measurements complement the optically obtained data. Results reveal that separation responds with time-lag to external disturbances, in agreement with unsteady stall data. Oscillating outer flows result in displacement of the point of separation and, under certain conditions, the Despard & Miller (1971) criterion was found to hold. Earlier theoretical models of separation are confirmed qualitatively and for the early stages of the transient phenomena. The findings provide physical insight and quantitative data that may help explain the phenomenon of unsteady stall and unsteady separation. © 1980, Cambridge University Press. All rights reserved.

Description: This paper examines the theory of the interaction of sound with a slit-perforated screen in the presence of a uniform, subsonic tangential mean flow on both sides of the screen. The sound induces vortex shedding from sharp edges of the screen. The coupling of this vorticity with the mean flow leads to a significant modification in the predicted acoustic properties as compared with those predicted by the classical treatments of Rayleigh (1897) and Lamb (1932). In particular a considerable portion of the incident acoustic energy can be lost during the interaction, and is convected away in the mean flow in the form of localized vortical disturbances. The analytical results provide theoretical support for the use of perforated plates to inhibit the onset of cavity resonances in, for example, cross-flow heat exchangers. © 1980, Cambridge University Press. All rights reserved.

Description: Calculated numerical results are presented for laminar buoyancy-induced flows driven by thermal transport to or from a vertical isothermal surface in cold pure and saline water wherein a density extremum arises. The present calculations specifically explore the consequences of temperature conditions wherein the buoyancy force reverses across the thermal region owing to the presence of a density extremum within the region. Such conditions commonly occur in terrestrial waters and in technological processes utilizing cold water. The linear approximation of density dependence on temperature, used in conventional analysis, is here replaced by a very accurate non-linear density equation of state for both pure and saline water. This permits an accurate treatment of such flows for bounding temperatures up to 20 °C at ambient salinity and pressure levels from 0 to 40 p.p.t. and 1 to 1000 bars, respectively. The results may be applied to the melting or slow freezing of a vertical ice surface in pure water as well as to a heated or cooled vertical isothermal surface in pure or saline water. For example, buoyancy force reversals arise for a vertical ice surface at 0 °C melting in fresh water between 4 °C and 8 °C at atmospheric pressure. Temperature conditions for which buoyancy force reversals occur are of special interest because of the resulting anomalous flow behaviour and low surface heat-transfer rates. The transition from conditions with no buoyancy-force reversal to those resulting in a large buoyancy-force reversal is accompanied by as much as 50% decrease in surface heat transfer. This produces a corresponding trend in the melt rate of a vertical ice surface in pure water. Sufficiently strong buoyancy force reversals are found to cause local flow reversal either at the edge of the flow layer or near the surface. Conditions are determined for which flow reversals occur at each of these locations. These local flow reversals are the precursors of convective inversion, that is, of the reversal of the net flow direction with changing ambient medium temperature. Limits on conditions for convective inversion are determined. Calculated transport is compared with previous experimental results, with good agreement throughout the several regions of such complicated flows. The calculations indicate that such flows are relatively very weak. However, their form may lead to early laminar instability. © 1980, Cambridge University Press. All rights reserved.

Description: One-dimensional bispectra are computed from the statistical theory of turbulence (using the Test Field Model) and are compared with experiments. For an inertial range, we obtain B(k1,p1) = k-3F, where B is the two-dimensional Fourier transform of with respect to e is the energy dissipation and F (= tan-1(k1/p1)) is an angular distribution of order unity, which is compared to measurements of planetary boundary-layer turbulence. We also compare theory to wind tunnel data, as reported by Helland et al. (1978). Finally, we discuss to what extent the bispectra give insight into the dynamics of the flow. The National Center for Atmospheric Research is sponsored by the National Science Foundation. © 1980, Cambridge University Press. All rights reserved.

Description: This paper describes an analysis of the forces induced by separation and vortex shedding from sharp-edged bodies in oscillatory flow at high Reynolds number. The analysis which is valid for the case of small oscillations of the fluid is compared with experimental data obtained at fairly low Keulegan–Carpenter numbers. © 1980, Cambridge University Press. All rights reserved.

Description: To account for our previous experimental results, the flow past a flat plate set parallel to a uniform stream is investigated on the basis of Oseen's equations of motion. We are mainly concerned with the wake region and calculate the velocity field in detail for the range of Reynolds numbers 20-3000, corresponding to the experiments. The calculation shows that there is a velocity overshoot in the velocity distribution near the trailing edge, with the same magnitude as in the experiments. Furthermore, we confirm the experimental fact that the velocity on the centre-line of wake recovers in proportion to the square root of the distance from the trailing edge in the near wake. The pressure field is also examined to see the stream wise pressure gradient in the near wake. © 1980, Cambridge University Press. All rights reserved.

