bluuebunny/crossref_metadata_2025 huggingface dataset:
The critical layer for internal gravity waves in a shear flow
Authors: Booker John R. , Bretherton Francis P.
Published: March 2006

Internal gravity waves of small amplitude propagate in a Boussinesq inviscid, adiabatic liquid in which the mean horizontal velocity U(z) depends on height z only. If the Richardson number R is everywhere larger than 1/4, the waves are attenuated by a factor $\exp\{-2\pi(R - \frac{1}{4})^{\frac{1}{2}}\}$ as they pass through a critical level at which U is equal to the horizontal phase speed, and momentum is transferred to the mean flow there. This effect is considered in relation to lee waves in the airflow over a mountain, and in relation to transient localized disturbances. It is significant in considering the propagation of gravity waves from the troposphere to the ionosphere, and possibly in transferring horizontal momentum into the deep ocean without substantial mixing.
Some experiments on equilibrium turbulent boundary layers in favourable pressure gradients
Authors: Herring H. J. , Norbury J. F.
Published: March 2006

A wind tunnel in which an arbitrary negative pressure gradient could be developed has been built for boundary-layer studies. The effects of selected pressure gradients on boundary layers grown on one of the walls of the tunnel were studied. It was possible to obtain equilibrium boundary layers of the type first suggested by Clauser: that is, layers whose non-dimensional velocity-defect distribution is invariant along the direction of flow. The velocity-defect distributions for two such boundary layers were established, corresponding to values of Clauser's dimensionless pressure-gradient parameter β of −0·35 and −0·53. These velocity profiles are compared with the profiles predicted theoretically by Mellor & Gibson (1966). The agreement between the two is very good.
Linear theory of rotating stratified fluid motions
Authors: Barcilon V. , Pedlosky J.
Published: March 2006

A linear theory for steady motions in a rotating stratified fluid is presented, valid under the assumption that ε < E, where ε and E are respectively the Rossby and Ekman numbers. The fact that the stable stratification inhibits vertical motions has important consequences and many features of the dynamics of homogeneous rotating fluids are no longer present. For instance, in addition to the absence of the Taylor-Proudman constraint, it is found that Ekman layer suction no longer controls the interior dynamics. In fact, the Ekman layers themselves are frequently absent. Furthermore, the vertical Stewartson boundary layers are replaced by a new kind of boundary layer whose structure is characteristic of rotating stratified fluids. The interior dynamics are found to be controlled by dissipative processes.
Detached shear layers in a rotating fluid
Authors: Hide R. , Titman C. W.
Published: March 2006

The occurrence of detached shear layers should, according to straightforward theoretical arguments, often characterize hydrodynamical motions in a rapidly rotating fluid. Such layers have been produced and studied in a very simple system, namely a homogeneous liquid of kinematical viscosity v filling an upright, rigid, cylindrical container mounted coaxially on a turn-table rotating at Ω0 rad/s about a vertical axis, and stirred by rotating about the same axis at Ω1 rad/s a disk of radius a cm and thickness b’ cm immersed in the liquid with its plane faces parallel to the top and bottom end walls of the container. By varying Ω0, Ω1 and a, ranges of Rossby number, the modulus of ε ≡ (Ω1 + Ω0)/½ (Ω1 + Ω0), from 0·01 to 0·3, and Ekman number, E ≡ 2v/a2(Ω1 + Ω0), from 10−5 to 5 × 10−4 were attained. Although the apparatus was axisymmetric, only when |ε| did not exceed a certain critical value, |εT|, was the flow characterized by the same property of symmetry about the axis of rotation. Otherwise, when |ε| > |εT|, non-axisymmetric flow occurred, having the form in planes perpendicular to the axis of rotation of a regular pattern of waves, M in number, when ε was positive, and of a blunt ellipse when ε was negative.

The axial flow in the axisymmetric detached shear layer, and the uniform rate of drift of the wave pattern characterizing the non-axisymmetric flow when ε is positive, depend in relatively simple ways on ε and E. The dependence of|εT| on E can be expressed by the empirical relationship |εT| = AEn, where A = 16·8 ± 2·2 and n = 0·568 ± 0·013 (= (4/7) × (1·000 − (0·005 ± 0·023))!), standard errors, 25 determinations. M does not depend strongly on E but generally decreases with increasing ε.
Balance of turbulence energy in the region of jet-flow establishment
Authors: Sami Sedat
Published: March 2006

To complete a previous investigation reported earlier and carried out in an air jet issuing from a 1·0ft. diameter nozzle into still air, additional measurements were made of one-point triple-velocity and two-point double-velocity correlations and the spatial mean gradients of the axial turbulent velocities. The one-point pressure/axial-velocity correlation term was evaluated by using piezo-electric ceramic elements in conjunction with total- and static-pressure probes. For the measurement of spatial mean gradients of the axial-velocity fluctuation, special probes with two parallel wires were constructed. The axial scale of turbulence was found to be almost twice as large as the other two scales. The tangential scale was the smallest. All terms of the integral and differential forms of the turbulence energy equation were evaluated throughout the region and the results are given in dimensionless form.
The dynamics of turbulence near a wall according to a linear model
Authors: Schubert Gerald , Corcos G. M.
Published: March 2006

