Showing papers in "Journal of Fluid Mechanics in 1968"

Gravity currents and related phenomena

Abstract: This paper presents a broad investigation into the properties of steady gravity currents, in so far as they can be represented by perfect-fluid theory and simple extensions of it (like the classical theory of hydraulic jumps) that give a rudimentary account of dissipation. As usually understood, a gravity current consists of a wedge of heavy fluid (e.g. salt water, cold air) intruding into an expanse of lighter fluid (fresh water, warm air); but it is pointed out in Q 1 that, if the effects of viscosity and mixing of the fluids at the interface are ignored, the hydrodynamical problem is formally the same as that for an empty cavity advancing along the upper boundary of a liquid. Being simplest in detail, the latter problem is treated as a prototype for the class of physical problems under study: most of the analysis is related to it specifically, but the results thus obtained are immediately applicable to gravity currents by scaling the gravitational constant according to a simple rule. In Q 2 the possible states of steady flow in the present category between fixed horizontal boundaries are examined on the assumption that the interface becomes horizontal far downstream. A certain range of flows appears to be possible when energy is dissipated; but in the absence of dissipation only one flow is possible, in which the asymptotic level of the interface is midway between the plane boundaries. The corresponding flow in a tube of circular cross-section is found in $3, and the theory is shown to be in excellent agreement with the results of recent experiments by Zukoski. A discussion of the effects of surface tension is included in 0 3. The two-dimensional energy-conserving flow is investigated further in Q 4, and finally a close approximation to the shape of the interface is obtained. In Q 5 the discussion turns to the question whether flows characterized by periodic wavetrains are realizable, and it appears that none is possible without a large loss of energy occurring. In $6 the case of infinite total depth is considered, relating to deeply submerged gravity currents. It is shown that the flow must always feature a breaking ‘head wave’, and various properties of the resulting wake are demonstrated. Reasonable agreement is established with experimental results obtained by Keulegan and others.

Distribution of local pressure and skin friction around a circular cylinder in cross-flow up to Re = 5 × 10 6

Abstract: In a large range of Reynolds numbers, 6 × 104 < Re < 5 × 106, the flow around single cylinders with smooth surfaces has been investigated. The high values of the Reynolds numbers were obtained in a test channel which could be pressurized up to 40 bar of static pressure. New experiments were performed to measure the local pressure and skin friction distribution around the cylinder. From these results the total drag, the pressure drag and the friction drag were calculated. By means of the skin friction distribution the position of the separation points, separation bubbles or transition points can be localized. These data allow one to define three states of the flow: the subcritical flow, where the boundary layer separates laminarly; the critical flow, in which a separation bubble, followed by a turbulent reattachment, occurs; and the supercritical flow, where an immediate transition from the laminar to the turbulent boundary layer is observed at a critical distance from the stagnation point. According to the total drag coefficient the values found in this paper connect the subcritical region represented by the measurements of Wieselsberger (1923) and Fage & Warsap (1930) with the supercritical range in which Roshko (1961) carried out his experiments.

On the equations of motion for mixtures of liquid and gas bubbles

Abstract: On the basis of previous work by the author, equations are derived describing one-dimensional unsteady flow in bubble-fluid mixtures. Attention is subsequently focused on pressure waves of small and moderate amplitude propagating through the mixture. Four characteristic lengths occur, namely, wavelength, amplitude, bubble diameter and inter-bubble distance. The significance of their relative magnitudes for the theory is discussed. It appears that for high gas content the dispersion is weak and then the conservation of mass and momentum lead to equations similar to the Boussinesq equations, describing long dispersive waves of finite amplitude on a fluid of finite depth. For waves propagating in one direction only, the corresponding equation is similar to the Korteweg–de Vries equation. It is shown that for mixtures of low gas content the frequency dispersion is in most cases not small. Finally, solutions of the Korteweg–de Vries equation representing cnoidal and solitary waves in a bubble–liquid mixture are given explicitly.