Description: A possible model of the flow in the neighbourhood of a point of reattachment of a supersonic laminar boundary layer consists of a three-tiered ‘ triple-deck ‘ structure in which the basic problem reduces to that of solving the incompressible boundary-layer equations in the lower deck, a region of lateral and streamwise extent O(R) and O(R), where R is a representative Reynolds number of the flow (Daniels 1979). The present paper describes a scheme for the numerical solution of this problem which provides evidence in support of the proposed model and quantitative values for the O(R) correction to the base pressure in the flow upstream and the O(R) correction to the position of reattachment relative to the point of intersection of the incoming shear layer and the wall. An important feature of the scheme is that it copes successfully with a flow which contains substantial reverse velocities. © 1980, Cambridge University Press. All rights reserved.

Description: Measurements of turbulent flow through a constricted tube with a contoured 75 % constriction have been performed with a laser-Doppler anemometer. One motivation for these studies was the problem of flow through arterial stenoses and consequently a range of relatively low Reynolds numbers (5000-15000) was employed. Velocity profiles, r.m.s. turbulence velocities, and energy spectra were recorded along with determinations of the wall pressure variation and length of recirculation region. Results show extremely high levels of turbulence and considerable turmoil within the so-called recirculation region. A method for determining the reattachment point in an intensely turbulent flow is given which should offer improvement over flow visualization techniques. Similarity in turbulence energy spectra measured at various radial positions is found at several axial locations, but does not occur in the region immediately downstream of separation. The downstream recovery of the turbulence velocity to upstream values takes place slowly. A secondary motivation of providing experimental data useful in evaluating turbulence models for computational fluid dynamics led to extensive measurements of radial and axial r.m.s. turbulence velocity components at a Reynolds number of 15000. © 1980, Cambridge University Press. All rights reserved.

Description: Large-Reynolds-number analysis is given for separated flows that have the characteristic feature of being confined in the transverse direction. Two principal limits of the governing equation are obtained depending on whether the streamwise length scale is of order one or of the order of the Reynolds number. The corresponding two types of separated flows are discussed. A method of calculation is given for the second type of flow. It employs an expansion in the eigenfunctions of the Poiseuille flow development and the problem is reduced to solving nonlinear first-order ordinary differential equations that have a tendency to decouple rapidly. The method is tested by a detailed comparison of the results of the present calculation with finite-difference solutions of the Navier-Stokes equations for a channel with sudden expansion. Applicability to other configurations is illustrated by solving the problem of the flow in a channel with a base. © 1980, Cambridge University Press. All rights reserved.

Description: The angular momentum A per unit horizontal distance of a train of periodic, progressive surface waves is a well-defined quantity, independent of the horizontal position of the origin of moment. The Lagrangian-mean angular momentum L consists of two parts, arising from the orbital motion and from the Stokes drift respectively. Together these contribute a positive sum, nearly proportional to the energy density (when the origin is taken in the mean surface level). If moments are taken about some point P not at the mean surface level, the angular momentum will differ by an amount proportional to the elevation of P. There is just one elevation for which the Lagrangian-mean angular momentum about P vanishes. This elevation is called the level of action. For infinitesimal waves in deep water the level of action is at a height above the mean surface equal to 1/2k, that is 1/4 times the wavelength. Just as for ordinary fluid velocities, the Lagrangian-mean angular momentum AL differs from the Eulerian-mean L, the latter being zero to second order. The difference between L and E is associated with the displacement of the lateral boundaries of any given mass of fluid. For waves of finite amplitude, an initially rectangular mass of fluid becomes ultimately quite distorted by the Stokes drift. Nevertheless it is possible to define a longtime average l.t. Ā and to calculate its numerical value accurately in waves of finite amplitude. In low waves, l.t. Ā is equal to L. Defining the level of action ya in the general case as l.t.Ā, where I is the linear momentum, we find that ya rises from 05k-1 for infinitesimal waves to about 06k-1 for steep waves. Thus ya is about the same as the height ymax of the wave crests above the mean level in limiting waves, a fact which may account for why steep irrotational waves can support whitecaps in a quasi-steady state. The same argument suggests that Gerstner waves (in which the particle orbits are theoretically circular) could not support whitecaps. © 1980, Cambridge University Press. All rights reserved.

