# Showing papers in "Journal of Fluid Mechanics in 1962"

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TL;DR: Kolmogorov and Oboukhov as discussed by the authors investigated the local structure of turbulence at high Reynolds number, based on Richardson's idea of the existence in the turbulent flow of vortices on all possible scales.

Abstract: The hypotheses concerning the local structure of turbulence at high Reynolds number, developed in the years 1939-41 by myself and Oboukhov (Kolmogorov 1941 a,b,c; Oboukhov 1941 a,b) were based physically on Richardson's idea of the existence in the turbulent flow of vortices on all possible scales l < r < L between the ‘external scales’ L and the ‘internal scale’ l and of a certain uniform mechanism of energy transfer from the coarser-scaled vortices to the finer.

2,682 citations

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TL;DR: In this article, the energy flux in a finite-depth gravity-wave spectrum resulting from weak non-linear couplings between the spectral components is evaluated by means of a perturbation method.

Abstract: The energy flux in a finite-depth gravity-wave spectrum resulting from weak non-linear couplings between the spectral components is evaluated by means of a perturbation method. The fifth-order analysis yields a fourth-order effect comparable in magnitude to the generating and dissipating processes in wind-generated seas. The energy flux favours equidistribution of energy and vanishes in the limiting case of a white, isotropic spectrum. The influence on the equilibrium structure of fully developed wave spectra and on other phenomena in random seas is discussed briefly.

1,220 citations

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TL;DR: In this article, an experimental investigation is described, in which principal emphasis is given to revealing the nature of the motions in the non-linear range of boundary-layer instability and the onset of turbulence, and it is demonstrated that the actual breakdown of the wave motion into turbulence is a consequence of a new instability which arises in the aforementioned three-dimensional wave motion.

Abstract: An experimental investigation is described in which principal emphasis is given to revealing the nature of the motions in the non-linear range of boundary-layer instability and the onset of turbulence. It has as its central purpose the evaluation of existing theoretical considerations and the provision of a sound physical model which can be taken as a basis for a theoretical approach. The experimental method consisted of introducing, in a two-dimensional boundary layer on a flat plate at ‘incompressible’ speeds, three-dimensional disturbances under controlled conditions using the vibrating-ribbon technique, and studying their growth and evolution using hot-wire methods. It has been definitely established that longitudinal vortices are associated with the non-linear three-dimensional wave motions. Sufficient data were obtained for an evaluation of existing theoretical approaches. Those which have been considered are the generation of higher harmonics, the interaction of the mean flow and the Reynold stress, the concave streamline curvature associated with the wave motion, the vortex loop and the non-linear effect of a three-dimensional perturbation. It appears that except for the latter they do not adequately describe the observed phenomena. It is not that they are incorrect or may not play a role in some aspect of the local behaviour, but from the over-all point of view the results suggest that it is the non-linear effect of a three-dimensional perturbation which dominates the behaviour. A principal conclusion to be drawn is that a new perspective, one that takes three-dimensionality into account, is required in connexion with boundary-layer instability. It is demonstrated that the actual breakdown of the wave motion into turbulence is a consequence of a new instability which arises in the aforementioned three-dimensional wave motion. This instability involves the generation of ‘hairpin’ eddies and is remarkably similar in behaviour to ‘inflexional’ instability. It is also shown that the results obtained from the study of controlled disturbances are equally applicable to ‘natural’ transition.

1,045 citations

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TL;DR: In this article, the second-order currents and changes in mean surface level which are caused by gravity waves of non-uniform amplitude are investigated, and the effects are interpreted in terms of the radiation stresses in the waves.

