Showing papers in "Journal of Fluid Mechanics in 1964"

Viscous and resistive eddies near a sharp corner

Abstract: Some simple similarity solutions are presented for the flow of a viscous fluid near a sharp corner between two planes on which a variety of boundary conditions may be imposed. The general flow near a corner between plane boundaries at rest is then considered, and it is shown that when either or both of the boundaries is a rigid wall and when the angle between the planes is less than a certain critical angle, any flow sufficiently near the corner must consist of a sequence of eddies of decreasing size and rapidly decreasing intensity. The ratios of dimensions and intensities of successive eddies are determined for the full range of angles for which the eddies exist. The limiting case of zero angle corresponds to the flow at some distance from a two-dimensional disturbance in a fluid between parallel boundaries. The general flow near a corner between two plane free surfaces is also determined; eddies do not appear in this case. The asymptotic flow at a large distance from a corner due to an arbitrary disturbance near the corner is mathematically similar to the above, and has comparable properties. When the fluid is electrically conducting, similarity solutions may be obtained when the only applied magnetic field is that due to a line current along the intersection of the two planes; it is shown that the effect of such a current is to widen the range of corner angles for which eddies must appear.

Saltation of uniform grains in air

Abstract: The interaction between a turbulent wind and the motion of uniform saltating grains of sand or soil, so massive as to fail to enter into suspension, is examined on the basis of two complementary hypotheses. The first asserts that the effect of the moving grains on the fluid outside the region to which saltation is confined is similar to that of solid roughness of height comparable with the depth of the saltation layer. The second requires the concentration of particles engaging in the saltation to adjust itself so that the shear stress exerted by the wind on the ground—different from that acting on the fluid outside the saltation layer by an amount accountable to the change in horizontal momentum suffered by the particles in their passage through the fluid—is just sufficient to maintain the sand-strewn surface in a mobile state.Existing experimental data on the wind profiles outside the saltation region and the horizontal flux of particles through it are shown to be consistent with these hypotheses.The second hypothesis implies a self-balancing mechanism for controlling the concentration of saltating particles. For if the concentration is too low the shear stress at the surface rises above the value required merely to secure mobility and more particles are encouraged to leave the surface; conversely, too large a concentration depresses the surface stress, and the consequent loss of surface mobility inhibits saltation and reduces th concentration of particles until equilibrium is restored.

The transition to turbulence in the wake of a circular cylinder

Abstract: The position of the region of transition to turbulence and the manner in which turbulence develops are investigated using a hot-wire anemometer to study the character of the flow in the wake of a circular cylinder. In the range of Reynolds numbers greater than 200 in which turbulent motion is developed, the region of transition to turbulence moves towards the cylinder with increasing Reynolds number. The manner of transition to turbulence appears to undergo a basic change as the region of transition moves from the periodic wake into the region of flow immediately behind the cylinder where the separated layers have not rolled into vortices, that is, as the Reynolds number increases from Roshko's transition range to his irregular range. When transition occurs in the periodic wake it is a result of distortion due to large-scale three-dimensional effects. Turbulence, when it develops in the separated layers, is preceded by two-dimensional Tollmien-Schlichting waves which eventually degenerate to turbulence by the action of small-scale three-dimensionalities.

On the inviscid instability of the hyperbolictangent velocity profile

Abstract: The Rayleigh stability equation of inviscid linearized stability theory was integrated numerically for amplified disturbances of the hyperbolic-tangent velocity profile. The evaluation of the eigenvalues and eigenfunctions is followed by a discussion of the streamline pattern of the disturbed flow. Here no qualitative distinction is found between an amplified and the neutral disturbance. But considering the vorticity distribution of the disturbed flow it is shown that in the case of amplified disturbances two concentrations of vorticity occur within a disturbance wavelength, while in the neutral case only one maximum of vorticity exists. The results are discussed with respect to the instability mechanism of free boundary-layer flow.

On cellular convection driven by surface-tension gradients: effects of mean surface tension and surface viscosity

Abstract: The onset of steady, cellular convection driven by surface tension gradients on a thin layer of liquid is examined in an extension of Pearson's (1958) stability analysis. By accounting for the possibility of shape deformations of the free surface it is found that there is no critical Marangoni number for the onset of stationary instability and that the limiting case of ‘zero wave-number’ is always unstable. Surface viscosity of a Newtonian interface is found to inhibit stationary instability. A simple criterion is found for distinguishing visually the dominant force, buoyancy or surface tension, in cellular convection in liquid pools.