Description: The fluid is incompressible, inviscid and non-diffusive. It has a uniform Brunt-Väisälä frequency, N, and is of constant depth, D. A body or wing moves horizontally through the fluid at velocity U in a straight line, exerting a vertical force during a given time interval. The force is constant, or oscillatory with frequency. The vertical average of the strain rate in a thin surface layer is calculated for a network of points behind the body. The linearized analysis is first applied with tank walls, then modified for remote walls and a vertical force of long duration. For moderately high velocity and forcing frequency (U/ND = 5, /N 4-16) the recurring internal wave pattern just behind the body is well established in one cycle of the oscillatory force. A tank width one or two times the depth gives good agreement between tank and no-wall calculations for the chosen examples. For a stationary wing (U/ND = 0) in a cubic tank with forcing frequency one-half the natural frequency (/N =) the strain rates after one cycle are 103 times greater than for the moving wing case. After five cycles the magnitudes are twenty times larger than after one cycle. Presumably these large increases are due to the continuous and efficient feeding of energy into a small fluid volume which occurs for the stationary wing. No-wall calculations for many cycles give amplitudes roughly one-half those for five cycles in the tank, showing the effect of escaping energy. The relation of these developments to stationary phase analysis and preferred directions is discussed. The stipulation of a rigid body in fluctuating motion leads to a much more difficult analysis. © 1980, Cambridge University Press. All rights reserved.

Description: A backwash vortex has been observed under waves running up a sloping bed. The vortex is formed at the edge of the swash zone when backwash turns back into an oncoming wave bore. The vortex eroded a bed consisting of glass beads, and formed a step which induced the formation of sand ripples. © 1980, Cambridge University Press. All rights reserved.

Description: Autorotation of an elliptic cylinder about an axis fixed perpendicular to a parallel flow is explained in this paper by means of numerical solutions of the Navier-Stokes equations. Potential-flow theory predicts, for constant angular velocity, half a period in which a torque supports rotation and half a period in which it opposes rotation, with vanishing torque in the average. This balance is disturbed by viscous-flow effects in such a way that, for a given angular velocity, vortex shedding either damps rotation or, under certain conditions, favours rotation. The proper interplay of those conditions, which include synchronization of vortex shedding and rate of rotation, results in auto-rotation. The numerical results for Re ≤ 400 are compared with experimental data for Re = 90000 from the literature. The agreement of the force coefficients and the large-scale flow patterns is surprisingly good. © 1980, Cambridge University Press. All rights reserved.

Description: The intradiurnal heating and cooling cycle of the mixed layer of a tropical ocean is investigated through the use of a ‘pseudo-two-dimensional’ numerical model. Particular emphasis is given to two-component diffusion resulting from dynamic instabilities in the water column. The conservation equations for salt and heat include the effects of solar heating, horizontal advection and turbulent fluxes at the sea surface, while wind mixing enters through the use of depth-dependent eddy diffusion coefficients resulting from the wave-orbital shear model of Kitaigorodskiy (1961). All inputs are treated as functions of time of day, or calculated via the bulk aerodynamic method. The entrainment fluxes of salt and heat due to the mechanical stirring of the wind and the fluxes due to molecular diffusion are treated as separate, their respective contributions being added to form the diffusion coefficients used in an alternating-direction explicit scheme to integrate the heat and salt equations. Near the surface, in the absence of strong solar heating (i.e. during the night), these two fluxes alone are insufficient to remove the near-surface static instabilities; thus the presence of some additional process is suggested. A dynamic stability analysis is carried out, based on the temperature and salinity gradients. The resulting Rayleigh numbers indicate the possibility of double-diffusive convection, whereby the vertical transfers of salt and heat may proceed at rates far greater than can be accounted for by molecular diffusion alone. Therefore, the molecular diffusions in the model are increased by a factor roughly proportional to the one-third power of the ratio of the local effective Rayleigh number to a critical Rayleigh number. The modified molecular diffusivities are then added to the eddy diffusion coefficients due to the wind, to form the total diffusion coefficients used in the numerical integrations. Comparisons are made between the model-generated profiles of temperature and the profiles observed in the ocean. The comparisons show reasonable agreement in the diurnal cycle of the heat wave at 1 m vertical resolution (except for the model-generated surface layer being too deep during the late afternoon hours). (Previous models typically predict only the temperature and thickness of a homogeneous layer.) The results obtained with the model are instructive in estimating the relative importance of the various mixing processes in the upper ocean. © 1980, Cambridge University Press. All rights reserved.

Description: A grid-generated ‘isotropic’ turbulent flow has been subjected to a spectrally local perturbation in the form of a high wavenumber, sinusoidal ripple in the mean velocity. It is introduced as the wake of a fine wire, low solidity screen (a ‘zither’), operating below its vortex-shedding Reynolds numbers. The perturbation appears most clearly downstream as a strongly periodic component in the transverse correlation of streamwise turbulent velocity, R11(r2). Its Fourier transform, E11(k2, a ‘one-dimensional spectrum’, shows a corresponding local ‘spike’. The downstream evolution of this perturbation has been chronicled for four different spectral locations. Their decays are approximately exponential. Measurements of the one-dimensional spectrum E11(k1) at several downstream stations show evidence of energy transfer to other wavenumber regions. The decay of the velocity correlation ripple is approximately consistent with the decrease in time of the narrow-band correlation function presented by Comte-Bellot & Corrsin (1971). It is also found that the linear perturbation response calculated by Kraichnan (1959) for a disturbance wavenumber larger than those of the incident turbulence shows fair agreement with this decay. © 1980, Cambridge University Press. All rights reserved.

