bluuebunny/crossref_metadata_2025_split huggingface dataset:
The motion of long bubbles in tubes
Authors: Bretherton F. P. 
Published: March 2006

A long bubble of a fluid of negligible viscosity is moving steadily in a tube filled with liquid of viscosity μ at small Reynolds number, the interfacial tension being σ. The angle of contact at the wall is zero. Two related problems are treated here.In the first the tube radius r is so small that gravitational effects are negligible, and theory shows that the speed U of the bubble exceeds the average speed of the fluid in the tube by an amount UW, where
$W \simeq 1\cdot 29(3 \mu U|\sigma)^{\frac {2}{3}}\;\;\; as\;\;\; \mu U|\sigma$
(This result is in error by no more than 10% provided $\mu U |\sigma \; \textless \;5 \times 10^{-3}\rightarrow 0$). The pressure drop, P, across such a bubble is given by
$P \simeq 3\cdot 58(3\mu U|\sigma)^{\frac {2}{3}}\sigma|r \; \; \;as\; \; \; \mu U|\sigma \rightarrow 0$
and W is uniquely determined by conditions near the leading meniscus. The interface near the rear meniscus has a wave-like appearance. This provides a partial theory of the indicator bubble commonly used to measure liquid flowrates in capillaries. A similar theory is applicable to the two-dimensional motion round a meniscus between two parallel plates. Experimental results given here for the value of W agree well neither with theory nor with previous experiments by other workers. No explanation is given for the discrepancies.In the second problem the tube is wider, vertical, and sealed at one end. The bubble now moves under the effect of gravity, but it is shown that it will not rise at all if
$\rho gr^2| \sigma \; \textless \; 0 \cdot 842,$
where ρ is the difference in density between the fluids inside and outside the bubble. If
$0 \cdot 842 \; \textless \; 1 \cdot 04,$ then $\rho gr^2| \sigma - 0 \cdot842 \simeq 1 \cdot 25 (\mu U|\sigma)^{\frac {2}{9}} + 2 \cdot 24(\mu U|\sigma)^{\frac {1}{3}},$
accurate to within 10%. Experiments are adduced in support of these results, though there is disagreement with previous work.
A theory for the core of a leading-edge vortex
Authors: Hall M. G. 
Published: March 2006

In the flow past a slender delta wing at incidence one can observe a roughly axially symmetric core of spiralling fluid, formed by the rolling-up of the shear layer that separates from a leading edge. The aim in this paper is to predict the flow field within this vortex core, given appropriate conditions at its outside edge.The basic assumptions are (i) that the flow is continuous and rotational, and (ii) that viscous diffusion is confined to a relatively slender subcore. In addition it is assumed that the flow is axially symmetric and incompressible. Together, these admit outer and inner solutions for the core from the equations of motion. For the outer solution the subcore is ignored, and the flow is taken to be inviscid (but rotational) and conical. The resulting solution consists of simple expressions for the velocity components and pressure. For the inner solution, which applies to the diffusive subcore, the flow is taken to be laminar, and certain approximations are made, some based on the boundary conditions and some analogous to those of boundary-layer theory. The solution obtained in this case is a first approximation, and has been computed.A sample calculation yields results which are in good qualitative and fair quantitative agreement with experimental measurements.
Analytical theory of turbulent diffusion
Authors: Roberts P. H. 
Published: March 2006

