ALBERT

Beschreibung: In studying the stability of the boundary layer with surface mass injection, a generalized version of the Orr–Sommerfeld equation was derived which takes account of the transverse velocity component in the mainflow. The new terms in the generalized Orr–Sommerfeld equation are inversely proportional to the Reynolds number. The resulting eigenvalue problem was solved numerically for a wide range of values of the mass injection intensity. It was found that the critical Reynolds number (based on the distance from the leading edge) decreases with increasing mass injection. The deviations between the critical Reynolds numbers from the generalized and conventional Orr–Sommerfeld equations have a different sign at low injection intensities from that at high injection intensities. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: It is demonstrated that the basic stratification in a fluid region subject to thermal forcing may be predicted rather simply for a fairly wide class of boundary conditions. Explicit solutions are derived in certain cases. A useful experimental method for maintaining a stratified system with arbitrarily specified vertical variation of density emerges from the analysis. A preliminary laboratory experiment has demonstrated the efficiency of this method. The restrictions on the validity of the theory involve a limitation on the thermal forcing of the fluid, which may be expressed as an upper limit on the thermal conductance of the boundary of the region. Furthermore, the buoyancy frequency characterizing the solution must be sufficiently large to give rise to a boundary-layer-type flow pattern. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The interactions between electrical tractions at the interface of a liquid jet and instability phenomena are studied with emphasis on effects due to interfacial charge relaxation. Charge relaxation causes the oscillatory growth of a perturbation. When viscous effects are small, small fields tend to decrease the growth rate of the axisymmetric mode, up to a point, and precipitate instability of the non-axisymmetric modes. Still larger field strengths increase the growth rates of asymmetric as well as axisymmetric modes. Instabilities characterized by highfrequency oscillations appear to persist even though the charge relaxation phenomena may be quite rapid. When, on the other hand, viscous effects predominate the only unstable disturbance form is the axisymmetric one, although the manner of growth may be oscillatory. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Centrifugally driven circulations in a rapidly rotating cylinder of fluid heated differentially in the vertical are considered. Boundary-layer solutions obtained previously are extended to include large diameter/height aspect ratios and a centrifugal acceleration of the same magnitude as that of gravity. The ratio of convective to conductive heat transfer is small in the region of parameter space considered. The effect of the circulations on the asymptotic stability of a fluid heat from below and subjected to Coriolis force is then considered. Away from the side wall of the cylinder the basic state circulation increases the critical Rayleigh number at which gravitational instabilities occur; however, a destabilization near the side wall is possible. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: A symposium on aerodynamic noise was held at Loughborough University from 14 to 17 September 1970 under the sponsorship of the Royal Aeronautical Society and the British Acoustical Society. The objective of the meeting was to focus attention on unsolved theoretical and experimental problems which will require attention over the next few years. Areas which were covered included jet noise, nonlinear acoustics, rotor noise, and diffraction theory. The symposium was successful in bringing together several new themes in aerodynamic noise research. The most significant of these were the existence of a degree of order in turbulent jet flows, and the dominant effect of inflow conditions on rotor noise radiation. In addition an improved and unified basis for jet noise theory seems to be evolving. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Measurements of turbulence energy diffusion and the spectral distributions of stress components in the core of turbulent pipe flow are presented. The results tend to confirm the proposal of Bradshaw (1967a, b) that an inertial subrange in the spectra can exist at quite modest laboratory Reynolds numbers. They also illuminate the inconsistencies in Laufer's (1954) measurements of dissipation and suggest that the fitting of a ⊟5/3 power law to the spectra may well provide the most accurate method of determining dissipation for Re ≳ 105. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: A von Kármán vortex street generated in the usual way was subjected to a deceleration, thereby changing the ratio of longitudinal to lateral spacing between the vortices. Distortion of the individual vortices followed which resulted in annihilation of concentrated vortex regions and creation of a stationary wake flow. This wake flow was itself dynamically unstable and developed into a new vortex street of a different frequency from the initial one. The breakdown of the initial vortex street is qualitatively explained by considering the convection of a concentrated vortex region due to the motion imposed by all the other vortices. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The theory of Hunt & Stewartson (1965) for MHD flow in a rectangular duct with conducting walls parallel and non-conducting walls perpendicular to the magnetic field is applied to the problem of electrically driven MHD flow in a rectangular annulus. It is assumed that the Hartmann number M is sufficiently great for secondary flow effects to be negligible. The experiment described here satisfied the conditions of the theory and thus provides a sensitive experimental check on Hunt & Stewartson's theory. The theory is found to agree with the experiments to within the accuracy of the asymptotic theory. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Stratified spin-up, the process of adjustment of a uniformly rotating stratified fluid to an abrupt change in the rotation of the container, is important in many geophysical contexts. An experimental study of this process is presented here for the case where a linearly stratified salt solution is enclosed in a cylindrical container whose rotation rate is changed by a small amount. Results are presented for a limited range of values of B, the internal Froude number, which measures the ratio of the frequencies due to buoyancy and rotation. The experimental study is augmented by a theoretical treatment of idealized models which clarify the more fundamental physical processes that occur. The response of a stratified fluid is faster than that of a homogeneous fluid but the adjustment is limited to layers near the bottom and top boundaries the thickness of which is determined by the value of B. A comparison of the experimental results with the theories of Holton, Walin and Sakurai is also made and it is shown that for the present physical arrangement (insulated side walls) the theories of the latter two authors agree much more closely with experiment than does the theory of Holton. However, all three theories tend to over-estimate the azimuthal displacement in the regions near the upper and lower boundaries where the spin-up is most rapid. The Sweet-Eddington circulation, which accompanies the ideal state of rigid-body rotation, can be significant under normal laboratory conditions and it was necessary to correct some of the spin-up results for this effect. The circulation in the vertical plane is described qualitatively. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The supersonic flight of a slender projectile in a fluid which may undergo internal non-equilibrium transformations is examined by a systematic perturbation scheme. In the frozen limit, the classical results of Whitham and his celebrated ‘rule’ are recovered. Unlike the classical theory, however, the shape of the nose shock can be expressed explicitly in terms of known functions when the relaxation decay length becomes of the same order as a characteristic length scale. The theoretically predicted shock angle, expressed as a function of the radial distance, is found to be in excellent agreement with the experimental measurements of Wegener and Klikoff. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Steady supersonic two-dimensional flows governed by the Navier–Stokes equations are considered. For flows past a thin body, the Oseen theory is shown to fail at large distances. An investigation of the equations bridging the linear and non-linear zones is made. From this, it follows that the resulting equations are a system of Burgers and diffusion equations. The Whitham theory is shown to result under the inviscid limit of our analysis. Various other limits are also obtained. An explicit expression for flows past a thin airfoil is given, and the flow past a double wedge is exhibited in terms of known functions. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: By assuming that an uncharged drop situated in a uniform electric field E retains a spheroidal shape while oscillating about its equilibrium configuration, two approximate equations of motion are derived for the deformation ratio γ expressed as the ratio a/b of the major and minor axis of the drop. Solutions of these equations of motion indicate that the stability of a drop of undistorted radius R and surface tension T depends upon E(R/T)½ and the initial displacement of γ from its equilibrium value. The predictions of the two equations are compared to assess the accuracy of the spheroidal assumption as applied to such a dynamical situation. The analysis is used to determine the stability criterion of a drop subject to a step function field. Finally, the limit of validity of the spheroidal assumption is discussed in terms of Rayleigh's criterion for the stability of charged spherical drops. By applying Rayleigh's criterion to the poles of a spheroidal drop, the stage at which the drop departs from spheroidal form to form conical jets was approximately determined. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Previous results concerning the effects of axial velocity on the motion of vortex filaments are reviewed. These results suggest that a slender-body force balance between the Kutta–Joukowski lift on the vortex cross-section and the momentum flux within the curved filament will give some insight into the behaviour of the filament. These simple ideas are exploited for both a single vortex filament and a vortex pair, both containing axial flow. The stability of a straight vortex filament containing an axial flow to long wave sinusoidal displacements of its centre-line is investigated and the stability boundary obtained. The effect of axial flow on the stability of a vortex pair is explored. It is shown that to lowest order (in the ratio of vortex core radius to distance between the vortices) the effect of axial flow is to reduce the self-induced rotation of a single filament and that this effect can be considered as a change in effective core radius. To the next order, travelling waves appear in the instability, the instability mode for the vortex pair becomes non-planar but the amplification rate of the instability is not affected. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Experiments are described in which velocities were measured ahead of a semi-infinite Rankine body moving parallel to a uniform magnetic field in a conducting fluid. The flow disturbance in front of the body is found to increase in length as N½, where N is the interaction parameter. In most of the experiments this parameter was varied from 4 to about 50. Measurements made along the axis of symmetry in the flow show that there is a relatively short region of stagnant fluid directly ahead of the body. The major part of the disturbance is found to consist of a much longer region in which the flow undergoes transition from conditions in the free stream to conditions near the body. Velocity profiles across the flow in this region show that for increased N, at a fixed distance ahead of the body, the velocity defect increases and the disturbance becomes more confined radially. Although the radial gradients in the flow increase with N, they are found to be much smaller than would be expected in a flow containing thin current layers. A physical model of the flow which has currents and pressures consistent with these results is discussed. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Interactions between short gravity waves and larger-scale flows are investigated in the two-scale approximation. The effect of the wave field on the mean flow is described by an interaction stress tensor and a surface mass transfer. The results are applied to Phillips’ and Longuet-Higgins’ model of short waves breaking on the crests of long carrier waves. It is found that the work done on the long waves by the interaction stresses (corresponding to Longuet-Higgins’ ‘maser’ mechanism of wave generation) is almost exactly balanced by the loss of potential energy arising from the mass transfer. The residual energy transfer leads to attenuation of the long waves, independent of their propagation direction relative to the short waves. Damping factors are estimated from the upwind–downwind ratios of radar backscatter cross-sections. It is found that interactions with waves shorter than 35cm yield attenuation rates about an order of magnitude smaller than the observed growth rates due to the wind. