WILLIAM N. GILL,
BE
Coverage of papers that emphasize the ~ e v e l o ~ ~and ent delineation of basic principle
Y
his one-year review covers papers published primarily during the period from January to December 1969. In constructiiig any review which deals with as wide-raiiging and dificult a subject as fluid dynamics, one has to determine a t the outset whether to concentrate o n breadth or depth of coverage. I n the present review, we have attempted to strike a balance in this regard. H o w ever, since we found it necessary to report approximately 700 references, some preference has been given to the breadth of coverage. I t is difficult to choose thosc papers to be included i n a revicw. I n general \YC have tried to restrict c o ~ e r a g c to papers that emphasize the development aiid delineation of basic principles rather than practical applications; such work is more likely to have permanent value. No doubt some very significant papcrs rclcvant to fluid dynamics have not been reported. lsofhermal Single-phase laminar Duct Flow
r\pplications of laser-Doppler techiiiqucs for flow measurements xvere described and reviewed by ;\iigus et a!. (74). This informative papcr includes 112 references. Berman and Santos (5‘4) used laser-Doppler methods to study laminar flow development in circular tubes dowiistreani from a plug made up of brass screens. They found that entrance region flow is dependent on the inlet profile shape and Reynolds number for Reynolds numbers less than 300. Fleming aiid Sparrow ( 8 A ) and \Viginton (22.4) describcd a geiieralization of a previous analytical model for studying flow in the hydrodynamic entrance region of ducts. This gencralizarion allows the longitudinal wlocity to depend on two cross-scctional coordinates and as drveloped (8.4 ) can be applied to ducts of arbitrary cross section. Results (84) for rectangular and triangular ducts are givcn as illustrations of the method and the hydrodynamics entrance lengths for several aeomcrries arc collated. Other studies of flow in the entrance region were Siven by Atkinson tf al. (2A) and Chen (GA4). .\tkinson et ul. presented cquations for the enrraiice length for low Reynolds number flows obtained from lincar combinations of the creeping flow and boundary layer solutions and also gave additional data for low Rrynolds number pipe flow I n a related study, Dickcrson and Rice (‘in) used the analysis of Langhaar for flow in the entrance of a tube to evaluate the discharge coefficient from very small diametcr laminar flow orifices. Cheii examined the effect of a first-order velocity slip boundary condition on the cntrance length in tubes. Tl’illiamson (23‘4) treated analytically the decay of symmetrical distorted vclocity profiles between parallel plates. Developing flow very near the fully developed region was examined by \\:ikon ( M A ) using the assumption of a small disturbance from parallel flow in linearizing the Xavier-Stokes equations. Solutions for limiting cases of large and small Iieyiiolds numbers are given and an approximate method for moderate Reynolds numbers is suggested. 108
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Unsteady developing flow in the entrance region of a circular tube was analyzed by .\vula ( 3 A ) usiiig the integral momentum equation. Velocity and prcssurc distributions arc obtained for the particular problem of a fluid initially at rest which is suddenly set into motion by a constant pressure gradient with a uniform inlet velocity. .\n cxact solution for unsteady pipe flow \rith a prescribed time dependent discharge rate rather than a prescribed time dependent prcssurc gradicnt was dcrived by Predebon et al. (768). For cases in which the series solution converges slowly, asymptotic solutions for large aiid sniall times are suggested and agree well with thc cxact solution for the case of a linearly acceleratcd discharge. Prakash (7511j generalized previous analytical work on time dependent flow i n annuli by solving thc case of a n arbitrary time dcpendcnt pressure gradient and arbitrary initial velocity u s i q the finite IIaiikel transform. More reports o n peristaltic pumping have appeared in the literature. Yih and Fuiig (25.4 ) generalized rheir previous Yvork to thc case of circular cylindrical tubes. I t is found t h a t the results are in qualitative agrcemcnt with those for a two-dimensional channel. T h e fluid incchanics of peristaltic pumping under the limiting conditions of a wave length which is large compared to the channel width and the flow is inertia free \vex described by Shapiro et al. (18A). Experimental and analytical results a r e givcn and two phenomena called reflux aiid trapping are described. Disagreement ovcr the occurreiicc of reflux between authors appears in ( 7 Z A ) . Ito ( 7 7 A ) continued the detailed study of steady laminar flow in curved circular tubes usiiig the Pohlhausen approximate approach, thus obtaining solutions for high Dean numbers where the flow can be considered as being made u p of a core region and a boundary layer region. T h e assumption that the ratio of the tube radius to the radius of curvature of the tubc is small is made in the analysis. I n a related study, Kawaguti (738)performed a numerical integration of the Navier-Stokes equations for channels with three different typcq of corners. Barnes and \Valters ( 4 4 ) pcrformed an experimental study of elastico-viscous liquid flow through curved tubes. Ir was found that the flow pattern is similar to that for Ncwtonian fluids and that thc flow rate is cnhanced by elasticity for laminar flow but retarded in turbulent flow. A new definition of the friction factor and a new characteristic length scale for annuli allowed Gormaii (roil) t o correlate friction factor cs. Reynolds number as a single line for all radius ratios. 11-allick and Savins ( 21.4) compared integral and diflcrcntial dcscriptions of the steady annular flow of a power law fluid and found the integral representation superior in both convenience and accuracy. Plane Couette flow of a power law fluid with a n imposed pressure sradient was solved numerically by Flumrrfclt et al. ( 9 A ) . Turian (204) solved the problem of plane Couette flow of a n Ellis fluid and established the critical stress above which steady tempera-
SUNG P. LIN, RICHARD J. NUNGE
ture and velocity fields cannot be established. Flow rate pressure drop correlations for a Sutterby fluid in rectangular and isosceles triangular ducts were obtained from theory and experiment by Mitsuishi and Aoyagi ( 1 4 4 ) . Terrill and Thomas ( 1 9 A )presented a n extensive analysis of the numerical and theoretical solutions for laminar Newtonian flow in a uniformly porous tube. I t is shown that two solutions exist for all values of injection as well as for suction. Non-Newtonian flow in a porous walled channel with movable wall was discussed by Rutkevich (77A). Bubble Dynamics
T h e literature on bubble dynamics for the period covered by this review concentrates primarily o n the growth or collapse problem with increased emphasis on microlayer evaporation as a dominant factor (under certain conditions) for growth on a heated surface. Theofanous et al. ( 2 5 B ) present solutions for bubble growth in constant and time-dependent pressure fields which take into account not only the effects of surface tension, inertia of the liquid, and heat conduction in the liquid, but also the effects of nonequilibrium a t the liquid-vapor interface, a n d the varying density of the vapor bubble. Comparisons of the solutions with experimental d a t a available for water and nitrogen indicate the importance of the latter two effects and support the analytical approach employed. High-speed frame photography is a very common technique used in obtaining experimental growth rate data. I n general, however, some error is always introduced into the time scale (an appreciable error a t small times) as a result of the inability to locate the moment a t which the bubble begins to grow. Many authors take this as the frame prior to the one in which the bubble first becomes visible, others extrapolate the experimental data backwards to reestimate time zero. Sernas and Hooper ( 2 4 B ) describe a simple technique that should minimize this error, and perhaps extend the usefulness of the photographic technique to lower time values. T h e method consists of applying streak photography, combined with the usual frame photography, using the streak films to estimate time zero. I t is claimed that the error in the zero time datum can be reduced to about 10 psec. Hospeti and Mesler ( 1 3 B ) report a n experimental study to determine that fraction of the volume of a bubble growing o n a heated surface that results from evaporation a t the bubble base
AUTHORS W. N . Gill is Professor and Chairman of Chemical Engineering, Robert Cole, E. J . Davis, and R. J . Nunge are Associate Professors of Chemical Engineering, and S. P. L i n is Associate Professor of Mechanical Engineering at Clarkson College of Technology, Potsdam, N . Y . 13676.
(microlayer vaporization). As might be anticipated, a comparison of the ratio of vapor volume formed a t the base to thr total volume of the bubble shows a dependency o n bubble shape; the ratio being smallest for spherical bubblw, largest for hemispherical, and intermediate for oblate bubbles. Further experimental evidence of the existence of thc microlayer is presented in a paper by Jawurek (15B). By use of a transparent heated surface, parallel monochromatic illumination from below is reflected as interference patterns by the microlayer. T h e resulting fringe patterns are recorded on a high-speed camera and provide details of the microlayer shape and thickness. Simultaneously, and o n the same film, normal bubble-growth pictures in profile are obtained. Cooper and Lloyd ( 4 B ) lend additional support to the microlayer hypothesis by measuring transient surface temperatures o n glass and ceramic heated plates. Experimental observations allow the microlayer thickness to be deduced, from which the rate of evaporation from the microlayer can be computed and compared with the growth rate of the vapor bubble. Additionally, the relative magnitudes of stresses owing to inertia, surface tension, viscosity, and gravitation are determined for a particular bubble and discussed in relation to bubble shape and microlayer formation. I n a second paper, Cooper (5B) presents a theory to determine the growth rate of a vapor bubble on a heated surface. T h e paper is significant in that it represents one of thc first attempts to predict growth rates from the microlayer hypothesis. T h e predictions of the analysis, with associated limitations o n the range of validity of the assumptions, together indicate conditions under which microlayers may be expected to have significant effects on bubble growth. Florschuetz et al. ( 7 8 ) have obtained extensive experimental data for growth rates of vapor bubbles in water, ethanol, and isopropanol a t small uniform superheats, and a t both normal and zero gravity conditions. A detailed comparison with Scriven’s exact solution shows good agreement under zero gravity conditions over the entire observation time, and satisfactory agreement a t normal gravity conditions up to the time when translational effects (resulting from buoyancy) become important. D u d a and Vrentas ( 6 B ) employ a perturbation series solution for isothermal bubble dissolution and bubble growth. T h e perturbation solutions are checked against numerical solutions of the finite difference forms of the exact equations. These solutions indicate that in many cases the previously presented numerical solutions contain significant errors presumably owing to inadequate finite-difference approximations. LeBlond ( 79B) considers the situation of gas diffusion from ascending gas bubbles. General qualitative rules are derived for the behavior of the volume of a n ascending spherical bubble and of the gas pressure within it. Quantitative results are presented for the Stokes law regime. A comparative study o n the experimental techniques in general use for the measurement of mass transfer from a single gas bubble was made by Garbarini and Chi Tien ( 7 I B ) . Instantaneous mass VOL. 6 2
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transfer coefficients obtained by the use of motion picture studies and by the measurement of pressure changes in the top of the bubble column, exhibited wide scattering. Discussions of the relative merits of the two methods are presented. Ishii and Johnson (7423) report measurements of the rates of mass transfer of carbon dioxide bubbles into water and into monoethanolamine solutions. Because of the design of the apparatus, a n appreciable wall effect was expected and included in the analysis of the results. T h e absorption rates of carbon dioxide, ethylene, and butene from single rising bubbles in water have been measured by Johnson et al. (76B). T h e results have been compared with theoretical equations for the rate of gas solution of the frontal surface of the bubble based on potential flow theory and Higbie theory. Akinscte and Lee ( 7 B ) employ a perturbation technique to investigate nonsimilar effects in the collapse of a n empty spherical cavity in water. The scheme applies only to the latter half of the collapse process when the cavity speed is highly supersonic. T h e results indicate that while the nonsimilar effects modify the distributions of flow parameters, the cavity trajectory is practically unchanged from the self-similar law. Very high pressures are shown to accompany cavity collapse even at moderately high collapse speeds. A study of collapsing cavities in liquids with chemical reaction is described by Fogler ( 3 B ) . T h e conversion is shown to vary with increasing initial cavity radius because of the interaction of collapse time and gas temperature within the cavity. T h e influence of a n increase in the number of molecules formed during the reaction upon collapse is described. In a second paper, Fogler (7OB) considers the influence of reacting gases on the motion of collapsing cavities in more detail. Cole and Rohsenow ( 3 B )present a correlation of bubble departure diameters for boiling of saturated liquids that covers a wide variety of fluids, and pressures ranging from 0.066 to 140 atmospheres. T h e main feature of the correlation is the use of a modified Jakob number that does not involve the wall superheat. Studies in bubble formation from orifices under constant flow conditions are presented by Ramakrishnan et al. (22B). ‘\ model based on a two-step mechanism is proposed. I n a second papcr, Satyanarayan et al. (23B) develop a model for bubble formation under conditions of constant pressurc. A third paper, authored by Khurana and K u m a r ( 7 7 B ) considers bubble formation from orifices, where both the pressure inside the chamber and flow rate into the bubble are time dependent. Kupferberg and Jameson ( 7 8 B ) propose a three-stage mechanism for bubble formation above a n orifice. Potential flow theory is used to derive equations for the motion of a spherically growing bubble, its radial expansion, and the corresponding dynamic pressures involved in such a bubbling system with a finite gas chamber volume. Potential theory is also used by McCann and Prince ( 2 0 B )for bubble growth a t a n orifice. Bubble size and frequency are determined as functions of hole gas velocity, chamber volume, and system physical properties. Hirose and Moo-Young ( 7 2 B ) investigate the drag and mass transfer characteristics of a gas bubble moving in power-law nonNewtonian fluids. &in approximate solution for the case of creeping flow around circulating bubbles shows that thc mass-transfer coefficient is enhanced for pseudoplastics and depressed for dilatants compared to the situation for Xewtonian fluids. T h e steady rise of a spherical cap bubble in a n infinite liquid under the action of buoyancy forces is examined by Parlange ( 2 7 B ) . IVasserman and Slattery ( 2 6 B ) investigate the effects of a trace quantity of surface-active agent on creeping flow past a bubble or droplet. T h e effect of vibrating surfaces on the dynamics of bubble motion is considered by Chon et ai. ( 2 B ) . Finch ( 8 R )proposes that pressure gradients under typical experimental conditions for cavitation nucleation studies may cause holes to drift and, hence, yield much lower tensile strengths than would be predicted theoretically in the absence of pressure gradients. Drop Formation a n d M o f i o n
