bluuebunny/crossref_metadata_2025 huggingface dataset:
An asymptotic theory of the jet flap in three dimensions
Authors: Tokuda Naoyuki
Published: March 2006

A uniformly valid asymptotic solution has been constructed for three-dimensional jet-flapped wings by the method of matched asymptotic expansions for high aspect ratios. The analysis assumes that the flow is inviscid and incompressible and is formulated on the thin airfoil theory in accordance with the well-established Spence (1961) theory in two dimensions.

A simple method emerges in treating the bound vortices along the jet sheet which forms behind the wing with the aid of the following physical picture. Three distinct flow regions—namely inner, outer and Trefitz—exist in the problem. Close to the wing the flow approximates to that in two dimensions. Therefore, Spence's solution in two dimensions applies. In the outer region a wing shrinks to a line of singularities from which the main disturbances of flow in this region arise. In particular, we find that the shape of the jet sheet, hence the strength of vortices, is now predetermined by the strength of the singularities there. Hence a complete flow field in the outer region can now be determined first by evaluating the flow due to various degrees of singularities along this line and then adding the effect of the jet bound vortices which is now known. Far removed from the wing, the well-known Trefftz region exists in which calculations of aerodynamic forces can be most easily done.

The result has been applied to various wing planforms such as cusped, elliptic and rectangular wings. The present result breaks down for rectangular wings. However, we can apply Stewartson's (1960) solution for lifting-line theory for semi-infinite rectangular wings, because, to this second-order approximation it is established that the jet sheet in the outer region makes no contribution to lift, with the direct contribution of the deflected jet at the exit being cancelled by the reduced circulation in the jet vortices. This result for the rectangular wing gives excellent agreement with the experiments made on a rectangular wing, while the result for elliptic wings underestimates them considerably.
The effect of stable thermal stratification on the stability of viscous parallel flows
Authors: Gage K. S.
Published: March 2006

A unified linear viscous stability theory is developed for a certain class of stratified parallel channel and boundary-layer flows with Prandtl number equal to unity. Results are presented for plane Poiseuille flow and the asymptotic suction boundary-layer profile, which show that the asymptotic behaviour of both branches of the curve of neutral stability has a universal character. For velocity profiles without inflexion points it is found that a mode of instability disappears as η, the local Richardson number evaluated at the critical point, approaches 0.0554 from below. Calculations for Grohne's inflexion-point profile show both major and minor curves of neutral stability for 0 < η [les ] 0.0554; for\[ 0.0554 < \eta < 0.0773 \]there is only a single curve of neutral stability; and, for η > 0.0773, the curves of neutral stability become closed, with complete stabilization being achieved for a value of η of about 0·107.
The effect of atmospheric conditions on plume rise
Authors: Slawson P. R. , Csanady G. T.
Published: March 2006

The buoyant rise of chimney plumes is discussed for relatively large distances from the source, where atmospheric turbulence is the dominant cause of mixing (rather than turbulence due to the plume's own upward motion). A simple theory is developed which shows a number of different shapes plumes can have under different atmospheric conditions (particularly in an unstable environment).

Experimental data are then presented, which were collected in the field near Toronto, with reasonably detailed supporting information on the atmospheric temperature structure. These (and earlier results) often show quite complex plume behaviour at large distances from the source, which can, however, be readily understood in terms of the simple theory presented.
On acoustic refraction by duct shear layers
Authors: Shankar P. N.
Published: March 2006

We consider the propagation of sound in a two-dimensional shear layer in a duct. Initially plane sound waves propagate in a subsonic flow whose steady velocity U(y) is in the axial direction and is a function of the normal co-ordinate alone. The initial-value problem in time and space is solved within the framework of a perturbation scheme. Far from the wave front and for large time the refractive effects are considerable. However, the present results indicate that standing wave type patterns lead to a strong dependence on axial location in the duct. The paper also points out the Weakness inherent in theories which attempt to treat the refraction problem without considering the source conditions.
Thermal convection with large viscosity variations
Authors: Torrance K. E. , Turcotte D. L.
Published: March 2006

