Showing papers in "Journal of Fluid Mechanics in 1961"

The motion of long bubbles in tubes

Abstract: 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 (This result is in error by no more than 10% provided ). The pressure drop, P, across such a bubble is given by 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 where ρ is the difference in density between the fluids inside and outside the bubble. If accurate to within 10%. Experiments are adduced in support of these results, though there is disagreement with previous work.

On the stability of heterogeneous shear flows

Abstract: Small perturbations of a parallel shear flow U(y) in an inviscid, incompressible fluid of variable density ρ0(y) are considered. It is deduced that dynamic instability of statically stable flows ( is the wave speed.

Experiments on the flow past a circular cylinder at very high Reynolds number

Abstract: Measurements on a large circular cylinder in a pressurized wind tunnel at Reynolds numbers from 10^6 to 10^7 reveal a high Reynolds number transition in which the drag coefficient increases from its low supercritical value to a value 0.7 at R = 3.5 × 10^6 and then becomes constant. Also, for R > 3.5 × 10^6, definite vortex shedding occurs, with Strouhal number 0.27.

Measurements of entrainment by axisymmetrical turbulent jets

Abstract: 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.

Note on a paper of John W. Miles

Abstract: The theorem X established by Miles in the preceding paper is here given a simpler and more general proof. Some further theoretical results concerning the stability of heterogeneous shear flows are also presented, in particular a demonstration that the complex wave velocity of any unstable mode must lie in a certain semicircle.

The transverse force on a spinning sphere moving in a viscous fluid

Abstract: The flow about a spinning sphere moving in a viscous fluid is calculated for small values of the Reynolds number. With this solution the force and torque on the sphere are computed. It is found that in addition to the drag force determined by Stokes, the sphere experiences a force FL orthogonal to its direction of motion. This force is given by .Here a is the radius of the sphere, Ω is its angular velocity, V is its velocity, ρ is the fluid density and R is the Reynolds number, . For small values of R, the transverse force is independent of the viscosity μ. This force is in such a direction as to account for the curving of a pitched baseball, the long range of a spinning golf ball, etc. It is used as a basis for the discussion of the flow of a suspension of spheres through a tube.The calculation involves the Stokes and Oseen expansions. A representation of solutions of the Oseen equations in terms of two scalar functions is also presented.

Deposition of a viscous fluid on the wall of a tube

Abstract: Measurements of the amount of fluid left behind when a viscous liquid is blown from an open-ended tube are described.

Equilibrium layers and wall turbulence

Abstract: 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.

Swimming of a waving plate

Abstract: The purpose of' this paper is to study the basic principle of fish propulsion. As a simplified model, the two-dimensional potential flow over a waving plate of finite chord is treated. The solid plate, assumed to be flexible and thin, is capable of performing the motion which consists of a progressing wave of given wavelength and phase velocity along the chord, the envelope of the wave train being an arbitrary function of the distance from the leading edge. The problem is solved by applying the general theory for oscillating deformable airfoils. The thrust, power required, and the energy imparted to the wake are calculated, and the propulsive efficiency is also evaluated. As a numerical example, the waving motion with linearly varying amplitude is carried out in detail. Finally, the basic mechanism of swimming is elucidated by applying the principle of action and reaction.

The changes in amplitude of short gravity waves on steady non-uniform currents

Abstract: The common assumption that the energy of waves on a non-uniform current U is propagated with a velocity (U + c) where cg is the group-velocity, and that no further interaction takes place, is shown in this paper to be incorrect. In fact the current does additional work on the waves at a rate γijSij where γij is the symmetric rate-of-strain tensor associated with the current, and Sij is the radiation stress tensor introduced earlier (Longuet-Higgins & Stewart 1960).In the present paper we first obtain an asymptotic solution for the combined velocity potential in the simple case (1) when the non-uniform current U is in the direction of wave propagation and the horizontal variation of U is compensated by a vertical upwelling from below. The change in wave amplitude is shown to be such as would be found by inclusion of the radiation stress term.In a second example (2) the current on the x-axis is assumed to be as in (1), but the horizontal variation in U is compensated by a small horizontal inflow from the sides. It is found that in that case the wave amplitude is also affected by the horizontal advection of wave energy from the sides.From cases (1) and (2) the general law of interaction between short waves and non-uniform currents is inferred. This is then applied to a third example (3) when waves encounter a current with vertical axis of shear, at an oblique angle. The change in wave amplitude is shown to differ somewhat from the previously accepted value.The conclusion that non-linear interactions affect the amplification of the waves has some bearing on the theoretical efficiency of hydraulic and pneumatic breakwaters.