Mikhaylenko et al. (75C) describe a thermocouple device for measuring the temperature of moving drops by interpreting oscillograph traces. The method consists essentially of passing a current through the thermocouple junction. Positive or negative pulses appear on the oscillograph, depending upon whether the ther110
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
mocouple junction is colder or hotter than the drops. T o eliminate conduction errors caused by the resulting temperature difference, the thermocouple junction is heated electrically until the pulses practically disappear, at which time the recorded thermal emf corresponds to the true drop temperature. Ruckenstein and Constantinescu (78C)employ a similarity transformation for solving the problem of mass transfer to a growing drop or bubble. Both radial and tangential velocity componcnts are considered and the method may be used either when the growth is due to heat transfer or to the flow rate of a liquid fed through a capillary. 11 model is presented by Huang and Kintner (7C) to account for reduced mass transfer to drops falling through a continuous phase containing a surface active agent. T h e reduction in mass transfer is said to be due to a reduction in available interfacial transfer area and to changes in both velocity and pattern of internal circulation. These are shown to be both functions of contact time and can be characterized. Hartland (6C)concludes that the shape of a fluid drop approaching a n interface does not change appreciably with time and is very close to the equilibrium dimensions. T h e limiting caws of a force balance around the drop for very small and very large drops are shown to agree with those previously established for both deformable and rigid interfaces. A theoretical method is given by Cox (3C) for the determination of the shape of a fluid drop in steady and unsteady flows by making an expansion in terms of the drop deformation. Examples given include the determination of the shape of a drop in shear and in hyperbolic flow when each is started impulsively from rest. An experimental study is presented by Mercier and Rocha (74C), using standard photographic equipment to follow the motion of large pulsating water drops falling into xylenc in both steady and unsteady state. T h e oscillations of a drop moving in another fluid medium havc becn studicd analytically by Subramanyam (73C). T h e drop deformation modifics the Lamb’s expression for frequency by including a correction term, while the viscous effects split the frequence into a pair of frcquencies-one lower and the other higher than Lamb’s. The deformation of drop suspended in a n immiscible liquid undergoing slow steady shear flow has been considered by Turner and Chaffey (20C) from a second-order theory. Woodmansee and Hanratty (24C)present a study of the critical conditions and the mechanism of atomization for the cocurrent flow of air and a liquid. High-speed motion pictures reveal that atomization occurs by the removal of small wavelets that exist on the top of flow surges (roll waves) in the liquid film. I t is suggested that atomization results from a Kelvin-Helmholtz instability whereby the destabilizing force is the pressure variation caused by the compression of the air streamlines a t the crests of wavelets. T h e effects of external vibrations on the disintegration of a liquid jet are described by \$-issema and Davies (ZZC). Conditions for the formation of uniformly sized drops are discussed and results presented to illustrate the effect of amplitude and frequency of vibration on the drop size and break-up length of the jet. Liquids of viscosities ranging from 60-1400 CPwere studied and the results compared with existing theories. Baryshev et al. ( Z C ) obtain a dimensionless equation making it possible to determine the liquid droplet size dispersed by a rapidly rotating flat disk in the presence of interactions with a gas stream. T h e mechanism of spray formation by colliding jets is described by TYalkden and Kell (27C) who suggested that water experiments give a reasonable indication of the atomization of molten “slag” for use as a heat transfer medium. i1 theoretical analysis is presented by Meister and Scheele (73C) for predicting the size of drops formed from a laminar cylindrical jet when one Newtonian liquid is injected through a nozzle into a second immiscible Newtonian liquid. T h e analysis couples stability theory with the requirement that the disturbance travel a t the same velocity as the jet interface. T h e disintegration processes occurring with charged liquid jets of distilled water have been examined by Huebncr ( 8 C ) . High-speed photographic techniques were used to determinc the cffect of charging on disintegration of the jet, the size distribution of the drops formed, and the velocities of the drops. A dye tracer technique was used by Letan and Kehat (7OC) to measure the residence time distribution of the drops in a spray column operating with a dispersed or a dense packing of drops. T h e results of a n investiga-
tion of the dispersion of various liquids i n tank extractors with and without mass exchange is given by Pebalk and Mishev (76C). I t is shown that the distribution of drop sizes is log-normal. McGinnis and Holman (72C) present a n experimental study of the splattering and bouncing of water, acetone, and alcohol droplets on a heated plate. An empirical comparison is made by Aylor and Bradfield ( I C )between the evaporation rates of film boiling droplets with and without applied voltage. Frazier and Hellier (5C) describe a n experimental method for the study of rapid vaporization of liquid droplets in high temperature gas streams. A mathematical model of a spherically symmetric quiescent droplet undergoing quasi-steady vaporization in the region of its thermodynamic critical point is investigated by Manrique and Borman (77C). T h e flame front model is used by Kotakc and Okazaki (OC) to numerically calculate the process of evaporation and combustion of a liquid droplet in still air. T h e near-equilibrium decompositional burning of monopropellant droplets in a stagnant atmosphere is examined by Fendell (4C). By using photographic techniques, Wood and Rosser (23C) have determined the size histories, ignition lags, and loci of ignition of small (100-300 p ) single, freely falling fuel droplets suddenly exposed to a hot, oxidizing atmosphere, as a function of initial droplet size. Ranger and Nicholls ( 7 7 C ) report experimental and analytical rrsults for the problem of liquid drop shattering. The purpose of the investigation was to establish the influence of various parameters on the rate of disintegration and o n the time required for breakup to occur. Cavitations
Brennen (ZO),in work based on the potential theory, obtained numerical solutions of axisymmetric fully cavitating flows around a disk and a sphere in different sizes of solid wall tunnel. Observed separation from the sphere occurs well downstream of the position predicted by theory because of the effect of viscosity. Measured drag is consistently smaller than the predicted theory. However, flows are found to be well modeled, especially for the case of disk. Air injection through a porous wall upstream of the cavity is used by Voorhees and Bertin ( 8 0 ) to simulate the injectants from a n oblative thermal protection system. Pressure distribution in the cavities and the boundary layer velocity profiles immediately upstream of the cavities are measured. Mdss injections tend to elongate the length of the cavity. H a h n ( 5 0 ) measured the velocity, temperature, and pressure profiles as well as the heat transfer ratio within the separated region of an annular cavity a t a freestream Mach number 7.3. Additional useful data on pressure and heat transfer have been obtained by Nestler, Saydoh, and Aurer ( 6 D ) for separated reattachment hypersonic turbulent flow over steps or cavities that are often present in reentry vehicle surface. Akinsete and Lee ( I D ) have calculated by means of a perturbation technique the motion of a n empty spherical cavity collapsing in water. However, the influence of surface tension viscosity and other dissipative effects on the motion are not considered. Flow inside triangular-shaped cavities resembling baffle cavities used i n liquid propellant rocket motors are experimentally studied by Torda and Pate1 (70). T h e experimental setup simulated radial and tangential modes of oscillations over the cavities. Fogler ( 3 0 ) found that the time required to reach a given radius during collapse increases with increasing magnitude of the heat reaction for a n exothermic reaction occurring in the gas phase; and for the endothermic case, the reverse is true. He also found that, in most instances, 957, of the conversion is accomplished during a 1yo change radius of the collapse. The reaction between the gas and the cavity is also discussed. T h e behavior of progressive waves moving through a circular pipe containing a rotating flow of water with a n axial cavity is studied experimentally by Hashimoto ( 4 0 ) . T h e problem is also studied semitheoretically with the aid of obtained data. Stratified Flows
Plane Couette flow of two superposed liquid layers with a viscosity gradient and discontinuity is used by Craik ( 2 E ) as an example for demonstrating the role of viscosity diffusion in contrast to the viscosity discontinuity in the stability ofplane Couette Aow with
viscosity stratification. Li ( Q E )found that, in a plane Couette flow of two superposed elastico-viscous liquids, the effect of elasticity on the stability of the flow is absent in the absence of viscosity stratification. Li (7023)also studied the instability of threelayer plane Couette flow and found that the instability is caused by viscosity discontinuity and by the resonance between the two interfacial waves. Related experimental studies of kinematic a n d dynamic characteristics of interface between streams of two fluids of different densities with stratified temperature are reported by Netyukhaylo and Sherenkov (73E). Data on transition flow from laminar to turbulent flow through steps of onset of long stable waves, instability of long waves, onset of short wave motion, and turbulent mixing are given in detail. Thorpe (2OE) also investigatcd the onset of Kelvin-Helmholtz instability in a long rectangular tube containing two immiscible fluids when the tube is tilted away from the horizontal. Theoretically predicted time a t which instability occurs compared quite well with the experiments, but not the wave length. This discrepancy may be due to nonlinear effects. Thorpe (7SE) has shown that many of the known analytic solutions of the equation for neutral disturbances to a stably stratified, inviscid shear flow actually belong to a wider family of solutions when a transformation to the hypergeometric differential equation is possible. I n considering a rectangular ocean basin of constant depth, Pedlosky (74E) has demonstrated with a linearized theory that the mid-ocean thermocline circulation, especially the problem of upwelling, is closely related to various boundary-layer phenomena a t the ocean’s rim. Conservation equations of mass, momentum, and energy for internal waves propagating o n an unsteady nonuniform current have been derived by Wang (27E)to the second order in the wave slope. These equations have been used to demonstrate in a two-liquid system that the internal waves are amplified in a n adverse current but suppressed in a n advancing current. I n a n attempt to expcrimentally reproduce the internal standing waves in a wedge with rigid upper and lower boundaries, Wunsch (22E) found instead that a propagating mode solution is more appropriate. T h e refraction of progressive internal waves on a sloping bottom is treated for the case of constant Brunt-Vaisala frequency and the effects of the viscosity are discussed. A more general problem of internal wave propagation in a n inhomogeneous fluid of nonuniform depth has been treated by Keller and Mow (623). Wunsch’s result is recovered as a special case. T h e theory is also applied to the problem of edge waves near a shoreline and trapped waves in a channel. An analysis by Kelley ( 7 E )based on the shallow water approximation and the two-fluid model indicates that the observed peaking of the incoherent energy of the mid-ocean surface displacement a t tidal frequency might be due, in part, to the transmission of deep-sea internal waves into shallow water as surface waves. A train of internal gravity waves in a stratified liquid exerts a stress o n the liquid, and induces changes in the mean motion of second order in the wave amplitude. Rrethcrton ( I E ) has determined to a first approximation such a mean motion for a slowly varying quasi-sinusoidal wave train and for propagating wave packets. T h e development of the turbulent layer by entrainment of the underlying stratified fluid through the action of a constant surface force is studied expcrimentally by K a t 0 and Phillips (8E). An empirical formula for the ratio of entrainment velocity to the friction velocity is given. T h e mechanism of the collapse of a homogeneously mixed circular turbulent region in a fluid with a uniform density stratification is studied expcrimentally by \Vu (23E). T h e energy density of the internal waves generated by the collapse is suggested to be peaked a t some particular frequency. Wu’s experimental results are confirmcd by numerical computation of Wessel (24E). A theory of the propagation of internal gravity waves of finite amplitude is given by Rarity (76E). Nonlinear internal gravity wave motion in a slightly stratified atmosphere is analyzed by Drazin (3E). Not a single solution for a large-amplitude threedimensional flow of a stratified fluid is yet known. However, Yih (2523)has recently found a class of steady solutions for gravity jets and channel expansion with or without an overlying stagnant layer. T h e underlying assumption for this class of solution is that the horizontal scales of the flow are much greater than the vertical scale (shallow water theory). The statically stable lee wave field VOL. 6 2
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produced by, and the consequent drag on, a semielliptical obstacle are determined by Huppert and Miles ( 4 E ) for a n inviscid upstream flow with a constant density gradient and constant kinetic energy per unit volume (Long's model). 4 general solution to the same flow over a n arbitrary two-dimensional obstacle is also obtained by the same authors (77E). If in the above model of steady flow, the upstream condition is such that the density stratification is sufficiently steep or/and the kinetic energy per unit volume is sufficiently small, then the flow is unstable and patches of turbulence appear. This is demonstrated both theoretically and experimentally by Russ (75E). T h e hydrodynamic thrust on a disk moving within a stratified fluid o n a steady path is calculated o n the basis of linearized theory by Mackinnon et al. (7223). Configurations of some associated phase surfaces arc also obtaincd. T h e power output for a vibrating cylinder in a stratified fluid of constant Brunt-Vaisala frequency is calculated by Hurley ( 5 E ) . Lighthill's theory for dispersive waves is used by Stevenson (77E) to obtain the phase configuration of axisymmetric internal waves generated by a n oscillating body traveling vertically in a stably stratified salt solution. Unlike the case of simple translation, the wave is unstationary with respect to the moving body and can appear in front of the body. Two-dimensional internal waves generated by a traveling oscillating cylinder are studied by Stevenson and Thomas (78E). Two-Phase Flow:
Fluid-Fluid