Recently Kraichnan (1959) has propounded a theory of homogeneous turbulence, based on a novel perturbation method, that leads to closed equations for the velocity covariance. In the present paper, this method is applied to the theory of turbulent diffusion and closed equations are derived for the probability distributions of the positions of marked fluid elements released in a turbulent flow.Two topics are discussed in detail. The first is the probability distribution, at timet, of the displacement of an element from its initial position. In homogeneous flows, this distribution is found to resemble that for classical diffusion but with a variable coefficient of diffusion which is proportional to$v^2_0 t$for$t \ll l|v_0$and which approaches a constant value [eDot ]lv0fortt[Gt ]l/v0(l= macroscale,v0= r.m.s. turbulent velocity).The second topic treated is the joint probability distribution of the displacements of two fluid elements. Particular attention is focused upon the probability distribution of relative displacement, i.e. Richardson's distance-neighbour function. This is found to be Gaussian for separationsrwhich are large ([Gt ]l). For smaller separations (r[Lt ]l), its behaviour at high Reynolds numbers is found to be quite well expressed in terms of a variable diffusion coefficientK(r,t), as suggested by Richardson (1926). For all but extremely short times,K(r,t) is found to depend only onrand on the form of the inertial range spectrumE(k). On assuming$E(r) \propto v^2_0 l(kl)^{- \frac {3}{2}}$as results from Kraichnan's approximation (1959), one finds$E(r) \propto v_0 l(r|l)^{ \frac {3}{2}}$. On the basis of similarity arguments of the Kolmogorov type, which give$E(r) \propto v^2_0 l(kl)^{- \frac {5}{2}}$, one finds$E(r) \propto v_0 l(r|l)^{ \frac {4}{3}}$as, in fact, Richardson originally proposed. The dispersionr2is proportional to$l^2(v_0 t|l)^4$on Kraichnan's theory; while$\langle r^2 \rangle \propto l^2 (v_0 t|l)^3$on the similarity theory. This illustrates that the behaviour of$\langle r^2 \rangle$is very sensitive to the spectrum.
Unsteady, viscous, circular flow
Authors: Sibulkin Merwin 
Published: March 2006

In this paper a study of the energy-transfer processes associated with the motion of a viscous, heat-conducting fluid is begun. The class of motions considered are unsteady, two-dimensional, vortical flows. After developing simplified equations of motion and energy appropriate to this type of flow in the low Mach-number limit, general solutions of the momentum equations are presented.The concept of a line impulse of angular momentum is introduced as an example of this class of motions for which a solution of the energy field is obtainable in closed form. The solution for the line impulse can be viewed as a combination of velocity, pressure, and temperature waves concurrently radiating from the origin of the impulse and decaying with time. Particular examples of the development of the energy field of the impulse in both liquids and gases are presented for selected values of Prandtl number. The energy-transfer processes are discussed in some detail, and the resulting differences in the energy fields for liquid gases are emphasized.
A simplified analysis of spherical and cylindrical blast waves
Authors: Friedman Manfred P. 
Published: March 2006

Investigations into the behaviour of the gas flowbehind spherical or cylindrical blasts have shown that secondary shocks arise within the original detonation gases. The secondary shock, at first weak, is carried outward with the expanding gases. Subsequently it strengthens and bends back toward the origin, arriving there with high intensity.By using some recently developed techniques in shock dynamics and extending them where necessary, a theory is developed by which the motion of the main shock wave, as well as the formation and subsequent motion of the secondary shock, are given by explicit formulae. In addition, a method for determining, also by explicit formulae, the location of the contact surface between the detonation gases and the outside atmosphere is given. The results of a specific problem, which has been solved by numerically integrating the total equations of motion, and has also been checked experimentally, are compared with the results of the present theory.
Measurements of entrainment by axisymmetrical turbulent jets
Authors: Ricou F. P. , Spalding D. B. 
Published: March 2006

A new technique is described for measuring the axial mass flow rate in the turbulent jet formed when a gas in injected into a reservoir of stagnant air at uniform pressure. The jet is surrounded by a porous-walled cylindrical chamber, and air is injected through the wall until the pressure in the chamber is uniform and atmospheric, a condition which is taken to signify that the ‘entrainment appetite’ of the jet is satisfied.Measurements made with the apparatus have allowed the deduction of an entrainment law relating mass flow rate, jet momentum, axial distance and air density, regardless of the injected gas, and including the effects of buoyancy. When the injected gas burns in the jet the entrainment rate is up to 30% lower than when it does not.
Stewartson's compressibility correlation in three dimensions
Authors: Cooke J. C. 
Published: March 2006

On the assumption of (a) zero heat transfer, (b) unit Prandtl number, (c) linear viscosity temperature relation, and (d) small cross-flow, a correlation between a compressible laminar boundary-layer flow and an incompressible flow with different external conditions is established.The correlation is applied to a known approximate method of solution for incompressible flow. For flow over a slender wing it is found that, so far as the effect on limiting streamlines—and hence on separation—is concerned, the fluid behaves as though it were incompressible but with pressure gradients multiplied by 1 + ½(γ − 1)M20 approximately, where M0 is the Mach number, supposed to be moderate in value. The effect on drag is also estimated.An attempt is made to assess the effect of variable Prandtl number Pr, and it is found that the above multiplying factor must be replaced by
$1+\frac {1}{2}Pr^{\frac{1}{2}}(\gamma - 1)M^2_0$
but that variations in Prandtl number have little effect on the previous estimate of drag.
Equilibrium layers and wall turbulence
Authors: Townsend A. A. 
Published: March 2006