Wave-induced shear instability in the summer thermocline

Abstract: The fine structure of the summer thermocline in the Mediterranean Sea around Malta, investigated by a new temperature-gradient meter and by photographs of dye tracers, is summarized. The principal internal feature of the thermocline is a series of thin, laminar-flow sheets of high static stability, separated by weakly turbulent layers of only moderate density gradient and a few metres thick. A mechanism for generating the patches of turbulence observed on these thermocline sheets is established by comparing dye photographs with a theory by Miles & Howard (1964).

The oscillations of a fluid droplet immersed in another fluid

Abstract: From an analysis of small oscillations of a viscous fluid droplet immersed in another viscous fluid a general dispersion equation is derived by which frequency and rate of damping of oscillations can be calculated for arbitrary values of droplet size, physical properties of the fluids, and interfacial viscosity and elasticity coefficients. The equation is studied for two distinct extremes of interfacial characteristics: (i) a free interface between the two fluids in which only a constant, uniform interfacial tension acts; (ii) an ‘inextensible’ interface between the two fluids, that is, a highly condensed film or membrane which, to first order, cannot be locally expanded or contracted. Results obtained are compared with those previously published for various special cases.When the viscosities of both fluids are low, the primary contribution to the rate of damping of oscillations is generally the viscous dissipation in a boundary layer near the interface, in both the free and inextensible interface situations. For this reason inviscid velocity profiles, which do not account for the boundarylayer flow, do not lead to good approximations to the damping rate. The two exceptions in which the approximation based on inviscid profiles is adequate occur when the interface is free and either the interior or exterior fluid is a gas of negligible density and viscosity.

The influence of molecular diffusivity on turbulent entrainment across a density interface

Abstract: The rate of mixing across a density interface between two layers of liquid has been measured in a laboratory experiment which allows a direct comparison between heat and salinity transports over the same range of density differences. Low Reynolds number turbulence was produced by stirring mechanically at a fixed distance from the interface, either in one or in both layers, and the results for these two sets of experiments are also compared. The measurements cover a factor of two in stirring rate and twenty in density. Over this range of conditions the ratio of entrainment velocity to stirring velocity can be expressed as functions of an overall Richardson number Ri, and in this form the results of the one and two stirred layer experiments are indistinguishable from one another. For density differences produced by heat alone, the functional dependence is close to Ri−1 except at small values of Ri where it approaches a finite limit. For experiments with a salinity difference across the interface, the mixing rate is the same as in the heat experiments at low values of Ri, but falls progressively below this as Ri is increased, with the approximate form .An interpretation of these results has been attempted, using a dimensional analysis and qualitative mechanistic arguments about the nature of the motion. The Ri−1 dependence implies a rate of change of potential energy proportional to the rate of working by the stirrer. The decreased mixing rates for salt have been attributed to a slower rate of incorporation of an entrained element into its surroundings by diffusion, which increases the tendency for it to return to the interface and dissipate energy in wave-like motions.

Non-linear capillary instability of a liquid jet

Abstract: A third-order theory has been developed to study capillary instability of a liquid jet. The result shows that the asymmetrical development of an initially sinusoidal wave is a non-linear effect with generation of higher harmonics as well as feedback into the fundamental. The growth of the surface wave is found to depend explicitly on the dimensionless initial amplitude of the disturbance and the dimensionless wave-number k of the wave. For the same initial disturbance, the wave is found to have a maximum growth rate at k = 0·7 in agreement with the linearized theory. For the same wave-number, the growth is proportional to the initial amplitude of the disturbance. The cut-off wave-number and the fundamental frequency (or growth rate for the unstable case) of the wave for a given k are found to be different from the linearized theory. Furthermore, at the cut-off wave-number, the present theory shows the disturbance experiences a growth which is proportional to t2. The excellent agreement between Donnelly & Glaberson's experiment and Rayleigh's linearized theory is found to be due to their method of measurement.