The influence of large variations of viscosity on convection in a layer of fluid heated from below has been investigated. Solutions for the flow and temperature fields were obtained numerically assuming infinite Prandtl number, free-surface boundary conditions, and two-dimensional motion of fixed horizontal wavelength. The effects of a temperature-dependent and a depth-dependent viscosity were each studied; calculations were also carried out using a temperature- and depth-dependent viscosity model appropriate to the earth's mantle.
The slow transverse motion of a flat plate through a non-diffusive stratified fluid
Authors: Janowitz G. S.
Published: March 2006

We consider the two-dimensional flow produced by the slow horizontal motion of a vertical plate of height 2b through a vertically stratified (ρ = ρ0(1 - βz)) non-diffusive viscous fluid. Our results are valid when U2 [Lt ] Ub/ν [Lt ] 1, where U is the speed of the plate and ν the kinematic viscosity of the fluid. Upstream of the body we find a blocking column of length 10−2b4/(Uν/βg. This column is composed of cells of closed streamlines. The convergence of these cells near the tips of the plate leads to alternate jets. The plate itself is embedded in a vertical shear layer of thickness (Uν/βg)1/3. In the upstream portion of this layer the vertical velocities are of order U and in the downstream portion of order Ub/(Uν/βg)1/3 ([Gt ] U). The flow is uniform and undisturbed downstream of this layer.
A rapidly varied flow phenomenon in a two-layer flow
Authors: Wilkinson D. L. , Wood I. R.
Published: March 2006

This paper examines a region of rapidly varied flow in a two-layered density stratified system with one layer flowing, and the other stationary. The analogous phenomenon in open channel hydraulics is the hydraulic jump. In density stratified flows the phenomenon is referred to as a density jump because it is generally accompanied by a change in density of the flowing layer.

It is shown there is a fundamental difference between the hydraulic jump and the density jump in that flow conditions on either side of a density jump are not uniquely related. A density jump with given flow conditions upstream has a range of possible states which may be attained downstream. The rate of entrainment of ambient fluid into a density jump, and the conditions downstream of the jump, are determined by the downstream control and the upsteam conditions. The particular case of a density jump controlled by a broad crested weir downstream is examined in detail.
Instabilities of convection rolls in a high Prandtl number fluid
Authors: Busse F. H. , Whitehead J. A.
Published: March 2006

An experiment on the stability of convection rolls with varying wave-number is described in extension of the earlier work by Chen & Whitehead (1968). The results agree with the theoretical predictions by Busse (1967a) and show two distinct types of instability in the form of non-oscillatory disturbances. The ‘zigzag instability’ corresponds to a bending of the original rolls; in the ‘cross-roll instability’ rolls emerge at right angles to the original rolls. At Rayleigh numbers above 23,000 rolls are unstable for all wave-numbers and are replaced by a three-dimensional form of stationary convection for which the name ‘bimodal convection’ is proposed.
Energy stability of the buoyancy boundary layer
Authors: Dudis Joseph J. , Davis Stephen H.
Published: March 2006

The critical value RE of the Reynolds number R is predicted by the application of the energy theory. When R < RE, the buoyancy boundary layer is the unique steady solution of the Boussinesq equations and the same boundary conditions, and is, further, stable in a slightly weaker sense than asymptotically stable in the mean. The critical value RE is determined by numerically integrating the relevant Euler–Lagrange equations. Analytic lower bounds to RE are obtained. Comparisons are made between RE and RL, the critical value of R according to linear theory, in order to demark the region of parameter space, RE < R < RL, in which subcritical instabilities are allowable.
The Magnus or Robins effect on rotating spheres
Authors: Barkla H. M. , Auchterlonie L. J.
Published: March 2006

Robins showed in 1742 that a transverse aerodynamic force on a rotating sphere could be detected by suspending it as a pendulum. Differences of periodic time in conical pendulum motion with spin and orbit parallel and opposed have been found to give a reasonably accurate measure of the lift coefficient, and the results shown extend knowledge of the effect down to a Reynolds number of 2 × 103 and up to a ratio of 12 between the peripheral and translational velocities.
On the reverse transition of a turbulent flow under the action of buoyancy forces
Authors: Steiner Alejandro
Published: March 2006