The dynamics of turbulent velocity fluctuations in and somewhat outside the viscous sublayer are examined by linearizing the equations of motion around the known mean velocity profile. The rest of the boundary layer is assumed to drive the motion in the layer by means of a fluctuating pressure which is independent of distance from the wall. The equations, which are boundary-layer approximations to the Orr-Sommerfeld equations, are thus treated as a non-homogeneous system and solved by convergent power series. The solutions which exhibit the strong role of viscosity throughout the layer considered provide a model endowed with many of the known features of turbulence near a wall. In particular, the phase angle between streamwise and normal fluctuations is found to be in plausible agreement with experiments. An important role is ascribed by the solutions to the displacement of the mean velocity by the normal fluctuations. The impedance of the layer is found to be anisotropic in that it favours fluctuations with a much larger scale in the streamwise than in the spanwise direction. For such disturbances, the ratio of turbulent intensity to the intensity of the pressure fluctuations approximates the experimental ratio. According to the solutions it is primarily the spanwise component of the pressure gradient which is responsible for the intense level of turbulence very near the wall. The model apparently underestimates the amplitude ratio of normal to streamwise components of the velocity.
The motion of a fluid due to a moving source of heat at the boundary
Authors: Davey A.
Published: March 2006

Experiments performed by Fultz et al. (1959) and Stern (1959) indicate that if a flame is rotated around the bottom exterior of a pan or of a circular cylindrical annulus containing water then the fluid acquires a net vertical component of angular momentum in a direction opposite to that of the flame's motion. Stern gave an analysis showing how fluid, bounded by two horizontal plates subjected to small two-dimensional sinusoidal travelling temperature perturbations, could acquire and maintain such momentum. An assumption in Stern's analysis was that the density variations were constant between the plates. Here we shall examine the importance of the relaxation of this assumption. Also, another model, in which the upper surface of the fluid is free, is analysed, because this model conforms more closely with the experiments of Fultz et al. and Stern.

For the problem of flow between two plates we shall show that there is a net flux of momentum in the direction opposite to the motion of the thermal field for all frequencies thereof. The same result holds in the case with a free surface when conditions are comparable with those in the experiments.

In both problems the phase lag of the thermal expansion wave with height produces a skewed thermal field. This field is the most important mechanism in producing velocity correlations. It is due to these correlations and the associated Reynolds stress that the fluid acquires its momentum.
Formation of vortex rings from falling drops
Authors: Chapman David S. , Critchlow P. R.
Published: March 2006

A study of the formation of vortex rings when a liquid drop falls into a stationary bath of the same liquid has been made. The investigation covered liquids with a wide range in surface tensions, densities and viscosities. The results confirm the reported existence of optimum dropping height from which the drop develops into a superior vortex ring. The optimum heights are analysed, by a photographic study, in terms of the liquid drop oscillation. It is found that vortex rings are formed best if the drop is spherical and changing from an oblate to a prolate spheroid at the moment of contact with the bath. A Reynolds number has been determined for vortex rings produced at optimum dropping heights; these numbers are approximately 1000. A possible mechanism for the ring formation is suggested.
The stability of a stratified fluid
Authors: Hinwood J. B.
Published: March 2006

For the flow of a stably-stratified fluid in the inlet region of a rectangular duct, it is shown experimentally that the upper and lower critical Reynolds numbers are functions of the interfacial Froude number F, and that if F is large they are lower than for a homogeneous flow. In stratified flows the disturbances leading to turbulent flow sometimes arise at the interface and lead to interfacial waves, whose wavelength at breaking is equal to the conduit depth.
Thermally induced boundary-layer flows in a rotating environment
Authors: Riley N.
Published: March 2006

We consider, in this paper, boundary-layer flows which are induced when an isothermal rigid-body rotation is disturbed by heating the fluid. The basic rigid-body rotation is sustained by one or more rotating planes at which the temperature differences are initiated. Conditions of both uniform and unsteady heating are discussed.
Visualization of an incompressible wake with base bleed
Authors: Wood C. J.
Published: March 2006

The wake behind a two-dimensional bluff body has been photographed in water in order to investigate the modifying effect of base bleed. With this technique it has been possible to measure velocities at chosen points and from these make simultaneous estimates of the base pressure and of the strength of individual vortices in the wake. The results suggest that the observed decay in the vortex structure of the wake is related to variations in the conditions of mixing between the base fluid and the external stream.
Two-dimensional incompressible fluid jet penetration
Authors: Hopkins D. F. , Robertson J. M.
Published: March 2006

The principal characteristics of jet penetration are the appearance of free stream-lines at the sides of the jet and of a dividing streamline, which separates the jet and penetrated fluid. Kinematic analysis of such flow via free-streamline theory and the notched hodograph is developed with one unspecified parameter, the ratio of jet to counterstream velocity in the steady flow case. The kinetics of the problem, appearing when the jet and penetrated (or counterstream) fluid differ in density, is simply related to the kinematic solution via the square root of the density ratio. Experiments, both steady state and transient with several liquids, are presented which generally verify the theory. The experiments also yield information on the magnitude of the parameter and indicate its variation with the density ratio.
The heated laminar vertical jet
Authors: Brand R. S. , Lahey F. J.
Published: March 2006