Vortex evolution in a round jet

Abstract: A study has been made of the varicose instability of an axisymmetrical jet with a velocity distribution radially uniform at the nozzle mouth except for a laminar boundary layer at the wall. The evolutionary phenomena of instability, such as the rolling up of the cylindrical vortex layer into ring vortices, the coalescence of ring vortex pairs, and the eventual disintegration into turbulent eddies, have been investigated as a function of the Reynolds number using smoke photography, stroboscopic observation, and the light-scatter technique.Emphasis has been placed on the wavelength with maximum growth rate. The jet is highly sensitive to sound and the effects of several types of acoustic excitation, including pure tones, have been determined.

A method of producing a shear flow in a stratified fluid

Abstract: When the end of a long closed horizontal tube containing a stratified fluid is raised, a laminar accelerating flow begins. The flow is two-dimensional in the central portion of the tube, and, in this region, it is predictable, at least until the onset of instability. The occurrence of instability, its nature, and the subsequent transition to turbulence, are described qualitatively. The apparatus may be used for the study of a variety of other internal hydraulic phenomena with applications to meteorology and oceanography.

The stability of thermally stratified plane Poiseuille flow

Abstract: In studying the stability of a thermally stratified fluid in the presence of a viscous shear flow, we have a situation in which there is an important interaction between the mechanism of instability due to the stratification and the Tollmien-Schlichting mechanism due to the shear. A complete analysis has been carried out for the Benard problem in the presence of a plane Poiseuille flow and it is shown that, although Squire's transformation can be used to reduce the three-dimensional problem to an equivalent two-dimensional one, a theorem of Squire's type does not follow unless the Richardson number exceeds a certain small negative value. This conclusion follows from the fact that, when the stratification is unstable and the Prandtl number is unity, the equivalent two-dimensional problem becomes identical mathematically to the stability problem for spiral flow between rotating cylinders and, from the known results for the spiral flow problem, Squire's transformation can then be used to obtain the complete three-dimensional stability boundary. For the case of stable stratification, however, Squire's theorem is valid and the instability is of the usual Tollmien—Schlichting type. Additional calculations have been made for this case which show that the flow is completely stabilized when the Richardson number exceeds a certain positive value.

Steady fluid flow in a precessing spheroidal shell

Abstract: The linear boundary-layer analysis by Stewartson & Roberts (1963) and by Roberts & Stewartson (1965) for the motion of a viscous fluid inside the spheroidal cavity of a precessing rigid body is extended to include effects due to the nonlinear terms in the boundary-layer equation. The most significant consequence is a differential rotation super-imposed on the constant vorticity flow given by the linear theory. In addition it is shown that a tidal bulge of the cavity forces a fluid motion similar to that caused by the precessional torque. The relevance of both effects for the liquid core of the earth is briefly discussed.

The behaviour of a stable salinity gradient heated from below

Abstract: The motions resulting when a linear, stable salt gradient is heated uniformly and at a steady rate from below are investigated theoretically and by laboratory experiment. A convecting, growing layer is first formed whose depth, temperature and salinity differences from the fluid above, are all increasing as t½. The way in which these quantities depend on the salinity gradient and heating rate is also predicted, and verified experimentally. A stability criterion is then developed which describes the breakdown of the diffusive boundary layer ahead of the advancing front, and leads to an expression for the thickness of the bottom layer when a second layer forms above it. The predicted form of dependence of layer thickness on the given parameters is again borne out by the experiments.