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.
Inertial-range transfer in two- and three-dimensional turbulence
Authors: Kraichnan Robert H.
Published: March 2006

A simple dynamical argument suggests that the k−3 enstrophy-transfer range in two-dimensional turbulence should be corrected to the form
\[
E(k) = C^{\prime} \beta^{\frac{21}{3}}k^{-3}[\ln (k/k_1)]^{-\frac{1}{3}}\quad (k \gg k_1),
\]
where E(k) is the usual energy-spectrum function, β is the rate of enstrophy transfer per unit mass, C′ is a dimensionless constant, and k1 marks the bottom of the range, where enstrophy is pumped in. Transfer in the energy and enstrophy inertial ranges is computed according to an almost-Markovian Galilean-in variant turbulence model. Transfer in the two-dimensional energy inertial range,
\[
E(k) = C\epsilon^{\frac{2}{3}}k^{-\frac{5}{3}},
\]
is found to be much less local than in three dimensions, with 60 % of the transfer coming from wave-number triads where the smallest wave-number is less than one-fifth the middle wave-number. The turbulence model yields the estimates C′ = 2·626, C = 6·69 (two dimensions), C = 1·40 (three dimensions).
Soret-driven thermosolutal convection
Authors: Hurle D. T. J. , Jakeman E.
Published: March 2006

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.
A numerical study of laminar separation bubbles using the Navier-Stokes equations
Authors: Briley W. Roger
Published: March 2006

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.
Some measurements of particle velocity autocorrelation functions in a turbulent flow
Authors: Snyder W. H. , Lumley J. L.
Published: March 2006

Particle velocity autocorrelations of single spherical beads (46·5 μhollow glass, 87 μ glass, 87 μ corn pollen, and 46·5 μ copper) were measured in a grid-generated turbulence. The hollow glass beads were small and light enough to behave like fluid points; the other types had significant inertia and ‘crossing trajectories’ effects. The autocorrelations decreased much faster for heavier particles, in contradiction to previous experimental results. The integral scale for the copper beads was 1/3 of that for the hollow glass beads. The particle velocity correlations and the Eulerian spatial correlation were coincident within experimental error when the separation was non-dimensionalized by the respective integral scale. The data generated by the hollow glass beads can be used to estimate Lagrangian fluid properities. The Lagrangian time integral scale is approximated byL/u′, whereLis the Eulerian integral scale andu′ is the turbulence intensity.
Magneto-fluid-mechanic pipe flow in a transverse magnetic field Part 2. Heat transfer
Authors: Gardner R. A. , Lykoudis P. S.
Published: March 2006

The present paper, part 2, consists of an experimental investigation of the influence of a transverse magnetic field on the heat transfer of a conducting fluid (mercury) flowing in an electrically insulated pipe subjected to a uniform heat flux at the wall. Mean temperature profiles and heat transfer data are presented which demonstrate that the magnetic field inhibits the convective mechanism of heat transfer through its damping of the turbulent velocity fluctuations.
The stability of immiscible liquid layers in a porous medium
Authors: Raghavan R. , Marsden S. S.
Published: March 2006

The stability of liquid layers in a porous medium under the action of viscous and surface forces is described. An extension of previous studies on the stability of a single interface in a porous medium is presented as the basis for solutions to many problems of practical interest where flow in porous media are involved.
A limitation on Long's model in stratified fluid flows
Authors: Segur Harvey
Published: March 2006

The flow of a continuously stratified fluid into a contraction is examined, under the assumptions that the dynamic pressure and the density gradient are constant upstream (Long's model). It is shown that a solution to the equations exists if and only if the strength of the contraction does not exceed a certain critical value which depends on the internal Froude number. For the flow of a stratified fluid over a finite barrier in a channel, it is further shown that, if the barrier height exceeds this same critical value, lee-wave amplitudes increase without bound as the length of the barrier increases. The breakdown of the model, as implied by these arbitrarily large amplitudes, is discussed. The criterion is compared with available experimental results for both geometries.
On the reattachment of a shock layer produced by an instantaneous energy release
Authors: Cheng H. K. , Kirsch J. W. , Lee R. S.
Published: March 2006