T h e literature o n fluid-fluid two-phase flow is still concerned primarily with the prediction of pressure drop, void fraction, and slip ratio. However, continued interest is exhibited in film behavior and wave motion in annular flows, dispersion in multiphase flows, and flow stability in parallel boiling channels. Additional topics of interest include flows in porous media, critical flows, and the effects of turbulence promotors. Brown et al. ( 8 F ) have developed a model to predict void fractions i n horizontal and vertical systems. T h e model accounts for the effect of buoyancy and by this method distinguishes between upward and downward flows. T h e prediction is based o n a single flow parameter and the slip velocity. Electrical conductivity determinations of void fractions for the tw-o-phase flow of potassium in tubes have been presented by Aladyev ( I F ) . T h e Armand equation is recommended for predicting this quantity. Additionally, friction factor d a t a were found to be well described by both the Armand and the Martinelli-Lockhart equation. T h e X-ray absorption method has been employed by Evangelisti and Lupoli ( 2 3 F ) to obtain void fraction information for forced convection boiling in a small annular channel, while Cimorelli and Evangelisti ( 7 Z F ) employ a n experimental technique based upon statistical correlation functions to determine the slip ratio in a tubular channel under adiabatic conditions. Sagan et al. ( 4 2 F ) report the results of a n experimental investigation of the friction loss for air-water and viscous air-solution mixtures in vertical circulator tubes a t low pressures (approximatcly 112 to 1 atmosphere). Formulas are presented for the range of parameters iniTestigated. X study of subcooled boiling pressure drop with water a t low pressure (below 100 psia) in horizontal tubes is presented by Bergles and Dormer ( 4 F ) . T h e data are correlared in graphical form and are reported to be in agreement with limited d a t a of other investigators. Styushin and Ryabinin (4727)present a method for calculating the hydraulic resistance in vapor generating tubes under conditions of low specific heat flux. Narasimhamurty and Vara Prasad ( 4 0 F )have performed studies o n the use of twisted tape turbulence promotors in two-phase gasliquid flow in horizontal pipes which indicate that such devices accentuate annular flow conditions, especially a t high velocities. Two-phase pressure drops were estimated satisfactorily by means of modified Lockhart-Martinelli parameters. I n a n apparently similar study, Zarnett and Charles (52F), using continuous spiral ribs fitted inside Lucite tubes, found the main effect of the rib was to move the gas phase away from the tube walls. Again pressure drops were successfully correlated with Lockhart-Martinelli parameters. Banerjee et ai. ( 3 F ) have conducted a n experimental study on cocurrent gas-liquid flow in helically coiled tubes, in which the 112
INDUSTRIAL A N D ENGINEERING CHEMISTRY
effects of tube diameter, coil diameter, helix angle, and liquid viscosity on flow patterns, pressure drop, and holdup have been investigated. T h e Lockhart-Martinelli approach adequately predicted pressure drop and holdup d a t a employing modified correlating parameters. Fedotkin ( 2 4 F )has derived a n expression for predicting the pressure d r o p across orifices for two-phase watersteam flows of high quality. An experimental investigation of the effect of heat flux and geometry o n the friction factor has been reported by Miropolskiy et al. (38F). \Yith all other conditions being equal, the friction losses in the three-rod vertical channel employed i n the investigation are lower than i n tubes. Kruzhilin and Labuntsov ( 3 5 F ) consider the effects of flow conditions and geometric factors on the critical heat input of a flow channel. Gill ( 2 6 F ) considers axial dispersion with time variable flow in multiphase systems. T h e analysis strongly suggests that experimental observations o n packed beds, or other multiphase systems, should be interpreted on the basis of the superficial, rather than the interstitial velocity. An experimental study of two-phase cocurrent downflow in packed beds is reported by Hochman and Effron ( 2 9 F ) . Liquid and gas phase residence time distribution measurements were made and were characterized by a n effective axial dispersion coefficient and mean residence time. Axial dispersion in both phases was found to be considerably greater than in single-phase flow. Increased gas-phase dispersion was attributed to bridging of liquid on the packing. Liquid phase behavior was satisfactorily described by a model that assumed exchange between free flowing and stagnant liquid. I n light of the previous reference ( 2 6 F ) , it is worth noting that the liquid and gas phase dispersion coefficients were correlated in terms of superficial gas and liquid velocities. M'oodmansee and Hanratty (57F)present a n experimental study of the thin liquid film that moves along the duct wall in the annular regime of gas-liquid flows by a technique that determines the base film height and velocity gradient a t the wall separately from the existing roll waves. A method was used by Sutey and Knudsen ( 4 8 F )to demonstrate the advantage of local instantaneous measurements in describing climbing film flow in a vertical annular duct. Experimental studies by Singh et al. (45F)for steamwater mixtures in cocurrent upward annular flows a t a pressure of 1000 psia, indicate that film flow rates are consistently higher than theoretical predictions; about one-third of the measured flow rates being twice as high. T h e results of studies on heat transfer and liquid boiling in a thin film formed over a surface by liquid supplied through a spray nozzle is presented by Kopchikov et QZ. (34F). New formulas and effects were obtained in the course of the investigation. Biasi et al. ( 5 F ) present a description of a heat-transfer annular dispersed flow in terms of nonequilibrium in the phase distribution. T h e results are compared with experimental burnout d a t a for uniform and nonuniform heating. Kholpanov ( 3 3 F ) has examined some aspects of two-phase film mass transfer in the absence of mass forces. T h e results are claimed to be novel in that the parametric equations contain expressions that take into account the physical properties of the contact phases. Periodic flow of thin liquid layers has also been investigated by Kholpanov ( 3 2 F ) , with allowance for the shear stress a t the liquid-gas interface. T h e results indicate that the flow friction of two-phase systems is a n important factor in calculating the characteristic parameters of wave motion of a liquid film. Cetinbudaklar and Jameson ( 7 7 F ) suggcst that flooding i n the countercurrent flow of liquid and gas in a vertical tube is caused by the appearance of infinitesimal waves moving with the interfacial velocity of the liquid. An analysis is presented to predict the critical gas velocity required to cause unstable waves, and is in excellent agreement with experimental d a t a for flooding velocity over a wide range of liquid viscosities. I n a n experimental investigation, Davis and Cooper ( 7 7 F ) have investigated the thermal entrance region for horizontal cocurrent air-water flow over a flat plate. Theory and experiment are compared in terms of dimensionless parameters that characterize the system. Linek and Mayrhoferova ( 3 7 F ) present a study of absorption rate on holdup and interfacial area in a heterogeneous gas-liquid system. T h e chemical method is used for determination of interfacial area. Hoffer and Rubin ( 3 0 F ) have experimentally studied stable foam systems to determine the flow regimes prevailing under
various conditions to find correlations between the relevant physical quantities in each flow regime, to find out what causes a certain flow regime to prevail under a certain set of conditions, a n d how the flow regime can be predicted from operating conditions. Holdup a n d liquid circulation in bubble columns was studied by Freedman a n d llavidson (25F). T h e work was carried out in three main parts; experimental work in which air was bubbled through d deep pool of liquid, a twodimensional bubble column in which the two-phase mixture was contained between two parallel plates (maldistribution of gas a t the base causing the liquid t o circulate), and a three-dimensional bubble column in which liquid circulation was studied by fitting a vertical tube immersed in the liquid, coaxially with the main column. Dolinskiy et al. (79F)report the results of a n experimental study of mass transfer coefficients in bubbling and the effects of gas bubbling rates, the height of the liquid layer, and the use of surfactants. T h e study was motivated by the frequent use of bubbling processes in the production of antibiotics. T h e liquid motion induced by ii chain of bubbles rising through a viscous liquid is investigated both experimentally a n d analytically by Crabtree and Bridgwater ( 7 3 F ) . T h e measured velocity profile a n d extra pressure drop (induced by the bubbling) is shown t o be in excellent agreement with a model of the flow, supposing the gas to be equivalent to a line force acting vertically upward along the axis a t the center of the cylinder. Brennen ( 7 F ) considers the problcm of cavitating flow pdst a spherical head form, where the air (originally dissolved in the water) a n d heat diffusing through the fluid toward the water-cavity interface must be balanced by the rate of entrainment of volume of air a n d vapor away from the cavity, in the wake. T h e water vapor or heat balance suggests that the temperature differences involved are likely to be virtually undetectable experimentally. TVitte (508’) considers the phenomenon in gas-liquid flows of a sudden change in structure from “jet flow” to “froth flow” accompanied by energy dissipation and pressure build-up. Upstream of this phenomenon, the gas is the continuous phase; downstream, the liquid is the continuous phase. T h e phenomenon, which has been called “mixing shock,” shows some similarity to a n d also some differences with the plane shock wave i n gas dynamics. T h e pressure pulse transmission in a flowing mixture provides the basis for a theoretical formulation of two-phase critical flow and sonic velocity by Moody (39F). T h e model predicts most available d a t a with homogeneous and separated phase patterns and suggests quality ranges for which each pattern applies. An analysis is presented by Raikin ( 4 7 F ) for the critical velocity of a steamwater mixture corresponding to a deterioration in heat transfer. Eigner ( 2 2 F ) has developed a logical consistent set of equations for isothermal, two-phase, two-component flows in which dissipation energies can be neglected, which allows the evaluation of existing extensive material 011 speed of sound measurements in two-phase flows. T h e propagation of infinitesimal sound waves in a liquid containing gas bubbles is considered by Crespo (74F). Relative motion of gas bubbles and liquid is explicitly allowed for, a n d it is shown that a significant error in the speed of waves may arise if the relative motion a n d fluctuation of mass fraction are neglected. A survey of theoretical a n d experimental studies on the discharge of saturated liquids a n d two-phase vapor-liquid streams from tubes a n d nozzles is presented by Dryndrozhik (ZOF). I t is shown that no agreement exists a t present about the definition of critical discharge velocity and about the factors o n which it depends. Glenn ( 2 7 F ) considers the radial source flow of uniform-size liquid particles into a vacuum. I n a second paper, Glenn ( 2 8 F ) studies the effect on the flow structure of recondensation of water vapor, initially generated by the flow described in the previous reference (27F). T h e problem was motivated by emergency dumping, leakage, etc. of various liquids, including propellants, which are stored onboard space vehicles. T h e first paper ( 2 7 F ) emphasizes the fluid mechanics of the vapor-particle interaction, whereas the second ( 2 8 F ) considers vapor recondensation resulting from the supersaturation experienced during the expansion process. A method for experimental determination of drop sizes in twophase flows by light scattering tcchniques is reported by Deych et al. (78F). Results are presented of experiments with d r o p
sizes behind a condensation shock in supersonic nozzles, designed by the method of characteristics for M = 2.0. Babarin et al. (2F) indicate the presence of two crisis mechanisms a t high steam qualities on the basis of tests with straight tubes. A device is suggested for intensifying turbulent transfer of liquid drops from the central water-steam nucleus of the flow t o the heated duct wall a n d thus increasing the critical heat flux from the boiling liquid. A number of papers in the current literature are concerned with multiphase reservoir flow problems, z.e., the flow of such fluids as oil, water, a n d gas through porous strata. Blair and Weinaug ( 6 F ) outline the development of a completely implicit difference analog for reservoir simulation, along with a Newtonian iterative method for solving the resulting nonlinear set of algebraic equations that arise a t each time step. A consolidated porous medium is mathematically modeled using networks of irregularly shaped, interconnected pore channels by Ehrlich a n d Crane ( Z I F ) . Mechanisms are described that form residual saturations during immiscible displacement both by entire pore channels being bypassed a n d by fluids being isolated by the movement of a n interface within individual pore channels. Snyder ( 4 6 F ) describes reservoir behavior by a set of differential equations that result from combining Darcy’s law a n d the law of conservation of mass for each phase in the system. T h e equations include the effects of viscous, capillary, and gravitational forces, a n d permit one phase to be soluble in the other. A study to develop a mathematical model that would simulate transient, two-phase flow of hydrocarbon mixtures in porous media under conditions that result in interphase mass transfer has been undertaken by Culham et al. (75F). Shutler ( 4 4 F ) presents a numerical simulation of the simultaneous flow of three phases in one dimension. I n cluded are the effects of three-phase relative permeabilities, capillary pressure, and temperature a n d pressure-dependent fluid properties. A technique for predicting the flow of oil, gas, a n d water through a petroleum reservoir is reported by Sheffield (43F). Gravitational, viscous, and capillary forces are considered, a n d all fluids are treatcd as being slightly compressible. Budnikov and Sergievskii ( 9 F ) examine the stability of a system of parallel boiling channels connected t o common headers a n d obtain the necessary and sufficient conditions of stability. I n a second paper, Budnikov and Sergievskii ( I O F ) extend the problem to include the effect of pressure drop in the heated part of the parallel channels, this effect had been considered negligible relative t o pressure drops in the unheated parts in the first reference ( 9 F ) . Davis ( 7 6 F ) employs impact tubes installed on the top a n d side walls of a tunnel containing cocurrent air-water stratified flow, to measure shear stress distribution in the air. This, together with a momentum balance on the gas phase a n d a knowledge of the pressure drop and average film thickness, allows the interfacial shear on the liquid film to be computed. T h e method is compared with other techniques, and the effects of neglecting side wall shear stress are determined. A study of saturated boiling heat transfer in narrow spaces is presented by Ishibashi and Nishikawa (3727). T h e main purpose of the study is to detect and prove the heat transfer characteristics of the coalesced bubble region. Various characteristics of the coalesced bubble region and isolated bubble region are compared, a n d a new type of correlating equation for the former is proposed. A theoretical analysis providing verification of the correlation is described. In other papers of interest, Kuznetskii ( 3 6 F ) derives a n expression for the enthalpy of a n incompressible turbulent liquid flow in a heated tube, the location of the boiling section being found in general form; Tavlarides and Gal-Or ( 4 9 F ) present a general analysis of multicomponent mass transfer with simultaneous reversible chemical reactions in multiphase systems. T h e results are said to be useful in predicting the yields a n d effluent concentrations of reacting dispersed systems as a function of the physical a n d operating variables. O n e interesting general conclusion from this analysis is that the error involved in assuming a uniform particle size in lieu of a distribution of sizes is small when calculating total average transfer rates. Two-Phase Flow:
Solid-Fluid
T h e literature on solid-fluid two-phase flows is concerned with VOL. 6 2
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the rheology of suspensions, particle size distributions, particle collisions and attrition, suspension by mechanical agitation, hydraulic conveying, and applications to \-arious countercurrent contacting devices. A symposium o n “the flow of fluid-solid mixtures” was held at the University of Cambridge from 24 to 28 March 1969, under thc auspices of the 1iiternatioIm.l Union of Theoretical and Applied Mechanics. There was no publication of the proceedings in full; hence, many of these papers will probably appear in the literature in the near future. h condensed account of the proceedings has been presented by Davidson, Pearson, and Vanoni (70G), all of whom were involved in the organization of the symposium. T h e rheology of concentrated suspensions is treated by Kambe (15G) where the dispersed particles are considered to behave as rigid bodies and the dispersing media are assumed to be Newtonian liquids. Even in this simplest case, the rheological properties are dependent upon the concentration of the dispersed particles, the system exhibiting Newtonian characteristics a t low concentrations, and non-Newtonian characteristics a t higher concentrations as a result of particle collisions and interference, and the possibility of coagulation. A study of the viscosity of concentrated suspensions is presented by Lee (20G). The study is primarily concerned with a study of hydrodynamic viscosity, i.e., that portion of the suspension viscosity that is indepcndent of rate of shear and that represents a limiting viscosity at high rates of shear. Some deviations of experimental data from theory have been discussed, and all the equations derived arc compared with the data. Leek et al. (27G) have studied the distribution of solid particles in a two-phase system by means of a spectrophotometcr and transparent phases of different absorptivities. The beta distribution gave a n excellent fit to the data, the parameters being a function of the size of the solid particles. A computer study of the effect of crystal breakage on particle size distribution in a continuous mixed suspension crystallizer has been reported by Randolph (37G). T h e amount of narrowing of the particle size distribution (as a result of breakage) was correlated in terms of empirical parameters in the breakage model. Stereophotogrammetry is used for the first time by Reddy et al. (32G) to measure the local particle velocity vcctor and its turbulent components in two-phase solid-gas flow. I t is demonstrated that the technique could be used for the direct determination of the radial concentration distribution in a pipe. Michael and Norey (2%)determine trajectories for small particles introduced upstream into a fluid flowing past a fixed sphere in order to calculatc the resulting particlc collision efficiency (collision with the sphere). T h e results are compared with those of other authors. Bahukha et al. ( 7 G ) present the results of a n experimental study on collision and coalescence of particles in a two-phase system. T h e study was motivated by the use of a continuous flow loop with uninterrupted material circulation for sintering of dispersed materials. h study of diffusion kinetics and catalyst attrition in cyclic processes is reported by \\’eisz (4ZG). I n catalytic systems where the catalyst is exposed alterhately and cyclically to different reaction environments, the mechanical particle attrition rate of the catalyst is dependent upon the regeneration kinetics. For the Thermofor Catalytic Cracking process, the particle attrition rate was materially reduced by modifying the diffusional kinetics of the coke combustion reaction. Byers and Calvert (9G) have investigated particle deposition from turbulent streams by means of a thermal gradient. Agreement with a mathematical model based on existing theories of thermophoresis was good, considering the accuracy with which the slip flow coefficients and constants are known. Narayanan et al. (26G) report an investigation of the suspension of solids by mechanical agitation. .in expression for the minimum velocity required to initiate the suspension of solids is compared with experimental values, and a suitable correction factor is incorporated in terms of the concentration of solids. The resulting correlation is claimed to agree well with a number of systems investigated. T h e purpose of a paper presented by Hughmark (74G) is to utilize the mass transfer equations for fixed particles in a flow field and settling particles in still fluids to estimate slip velocities to correspond to the experimental mass transfer coefficient data for agitated vessels. Correlations are then developed 114
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
to represent these slip velocities. Brian et al. (6G) report data for heat transfer from water to melting ice spheres and for mass transfer in the case of dissolving spheres of pivalic acid suspended in water agitated in a stirred vessel. This and other data are used to develop a correlation involving Nusselt and Prandtl or Schmidt numbers together with a dimensionless group involving agitation power. Slater (38G) presents a review article on the variety of continuous countercurrent liquid/solid contacting devices. Existing equipment is comparcd on the basis of chemical performance and throughput. Experimental observations of solid-liquid mixedphase flow through horizontal and vertical pipes are reported by Toda et al. (47G). Velocity profiles, pressure drop, and particle velocity were determined. :\n abbreviated article on the same subject is also presented elsewhere by Toda et al. (40G)which is a translation of the original Japanese article (see 40G). T h e topic of pneumatic transport is discussed by Owen (29G) in an annotated version of a survey presented at the symposium at Cambridge, mentioned in the earlier part of this section. Attcntion is drawn to the difficult problem of dynamically scaling a two-phase flow, and to the different types of interaction between the phases that can occur in a pipe according to its size, thc gas velocity through it, and the characteristics of thc particles. LeFroy and Davidson (22G) report measurements of maximum spoutablc bed depth and of overall pressure drop in spouted beds, defined as a bed of particles contained in a vertical hopper and operated by a gas jet entering at the base. The authors advise caution when using the developed theory beyond the range of thc present experiments. A study of particle mixing by percolation is presented by Bridgwater et a!. (7G). One limiting case, the percolation and radial dispersion of small spheres through a packed bed of larse spheres, is considered and shown to conform to a diffusional mechanism. T h e process is claimed to be analogous to the dispersion of a tracer in flow through a packed bed in the turbulent regime, the analog being employed to correlate the results in a plot of Pcclet number against coefficient of restitution. McDougall (23G) proposes a compound flow mechanism for the discharge of solids from a hopper as a means of ensuring reliable discharge performance by means of air flows to assist the flow of solids. Holland et al. (73G) consider fluid drag effects in the discharge of granules from hoppers. Fluid drag forces becomc significant as the granule diameter is reduced. Further, these forces account for the reduced flow rate of fine powders from hoppers and for the increased flow rates produced by suction and airinjection techniques. A photographic study by Bosley et al. ( 5 G ) of granule discharge from model hoppers, reveals the effect of hopper shape, granule size, granule density, and hopper sizc upon the velocity profiles in hopper discharge. Bennett et al. (4G) describe the use of a hydraulic fluid, such as water, to convey an extremely concentrated solids-liquid mixture from a fecd tank to a reaction vessel. The technique is claimed to represrnt a simple but accurate method of hydraulic metering. MeDougall and Knowles (24G)consider the flow of particles through orifices and sugeest that the movement involves two different flow mechanisms according to whether the particles are in contact with each other or not. A study concerned with thc flow of granular particles through a vibrating orifice is presented by Yoshida and Kousaka (44G). Sinusoidal vibrations (in the vertical direction) were carried out a t frequencies of 18, 29, and 40 cps. Khitrin and Mosse (76G)have studied the concentration of solids in a dust suspension flow. T h e unsteady dust concentration in a high speed flow may exceed the calculated value for steady motion by 10-20 times. A set of differential equations is derived by Kudryavtseva (78G) for motion of a powder-laden flow in a pipe of variable cross scction with allowance for gas compressibility and heat transfer between the components. An experimental study of the dynamics of an air-dust jet is reported by Laats (1SG). T h e experimental data show that the tempcrature fields in the jet cross sections in dimensionless coordinates may be correlated by a single universal curve for various sections and various initial concentrations. Komov (77G) has madc an experimental investigation of the influence of solid particles in an air stream flowing in a Lava1 nozzle. Shadow photographs are
presented of the compression shocks a t the outlet section of the nozzle. A theoretical analysis by Saks (34G) of the turbulent motion of air-solid mixtures in a pipe indicates a n additional resistance owing to dissipation of the energy of turbulent fluctuations by the solid particles. Orlov (3OG) investigates certain properties of the equations of motion and the boundary conditions for a slowly flowing suspension with allowance for sedimentation. T h e method obtained is applied to the calculation of the sedimentation boundary layer near a n inclined plate. T h e appearance of a new inertial term in the transfer equations is reported by El’perin ( I I G ) as a result of nonuniform motion in a gas suspension flow. Zabrodskii and Parnas (45G) present calculations which indicate that the use of resonance oscillations intensify interphase heat and mass transfer in gas suspensions. Losscs of available energy in gas suspensions as a result of heat transfer between the gaseous a n d solid phascs arc derivcd by Yasnikov and Gal’pcrin ( 4 3 G ) utilizing the methods of the thermodynamics of irreversible processes. Simpson et al. (36G) present a n analytical treatment of heat transfer to a two-phase mixturc of well-dispersed subliming particles and vapor, flowing over a heated surface. An analytical and experimcntal study of the {low properties of suspensions with high solids concentration is reported by Gay et al. (72G). T h e correlation equations are based o n an analysis that considers the particle-particle interaction in a settled suspension, and are claimed to fit all systems investigated. Navari et al. (27G) have tested and developed a suspension with properties similar to those of whole blood in the mammalian body, providing a possible tool for flow visualization under ideal conditions. An exact similarity solution is given by Zung (4GG) for the flow of a fluid-particle suspension over an infinitely large disk rotating a t constant velocity. Numerical solutions of the resulting ordinary differential equations provide velocity distributions for both fluid and solid phases and density distributions for the solid. A description of the types of mixing equipment used to carry out the process of agglomeration is presented by Sirianni e , al. (37G). A number of specific applications are considered, including the upgrading of coking coals by selective agglomeration, the pelletization of soluble salts, the formation of highly spherical bodies from various materials, and the beneficiation of ores. Briller and Robinson (8G) report a simple, reliable means for measuring particle diffusivity in a particle-fluid system in the core of a duct. Particles are injected into the midstream of the duct, and the diffusivity is calculated from measurements of the standard dwiations of the particle concentration profiles a t various downstream distances from the injector. T a m (3QG)develops a formula for the drag exerted on a cloud of spherical particles of a given particle size distribution in low Reynolds number flow. Mean flow equations, obtained by appealing to the “randomness” of the particle cloud are found to be i n a form resembling a generalized version of Darcy’s empirical equation for the motion of fluid in a porous medium. An extensive investigation was performed by Reddy and Pei (33G) of the particle dynamics of narrow spherical glass beads transported vertically by a turbulent air strcam in a 10-cm inside diameter pipe. T h e investigation was undertaken to study the effects of particle diameter and concentration and the gas mean velocity on the particle velocity, the gas-phase velocity, the slip velocity between the phases, and the axial pressure gradient along the pipe. An analytical and experimental investigation is presented by Neilson a n d Gilchrist (28G)of the trajectories of particles entrained by the gas flow in nozzles. T h e investigation was part of a study of erosion damage in rocket motor tail nozzles, attributable to the action of particles produced in the combustion chamber. Bena et al. (3G) use a theoretical analysis to show that aerodynamic classification in a n elutriation column c a n be used to determine relationships useful in calculation of free-fall velocity. A variabledensity model of the pipeline flow of suspensions is reported by Shook and Daniel (35G). Friction factors are found to depend upon the distribution of solid particles within the conduit as well as the Reynolds number of the flow. Basina and Yugay (2G) consider theoretical solutions of thc problem of motion of a burning particle in swirling flow, taking into account the effect of combus-
tion on the drag coefficient. T h e results of calculations are compared with experimental data. Two-Phase Flow:
Fluidized Beds