On the behaviour of liquid dispersions in mixing vessels

Abstract: The present paper is concerned with the conditions of flow in tanks containing stirred fluids An attempt is made to apply the theoretical concepts of local isotropy to explain the behaviour of liquid in liquid dispersions, subjected to turbulent agitation Relations describing quantitatively the influence of turbulence on both break-up and coalescence of individual droplets are derived and are compared with experimental evidence A special type of dispersion is described in which droplet size is controlled by the prevention of coalescence due to turbulence The dependence of droplet size on energy dissipation per unit mass, as predicted by the theory of local isotropy, is put to an experimental test using geometrically similar vessels of different sizesThough the results are not entirely conclusive, experimental evidence suggests that the hypothesis of locally isotropic flow may be applicable to the flow conditions described in the paper, and that statistical theories of turbulence can be of practical value in estimating droplet sizes in agitated dispersions

The two-dimensional turbulent wall-jet

Abstract: With the aid of a hot-wire anemometer, the mean velocity distribution of the incompressible, turbulent, plane wall-jet has been examined in some detail. As previously reported, this fully developed boundary layer belongs to the class of self-preserving flows. Over the entire range of experimental conditions, a single velocity scale , the Boussinesq exchange coefficient e, and the shear stress at the wall have been evaluated. The Reynolds number based on the maximum velocity and the thickness of the boundary layer varied from 22,000 to 106,000.

Boundary-layer flow at a saddle point of attachment

Abstract: This paper is a study of the flow of a viscous incompressible fluid in the immediate neighbourhood of a saddle point of attachment, near which the external flow is irrotational with components {ax, by, −(a + b)z}, where a > 0 > b. It is shown that the flow is of a boundary-layer character, and that part of the boundary-layer flow is reversed when b/a < − 0·4294.On the assumption that such flows are physically plausible, the problem may be solved for all values of b/a [ges ] −1. Even in the limiting case b/a = −1, an effect of the boundary layer is everywhere to draw fluid towards the wall, so that vorticity is still convected towards the wall.Numerical solutions have been computed, and some of the results are presented in the tables and diagrams.

Diffraction of strong shocks by cones, cylinders, and spheres

Abstract: This paper presents experimental investigations of the diffraction of plane strong shocks by several cones, a cylinder, and a sphere. The diffraction pattern, in particular the loci of Mach triple points and the shape of the diffracted shocks are compared with theoretical results obtained from a diffraction theory proposed by Whitham (1957, 1958, 1959). The agreement between theory and experiment is shown to be good. Also given are extensive numerical results supplementing Whitham's papers, and theortical considerations applying Whitham's theory to very blunt bodies.

The mechanism of transition in the wake of a thin flat plate placed parallel to a uniform flow

Abstract: A study was made of the laminar-turbulent transition of a wake behind a thin flat plate which was placed parallel to a uniform flow at subsonic speeds. Experimental results on the nature of the velocity fluctuations have made it possible to classify the transition region into three subregions: the linear region, the non-linear region and the three-dimensional region.In the linear region there is found a sinusoidal velocity fluctuation which is antisymmetrical with respect to the centre-line of the wake. The frequency of fluctuation is proportional to the power of the free-stream velocity, and the amplitude increases exponentially in the direction of flow. The behaviour of small disturbances in the linear region was investigated in detail by inducing velocity fluctuation with an external excitation—actually sound from a loudspeaker. Solutions of the equation of a small disturbance superposed on the laminar flow were obtained numerically and compared with the experimental results. The agreement between the two was satisfactory.When the amplitude of fluctuation exceeds a certain value, the growth rate deviates from being exponential due to non-linear effects. Although velocity fluctuations in the non-linear region are still sinusoidal and two-dimensional, the experimental results on the distributions of amplitude and phase indicate that the flow pattern may be described by the model of a double row of vortices. This configuration lasts until three-dimensional distortion takes place in the final subregion, the three-dimensional region, in which the fluctuation loses regularity and gradually develops into turbulence without being accompanied by abrupt breakdown or turbulent bursts.