Axial flow in trailing line vortices

Abstract: A characteristic feature of a steady trailing line vortex from one side of a wing, and of other types of line vortex, is the existence of strong axial currents near the axis of symmetry. The purpose of this paper is to account in general terms for this axial flow in trailing line vortices. the link between the azimuthal and axial components of motion in a steady line vortex is provided by the pressure; the radial pressure gradient balances the centrifugal force, and any change in the azimuthal motion with distance x downstream produces an axial pressure gradient and consequently axial acceleration.It is suggested, in a discussion of the evolution of an axisymmetric line vortex out of the vortex sheet shed from one side of a wing, that the two processes of rolling-up of the sheet and of concentration of the vorticity into a smaller cross-section should be distinguished; the former always occurs, whereas the latter seems not to be inevitable.In § 4 there is given a similarity solution for the flow in a trailing vortex far downstream where the departure of the axial velocity from the free stream speed is small. The continual slowing-down of the azimuthal motion by viscosity leads to a positive axial pressure gradient and consequently to continual loss of axial momentum, the asymptotic variation of the axial velocity defect at the centre being as x−1 log x.The concept of the drag associated with the core of a trailing vortex is introduced, and the drag is expressed as an integral over a transverse plane which is independent of x. This drag is related to the arbitrary constant appearing in the above similarity solution.

The structure of the turbulent pressure field in boundary-layer flows

Abstract: The paper is discussion of measurements of the statistical properties of the pressure field at the wall of turbulent attached shear flows. These measurements have been made only in part by the author. A preliminary discussion is given of the important limitations imposed by the imperfect space resolution of contemporary pressure transducers. There follows a discussion of the appropriate scales of the pressure field. It is shown that measurements of the longitudinal cross-spectral densities lead to similarity variables for the space-time covariance of the pressure and for the corresponding spectra. The existence of these similarity variables may be due to the dispersion of the sources of pressure by the mean velocity gradient. Such a mechanism is illustrated by a simple model. Lateral cross-spectral densities also lead approximately to similarity variables.Computations based directly upon detailed pressure-velocity correlation measurements by Wooldridge & Willmarth reveal that an important part of the pressure at the wall of a boundary layer is contributed by source terms which are quadratic in the turbulent velocity fluctuations; the interaction of the mean strain rate with normal velocity fluctuations, being in effect limited to a region very near the wall, supplies a dominant contribution only at high frequencies and its scales, downstream convective speed and convective memory are markedly smaller than those of the observed wall pressure.The inner part of the Law of the Wall region (y* [les ] 100) seems to be substantially free of pressure sources and within that region (a) the pressure can be given in terms of its boundary value, and (b) the local velocity field is dependent upon but unbale to affect appreciably the turbulent pressures.

Surface tension and buoyancy effects in cellular convection

Abstract: The cells observed by Benard (1901) when a horizontal layer of fluid is heated from below were explained by Rayleigh (1916) in terms of buoyancy, and by Pearson (1958) in terms of surface tension. These rival theories are now combined. Linear perturbation techniques are used to derive a sixth-order differential equation subject to six boundary conditions. A Fourier series method has been used to obtain the eigenvalue equation for the case where the lower boundary surface is a rigid conductor and the upper free surface is subject to a general thermal condition. Numerical results are presented. It was found that the two agencies causing instability reinforce one another and are tightly coupled. Cells formed by surface tension are approximately the same size as those formed by buoyancy. Benard's experiments are briefly discussed.

The radial spread of a liquid jet over a horizontal plane

Abstract: When a smooth jet of water falls vertically on to a horizontal plane, it spreads out radially in a thin layer bounded by a circular hydraulic jump, outside which the depth is much greater. The motion in the layer is studied here by means of boundary-layer theory, both for laminar and for turbulent flow, and relations are obtained for the radius of the hydraulic jump. These relations are compared with experimental results. The analogous problems of two-dimensional flow are also treated.

Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile

Abstract: An investigation is carried out to determine the conditions marking the onset of convective motion in a horizontal fluid layer in which a negative temperature gradient occurs somewhere within the layer. In such cases, fluid of greater density is situated above fluid of lesser density. Consideration is given to a variety of thermal and hydrodynamic boundary conditions at the surfaces which bound the fluid layer. The thermal conditions include fixed temperature and fixed heat flux at the lower bounding surface, and a general convective-radiative exchange at the upper surface which includes fixed temperature and fixed heat flux as special cases. The hydrodynamic boundary conditions include both rigid and free upper surfaces with a rigid lower bounding surface. It is found that the Rayleigh number marking the onset of motion is greatest for the boundary condition of fixed temperature and decreases monotonically as the condition of fixed heat flux is approached. Non-linear temperature distributions in the fluid layer may result from internal heat generation. With increasing departures from the linear temperature profile, it is found that the fluid layer becomes more prone to instability, that is, the critical Rayleigh number decreases.

On the deformation and drag of a falling viscous drop at low Reynolds number

Abstract: The motion at low Reynolds number of a drop in a quiescent unbounded fluid is investigated theoretically by means of a singular-perturbation solution of the axisymmetric equations of motion. Special attention is paid to the deformation of the drop. It is shown that for small values of the Weber number W e the drop will first deform exactly into an oblate spheroid and then, with a further increase in W e, into a geometry approaching that of a spherical cap. These results are quite insensitive to the ratio of the viscosities of the two fluid phases. The first-order effect of the deformation on the drag of the drop is also included in the analysis.

Forces on a sphere accelerating in a viscous fluid

Abstract: A detailed equation is proposed for the force exerted on a sphere that accelerates rectilinearly in an otherwise still fluid. In addition to the buoyant force, the fluid exerts forces that depend on (a) the velocity of the sphere, (b) the acceleration of the sphere and (c) the history of the motion. The equation reduces to the known theoretical solution for low velocity and large acceleration.The proposed equation was tested and found most satisfactory for a particular case in which the velocity was not small, viz. the case of simple harmonic motion along a straight line. The acceleration (added mass) and history coefficients in the equation were evaluated experimentally. They were found to depend on the ratio of the convective acceleration to the local acceleration as measured by the parameter V2/aD, in which V, a and D are the velocity, acceleration and diameter of the sphere, respectively. The Reynolds numbers varied from 0 to 62 during the tests.

The drag of a compressible turbulent boundary layer on a smooth flat plate with and without heat transfer

Abstract: The theoretical treatments given by earlier authors are classified, reviewed and where necessary extended; then the predictions of twenty of these theories are evaluated and compared with all available experimental data, the root-meansquare error being computed for each theory The theory of van Driest-II gives the lowest root-mean-square error (110%)A new calculation procedure is developed from the postulate that a unique relation exists between cfFc and RFR where cf is the drag coefficient, R is the Reynolds number, and Fc and FR are functions of Mach number and temperature ratio alone The experimental data are found to be too scanty for both Fc and FR to be deduced empirically, so Fc is calculated by means of mixing-length theory and FR is found semi-empirically Tables and charts of values of Fc and FR are presented for a wide range of MG and TS/TG When compared with all experimental data, the predictions of the new procedure give a root-mean-square error of 99%

The production and diffusion of vorticity in duct flow

Abstract: Secondary flows in non-circular ducts are accompanied by a longitudinal component of vorticity. The equation of motion defining this component in a turbulent flow is composed of three terms giving the rates of production, diffusion and convection. Since the expression for production is the second derivative of Reynolds strees components, longitudinal vorticity cannot exist in laminar flow. For turbulent flow in a square duct the Reynolds stress tensor is examined in detail. Symmetry requirements alone provide relationships showing that the production is zero along all lines of symmetry. General characteristics of flow in circular pipes are sufficient to indicate where the production must be greatest. Experimental measurements verify this result and define the point density of production, diffusion and convection of vorticity. Data also indicate that the basic pattern of secondary flow is independent of Reynolds number, but that with increasing values of Reynolds number the flows penetrate the corners and approach the walls. A similar experimental investigation of a rectangular duct shows that the corner bisectors separate independent secondary flow circulation zones. Production of vorticity is again associated with the region near the bisector. However, there is some evidence that the secondary flow pattern is not so complex as inferred from the distortion of the main longitudinal flow.