T h e literature on fluidized beds is again concerned with bubble properties, gas mixing, particle motion, pressure and density fluctuations, bed expansion, holdup, pressure drop, and minimum fluidizing velocity. T h e number of papers reported on in this section has diminished somewhat because of a n expanded section on solid-fluid two-phase flow and a desire to minimize duplication between the section on bubble dynamics and that on bubble properties in fluidized beds. G e n e r a l theory, h y d r o d y n a m i c stability, a n d viscosity. Krambeck et al. ( 3 0 H ) propose a mathematical model for gasfluidized beds that allows for a randomly fluctuating flow pattern. A simplified version of the model indicates the effect of fluctuating flow to be similar to that of stagnancy in steady systems. Although the effect is apparently inconsistent with the usual steady-state models, some published d a t a on conversion in fluidized beds is stated to exhibit this effect. Theoretical equations predicting the extent of axial dispersion of a flowing gas in fluidized beds are derived by Yoshida et al. (68H) from the recciitly proposed bubbling bed model. Prcdicted dispersion values are comparcd with experiment and thc results are discussed. Vakhrushev and Basov (65H) present both theoretical and experimental considerations of the factors leading to and effecting heterogeneous fluidization. Distribution functions are obtained by Taganov and liomankov ( 5 9 H ) for the axial velocities of movement of the phascs in a fluidized bed. I n a second paper (GOH), the same authors compare the distribution functions with experimental data, and the procedure used in the investigation is. described. Strel’tsov and Shlyakhtov (55H) demonstrate that it is possible to use a simplified exponential relationship for calculation of the actual conditions of a chemical reaction in a fluidized bcd. A study of a series of catalytic processes in a fluidized catalyst bed arc presented by Mukhlenov (39H). A number of equations are then recommended for use in the design of catalytic equipmcnt. Lyall ( 3 2 H ) describcs the various photographic techniques used to reveal particle flow patterns and bubble distributions in both two- and thrce-dimensional fluidized beds. Tamarin ( 6 1 H ) considers the laws of propagation of an instantaneous heat pulse and obtains a function representing its effect in a fluidized bed in the presence of heat losses owing to gas flow and through the walls of the container. A mechanism of heat transfer between fluidized beds and wall surfaces is proposed by Yoshida et al. (6QH). T h e mechanism utilizes the film-penetration theory for mass transfer and considers both steady-state heat conduction through a n emulsion layer a t the wall and unsteady-state absorption of heat by emulsion elements. Experimcntal results for radiant wall to bed heat transfer in fluidized beds is reported by Szekely and Fisher (57H). T h e results were interpreted in trrms of a model postulating radiant heat transport from source to particles during their finite residence time a t the wall. From the analysis, a n estimate was made of the conditions under which radiation effects are significant in a real system where radiation and convection occur simultaneously. Verteshev et al. ( 6 6 H ) consider horizontally moving fluidized beds, which are claimed to have a number of advantages over nonmoving beds, and derive a n equation expressing the relationship between the rate of horizontal movement, physical properties, hydrodynamic conditions, and geometric characteristics of the apparatus. Stankevich et al. ( 5 3 H )present the results of a n investigation of the outflow of fine-grained material, fluidizcd by a gas, across a circular vertical baffle that is a structural elcment of multistage counterflow fluidized-bed contact apparatus. Prokhorenko et al. ( 4 4 H ) study the motion of particles in a fluidized bed, and the influence of the bed structure as a whole, by analyzing the system of differential equations describing the particle motion in the bed using the Lyapunov stability criterion. Experimental measurements of propagation properties of disturbances in liquid fluidized beds are reported by Anderson and Jackson (2H).Comparison of the results with the predictions of stability theory appears to be encouraging. VOL. 6 2
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Finnerty e t al. ( 7 4 H ) measured wave length and frequency for waves a t the top surface of air-fluidized beds. The data indicate the Rice-Wilhelm model to be a useful approximation to gasfluidized beds -of light to moderate particle densities. Further verification was provided by good agreement between bed viscosities, predicted by the model, and viscometer measurements on the catalyst bed. -1verag.e bed viscosity and bed expansion data have been obtained by Hetzler and IVilliams ( 2 6 H ) for homogeneous, water-fluidized beds of uniform glass spheres. T h e data are correlated in terms of a free volume theory and arc shown to predict the viscosity of liquid-fluidized and gas-fiuidized systems, as measured by other workers, with no change of parameters. Bubble properties. Miyauchi and Morooka ( 3 6 H ) have determined the radial distribution of bubble properties and holdup, in addition to carrying out heat transfer experiments in a fluidizedbed contactor. O n the basis of the results of observations of the flow state, the reaction yield was calculated for first-order irreversible reactions. i\n electroresistivity probe has been used by Park et al. ( 4 3 H ) to yield information on bubble properties (frequency, volume fraction, size and size distribution, and rising lselocity) as a function of fluidization level, particle size, and position within the bed. T h e properties of freely rising bubbles in unbaffled fluidized beds is considered by Grace and Harrison (24H). T h e rising velocity is found to bc greater for a swarm of bubblcs than for a single isolated bubble aiid is attributed to coalescence. Xdditionally, bubble throughflow velocities are found to be greater than predicted by availablc theories. .\t least part of this discrepancy is due to modifications of bubble shape and high local concentrations of bubbles. Godard and Richardson ( 2 Z H ) in an experimental study of freely rising bubbles confirm the findings of Grace and Harrison ( 2 4 H ) that bubblcs present in high concentrations rise a t much greater velocities than isolated bubbles of the same volume. I n a second paper, Godard and Richardson ( 2 7 H ) measure the collapse rates of small bubbles injected into particulate gas fluidized beds. Determinations are also made of the minimum stable bubble volume in a bed fluidized at a velocity equal to twice the minimum fluidizing velocity. Gabor (1GH) considcrs a chain of rising bubbles in order to investigate interaction effects on the fluid dynamics of bubbles in a fluidized bed. In an investigation of the hydrodynamics of a bubble bcd, Efrcmov and Vakhrushev ( 7 3 H ) find the bubble diameter to be practically independent of the gas velocity, increasing with an increase of thc free cross-section, and depending only slightly on the orifice diamctcr. I\ study of gas transfer between a bubble and the continuous phase in a gassolid fluidized bcd is reported by Toe1 et al. ( 6 3 H ) . Rubbles of N O aand S O nwere employed. Gas mixing a n d models. Oigenblik et ai. ( 4 2 H ) consider the modeling of catalytic proccsses in a fluidized bed on the basis of experiments with tracers. T h e objective of the work w-as to determine the Coefficients of interphase transfer o n the basis of experiments with the washour of adsorbed or unadsorbed tracers. Physical models of fluidized systems are discussed by Gel’perin et al. ( 7 9 H ) . One such model combines a spouting, fluidizing, and cyclically shifting bed to yield a combination of pulsating unit volumes, cach volume element of the bed bciiig dynamically unstable. Furusaki ( 7 5 H ) considers the effect of adsorption in fluidized-bed reactors and shows that when the mixing of the gases in the axial direction is great, the volume cffcct can bc ignored and the homogeneous-phase model can be used with some modifications. Goikhman e l nl. 123H) present experimental results o n the effect of floating spherical packing on the distribution of the gas residence time in a fluidized bed. Gel’perin et ai. ( 7 7 H ) consider the use of a column containiqg fluidized ball packing for achieving higher tray performance. Particle motion and mixing. Svcto7arol-a and Burovoi (56“) develop mathematical models for drying proccsscs in a fluidized bed. T h e models may be applied to calculations on drying equipment involving a fluidized bed and to thc development of an automatic process control system using computing techniques. T h e description of the systems is obtained by use of a five-dimensional phase space and allows for a finite rate of particle mixing, variation of grain size spectrum, kinetics, entrainment, and precipitation of particles from the fluidized bed. T h e results of an experimental study of heat transfer in a pulsating fluidized bed are reported by Sapozhnikov and Syromyatnikov ( 4 8 H ) . T h e data 116
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arc analyzed in terms of the effect of pressure, particle diameter, and vibration frequency on the heat transfer coefficient. Taganov and Romankov ( 6 0 H ) consider the statistical characteristics of heat and mass transfer in a fluidized bed. Interpolation formulas of simple structure are proposed by Strel’tsov (54H) for describing the composition and reaction rate of solid particles with a gas as functions of time. External mass transfer is taken into account, resistance to chemical reaction, arid diffusional transfer of material through a film of solid product. Shakhova and Gorelik ( 5 0 H ) present results of a n experimental investigation of the thermal relaxation time of particles in a fluidized bed. Values of the coefficient of heat transfer from particles to ga’s are determined. .\ method for checking the “principle of independence” for the process of growth of a collection of polydisperse particles is considered by Razumovskii et al. ( 4 7 H )for the example of a process in a fluidized bed. T h e principle of the approach, however, is claimed to be general provided that certain conditions are satisfied. I n a second paper, Razumovskii and Strel’tsov (46H) use a statistical analysis of the operation of a fluidized bed during the crystallization of salts from solution to derive expressions for the distributions of the number, surface area, aiid mass of the crystals with respect to size. Mogan e t al. ( 3 7 H ) report a method of prediction of the porosities of high-pressure gaseous fluidization systems. A 16.67, standard deviation was found in the values of the calculated porosities when compared w-ith the measured values. T h e porosities were measured a t 331 superficial velocities in fluidized beds of cracking catalyst, using hydrogen, nitrogen, and argon at pressures of 300 to 800 psig as the fluidizing media. T h e fluctuation of the number of particles in a bed consisting of glass pellets fluidized by water has been experimentally investigated by Kislykh (28H). T h e relative fluctuations are compared with analysis. 4 study of gas-solids mixing and heat transfer in incipiently fluidized beds of nonuniform cross-sectional area is reported by Miller and Edwards (35H). The solids mixing studies were performed a t isothermal conditions to determine if incipient fluidization could be achieved in beds of variable cross-section, and if flow patterns were significantly different from that of a constant cross-sectional bed. I n the heat transfer studies, heat transfer coefficients ranging from 2 to 15 Btu/hr sq f t O F were found between a n internal heating element and the bed. Baskakov and Mityushiii (5H) report an experimental investigation of the hydrodynamics of a fluidized bed near a submerged plate. T h e character and direction of particles, as well as the velocities of individual particles a t the chamber wall werc determined visually. Axial dispersion of the liquid phase in a turbulent-bed contactor has been investigated by Chen and Douglas ( 7 0 H ) . T h e contacting technique involves countercurrent flow of gas and liquid through a bed of low-density spheres, the spheres being set in motion by the upward flow of gas. A method for determining the mixing coefficient in a fluidized bed by a mathematical model of the process is presented by Rurovoi and Svetozarova ( 9 H ) . T h e distribution of vertical and horizontal particle velocity in a fluidized bed has been investigated by Akhromenkov and Kruglov ( 7 H )utilizing a labeled atom method. Pressure and density fluctuations, b e d expansion, entrainment, pressure drop, minimum fluidizing velocity, a n d denexamines the effect of bed height, sity distribution. Kozin (29N) dimensions, and depth of surface submergence on the frequency and amplitude of pressure fluctuations in a fluidized bed. Quickresponse ineinbrane pressure transducers were employed. Equations are derived by Soroko et al. (52”) for calculating the required height of space above a fluidized bed and, thus, the total height of fluidized-bed contact equipment. Matsen et al. ( 3 3 H ) have developed an expression for the maximum height of a fluidizcd bed in a slug flow that is supported by over 60 sets of experimental data. Stabilization of a bed of dielectric particles against fluidization by a n electric field, is reported by Katz and Sears ( 2 7 H ) . Under suitable conditions, 100yG bed expansion without diffusivc particle motion or bubble formation was obtained using silica gel particles. Additionally, glass bead and silica gel particle beds were observed to behave as packed beds with flow rates of the fluidizing gas up to 1 5 times the normal incipient fluidization rate. Black and Boubel ( 8 H )have investigated the factors contributing
to removal efficiencies of small-diameter aerosols in a bed of fluidized glass shot. An examination of the dynamic structure of solid-phase equilibrium during fluidization is made by Todes and Tsitovich (6214). A mathematical description is obtained for the entrainment of the solid phase. The inflection in the entrainment curves is explained by the two-phase nature of the process, dense phase-sublayer space-entrainment. T h e entrainment of shale flotation concentrate and polyvinyl chloride under fluidization conditions is investigated by Tsitovich (64H). Mayak and Matrozov ( 3 4 H )propose a model for calculation of the hydraulic resistance of grid plates with a large free section a t the high gas velocities characteristic of operation of the columns with fluidized packing. T h e resistance of plates with fluidized spherical packing, a t large densities of the latter, c a n be represented in the form of a sum of the resistance of the grid plate and the resistance created by the additional amount of liquid retained by the packing. Some features of the operation of a dense bed a t high velocities are examined by Chukin and Kuznetsov ( I I H ) . T h e maximum permissible gas velocities a t which the bed of loose material ceases to descend and becomes suspended in the shaft are obtained, as is the flow friction in thc moving and stationary beds. Baskakov and Gal’perin ( 4 H ) analytically investigate the problem of the critical resistancr and critical fluidization velocity of fine-grained material in conical equipment. T h e use of a fixed bed-of random loose porosity is used as a datum by Eastwood et al. ( 7 2 H ) for correlating the flow through denser beds and for predicting the minimum fluidizing velocity. Godard and Richardson (20H) employ the correlations of Carman and of Ergun for pressure drop for flow of fluids through fixed beds, to express the ratio of the free falling velocity of an individual particle to the minimum fluidizing velocity for a bed of uniform particles as a function of the Galileo number. Experimental verification that air classification in a fluidized column can be used to establish a n equation for thc free-fall velocities of the particles has been reported by Bena et al. ( 7 H ) . General two-phase theory is applied to the special case of a liquid and a granular solid flowing together up a vertical tube by Leung et al. (31H). T h e question of whether the solids flow in the particulate fluidized state or as a packed bed is considered, and a prediction of the transition from fluidized to packed bed flow in terms of the mean velocity of the mixture arid the concentration of solids in the entering slurry is presented. The superficial gas velocity required to suspend a given amount of solid particles in a liquid is considered by Narayanan et al. (47H). Fundamental aspects of the fluid mechanics of a single particle are employed. Basov et al. ( 6 H ) employ radioscopy of a fluidized bed in three directions, in order to permit determination of the nonuniformity of the density distribution over the cross-section of the apparatus. Miscellaneous operations. Moncman (38H) presents both theoretical and practical treatments of the factors connected with average retention time in a continuous fluidized dryer, and of the degree of drying that can be achieved in such installations. T h e drying of sodium sulfate solutions was studied by Bakhshi and Chai (3H). A correlation between the bulk density and the operating variables is reported. Mullin and Gaska (40H) present a study of the nucleation, growth, and dissolution characteristics of potassium sulfate in a laboratory scale fluidized bed crystallizer. The importance of measuring nucleation data for crystallizer design purposes under the appropriate conditions is clearly demonstrated. Gel’perin et al. ( 7 8 H ) consider heat transfer between a fluidized bed and staggered bundles of horizontal tubes. Experimental data on the maximum heat transfer coefficients for five- and ninerow bundles of tubes arc correlated in the form of a n empirical formula. A study by Gutfinger and Chen (25H) presents a theoretical relationship between coating thickness on a n object immersed in a fluidized bed of coating material and the physical properties of the system. I t is found that the heat transfer coefficient is a major factor in the fluidized bed coating process. Since coating d a t a reported in the literature do not generally report the heat transfer coefficient, a graphical method for estimating the coefficient was developed where final thickness d a t a are reported. Pulsifer et al. (45H) investigate the use of a n electrofluid reactor for the production of synthesis gas from coal char and steam. T h e experimental results of a study of the methane conversion process