The boundary-layer equations for the steady laminar flow of a vertical jet, including a buoyancy term caused by temperature differences, are solved by similarity methods. Two-dimensional and axisymmetric jets are treated. Exact solutions in closed form are found for certain values of the Prandtl number, and the velocity and temperature distribution for other Prandtl numbers are found by numerical integration.
The effect of heating rate on the stability of stationary fluids
Authors: Currie I. G.
Published: March 2006

A horizontal fluid layer whose lower surface temperature is made to vary with time is considered. The stability analysis for this situation shows that the criterion for the onset of instability in a fluid layer which is being heated from below, depends on both the method and the rate of heating. For a fluid layer with two rigid boundaries, the minimum Rayleigh number corresponding to the onset of instability is found to be 1340. For slower heating rates the critical Rayleigh number increases to a maximum value of 1707·8, while for faster heating rates the critical Rayleigh number increases without limit.

Two specific types of heating are investigated in detail, constant flux heating and linearly varying surface temperature. These cases correspond closely to situations for which published data exist. The results are in good qualitative agreement.
Magnetohydrodynamic flow at a rear stagnation point
Authors: Leibovich S.
Published: March 2006

An exact rear stagnation point solution is sought for a viscous, incompressible conducting fluid in the presence of a magnetic field. It is found that a steady solution exists only if [Nscr ] ≥ 2, where [Nscr ] = σB2/ρα is the interaction parameter of the flow based upon the normal component of the magnetic field at the wall. Here α is a positive rate of strain, which, for a finite body with length l and velocity U0, is of the order of U0/l.

The steady solution is found, and from its existence it is inferred that separation of the viscous boundary layer does not begin at the rear stagnation point when [Nscr ] ≥ 2, and that such separation can be prevented. This supports theoretical work by Moreau (1964) and experimental work of Tsinober, Shtern & Shcherbinin (1963).

When [Nscr ] < 2, the flow is necessarily unsteady, and in this case an asymptotic analysis (as t → ∞) similar to that of Proudman & Johnson (1962) is undertaken. For [Nscr ] < 1, the magnetic and non-magnetic flows are qualitatively alike, in that there is a growing inviscid region dominated by eddies, and an ultimately steady layer at the wall representing a viscous forward stagnation point flow.

For 1 < [Nscr ] < 2, the inviscid region again grows with time, but no eddies appear. It is thus suggested that for this range of [Nscr ] separation occurs without reversed flow.
On finite amplitude oscillations in laminar mixing layers
Authors: Stuart J. T.
Published: March 2006

In the first part of the paper, a mixing layer of tanhyform is considered, and twodimensional solutions of the non-linear inviscid equations are found representing periodic perturbations from the neutral wave of linearized stability theory. To second order in amplitude the solutions are equivalent to the equilibrium state calculated by Schade (1964), who studied the development of perturbations in time and found an evolution towards that equilibrium state. The present calculation, however, is taken to fourth-order in amplitude. It is noted that the solutions presented in this paper are regular, even though viscosity is ignored; and the relationships to the singular (if inviscid) time-dependent solutions of Schade are explained. Such regular, inviscid solutions have been found only for odd velocity profiles, such as tanhy.

Although the details of the velocity distributions are not of the form observed experimentally, it is shown that the amplitude ratios of fundamental and first harmonic, for a given absolute amplitude, are comparable with those observed.

In part 2 some exact non-linear solutions are presented of the inviscid, incompressible equations of fluid flow in two or three spatial dimensions. They illustrate the flows of part 1, since they are periodic in one co-ordinate (x), have a shear in another (y) and are independent of the third. Included, as two-dimensional cases, are (i) the tanh y velocity distribution for a flow wholly in the x-direction, (ii) the well-known solution for the flow due to a set of point vortices equi-spaced on the axis, and (iii) an example of linearized hydrodynamic (Orr-Sommerfeld) stability theory. The flows may involve concentrations of vorticity. In three-dimensional cases the z component of velocity is even iny, whereas the x component is odd. Consequently, the class of flows represents, in general, small or large periodic perturbations from a skewed shear layer. Time-dependent solutions, representing waves travelling in the x direction may be obtained by translation of axes.
Equations governing steady three-dimensional large-amplitude motion of a stratified fluid
Authors: Yih Chia-Shun
Published: March 2006

The exact equations governing three-dimensional motion of an inviscid non-diffusive incompressible fluid stratified in density or of an inviscid non-diffusive gas stratified in entropy are given and briefly discussed.
Ship waves in a stratified ocean
Authors: Crapper G. D.
Published: March 2006

This paper considers the effect of a pressure point moving over the surface of a fluid which is composed of two layers of different densities, the lower layer being of infinite depth. The method is the same as that used by Crapper (1964) for the same problem with a uniform fluid. The resulting waves are in two sets, the first being the normal surface wave for the uniform ocean, with a slight change in amplitude, and the second an internal wave. The crests of the internal wave have a pattern similar to that of the surface wave for a shallow uniform fluid.
On the steady motions produced by a stable stratification in a rapidly rotating fluid
Authors: Barcilon V. , Pedlosky J.
Published: March 2006

The equilibrium state of a rapidly rotating fluid, heated uniformly from above and cooled uniformly from below while contained in a cylinder with insulated side-walls is studied.