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,$\dot{y}_{sh}/a_{\infty}\rightarrow \infty $, and a high shock-compression ratio, ρs/ρ∞∼ 2γ/(γ − 1) ≡ ε−1[Gt ] 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 [Lt ]t[Lt ] ε−2/3(1+ν), the oscillation frequency is shown to be (1 + ν) (2π)−1t−½(1−ν)and the amplitude varies ast−¼(3+ν)matching the earlier results of Chenget 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.
Transition to a wavy vortex régime in convective flow between inclined plates
Authors: Hart J. E.
Published: March 2006

Experimental results concerning the supercritical behaviour of instabilities occurring in the thermal convective flow between inclined plates are presented. For inclination δ between 90° and 10° (δ = 0° is the vertical orientation), with the bottom plate hotter, it is known that the primary instabilities are longitudinal convective rolls with axes oriented up the slope. These rolls are found to become wavy at a supercritical Rayleigh number Ra(δ, Pr), with an upslope wavelength which seems to be related directly to the wavelength of the original rolls. Measurements indicate that these transitions take place at Reynolds numbers which are probably too small for the process to be attributed to a shear instability. It is thought that the cross-slope derivative of the buoyancy force, which exists because of the tilted geometry, is important in generating the required vorticity. The transition is crucial to the development of violent unsteadiness. The breaking of the waves leads to turbulence at much lower Rayleigh numbers than those required in convection between horizontal plates. The transition to wavy vortices appears to be very similar to that which occurs for Taylor vortices in cylindrical Couette flow.
The ‘bursting’ phenomenon in a turbulent boundary layer
Authors: Rao K. Narahari , Narasimha R. , Narayanan M. A. Badri
Published: March 2006

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.
Magnetohydrodynamic duct flow in a uniform transverse magnetic field of arbitrary orientation
Authors: Alty C. J. N.
Published: March 2006

The paper presents an approximate analysis for high Hartmann number of the flow of an electrically conducting, incompressible fluid in a duct of square crosssection, having one pair of opposite walls insulating, and the other pair perfectly conducting and inclined at arbitrary orientation to a uniform transverse magnetic field. The flow is considered to be either pressure-driven with the two perfectly conducting electrodes short-circuited together or electrically driven by a potential difference applied between these electrodes in the absence of axial pressure gradient. The paper describes experiments on the pressure-driven, short circuited case using mercury in copper ducts to investigate the variation of the streamwise pressure gradient and of the potential distribution along one insulating wall with orientation, magnetic field and flow rate.

At general orientations the analysis suggests and the experiments confirm the existence of regions of stationary fluid in the corners of the duct, together with viscous shear layers parallel to the magnetic field. For the case in which the electrodes are parallel to the magnetic field the experimental results for the pressure gradient, but not those for the potential distribution, agree reasonably well with Hunt & Stewartson's (1965) asymptotic solution. Both pressure gradient and potential results agree closely with the analysis by Hunt (1965) of the case in which the electrodes are perpendicular to the magnetic field.
The method of multiple scales and non-linear dispersive waves
Authors: Nayfeh Ali Hasan , Hassan Sayed D.
Published: March 2006

The method of multiple scales is used to analyze three non-linear physical systems which support dispersive waves. These systems are (i) waves on the interface between a liquid layer and a subsonic gas flowing parallel to the undisturbed interface, (ii) waves on the surface of a circular jet of liquid, and (iii) waves in a hot electron plasma. It is found that the partial differential equations that govern the temporal and spatial variations of the wave-numbers, amplitudes, and phases have the same form for all of these systems. The results show that the non-linear motion affects only the phase. For the constant wave-number case, the general solution for the amplitude and the phase can be obtained.
A non-linear instability theory for a wave system in plane Poiseuille flow
Authors: Stewartson K. , Stuart J. T.
Published: March 2006

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.
Contained non-homogeneous flow under gravity or how to stratify a fluid in the laboratory
Authors: Walin Gösta
Published: March 2006