Abstract: This paper studies the second-order currents and changes in mean surface level which are caused by gravity waves of non-uniform amplitude. The effects are interpreted in terms of the radiation stresses in the waves.The first example is of wave groups propagated in water of uniform mean depth. The problem is solved first by a perturbation analysis. In two special cases the second-order currents are found to be proportional simply to the square of the local wave amplitude: (a) when the lengths of the groups are large compared to the mean depth, and (b) when the groups are all of equal length. Then the surface is found to be depressed under a high group of waves and the mass-transport is relatively negative there. In case (a) the result can be simply related to the radiation stresses, which tend to expel fluid from beneath the higher waves.The second example considered is the propagation of waves of steady amplitude in water of gradually varying depth. Assuming no loss of energy by friction or reflexion, the wave amplitude must vary horizontally, to maintain the flux of energy constant; it is shown that this produces a negative tilt in the mean surface level as the depth diminishes. However, if the wave height is limited by breaking, the tilt is positive. The results are in agreement with some observations by Fairchild.Lastly, the propagation of groups of waves from deep to shallow water is studied. As the mean depth decreases, so the response of the fluid to the radiation stresses tends to increase. The long waves thus generated may be reflected as free waves, and so account for the 'surf beats’ observed by Munk and Tucker.Generalle speaking, the changes in mean sea level produced by ocean waves are comparable with those due to horizontal wind stress. It may be necessary to allow for the wave stresses in calculating wind stress coefficients.

959 citations

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TL;DR: In this article, the authors introduce the notion of small-scale and large-scale turbulence, a set of disturbances, the scales of which do not exceed the distance to the wall and for which the hypothesis of three-dimensional isotropy is valid in a certain rough approximation.

Abstract: The spectrum of atmospheric turbulence is very broad by comparison with spectra in wind tunnels We introduce the notion of small-scale and large-scale turbulence Small-scale turbulence consists of a set of disturbances, the scales of which do not exceed the distance to the wall and for which the hypothesis of three-dimensional isotropy is valid in a certain rough approximation Large-scale turbulence is essentially anisotropic; the horizontal scale in the atmosphere is much larger than the vertical one, the latter being confined to a certain characteristic height H The horizontal scale varies widely according to the external conditions and characteristics of the medium

867 citations

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TL;DR: In this paper, the authors investigated how far the orbit of a particle of more general shape in a non-uniform shear in the presence of rigid boundaries may be expected to be qualitatively similar.

Abstract: According to Jeffery (1923) the axis of an isolated rigid neutrally buoyant ellipsoid of revolution in a uniform simple shear at low Reynolds number moves in one of a family of closed periodic orbits, the centre of the particle moving with the velocity of the undisturbed fluid at that point. The present work is a theoretical investigation of how far the orbit of a particle of more general shape in a non-uniform shear in the presence of rigid boundaries may be expected to be qualitatively similar. Inertial and non-Newtonian effects are entirely neglected.The orientation of the axis of almost any body of revolution is a periodic function of time in any unidirectional flow, and also in a Couette viscometer. This is also true if there is a gravitational force on the particle in the direction of the streamlines. There is no lateral drift. On the other hand, certain extreme shapes, including some bodies of revolution, will assume one of two orientations and migrate to the bounding surfaces or to the centre of the flow. In any constant slightly three-dimensional uniform shear any body of revolution will ultimately assume a preferred orientation.

813 citations

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TL;DR: In this paper, Segre and Silberberg showed that a rigid sphere transported along in Poiseuille flow through a tube is subject to radial forces which tend to carry it to a certain equilibrium position at about 0.6 tube radii from the axis, irrespective of the radial position at which the sphere first entered the tube.

Abstract: It is shown that a rigid sphere transported along in Poiseuille flow through a tube is subject to radial forces which tend to carry it to a certain equilibrium position at about 0.6 tube radii from the axis, irrespective of the radial position at which the sphere first entered the tube. It is further shown that the trajectories of the particles are portions of one master trajectory and that the origin of the forces causing the radial displacements is in the inertia of the moving fluid. An analysis of the parameters determining the behaviour is presented and a phenomenological description valid at low Reynolds numbers is arrived at in terms of appropriate reduced variables. These phenomena have already been described in a preliminary note (Segre & Silberberg 1961). The present more complete analysis confirms the conclusions, but it appears that the dependence of the effects on the particle radius go with the third and not the fourth power as was then reported.It is also shown that the description of the phenomena becomes more complicated at tube Reynolds numbers above about 30.