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The relaxation of an initially non-uniform gas to equilibrium is studied within the framework of the kinetic theory of gases. The macroscopic gas properties are taken to depend on one spatial dimension as well as the time. The amplitude of the non-uniformity is assumed to be small with a length scale large compared with the mean free path, and the Krook model of the Boltzmann collision integral is employed. By applying multi-time scale perturbation methods to this reduced problem, uniformly valid analytical solutions for the macroscopic velocity, density and temperature are obtained. The macroscopic equations appropriate to each stage of the relaxation process are obtained in a straightforward and unambiguous manner. The distribution function obtained is shown to be a re-expansion of the Chapman–Enskog solution of the Krook equation, with additional terms accounting for the relaxation of the initial conditions to a near equilibrium form. The results indicate that the uniformly valid frst approximation to the macroscopic velocity, density and temperature can be obtained from the Navier–Stokes equations, but that no purely macroscopic set of equations will suffice for the determination of higher approximations. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The effect of blowing through a porous rotating disk on the flow induced by this disk is studied. For strong blowing the flow is almost wholly inviscid. First-order viscous effects are encountered only in a thin layer at some distance from the disk. The results of an asymptotic analysis are compared with numerical integrations of the full equations and complete agreement is found. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: An experimental investigation was conducted in a circular pipe to examine the influence of a transverse magnetic field on the structure of turbulent shear flow of a conducting fluid (mercury). In the present paper, part 1, mean velocity profiles, turbulence intensity profiles, velocity fluctuation spectra, axial pressure drop profiles, and skin friction data are presented which quantitatively exhibit the Hartmann effect and damping of the velocity fluctuations over a broad range of Reynolds numbers and magnetic fields. The results of heat transfer experiments will be reported by the authors in the following paper, part 2. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Experiments were conducted in an ascending laminar flow through a vertical pipe under combined free and forced convection at constant heat flux through the wall. Mean velocity and temperature profiles were measured with a hot-wire probe. This velocity profile which is deformed by the buoyancy forces, enabled us to compute the reduced acceleration parameter. The profiles obtained showed that the value of the parameter at which reverse transition takes place is approximately the same as that found in isothermal boundary-layer flow. By measuring the autocorrelation function of the velocity after the reverse transition it was shown that the flow in the boundary layer becomes laminar as well as fluctuating and that it oscillates with a predominating period. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The 23rd EUROMECH colloquium on finite-amplitude and diffusive effects in acoustics was held in Rapperswil (Switzerland), 5–7 April 1971. There were 41 participants from 7 countries, and the authors were the chairmen of the meeting organization. References quoted in this report give the titles of the talks and sources for further details of the work described at the meeting; there will be no other publication of the proceedings. The subject matter of this meeting was more strongly restricted than is indicated by its title, inasmuch as papers motivated solely by sonic-boom research problems were not included. Included in particular were problems of acoustic damping by relaxation, dust, moisture, etc.; damping in ducts; effects of turbulence; acoustic streaming; and thermo-acoustic effects. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Waves generated at the stern of a ship must travel through the ship's wake. The effect of the mean flow in the wake refracting the waves is calculated by using a much simplified model. It is found that the waves diverging from the stern of a ship may differ considerably from the bow waves, in qualitative agreement with observation. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The velocity fields in and around a deformed drop suspended in an arbitrary (albeit Stokesian) unbounded flow field are solved. The usefulness of the solution is demonstrated by solving the drag force and lateral migration of a drop suspended in an unbounded Poiseuillian field. It is demonstrated that, due to the deformation of the drop, there exists a radial component of the settling velocity. The direction of the radial migration depends primarily on the product UHR (the Hadamard-Rybczynski terminal settling velocity) by U0 (the maximum Poiseuillian velocity). A positive product results in a lateral migration away from the location of maximum velocity; the converse also holds. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Benjamin (1970) has calculated the upstream influence in open-channel flow, and has argued that a similar effect occurs when a body is moved along the axis of a tube of uniformly rotating fluid. In the present paper Benjamin's work is extended to the case of interfacial waves in a two-fluid system. It is shown that there are certain special flows for which the upstream influence vanishes. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Galerkin (spectral) methods for numerical simulation of incompressible flows within simple boundaries are shown to possess many advantages over existing finite-difference methods. In this paper, the accuracy of Galerkin approximations obtained from truncated Fourier expansions is explored. Accuracy of simulation is tested empirically using a simple scalar-convection test problem and the Taylor–Green vortex-decay problem. It is demonstrated empirically that the Galerkin (Fourier) equations involving Np degrees of freedom, where p is the number of space dimensions, give simulations at least as accurate as finite-difference simulations involving (2N)p degrees of freedom. The theoretical basis for the improved accuracy of the Galerkin (Fourier) method is explained. In particular, the nature of aliasing errors is examined in detail. It is shown that ‘aliasing’ errors need not be errors at all, but that aliasing should be avoided in flow simulations. An eigenvalue analysis of schemes for simulation of passive scalar convection supplies the mathematical basis for the improved accuracy of the Galerkin (Fourier) method. A comparison is made of the computational efficiency of Galerkin and finite-difference simulations, and a survey is given of those problems where Galerkin methods are likely to be applied most usefully. We conclude that numerical simulation of many of the flows of current interest is done most efficiently and accurately using the spectral methods advocated here. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: A new theory is developed for the wake far downstream of a cylindrical body of height h, placed with its generators perpendicular to the flow on a surface above which there is a boundary layer of thickness δ. If the streamwise (x) velocity in the wakeis (U + u), then assuming (h/δ) is small enough that the velocity profile in the boundary layer may be regarded as U = αy, and assuming |u| ≪ U, linear differential equations governing u are derived. It is found that a constant along the wake is [formula omited] This result can be used to find an order of magnitude estimate for u, because I is related to the forces on the body producing the wake by the approximate formula [formula omited] where C1 is that component of the couple on the body produced by pressure and viscous stresses in the x direction. For the particular case of a small hump on the boundary of height h and length b, such that h ≪ b, the above relation is shown to be exact. The perturbation velocity in the wake is found to have a similarity solution [formula omited] the physical implications of which are discussed in detail. The relevance of the theory to the problem of transition behind a trip wire is also mentioned. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The flow past a circular cylinder moving close to a free surface at high Froude number is investigated by the method of matched asymptotic expansions. In contrast with the linearized solution in which the dimensionless depth of immersion h = h′g/U′2 is kept constant, in the present analysis h → 0 as Fr → ∞. The inner flow model is that of a non-separated non-linear gravity-free flow past a doublet, while the linear outer solution is that of a singularity a t the free surface. At deep submergence the solution coincides with the linearized solution. At moderate immersion depths the linearized solution is still valid, provided that the depth is replaced by an effective depth, larger than the actual one. For a body close to the free surface the non-linear solution differs significantly from the linearized solution. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The modified Oseen method is extended to provide a description of the boundary layers which accompany certain swirling flows over a rigid boundary in a rotating container. By comparison with known results it is shown that a refined procedure has errors of the order of 1% when the inviscid flow is a rigid body rotation; it is anticipated that, for the more interesting flows, the error is of the order of 30%. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: This paper describes some laboratory and numerical experiments made on the longitudinal dispersion in an open channel flow. Particular attention has been paid to the initial stages of the process. Physical arguments suggest that the streamwise dispersion of a line of marked fluid elements across a two-dimensional turbulent shear flow occurs in three distinct stages. These stages are identified by a change in the form of the distribution of marked fluid elements in the flow direction. The skewed distribution of the first stage is readily identified by a constant value (approximately 1-1) for the ratio of the peak velocity (V,) of the distribution to the mean-flow velocity [formula omited] experiments using dyed fluid, made at this stage of the process, have revealed six identifiable features of the suggested distribution. The distributions suggested for the second and the third stage are consistent with the experimental findings of Elder (1959) for the second stage and Taylor (1954) for the third stage. An attempt has been made to simulate the process numerically using a Markovian model. The results of the simulation confirm features suggested by physical arguments and are in agreement with the open channel experiments. The Lagrangian autocorrelation function is found to be related to the Lagrangian velocity-history of marked fluid released from extreme positions on the flow cross-section. The correlation function, as expressed in terms of the velocity-history function provided by the numerical simulation, is [formula omited] where u* is the friction velocity and U(y′) is the temporal mean velocity at a (non-dimensionalized) distance y′ from the flow boundary. In an open channel flow at a Reynolds number (based on friction velocity and channel depth) of 500, the numerical simulation provides the value of b = 0·536. The results of an experiment, in which the three-dimensional motion of small neutrally buoyant spheres was recorded in many small discrete time intervals, corroborate the theoretical suggestions and simulation results. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Some experiments are described in which steady-state shearing flows are developed in stratified brine solutions contained in a cyclically continuous tank of rectangular cross-section. Over the range of overall Richardson numbers studied, the results suggest that whenever turbulent layers are present on either side of a region of fluid with a gravitationally stable density gradient, they cause erosion of this region to occur. The erosion leads to the formation of two homogeneous layers separated by a thin layer of strong density and velocity gradients. The gradient Richardson number, computed by using the velocity and density gradients in this transition layer, tends to have a value of order one. If we define an overall Richardson number Ri* by averaging the velocity and density gradients over the entire depth of fluid in the tank, we find that the non-dimensional buoyancy flux, Q, is functionally related to Ri* by an equation of the form Q = C1(Ri*)⊟1 where C1 is a constant, approximately, and Ri* ranges in value between one and thirty. To check the effect of a large variation of the molecular diffusivity coefficient on flow conditions, we ran a limited number of experiments with thermally stratified fluid. Over a restricted range, 1·0 ≪ Ri* ≪ 5·0, velocity profiles very similar to those measured in the brine-stratified experiments at like values of Ri* were obtained. This suggests that the coefficient of molecular diffusion is not an important parameter in either type of experiment. Other experiments, made in the same apparatus, describe the entrainment by a turbulent, homogeneous layer of an initially quiescent layer of fluid with a linear density gradient. The depth of the turbulent layer, D, increases with time, t, according to the relation. [formula omited] This result is consistent with that found by Kato & Phillips (1969), although the turbulent layer in the present experiment is generated in a different manner. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: A thermal boundary layer is established by heating a vertical plate in a dielectric liquid. An alternating voltage is applied between the heated plate and another plate which is not parallel to the heated plate. This voltage produces a non-uni-form electric field which in turn produces electrical forces acting on the gradients in dielectric permittivity which result from the temperature gradients. These electrical forces alter the boundary layer. In this paper approximate equations are developed which allow one to calculate the boundary-layer,thickness, velocity, and Nusselt numbers for the boundary layer in the presence of the non-uniform electric field. Numerical calculations show that the heat-transfer coefficient can be either increased or decreased by the non-uniform field, depending on whether the field is strongest at the top or bottom of the plates and also on the field strength. Experiments were performed which demonstrate the change in heat transfer caused by the non-uniform field. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The thermally driven motion of water contained in a rotating annulus of square cross-section and having a free surface is investigated by numerical integration of the three-dimensional non-linear Navier–Stokes equations. The nature of steady wave flow is examined in detail and a comparison made with the corresponding axisymmetric solution in parameter space. The steady wave solution proves to be consistent kinematically, dynamically and energetically with Lorenz's hypothesis that the wave can be attributed to the baroclinic instability mechanism. The deviaboric The deviation from the zonal mean. The wave motion is almost completely independent of the side boundary layers which make little contribution to the characteristics and energetics of the deviakoric flow. These side layers are approximately axisymmetric and appear qualitatively indistinguishable from their counterparts in the axisymmetric solution. However, significant Ekman layer features appear in the deviatoric wave structure. Away from the boundaries the dynamical balance of terms is hydrostatic and quasi-geostrophic with changes of vertical vorticity influenced by stretching and viscous diffusion. Heat conduction is completely unimportant except in the side boundary layers. The angular momentum transport by the deviatoric motion is largest at the free surface and is mainly against the angular momentum gradient. A strong outward deviatoric flux of momentum is found in the Ekman layer. The dissipation of deviatoric kinetic energy occurs in the Ekman layer and jet whilst most of the dissipation of the mean kinetic energy occurs in the boundary layer of the inner wall. The large differences between the axisymmetric and zonal mean states indicate that linear baroclinic instability analysis of the axisymmetric state is not strictly relevant to an understanding of the wave formation. The character of the wave suggests that the mean environment with which the deviatoric wave interacts is the wave-present zonal mean state. Only a non-linear finite amplitude baroclinic instability analysis (as yet undeveloped) could possibly explain the wave formation. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: An apparatus that approximates a two-dimensional, infinite train of peristaltic waves yields measurements of mean flow, of mean pressure rise, and of pressure-time pulses at fixed locations. In addition, visual observations of ‘reflux’ and ‘trapping’, using dyed fluid, are shown. The inertia-free range extends up to a Reynolds number of about 1. In this range, the theory of Shapiro, Jaffrin & Weinberg (1969) is confirmed with respect to mean pressure vs. mean flow, pressure vs. time, reflux, and trapping. The controversy regarding the criterion of material reflux is settled in favour of the Lagrangian time-mean velocity rather than the Eulerian time-mean velocity. Experiments at higher Reynolds numbers show that the second-order expansion theory of Jaffrin (1971) is valid up to a Reynolds number of about 10. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Davey, Di Prima & Stuart's (1968) double amplitude expansion for disturbances in flow between concentric cylinders is formulated in matrix notation. The stability of the secondary equilibrium (Taylor-vortex) flow is calculated using fifth-order terms in amplitude, and using the full equations rather than the small-gap approximation. Qualitative confirmation is found of instabilities to the Taylor-vortex flow to non-a.xisymmetric disturbances at about 10 % above the first critical Taylor number. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: An axisymmetric flow of a rotating fluid into a point sink was studied experimentally. The type of motion is mainly controlled by the value of the Rossby number R, a ratio of inertial and Coriolis forces. Experimental investigation shows that at a sufficiently large value of R the fluid motion resembles potential flow. However, as R falls below a critical number the withdrawal of the fluid starts to be selective. The flow field then divides into two regions; namely, a central flowing core and an almost stagnant region surrounding it. It is observed that at a Rossby number below the critical value the flow field, induced by a sudden start of discharge at the sink, experiences several distinct stages during the course of each run. At the initial moment the flow exhibits a feature of potential flow. During the second stage, it develops into a state of selective withdrawal with an inviscid profile of a flowing core, which is the main interest of the present study. In the third stage, due to the unavoidable influence of the free or solid surface at the upstream, flow undergoes another change. The flowing core becomes a fast-spinning jet, in which the viscous force becomes important. It is also found that once selective withdrawal begins, the angular velocity and flow rate of the flowing core differ substantially from the basic rotation and the actual discharge at the sink. During this second stage of flow development, the flowing core tends to adjust itself such that the intrinsic Rossby number R′(= Wmax/2Ωcδc) based on the properties of the flowing core, virtually remains constant for all values of R below the critical value. This constant value of R′ is found to be about 0·36. The critical value of R which marks the beginning of the selective withdrawal is found to be in the neighbourhood of 0·26. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Complementary variational formulations are developed for the scattering of a gravity wave by a circular dock. These formulations, which are based on assumed distributions of the radial velocity and the potential, respectively, on the projection of the cylindrical boundary, yield lower and upper bounds to an impedance parameter that determines the difference between the scattered wave for the dock and the corresponding wave for a circular cylinder. Numerical results, using trial functions based on the incident wave, are compared with the results implied by a Galerkin solution (Garrett 1971). The maximum errors in the variational approximations to the total scattering cross-section are found to be of the order of 2% for a typical depth/radius ratio, draft/depth ratios of 0, ½ and 1, and all wavelengths. The axisymmetric component of the scattering cross-section is found to be very close to the value for scattering by a circular cylinder (dock extending to bottom). The intensity of the scattered wave on the forward axis for long wavelengths and a certain range of the geometric parameters is significantly less than that for a circular cylinder, and may vanish for critical combinations of these parameters. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The statistical initial-value problem for a class of weakly coupled waves whose linear dispersion relation is Ω ∞ ± |k| is examined. It is found that in two and higher dimensions a natural asymptotic closure is possible. The redistribution of energy is achieved by means of two mechanisms; the first by a resonance between collinear wave vectors; the second by a local transfer between adjacent rays. The entropy functional is ∫ log n(k) dk and corresponds to particles obeying Bose–Einstein statistics. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Expansions are obtained for the large Prandtl number structure of the laminar natural convection boundary layer, together with its linear stability characteristics, for the case of a uniform-heat-flux semi-infinite vertical plate. The primary source of instability is shown to arise from a temperature-coupling effect associated with the inner heated region of the boundary layer. Based upon an empirical correlation between the results of linear stability theory and experimentally determined regimes of the turbulent-transition process, it is shown that the flow can be expected to become turbulent before the outer vorticity region of the laminar boundary layer is fully established. The results are generalized to the isothermal plate case. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The effect of a uniform and parallel magnetic field upon the stability of a free shear layer of an electrically conducting fluid is investigated. The equations of the velocity and the magnetic disturbances are solved numerically and it is shown that the flow is stabilized with increasing magnetic field. When the magnetic field is expressed in terms of the parameter N (= M2/R2), where M is the Hartmann number and R is the Reynolds number, the lowest critical Reynolds number is caused by the two-dimensional disturbances. So long as 0 ≤ N ≤ 0·0092 the flow is unstable at all R. For 0·0092 ≪ N ≤ 0·0233 the flow is unstable at 0 ≪ R ≪ Ruc where Ruc decreases as N increases. For 0·0233 ≪ N ≪ 0·0295 the flow is unstable at Rlc ≪ R ≪ Ruc where Rlc increases with N. Lastly for N 〉 0·0295 the flow is stable at all R. When the magnetic field is measured by M, the lowest critical Reynolds number is still due to the two-dimensional disturbances provided 0 ≤ M ≤ 0·52, and Rc is given by the corresponding Rlc. For M 〉 0·52, Rc is expressed as Rc = 5·8M, and the responsible disturbance is the three-dimensional one which propagates at angle cos⊟1(0·52/M) to the direction of the basic flow. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The governing equations for the problem of linearized flow through a normal shock wave in an emitting, absorbing, and scattering grey gas are reduced to two linear coupled integro-differential equations. By separation of variables, these equations are further reduced to an integral equation similar to that which arises in neutron-transport theory. It is shown that this integral equation admits both regular (associated with discrete eigenfunctions) and singular (associated with continuum eigenfunctions) solutions to form a complete set. The exact closed-form solution is obtained by superposition of these eigen-functions. If the gas downstream of a strong shock is absorption–emission dominated, the discrete mode of the solution disappears downstream. The effects of isotropic scattering are discussed. Quantitative comparison between the numerical results based on the exact solution and on the differential approximation are presented. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The effects of sinusoidal protrusions on steady laminar free convection between vertical walls is investigated in this paper. Numerical results are presented for various values of the size and spacing of the protrusions. In particular optimum values are found which yield maximum wall heat-transfer coefficients. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: This paper is intended to evaluate the wall effects in the pure-drag case of plane cavity flow past an arbitrary body held in a closed tunnel, and to establish an accurate correction rule. The three theoretical models in common use, namely, the open-wake, Riabouchinsky and re-entrant-jet models, are employed to provide solutions in the form of some functional equations. From these theoretical solutions several different rules for the correction of wall effects are derived for symmetric wedges. These simple correction rules are found to be accurate, as compared with their corresponding exact numerical solutions, for all wedge angles and for small to moderate ‘tunnel-spacing ratio’ (the ratio of body frontal width to tunnel spacing). According to these correction rules, conversion of a drag coefficient, measured experimentally in a closed tunnel, to the corresponding unbounded flow case requires only the data of the conventional cavitation number and the tunnel-spacing ratio if based on the open-wake model, though using the Riabouchinsky model it requires an additional measurement of the minimum pressure along the tunnel wall. The numerical results for symmetric wedges show that the wall effects in-variably result in a lower drag coefficient than in an unbounded flow at the same cavitation number, and that this percentage drag reduction increases with decreasing wedge angle and/or with decreasing tunnel spacing relative to the body frontal width. This indicates that the wall effects axe generally more significant for thinner bodies in cavity flows, and they become exceedingly small for sufficiently blunt bodies. Physical explanations for these remarkable features of cavity-flow wall effects are sought; they are supported by the present experimental investigation of the pressure distribution on the wetted body surface as the flow parameters are varied. It is also found that the theoretical drag coefficient based on the Riabouchinsky model is smaller than that predicted by the open-wake model, all the flow parameters being equal, except when the flow approaches the choked state (with the cavity becoming infinitely long in a closed tunnel), which is the limiting case common to all theoretical models. This difference between the two flow models becomes especially pronounced for smaller wedge angles, shorter cavities, and with tunnel walls farther apart. In order to gauge the degree of accuracy of these theoretical models in approximating the real flows, and t o ascertain the validity of the correction rules, a series of definitive experiments was carefully designed to complement the theory, and then carried out in a high-speed water tunnel. The measurements on a series of fully cavitating wedges at zero incidence suggest that, of the theoretical models, that due to Riabouchinsky is superior throughout the range tested. The accuracy of the correction rule based on that model has also been firmly established. Although the experimental investigation has been limited to symmetric wedges only, this correction rule (equations (85), (86) of the text) is expected to possess a general validity, at least for symmetric bodies without too large curvatures, since the geometry of the body profile is only implicitly involved in the correction formula. This experimental study is perhaps one of a very few with the particular objective of scrutinizing various theoretical cavity-flow models. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Upon review of past experimental results and theoretical efforts it is apparent that the mechanism by which combustion noise is generated is not well understood. A theory of combustion noise is developed in this paper which follows rigorously from the principles of fluid mechanics. Lighthill's approach, used in his studies of aerodynamic noise, is closely followed in the present work. The sound radiated from open, turbulent flames is found to depend strongly upon the structure of such flames; at present their structure is not well known. However, meaningful bounds and scaling rules for the sound power output and spectral content are derived based upon the present limited knowledge. A framework is developed which explains past experimental work and the origin of combustion noise. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: We investigate the behaviour of infinitesimal perturbations introduced into an unstably stratified horizontal Couette flow. We assume that the fluid is Boussinesq and contained in an infinite conducting rectangular channel which is uniformly heated from below. The sidewalls are rigid and the Couette flow is generated by moving them with equal and opposite velocities along the channel. The top and bottom are assumed to be free so that we can separate variables. Without shear, the preferred modes of convection closely resemble transverse ‘finite rolls’ (Davies-Jones 1970). Shear increases the critical wavelength so that the preferred modes become longitudinally elongated cells, or even longitudinal rolls in some cases. The critical Rayleigh number increases quite rapidly at fist with Reynolds number, but at higher Reynolds numbers it levels off to a constant value (which cannot be greater than the shear-independent Rayleigh number at which longitudinal disturbances fist become unstable). We also find that the disturbances are tilted in the same direction as the shear, and that the marginally stable ones transfer kinetic energy from the mean flow to the perturbations. Except at low Reynolds numbers, the long wave perturbations gain more energy through the conversion of mean flow kinetic energy than through the release of potential energy, even though the instability is convective in origin. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The steady axially symmetric incompressible flow past a sphere is investigated for Reynolds numbers, based on the sphere diameter, in the range 0·1 to 40. The formulation is a semi-analytical one whereby the flow variables are expanded as series of Legendre functions, hence reducing the equations of motion to ordinary differential equations. The ordinary differential equations are solved by numerical methods. Only a finite number of these equations can be solved, corresponding to an approximation obtained by truncating the Legendre series at some stage. More terms of the series are required as R increases and the present calculations were terminated at R = 40. The calculated drag coefficient is compared with the results of previous investigations and with experimental data. The Reynolds number at which separation first occurs is estimated as 20·5. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: A doubly-infinite sloping flat plate, initially at rest in a slightly diffusive viscous density-stratified fluid, starts to move impulsively with a constant velocity along the line of greatest slope. The resulting flow is found to be an unsteady motion superimposed on a steady diffusion-induced flow, which is present throughout. Laplace transform methods give solutions which are valid either in an essentially non-diffusive outer layer or in a diffusive inner layer. The impulsive start sets up oscillations in the outer layer. These gradually die out, and a steady diffusive flow develops. A glass plate was towed vertically through stratified brine, into which aluminium particles were introduced. The flow velocities deduced from the particle motions confirmed the theoretical predictions. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Recently Cheng & Akiyama (1970) published a numerical analysis of laminar flow in curved channels of square and rectangular section. Experimental results are presented here for flow in curved channels of square section. The channels were toroidal in shape, and the flow was driven electromagnetically. Various ratios of the channel dimension d to the channel radius of curvature, R, were used to investigate the dependence of friction factor, f, on the Dean number K, and the Reynolds number, Re. For 5 × 102 ≪ K ≪ 7 × 104 the formula (fRe) = 1·51 K½ was found to fit all the results, although R/d was varied from 17·5 down to the low value of 1·75. At lower values of K the analysis of Cheng & Akiyama was approximately validated. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The initial-value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given. For values of the Reynolds number slightly greater than the critical value, above which perturbations may grow, the asymptotic solution is used as a guide in the choice of appropriate length and time scales for slow variations in the amplitude A of a non-linear two-dimensional perturbation wave. It is found that suitable time and space variables are εt and ε½(x+a1rt), where t is the time, x the distance in the direction of flow, ε the growth rate of linearized theory and (⊟a1r) the group velocity. By the method of multiple scales, A is found to satisfy a non-linear parabolic differential equation, a generalization of the time-dependent equation of earlier work. Initial conditions are given by the asymptotic solution of linearized theory. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Stability curves are computed for both spatially and temporally growing disturbances in a stratified mixing layer between two uniform streams. The low Froude number limit, in which the effects of buoyancy predominate, and the high Froude number limit, in which the effects of density variation are manifested by the inertial terms of the vorticity equation, are considered as limiting cases. For the buoyant case, although the spatial growth rates can be predicted reasonably well by suitable use of the results for temporal growth, spatially growing disturbances appear to have high group velocities near the lower cutoff wave-number. For the inertial case, it is demonstrated that density variations can be destabilizing. More precisely, when the stream with the higher velocity has the lower density, both the wave-number range of unstable disturbances and the maximum spatial growth rate are increased relative to the case of homogeneous flow. Finally, it is shown how the growth rate of the most unstable wave in the inertial case diminishes as buoyancy becomes important. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The stability of the flow induced by an impulsively started inner cylinder in a Couette flow apparatus is investigated by using a linear stability analysis. Two approaches are taken; one is the treatment as an initial-value problem in which the time evolution of the initially distributed small random perturbations of given wavelength is monitored by numerically integrating the unsteady perturbation equations. The other is the quasi-steady approach, in which the stability of the instantaneous velocity profile of the basic flow is analyzed. With the quasi-steady approach, two stability criteria are investigated; one is the standard zero perturbation growth rate definition of stability, and the other is the momentary stability criterion in which the evolution of the basic flow velocity field is partially taken into account. In the initial-value problem approach, the predicted critical wavelengths agree remarkably well with those found experimentally. The kinetic energy of the perturbations decreases initially, reaches a minimum, then grows exponentially. By comparing with the experimental results, it may be concluded that when the perturbation kinetic energy has grown a thousand-fold, the secondary flow pattern is clearly visible. The time of intrinsic instability (the time at which perturbations first tend to grow) is about ¼ of the time required for a thousandfold increase, when the instability disks are clearly observable. With the quasi-steady approach, the critical times for marginal stability are comparable to those found using the initial-value problem approach. The predicted critical wavelengths, however, are about 1½ to 2 times larger than those observed. Both of these points are in agreement with the findings of Mahler, Schechter & Wissler (1968) treating the stability of a fluid layer with time-dependent density gradients. The zero growth rate and the momentary stability criteria give approximately the same results. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Using a hot wire in a turbulent boundary layer in air, an experimental study has been made of the frequent periods of activity (to be called ‘bursts’) noticed in a turbulent signal that has been passed through a narrow band-pass filter. Although definitive identification of bursts presents difficulties, it is found that a reasonable characteristic value for the mean interval between such bursts is consistent, at the same Reynolds number, with the mean burst periods measured by Kline et al. (1967), using hydrogen-bubble techniques in water. However, data over the wider Reynolds number range covered here show that, even in the wall or inner layer, the mean burst period scales with outer rather than inner variables; and that the intervals are distributed according to the log normal law. It is suggested that these ‘bursts’ are to be identified with the ‘spottiness’ of Landau & Kolmogorov, and the high-frequency intermittency observed by Batchelor & Townsend. It is also concluded that the dynamics of the energy balance in a turbulent boundary layer can be understood only on the basis of a coupling between the inner and outer layers. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Axisymmetric particles in zero Reynolds number shear flow execute closed orbits. In this paper we consider the role of small Brownian couples in establishing a steady-state probability distribution for a particle being on any particular orbit. After presenting the basic equations, we derive an expression for the equilibrium distribution. This result is then used to calculate some bulk properties for a suspension of such particles, and these predicted properties are compared with available experimental observation. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The development of optical methods for the quantitative study of the fluctuating properties of turbulent flows can provide a supplement to conventional hot-body anemometry techniques. In particular, the study of high-speed flows by hot-wire or hot-film anemometry is often difficult owing to the presence of temperature and velocity fluctuations in the flow, thereby complicating the correct interpretation of measured signals. In addition, restrictions are placed on the application of such anemometers by their physical strength, frequency-response characteristics and the introduction of disturbances by the measuring probes into the flow. The operation of an optical detection system depends primarily on the mechanism by which the detected radiation intensity is modulated by the flow. Methods which have been used successfully include scattering or absorption of incident light by tracer constituents or particles, the absorption or emission of infra-red radiation by the flow and quantitative adaptations of the schlieren and interferometer systems which are sensitive to the flow density structure. All these systems detect a summation of signals from different parts of the flow and in consequence it is necessary to consider in detail the relation of the integrated signal to the local properties of the flow. This paper deals in particular with the application of the schlieren principle to an axisymmetric turbulent jet. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The gravitational instability of a horizontal fluid layer with a univariant phase transition is considered. It is found that the layer can be unstable even when the less dense phase lies above the dense phase and can be stable in the opposite case. Applications of the theory to convection with phase transitions in astrophysical and geophysical problems are briefly discussed. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: According to the Lighthill acoustic analogy, the sound induced by a region of turbulence is the same as that due to an equivalent distribution of quadrupole sources within the fluid. It is known that the presence of scattering bodies situated near such multipoles can convert some of their intense near field energy into the form of sound waves whose amplitude is far greater than that of the incident field. Calculations are here presented to determine the extent of this conversion, for hard and soft bodies of various shapes, making use of the reciprocal theorem to recast the problem into one of finding the field, near the obstacle, induced by an incident plane wave. If the obstacle is small compared with a wavelength, then its presence is equivalent to an additional dipole (or source) whose greater efficiency as a sound radiator implies that the familiar intensity law I ∝ U8, for far field intensity I against typical turbulence velocity U for an unbounded flow, is replaced by I ∝ U6 (or I ∝ U4) for a hard (or soft) body. For the situation where the scatterer is large compared with wavelength, the prototype problem of a wedge of exterior angle (p/q)π is shown to yield an intensity law I ∝ U4+2q/p for both hard and soft surfaces. This result is shown to hold for the more general ‘wedge-like’ surfaces, whose dimensions are large scale and whose edges may be smoothed out on a small scale, compared with wavelength. The method used involves the matching of an incompressible flow, on the fine scales typical of the edge geometry, to an outer flow determined by the large scale features of the surface. Favourable comparisons are made with previous results pertaining to the two-dimensional semi-infinite duct and to the half-plate of finite thickness. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The behaviour of a strong shock wave, which is initiated by a point explosion and driven continuously outward by an inner contact surface (or a piston), is studied as a problem of multiple time scales for an infinite shock strength,[formula omited], and a high shock-compression ratio, ρs/ρ∞ ∼ 2γ/(γ ⊟ 1) ≡ ε⊟1 〉 1. The asymptotic analyses are carried out for cases with planar and cylindrical symmetry in which the piston velocity is a step function of time. The solution shows that the transition from an explosion-controlled régime to that of a reattached shock layer is characterized by an oscillation with slowly-varying frequency and amplitude. In the interval of a scaled time 1 ≪ t ≪ ε⊟2/3(1+ν), the oscillation frequency is shown to be (1 + ν) (2π)⊟1t⊟½(1⊟ν) and the amplitude varies as t⊟¼(3+ν) matching the earlier results of Cheng et al. (1961). The approach to the large-time limit, ε1/(1+ν)t → ∞ is found to involve an oscillation with a much reduced frequency, ¼π(1+ν)ε⊟½t⊟1, and with an amplitude decaying more rapidly like ε⊟⅘t⊟½(4+3ν); this terminal behaviour agrees with the fundamental mode of a shock/acoustic-wave interaction. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The suggestion made by the authors in a previous paper (Hurle & Jakeman 1969) that the Soret effect could give rise to overstable solutions of the thermosolutal Rayleigh–Jeffreys problem is investigated theoretically and experimentally. Oscillatory instability is shown to occur in initially homogeneous layers of water-methanol mixtures when they are heated from below. This instability triggers a finite-amplitude steady mode. The magnitude and sign of the Soret coefficient was changed by varying the composition of the mixture; as predicted, overstable modes were observed when the sign of the coefficient was such as to produce a stabilizing contribution to the density gradient. The observed critical Rayleigh numbers and temporal frequencies are consistent with theory. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: A solution to the hypersonic small disturbance equations is obtained for a class of two-dimensional bodies supporting logarithmic shock waves by reducing the partial differential equation to an ordinary differential equation. The body shape is calculated and shown not to be logarithmic if ≠ 1, where is the ratio of the specific heats of the gas. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: In this paper, an attempt has been made to model the dynamics of ciliary propulsion through the concept of an ‘envelope’ covering the ends of the numerous cilia of the microscopic organism. This approximation may be made in the case when the cilia are close together, as can occur in the case of the symplectic metachronal wave (i.e. the wave travels in the same direction as the effective beat). For simplicity, a spherical model has been chosen, and the analysis which follows is a correction to Lighthill's (1952) paper on squirming motions of a nearly spherical organism. The velocity and efficiency compared to the work done in pushing an inert organism are obtained, and compared to that of a ciliated organism. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The flow in a two-dimensional laminar separation bubble is analyzed by means of finite-difference solutions to the Navier-Stokes equations for incompressible flow. The study was motivated by the need to analyze high-Reynolds-number flow fields having viscous regions in which the boundary-layer assumptions are questionable. The approach adopted in the present study is to analyze the flow in the immediate vicinity of the separation bubble using the Navier-Stokes equations. It is assumed that the resulting solutions can then be patched to the remainder of the flow field, which is analyzed using boundary-layer theory and inviscid-flow analysis. Some of the difficulties associated with patching the numerical solutions to the remainder of the flow field are discussed, and a suggestion for treating boundary conditions is made which would permit a separation bubble to be computed from the Navier-Stokes equations using boundary conditions from inviscid and boundary-layer solutions without accounting for interaction between individual flow regions. Numerical solutions are presented for separation bubbles having Reynolds numbers (based on momentum thickness) of the order of 50. In these numerical solutions, separation was found to occur without any evidence of the singular behaviour at separation found in solutions to the boundary-layer equations. The numerical solutions indicate that predictions of separation by boundary-layer theory are not reliable for this range of Reynolds number. The accuracy and validity of the numerical solutions are briefly examined. Included in this examination are comparisons between the Howarth solution of the boundary-layer equations for a linearly retarded freestream velocity and the corresponding numerical solutions of the Navier-Stokes equations for various Reynolds numbers. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The field radiated by an acoustic monopole in the presence of an infinite membrane, or plate, is studied, with emphasis on the case when fluid loading effects are small and when a free wave in the surface has supersonic phase speed relative to the fluid. Coupling between fluid and surface is then specified by a Mach angle θM and by a fluid loading parameter ε, with ε ≪ 1. Asymptotic expressions for the field are derived which are uniform in the observation angle θ, measured from the surface. Previous descriptions have suggested the formation of a strong two-dimensional beaming effect along the surface of the Mach cone θ = θM. Here it is shown that this effect is a spurious consequence of nonuniform asymptotics. A beam is indeed formed, and persists without attenuation or distortion to large distances k0R ∼ ε⊟2. However, the beam amplitude is small compared with that of the three-dimensional reflected field, while at distances k0R 〉 ε⊟2 only the reflected wave survives. Some interesting features of the reflexion coefficient and of the field near to the membrane are also discussed. In particular, it is shown that the pressure field generated by a subsonic surface wave is also confined to a conical zone, the transition across the generators of the cone being described by Fresnel functions of a familiar kind. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: This paper discusses a general theory of wave propagation through a random medium whose random inhomogeneities are confined to small deviations from the mean. The theory is initially worked out in detail for the propagation of transverse waves along an infinite stretched string whose density is a random function of position. The manner in which the mean wave profile is modified by scattering from the density inhomogeneities is discussed in great detail, with particular emphasis on physical interpretation. The general theory of wave propagation in arbitrary dispersive or non-dispersive media is then discussed, and it is shown how the theory may be extended to wave propagation problems involving scattering from rough boundaries. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The transient situation which follows when a plane normal shock wave is reflected (from a coplanar wall) into its own relaxation zone is examined theoretically. Approximate inner and outer solutions for the wall-pressure history are employed to establish the timewise variations in the thermodynamic state of the gas adjacent to the wall. Results for chemically relaxing O2 and vibrationally relaxing CO2 are compared with previous numerical solutions based on the method of characteristics, and the agreement is found to be excellent. The approximate technique is simple and requires only a minimum of computing time. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Further evidence of a universal maximum drag reduction asymptote is presented. In the elastic sublayer model, inferred therefrom, the mean velocity profile during drag reduction is approximated by three zones: the usual viscous sublayer, an elastic sublayer where the mixing-length constant is derived from the maximum drag reduction asymptote, and an outermost region with Newtonian mixinglength constant. Upon integration the model yields a friction factor relation, parametric in elastic sublayer thickness, which properly reproduces the known features of turbulent dilute polymer solution flow. The dependence of elastic sublayer thickness upon flow and polymeric parameters is inferred from experimental data revealing two hitherto unknown relationships: namely that on Prandtl co-ordinates, 1/f½ vs. log Re f½ the difference in slope between a polymer solution and solvent is proportional to the square root of molar concentration and to the three-halves power of backbone chain links in the macromolecule. The proportionality constant in the preceding relationship is approximately the same for several different polymer species of carbon-carbon or similar skeletal structure in various thin solvents; there is an indication that this constant further depends upon the product of solvent viscosity times the cube of the effective bond length per chain link of the polymer species. Some recent results regarding the onset of drag reduction are also summarized. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: An experiment with the pressure-driven flow down a long rotating channel is described. For zero rotation the flow is quasi-parabolic, laminar, and one-dimensional up the channel. With slight rotation Ω there is a weak double-vortex secondary circulation aligned with the channel. At intermediate Ω there exists an instability in the form of logitudinal rolls of non-dimensional wavenumber 5. The instability disappears at high rotation rates. The general stability problem for a rotating zonal flow [formula omited] is considered theoretically. For perturbations independent of the co-ordinate in the direction of the flow, the problem is exactly analogous to the stability problem of a temperature-stratified fluid where the stratification [formula omited] corresponds to the quantity[formula omited] This analogy extends to much more general mean fields (e.g. non-linear or time dependent) than does the oft-quoted analogy between thermal convection and cylindrical Couette flow. The instability theory is in qualitative agreement with the experiment. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The axial and radial gradients of the tangential velocity distribution are calculated from prescribed secondary flow functions on the basis of a zero-order approximation to the momentum equations developed by Lewellen. It is shown that secondary flow functions may be devised which meet pertinent physical requirements and which at the same time lead to realistic tangential velocity gradients.The total-temperature distribution in both the axial and radial directions is calculated from such secondary flow functions and corresponding tangential velocity results on the basis of an approximate turbulent energy equation. The method employed for the solution of this equation stresses the equivalence of the vortex tube to counter-current systems with transverse diffusion such as distillation columns and heat exchangers.An availability function is derived that permits the evaluation of vortex tube performance on the basis of velocity data.Turbulent diffusivities resulting from the quantitative use of the tangential velocity approximation are shown to agree with those derived from the total-temperature calculations.