The circulations which are produced by the resulting stratification are studied over a wide range of parameters and it is shown that many of the features of the linear theory of rotating stratified fluid flows found in earlier studies reappear in this non-linear problem.

These include the gradual disappearance of Ekman layer suction and O(1) Ekman layers as the stratification increases, and the determination of the interior flow by the side-wall boundary layers in conjunction with the Ekman layers.

It is suggested therefore, that studies of rotating stratified flows which are unbounded laterally may frequently be defective and lead to solutions which are not the limit of any physically realizable experiment.
A non-linear theory of turbulence onset in a shear flow
Authors: Hasen E. M.
Published: March 2006

Turbulence onset is considered in viscous incompressible flows. Development of fluctuations in an infinite flow with a constant gradient of mean velocity ∂U1/∂x2 = const., U2 = U3 = 0 is rigorously treated.

It is shown that two-dimensional eddy fluctuations, with infinitesimal initial amplitude A and scale of initial eddies l, increase in this flow so that the maximum ratio max ε(t)/ε(0) of their energy at the moment t to the initial energy exceeds any prescribed value as the Reynolds number R = (∂U1/∂x2) l2/v increases. The analysis of the non-linear equations obtained in the paper which describe development of fluctuations with a finite amplitude leads to the conclusion that there exists a’ stability barrier’ Ã (R) for the initial amplitude of eddy fluctuations. If A < Ã (R), then fluctuations decay as t → ∞, and if A > Ã(R) the energy of fluctuations does not decay. As R → ∞, Ã(R) → 0 according to the inequalities
\[
l(\partial U_1/\partial x_2)K_2/R^{\frac{2}{3}}\leqslant \tilde{A}(R)\leqslant l(\partial U_1/\partial x_2)\,K_1/R^{\frac{2}{3}}.
\]
It is shown that the non-linear mechanism of preventing turbulence from decay involves generation of large-scale turbulent oscillations which then transmit energy to small-scale motions.

The described mechanism of turbulence onset from small eddies in shear flows appears to be of universal character. It is interesting that several qualitative characteristics of turbulence observed in various shear flows can be rigorously deduced even in a model where disturbances remain two-dimensional.
Shock curvature and gradients at the tip of pointed axisymmetric bodies in non-equilibrium flow
Authors: Sedney Raymond , Gerber Nathan
Published: March 2006

The shock curvature and flow variable gradients at the tip of a pointed body caused by non-equilibrium effects are considered. Co-ordinates introduced by Chester (1956) are used since they offer a convenient way of treating the boundary conditions. The desired functions are obtained by solving numerically a system of linear ordinary differential equations. These equations have a singularity; the nature of the singularity is found analytically, and its numerical treatment is discussed. The specific non-equilibrium effect considered is vibrational relaxation in a pure diatomic gas. Representative results are given for flow of N2over a cone for a comprehensive range of Mach number and cone angle. There is a point analogous to the Crocco point. The exact results are compared with predictions from (i) a hypersonic, small disturbance theory; (ii) the application of an integral method; (iii) characteristic calculations. In an appendix, a comparative discussion is given of results for frozen flow over ogival bodies.
Convection in horizontal layers with internal heat generation. Experiments
Authors: Tritton D. J. , Zarraga M. N.
Published: March 2006

A largely qualitative experimental investigation has been made of cellular convection patterns produced by the instability of a horizontal layer of fluid heated uniformly throughout its body and cooled from above. The fluid was aqueous zinc sulphate solution, and the heating was produced by an electrolytic current. The flow was visualized and photographed using polystyrene beads which, because of a differential expansion, came out of suspension in regions where the fluid was hotter or colder than average.

The development of the cellular patterns as the Rayleigh number (a modified form) was varied was in many respects similar to that for Bénard convection. There were, however, two striking differences. First, the fluid descended in the centres of the cells and ascended at the peripheries; this was probably associated with the asymmetry between the heating and cooling. Secondly, and more surprisingly, the horizontal scale of the convection patterns was unusually large (except at the lowest Rayleigh numbers); the distance between rising and falling currents was observed to reach typically five times the depth of the layer.

This last observation may be relevant to theories of convection within the Earth's mantle, to mesoscale convection in the atmosphere, and to patterns formed in ice.
Convection in horizontal layers with internal heat generation. Theory
Authors: Roberts P. H.
Published: March 2006

A theoretical study has been made of an experiment by Tritton & Zarraga (1967) in which eonvective motions were generated in a horizontal layer of water (cooled from above) by the application of uniform heating. The marginal stability problem for such a layer is solved, and a critical Rayleigh number of 2772 is obtained, at which patterns of wave-number 2·63 times the reciprocal depth of the layer are marginally stable.