in a fluid catalyst bed is reported by Sechenov et al. (4QH). An optimization study aimed a t producing the maximum amount of gasoline, with due regard to limiting the yield of coke in a fluidizedbed catalytic cracking process is presented by Shumskii (57H). M’itte (67H) reports a n experimental study of fluidized-bed combustion of graphite-base nuclear reactor fuels. T h e experimental results from fluidized-bed combustion studies on both unirradiated and irradiated fuel specimens compared favorably with predictions. Zielke et al. (70H) consider the fluidized combustion process for regeneration of spent zinc chloride catalysts. Hydrodynamic Stability
Applications of linear and nonlinear stability analysis continue to be numerous in the literature, with an increasing interest in the latter. T o include in this review the considerable amount of work associated with plasma instability would be to introduce a number of specialized topics that extend beyond the theme of hydrodynamic stability, so only those papers dealing with hydromagnetic and electrical phenomena that are construed to be of some general interest are included. I n the continuing effort to understand and predict the limits of stability of Poiseuille flow, Davey and Drazin (71) numerically studied the stability of Poiseuille flow in a pipe with respect to small axisymmetric disturbances, showing that the flow is stable to such disturbances a t all Reynolds numbers, and Pekeris and Shkoller (401, 471) studied the stability of plane Poiseuille flow to finite amplitude periodic disturbances. I n their second paper of the pair, Pekeris and Shkoller presented the neutral stability curves obtained in their analysis. Dowel1 ( I O Z ) examined the stability of plane Poiseuille flow to both infinitesimal and large disturbances using the full nonlinear Navier-Stokes equations. T h e results of the large disturbance analysis were negative, b u t he obtained the limit cycle oscillations of a n unstable flow for small disturbanccs. Craik (51)and Li (292)analyzed flows of the Couette type. Craik found that for plane Couette flow with a viscosity stratification the flow may be stable or unstable a t small Reynolds numbers for disturbances with long wavelength-to-fluid depth ratios depending upon the viscosity gradient. Li studied plane Couette flow of three superposed layers where, it was found, the instability depends upon the two modes of waves a t the two surfaces of discontinuity. I n a study of film flow down an inclined plate Lin (371) used previously developed nonlinear stability analysis to establish that supercritically stable wave motion of finite amplitude may develop. Some mathematical properties of the Orr-Sommerfeld equation, which occurs so frequently in stability analysis, were studied by Yih (572), Joseph (21Z), and Thorpe (471). Yih extended an earlier paper by Joseph on the bounds for the complex wave velocity for channel flow, and Joseph examined eigenvalue bounds for the Orr-Sommerfeld equation for parallel flow in the boundary layer and in round pipes. Thorpe showed that many of the analytic solutions of the stability equation for a stably stratified, inviscid, parallel shear flow belong to a wider family of solutions that can be written in terms of the hypergeometric function. Seven papers deal with the stability of rotating flows. DiPrima and Rogers (92) discussed computing problems related t o the nonlinear stability of the flow between concentric rotating cylinders, and Versteegen and Jankowski (522) experimentally studied the stability of viscous flow between eccentric rotating cylinders. For counterrotation of the cylinders, a new flow phenomenon different from the usual vortex pattern was observed for large eccentricity. I n a technical note, Leibovich (272) showed that earlier work on the stability of nondissipative swirling flows between concentric cylinders c a n be extended to density stratified flows, and he established a sufficient condition for stability to axisymmetric disturbances in terms of a modified Richardson number. Chang and Sartory (41) examined the hydromagnetic stability of dissipative flow between rotating permeable cylinders with respect to oscillatory axisymmetric disturbances, and they obtained asymptotic results. I n a short paper dealing with flow over a rotating cylinder Nayfeh (361) investigated the effect of centrifugal forces owing to spinning on the stability of the liquid layer away from the stagnation region. In another vein, Pedley (392) studied the stability of fully developed viscous flow in a rotating VOL. 6 2
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pipe. For small disturbances, the flow was found to be unstable for Reynolds number greater than 82.9, the unstable disturbances taking the form of growing spiral waves. Other papers dealing with the stability of rotating flows are by McConaghy and Finlayson (341), Kent, .Ten, and Chen (221), and Calvert and Melcher (31). T h e former showed that oscillatory convective instability occurs in a thin rotating fluid layer when convection is caused by surface-tension gradients a t a free surface. Kent et al. analyzed the Kelvin-Helmholtz instability in the region of the radial boundary of a rotating plasma, and Calvert and Melcher studied the stability and dynamics of a circular cylindrical column of inviscid dielectrophoretic liquid undergoing rigid body rotation in a n electric field. Several papers can be classified under the stability of shcar flows. 'I'horpe (481) experimentally studied the instability of stratified shear flows by tilting a long rectangular tube containing two immiscible fluids, and he compared the observations and theoretical analysis of the conditions for the onset of Kelvin-I-Ielmholtz instability. T h e wavelength of the unstable waves was greater than predicted, b u t the time a t which the instability occurs was well predicted. Loer (321)used a simplified form of the NavicrStokes equations in his numerical study of the instability of a boundary layer flow along a flat plate, and Laiidahl (261)reported the development of a series of computer programs for the calculation of the stability of channcl and boundary layer flows. Lessen, Harpavat, and Zieii (281) analyzed the linear instability of threedimensional laminar ancl turbulent shear layers, obtaining neutral stability curves for the flow of two semi-infinite streams. Michalke (351)examined the stability of spatially growing three-dimensional disturbances in a free shear layer to determine if three-dimensional disturbances are less stable than two-dimensional disturbances, H e performed numerical calculations for the special case of a hyperbolic tangent velocity profile, finding that for spatially growing disturbances the amplification of three-dimensional disturbances is smaller than for two-dimensional ones. In a pair of papers on incompressible jets K O and Lessen (241, 251) analyzed the stability of a free laminar boundary between parallel streams and the stability of a viscous, laminar plane jet. I n the former case the Aow was found to be stable unless the theory is corrected for the nonparallelism that exists a t low Reynolds numbers, and in the second paper a minimum critical Reynolds number of 4.03 was found for parallel flow and 12.4 when nonparaiielism was considered. There appears to be an increasing interest in the stability of nonNewtonian fluid flows, espfcially the flow of viscoelastic materials. F6nyad and Gyr (731) cxtendcd an earlicr paper of Gupta arid analyzed the effects of surface and shearwave perturbations on the flow of anisotropic viscoelastic liquid films. Their preliminary findings were that the flow is stable for small wave numbers and unstable for large ones. Feinberg and Schowalter ( 7 7 1 ) employed a n energy method to determine sufficient conditions for the stability of fluid motions that are described by the infinitesimal theory of viscoelasticity. T h e energy method avoids the restrictions of linearization and Newtonian behavior; however, the results of the analysis are limited to fluids having a Newtonian memory ( ; , e . , no memory). In yet another paper on viscoelastic fluids Vest and Arpaci (531) studied the overstability of a viscoelastic fluid layer heated from below. Linear stability analysis was used by Makarov, hlartinson, and Pavlov (331) to derive neutral stability curves for the planar flow of a non-New-tonianpowcr law fluid. T h e stability of two superposed elastico-viscous liquids in plane Couette flow was examined by Li (301). T h e stability effects of the elasticity in the liquid exist if there is viscosity stratification, b u t are absent if the viscosity is uniform. Some extensions or modifications to the theory of Taylor instability have been made by Daly (61),Kiang (231),and Nayfeh (371), and Raco and Peskin (421)have examined the effects of a transverse electrostatic field 011 the stability of a plane fluid interface. By numerical methods, Daly confirmed the variation of the linear growth rate with surface tension coefficient predicted analytically by Chandrasekar, and in the nonlinear regime he showed how the surface tension provided the mechanism for drop separation from a Rayleigh-Taylor spike. Kiang developed solutions for the non118
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linear oscillation associated with inviscid Taylor instability. I n a related paper, Nayfeh examined nonlinear inviscid stability of the interface between two superposed fluids, one bounded by a solid boundary, the other in contact with a semi-infinite gas. 11s in the paper of Kiang, solutions for the instability problem arc developed for wave numbers abovc the cut-off wave number, the wave number below which disturbances grow in amplitude. Kaco and Peskiii showed how the clectrical conductivity, surface charge, frequency of the applied ficld, viscosity, surface tension, and other parameters affect the stability of the interface between two low viscosity fluids. I n addition to the papers dealing with the umal linear and nonlinear stability analyses, several papers dealing with the transition from laminar to turbulent flow were published in 1969. Goldstcin, .idrian, and Krrid (761) used a laser-Doppler instrument to study transition and turbulent flow of dilute aqueous polymcr solutions, findiiig that the addition of Polyox to water results in unsteady flow at lower Reynolds number than for pure water. It-ortniann (5GZ) used the tellurium method to study the devclopmeiit of transition flow downstream of Gortlcr vortices. He was ablc to distinguish succcssive modcs of instability by visualization. Ffoowcs It'illiams, Rosenblat, and Stuart (121) and Shabrin (451) also studied laminar-to-turbulent flow transition. T h e latter used the energy method to analyze the transition. Reshotko 1431) rather qualitatively discussed the use of dinlensional analysis to determine thc significant parameters in the evaluation of transition data for boundary layer flows. T h e problems of convectivc stability owing to rithcr thermal gradients, concentration sradients, surface tension gradients, or electrical and magnetic forces have receivcd a considerablc amount of attention recently. T h e paper by McConaShy and Finlayson (,?41)on surface tension driven oscillatory instability has alrcady been rnentioncd. Gross and Hixson (791) analyzed Marangoni instability with nnsteady diffusion in the undisturbed state, showing numerically that a system exhibiting resistancc to transfer a t the interface may be unstable with transfer in cithcr direction, a result supported by experiments. Variations on the theme of natural convection produced by a temperature gradient have becii published by Gunness and Gebhart (201), Gebhart ( 7 m , Turnbull ( & I ) , Goldstein and Graham (171),Vcst and Arpaci (541),and Unriy and Niessen (511). Gunness and Gcbhart, by means of perturbation analysis, analyzed the stability of transient natural convection between two infinite parallel vertical plates, and Gebhart provided a survey of natural convection flow, instability, a n d transition. T h e effect of dielcctrophoretic forces on classical RCnard instability w-as examined by Turnbull who established that the exchange of stabilities can occur for certain boundary conditions. Goldstein and Graham measured the critical Iiaylcigh number for the onset of thermal convection in a horizontal fluid layer heated from below and bounded above aiid below by zero shear boundaries, and the results were found to agree with Rayleigh's solution. Theoretical predictions of the critical Rayleigh number associated \rith natural convection in a vertical slot with isothermal side w-alls a t diffcrent temperaturcs were verified experimentally by Vest and Arpaci, and Unny and Kiessen examined the somewhat similar problem, h u t with both vertical and horizontal temperature gradients imposed. Both surface tension effects and buoyancy cffec ts were considered by Davis (81) in his analysis of a modified B h a r d problem. I-le used the energy method to determine the qualitative behavior and to compute thr optimal stability boundary. Westbrook (551) used small pcrturbation methods to study the stability of corivrctivc flow in a porous medium, and lie obtained a criterion for stability in terms of a. Reynolds number and a Rayleigh number. I n a paper more related to the stability of stratified flow than to natural conbection Stern (461)examined the instability of salt fingers with respect to internal gravity waves, comparing relationships developed in the analysis with laboratory measurements and ocean observations. As the emphasis on environmental problems increases, onc: can expect a n increase in the research on stability prohlcms related to oceanography and atmospheric sciences, but in 1969 the pnblishrd work in these areas was meager. Brown (71, 21) numerically analyzed the hydrodynamic stability characteristics of planetary
zonal flows, and he used the numerical model to further examine special instability phenomena caused by vertical walls that specify the lateral boundary conditions a t the northern and southern boundaries of the atmosphere. Green (782)used a geophysical model to examine BCnard convection, and he discussed the position of the lower oceanic boundary of a convective layer. Two papers dealing with electrohydrodynamic stability are by Gokhale (751)and Turnbull and Melcher (502). Gokhale analyzed the effects of viscosity and electrical conductivity of mercury on the axisymmetric disturbances of a current-carrying annular cylinder of mercury. Serious discrepancies between theory and experimcnt were found. Turnbull and Melcher established a theoretical stability criterion for a n initially static, stratified fluid subject to a n electric stress, and in the limit of a perfectly insulating fluid several characteristics of this electrohydrodynamic RayleighTaylor instability were analyzed. T w o papers that are isolated from the arbitrary groupings discussed above are those of Pearson and Matovich (381)and Seleznev and Tsiklauri (441). T h e former investigators considered the stability of a nonuniform, axially symmetric extensional flow of a Newtonian fluid with respcct to small disturbances. Analytic results for the disturbance amplification factor were obtained. T h e latter authors examined the stability of a two-phase system.