Effect of a stabilizing gradient of solute on thermal convection

Abstract: A stabilizing gradient of solute inhibits the onset of convection in a fluid which is subjected to an adverse temperature gradient. Furthermore, the onset of instability may occur as an oscillatory motion because of the stabilizing effect of the solute. These results are obtained from linear stability theory which is reviewed briefly in the following paper before finite-amplitude results for two-dimensional flows are considered. It is found that a finite-amplitude instability may occur first for fluids with a Prandtl number somewhat smaller than unity. When the Prandtl number is equal to unity or greater, instability first sets in as an oscillatory motion which subsequently becomes unstable to disturbances which lead to steady, convecting cellular motions with larger heat flux. A solute Rayleigh number, Rs, is defined with the stabilizing solute gradient replacing the destabilizing temperature gradient in the thermal Rayleigh number. When Rs is large compared with the critical Rayleigh number of ordinary Benard convection, the value of the Rayleigh number at which instability to finite-amplitude steady modes can set in approaches the value of Rs. Hence, asymptotically this type of instability is established when the fluid is marginally stratified. Also, as Rs → ∞ an effective diffusion coefficient, Kρ, is defined as the ratio of the vertical density flux to the density gradient evaluated at the boundary and it is found that κρ = √(κκs) where κ, κs are the diffusion coefficients for temperature and solute respectively. A study is made of the oscillatory behaviour of the fluid when the oscillations have finite amplitudes; the periods of the oscillations are found to increase with amplitude. The horizontally averaged density gradients change sign with height in the oscillating flows. Stably stratified fluid exists near the boundaries and unstably stratified fluid occupies the mid-regions for most of the oscillatory cycle. Thus the step-like behaviour of the density field which has been observed experimentally for time-dependent flows is encountered here numerically.

On the instability of Taylor vortices

Abstract: It is known experimentally that laminar circular Couette flow between two concentric circular cylinders, the outer of which is fixed, becomes unstable when the speed of the inner cylinder is high enough. The flow is then replaced by a new circumferential flow with superimposed toroidal (or Taylor) vortices spaced periodically along the axis. At a higher speed still the new flow develops another instability, and is replaced by a flow in which the axially periodic vortices are simultaneously periodic travelling waves in the azimuth.In the present paper an attack is made on the problem of instability of the Taylor-vortex flow against perturbations which are periodic both in the axial and azimuthal co-ordinates and, moreover, travel with some phase velocity in the latter. Subject to a number of assumptions and approximations, which are detailed in the paper, it is found that the Taylor-vortex flow is stable against perturbations with the same axial wavelength and phase, but unstable against perturbations differing in phase by ½π. After instability the new flow no longer has planes separating neighbouring vortices, but has wavy surfaces travelling in the azimuth. This feature is in accord with much (though not all) of the experimental evidence.The critical Taylor number (proportional to the square of the speed) at which the Taylor vortices become unstable is found theoretically to be about 8% above the value for which Taylor vortices first appear. This must be compared with a value in the range 5-20% for the experiments which our work models most closely. The azimuthal wave-number given a slight preference by theory is 1, in agreement with those experiments.

On the flow between a rotating and a stationary disk

Abstract: The analysis and experiments in this paper are restricted to the flow between two coaxial, infinite disks, one rotating and one stationary. The results of numerical calculations show that many solutions can exist for a given Reynolds number Ωl2/v (Ω is the angular velocity of the rotating disk and I is the spacing between the two disks). Out of a greater number of possible solutions, three solution branches have been identified; the branches correspond to one-, two- and three-flow cells in the meridional plane.The one-cell branch has been accorded detailed treatment. Within this branch there are two subbranches. The first, now well documented in the literature, includes solutions from zero to infinite Reynolds number. The latter limiting case is characterized by an inward-flowing boundary layer on the stationary disk and an outward-flowing boundary layer on the rotating disk. In between is a core flow rotating with a constant angular velocity. The second sub-branch of the single-cell flows, apparently unknown heretofore, begins with an infinite Reynolds number, decreases to a minimum and then increases to an infinite Reynolds number again. The first infinite Reynolds number limit again corresponds to two boundary-layer flows separated by a core flow with constantangular velocity opposite in direction to the angular velocity of the rotating disk. The second limiting case of infinite Reynolds number is the free-disk solution of von Karman (1921). Asymptotic solutions have been obtained which more fully describe the nature of this flow as the Reynolds number increases.The second part of the paper presents experimental measurements corresponding to the Reynolds number range 0–100. Profiles were measured with a hot-wire anemometer. The measurements are in agreement with the first, one-cell branch of solutions. A semi-quantitative evaluation of edge effects is obtained.