813 citations

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TL;DR: In this paper, the effect of dust on the stability of a gas is described by two parameters; the concentration of dust and a relaxation time τ which measures the rate at which the velocity of a dust particle adjusts to changes in the gas velocity and depends upon the size of the individual particles.

Abstract: The equations describing the motion of a gas carrying small dust particles are given and the equations satisfied by small disturbances of a steady laminar flow are derived. The effect of the dust is described by two parameters; the concentration of dust and a relaxation time τ which measures the rate at which the velocity of a dust particle adjusts to changes in the gas velocity and depends upon the size of the individual particles. It is shown that if the dust is fine enough for τ to be small compared with a characteristic time scale associated with the flow, then the addition of dust destabilizes a gas flow; whereas if the dust is coarse so that τ is relatively large, then the dust has a stabilizing action.For plane parallel flow, it is shown that the stability characteristics for a dusty gas are still determined by solutions of the Orr-Sommerfeld equation, but with the basic velocity profile replaced by a modified profile which is in general complex. A simple, although unrealistic, example is used to illustrate some features of the action of dust. It is intended to describe the solution of the modified Orr-Sommerfeld equation for plane Poiseuille flow in a later paper.

721 citations

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TL;DR: In this paper, it is shown that vortex breakdown is not a manifestation of instability or of any other effect indicated by study of infinitesimal disturbances alone, but instead a finite transition between two dynamically conjugate states of axisymmetric flow, analogous to the hydraulic jump in open-channel flow.

Abstract: The phenomenon examined is the abrupt structural change which can occur at some station along the axis of a swirling flow, notably the leading-edge vortex formed above a delta wing at incidence. Contrary to previous attempts at an explanation, the belief demonstrated herein is that vortex breakdown is not a manifestation of instability or of any other effect indicated by study of infinitesimal disturbances alone. It is instead a finite transition between two dynamically conjugate states of axisymmetric flow, analogous to the hydraulic jump in openchannel flow. A set of properties essential to such a transition, corresponding to a set shown to provide a complete explanation for the hydraulic jump, is demonstrated with wide generality for axisymmetric vortex flows; and the interpretation covers both the case of mild transitions, where an undular structure is developed without the need arising for significant energy dissipation, and the case of strong ones where a region of vigorous turbulence is generated. An important part of the theory depends on the calculus of variations; and the comprehensiveness with which certain properties of conjugate flow pairs are demonstrable by this analytical means suggests that present ideas may be useful in various other problems.

656 citations

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TL;DR: In this paper, the frequency and amplification rates for a disturbance growing with respect to time are compared with those of a spatially growing wave having the same wave number, and it is shown that the frequencies are equal to a high order of approximation.

Abstract: The frequency and amplification rates for a disturbance growing with respect to time are compared with those of a spatially-growing wave having the same wave-number. For small rates of amplification it is shown that the frequencies are equal to a high order of approximation, and that the spatial growth is related to the time growth by the group velocity.

632 citations

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TL;DR: In this article, the authors used a hot film flowmeter in the sea and presented experimental measurements of the downstream component of turbulent velocity in a tidal channel, where the Reynolds number of the flow is about 108 and the scale of the turbulence is so large that a ship is carried about to a considerable extent by the energy-containing eddies.

Abstract: This paper describes the use of a hot film flowmeter in the sea and presents experimental measurements of the ‘downstream’ component of turbulent velocity in a tidal channel. The Reynolds number of the flow is about 108 and the scale of the turbulence is so large that a ship is carried about to a considerable extent by the energy-containing eddies. Under these conditions, a velocity measuring probe attached to a ship cannot be used for reliable measurements in the energy-containing range of the spectrum. It is possible, however, to observe the intertial and dissipation ranges. Records have been made at various stages of the tide. The one-dimensional spectra are found to be proportional to for several decades in k as predicted by Kolmogoroff, and a value is given for Kolmogoroff's constant. In the dissipation range there is close agreement with both Kovasznay's theory and Heisenberg's theory. These two theories are not very different in the low wave-number end of the range and the observations do not extend to high enough wave-numbers to distinguish between them.