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.
On the interpretation of some longitudinal dispersion experiments
Authors: Chatwin P. C.
Published: March 2006

Experiments to determine the value of Taylor's longitudinal diffusivity in turbulent flows in open channels and circular pipes have produced results which are in many cases inconsistent with one another and with theoretical estimates due respectively to Elder (1959) and Taylor (1954). Neither is there general agreement on when Taylor's theory becomes applicable. In an attempt to clarify the discrepancies two well-known sets of experiments by Fischer (1966) and Taylor (1954) are re-examined by using a natural procedure which, it is argued, has certain advantages over more usual methods. It is shown that Fischer's observations in an open channel were not made at a sufficient distance downstream from the point of injection for Taylor's theory to apply but that they are consistent with a description of the early stages of the dispersion process due to Sullivan (1968). It is subsequently argued that these observations suggest that Elder's estimate of the diffusivity is too low for two reasons. The first is the error caused by assuming the existence of an eddy diffusivity calculated by means of Reynolds analogy and the second is the neglect of the viscous sublayer. On the other hand it is shown that some of Taylor's observations in a circular pipe are consistent with his theory, although the values of the diffusivity which best fit the data are about 25% higher than Taylor's estimate, and it is suggested that this is for the same reasons as in the open channel. The paper concludes with a discussion of the effect of the viscous sublayer on the value of the longitudinal diffusivity. Partly on the basis of an approximate model it is argued that theoretical calculations which ignore the viscous sublayer are too low by amounts which depend on the Reynolds and Schmidt numbers and can be of the order of 20%.
Calculation of the steady flow past a sphere at low and moderate Reynolds numbers
Authors: Dennis S. C. R. , Walker J. D. A.
Published: March 2006

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 asRincreases and the present calculations were terminated atR= 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.
Electrohydrodynamic stability: effects of charge relaxation at the interface of a liquid jet
Authors: Saville D. A.
Published: March 2006

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.
Generation of surf beat by non-linear wave interactions
Authors: Gallagher Brent
Published: March 2006

Non-linear interactions among wind-generated gravity waves transfer energy to low frequency waves in a coastal zone. A transfer function is derived for a straight coastline of constant bottom slope. This model is applied to three actual cases, and numerical evaluation of the energy transfer produces low frequency spectra which are compared with observations.
The effect of sinusoidal protrusions on laminar free convection between vertical walls
Authors: Watson A. , Poots G.
Published: March 2006

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.
Bed forms due to mass transfer in turbulent flows: a kaleidoscope of phenomena
Authors: Allen J. R. L.
Published: March 2006

This paper gives a brief account and interpretation of bed forms produced during mass transfer from beds of hardened Plaster of Paris (calcium sulphate) exposed to turbulent streams of water. Defects within or upon the initially plane plaster beds perturbed the profile of mass-transfer (solution) rate as the result of the setting up of separated flows. Those defects that exceeded a certain critical dimension set by flow conditions grew in length, breadth and depth into heel-shaped hollows of parabolic plan calledflutes.The continued growth and eventual interference of coextensive flutes gave rise to a bed withscallops, an assemblage of intersecting, saucer-shaped depressions. Defects which fell below the critical dimension set by flow conditions grew initially in length and breadth to give long, narrowgrooveslying parallel with flow, but after a sufficient time were erased so that the bed reverted to plane. Many plane beds became fashioned into continuous systems of very longfurrows and ridgesaligned parallel with flow and coupled with paired longitudinal vortices resembling Taylor–Görtler vortices klinematically. The dimensions and orientation of this structure suggested that Klineian streaks, which on rigid beds occur randomly in space and time, had become stabilized in space in the new context of the deformable beds. The furrows commonly experienced a higher-order instability which resulted in the production ofsecondary flutesandscallopssimilar to those resulting from pre-existing defects. All of the bed forms described are associated with separated flows, and it is suggested that separation of flow may be a phenomenon necessary for their maintenance and, in the case of flutes, perhaps also for their origin.
Transmission of water waves through small apertures
Authors: Tuck E. O.
Published: March 2006