The remainder of the paper is devoted to the finite amplitude convection which ensues when the Rayleigh number, R, exceeds 2772. The theory is approximate, the basic simplification being that, to an adequate approximation, Fourier decompositions of the convective motions in the horizontal (x, y) directions can be represented by their dominant (planform) terms alone. A discussion is given of this hypothesis, with illustrations drawn from the (better studied) Bénard situation of convection in a layer heated below, cooled from above, and containing no heat sources. The hypothesis is then used to obtain ‘mean-field equations’ for the convection. These admit solutions of at least three distinct forms: rolls, hexagons with upward flow at their centres, and hexagons with downward flow at their centres. Using the hypothesis again, the stability of these three solutions is examined. It is shown that, for all R, a (neutrally) stable form of convection exists in the form of rolls. The wave-number of this pattern increases gradually with R. This solution is, in all respects, independent of Prandtl number. It is found, numerically, that the hexagons with upward motions in their centres are unstable, but that the hexagons with downward motions at their centres are completely stable, provided R exceeds a critical value (which depends on Prandtl number, P, and which for water is about 3Rc), and provided the wave-number of the pattern lies within certain limits dependent on R and P.
Extremum principles for slow viscous flows with applications to suspensions
Authors: Keller Joseph B. , Rubenfeld Lester A. , Molyneux John E.
Published: March 2006

Helmholtz stated and Korteweg proved that of all divergenceless velocity fields in a domain, with prescribed values on the boundary, the solution of the Stokes equation minimizes the rate of viscous energy dissipation. Hill & Power and also Kearsley proved the corresponding reciprocal maximum principle involving the stress tensor. We prove generalizations of both these principles to the flow of a liquid containing one or more solid bodies and drops of another liquid. The essential point in doing this is to take account of the motion of the solids or drops, which must be determined along with the flow. We illustrate the use of these principles by deducing several consequences from them. In particular we obtain upper and lower bounds on the effective coefficient of viscosity and a lower bound on the sedimentation velocity of suspensions of any concentration. The results involve the statistical properties of the distribution of suspended particles or drops. Graphs of the bounds are shown for special cases. For very low concentrations of spheres, both bounds on the effective viscosity coefficient are the same, and agree with the results of Einstein and Taylor.
On the generation of surface waves by shear flows. Part 5
Authors: Miles John W.
Published: March 2006

Laboratory and field measurements of the generation of gravity waves by turbulent winds imply that a theoretical model based on laminar flow may be adequate on a laboratory, but not an oceanographic, scale. This suggests that the significance of wave-induced perturbations in the turbulent Reynolds stresses for momentum transfer from wind to waves must increase with an appropriate scale parameter. A generalization of the laminar model is constructed by averaging the linearized equations of motion for a turbulent shear flow in a direction (say y) parallel to the wave crests of a particular Fourier component of the surface-wave field. It is shown that the resulting, mean momentum transfer to this component comprises: (i) a singular part, which is proportional to the product of the velocity-profile curvature and the mean square of the wave-induced vertical velocity in the critical layer, where the mean wind speed is equal to the wave speed; (ii) a vertical integral of the mean product of the vertical velocity and the vorticity ω, where ω is the wave-induced perturbation in the total vorticity along a streamline of the y-averaged motion; (iii) the perturbation in the mean turbulent shear stress at the air-water interface. The equation that governs the advection of the vorticity ω under the action of the perturbations in the turbulent Reynolds stresses is derived. Further theoretical progress appears to demand some ad hoc hypothesis for the specification of these turbulent Reynolds stresses. Two such hypotheses are discussed briefly, but it does not appear worth while, in the absence of more detailed experimental data, to carry out elaborate numerical calculations at this time.
Incipient instability in free convection laminar boundary layers
Authors: Polymeropoulos Constantine E. , Gebhart Benjamin
Published: March 2006

This paper presents the results of an experimental investigation of the behaviour of artificial disturbances produced by an oscillating ribbon in the free convection boundary layer over a vertical uniform flux plate. The boundary layer was observed using a Mach Zehnder interferometer. The observations resulted in the approximate location of a neutral curve which is in good agreement with calculations from the linear stability theory.
The Toms phenomenon: turbulent pipe flow of dilute polymer solutions
Authors: Virk P. S. , Merrill E. W. , Mickley H. S. , Smith K. A. , Mollo-Christensen E. L.
Published: March 2006

Drag reduction caused by dilute, distilled water solutions of five polyethylene oxides, molecular weights from 80,000 to 6,000,000, in turbulent pipe flow was studied experimentally in 0·292 and 3·21 cm ID pipes. It was found that the onset of drag reduction occurs at a well-defined wall shear stress related to the random-coiling effective diameter of the polymer. Laminar to turbulent transition is not, in general, delayed. The extent of drag reduction induced by a homologous series of polymers in a given pipe is a universal function of concentration, flow rate, and molecular weight. The maximum drag reduction possible is limited by an asymptote that is independent of polymer and pipe diameter. Flow structure measurements in a single polymer solution, 1000 ppm of molecular weight 690,000, showed that the mean flow follows an ‘effective slip’ model. In this, the mean velocity profile consists of a ‘Newtonian plug’ convected along at an additional, ‘effective slip’ velocity. The turbulent flow structure follows the ‘effective slip’ model towards the pipe wall, but is significantly different from Newtonian towards the pipe axis; in particular, the inertial subrange observed in isotropic Newtonian turbulence was absent in an energy spectrum taken on the pipe axis in the polymer solution.
Interferometric measurement of temperature gradient reversal in a layer of convecting air
Authors: Gille John
Published: March 2006