Description: An experimental study of the motion of small water droplets in a shock tube is reported. Droplet displacement data were obtained by means of reflected-light stroboscopic illumination for droplet diameters in the range 87–575 μm, and for shock strengths, ΔP/P1, in the range 0·0018–0·3. The displacement data are fitted by means of best-fit polynomials in time, which are used to compute droplet velocities, accelerations, and drag coefficients. All of our drag coefficient data have values which are larger than the steady drag at the same Reynolds numbers. The differences are attributed to time changes of the relative fluid velocity Ur. This may affect the size of the recirculating region and, therefore, the drag. In particular, it is argued that the drag is larger or smaller than the steady drag, depending on whether the dUr/dt is negative or positive, respectively. Our experiments, which were performed for dUr/dt 〈 0, confirm this expectation. Furthermore, it is shown that the difference between steady and transient drag coefficients, at the same Reynolds number, depends only on the value of a parameter A = (ρp/ρ0−1)(D/U2r)(dUr/dt). Here ρp and ρ0 are the densities of the droplets and of the surrounding gas, respectively, and D is the droplet diameter. In fact, in the Reynolds number range 3·2 〈 Re 〈 77, where multiple data are available having the same value of Re but having different values of A, the drag data can be expressed as CD = CDS(Re) – KA, where CDS(Re) is the steady drag at the instantaneous Reynolds number Re, and K is a constant of order 1. © 1980, Cambridge University Press. All rights reserved.

Description: Howells' model of the scalar spectrum for isotropic turbulent flow is simplified using a restriction to large Péclet number and statistically stationary turbulence at high wavenumbers, and is generalized by introducing Batchelor's constant as a free parameter. The resulting model is compared with data from the atmospheric surface layer, ocean, and liquid mercury. It is found that Howells' model, which is applicable for arbitrary Prandtl number (Pr), does not compare well with data for large and intermediate Pr at wavenumbers higher than those in the inertial-convective range. The model implies that the inertial-diffusive range asymptotic form, cannot appear even for Pr as small as that of mercury (Pr = 0·018). © 1980, Cambridge University Press. All rights reserved.

Description: A physical explanation is suggested for how the instability of certain fluid systems may be provoked by the addition of a ‘bottom-heavy’ density gradient. It is shown that in all the recent examples of this behaviour the stratification shifts the oscillation frequency at the marginal state towards the diffusion rate associated with the driving mechanism for the instability, and this allows a more effective release of the available energy. When the driving mechanism is an adverse temperature gradient, for example, the frequency shift induced by a bottom-heavy solute distribution can increase the temperature change that a vertically-displaced fluid parcel acquires during each halfcycle, thereby enhancing the thermal buoyancy forces which drive the instability. © 1980, Cambridge University Press. All rights reserved.

Description: An initial thermal stratification may be swept out of a valley if a cross-wind is strong enough: the valley is then said to be ventilated. Numerical and laboratory experiments indicate that the critical parameter is the Froude number, where u is the mean horizontal velocity above the crest of the valley sides, N is the Brunt–Väisälä or bouyancy frequency of the thermal stratification in the valley air and h is the height of the valley walls. Ventilation occurs whenever the Froude number exceeds a value of 1·3. © 1980, Cambridge University Press. All rights reserved.

Description: We continue the experiments of Chaudhary & Maxworthy (1980) to include the behaviour at and beyond the point along the jet at which individual drops interspersed with small satellite drops begin to form. In particular, we show how the perturbation wavenumber and amplitude for a fundamental disturbance alone combine to give a variety of modes of breakup into drops and a variety of patterns of drop recombination further downstream. We then show how the addition of a third harmonic to this fundamental can be used in a variety of parameter combinations to control the behaviour of the satellite drops and in some cases eliminate them completely. © 1980, Cambridge University Press. All rights reserved.