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TL;DR: In this article, the authors studied the stability of steady axisymmetric parallel flows of uniform fluid in the absence of rigid boundaries, and they focused on the stability characteristics in the limit of large Reynolds number and determined the (integral) value of n at which the growth rate is a maximum in these simpler circumstances.

Abstract: This paper is a contribution to the mathematical analysis of the stability of steady axisymmetric parallel flows of uniform fluid in the absence of rigid boundaries. A jet at sufficiently high Reynolds number for the angle of viscous spreding to be small is the typical example of the primary flows considered, and the theoretical velocity profile far downstream in such a jet is kept in mind continually. It is obvious from experience that such jets are unstable, presumably to infinitesimal disturbances, but there is little observational data about the critical Reynolds number or the mode of disturbance that grows most rapidly at a given Reynolds number.The typical small disturbance considered is a Fourier component with sinusoidal dependence on both ax and nϕ (x, r, ϕ are cylindrical polar co-ordinates). There is no analogue of Squire's theorem for two-dimensional primary flows, and both α and n are essential parameters of the disturbance. We have concentrated on the stability characteristics in the limit of large Reynolds number, and have aimed in particular at determining the (integral) value of n at which the growth rate is a maximum in these simpler circumstances.A number of general results for inviscid fluid are established, many of them analogues of corresponding results for two-dimensional primary flow. A necessary condition for the existence of amplified disturbances is that should have a numerical maximum a t some point in the fluid; this condition is satisfied for all n in the case of a cylindrical shear layer or ‘top-hat’ jet profile (for which a complete solution of the disturbance equation can be obtained), and for n > 1 in the case of a ‘far-downstream’ jet profile. The wave speed cr of a neutral disturbance is equal to the value of U either a t the point where dQ/dr = 0 or at r = 0. In the latter case the eigen-function (if one exists) is singular at the axis in general; the former case is presumably relevant to the ‘upper branch’ of the curve of neutral stability (for given n). The Reynolds stress due to the disturbance acting across a cylindrical surface is examined. Here, as in some other contexts, it is useful to consider components of velocity parallel and perpendicular to a circular helix on which the phase of the disturbance wave is constant. For a neutral disturbance the component of disturbance velocity parallel to the local wave helix is infinite a t the critical point where U = cr, (corresponding to the known singularity for a three-dimensional disturbance to two-dimensional flow), and there is a peak in the Reynolds stress there.It is shown from the form of the disturbance equation that there is an upper limit to the value of n (≠ 0) for a neutral (inviscid) disturbance with cr equal to the value of U at the point where Q′ = 0. In the case of a jet with a ‘far-downstream’ profile, only the value n = 1 satisfies this restriction; thus only the sinuous mode n = 1 can yield amplified disturbances in an inviscid fluid. A numerical investigation shows that for this profile the wave-number of the neutral disturbance with n = 1 is α = 1·46.

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TL;DR: In this paper, an apparatus is described for determining particle passages through any selected point on a tube cross-section, based on the simultaneous blocking out of two mutually perpendicular light beams by a particle passing through their common region.

Abstract: An apparatus is described for determining particle passages through any selected point on a tube cross-section. The method depends on the simultaneous blocking out of two mutually perpendicular light beams by a particle passing through their common region. The coincidence is registered and counted electronically. At higher particle concentrations coincidences are also registered arising from a pair-wise occupation of the light beams by two particles. An analysis is presented showing that these pair coincidences can be allowed for exactly in terms of experimentally measurable quantities.The reliability and reproducibility of the method is discussed and illustrated by examples from sphere suspensions in Poiseuille flow.

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TL;DR: In this paper, the effect of viscous dissipation in natural convection is considered for vertical surfaces subject to both isothermal and uniform-flux surface conditions, and the first temperature perturbation function is calculated for Prandtl numbers from 10−2 to 104.