Experiments in which a horizontal layer of convecting air is probed by one beam of a Michelson interferometer are described. When the localized interference fringes are horizontal, they indicate that the beam is wide enough to provide a suitable horizontal average. When the fringes are oriented nearly vertically, quantitative temperature measurements may be made. Results are presented for ratios (λ) of Rayleigh number to critical Rayleigh number of 1·48, 3·81 and 16·0. The temperature profile becomes more distorted from linear as the Rayleigh number is increased. An isothermal central region and thermal boundary regions each occupy one-third of the layer at λ = 3·81. By λ = 16·0 each boundary region occupies only one-quarter of the layer thickness, and the central region shows a reversed gradient. No full calculation is presently available to compare to the measurements. However, the shape assumption, using the first eigenfunctions of the linear stability problem, predicts the profile rather well for λ = 1·48 and 3·81 if Nusselt number agreement is imposed.
Laminar and transitional flow of drilling muds and various suspensions in circular tubes
Authors: Fur Bernard Le , Martin Madeleine
Published: March 2006

Most suspensions exhibit a rheological behaviour which cannot be represented by either Bingham's or Ostwald–De Waele's law. In studying such cases a very simple expression with only three parameters may be used. Starting with an intermediate law of this sort, this paper gives velocity profiles and head losses in laminar flow, which have been computed and plotted on diagrams in non-dimensional co-ordinates.

It has been found that transition flow rates in circular tubes for data taken from the literature and from experiments conducted on drilling muds at the Institut Français du Pétrole, are efficiently predicted by an empirical criterion (Ryan & Johnson 1959) which establishes a relation between a generalized Reynolds number and a generalized Hedström number.
The internal wave pattern produced by a sphere moving vertically in a density stratified liquid
Authors: Mowbray D. E. , Rarity B. S. H.
Published: March 2006

Experiments were conducted to test the linear theory of internal gravity waves produced in a stably stratified liquid by the steady vertical motion of a sphere. The measurements of the phase configuration in a medium whose density increased linearly with depth were made by means of a Toepler-schlieren system. The agreement between observation and prediction was found to be good.
Oscillatory free convection from an infinite horizontal cylinder
Authors: Merkin J. H.
Published: March 2006

The boundary layer on a horizontal cylinder caused by free convection, when the temperature of the cylinder is oscillating harmonically, is considered in a way which is analogous to the problem of a cylinder performing high-frequency oscillations along its diameter. It is found that, outside the thin boundary layer on the cylinder, a steady flow is induced. An outer steady boundary layer, matched with the inner boundary layer, is required in order to be able to satisfy the conditions which apply at large distances from the cylinder.
On the occurrence of cellular motion in Bénard convection
Authors: Palm E. , Ellingsen T. , Gjevik B.
Published: March 2006

The interval of Rayleigh numbers in Bénard convection corresponding to cellular motion is determined in the case of free-free boundaries, rigid-free boundaries and rigid-rigid boundaries, taking into account the variation of the kinematic viscosity with temperature. Neglecting the effect of surface tension, it is shown that this interval is largest for the rigid-rigid case. The most important feature from the obtained formula (6.1) is, however, that the interval is extremely dependent on the depth of the fluid layer. To obtain a cellular pattern it is therefore necessary to have very small fluid depths. For example, with Silicone oil and a fluid depth of 7 mm, cellular motion will, according to the theory, be observed for Rayleigh numbers larger than the critical value and less than 1·07 times the critical value. For a fluid depth of 5 mm, however, the formula (6.1) gives that cellular motion will be observed for Rayleigh numbers up to 1·54 times the critical value.
The instability of evaporating charged drops
Authors: Abbas M. A. , Latham J.
Published: March 2006

These studies constitute an extension of the work of Doyle, Moffett & Vonnegut (1964).

Experiments showed that the evaporation of a charged droplet of water, aniline or toluene supported in a vertical electric field is accompanied by no discernible loss of charge and a consequent increase in electrical pressure in the drop surface owing to the decrease in radius. When the relationship between drop chargeQand radiusRbecomes consistent with the Rayleigh criterion the drop disintegrates at its uppermost point, where the electrical pressure is a maximum, to eject about 25% of its mass in the form of highly charged droplets. The measured relationship between the quantity of ejected charge, ΔQ, the mass loss, ΔM, andRwas close to that derived theoretically from Rayleigh's equation, indicating that the value of ΔQfor a particular ΔMandRis approximately the theoretical minimum and that an individual drop may undergo a sequence of disintegrations, as was observed. The range of droplet radii studied by means of this technique was 30–200 μ.