Description: This paper presents the first ‘exact’ solutions to the creeping-flow equations for the transverse motion of a sphere of arbitrary size and position between two plane parallel walls. Previous solutions to this classical Stokes flow problem (Ho & Leal 1974) were limited to a sphere whose diameter is small compared with the distance of the closest approach to either boundary. The accuracy and convergence of the present method of solution are tested by detailed comparison with the exact bipolar co-ordinate solutions of Brenner (1961) for the drag on a sphere translating perpendicular to a single plane wall. The converged series collocation solutions obtained in the presence of two walls show that for the best case where the sphere is equidistant from each boundary the drag on the sphere predicted by Ho & Leal (1974), using a first-order reflexion theory, is 40 per cent below the true value when the walls are spaced two sphere diameters apart and is one order-of-magnitude lower at a spacing of 1.1 diameters. © 1980, Cambridge University Press. All rights reserved.

Description: Under some conditions of temperature and flow an ice-water interface in the presence of a turbulent stream has been observed to be unstable. In this paper the source and the conditions for the instability were investigated for a well-defined turbulent boundary-layer flow. It was found that the instability resulted from the interaction that occurs between a wavy surface and a turbulent flow over it. Such an interaction results in a heat transfer variation which is 90 to 180 degrees out of phase with the surface wave shape – a result which is consistent with the calculations of Thorsness & Hanratty (1979a, b). The main factor controlling damping of the instability at an ice-water interface was found to be the rate at which heat is conducted away from the interface into the ice. In the past it has been found that when an ice layer is melting, that is when the heat conduction in the ice is small, the ice surface is highly unstable. In the present study it was found that for a sufficiently large temperature ratio (Tf−Tw)/(T∞−Tf), a steady-state ice layer is also unstable. Furthermore it is predicted, from the present observations, that a growing ice layer with a ratio of ice-side to water-side heat fluxes of up to 2.3 could be unstable. Under sufficiently unstable conditions waves on the ice surface grow to an amplitude at which flow separations occur near the wave crests. This results in a ‘rippled’ ice surface pattern very similar to the patterns observed on mobile bed surfaces (Kennedy 1969) or surfaces which are being dissolved into a flowing stream (Allen 1971). The development of a ‘rippled’ ice surface results in a very substantial increase in the mean heat-transfer rate which would have an important influence on predictions of ice formation in the presence of a turbulent stream. © 1980, Cambridge University Press. All rights reserved.

Description: The subject is the small-scale structure of a magnetic field in a turbulent conducting fluid, ‘small scale’ meaning lengths much smaller than the characteristic dissipative length of the turbulence. Philip Saffman developed an approximation to describe this structure and its evolution in time. Its usefulness invites a closer examination of the approximation itself and an attempt to place sharper limits on the numerical parameters that appear in the approximate correlation functions, topics to which the present paper is addressed. A Lagrangian approach is taken, wherein one makes a Fourier decomposition of the magnetic field in a neighbourhood that follows a fluid element. If one construes the viscous-convective range narrowly, by ignoring magnetic dissipation entirely, then results for a magnetic field in two dimensions are consistent with Saffman's approximation, but in three dimensions no steady state could be found. Thus, in three dimensions, turbulent amplification seems to be more effective than Saffman's approximation implies. The cause seems to be a matter of geometry, not of correlation times or relative time scales. Strictly-outward spectral transfer is a characteristic of Saffman's approximation, and this may be an accurate description only when dissipation suppresses the contributions from inwardly directed spectral transfer. In the spectral region where dominance passes from convection to dissipation, one can generate expressions for the parameters that arise in Saffman's approximation. Their numerical evaluation by computer simulation may enable one to sharpen the limits that Saffman had already set for those parameters. © 1980, Cambridge University Press. All rights reserved.

Description: Calculations of the flow field under laminar conditions in a helical semicircular duct have been made by numerically solving the Navier–Stokes equations. With the flat wall of the duct being the outer wall, the solution of the momentum equations for Dean numbers below 105 gave, for the secondary flow, twin counter-rotating vortices of Taylor–Goertler type. However, above a Dean number of Dn = 105, two solutions were possible. One solution was similar to that obtained for Dn 〈 105. The other solution revealed four vortices for the secondary flow. For Dn 〉 105, convergence to either flow pattern depended on the initial guess used in the numerical solution. Flow visualization confirmed the possibility of the presence of both types of secondary flow patterns. © 1980, Cambridge University Press. All rights reserved.

Description: European Mechanics Colloquium number 119 was held at Imperial College on 16–18 July 1979, when the subject of vortex shedding from bodies in unidirectional flow and oscillatory flow, was discussed. A wide range of experimental work was presented including low-Reynolds-number flows around circular cylinders, the influence of disturbances on bluff body flow, the measurement of fluctuating forces and the influence of oscillations of the stream. About a third of the 33 papers presented concentrated on theoretical aspects and the majority of these were concerned with the ‘method of discrete vortices’. © 1980, Cambridge University Press. All rights reserved.