A study of the behaviour of a thin sheet of moving liquid

Abstract: An investigation has been carried out on the behaviour of a thin sheet of liquid in connexion with the new method of lacquer application known as ’curtain coating’ A method is described for measuring the velocity of the sheet and an equation deduced for calculating this for a liquid of any viscosity The minimum liquid flow rate required to maintain a stable sheet is discussed, and some effects of the impingement of the sheet on a rapidly moving surface are described

A study of natural convection above a line fire

Abstract: The behaviour of a natural convection plume above a line fire is studied both theoretically and experimentally. In the theoretical treatment, a turbulent plume above a steady two-dimensional finite source of heated fluid in a uniform ambient fluid is investigated. By the use of the lateral entrainment assumption, a quadrature solution has been obtained for each of two separate ranges of a source Froude number, F 1. In neither of these cases can the finite width line source be accurately represented by an equivalent mathematical line source at a lower level. Only the special case, F = 1, can be so represented and its solution is discussed.In the experimental treatment, hot gases, resulting from the burning of a liquid fuel in a long channel burner, are driven upwards by buoyancy and gradually cooled down by the entrainment of ambient air. The average temperature along lines parallel to the channel burner was measured by a piece of resistance wire. For the case of a non-luminous diffusion flame, the effective radiation loss to the surroundings was assumed to be negligible, and,by a comparison of the energy flux supplied from the fuel and the energy flux contained in the plume, the characteristic turbulence entrainment coefficient is determined. By the alternate use of either an absorbing or a reflecting surface for the table-top surrounding a luminous flame, a measurement was made of the energy radiated from the flame that was intercepted by the fire surroundings and subsequently returned to the buoyancy plume by heating the ingested air. These measurements agree with estimates computed from such data as are available. The experimental results relating to the behaviour of the convection plume agree closely with the theoretical predictions in all cases.

The flow of thin liquid films between rollers

Abstract: When rollers, placed horizontally and side by side so that each is half immersed in a tank of liquid, rotate in opposite directions, liquid is carried through the gap between them and divides to form a sheet over each roller. At low speeds the sheets are of uniform thickness across the width of the rollers, but at higher speeds they are regularly ridged owing to alternate increase and decrease in thickness. Preliminary observations led to the development of an approximate theoretical treatment of the even-flow regime and the critical conditions when the ribbed flow is about to begin. Results of this work are in full agreement with detailed experimental results.

On the dynamics of unsteady gravity waves of finite amplitude Part 2. Local properties of a random wave field

Abstract: 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).

A non-linear theory for oscillations in a parallel flow

Abstract: Three-dimensional periodic oscillations in the shear flow region between two parallel streams are considered up to that second order of the oscillation amplitude. It is shown that, as an integral part of the oscillation, there is a mean secondary flow in the nature of a longitudinal vortex. Despite the dissimilarity in the profile of the basic flows, several of the principal features of the calculated results can be compared with those observed for the Blasius flow by Schubauer and Klebanoff & Tidstrom at the National Bureau of Standards.

Gaskinetics and gasdynamics of orifice flow

Abstract: The paper gives the result of a study on the efflux of gases through circular apertures. The problem is considered as an example of a transition from the gasdynamic to the gaskinetic regime.The mass flow of helium, argon and nitrogen was measured for a range of upstream pressures corresponding to (mean free path)/(aperture diameter) from about 50 to 5 × 10−3; within this range the transition from molecular effusion to inviscid, transonic flow takes place. The theory for the two asymptotic limits is discussed and first-order corrections to the free molecular and inviscid limit formulae are given.