On high-frequency oscillatory viscous flows

Abstract: The equations governing high-frequency oscillatory viscous flows are investigated through the separation of the steady and the unsteady parts All Reynolds number ranges are studied and the orders of magnitude of the steady streaming produced by the Reynolds stresses are establishedThe oscillating circular cylinder at low Reynolds numbers is studied through the method of inner and outer expansions Steady recirculating cells exist near the cylinder The results compare very well with experiments Analytic expressions for the streamfunction and the drag coefficient are obtainedThe oscillating flow towards an infinite plate is investigated in detail The steady streaming is caused by the steady component of the Reynolds stress The pressure gradient always causes reverse flow near the solid boundary

Viscoelastic properties of fine-grained incompressible turbulence

Abstract: A number of shear-flow phenomena can be explained qualitatively if turbulence is regarded as a continuous viscoelastic medium with respect to its action on a mean field. Conditions are sought under which the analogy is quantitative, and it is found that the turbulence must be fine-grained and the mean field weak. For geometrical convenience the turbulence is assumed to be nearly homogeneous and isotropic so that body forces are required to maintain it. The turbulence is found to respond initially to an arbitrary deformation as an elastic medium, in which Reynolds stress is linearly proportional to strain. Three processes that cause the resulting Reynolds stress to relax are distinguished: viscous diffusion, body-force agitation and non-linear scrambling. It is argued that, regardless of which process dominates, Reynolds stress evolves in a continuously changing mean field according to a viscoelastic constitutive law, relating stress to deformation history by means of a scalar memory function. The argument is carried through analytically for weak turbulence, in which non-linear scrambling is negligible, and the memory function is computed in terms of the wave-number-frequency spectrum of the background turbulence. In the course of the analysis, a new type of Reynolds stress arises related to the passage of the turbulence through its sustaining environment of body forces. It is found that the mean field must be surprisingly weak for this ‘translation stress’ to be negligible. Applications of the viscoelasticity theory of turbulent shear flow are discussed in which body forces and therefore translation stress are absent.

The spectrum of temperature fluctuations in turbulent flow

Abstract: : Temperature and velocity fluctuations have been recorded in the open sea and in a tidal channel, and power spectra have been determined from the records. The one-dimensional spectra of temperature fluctuations are found to have an inertial subrange. At larger wave-numbers the data can be fitted by Batchelor's spectrum function for the viscous-convective range. The spectra are inconsistent with the form proposed by Pao for the vicsous-convective range. Estimates are given for the constants in Batchelor's spectrum function, but these depend upon knowledge of the rate of dissipation of kinetic energy, which is determined from the velocity spectra. There is doubt about the validity of some of the velocity spectra, and in other cases there is reason to suspect that the turbulence is not locally isotropic. (Author)