Resonant oscillations in closed tubes

Abstract: An investigation is made of the disturbances produced in a closed, gas-filled tube by the oscillations of a piston at one end, when the piston oscillates at near resonant frequencies. Within a well-defined frequency band around each resonant frequency, shock waves appear in the solution; outside this interval the oscillations are continuous, but not purely sinusoidal.The solution includes the effects of compressive viscosity, and of shear viscosity in the boundary layer at the walls of the tube. For typical laboratory conditions the effect of compressive viscosity is found to be quite small (giving a shock thickness of the order of 10−4in.). The boundary layer effect can be more significant, though the most important modification required of the usual acoustic theory is found to arise from the non-linear terms.

Turbulence in the noise-producing region of a circular jet,

Abstract: The flow in the noise-producing region of a circular jet is found to be dominated by a group of large eddies, containing nearly a quarter of the turbulent shear stress in the quasi-plane region of the shear layer: their contribution to the shear stress decreases as the effects of axisymmetry become noticeable at more than about two diameters downstream of the nozzle. These large eddies appear to be almost entirely responsible for the irrotational fluctuations near the nozzle, which, for this and other reasons, are larger relative to the reference dynamic pressure than in other shear flows. As a consequence of this, the convection velocity near the high- and low-velocity edges of the flow is biased towards the mean velocity in the high-intensity region. The dominance of the large eddies therefore explains the measurements of near-field pressure fluctuations by Franklin & Foxwell (1958), and of convection velocity by Davies, Barratt & Fisher (1963) and the present authors. The strength of these large eddies, compared with those in the boundary layer or wake, is remarkable.The large eddies appear to be mixing-jets similar to those found by Grant (1958) in the wake, but with their projection in the (y, z)-plane inclined at about 45° to the y (radial) axis instead of lying along the y-axis as in the wake.It is suggested that the augmentation of these large eddies by artificial means could be used to increase the mixing rate and permit the reduction of jet noise by means of acceptably short ejector shrouds.The medium-scale motion is found to be far from isotropic in scales, although the two scales associated with a given vorticity component are more nearly equal. This phenomenon is also noticeable in the wake.It is found that the departure from self-preservation, which starts when the shear layer thickness is no longer small compared with the nozzle radius, does not grossly affect the region of high turbulence intensity and maximum noise production until this region itself is no longer small compared with the radius. The maximum shear stress seven diameters downstream of the exit is still 70% of its value near the exit, and the non-dimensional mean velocity gradient is practically unchanged.

An experimental investigation of the steady separated flow past a circular cylinder

Abstract: The steady separated flow past a circular cylinder was investigated experimentally. By artificially stabilizing the steady wake, this system was studied up to Reynolds numbers R considerably larger than any previously attained, thus providing a much clearer insight into the asymptotic character of such flows at high Reynolds numbers. Some of the experimental results were unexpected. It was found that the pressure coefficient at the rear of the cylinder remained unchanged for 25 [les ] R [les ] 177, that the circulation velocity within the wake approached a non-zero limit as the Reynolds number increased, and that the wake length increased in direct proportion to the Reynolds number.

Study of rarefied shear flow by the discrete velocity method

Abstract: The application of a simple discrete velocity model to low Mach number Couette and Rayleigh flow is investigated. In the model, the molecular velocities are restricted to a finite set and in this study only eight equal speed velocities are allowed. The Boltzmann equation is reduced by this approximation to a set of coupled differential equations which can be solved in closed form. The fluid velocity and shear stress in Couette flow are in approximate accord with those of Wang Chang & Uhlenbeck (1954) and of Lees (1959) over the complete range of Knudsen number. Similarly, the Rayleigh flow solution is remarkably like those found by other investigators using moment methods.

Note on a heterogeneous shear flow

Abstract: Goldstein (1931) has considered the stability of a shear layer within which the velocity and the density vary linearly and outside which they are constant. Rayleigh (1880, 1887) had found that the corresponding, homogeneous shear flow is unstable in and only in a finite band of wave-numbers. Goldstein concluded that a small density gradient renders the flow unstable for all wave-numbers. This conclusion appears to depend on the acceptance of all possible branches of a multi-valued eigenvalue equation, and it is shown that the principal branch of this eigenvalue equation yields one and only one unstable mode if and only if the wave-number lies in a band that decreases from Rayleigh's band to zero as the Richardson number increases from 0 to ¼.