Description: The flow near the mouth of an open tube is examined, experimentally and theoretically, under conditions in which resonant acoustic waves are excited in the tube at the other end. If the edge of the tube is round, separation does not occur at high Strouhal numbers, which enables us to verify theoretical predictions for dissipation in the boundary layer and for acoustic radiation. Observation with the aid of schlieren pictures shows that in the case of a sharp edge vortices are formed during inflow. The vortices are shed from the pipe during outflow. Based on these observations a mathematical model is developed for the generation and shedding of vorticity. The main result of the analysis is a boundary condition for the pressure in the wave, to be applied near the mouth. The pressure amplitudes in the acoustic wave measured under resonance are compared with theoretical predictions made with the aid of the boundary condition obtained in the paper. © 1980, Cambridge University Press. All rights reserved.

Description: Boundary-layer structure of prograde and retrograde rotating flows past a cylinder on a β-plane is investigated. It is found that β inhibits boundary-layer separation for prograde flows but it exerts no influence on the boundary-layer structure for retrograde flows. The results agree with the few available experimental observations. © 1980, Cambridge University Press. All rights reserved.

Description: The waves propagating from an oscillating plane piston into a vibrationally relaxing gas are calculated by an exact numerical method ignoring viscosity and heat conduction. Secondary effects due to the starting of the piston from rest and to acoustic streaming can be eliminated from the calculated flows, leaving a truly periodic progressive wave which can be analysed and compared with approximate solutions. It is found that for moderate amplitude waves nonlinearity is only important as a convective effect which produces higher harmonics, whereas dissipation is adequately described by linear theory. © 1980, Cambridge University Press. All rights reserved.

Description: The nonlinear evolution and breakdown of laminar flow in the boundary layer on a flat plate is examined with the aim of making a closer comparison of theory and experiment than has been attempted previously. The importance of three-dimensionality is emphasized. It is concluded that many features of the nonlinear instability are consistent with existing linear and weakly nonlinear theories even as breakdown is approached. The development of the secondary instability, or ‘spike’, is also considered and suggestions for an improved theory of its growth are made. © 1980, Cambridge University Press. All rights reserved.

Description: Steady rotating flow over topography in a periodic channel is examined, with emphasis on the interaction of waves, topography and mean flow. A simple quasi-linear theory is presented that features an implicit equation relating the net zonal flow to the forcing and topography. A good description of the dynamics is obtained, even when resonant Rossby waves appear. Multiple solutions for given external parameters are predicted in some cases, and confirmed by comparison with a fully nonlinear numerical model. The nonlinear results also indicate that the zonally averaged shear can be important when topographic effects or Rossby numbers are large. With this factor taken into account the theory gives good agreement with the fully nonlinear model, as long as eddy–eddy interactions are minor. The theory is relevant to the dynamics of planetary waves in the atmosphere, and may also be applied to some oceanic problems. © 1980, Cambridge University Press. All rights reserved.

Description: The linear stability of the spiral motion induced between concentric cylinders by an axial pressure gradient and independent cylinder rotation is studied numerically and experimentally for a wide-gap geometry. A three-dimensional disturbance is considered. Linear stability limits in the form of Taylor numbers TaLare computed for the rotation ratios µ = 0, 0.2, and — 0.5 and for values of the axial Reynolds number Re up to 100. Depending on the values of µand Re, the disturbance which corresponds to TaLcan have a toroidal vortex structure or a spiral form. Aluminium-flake flow visualization is used to determine conditions for the onset of a secondary motion and its structure at finite amplitude. The experimental results agree with the predicted values of TaLfor µ ≥ 0, and low Reynolds number. For other cases in which agreement is only fair, apparatus length is shown to be a contributing influence. The comparison between experimental and predicted wave forms shows good agreement in overall trends. © 1981, Cambridge University Press. All rights reserved.

Description: Convection flows have been systematically observed in a layer of fluid between two isothermal horizontal boundaries. The working fluid was a nematic liquid crystal, which exhibits a liquid-liquid phase change at which latent heat is released and the density changed. In addition to ordinary Rayleigh–Béxnard convection when either phase is present alone, there exist two distinct types of convective motions initiated by the unstable density difference. When a thin layer of heavy fluid is present near the top boundary, hexagons with downgoing centres exist with no imposed thermal gradient. When a thin layer of light fluid is brought on near the lower boundary, the hexagons have upshooting centres. In both cases, the motions are kept going once they are initiated by the instability due to release of latent heat. Relation of the results to applicable theories is discussed. © 1981, Cambridge University Press. All rights reserved.