The absence of disintegrations during the evaporation of much larger charged drops suspended from insulating fibres was attributed to corona discharge.
The interfacial stability of a ferromagnetic fluid
Authors: Cowley M. D. , Rosensweig R. E.
Published: March 2006

A normal magnetic field has a destabilizing influence on a flat interface between a magnetizable and a non-magnetic fluid. Stabilizing influences are provided by interfacial tension and gravity if the lighter fluid is uppermost. The critical level of magnetization for onset of the instability is derived for a fluid having a non-linear relation between magnetization and magnetic induction. Experiments using a magnetizable fluid, which contains a colloidal suspension of ferromagnetic particles, at interfaces with air and water are made and cover a wide range of density differences. Measurements confirm the prediction for critical magnetization, and it was found that, after onset, the interface took a new form in which the elevation had a regular hexagonal pattern. The pattern was highly stable, and the measured spacing of peaks agreed reasonably with that derived from the critical wave-number for the instability of a flat interface.
The structure of turbulent boundary layers
Authors: Kline S. J. , Reynolds W. C. , Schraub F. A. , Runstadler P. W.
Published: March 2006

Extensive visual and quantitative studies of turbulent boundary layers are described. Visual studies reveal the presence of surprisingly well-organized spatially and temporally dependent motions within the so-called ‘laminar sublayer’. These motions lead to the formation of low-speed streaks in the region very near the wall. The streaks interact with the outer portions of the flow through a process of gradual ‘lift-up’, then sudden oscillation, bursting, and ejection. It is felt that these processes play a dominant role in the production of new turbulence and the transport of turbulence within the boundary layer on smooth walls.

Quantitative data are presented providing an association of the observed structure features with the accepted ‘regions’ of the boundary layer in non-dimensional co-ordinates; these data include zero, negative and positive pressure gradients on smooth walls. Instantaneous spanwise velocity profiles for the inner layers are given, and dimensionless correlations for mean streak-spacing and break-up frequency are presented.

Tentative mechanisms for formation and break-up of the low-speed streaks are proposed, and other evidence regarding the implications and importance of the streak structure in turbulent boundary layers is reviewed.
The equilibrium and stability of a translating cavity in a liquid
Authors: Eller Anthony , Flynn H. G.
Published: March 2006

A strong sound field in a liquid may generate small cavities that move rapidly through the liquid. This paper is an analysis of the equilibrium states, and the conditions for their stability, of a cavity in translational motion through an inviscid, incompressible liquid. A principal conclusion is that translating cavities have a stable size and shape for a wide range of conditions. The equilibrium shape of a stable cavity is approximately an oblate spheroid. It is also found that the presence of translational motion contributes an outward dynamic pressure that tends to enlarge the cavity. As a result, the equilibrium radius of a translating cavity is greater than the equilibrium radius of the same cavity at rest.
Dual solutions of the boundary-layer equations at a point of attachment
Authors: Schofield D. , Davey A.
Published: March 2006

This paper indicates the existence of a dual solution of the boundary-layer equations for the flow near a nodal point of attachment of the external flow; especial note is made of the non-axisymmetric boundary-layer flow when the external flow is axisymmetric.
Finite amplitude convective cells and continental drift
Authors: Turcotte D. L. , Oxburgh E. R.
Published: March 2006

A solution is obtained for steady, cellular convection when the Rayleigh number and the Prandtl number are large. The core of each two-dimensional cell contains a highly viscous, isothermal flow. Adjacent to the horizontal boundaries are thin thermal boundary layers. On the vertical boundaries between cells thin thermal plumes drive the viscous flow. The non-dimensional velocities and heat transfer between the horizontal boundaries are found to be functions only of the Rayleigh number. The theory is used to test the hypothesis of large scale convective cells in the earth's mantle. Using accepted values of the Rayleigh number for the earth's mantle the theory predicts the generally accepted velocity associated with continental drift. The theory also predicts values for the heat flux to the earth's surface which are in good agreement with measurements carried out on the ocean floors.
Potential flow about two-dimensional hydrofoils
Authors: Giesing Joseph P. , Smith A. M. O.
Published: March 2006

This paper describes a very general method for determining the steady two-dimensional potential flow about one or more bodies of arbitrary shape operating at arbitrary Froude number near a free surface. The boundary condition of zero velocity (solid wall) or prescribed velocity (suction or blowing) normal to the body surface is satisfied exactly, and the boundary condition of constant pressure on the free surface is satisfied using the classic small-wave approximation. Calculations made by the present method are compared with analytic results, other theoretical calculations and experimental data. Examples for which no comparison exists are also presented to illustrate the capability of the method.
Axially symmetric potential flow around a slender body
Authors: Handelsman Richard A. , Keller Joseph B.
Published: March 2006