The statistical dynamics of homogeneous turbulence

Abstract: The steady distribution function for homogeneous turbulence is studied starting from Liouville's equation, modified by the introduction of an instantaneously fluctuating external force, which acts as a random source of energy. A new technique for solving Liouville's equation is presented giving a systematic development of the concepts of turbulent diffusion and turbulent viscosity. It amounts to a consistent generalization of the random phase approximation. When the rate of input of energy into the kth Fourier component uk has a power form h|k|−α, the functional form of the mean value 〈 uku−k 〉 can be determined exactly in the limit of large Reynolds number; it is .

Resonant interactions between waves. The case of discrete oscillations

Abstract: The mathematical basis for resonance is investigated using a model equation describing one-dimensional dispersive waves interacting weakly through a quadratic term. If suitable time-invariant boundary conditions are imposed, possible oscillations of infinitesimal amplitude are restricted to a discrete set of wave-numbers. An asymptotic expansion valid for small amplitude shows that oscillations of different wave-number interact primarily in independent resonant trios. Energy is redistributed between members of a trio over a characteristic time inversely proportional to the amplitude of the oscillations in a periodic manner. The period depends on the initial conditions but is in general finite. Cubic interactions through resonant quartets are also discussed. The methods used are valid for a fairly wide class of equations describing weakly non-linear dispersive waves, but the expansion procedure used here fails for a continuous spectrum.

The unsteady flow within a spinning cylinder

Abstract: A theoretical analysis is given of the unsteady flow of a liquid within a cylinder of finite length started suddenly so as to spin about its axis. It is found that a secondary flow, caused by the end walls of the cylindrical container, has a strong effect on the generation of spin in the liquid. In the vicinity of the end walls the fluid motion is characterized by a boundary-layer flow, which can be either laminar or turbulent. The fluid within the boundary layers rotates faster than that at a large distance from the end walls, and therefore is thrown, by centrifugal forces, radially outwards. The radial outflow in the boundary layer creates a slow secondary motion within the spinning liquid. Due to the secondary flow, the transport of angular momentum from the walls to the interior is accomplished by convection rather than diffusion. A treatment is given for both laminar and turbulent end-wall boundary layers. The theoretical results are compared with experimental observations and good agreement is found.

On convection velocities in turbulent shear flows

Abstract: The problem of defining an effective velocity of convection for turbulent fluctuations in a shear flow is considered, and the definitions adopted by various workers are discussed. An experiment in the shear layer of a circular jet has shown that the usual definitions, based on peaks of the space-time correlations of the fluctuations, yield convection velocities whose magnitudes depend on the value of space or time separation chosen. An alternative approach shows that, by considering the turbulent field as a superposition of harmonic travelling waves, a wave-number/velocity spectrum can be defined that lends itself to the definition of a wave-number-dependent convection velocity and an overall convection velocity, both of which have real physical significance. An experimental technique is described for obtaining the spectrum, and results are presented for one position in the shear layer of the jet.

Correlation measurements in a non-frozen pattern of turbulence

Abstract: The properties of a turbulent flow are often described in terms of velocity correlations in space, in time, and in space-time. In this paper the interpretation of velocity correlation measurements which are made in a region of highintensity turbulence is considered in some detail. Under these conditions it is shown that some account must be taken of the effects of both mean and fluctuating shear stresses which are continuously modifying the turbulent structure. For an almost frozen pattern, for example, in the turbulence behind a grid, the turbulent convection velocity is amost equal to the mean flow velocity, while the space correlation and auto-correlation of the velocity fluctuations are simply related through this velocity. In contrast to this, when the intensity is high, the convection velocity may differ considerably from the mean velocity, while it is shown that different turbulent spectral components appear to travel at different speeds. This means that the turbulent spectrum and the turbulent space scales are no longer simply related. For example, the high-frequency spectral components may be ascribed to both the high-velocity eddies and the small wave-number components acting together.Experimental results are presented which indicate the conditions under which the assumption of a frozen pattern leads to uncertainties in the subsequent interpretation of the measurements. The measurements also show that the observed difference between the mean and the convection velocity may be qualitatively explained in terms of the skewness of the velocity signals.