Description: The stability to three-dimensional disturbances of three classical steady vortex configurations in an incompressible inviscid fluid is studied in the limit of small vortex cross-sectional area and long axial disturbance wavelength. The configurations examined are the single infinite vortex row, the Karman vortex street of staggered vortices and the symmetricvortex street. It is shown that the single row is most unstable to a two-dimensional disturbance, while the Karman vortex street is most unstable to a three-dimensional disturbance over a significant range of street spacing ratios. The symmetric vortex street is found to be most unstable to three-dimensional or two-dimensional symmetric disturbances depending on the spacing ratio of the street. Short remarks are made concerning the relevance of the calculations to the observed instabilities in free shear layer, wake and boundary-layer type flows. © 1982, Cambridge University Press. All rights reserved.

Description: In this paper we present a rather personal view of the important developments in double-diffusive convection, a subject whose evolution has been the result of a close interaction between theoreticians, laboratory experimenters and sea-going oceanographers. More recently, applications in astrophysics, engineering and geology have become apparent. In the final section we attempt to draw some general conclusions and suggest that further progress will again depend on a close collaboration between fluid dynamicists and other scientists. © 1981, Cambridge University Press. All rights reserved.

Description: This article gives a review of six areas of current activity and importance in aero-acoustics, including (i) the generation of sound and vorticity by vorticity and sound, respectively, (ii) the basis for, and consequences of, the application of a Kutta condition in unsteady leading- and trailing-edge flows, and (iii)the suppression or amplification of broadband hydrodynamic and acoustic fields in a jet under the influence of weak discrete tone forcing. The intention is also to promote acceptance once again of acoustics as a serious branch of fluid mechanics. © 1981, Cambridge University Press. All rights reserved.

Description: Two topics are discussed in order to illustrate the author's own enjoyment of fluid mechanics. The first and longer discourse is about splashes. It makes no attempt at completeness but includes a little new research. The second part deals briefly with many variations on the theme of flow in pipes. © 1981, Cambridge University Press. All rights reserved.

Description: This is in no way intended as a review of turbulence - the subject is far too big for adequate treatment within a reasonably finite number of pages; the monumental treatise of Monin & Yaglom (1971, 1975) bears witness to this statement. It is rather a discourse on those aspects of the problem of turbulence with which I have myself had contact over the last twenty years or so. My choice of topics therefore has a very personal bias - but that is perhaps consistent with the style and objectives of this rather unusual issue of JFM. I have approached the dynamical problem of turbulence via two simpler (but nevertheless far from trivial) problems - viz the convection and diffusion of a passive scalar field and of a passive vector field by turbulence of known statistical properties. Particular emphasis is given to the method of successive averaging (a simplified version of the renormalization-group technique) which seems to me to have considerable potential. The difficulty of extending this method to the dynamical problem is discussed. In a final section (§6) I have allowed myself the luxury of discussing a somewhat esoteric topic - the problem of inviscid invariants and their relationship with the topological structure of a complex vorticity field. The helicity invariant is the prototype; it is identifiable with the Hopf invariant (1931) and it may be obtained from appropriate manipulation of Noether's theorem (Moreau 1977). A suggestion is made concerning possible measurement of this fundamental measure of ‘lack of reflexional symmetry’ in a turbulent flow. © 1981, Cambridge University Press. All rights reserved.

Description: A formalism that accounts for inertial and diffusive effects in the dynamics of a dilute gas-particle suspension is introduced. The treatment is purely deterministic away from a very thin Brownian diffusion sublayer, while, within the sublayer, inertial effects are small, permitting a near-equilibrium expansion in powers of the Stokes number (particle relaxation time divided by flow characteristic residence time). This expansion provides phenomenological expressions for theparticle velocity including two terms: the standard Brownian diffusion, and an additional inertial drift velocity which is closely related to the pressure diffusion term of the Chapman-Enskog expansion. As an example, the general formalism is applied in detail to the case of Stokes flow about a sphere, and sketched for the similar case of a cylinder. Two competing mechanisms are seen to affect the total rate of particle capture by the sphere: (i) the stagnation-point region is considerably enriched in particles owing to the high compressibility of the particle phase, which leads to locally enhanced deposition; (ii) centrifugal forces tend to deplete the Brownian diffusion sublayer of particles, reducing diffusion rates away from the stagnation point to the surface. The first effect is seen to dominate over the second except in a very narrow zone of small Stokes numbers. Our method bridges the gap between Levich's solution for the ‘pure-diffusion’ limit and Michael's treatment in the ‘pure-inertia’ limit. © 1982, Cambridge University Press. All rights reserved.

Description: Arguments are presented to show that the concept of gradient diffusion is inapplicable to mixing in turbulent shear layers. A new model is proposed for treating molecular mixing and chemical reaction in such flows at high Reynolds number. It is based upon the experimental observations that revealed the presence of coherent structures and that showed that fluid elements from the two streams are distributed unmixed throughout the layer by large-scale in viscid motions. The model incorporates features of the strained flame model and makes use of the Kolmogorov cascade in scales. Several model predictions differ markedly from those of diffusion models and suggest experiments for testing the two approaches.† Present address: Dept of Aeronautics, University of Washington, Seattle, WA 98195. © 1982, Cambridge University Press. All rights reserved.