Abstract: The effect of viscous dissipation in natural convection is appreciable when the induced kinetic energy becomes appreciable compared to the amount of heat transferred. This occurs when either the equivalent body force is large or when the convection region is extensive. Viscous dissipation is considered here for vertical surfaces subject to both isothermal and uniform-flux surface conditions. A perturbation method is used and the first temperature perturbation function is calculated for Prandtl numbers from 10−2 to 104. The magnitude of the effect depends upon a dissipation number, which is not expressible in terms of the Grashof or the Prandtl number.

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TL;DR: In this article, the authors measured the turbulent pressure field at the wall beneath a thick (5-inch) turbulent boundary layer produced by natural transition on a smooth surface and found that the root-mean-square wall pressure was 2.19 times the wall shear stress.

Abstract: Measurements of the turbulent pressure field at the wall beneath a thick (5-inch) turbulent boundary layer produced by natural transition on a smooth surface are reported. The data include the mean-square pressure, parallel to the stream, and spatial correlation of the pressure transverse to the stream.The root-mean-square wall pressure was 2.19 times the wall shear stress. The power spectra of the pressure were found to scale with the free-stream speed and the boundary-layer displacement thickness. A few tests with a rough surface showed that the increase in root-mean-square wall pressure was greater than the increase in wall shear stress.The space-time correlation measurements parallel to the stream direction exhibit maxima at certain time delays corresponding to the convection of pressure-producing eddies at speeds varying from 0.56 to 0.83 times the stream speed. The lower convection speeds are measured when the spatial separation of the pressure transducers is small, or when only the pressure fluctuations at high frequencies are correlated. Higher convection speeds are observed when the spatial separation of the pressure transducers is large, or when only low frequencies are correlated. The result that low-frequency pressure fluctuations have the highest convection speed is in agreement with the measurements of Corcos (1959, 1962) in a fully turbulent tube flow. Analysis of these measurements also shows that both large- and small-scale pressure-producing eddies decay after travelling a distance proportional to their scale. More precisely, a pressure-producing eddy of large or small wavelength λ decays and vanishes after travelling a distance of approximately 6λ.The transverse spatial correlation of the wall-pressure fluctuations was measured and compared with the longitudinal scale. Both the transverse and the longitudinal scale of the pressure fluctuations were of the order of the boundary-layer displacement thickness. The transverse and longitudinal scales of both large- and small-scale wall-pressure fluctuations were also measured and were also found to be approximately the same.

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TL;DR: In this paper, some general stability criteria for non-dissipative swirling flows are derived, and extended to the case of an electrically conducting fluid in the presence of axial magnetic field and current.

Abstract: Some general stability criteria for non-dissipative swirling flows are derived, and extended to the case of an electrically conducting fluid in the presence of axial magnetic field and current. In particular, it is shown that the analogy between a rotating and a stratified fluid holds in this case, and that an important determinant of stability is a Richardson number'' based on the analog of the density gradient and the shear in the axial flow. (auth)

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TL;DR: In this article, it is shown that a similarity solution can be obtained once it is recognized that the velocity of the front may be less than that of the steady plume, while the solutions for these two cases cannot be matched directly, since the dependence of velocity on height is different.

Abstract: The advancing front of a buoyant plume which is being established in uniform surroundings has some properties in common with the plume, while in other respects it behaves more like a ‘thermal’ released from rest. The solutions for these two cases cannot be matched directly, since the dependence of velocity on height is different. It is shown here that a similarity solution, which is consistent with the equations describing both parts of the flow, can be obtained once it is recognized that the velocity of the front may be less than that of the steady plume. The cap moves with a constant fraction of the plume velocity at the same level, and the total buoyancy in the cap is increasing, so a modification of the simple relations for thermals is required.This prediction is verified experimentally, and numerical values for the ratio of the velocities and the rate of increase of the cap radius with height determined. The extreme front of plume cap advances at about 0·6 times the mean velocity on the axis of a steady plume, and it spreads at just over half the angle of a thermal. This implies a smaller rate of entrainment and therefore a smaller rate of dilution per unit height compared with a thermal, especially since about half of the fluid mixed into the cap comes from the plume below. The model allows one to estimate the time necessary for convection from a known steady source on the ground to lead to the formation of a cloud.