Pressure-forcing of tightly fitting pellets along fluid-filled elastic tubes

Abstract: Some insight into the behaviour of tightly fitting solid pellets, which may be deformable, and are being forced by a pressure difference to move slowly along a distensible tube filled with viscous fluid, is sought by theoretical study of a simple axisymmetric model (§2). In this, the pellet's clearance in the tube is taken to be a small fraction of the tube radius; the fraction may, at a pressure characteristic of that ahead of the pellet, be either positive or negative. Even if it is positive, the tube may still be distended (or the pellet compressed, or both) as the pellet passes, because the thickness of lubricating film generated may exceed the clearance. Naturally, still greater elastic deformation can occur in the case of negative clearance.Highly simplified elastic properties are assumed; with an eye on tubes occurring in physiological systems (with Poisson's ratio close to 0.5), the local distension of the tube is taken to vary linearly with the local excess pressure; as a still cruder approximation, a similar relation for local reduction of pellet radius is assumed. A parabolic approximation to the pellet's undistorted meridian section, in the region where the lubricating film is thin, is also assumed, leading to a simple relation between pressure and local film thickness which is used, together with Reynolds's lubrication equation, to evaluate both. An arbitrary constant, the rate of leakback of fluid past the pellet, is determined by the condition that the pressure difference forcing the pellet must just balance the skin-frictional resistance to its motion.The problem is non-dimensionalized (§3) and reduced to that of finding a particular solution of a differential equation containing a certain parameter L. In addition to numerical solutions for particular values of L (§6), perturbation solutions for both small and large L are obtained (§§4 and 5), to give mathematical and physical insight; the perturbation for large L (corresponding to negative clearance) is highly singular, requiring the matching of approximate solutions different in each of six different layers.A striking feature of the solutions is a necking of the gap between pellet and tube behind the pellet. This is so pronounced in the case of negative clearance (figure 2) that it might give the false impression that the pellet was being propelled by peristaltic contraction of the tube instead of by fluid pressure gradient. The physical reason for this is elucidated (§6).In the case of positive clearance, rather small pressure differences suffice, on this theory, to propel the pellet, because different parts of the lubricating layer act on it with frictional resistances of different signs, which almost cancel out. By contrast, for negative clearance, the resistance becomes a large multiple of that found in a purely fluid-filled tube of length and mean velocity equal to that of the pellet. This multiple increases, and the film thickness correspondingly decreases (figure 7), as the pellet velocity decreases.One physiological system on which the model may throw some light is the narrow capillary with red blood cells being squeezed through it in single file, lubricated by plasma (§1 and 8). At the higher flow speeds, around 0.1 mm/s, the lubricating film, predicted to be about 0·2μm thick, appears likely to play a significant role in mass transfer to and from the tissue spaces. At much lower speeds, predicted film thicknesses are so small that any of a number of mechanisms, including loss of fluid through the porous capillary wall due to the local excess pressure in the layer, might cause movement to ‘seize up’ altogether.

Reversion of turbulent to laminar flow

Abstract: It has been shown experimentally that quite large departures occur from the universal inner-law velocity distribution in the presence of severe favourable pressure gradients in turbulent boundary layers and that these departures are associated with the tendency for the turbulent boundary layer to revert to a laminar state. From the measurements a criterion for the onset of reverse transition has been deduced in terms of the mean shear-stress gradient in the wall region of the flow. Experiments in fully developed pipe and channel flows suggest that the proposed criterion may be quite generally applicable to all fully turbulent shear flows.

The form drag of two-dimensional bluff-plates immersed in turbulent boundary layers

Abstract: Measurements of the distributions of pressure on a bluff flat plate (fence) have been correlated with the characteristics of the smooth-wall boundary layer in which it is immersed. For zero pressure-gradient flows, correlations are obtained for the variation of form drag with plate height h which are analogous in form to the ‘law of the wall’ and the ‘velocity-defect law’ for the boundary-layer velocity profile. The data for adverse pressure-gradient flows is suggestive of a ‘law of the wake’ type correlation. Pressures on the upstream face of the bluff-plate are determined by a wall-similarity law, even for h/δ > 1, and are independent of the pressure-gradient history of the flow; the separation induced upstream is apparently of the Stratford-Townsend type. The effects of the history of the boundary layer are manifested only in the flow in the rear separation bubble, and then only for h/δ > ½. The base pressure is also sensitive to free-stream pressure gradients downstream of the bluff-plate. The relative extent of upstream influence of the bluff-plate on the boundary layer is found to increase rapidly as h/δ decreases. One set of measurements of the mean flow field is also presented.

Static menisci in a vertical right circular cylinder

Abstract: The solution of the differential equation describing the equilibrium meniscus in a vertical right circular cylinder is obtained over the entire range of contact angles and Bond numbers (dimensionless ratios of gravitational to capillary forces) for which a stable meniscus exists. The first few terms of the asymptotic series valid for Bond numbers of small and large magnitude are given, and the numerical solution for intermediate values is computed. The behaviour of the solution as a function of contact angle and Bond number is depicted graphically.