Beschreibung: This paper is the report of an experimental study of heat transfer from fine horizontal wires of various lengths to fluids having various Prandtl numbers. Transport characteristics were determined for the full spectrum of processes from natural, through mixed, to forced convection. The experimental results are compared with those of past analysis and with correlations. The very high level of accuracy and reproducibility of the data makes possible the detection of very small effects. The influence of cylinder length on transport is shown for both of the asymptotic régimes of natural and of forced convection. Transport in the mixed-convection régime is determined and limits of that régime are estimated.

Beschreibung: Two problems representative of the gravity flow of granular materials are considered in the context of a theory presented by Goodman (1970). The problems consist of steady fully-developed flow of a granular material down an inclined plane and between vertical parallel plates. It is shown that the dynamical behaviour of these materials is quite different from that of a viscous fluid. For the inclined flow problem, the normal stresses are not only unequal but vary non-linearly with depth. Also the maximum value of the mass flux distribution does not necessarily occur at the upper surface. For the vertical channel-flow problem, the material behaves somewhat like a Bingham fluid in that a plug region exists in the central part of the channel. The interesting feature of this problem is that the concentration of material volume in the shearing region outside the plug may either increase or decrease from the plug to the channel wall, depending on the boundary conditions. Experimental evidence for these phenomena in real granular materials is cited. The results of this investigation suggest that the gravity flow of granular materials is essentially governed by two factors-a material characteristic length, which is possibly related to the grain size, and the externally imposed constraints such as the gravity field or the pressure exerted upon the granular material from the confining plates. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Drag reduction caused by dilute, distilled water solutions of four polyethylene-oxides and one polyacrylamide, molecular weights respectively 0·1 × 106 to 8 × 106 and 13 × 106, was studied experimentally in one smooth and three sand-roughened pipes, relative roughnesses (R/k) 15, 23, 35, all of about 0·34, in inside diameter. The onset of drag reduction in the rough pipes occurred at the same wall shear stress as in smooth; the onset wall shear stress was essentially independent of polymer concentration, varied inversely as the square of polymer radius of gyration and was unaffected by the flow régime, hydraulically smooth, transitional or fully rough, during which onset occurred. Following onset a flow régime was observed wherein the fractional slip, i.e. fractional increase in mean velocity relative to solvent at a given friction velocity, obtained with a given polymer solution in a rough pipe was the same as the fractional slip in the smooth pipe despite marked differences in the respective rough and smooth friction factors. This ‘effectively smooth’ régime prevailed for values of non-dimensional roughness k+* 〈 k+ 〈 k+es from onset, k+*, to an upper limit given by k+es ∼ 50 for all of the present experiments. For k+ 〈 k+es, the fractional slip in the rough pipes was always less than that corresponding to smooth and was a function of relative roughness as well as flow and polymeric parameters. The maximum drag reduction possible in the rough pipes was limited by an asymptote which was independent of polymeric parameters. Under asymptotic conditions, friction factors in all the rough pipes identically obeyed the smooth pipe friction factor relation for k+ 〈 12; the onset of roughness at k+ ∼ 12 indicated that the maximum viscous sublayer thickness attained during drag reduction was approximately 2½ times Newtonian. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: When a layer of particles moves through a viscous liquid it experiences forces which tend to disrupt the layer into clusters of particles separated by open channels. A theoretical description of this process is presented and a viscous instability is predicted. The spatial growth of the instability is approximated by eγz, where [ gamma = {extstylefrac{3}{2}} a/d^2, ] where a is the particle radius and d is the average distance between particles. This result implies that any initial irregularity in a uniform particle distribution will be amplified by viscous forces alone. Significant amplification will occur when the particle has drifted a small multiple of the separation distance, if this separation is not much greater than the particle diameter. Thus, any initially uniform particle layer will form clusters as it drifts through a viscous fluid. The distance in which this clustering occurs will be unaffected by changes in the particle velocity, as long as the Reynolds number remains small. The preferred form of irregularity will consist of small clusters separated by individual particles which trail some distance behind. Experimental verification of these conclusions is presented.