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TL;DR: In this paper, the effect of damping on boundary-layer stability has been investigated and it has been shown that damping is associated with a decrease of the total kinetic and elastic energy of the fluid and the wall, so that any dissipation of energy in the wall will only increase the wave amplitude.

Abstract: The stability of small two-dimensional travelling-wave disturbances in an incompressible laminar boundary layer over a flexible surface is considered. By first determining the wall admittance required to maintain a wave of given wave-number and phase speed, a characteristic equation is deduced which, in the limit of zero wall flexibility, reduces to that occurring in the ordinary stability theory of Tollmien and Schlichting. The equation obtained represents a slight and probably insignificant improvement upon that given recently by Benjamin (1960). Graphical methods are developed to determine the curve of neutral stability, as well as to identify the various modes of instability classified by Benjamin as ‘Class A’, ‘Class B’, and ‘Kelvin-Helmholtz’ instability, respectively. Also, a method is devised whereby the optimum combination of surface effective mass, wave speed, and damping required to stabilize any given unstable Tollmien-Schlichting wave can be determined by a simple geometrical construction in the complex wall-admittance plane.What is believed to be a complete physical explanation for the influence of an infinite flexible wall on boundary-layer stability is presented. In particular, the effect of damping in the wall is discussed at some length. The seemingly paradoxical result that damping destabilizes class A waves (i.e. waves of the Tollmien-Schlichting type) is explained by considering the related problem of flutter of an infinite panel in incompressible potential flow, for which damping has the same qualitative effect. It is shown that the class A waves are associated with a decrease of the total kinetic and elastic energy of the fluid and the wall, so that any dissipation of energy in the wall will only make the wave amplitude increase to compensate for the lowered energy level. The Kelvin-Helmholtz type of instability will occur when the effective stiffness of the panel is too low to withstand, for all values of the phase speed, the pressure forces induced on the wavy wall.The numerical examples presented show that the increase in the critical Reynolds number that can be achieved with a wall of moderate flexibility is modest, and that some other explanation for the experimentally observed effects of a flexible wall on the friction drag must be considered.

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TL;DR: In this paper, the Hamilton-Cayley theorem is used to define a theory of the stress tensor, which is a function of the components of the rate of deformation tensor and a symmetric tensor describing the microscopic structure of a fluid.

Abstract: A theory is proposed in which the stress tensor is a function of the components of the rate of deformation tensor and a symmetric tensor describing the microscopic structure of a fluid. The expression for the stress tensor can be written in closed form using results from the Hamilton-Cayley theorem. This theory is shown to contain Prager's theory of dumbbell suspensions as a special case. By limiting the type of terms in the constitutive equations, various stress components can be evaluated for simple shear. These exhibit non-Newtonian behaviour typical of certain higher polymer solutions. Some of the results of the anisotropic fluid theory are compared with experimental measurements of normal stress and apparent viscosity. Certain high polymers in solution show good agreement between theory and experiment, at least for low enough values of the rate of shear.

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TL;DR: In this article, a cylindrical vortex, formed in a long tube, was used to study the "vortex breakdown" that has been previously reported in investigations of the flow over slender delta wings.

Abstract: This paper describes an experiment in which a cylindrical vortex, formed in a long tube, was used to study the ‘vortex breakdown’ that has been previously reported in investigations of the flow over slender delta wings. By varying the amount of swirl that was imparted to the fluid before it entered the tube, it was found that the breakdown was the intermediate stage between the two basic types of rotating flows, that is, those that do and those that do not exhibit axial velocity reversal. In addition, it was shown that an unusual flow pattern was established after the breakdown and that certain features of this pointed to it being a ‘critical’ phenomenon. The tests were concluded by measuring the swirl angle distribution a short distance ahead of the breakdown and comparing these results with the prediction of Squire's theory (1960).

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TL;DR: The instability of annular coating of liquid on a wire or on the inside of a small tube subject to capillary forces at its free surface is discussed in this article, where it is shown that for given values of s/a, the ratio of the two radii of the annulus, and S ≡ρTa/μ2, the reciprocal of the square of the Ohnesorge number, there is a disturbance of a certain wavelength (2πa/λ*) which grows more rapidly than disturbances of any other wavelength.

