ms
FLUID DYNAMICS THOMAS B A R O N Shell Development Co., Emeryville, Calif. A. K. OPPENHEIM University of California, Berkeley, Calif.
This review has been compiled to indicate only major trends i n research. In contrast to the well-understood principles of viscous flow, other branches of fluid dynamics, notably turbulence, are still in the-state of flux.
which define the transition regionthe third stop on our tour. Before exploring the remaining turbulent flow regime, we inspect the structures of vortex flow and HE chemical engineer’s most important contribution to inthose of unsteady flow phenomena. dustry is the increasing utilization of flow systems. This In considering turbulent flow, an essentially new difficulty apinvolves a detailed study of flow conditions. Important pears. Small causes produce large effects because of the nature of progress in chemical technology is made by applying the results the inertia terms in the equations of motion and the flow is obtained from this study to more detailed design specifications governed by small unknown perturbations on the average as well as by expanding the sphere of design conditions from normal temperatures and pressures and moderate velocities and conboundary conditions. After reviewing the types of flow regimes, our schedule leads tact times toward the relatively unexplored regions of temperatures of thousands of degrees, pressures of a few millimeters or to an inquiry into the effects of interaction between flow and boundaries. The next stop is, therefore, a t the boundary layer thousands of atmospheres, velocities several times the velocity of laboratory concerned with stationary boundaries, followed by a sound, and correspondingly low contact times of a few millisecvisit to the multiphase laboratory where the problems are charonds. The challenge of better design and extreme conditions can acterized by the changing positions of the boundaries as deterbe met only by emphasizing fundamental aspects. This characmined by the equations of motion. terizes a universal trend in modern technology and results in partial merging of engineering sciences. The problems mentioned so far are, in general, unrestricted by Thus, fluid mechanics supplies a common ground for the aerothe nature of the substance. The effects of compressibility, nautical, mechanical, and chemical engineer, and its literature is however, are of a sufficient importance t o merit special consideranumerous and widely scattered. The references cited in this retion. Papers dealing explicitly with this subject are discussed a t the next stop. view represent less than 50% of the papers reviewed and have been selected to illustrate primarily major trends of research. The occurrence of chemical reactions adds special complicaMany of these are of immediate interest t o chemical engineers; tions to flow problems. The interaction between the chemical others are part of the foundation on which future work of practiand physical effects is especially severe in the case of rapid reaccal importance may be based; still others are of interest because tions releasing large amounts of energy. Thus, the most importhey represent useful analytical techniques. Since most readers tant application of the analysis of combined chemical and flow of this paper are chemists and chemical engineers and because of processes are to problems involving combustion. This class of problems, considered from an over-all rather than from a strictly the comprehensive fluid dynamics review of Weintraub and Leva in the January, 1952, issue of this journal, all references to physico-chemical point of view, is properly termed dynamics of Chemical Engineering Progress and INDUSTRIAL AND ENGINEERING combustion-the last stop on our tour. CHEMISTRY are omitted from this article. These sources will be Our tour has been scheduled to highlight the topics of principal covered, however, in next year’s review. interest t o the chemical engineer. However, any survey of fluid We propose t o take our readers on a “Cook’s tour” through the mechanics would be incomplete without at least a brief reference workshops of fluid mechanics. We imagine that the whole field to aerodynamics, and this shall be included in our farewell address. At the same time we shall hand out a short list of recently consists of structures built in various laboratories. I n some of these, structures are already erected while in others foundations published books and reviews of general character. are being laid, the outlines of eventual results being barely disViscous Flow cernable. We shall explore these structures in decreasing order of completion. Theories of Viscosity. Theories of viscosity form the basis The foundations of fluid mechanics are the equations of motion, for the correlation and extrapolation of data. The generally These are solvable only for cases where the inertia terms are accepted foundation for the evaluation of viscosity of gases is the negligible. The remaining terms represent the balance between kinetic theory as developed by Enskog, Chapman, and Cowling. pressure gradients existing in the fluid and viscous shear stresses. For simple gases at low pressures the elementary mean free Thus the first stop on the schedule is the viscous flow laboratory. path theory suffices. Truesdell (S6A) describes the means by Here we shall observe work on laminar flow, flow associated with which various authors have approximated the relationship besuspended particles, and flow through porous media. tween the viscosity and the mean free path. The dependence of In considering these flows, the fluid can be treated as a conviscosity on the state is treated b y the method of dimensional tinuum as long as the characteristic scale of the apparatus is analysis, and approximation formulas for gases and liquids are orders of magnitude larger than the mean free path of the molediscussed. cules. Since this restriction does not apply to an important class The kinetic theory of liquids has not yet progressed to a stage of problems, a side trip to the workshop of rarefied gas flow is of where practical applications could be made, chiefly because of logical interest. mathematical difficulties. Consequently, the theories of visThe next step is to inquire into the limits of the viscous flow cosity of liquids are based on simplified models which only occarange. These are determined by the conditions of stability, sionally yield quantitative agreement with experiment. Andrade
T
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INDUSTRIAL AND ENGINEERING CHEMISTRY