A study of the sprays formed by impinging jets in laminar and turbulent flow

Abstract: A study has been made of the factors influencing the break-down of sheets formed by the impingement of two liquid jets. It is shown that disintegration generally results from the formation of unstable waves of aerodynamic or hydrodynamic origin. While the characteristics of the former waves are fairly well understood, little is known about the latter. The results of this study indicate that hydrodynamic (or ‘impact’) waves are generated when the Weber number of each jet (ρlDV2 sin 2θ/γ, where ρl is the liquid density, D the jet diameter, V the mean jet velocity, θ the half-angle of impingement, and γ is the surface tension) is above a critical value, and that their formation is independent of the Reynolds number. Drop sizes have been measured and are shown to be critically dependent upon the mechanism of disintegration.

Experiments on jet flows and jet noise far-field spectra and directivity patterns

Abstract: The results of measurements of far-field sound emitted from jets are reported. The narrow-band power spectral density of the sound in the far field was measured for three jet diameters, three Mach numbers, and five angular positions. The intensity distribution of mean-square pressure fluctuation in the far field in several wide frequency ranges were also measured. The similarity relations found from the experiments are reported.

Magnetohydrodynamic free convection

Abstract: The flow of an electrically conducting fluid up a hot vertical plate in the presence of a strong magnetic field normal to the plate is considered. A solution is developed based on the idea of matching ‘outer’ and ‘inner’ solutions in the moving layer of fluid. An approximate Pohlhausen method of solution is also given which yields results in fairly good agreement with the exact analysis.

The conservation equations for a non-equilibrium plasma

Abstract: The paper is concerned with formulation of the gas dynamic conservation equations for the individual species in a non-equilibrium partially ionized gas mixture. As an example, the conservation equations for the electrons and the overall conservation equations are developed for a three component plasma consisting of electrons, singly-ionized positive ions and neutral atoms. Non-elastic collisions are represented by the collisional-radiative decay mechanism of Bates, Kingston & McWhirter (1962a, b). Maxwellian velocity distributions are assumed, but the electrons are allowed to have a temperature different from the heavier particles and to drift relative to them. Particular attention is given to the electron energy balance equation which differs from that used by other investigators.

Turbulent flow characteristics of viscoelastic fluids

Abstract: : The turbulent flow characteristics of viscoelastic fluids are investigated quantitatively. The outstanding property of these flow fields is seen to be a very pronounced suppression of turbulence, accompanied by major reductions in the turbulent drag coefficients. Careful measurements of the rheological properties of the several fluids used suggest that the observed turbulence suppression is a function of the ratio of the elastic to the viscous forces developed in the fluid. An empirical correlation of the results, based upon this observation, is proposed; the present data, while indicative, are not sufficiently extensive to verify conclusively the existence of a quantitative correlation. In a number of respects, the observed reduction in drag is similar to that which may be obtained through promotion of 'slip' at the tube wall or by addition of particulate matter. It is shown that slip phenomena are clearly distinct from those studied in the present work but that particulate effects (albeit of much greater magnitude than observed heretofore) cannot be ruled out as contributory mechanisms. Further studies are thus required to determine the relative importance of continuum (viscoelastic) and particulate effects. (Author)

The behaviour of clusters of spheres falling in a viscous fluid Part 1. Experiment

Abstract: The sedimentation of small clusters of uniform spheres, falling freely through a viscous liquid, has been studied with Reynolds numbers (based on diameter of the sphere and its velocity of free fall in the unbounded fluid) of individual spheres ranging from 10−4 to 10. The fall velocity of a cluster is, in all cases, greater than that of individual spheres, the more so when the spheres are closer together. Two spheres falling side-by-side rotate inwards and separate as they fall if Re > 0·05, but no rotation nor separation is observed for Re 1 fall vertically one behind the other, the rear sphere is accelerated into the wake of the leader, rotates, round it and separates from it when the line of centres is horizontal. If two spheres of unequal size but the same individual terminal velocity fall together, the smaller always travels faster than the larger. When three similar equally spaced spheres are dropped in a horizontal line, they interchange positions but do not separate when 0·06 7, or the cluster contains 7 or more equal spheres, it shows no tendency to form a regular polygon but breaks up into two or more groups. A regular heptagon, and a hexagon with an additional sphere at its centre, are also unstable.