Laboratory studies of wind–wave interactions

Abstract: The present study consists of wind profile surveys, drift current measurements and water surface observations for a wide range of wind velocities in a wind–wave tank. It is confirmed that the velocity distribution essentially follows the logarithmic law near the water surface and the velocity-defect law toward the outer edge of the boundary layer. The wind stresses and surface roughnesses calculated from these distributions are divided into two groups separated by the occurrence of the wave-breaking phenomenon. For low wind velocities the surface roughness is dictated by ripples, and the wind-stress coefficient varies with U0−½, where U0 is the free-stream wind velocity. The surface roughness is proportional to the average height of the basic gravity wave at higher wind velocities; the stress coefficient is then proportional to U0. In addition, it is found that Charnock's expression (k ∝ u*2/g) holds only at high wind velocities, and that the constant of proportionality determined from the present experiment correlates very well with field observations. A new technique, involving the use of various-sized surface floats to determine the drift current gradient and the surface drift current, has been developed. A good agreement is shown between the gradients obtained from the measured currents and those determined from the wind stresses. Finally, the wind-stress coefficient is shown to be larger than the friction coefficient for turbulent flow along a solid rough surface; the difference is shown to be the wave drag of the wind over the water surface.

The motion of a spherical liquid drop at high Reynolds number

Abstract: The steady motion of a liquid drop in another liquid of comparable density and viscosity is studied theoretically. Both inside and outside the drop, the Reynolds number is taken to be large enough for boundary-layer theory to hold, but small enough for surface tension to keep the drop nearly spherical. Surface-active impurities are assumed absent. We investigate the boundary layers associated with the inviscid first approximation to the flow, which is shown to be Hill's spherical vortex inside, and potential flow outside. The boundary layers are shown to perturb the velocity field only slightly at high Reynolds numbers, and to obey linear equations which are used to find first and second approximations to the drag coefficient and the rate of internal circulation.Drag coefficients calculated from the theory agree quite well with experimental values for liquids which satisfy the conditions of the theory. There appear to be no experimental results available to test our prediction of the internal circulation.

Effect of waves at a gas-liquid interface on a turbulent air flow

Abstract: Air flowing over a liquid surface encounters an increased resistance if waves are present. The relation of this increased resistance to the properties of the waves has been studied. Air and a liquid flowed co-currently in an enclosed channel which is 12 in. wide and 1 in. high and which is long enough so that flow in the air and the liquid and the interfacial structure are fully developed. The drag on interfaces with three-dimensional wave structures was found to increase with the square of the gas velocity and to depend more on the height of the waves than on other parameters characterizing the interface. The ratio of the equivalent sand roughness to the root-mean-square of the fluctuations in the height of the liquid film is approximately equal to 3 √2. The velocity profiles in the gas were found to be different from what has been reported for flows over sand roughened surfaces.

On standing internal gravity waves of finite amplitude

Abstract: Two-dimensional internal gravity waves in a rectangular container are examined theoretically and experimentally in (a) fluids which contain a single density discontinuity and (b) fluids in which the density gradient is everywhere continuous. The fractional density difference between the top and bottom of the fluid is small.Good agreement is found between the observed and calculated wave profiles in case (a). Unlike surface standing waves, which tend to sharpen at their crests as the wave amplitude increases, and which eventually break at the crests when fluid accelerations become equal to that of gravity, internal wave crests are found to be flat and exhibit no instability. In the case (a) breaking is found to occur at the nodes of the interfacial wave, where the current shear, generated by the wave itself, is greatest. For sufficiently large wave amplitudes, a disturbance with the form of a vortex but with direction of rotation reversing twice every cycle, grows at the wave node and causes mixing. This instability is found to be followed by the generation of cross-waves, of which two different forms are observed.Several modes of oscillation can be generated and are observed in a fluid with constant density gradient. The wave frequencies and shape are well predicted by theory. The experiments failed to establish any limitation of the possible wave amplitudes.