A method of solution is proposed for flows through small apertures in otherwise impermeable barriers. This method, which is an application of the method of matched asymptotic expansions, is used to solve a specific water-wave problem, yielding an approximate formula for the transmission coefficient.
Free-surface gravity flow past a submerged cylinder
Authors: Dagan G.
Published: March 2006

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.
Infinite models for ciliary propulsion
Authors: Blake J. R.
Published: March 2006

This paper discusses two infinite length models (planar and cylindrical) for ciliary propulsion of microscopic organisms. Through the concept of an extensible envelope (instantaneous surface covering the numerous cilia) over the organism, we consider a small amplitude analysis for the velocity of propulsion. A comparison of the velocities of propulsion for the infinite models reveals that they are just over twice that obtained for the finite spherical model (Blake 1971). This indicates that planeness is more important than finiteness, as the solution for typical micro-organisms (e.g. disk-shaped) should occur somewhere between these two models. The maximum velocity of propulsion obtained is one quarter of the wave velocity; that inferred for Opalina, the organism modelled, is nearly one fifth of the wave velocity. Associated shapes of the surface and paths of movement of the tips of the cilia are illustrated.
The dynamics of a deformable drop suspended in an unbounded Stokes flow
Authors: Haber S. , Hetsroni G.
Published: March 2006

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.
Viscous incompressible flow between concentric rotating spheres. Part 1. Basic flow
Authors: Munson B. R. , Joseph D. D.
Published: March 2006

The steady motion of a viscous fluid contained between two concentric spheres which rotate about a common axis with different angular velocities is considered. A high-order analytic perturbation solution, through terms of order Re7, is obtained for low Reynolds numbers. For larger Reynolds numbers an approximate Legendre polynomial series representation is used to reduce the governing system of equations to a non-linear ordinary differential equation boundary-value problem which is solved numerically. The resulting flow pattern and the torque required to rotate the spheres are presented for various cases considered.
Viscous incompressible flow between concentric rotating spheres. Part 2. Hydrodynamic stability
Authors: Munson B. R. , Joseph D. D.
Published: March 2006

The energy theory of hydrodynamic stability is applied to the viscous incompressible flow of a fluid contained between two concentric spheres which rotate about a common axis with prescribed angular velocities. The critical Reynolds number is calculated for various radius and angular velocity ratios such that it is certain the basic laminar motion is stable to any disturbances. The stability problem is solved by means of a toroidal–poloidal representation of the disturbance flow and numerical integration of the resulting eigenvalue problem.
Instability and breakup of charged liquid jets
Authors: Huebner A. L. , Chu H. N.
Published: March 2006

Drops formed from the breakup of charged cylindrical liquid jets have been shown to be smaller than those formed in the unelectrified case. This change has been related to electrically induced instabilities on the jet. Small-perturbation analysis was used to formulate the Lagrangian and write the differential equation for the jet. Non-dimensionalized plots of the solutions exhibit the stabilizing or destabilizing influences of surface tension and electric effects, and allow these influences to be related back to liquid physical properties. Drop diameter to be expected from breakup of the electrified jet was calculated as a function of jet diameter, physical properties of the liquid, jet electrification, and the mode of instability dominating the breakup process.
Upstream influence in a two-fluid system
Authors: Keady G.
Published: March 2006

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.
Finite-amplitude and diffusive effects in acoustics: a report on EUROMECH 23
Authors: Rott N. , Thomann H.
Published: March 2006

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.
An experimental investigation of turbulent stratified shearing flow
Authors: Moore Michael J. , Long Robert R.
Published: March 2006

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.
\[
D^3\propto t.
\]
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.
A pulsed-wire technique for velocity measurements in highly turbulent flows
Authors: Bradbury L. J. S. , Castro I. P.
Published: March 2006

The velocity measuring technique described in this paper consists of measuring the time of flight of a tracer of heated air from an electrically pulsed wire to one of two sensor wires which are operated as resistance thermometers. These sensor wires are at right angles to the pulsed wire and are placed one on either side of the pulsed wire. The instrument may be used in highly turbulent flows including regions in which flow reversals occur. The paper discusses the theoretical behaviour of the probe and the results of some calibration experiments.
Effect of a non-uniform alternating electric field on the thermal boundary layer near a heated vertical plate
Authors: Turnbull Robert J.
Published: March 2006