Abstract: The instability of an annular coating of liquid on a wire or on the inside of a small tube subject to capillary forces at its free surface is discussed. It was found that for given values of s/a, the ratio of the two radii of the annulus, and S ≡ρTa/μ2, the reciprocal of the square of the Ohnesorge number, there is a disturbance of a certain wavelength (2πa/λ)*, which grows more rapidly than disturbances of any other wavelength. One would therefore expect the liquid to break up into a regular pattern of drops with spacing given by this wavelength. The dependence of (2πa/λ)* on s/a and S has been calculated and is presented graphically. Experimental observations on drop formation on wires and in tubes which agree with the calculations are given.

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TL;DR: In this paper, the authors used a digital computer to estimate the shape and amplitude of toroidal vortices in flow between concentric rotating circular cylinders, and showed that the equilibrium amplitude is almost proportional to 1 − m, where m is the ratio of the angular speeds of the inner and inner cylinders.

Abstract: In flow between concentric rotating circular cylinders, it was shown by Taylor (1923) that instability may occur in the form of toroidal vortices spaced regularly along the axis. When the vortex motion occurs additional torque is required to keep the cylinders in motion at given speeds. Stuart (1958) used an energy-balance method, in the case when the annular gap is small compared with the radius, to estimate the additional torque and the associated finite amplitude attained by the vortices. He included the effect of distortion of the mean motion, but ignored the generation of harmonics of the fundamental mode and the distortion of the velocity associated with the fundamental mode. It is now known that these are not valid mathematical approximations and a rigorous perturbation expansion is developed here to remedy the deficiency. The analysis is valid for any gap width and any angular speeds of the containing cylinders, but requires the amplification rate of the disturbance to be small.Numerical results using a digital computer are obtained for the shape and amplitude of the vortices in three cases: (i) when the outer cylinder has twice the radius of the inner one and is kept at rest, (ii) when the gap is small and the cylinders rotate with nearly the same speeds, and (iii) when the gap is small and the outer cylinder is kept at rest. The equilibrium amplitude obtained in the last case is substantially the same as that found by Stuart.The results for cases (i) and (iii) give close agreement with the experimental values obtained by Taylor (1936) and Donnelly (1958) for the torque required to keep the inner cylinder rotating with constant speed while the outer one is at rest, for a certain range of speeds. In the small-gap problem it is shown that the equilibrium amplitude is almost proportional to 1 − m, where m is the ratio of the angular speeds of the outer and inner cylinders.

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TL;DR: In this paper, a general theory for the effect of wall proximity on the Stokes resistance of an arbitrary particle was proposed, where the motion of the particle is parallel to a principal axis of resistance.

Abstract: A general theory is put forward for the effect of wall proximity on the Stokes resistance of an arbitrary particle. The theory is developed completely for the case where the motion of the particle is parallel to a principal axis of resistance. In this case, the wall-effect correction can be calculated entirely from a knowledge of the force experienced by the particle in an unbounded fluid, providing (i) that the wall correction is already known for a spherical particle and (ii) that the particle is small in comparison to its distance from the boundary. Experimental data are cited which confirm the theory. The theory is extended to the wall effect on a particle rotating near a boundary.

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TL;DR: In this article, it was shown experimentally that the amount of viscous fluid left on the walls of a horizontal tube when it is expelled by an inviscid fluid, reaches an asymptotic value of 0.60 of the amount required to fill the tube, when the parameter μU/T is increased, μ and T being the coefficients of viscosity and interfacial surface tension respectively, and U the velocity of the interface between the two fluids.

Abstract: Two problems are considered. First, it is shown experimentally that the amount of viscous fluid left on the walls of a horizontal tube, when it is expelled by an inviscid fluid, reaches an asymptotic value of 0.60 of the amount required to fill the tube, when the parameter μU/T is increased, μ and T being the coefficients of viscosity and interfacial surface tension respectively, and U the velocity of the interface between the two fluids. Secondly, by neglecting the inertia terms in the equations of motion and the effect of gravity, a theory for the passage of this type of bubble is presented, together with experimental results in support of the theory. It is shown that such a solution is only valid under certain other restrictions, and then only to within half a tube diameter of the nose of the bubble.