Description: Observations of inertial waves generated by uniform horizontal flow over ridges and truncated axisymmetric obstacles in a homogeneous fluid rotating about a vertical axis are discussed and compared with linear theory. The dependence of the flow on obstacle shape, Ro, H, E and e is investigated. Here Ro = U/2ΩL is the Rossby number, H = Ro(D/L), E = v/2ΩL2 is the Ekman number, and e = h/L is the non-dimensional height of the obstacle, where U is the basic velocity Ω is the angular frequency, L is a streamwise length, D is the depth of the fluid, h is the height of theobstacle, and v is the kinematic viscosity. Previous linear analysis of this problem has been for the limit H fixed, Ro→ 0, referred to here as the small-Ro limit. However, it is shown that certain linear terms neglected in the small-Ro limit can be important for finite Ro, and are included in the analysis given here. The observed flow is then well described by linear theory for H/ e 1, particularly in the case of two-dimensional flow over a ridge. However, for H/e 1 the flow field is dominated by a vertical columnar motion, which is not adequately described by the analysi. © 1982, Cambridge University Press. All rights reserved.

Description: The robustness of localized states that transport energy and mass is assessed by a numerical study of the Euler equation in two space dimensions. The localized states are the translating ‘F-states’ discovered by Deem & Zabusky. These piecewise-constant dipolar (i.e. oppositely-signed or) vorticity regions are steady translating solutions of theEuler equations. A new adaptive contour-dynamical algorithm with curvature-controlled node insertion and removal is used. The evolution of one F-state, subject to a symmetric-plus-asymmetric perturbation is examined and stable (i.e. non-divergent) fluctuations are observed. For scattering interactions, coaxial head-on (or on) and head-tail (or on) arrangements are studied. The temporal variation of contour curvature and perimeter after F-states separate indicate that internal degrees of freedom have been excited. For weak interactions we observe phase shifts and the near recurrence to initial states. When two similar, equal-circulation but unequal-area F-states have a head-on interaction a new asymmetric state is created by contour ‘exchange’. There is strong evidence that this is near to a F-state. For strong interactions we observe phase shifts, ‘breaking’ (filament formation) and, for head-tail interactions, merger of like-signed vorticity regions. © 1982, Cambridge University Press. All rights reserved.

Description: The onset of double-diffusive convection is discussed for a layer of fluid in which the vertical salinity gradient varies with depth and for which the thermal and saline Rayleigh numbers R and Rs are large. These conditions are similar to those that exist in a solar pond prior to the onset of any instability. It is shown that when convection occurs it takes the form of an overstable mode and is essentially confined to a narrow region of vertical extent Rs- 1/4 x depth of the fluid layer, centred at the critical depth where the salt gradient is smallest. The leading terms in asymptotic expansions of the ratio R/Rs, the frequency of oscillation p and the horizontal wavenumber a are determined for Rs 1. The results predicted by the theory are shown to be in good agreement with numerical results and with observations of solar ponds. © 1982, Cambridge University Press. All rights reserved.

Description: The first volume of the Journal of Fluid Mechanics contained nine articles (of 39) on shock waves. Some of these pioneered new branches of fluid mechanics. Others dealt with older problem areas. Surprising is one’s realization that important elements of all topics are still of current interest. The subjects treated were shock structure, diffraction, refraction, waves in supersonic and hypersonic flows, large-amplitude acoustic and blast waves, and astrophysical processes. The subsequent addition of work on chemically reactive flows, radiating and laser-induced shocks, the effects of electric and magnetic fields on shock propagation in ionized media and the development of computer-based methods of analysis have greatly broadened the scope of shock wave investigations during the ensuing twenty-five years. The paper traces some of the principal lines of investigation from early motivations to the present state of understanding and application. Motivation is not often consciously expressed in the scientific literature. Usually an external motivation in terms of identifiable needs for better understanding for the solution of practical problems can be identified; though much excellent work must be ascribed to that ubiquitous trait curiosity. The topics covered in this article were chosen as representative of the basic elements of shock wave interactions and effects. They are: shock structure, refraction, diffraction, shocks in liquid helium, and condensation and liquefaction shocks. The paper closes with an assessment of how approximate and computational methods developed for handling complex flow problems fare when applied to some of the basic shock interactions considered here. Most of the emphasis will be on shock waves in gases, for which knowledge of an equation of state has been key to the significant advances made during the last twenty-five years. For liquids and solids, shock waves have been used the other way around; to study state properties. © 1981, Cambridge University Press. All rights reserved.