Turbulence
New models for turbulence, and new schemes employing existing models have appeared in the literature. Nee and Kovasznay ( 6 4 5 ) proposed a n eddy viscosity model that is based on the physics of the turbulent motion. A rate equation for the mixing length, in which the form of the generation and decay terms are obtained from physical requirements and the equations of motion, forms a closed system which can be solved for specific applications. Laufer (485)showed that the mean flow field for free turbulent layers of variable density can be found from the corresponding constant density case without using a compressible eddy viscosity. A new physical model of statistically isotropic turbulence, constructed from the vorticity equation and representing the turbulence as a superposition of individual vortex sheets, was proposed by Parker (735). The model contains one free parameter and yiclds good comparisons with experiment throughout the inertial and viscous ranges for a single value of the parameter. Bodner (95)formulated a theory for homogeneous turbulence using a time-varying Wiener-Hermitc basis. T h e development is similar to that given previously by Meecham and coworkers except that the time dependence is handled differently so that the main objection to previous efforts is removed. Burgers’ model has been employed by several workers. Meecham and Su (605) used the first two terms of the Wiener-Hermite expansion and integrated the equations more completely than had been done previously. Jeng ( 4 3 J ) modified the model by introducing a n artificial random force. Burgers’ one-dimensional turbulence model retains the nonlinear and dissipative characteristics of the Navier-Stokes equations and, hence, studying it may yield results of general consequences in turbulent flows. For example, Jeng found that the velocity correlation function is similar to the result obtained from experiment. Tatsumi ( 9 6 5 )expanded the velocity field in Burgers’ model into a series of nonlinear sawtooth waves, each of which is a solution to the model as the Reynolds number becomes large. T h e energy spectrum and decay compared well with previous analytical work. Lundgren ( 5 6 5 )solved a model previously proposed for the onepoint turbulent distribution function for a class of rectilinear flows. Closure was obtained by assuming a relaxation model for the pressure fluctuation term. Panchev presented a generalization of Kovasznay’s spectral theory ( 7 2 5 )and also solved analytically the spectral equation for nonstratified shear flow in the equilibrium range (775). Grid-generated turbulence continues to be a prime area of research because of its isotropic, or nearly isotropic, nature. Portfors and Keffer ( 7 9 J )measured turbulence intensities and reported that within the limits of their measurements, the flow became isotropic by a distance of about 30 mesh lengths. O n the other hand, Uberoi and Wallis (7075)found, from similar measure-
ments, that isotropy is not satisfied, but the energy-dissipating eddies are more nearly isotropic than energy-containing eddies. Furthermore, the deviations from isotropy were dependent on the grid geometry. A novel feature of the experimental study of grid-generated turbulence reported by Van Atta and Chen (102J) is the use of digital harmonic analysis with the fast-Fourier-transform method to compute correlation functions and spectra. I t was found that isotropy was approximately satisfied. Corrsin and Karweit ( 2 7 5 ) measured the growth of fluid material lines in grid generated turbulent water flow using a hydrogen bubble technique. This work has implications in the area of vorticity production. Deissler (23J) developed a n equation for the direction of the maximum turbulent vorticity in a turbulent shear flow and found that the direction is a t a 45” angle t o the main flow only for isotropic turbulence. Many papers concerned with the delineation of turbulent flow by numerical solutions resulted from T h e International Symposium on High-speed Computing in Fluid Dynamics held in Monterey, California, in August 1968. Batchelor ( 7 J ) , Hirt (395), Orszag (705), and Schonauer ( 8 6 J )described the difficulties of numerically integrating the instantaneous Navier-Stokes equations for three dimensions, especially as the Reynolds number becomes large, T w o dimensional schemes can be handled, and it has been suggested ( 7 5 ) that there are enough common general properties of two- and three-dimensional turbulence to justify using the two-dimensional case as a means of testing the soundness of hypotheses about turbulent flow. Hirt pointed out that some of the computational difficulties for high Reynolds number flows can be avoided by examining only the mean effects of turbulence. A finite difference scheme for accomplishing this is presented. Orszag (705) also presented two techniques for the numerical solution of the Navier-Stokes equations and indicated that results for three-dimensional flow a t moderate Reynolds numbers c a n be computed. Lilly (525) performed a direct numerical integration of the equations for two-dimensional turbulence and concluded that the principal features predicted by Kraichnan’s model were borne out. T h e difficulties associated with turbulence measurements in liquid mercury using hot film sensors were described by Malcolm ( 5 7 5 ) . An equation for the determination of turbulence intensity is given. Hill and Sleicher (385) generalized previous equations for describing the errors encountered in turbulence measurements with inclined hot wires. T h e cosine law of direct sensitivity is assumed to be valid, and the results are for weak turbulence. Frantisak et al. ( 2 7 5 ) used a photochromic dye dissolved in test fluid, irradiated by ultraviolet light, for quantitive measurements of turbulence in the entire cross section of a tube. T h e results establish the usefulness of the technique as a nondisturbing tracer. T h e effect of turbulent fluctuation in an optically active fluid medium was investigated analytically by Sutton (945). T h e results are applicable to the use of large aperture optics in turbulent wind tunnels. Schlieren photographs of turbulent flow were examined by Thompson and Taylor (7005)who considered the relationship of the statistical parameters that describe the turbulent density variations to the intensity variations on the photograph. T h e diffusion of a passive scalar in a homogeneous turbulent flow was treated by Saffman (855)using the truncated WienerHermit expansion of the velocity and concentration fields. For diffusion from a point source, a result consistent with the earlier work of G. I. Taylor was obtained, thus indicating that the approach gives reasonable results. Christiansen ( 79J)studied the mixing of two miscible liquids in turbulent pipe flow using lasergenerated light as the measuring tool. T h e results includc the spectrum for spatial variations in concentration and the dependence of this spectrum on the decay time of the concentration variations. O’Brien (685) continued his work on chemical reactions occurring in a turbulent flow. H e proposed a form of statistical independence for a species undergoing reaction and simultaneously experiencing turbulent convection and molecular diffusion. T h e bases for this hypothesis are independent of the order or classification of the reaction, so that the results for the case studied should have general implications. Wall region. Further studies of the nature of the viscous subVOL. 6 2
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layer i n turbulent flow have been reported. Corino and Brodkey (20.7) performed visual and photographic observations of solid particles of colloidal size in a liquid flowing turbulently through a tube. T h e viscous sublayer was continuously disturbed by smallscale velocity fluctuations and periodically disturbed by fluid elements penetrating from regions farther from the wall. Periodic ejections of fluid elements from a thin region adjacent to the sublayer toward the center line occurred. When these elements entered the shear zone, intense and chaotic velocity fluctuations were created. These ejections and related fluctuations are believed to be a factor in generating and maintaining turbulence. Popovich (7885)analyzed experimental data on the viscous sublayer by a statistical approach and concluded that turbulent pipe flow is statistical in nature u p to the wall itself. T h e wall shear stress is distributed according to the Maxwell-Boltzmann function, while Helmert’s distribution appears to describe the sublayer thickness. T h e effect of a streamwise pressure gradient on the velocity profile in the viscous sublayer along a smooth wall in two-dimensional flow was estimated by McDonald (58J). T h e analysis allows the assumption that the stress and pressure gradients are equal in the sublayer to be relaxed. T h e results include new additive constants for use in Townsend’s modified law of the wall, and the new results improve thc comparison with experiment in a n adverse pressure gradient. McDonald ( 5 9 J ) has also provided a n indirect check of the modified law of the wall for strongly retarded boundary layers. Sirkar and Hanratty ( 9 0 5 ) reported experimental mass transfer measurements in a pipe t h a t allowed the determination of the root-mean-square value of the fluctuating velocity component in the transverse direction close to the wall. These results suggest that the eddy diffusivity is proportional to the cube of the distance from the wall as the wall is approached in high Schmidt number systems. This result was demonstrated conclusively by Notter and Sleicher ( 6 7 J )using a n analytical technique that the authors feel has potential in solving other turbulence problems. Squire ( 9 3 J ) has analyzed existing experimental d a t a for compressible boundary layers with air iiijection a t the wall to find the parameters that occur in the law of the wall for this system. T h e absolute values obtained for the parameters are subject t o error because of the lack of accurate skin friction measurements, b u t the trends with Mach number and injection rate are defined. Randhava ( 8 2 J ) has given a simple result for predicting the velocity distribution in the inner wall region of concentric annuli. T h e proposed relationship is a logarithmic distribution with constants depending upon the radii of the annulus and the radius of maximum velocity. Further analytical work o n turbulent flow models for the wall region was presented by Buyevich ( 7 6 J ) . nnother paper ( 5 3 J ) concerned with experimental verification of the law of the wall will be discussed in the Duct Flows section. Boundary layers. .A state-of-the-art conference on the prediction of turbulent boundary layers was held a t Stanford University in 1968 and has been reported by Kline et al. ( 4 5 5 ) . T h e proceedings of this conference have been published (8GJ). Included in the proceedings are d a t a for 33 standardized flows against which new prediction methods may be tested, the results of 28 existing prediction methods, evaluations of the flows, descriptions of thc underlying theories, discussions of the statils of knowledge on flow structure and separation, and summaries of additional research needs. I n all, the proceedings are a necessity for those concerned with predicting turbulent boundary layers. I n addition to this, analyses for the prediction of iiicompressible turbulent boundary layers on smooth surfaces have now been given by several workers (3GJ, 6 6 J , 775, 95J, 7065). IYhite ( 7 0 6 J ) proposed a n integral method t h a t results in a single equation to be solved for the skin friction coefficient with the Reynolds number as the only parameter. T h e use of a shape factor involved in other integral techniques is avoided. T h e calculations appear to be much simpler than the von K a r m a n method and yield accurate results and, in addition, the procedure contains a built-in separation criterion. T h e entrainment theory was modified by Nicoll and Ramaprian ( 6 6 J ) for predictions in an adverse pressure gradient. Pletcher (77J)developed a n explicit finite difference method that 120
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is claimed to be always stable and has as its only empiricism the eddy length distribution. Results show good agreement with d a t a for favorable and unfavorable pressure gradients and for flat plates with blowing. Gaddis and Lamb ( 2 8 J ) developed a two-region boundary layer model made up of a n inner region dominated by viscous stresses and small eddy scales and a n outer region that has characteristics similar to free turbulence. T h e two regions are joined by continuity of shear stress and eddy viscosity. T h e case of variable density is also discussed, and the reference density is taken as that a t the boundary between the two regions. Brown and Joubert ( 7 4 J )described a floating element skin friction meter designed for use in turbulent boundary layers ivith a d verse pressure gradients. T h e theory and limitations of the device and other means of obtaining the samc information are discussed, and comparison of the measurements in a two-dimensional boundary layer using Preston tubes and the new device are given. A single equation for a continuous eddy viscosity obtained from d a t a for turbulent boundary layers on flat plates was reported by Chi and Chang ( 7 8 J ) ,while Bull ( 7 5 J )has given a functional representation of the mean shear. An analogy between flows over curved or rotating boundaries and flowswith buoyancy in turbulent flow allowed Bradshaw ( I I J ) to improve the prediction of turbulent boundary layers on curved surfaces even though the analogy is not as close in turbulent flow as it is in laminar. Lissaman and Harris (54J)reported the results of a n experiment on the effect of a compliant wall on the turbulent boundary layer. No major effect on the drag of order of magnitude greater than lOyowas obtained for any of the parameters tested. Using a n approximate method for finding the Reynolds analogy factor for turbulent boundary layers with pressure gradient, Tetervin ( 9 9 J ) concluded that the factor increases in flows with adverse pressure gradients. Laminarization or reversion of turbulent to laminar flow has been investigated both analytically and experimentally. Three experimental studies of accelerated incompressible boundary layer flows were reported, two using nozzles ( 3 J , 3 5 J ) and the third a convergent channel ( 5 J ) . I n general, these studies demonstrated a strong effect of flow acceleration on an originally turbulent boundary layer. Measured velocity profiles showed that the wall region adjusts to laminar conditions earlier than the outer region. Criterion for laminarization have been given ( 3 J , 5 J ) , and Bradshaw ( 7 O J ) has demonstrated a general Reynolds number criterion t h a t was deduced from physical arguments. Launder and Jones (50J) presented theoretical solutions for the turbulent boundary layer in convergent channels that shift toward the laminar solution as the acceleration parameter is increased. Incompressible boundary layers on rough surfaces have also been investigated. Generally good agreement between prediction and experiment was obtained by Dvorak ( 2 4 J ) for the growth of a turbulrnt boundary layer with favorable, zero: and unfavorable pressurc gradients, T h e development of a turbulent boundary layer with several abrupt changes in surface roughness was examined by Blom and Wartena (81)for a n application in micrometeorology since it was concluded that the turbulent boundary layers in the atmosphere seldom have adapted to changes in the surface roughness before another change occurs. Perry et al. ( 7 6 J ) experimented on a surface roughness characterized by a smooth surface with a series of depressions or grooves. A s contrasted with the usual surface roughness, the friction factorReynolds number relationship in pipe flow for this particular roughness depends on the pipe diameter rather than the roughness scale. For flow over flat plates with either type of roughness, the results correlate with the same parameters. Spaid ( 9 2 J ) examined drag d a t a for small protuberances immersed in a turbulent boundary layer for both incompressible and compressible Aow and concluded t h a t a drag coefficient based on the average dynamic pressure is useful in correlating the data. h more general numerical integration of the time-averaged equations of motion for three-dimensional incompressible turbulent boundary layers than presently exists was performed by Nash ( 6 3 J ) . T h e shear stress is found by a separate integration of the energy equation into which empirical functions carried over from two-
dimensional studies were incorporated, and this is a source of uncertainty. A plea is made by the author for experimental results in a more suitable form for comparison with prediction. Turbulent shear layer behavior was analyzed by Lamb, Greenwood, and Gaddis (475) and by Shamroth ( 8 9 5 ) . L a m b et al., using a linearized longitudinal equation of motion different from the classical Oseen linearization, were able to obtain results for velocities in the midpart of a free shear layer in agreement with predictions of more elaborate nonlinear analyses. Shamroth developed integral methods of solution for the problem of the plane turbulent near wake, which avoid a nonphysical constraint usually encountered, and he discussed the constraint in some detail. T o the extensive literature on turbulent boundary layer flows h i t h mass injection can be added the work of Squire ( 9 3 5 ) , Ginzburg and Krest’yaninova ( 3 7 5 ) and Isaacson and AlSaji ( 4 Z J ) . Squire analyzed the experimental data available to synthesize information on the parameters that occur in the law of the wall as deduced from mixing-length theory. H e showed that the trends of these parameters with Mach number (up to Mach 6.5) and injection mass flow are clearly defined. Ginzburg and his associate analyzed the problem of the turbulent boundary layer in the presence of a pressure gradient with mass injection using a semiempirical theory based on a two-layered model. I n a short technical note, Isaacson and AlSaji developed expressions for the mean temperature distribution in a two-dimensional boundary layer developed initially over a nontranspiring surface with uniform mass injection of a downstream station. Their analysis represents a n extension of existing turbulent boundary layer theory. Repukhov and Bogachuk-Kozachuk ( 8 3 5 ) and Warner and Crabtree (7055)were concerned with the heat transfer associated with thin films. T h e Russidns compared experimental and theoretical results for multislot film cooling in which hot gas streams are separated from a heat transfer surface by a thin layer of gaseous coolant supplied through wall slots. Equations for the effectiveness of the film cooling previously published were found to be in satisfactory agreement with the author’s experimental data. Warner and Crabtree, on the other hand, provided experimental data on the d o c i t y and temperature profiles in the turbulent boundary layer above a film of distilled water. T h e measured boundary layer was found to bc 30 to 507, thicker than corresponding boundary layers for smooth dry surfaces. Boundary layers associated with hypersonic flow were studied by Hoydysh and Zakkay ( 4 0 5 ) ,Vereshchagina-Myshkova (7035), and Lee and Cheng (57J). T h e first two papers deal with turbulent boundary layers, b u t the third involves a laminar boundary layer. T h e experimental study of the Hoydysh and Zakkay were conducted a t Mach 5.75 with large adverse pressure gradients on flared axisymmetric surfaces. Boundary layer profiles of static pressure, pitot pressure, stagnation temperature, and transient heat transfer rates were measured. Correlation of the velocity profiles in the incompressible plane indicated that the law of the wall applied. Vereshchagina-Myshkova analyzed the free j e t boundary layer during equilibrium gas dissociation, assuming a conventional boundary layer that does not affect the external flow field of inviscid gas. Lee and Cheng provided a n asymptotic analysis of the Navier-Stokes equations in von Mises’s variables, which allows a three-region structure, t o study the hypersonic strong interaction problem. New and different numerical results for a n insulated and a cold flat plate were obtained. Measurements of the density and density fluctuations in a compressible turbulent boundary layer were reported by Wallace (1045). T h e results, obtained by using a n electron-beam luminescence probe, were used to compute velocity and total enthalpy profiles which ’were then compared with existing hypotheses. Harvey and Bushnell ( 3 6 5 ) employed the Wallace d a t a to estimate the intensity of velocity fluctuations and to evaluate a mixing length approach for computing the magnitude and trends of the velocity and density fluctuations. Hebbar and Paranjpe ( 3 7 J ) formulated a n explicit expression for the coefficient of skin friction in compressible turbulent boundary layers t h a t is valid over a wide range of Reynolds numbers for general heat transfer conditions. T h e usefulness of the result is demonstrated.