In turbulent flow past rigid boundaries, there can be distinguished regions close to the wall in which the local rates of energy production and dissipation are so large that aspects of the turbulent motion concerned with these processes are determined almost solely by the distribution of shear stress within the region and are independent of conditions outside it. These regions are here called equilibrium layers because of the equilibrium existing between local rates of energy production and dissipation. Three kinds of equilibrium layer have been studied experimentally, the constant-stress layer, the transpiration layer and the zero-stress layer, but there are other possible forms. One that is of importance in the theory of self-preserving flow in boundary layers and in diffusers is the ‘linear-stress’ layer in which the stress increases linearly with distance from the wall. The properties of these various equilibrium layers are considered and the distributions of mean velocity are derived from the equation for the turbulent kinetic energy and certain assumptions of flow similarity.The theory of self-preserving wall flow, usually expressed as a combination of the law of the wall and the defect law, assumes compatibility between the outer flow and the equilibrium layer, and the course of development depends on the kind of equilibrium layer. Earlier work by the author, which assumed the defect law, is only valid if the whole of the equilibrium layer is a constant-stress layer and this is not true in strong adverse pressure gradients. A consistent theory is developed for these flows by assuming a ‘linear-stress’ layer, and the solutions show the relation between flows of finite stress and of zero stress and provide a plausible explanation of the phenomenon of downstream instability observed by Clauser. Self-preserving flow in wedges is treated on similar lines.
On the dynamics of unsteady gravity waves of finite amplitude Part 2. Local properties of a random wave field
Authors: Phillips O. M. 
Published: March 2006

Expressions in closed form are derived for a number of local properties of a random, irrotational wave field. They are: (i) the mean potential and kinetic energies per unit projected area; (ii) the energy balance among the processes of energy input from the surface pressure fluctuations, rate of growth of potential and kinetic energy and horizontal energy flux; and (iii) the partition between potential and kinetic energy. These expressions are mainly in terms of quantities measured at the free surface, which are therefore functions of only two spatial variables (x, y) and of time t.Approximations for these expressions can be found simply by subsequent expansion methods; the fourth order being the highest for which the assumption of irrotational motion is appropriate in a real fluid. It is shown that the mean product of any three first-order quantities is of fourth or higher order in the root-mean-square wave slope, and this result is applied in estimating the magnitude of some higher order effects. In particular, the skewness of the surface displacement is of the order of the root-mean-square surface slope, which has been confirmed observationally by Kinsman (1960).
The development of three-dimensional disturbances in an unstable film of liquid flowing down an inclined plane
Authors: Benjamin T. Brooke 
Published: March 2006

On the basis of results from a previous paper, expressions are found for the phase velocity and amplification rate of a wave travelling obliquely to the direction of flow. This wave comprises the general harmonic component of three-dimensional small disturbances, and accordingly a double Fourier integral is introduced to represent a bounded disturbance whose initial distribution over the free surface may be arbitrarily prescribed. Hence an asymptotic approximation is derived for a disturbance which is initially concentrated around a point on the free surface. Several distinctive properties of a localized unstable disturbance are noted: for instance, it lies mainly within an elliptical region whose area increases linearly with time as it moves downstream and which is modulated by long-crested waves. An experimental observation of a growing disturbance on an unstable film is recorded, and its main features are seen to be in agreement with the theory.In so far as linearized perturbation theory remains applicable, the effects investigated are common to a wide class of parallel and nearly parallel laminar flows. In the final part of the paper the method used to analyse the instability of a film is generalized in order to reveal the connexion between this and other problems; this aim is achieved by demonstrating collective properties of the complete class of flows in question, but particular reference is made to the example of laminary boundary layers and Poiseuille flow between parallel planes.
The effect of a magnetic field on the flow of a conducting fluid past a circular disk
Authors: Chester W. , Moore D. W. 
Published: March 2006