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TL;DR: In this article, it was shown that when k1 and k2 are parallel or antiparallel, the interaction vanishes unless k1 = k2 or k2 = k1.

Abstract: In a previous paper, Phillips (1960) showed that two or three trains of gravity waves may interact so as to produce a fourth (tertiary) wave whose wave-number is different from any of three primary wave-numbers k1, k2, k3, and whose amplitude grows in time. Such resonant interactions may produce an appreciable modification of the spectrum of ocean waves within a few hours. In this paper, by a slightly different method, the interaction is calculated in detail for the simplest possible case: when two of the three primary wave-numbers are equal (k3 = k1).It is found that, when k1 and k2 are parallel or antiparallel, the interaction vanishes unless k1 = k2. Generally, if θ denotes the angle between k1 and k2, the rate of growth of the tertiary wave with time is a maximum when θ [eDot ] 17°; the rate of growth with horizontal distance is a maximum when θ [eDot ] 24°. The calculations show that it should be possible to detect the tertiary wave in the laboratory.

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TL;DR: The transfer of energy from wind to short surface waves through the viscous Reynolds stress in the immediate neighbourhood of the surface is explored in this paper, where the resonance between the Tollmien-Schlichting waves for a given wind profile and the free-surface waves is shown to be an important possibility.

Abstract: The transfer of energy from wind to short surface waves through the viscous Reynolds stress in the immediate neighbourhood of the surface is explored. Resonance between the Tollmien-Schlichting waves for a given wind profile and the free-surface waves is shown to be an important possibility. Numerical results are given for water wave generation by a wind having a velocity profile that is linear in the neighbourhood of the surface and asymptotically logarithmic.

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TL;DR: In this paper, a solution to the problem of the steady one-dimensional flow of an incompressible, viscous, electrically conducting fluid through a circular pipe in the presence of an applied (transverse) uniform magnetic field was obtained.

Abstract: The solution is obtained to the problem of the steady one-dimensional flow of an incompressible, viscous, electrically conducting fluid through a circular pipe in the presence of an applied (transverse) uniform magnetic field. A no-slip condition on the velocity is assumed at the non-conducting wall. The solution is exact and thus valid for all values of the Hartmann number. Excellent agreement exists between the present theoretical results and the experimental values obtained by Hartmann & Lazarus (1937) in the low to medium Hartmann number range. The high Hartmann number case is treated by Shercliff (1962) in the following paper.

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TL;DR: In this article, the energy-sharing mechanism involved in phase velocity effects and possible resonances associated with the interactions of gravity waves is discussed from a different viewpoint, which demonstrates more clearly the energy sharing mechanism involved.

Abstract: In earlier papers Phillips (1960) and Longuet-Higgins (1962) have investigated phase velocity effects and possible resonances associated with the interactions of gravity waves. In this note the problem is discussed from a different viewpoint which demonstrates more clearly the energy-sharing mechanism involved. Equations governing the time dependence of the resonant modes are obtained, rather than the initial growth rate as has been found previously.

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TL;DR: The formation of vortex streets in the wake of two-dimensional bluff bodies can be explained by considering the non-linear interaction of two infinite vortex sheets, initially a fixed distance, h, apart, in an inviscid incompressible fluid as discussed by the authors.

Abstract: The formation of vortex streets in the wake of two-dimensional bluff bodies can be explained by considering the non-linear interaction of two infinite vortex sheets, initially a fixed distance, h, apart, in an inviscid incompressible fluid. The interaction of such sheets (represented in the calculation by rows of point-vortices) is examined in detail for various ratios of h to the wavelength, a, of the initial disturbance. The number and strength of the concentrated regions of vorticity formed in the interaction depend very strongly on h/a. The non-linear interaction of the two vortex sheets explains both the cancellation of vorticity and vortex-street broadening observed in the wakes of bluff bodies.