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.
Non-linear wave-number interaction in near-critical two-dimensional flows
Authors: Diprima R. C. , Eckhaus W. , Segel L. A.
Published: March 2006

This paper deals with a system of equations which includes as special cases the equations governing such hydrodynamic stability problems as the Taylor problem, the Bénard problem, and the stability of plane parallel flow. A non-linear analysis is made of disturbances to a basic flow. The basic flow depends on a single co-ordinate η. The disturbances that are considered are represented as a superposition of many functions each of which is periodic in a co-ordinate ξ normal to η and is independent of the third co-ordinate direction. The paper considers problems in which the disturbance energy is initially concentrated in a denumerable set of ‘most dangerous’ modes whose wave-numbers are close to the critical wave-number selected by linear stability theory. It is a major result of the analysis that this concentration persists as time passes. Because of this the problem can be reduced to the study of a single non-linear partial differential equation for a special Fourier transform of the modal amplitudes. It is a striking feature of the present work that the study of a wide class of problems reduces to the study of this single fundamental equation which does not essentially depend on the specific forms ofthe operators in the original system of governing equations. Certain general conclusions are drawn from this equation, for example for some problems there exist multi-modal steady solutions which are a combination of a number of modes with different spatial periods. (Whether any such solutions are stable remains an open question.) It is also shown in other circumstances that there are solutions (at least for some interval of time) which are non-linear travelling waves whose kinematic behaviour can be clarified by the concept of group speed.
On geostrophic motion of a non-homogeneous fluid
Authors: Hide R.
Published: March 2006

A general theoretical relationship between the (three-dimensional) vcLacity field and density field is established for geostrophic flow of a non-homogeneous fluid. The practical value of the relationship – which reduces to the celebrated two-dimensional theorem due to Proudman and Taylor when the fluid is homogeneous – is illustrated by means of three examples of flows in rapidly rotating fluids, namely (i) baroclinic waves in laboratory systems and in the atmosphere, (ii) the gyroscopic ‘steering’ processes in Jupiter's atmosphere that are implied by the ‘Taylor column’ theory of the Great Red Spot, and (iii) certain striking properties of ocean currents revealed by recent observations in the Atlantic and Mediterranean.
The propagation of large amplitude tsunamis across a basin of changing depth Part 1.Off-shore behaviour
Authors: Varley E. , Venkataraman R. , Cumberbatch E.
Published: March 2006

A theory is presented which describes the propagation of large amplitude tsunamis across a basin of variable depth in the limit when this depth is varying slowly on a scale deiined by the wavelength. In part 1 only the off-shore behaviour is considered; in part 2 some features of the final run up are described.

The technique used is to regard the wave as a slowly modulated simple wave with a slowly changing Riemann invariant. One of the most significant results is that over distances where the effect of depth variation modulates the amplitude of the wave, but does not disperse it, the variations of the amplitudes of the flow variables, such as maximum surface elevation, can be calculated as functions of the undisturbed depth without knowing how this depth varies in distance and without knowing the wave profile. These variations are fully calculated.

The work continues the investigation on large amplitude acoustic pulses in stratified media described in an earlier paper by Varley & Cumberbatch (1970). It is a generalization of Whitham's work (1953) on the sonic boom.
Aerodynamic excitation of the harmonium reed
Authors: Hilaire Arthur O. St , Wilson Theodore A. , Beavers Gordon S.
Published: March 2006

An investigation of the mechanism responsible for the self-excited oscillations of a harmonium reed is presented. Experiments show that the amplitude of vibration of the reed grows exponentially, and measurements of the growth rate as a function of the flow past the reed are reported. Flow visualization studies lead to the conclusion that jet or wake instability is not important in exciting the reed vibration. An analysis of the flow around the reed as an unsteady potential flow results in the evaluation of the aerodynamic forces exciting the reed. The analysis shows that the pressure which excites the reed motion is of the order of $\rho U_0\dot{a}$, where U0 is the flow velocity and $\dot{a}$ is the reed velocity.