Axially symmetric potential flow about an axially symmetric rigid body is considered. The potential due to the body is represented as a superposition of potentials of point sources distributed along a segment of the axis inside the body. The source strength distribution satisfies a linear integral equation. A complete uniform asymptotic expansion of its solution is obtained with respect to the slenderness ratio ε½, which is the maximum radius of the body divided by its length. The expansion contains integral powers of ε multiplied by powers of log ε. From it expansions of the potential, the virtual mass and the dipole moment of the body are obtained. The flow about the body in the presence of an axially symmetric stationary obstacle is also determined. The method of analysis involves a technique for the asymptotic solution of integral equations.
Turbulence measurements with inclined hot-wires Part 1. Heat transfer experiments with inclined hot-wire
Authors: Champagne F. H. , Sleicher C. A. , Wehrmann O. H.
Published: March 2006

The measurement of the turbulent shear stresses and normal and bi-normal intensities with a hot-wire anemometer requires that the directional sensitivity of the hot-wire be known. Normal component or cosine law cooling is generally assumed, although for finite wire lengths the non-uniform wire temperature must cause a deviation from the cosine law.

Careful heat transfer measurements from wires inclined and normal to the flow were taken for several values of the Reynolds number, the length-to-diameter ratio of the wire, the overheat ratio and for several support configurations. All experiments were performed in air at low subsonic velocities, i.e. M < 0·1. The measurements indicate that the heat loss from an inclined wire is larger than that from a wire normal to the flow with the same normal component of velocity. The data were correlated by
\[
U^2_E(\alpha) = U^2(0)(\cos^2\alpha + k^2\sin^2\alpha),
\]
where UE(α) is the effective cooling velocity at the angle α between the normal to the wire and the mean flow direction and U(0) is the velocity at α = 0. The value of k was found to depend primarily upon the length-to-diameter ratio ([lscr ]/d) of the wire. For platinum wires k is approximately 0·20 for [lscr ]/d = 200, decreases with increasing [lscr ]/d, and becomes effectively zero at [lscr ]/d = 600.

To aid in interpreting the heat transfer data, measurements of the temperature distribution along inclined and normal wires were made with a high sensitivity infra-red detector coupled to a high resolution microscrope with reflective optics. The measurements indicate that inclined wires and normal wires have nearly identical end conduction losses, although the temperature distribution on an inclined wire is slightly asymmetrical. Therefore, the deviation from the cosine law is caused by an increase in the convection heat loss, and this increase is attributed to the tangential component of velocity.
Turbulence measurements with inclined hot-wires Part 2. Hot-wire response equations
Authors: Champagne F. H. , Sleicher C. A.
Published: March 2006

Hot-wire response equations to include the effects of the tangential velocity component as well as the non-linearities caused by high intensity turbulence are derived for linearized constant temperature operation. For low intensity turbulence similar equations are derived for constant current operation. The equations are applied to an X-wire array to determine the errors in selected turbulence quantities which arise from the assumption of cosine law cooling. The error depends upon the quantity measured, the method of operation, and [lscr ]/d. For [lscr ]/d = 200 the error ranges from 0 to 17%.
Velocity measurements in axisymmetric jets of dilute polymer solutions
Authors: White D. A.
Published: March 2006

Experiments are described in which the velocity profiles of axisymmetric jets are measured in water and in dilute solutions of water-soluble polymers. The ambient fluid and that entering through the nozzle were homogeneous. The similarity characteristics of velocity profiles including a linear variation of centre-line velocity with the inverse of the distance from the nozzle were unaltered by these substances. On adding large polyethylene oxide molecules to the system the jet spread at an increased angle, possibly due to the entrainment rate being enhanced by an increase in turbulent elasticity for polyethylene oxide solutions. It is also postulated that additives which had no effect on the velocity were not large enough to influence the small microscale eddies in which the energy to activate an entrainment cycle is stored.
Surface-wave scattering matrix for a shelf
Authors: Miles John W.
Published: March 2006

The diffraction of gravity waves at a discontinuity in depth is described by a scattering matrix that relates the asymptotic, plane-wave fields (each of which may contain waves travelling towards and away from the discontinuity) on the two sides of the discontinuity. Plane-wave and variational approximations for the elements of this scattering matrix are developed. These approximate results are tested by comparison with the more accurate results obtained by Newman for an infinite step. The plane-wave approximation to the magnitude of the transmission coefficient is within 5% of Newman's result for all wavelengths, but the corresponding approximation to the reflexion coefficient is satisfactory only for rather long wavelengths. The variational approximations to the complex transmission and reflexion coefficients agree with Newman's results, within the accuracy with which his graphs can be read, for all wavelengths. The variational approximations also are used to determine the effects of trapped modes on the resonant width of a shelf that terminates at a vertical cliff.
A shear-free turbulent boundary layer
Authors: Uzkan T. , Reynolds W. C.
Published: March 2006

A simple wall-turbulence interaction has been studied experimentally. In the idealized model an infinite flat plate is suddenly inserted into a pre-existing field of homogeneous isotropic turbulence, and subsequent changes in the turbulence field examined. The experiment involved passing grid-produced turbulence over a wall moving at the mean speed. Mean velocity gradients vanish in both the model and experiment, and hence production of new turbulence is absent. This allowed the inhibiting effects of the wall to be studied separately. The growth of the ‘inhomogeneity layer’ into the impressed turbulence field and other statistical features of the turbulence were measured.