The motion of a thin oil sheet under the steady boundary layer on a body

Abstract: A solution is obtained for the motion of a thin oil sheet, of non-uniform thickness, under a boundary layer. The following points are deduced: (a) The oil flows in the direction of the boundary-layer skin-friction, except near separation, where the oil tends to indicate separation too early. These conclusions are independent of oil viscosity. (b) The effect of the oil flow on the boundary-layer motion is very small.The application of the results to the interpretation of oil-flow patterns is briefly considered.

A vortex in an infinite viscous fluid

Abstract: A solution is given for a viscous vortex in an infinite liquid. Similarity arguments lead to a reduction of the equations of motion to a set of ordinary differential equations. These are integrated numerically. A uniform feature is the constant circulation K outside the vortex core, which is also a viscous boundary layer. The circulation decreases monotonically towards the axis. The axial velocity profiles and the radial velocity profiles have several characteristic shapes, depending on the value of the non-dimensional momentum transfer M. The solution has a singular point on the axis of the vortex. The radius of the core increases linearly with distance along the axis from the singularity, and, at a given distance, is proportional to the coefficient of viscosity and inversely proportional to K.Finally, a discussion is given to indicate that intense vortices above a plate, like the confined experimental vortex, or above the ground, like the atmospheric tornado and dust whirl, will not resemble the theoretical vortex except, possibly, far above the plate.

An experimental investigation of the oscillating lift and drag of a circular cylinder shedding turbulent vortices

Abstract: The oscillating lift and drag on circular cylinders are determined from measurements of the fluctuating pressure on the cylinder surface in the range of Reynolds number from 4 × 103 to just above 105.The magnitude of the r.m.s. lift coefficient has a maximum of about 0.8 at a Reynolds number of 7 × 104 and falls to about 0.01 at a Reynolds number of 4 × 103. The fluctuating component of the drag was determined for Reynolds numbers greater than 2 × 104 and was found to be an order of magnitude smaller than the lift.

A theory for the core of a leading-edge vortex

Abstract: 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.

The wave drag of wind over water

Abstract: It is shown to be probable that a large proportion of the drag exerted by a water surface on the wind is in the form of wave drag. As a result the usual relation between the wind profile and the surface stress and roughness length are modified. In particular, close to the surface the relation between the transport of momentum and that of heat and water vapour are different from that obtaining over a rough solid surface.

The reactant concentration spectrum in turbulent mixing with a first-order reaction

Abstract: The power specturm of a passive scalar contaminant undergoing a first-order chemical reaction and isotropic turbulent mixing is deduced for three different spectral ranges: (i) the inertial-convective range; (ii) the viscous-convective and viscous-diffusive ranges for very large Schmidt number; (iii) the inertial-diffusive range for very small Schmidt number. The analysis is restricted to stationary, locally isotropic fields, and to systems so dilute that the heat of reaction has no effect on the reaction rate.

A simplified analysis of spherical and cylindrical blast waves

Abstract: 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.

The amplification of a weak applied magnetic field by turbulence in fluids of moderate conductivity

Abstract: The effect of turbulence on an applied magnetic field is considered in the case when the magnetic Reynolds number Rm is large compared with unity but small compared with the ordinary Reynolds number R of the turbulence. When the applied field is sufficiently weak, it is argued that its effect on the velocity field is negligible. The equation for the field is then linear and its spectrum may be obtained throughout the equlibrium range of wave-numbers. It appears that the spectrum increases as times the electrical conductivity of the fluid. The effect of magnetic forces when these are not negligible is also tentatively considered.

Photographic evidence of the formation and growth of vorticity behind plates accelerated from rest in still air

Abstract: The shedding of vorticity from the edges of a plate, which is accelerated normal to itself from rest in still air, is investigated experimentally by means of a new flow-visualization technique using spark-lighted shadowgraphs and heated air, benzene or other vapours to introduce sufficient changes in density along the surface of discontinuity. The photographs show small-scale undulations and the formation of a number of centres of vorticity along the well known big-scale vortex sheets which roll up along their edges.