Beschreibung: The investigation of laminar free convective plumes in an otherwise stationary environment has formed the basis of numerous investigations, initiated by Zeldovich (1937). For the non-rotating environment alone the authors have been able to locate twenty-nine papers: many of these repeat work previously undertaken. There are, however, two cases of some technological significance which have so far not been considered: (i) the plume in an otherwise quiescent environment for a fluid of very large Prandtl number, of importance in the heating of reservoirs of viscous fluid such as fuel oil; and (ii) the case of vanishingly small Prandtl number, of application to liquid metal-cooled nuclear reactors. Both of these cases have some theoretical interest, as will be shown. Their analysis leads to singular asymptotic perturbations and hence to matched-expansions techniques. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Wave-induced oscillations in harbours of constant depth but arbitrary shape in the horizontal plane connected to the open-sea are investigated both theoretically and experimentally. A theory termed the ‘arbitrary-shape harbour’ theory is developed. The solution of the Helmholtz equation is formulated as an integral equation which is then approximated by a matrix equation. The final solution is obtained by equating, at the harbour entrance, the wave amplitudes and their normal derivatives obtained from the solutions for the regions outside and inside the harbour. Special solutions using the method of separation of variables for the region inside circular and rectangular harbours are also obtained. Experiments were conducted to verify the theories. Four specific harbours were investigated: two circular harbours with 10° and 60° openings respectively, a rectangular harbour, and a model of the East and West Basins of Long Beach Harbour, California. In each case, the theoretical results agreed well with the experimental data. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: An analytical solution is developed to describe the steady, closed streamline velocity field within a cylindrical cavity with a uniformly translating wall at low Reynolds numbers. The solution has application for the case of two-phase flow in a tube where regions of fluid are segmented by a moving train of bubbles or plugs, such as in the pulmonary and peripheral capillaries of the body where segments of plasma are trapped between red blood cells. The mathematical approach presented in this study can in principle be useful in the analysis of a wide class of closed-streamline creeping-flow problems. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: This paper describes some experiments in swirling flows in a diverging cylindrical tube in which various types of vortex breakdowns were observed. In one set of experiments, the position of the breakdown, axial component of the velocity of the vortex core, swirl angle distribution ahead of the breakdown, and the pressure distribution along the tube were determined for various flow rates and for various values of circulation imparted to the fluid (water). Basically, three types of vortex breakdown were observed, viz. mild (double helix) breakdown, spiral breakdown (followed by turbulent mixing), and axisymmetric breakdown (followed by a thicker vortex core, then a spiral breakdown, and finally by turbulent mixing). The type and the location of the stationary breakdowns were found to be dependent, for the particular vortex tube used, upon the Reynolds and circulation numbers of the flow. In a spiral breakdown, the vortex core filament maintained the same sense of rotation as the upstream fluid elements. In an axisymmetric breakdown, the bubble included an inclined vortex-ring whose axis gyrated about the axis of the tube. In a second set of experiments, the response of the abrupt structural change along the axis of flow to gradual and abrupt changes in the upstream and downstream flow conditions was examined. The axisymmetric breakdown responded in a manner analogous to the hydraulic jump in open-channel flow before if reached a new stationary position along the axis of the tube. The observations reported and the evidence presented herein revealed that the axisymmetric breakdown is a finite transition between two sequent states of flow as proposed by Benjamin (1962, 1965, 1967) on theoretical grounds. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: The hydrodynamic theory of the resonant cylinder (Gans 1970) is extended to include the effects of a magnetic field parallel to the rotation axis. The linear response is modified by a change in boundary-layer suction and a change in the resonant length. These effects are of equal importance. The theory is valid for small container conductivity and for amplitudes such that the cube of the amplitude is less than the dimensionless precession rate. The importance of container conductivity is assessed. The free modes of the system are given in an appendix. These modes move both east and west.Experimental apparatus capable of producing magnetic Reynolds numbers of the order of 20 for indefinite lengths of time is described. The apparatus was used to assess the linear theory, though not designed for this purpose. Experiments beyond the range of linear theory are described. The results show finite amplitude effects similar to those previously observed in precessing spheroids in the absence of magnetic effects (Malkus 1968). Additional structure attributable to magnetic effects is observed.