Theoretical and experimental studies of the development of the turbulent boundary layer o n a disk rotating in free air were supplied by Cham and Head ( 7 7 5 ) . They analyzed the phenomenon using circumferential and radial momentum integral equations, and a n entrainment equation adapted from earlier work by Head. T h e measurements and calculations were found to be in excellent agreement. Experimental measurements of the reattachment of a supersonic jet with a turbulent boundary layer abruptly expanding into a n axisymmetric parallel diffuser were reported by Mukerjee and Martin ( 6 2 5 ) . A reattachment criterion correlated their data, and they found that disturbances downstream of reattachment d o not affect the upstream region. An assortment of studies o n turbulent boundary layers was published in Heat Transfer-Sovzet Research. Girol’ (335)examined the flow past a plane porous wall, cooled by a gas blown through the porous surface. H e used a momentum-integral approach to analyze the problem. A somewhat similar problem was analyzed by Kocheryzhenkov ( 4 6 J )using a previously developed approach to study the turbulent boundary layer on a porous plate through which a coolant is supplied. T h e main flow was considered to be a gas mixture whose components are reactive species. Ginzburg ( 2 9 5 )also examined the turbulent boundary layer flow of gas mixtures using semiempirical approximate methods of solution, and in a paper with Skurin ( 3 Z J ) he used semiempirical methods to analyze the turbulent boundary layer in a transverse magnetic field. A group of papers deal with atmospheric or planetary boundary layer phenomena. Arya and Platc ( 7 J )modeled the stably stratified atmospheric boundary layer by experimentally studying the boundary layer developing over a cold plate in a wind tunnel. They found that the mean flow and turbulence characteristics in the near wall region of the stratified boundary layer are well desci ibed by previously published similarity theory. Taylor (97J, 9 8 J ) studied characteristics of planetary boundary layer flow theoretically. I n the first reference, he used a wind spiral model, previously proposed, to represent the flow above a surface of uniform roughness in the planetary boundary layer. An inconclusive attempt t o determine the applicability of the mixing length model used was made. I n the second paper, a continuation of the first, he examined the effects of a change in surface roughness on the planetary boundary layer. Three papers concentrate on the Ekman layer, which is associated with geostrophic flows. Hsueh ( 4 7 J )analyzed the structure of a buoyant Ekman layer in a vertically stratified fluid, obtaining a solution of the steady-state baroclinic boundary layer over a two-dimensional terrain. T h e boundary layer thickness was found to depend o n both the stability described in the paper and the terrain slope. Csanady ( Z Z J ) analyzed diffusion in an Ekman layer using the classical diffusion equation with a constant eddy diffusivity and the classical Ekman layer velocity profile. H e solved the governing equation by the concentration-moment method to obtain information on the motion and spread of a cloud. Olsen ( 6 9 5 ) analyzed the development of the Ekman layer to estimate the distance from a leading edge a t which the Ekman boundary layer solution is valid. Duct flows. Interest in the prediction of turbulent flow in simple geometries such as straight tubes and channels continues, thus attesting to the importance of such configurations. New analytical and experimental results concerned with the mean velocity distribution in steady flow have been reported. Lindgren and Chao (53J)employed a specially designed apparatus to make hot film measurements in turbulent tube flow u p t o the surface of the tube wall. These measurements provide a n excellent verification of the “law of the wall” and the “law of the wake.” Other new experimental results for the mean velocity in the core region of pipe flow by Brinkworth and Smith ( 7 2 J )led them to propose a revised expression for the mean velocity for y + > 30. This expression, valid for Reynolds numbers in the range 50,000 to 350,000 is logarithmic in Reichardt’s position parameter. Quarmby (875)has given a theoretical analysis of fully developed tube and channel flow, which takes into account molecular viscosity and the variation of the shear stress across the duct. Both DeisVOL. 6 2
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sler’s eddy diffusiviry and voii Karman’s shear stress distribution are employed, and unknowns in the result are determined from the knoivn relationship between the friction factor and the Reynolds number. Univcrsal constants in the analysis are found to vary either with Reynolds number or position. .\nother analytical work in which new expressions for thc mean velocity and eddy diffusivity are presented has been presented by Zagustin and Zagustin (108J). Measurements of skin friction and mean velocity in fully developed tube and channel flow as reported by Patel and FIead (74J)showed that the three criteria, which have been used to determine when the flow is fully developed, become applicable ar different Reynolds numbers. Furthermore, the results appear to depend upon whether the flow is axisymmetric or two-dimensional, thus making a precise definition of fully developed turbulent flow dificult. Morrison aiid Kronauer ( 6 7 J ) have obtained a structural similarity hypothesis from data taken on fully dcveloped tube i i o ~ v . T h e d a t a , longitudinal and transverse correlations of the longit u d i n d component of turbulence, were transformad to sho\v the distribution of turbulence intensity among waves of different size a n d inclination. Significanrly, a geometric similarity for the waves in the region of logarithmic velocity distribution was found a n d extended to the sublayer region. Turbulent swirling flow in ducts has been investigated by severai workers. Rochino and Laran (84J)employcd Taylor‘s modified vorticity transport theorem and von Karman’s similarity hypothcsis extended to three-dimensional AOWS t o describe the flow. Numerical results compared wcll with cxistinS experiments. Backshall arid Landis ( 4 J ) reported new cxperimcntal measurements and sho\ved that the total velocity was well approximated by the logarithmic \docity profile. T h e phenomenon of reversiiip axial flow in tubes was studied experimentally, and static pressure and velocity profilcs irere mcasurcd by King p i ai. ( g 4 J ) . JVolf el ~ l f 707J) . also reported cxpcrimcntal data 011 the decay of swirl in a tube and found that the decay decreascs with increasing axial Reynolds number and is independent of initial sxvirl angle. .\ rather extensive study of unsteady turbulent flow in a tube. which includes both analytical a n d experimental Lvork, was rcported by Brown rt oi. 17.3J). For small amplitude disturbances superiniposed on a gross turbulent flow. a tirnc-invariant eddy viscosity predicts the behavior well. T h e question of separation of the transverse locations of vanishiiig turbulent shear and zero mean velocity gradient in asymmetrical main flow fielcis was investigated in a number of recent paprrs. This separation zone has been t r r m r d the “zone of opposing shear” aiid the “zone of eiieray reversal” because of its characteristics. Eriaii (255)reported its cxistclicc in a ~ v a l ljet issuing into a passage that is first convergent and then divcrgcnt. Eskinazi and Erian ( 2 6 J )presented a n analysis of the possibility of rhe esistrnce of such regions whcrein the production term in the mechanical ciiergy balance becomes negative: that is, i t has the same sign as the dissipation term. T h e existence of such zones negatrs the use of a proportionality bct\vcen the eddy diffusi\-ity and the mean velocity gradiciit. T h e relationship may be modi: altcrnatcly, fied to include second-order tcrms empirically ( 2 f i J ) or by physical arguments, such as those employed by Launder ( 4 9 J ) concerning gradients of the length scale. T h e inclusion of secondorder terms allows the turbulent shear stress to be finite at locations where the mean d o c i t y gradient vanishes. Drag reduction and related phenomena. Patterson et al. (75J)presented a useful review of drag reduction work that contains 89 references. Efforts aimed at explaining drag reduction and correlating drag reduction d a t a continue. T h e charactcrization of turbulence phenomena in drag reducing systems is a step forward in understanding the mechanism of drag reduction, and in this vein a number of workers have reporred velocity distribution and turbulence intensity measurements. Seyer and Metzner (87J)used a n air bubble technique to measure the axial aiid radial components of the instantaneous velocity in two different types of drag reducing systems. It was determined that for solutions exhibiting drag reduction a t all Reynolds numbers, the Aow may be transitional a t Reynolds numbers u p to 105. Furthermore, the turbulent core velociry is flatter than that 122
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for Nekvtonian fluids, apparently because of increased isotropy. I\ theoretical analysis given in this paper leads to the propcr form of the similarity laws previously found empirically. Scycr and 1,fetzner !87J,88J)also point out that with prescntly available materials little or no drag reduction can bc achieved in large pipclines a t levels of polymer additive that are economical. Scw additives having large relaxation times are needed. 1-Towever, equivalent drag reduction may be obtained (88J)on addition of polymers to the boundary layer insidc large diamctcr pipes. ,\nother study of velocity disrribution in turbulcnt drag reduciiig systems is reported by Nicodemo et 01. (cj5J)whcrein it W B S also found that the core region velority distribution is flatter. Thcsr d a t a gave no indication of boundary layer thickening as a mechanism for drag rcduction. Turbuleiicc intensities arc discussed in scvcral papers !.74J, G 5 J , 87J). T h c most accurate measurements hcrc, made Lvith laser-Doppler equipment (.3?J), indicate that the addition of polymer to Lvater does not sigiiilicantly alter the level of longitudinal turbuleiice iiitciisity at the center line. .\xi experiment t o test rhc rolc played by adsorbed polymrr molccules in drag reduction is reported by Little i i S J ) . This consisrcd of a continuous displacemeiit of the drag reclucinS solution with thc basc solvent in a pipe and a n observation of the time nccdcd to reach a constant pressure gradicnt. T h e results werc inconclusive, but pointed out that this time can be greatly influenced by viscosity and migration of the drag reducing fluid into prc%urc t a p connections. S e w drag reducing data arc reported (ZJ, 6 J , 87J, 9 7 J ) . ,\starita et 01. ( 2 J ) present information for fivc concentrations of water solvciir polymer and rhrcc rube diamctcrs. \I simple diniensioiilcss correlation of the results is given b u t is restricted b r cause of the single polymer used. Several acrylamide basc polymers in water and brine solutions werc studicd by Sitaramaiah and Smith ( S 7 J ) aiid compared with polyethylene oxide solutions. Tests showed the polyacrylamides werc incompatible with brine, \vhilc the poly OX polymers could be used advantageously. Flow Past Spheres, Cylinders, and Special Shapes
O n e of the central problems i n the history of fluid mechanics has bccii the flow about common geometrical shapes such as t h e sphere, the circular cylinder, and the airfoil. Although the existing. litcrature o n the subject is very cxtcnrive, the annual contribution to that literature is still large, as partially evidenced by the Proceedings of a n InternationaI Symposium on High-speed Computing in Fluid Dynamics sponsored by the International Union of Theoretical and .\pplied Mechanics. T h e symposium proceedings of 1968 were published as a supplement to I’hjsics OJ’ Fluids in December 1969 and contained a large number of papers dealiiig with flow around submergcd objects a n d other topics covered in this review. I t is convenient to organize this portion of the survey of recent publications according to the geometry i n question. .\mong the computing symposium papers were thc studies of Hamiclec and R a a l (13R),Takami arid Keller (&)OK), and Takaisi ( 3 9 K ) , all of which are numerical studies of steady flo\r around a circular cylinder. T h e first two authors obtained solutions for Reynolds numbers from 1 to 500, presenting thc results as vorticity and stream function distributions, and they showed cxccllent agreement in their comparisons of drag coefficients, pressure distributions, and vortex dimensions with experimental and available theoretical results. Takami and Keller studied a more limited range of Reynolds numbers (from 1 to 60)> and excellent agrcement of their results with recent experiments was found. A low Reynolds number range was examined numerically by Takaisi, and he reported that the calculated drag coefficients approach values obtained from Imai’s formula rather than those based on Oseen’s theory. I n a pair of papers dealing with flow around a cylinder Son and Hanratty (37K, 38K)provided a numerical solution for Reynolds numbers of 40: 200, and 500 and reportcd cxperimental measurements of velocity gradients a t the wall for Reynolds numbers from 3 X 103 to 106. I n their experimental work, they used the electrochemical technique previously used by Hanrarty and others to obtain local velocity gradients. In yet
another paper o n the steady flow around a cylinder, Underwood ( 4 4 K ) used a semianalytical method to obtain solutions. By reducing the partial differential equations to ordinary differential equations, which h e solved numerically, Underwood confirmed previous analyses and showcd favorable comparisons with available experimental d a t a for Reynolds numbers from 0.4 t o 10.0. With this great duplication of effort, it can be safely said that steady flow around a circular cylinder has been amply studied in the low and intermediate Reynolds number range. Time-dependent flow past a circular cylinder was studied numerically by J a m and R a o ( 7 7 K ) and by T h o m a n and Szewczyk ( 4 1 K ) . T h e former authors reported the flow pattern, vorticity distribution, pressure distribution, and the drag for Reynolds numbers fiom 40 to 200, and the latter presented solutions for Reynolds numbers from 1 to 3 X 105 for the flow started impulsively from rest. Both papers consider the limiting steady state or the K a r m a n vortex street, whichever is the appropriate limit for the Reynolds number in question. Unsteady flows around cylinders of a slightly different nature were studied by Tuck ( 4 3 K ) and by Mei and Currie (22K). Tuck analyzed the problem of small oscillations of a cylinder of general cross-section in terms of integral equations, which he solved numerically for the special case of a ribbon of zero thickness. Mei and Currie measured the location of the boundary-layer separation point o n a circular cylinder for a stationary cylinder and for a n oscillating cylinder. T h e results for a stationary cylinder agreed well with previously published results, and for the vibrating cylinder the range of angular displaccrnent of the separation point was found to depend upon the vibration amplitude and frequency. Stokes flow around a finite length cylinder confined between parallel plates was analyzed by Lee and Fung ( 7 9 K ) by writing the solution as a series whose terms satisfy the equations of motion and continuity and the no-slip condition on the plates. T h e numerical results show that the coefficient of resistance to flow decreases rapidly as the platc spacing to the cylinder diameter increases. Pich ( 2 7 K ) analyzed the drag of a cylinder, the size of which is comparable with the m e a n free path of gas molecules, the application being the physics of aerosols. T h e resulting equation is compared with experimental data. Additional papers dealing with flow around a circular cylinder are the experimental studies of Bearman ( 3 K ) and Felsing and Moller ( 7 2 K ) , and the short note by Savage (34K)o n free convection flows about inclined cylinders. Bearman examined narrowband vortex shedding over the Reynolds number range 106 to 7.5 X 105 and presented the spectra of velocity fluctuations in the wake for several Reynolds numbers. Felsing and Moller investigated the characteristics of a wall j e t attached to a circular cylinder with radial injection that produces separation and redirection of the wall jet, and Savage pointed out the existence of certain classes of three-dimensional free convection flows in which the equations are separable, particularly the flow about cylinders inclined to t h r horizontal. Another set of papers deal with flow past a sphere or variations of that theme. Chester, Breach, and Proudman ( 5 K ) obtained higher order approximations for the Stokes drag, and Rimon and Cheng ( 2 9 K ) numerically calculated solutions of the transient uniform flow around a sphere for intermediate Reynolds numbers (1 5 R e _