Finite amplitude convection with changing mean temperature. Part 1. Theory

Abstract: When a horizontal layer of fluid is heated from below and cooled from above with the mean temperature and physical parameters of the fluid constant, the two-dimensional roll is known to be the stable solution near the critical Rayleigh number. In this study, with the mean temperature changing steadily at a rate η, the Rayleigh number and the velocity and temperature fields governed by the Boussinesq equations are expanded in two parameters: η, and the amplitude e. Hexagons are shown to be the stable solution near the critical Rayleigh number. The direction of the motion depends upon the sign of η. A finite amplitude instability is possible with an associated hysteresis in the heat flux as the critical Rayleigh number is approached from below or from above.

Reflexion and stability of waves in stably stratified fluids with shear flow: a numerical study

Abstract: A numerical examination has been made of the reflectivity of critical levels with low Richardson number to internal gravity waves propagating in stratified fluids with shear. At sufficiently low positive Richardson numbers the reflected wave may actually be stronger than the incident.The normal mode instabilities of three simple models have also been computed. The results are presented in three dimensions: Richardson number, horizontal wave scale and real wave frequency.

The distortion of turbulence by irrotational plane strain

Abstract: : The experiments extend those of Townsend which form the basis of his model of free turbulence. Here straining is carried to a strain ratio of 6:1, while Townsend's straining went only to 4:1. Two kinds of distorting ducts are used to produce the uniform mean strain applied to initially nearly isotropic grid turbulence. The results differ from Townsend's in that (1) a considerably higher degree of anisotropy is achieved, Townsend's measure of anisotropy attaining values up to 0.6, rather than the maximum of 0.42 he found; (2) there is no evidence that an equilibrium structure is attained; and (3) the strained turbulence rapidly becomes less anisotropic when the straining ceases. It is found to be possible to predict the variation of the total turbulence energy using rapid-distortion theory with a correction for decay. However, the individual components cannot be accurately predicted in this way. (Author)

The stability of steady and time-dependent plane Poiseuille flow

Abstract: The linear stability of plane Poiseuille flow has been studied both for the steady flow and also for the case of a pressure gradient that is periodic in time. The disturbance streamfunction is expanded in a complete set of functions that satisfy the boundary conditions. The expansion is truncated after N terms, yielding a set of N linear first-order differential equations for the time dependence of the expansion coefficients.For the steady flow, calculations have been carried out for both symmetric and antisymmetric disturbances over a wide range of Reynolds numbers and disturbance wave-numbers. The neutral stability curve, curves of constant amplification and decay rate, and the eigenfunctions for a number of cases have been calculated. The eigenvalue spectrum has also been examined in some detail. The first N eigenvalues are obtained from the numerical calculations, and an asymptotic formula for the higher eigenvalues has been derived. For those values of the wave-number and Reynolds number for which calculations were carried out by L. H. Thomas, there is excellent agreement in both the eigenvalues and the eigenfunctions with the results of Thomas.For the time-dependent flow, it was found, for small amplitudes of oscillation, that the modulation tended to stabilize the flow. If the flow was not completely stabilized then the growth rate of the disturbance was decreased. For a particular wave-number and Reynolds number there is an optimum amplitude and frequency of oscillation for which the degree of stabilization is a maximum. For a fixed amplitude and frequency of oscillation the wave-number of the disturbance and the Reynolds number has been varied and a neutral stability curve has been calculated. The neutral stability curve for the modulated flow shows a higher critical Reynolds number and a narrower band of unstable wave-numbers than that of the steady flow. The physical mechanism responsible for this stabiIization appears to be an interference between the shear wave generated by the modulation and the disturbance.For large amplitudes, the modulation destabilizes the flow. Growth rates of the modulated flow as much as an order of magnitude greater than that of the steady unmodulated flow have been found.