In the previous paper (Chester 1961) it was shown that, for large values of the Hartmann number, the asymptotic solution for the flow past a body of revolution has a discontinuity on the surface of a cylinder which circumscribes the body. The flow in the region of this discontinuity is now investigated in more detail when the body is a circular disk broadside-on to the flow. It will be shown that there is actually a region of transition whose thickness is O(|x|½/M½), where x is the axial distance from the disk and M is the Hartmann number. This region is thin near the disk, but gradually thickens until it merges into the over-all flow field for x = O(M).The leading terms in the expression for the drag are given by
$\frac{D}{D_s} = \frac{M \pi }{8} \left( 1 + \frac{2}{M} \right) $,
where Ds is the Stokes drag.
On the theory of the wind-driven ocean circulation
Authors: Carrier G. F. , Robinson A. R. 
Published: March 2006

A surface distribution of stress is imposed on an ocean enclosed by two continental boundaries; the resulting transport circulation is studied between two latitudes of zero surface wind-stress curl, within which the curl reaches a single maximum. Under the assumption that turbulent transfer of relative vorticity has a minimum effect on the mean circulation, inviscid flow patterns are deduced in the limit of small transport Rossby number. Inertial currents, or naturally scaled regions of high relative vorticity, occur on both the eastern and the western continental coasts. Limits on the relative transports of the currents are obtained and found to depend on the direction of variation of the wind-stress curl with latitude, relative to that of the Coriolis accelerations. The most striking feature of the inviscid flow is a narrow inertial current the axis of which lies along the latitude of maximum wind-stress curl. All eastward flow occurs in this midlatitude jet.A feature of the flow which cannot remain essentially free of turbulent processes is the integrated vorticity relationship, since the imposed wind-stress distribution acts as a net source of vorticity for the ocean. Heuristic arguments are used together with this integral constraint to deduce the presence and strength of the turbulent diffusion which must occur in the region of the mid-latitude jet. It is further inferred that the turbulent meanders of the jet must effect a net meridional transport of relative vorticity.
Resonant interactions between two trains of gravity waves
Authors: Longuet-Higgins M. S. 
Published: March 2006

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.
The shape and stability of a bubble at the axis of a rotating liquid
Authors: Rosenthal D. K. 
Published: March 2006

The shape of a bubble of one liquid inside a denser body of liquid which rotate rigidly together is determined, the effect of gravity being neglected. When the anular velocity of the liquids is zero the bubble assumes a spherical form, and with increasing angular velocity the bubble flattens at the equator and the length increases. It is found that the length of the bubble is asymptotically proportional to the four-thirds power of the angular velocity. If the speed of the rotation is held constant and the volume is increased, then the bubble elongates, the radius approaches a limiting value, and the bubble length increases almost linearly with the volume. This result suggests a method whereby the interfacial surface tension can be measured.In the second part of the paper the stability of a long bubble subjected to small amplitude axisymmetric disturbances sinusoidal in the axial direction is investigated. The relation between the wave-number and angular velocity for neutral stability is elliptic. When account is taken of the decrease in the radius of the undisturbed bubble with increase in the angular velocity, it is found that the bubble is stable to all wave-lengths provided the radius attains at least 63% of the limiting value. A criterion is then found for the minimum length of the bubble consistent with stability.
The barotropic stability of the mean winds in the atmosphere
Authors: Lipps Frank B. 
Published: March 2006

This paper considers the stability of a barotropic current on a beta earth. The motion is assumed to be horizontal, non-divergent and barotropic. The current is taken to be of the formU(y)=Asech2by+B. The perturbations are required to approach zero asyapproaches ± ∞. We introduce the non-dimensional wave-numberland a parameter χ, which is a measure of the rotation effect. χ is inversely proportional to β.There are only two kinds of perturbations: symmetric disturbances (those with maximum amplitude aty= 0) and antisymmetric disturbances (those with zero amplitude aty= 0). We find the neutral curve in the (χ,l2)-plane for both types of disturbances. The rates of amplification in the immediate vicinity of the neutral curves are also found. It is seen that the beta effect, which is due to the earth's rotation, tends to stabilize the current. For the symmetric disturbances we find a band of unstable wavelengths when χ > 1/2; and for large χ the estimated curve of the maximum value of the imaginary part of the phase velocity is asymptotic to the lower branch of the neutral curve. The antisymmetric disturbances are more stable than the symmetric disturbances.
On the stability of laminar flow of a dusty gas
Authors: Saffman P. G. 
Published: March 2006

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.