Beschreibung: The flow in a pipe of circular cross-section which is coiled in a circle is studied, the pressure gradient along the pipe varying sinusoidally in time with frequency ω. The radius of the pipeais assumed small in relation to the radius of curvature of its axisR. Of special interest is the secondary flow generated by centrifugal effects in the plane of the cross-section of the pipe, and an asymptotic theory is developed for small values of the parameter β = (2ν/ωa2)½, where ν is the kinematic viscosity of the fluid. The secondary flow is found to be governed by a Reynolds number$R_s = overline{W}^2a/R omega u$, where$overline{W}$is a typical velocity along the axis of the pipe, and asymptotic theories are developed for both small and large values of this parameter. For sufficiently small values of β it is found that the secondary flow in the interior of the pipe is in the opposite sense to that predicted for a steady pressure gradient, and this is verified qualitatively by an experiment described at the end of the paper.

Beschreibung: 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.

Beschreibung: 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.

Beschreibung: 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.

Beschreibung: 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.

Beschreibung: 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.

Beschreibung: 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.

Beschreibung: 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.

Beschreibung: The present study is to investigate the structure of space-time correlations of bursting motions, such as ejections and sweeps in an open-channel flow, by a new conditional sampling analysis of the instantaneous velocity and Reynolds-stress signals measured simultaneously by two dual-sensor hot-film probes. One probe was fixed near the edge of the buffer layer, while the other probe was moved in the streamwise, vertical and spanwise directions. The sorted instantaneous Reynolds-stress signals obtained from the fixed probe were used as a detecting information of the occurrences of ejection or sweep events. The streamwise and vertical spatial characteristics of the ejection-sweep motions, and their convection process are investigated in detail. Also, the spanwise spatial properties of the high- and low-speed streaks in the bursting motions are examined experimentally by the present conditional sampling method. Next, a qualitative model is proposed which attempts to explain the space-time structures of the bursting phenomenon, on the basis of the above anemometry information and other visual observations. © 1981, Cambridge University Press. All rights reserved.

Beschreibung: In a high-speed subsonic jet impinging on a flat plate, the surface pressure fluctuations have a broad spectrum due to the turbulent nature of the high-Reynolds-number jet. However, these pressure fluctuations dramatically change their pattern into almost periodic waves, if the plate is placed close to the nozzle (x0/d 〈 7.5). In the present study extensive measurements of the near-field pressure provide solid support for the hypothesis that a feedback mechanism is responsible for the sudden change observed in the pressure fluctuations at the onset of resonance. The feedback loop consists of two elements: the downstream-con vected coherent structures and upstream-propagating pressure waves generated by the impingement of the coherent structures on the plate. The upstream-propagating waves and the coherent structures are phase-locked at the nozzle exit. The upstream-propagating waves excite the thin shear layer near the nozzle lip and produce periodic coherent structures. The period is determined by the convection speed of the coherent structures, the speed of the upstream-propagating waves as well as the distance between the nozzle and the plate. An instability process, herein referred to as the collective interaction’, was found to be critical in closing the feedback loop near the nozzle lip. © 1981, Cambridge University Press. All rights reserved.

Beschreibung: We examine the Galerkin (including single-mode and Lorenz-type) equations for convection in a sphere to determine which physical processes are neglected when the equations of motion are truncated too severely. We test our conclusions by calculating solutions to the equations of motion for different values of the Rayleigh number and for different values of the limit of the horizontal spatial resolution. We show how the gross features of the flow such as the mean temperature gradient, central temperature, boundary-layer thickness, kinetic energy and temperature variance spectra, and energy production rates are affected by truncation in the horizontal direction. We find that the transitions from steady-state to periodic, and then to aperiodic convection depend not only on Rayleigh number but also very strongly on the horizontal resolution of the calculation. All of our models are well resolved in the vertical direction, so the transitions do not appear to be due to poorly resolved boundary layers. One of the effects of truncation is to enhance the high-wavenumber end of the kinetic energy and thermal variance spectra. Our numerical examples indicate that, as long as the kinetic energy spectrum decreases with wavenumber, a truncation gives a qualitatively correct solution. © 1981, Cambridge University Press. All rights reserved.

Beschreibung: Experiments are described which show that large-scale coherent structures exist in the wakes behind three-dimensional blunt bodies. Using a ‘flying hot-wire’ apparatus, the vortex-shedding cycle has been described by phase-averaged vector fields allowing an animation of large-scale motions to be produced. It is found that the large-scale structures retain their identity for long streamwise distances and contribute significantly to the Reynolds stress. Using critical-point theory as described recently by Perry, Lim & Chong (1980), the effect of phase ‘jitter’ on ‘washout’ has been analysed. Furthermore, it is found from critical-point theory that the large-scale motions possess the same geometrical features as the low-Reynolds number (unsteady laminar flow) wake results of Perry & Lim (1978) and Perry et al. (1980). © 1981, Cambridge University Press. All rights reserved.

Beschreibung: A method is presented for integrating numerically the equations of motion for a compressible free shear layer developing from a boundary-layer profile of arbitrary shape. Sutherland's law is used to determine the coefficient of viscosity and the Prandtl number is taken as 0·72. Calculated results are reported for free-stream Mach numbers ranging from 0 to 10 and for stagnation-enthalpy ratios ranging from 0 to 5·0. The effects of varying the initial boundary-layer profile and of a discontinuity in temperature at the origin are also studied. The results include graphs showing the development of dividing-streamline velocity, of local Nusselt number, and of dividing-streamline location. © 1971, Cambridge University Press. All rights reserved.

Beschreibung: Numerical solutions are presented for steady two-dimensional symmetric flow past a parabolic cylinder in a uniform stream parallel to its axis. The solutions cover the range R = 0·25 to ∞, where R is the Reynolds number based on the nose radius of the cylinder. For large R, the calculated skin friction near the nose of the cylinder is compared with known theoretical results obtained from second-order boundary-layer theory. Some discrepancy is found to exist between the present calculations and the second-order theory. For small R, it is possible to obtain a reasonably consistent check with a recent theoretical prediction for the limit of the skin friction near the nose of the cylinder as R → 0. © 1971, Cambridge University Press. All rights reserved.