(2A) discusses such a model based on the concept of the slowly moving oscillator. The theory is used to predict the viscosity of certain molten metals, the results differing by, a t most, 40% from experiment. The dependence of viscosity of liquids on molecular weight and free space is discussed by Doolittle ( f O A ) . Cumper and Alexander (BA) study the viscosity and rigidity of gelatin solutions as a function of pH, ionic strength, and temperature, and conclude that in concenbrated systems the viscosity depends primarily on coulo~nbicforces between ionized groups on adjacent chains. L a m i n a r Flow. The analysis of laminar flow under normal conditions is based on the solution of the Navier-Stokes equations for various boundary conditions. Such solutions for somewhat unusual boundary conditions are given by Ackeret ( 1 ~ 4 ) . An interesting extension of the earliest solution of Savier-Stokes equations obtained by Poiseulle to study the flow of blood is given by Morgan (%$A),who analyzes the steady laminar flow of a viscous incompressible fluid in an elastic tube. He demonstrates that the tube diameter decreases due to axial pressure gradient and evaluates the change of the diameter as a €unction of mass flow rates. The approximate solutions are better than those published earlier by Rashevsky [Bull. Math. Biophys., 7,35 (194513. Krainer and Stanitz (18A) treat the two-dimensional shear flow in a 90” elbon. Mochieuki (23A) considers the laminar incompremible flow a t the inlet to a circular pipe. B y introducing a number of very flexible assumptions, he obtains results fitting Nilruradse’s experimental curves much better than previous solutions. A design analysis of a thrust bearing a t the end of R vertical shaft is described by Heinrich (f5A). h somewhat unusual treatment of heat transfer in laminar f l o ~ is given by Ohji (88A), ~ 4 1 0obtains an approximate temperature distribution satisfying the differential equation a t the centerline arid the condition of constant temperature at the wall. Constant thermal conductivity and parabolic velocity profiles are assumed. An important effect, not considered in this paper nor in the classical work of Graetz, is the variat,ion of viscosity due to temperature gradient. This results in the distortion of velocjty profiles which, in turn, influences temperature gradients, etc. I t is probable that exact solution of this problem can be obtained only by an iterative method. Heat transfer coefficients calculated from the Graetz solution are valid only for a constant tube wall temperature although, for want of a better approximation, they are frequently used even when the wall temperature is varying. In this connection, Hahneniann’s paper ( I d a ) is of considerable importance. A heat transfer coefficient is defined and calculated when the wall temperature varies linearly along the axis. Heat transfer in laminar flon. over a rotating plate is considered by AIillsaps and Pohlhausen ( 8 2 A ) . The problem of submerged laminar jet.s is treated by Rumer (314)and Squire (339). A fascinating intellectual exercise of some interest to bio1ogist)s is provided by the recent ivorks of Taylor (34A),Lighthill (20L4), and Gadd ( I f A ) concerning the propulsion of micro-organisms and small deformable bodies in laminar flow. A few papers concerning the fundamental aspects of experimental techniques in laminar flow deserve mention, Rosenberg (3OZ4)considers the use of doubly refracting solutions in the investigation of fluid flow phenomena; Ybrahim and Kabiel (%A ) describe the theory of an oscillat’ing cylinder viscometer; and Dean (QA)presents an analysis of the pressure signal given by a half-pitot tube set flush against the wall. N o n - N e w t o n i a n Flow. In non-Newtonian flow the viscosity, instead of being constant, depends on shear stress. Beyer and Towsley(5A)extend the Spencer-Dillon flow cquation [ J . Colloid Sci., 4, 241 (1949)] by dimensional analysis to a form suitable for a rectangular channel. Krieger and Maron (19A ) attempt t,o determine directly the flow curves of non-Newtonian fluids. Magnusson ( 2 f A ) considers the dependence of power
Vol. 45, No, 5
consumption on the effective viscosity when agitating liquids of viscous structure. Whitfield and Baron (37A) describe an experiment to determine the constants in Eyring’s equation for the viscosity of non-Nervtonian fluids. Flow of Suspensions. The flow of suspensions is of interest not onlyin the transportation of suspensions but also in t,he operations of sett’ling and fluidization. When the density of the suspended material is of the same or lower order of magnitude than the density of the suspending liquid, the suspension can be treated as a single phase with an appropriate effective viscosity. The well-lmown Einstein equation applies only in the dilutc range-say up to 3% v. of Buspension. Current research on this subject is devoted to higher concentration ranges. A rather novel approach is presented by Simha ( 3 2 S ) , whose earlier work indicated that a term equal to 14.1 times the square of the volume fraction of suspension has to be addcd to the Einstein terms. Experiment, however, indicates that the constant should be 7 or 10. Simha’s latest approach is consistent with modern trends in the theory of liquids. Each particle of suspension is assigned a certain free volume enclosed in a spherical shell whose influence takes the place of the effects of all the other suspended particles on the particle in question. The results do not permit a quantitative evaluation of the viscosity of a concentrated suspension. However, it is shown that the hydrodynamic interaction gives an average effect that is equivalent to a geometric crowding factor. Simha’s main contribution is that his treatment bridges the gap between the hydrodynamic and purely geometrical theories. Brinkman ( 6 A )revives the argument in which a suspension is regarded as having been generated by the addition of particle upon particle, before each addition the suspension being regarded as a pure liquid whose viscosity is used in the Einstein equation. Strictly speaking, this argument would be valid only if each particle were orders of magnitude larger than the one preceding it. Brinkman, however, feels that this restriction is not important and demonstrates satisfactory agreement between his equation and experiment. The reason for this agreement may n-ell bc due to an effect pointed out by Roscoe (%?A), who makes the interesting obPervation that a concentrated suspension consists of singlets, doublets, and other multiplets, and is, therefore, a heterogeneous rather than homogeneous mixture. F l o w t h r o u g h Poraus M e d i a . Flo~vthrough porous media may be regarded as the limiting case of flow through suspensions as the void space approaches its minimum value. I t is of special interest a t present in connect,ion with the problem of secondary recovery from oil fields. A short revicrv of the flow of gases through porous inedia i:: given by Nielsen ( 2 6 A ) . Klute (Ira) suggests a numerical method for solving the flow equation for water in unsaturated materials. Darcy’s law is combined with t,he equation of continuity; an iterative process is then used to obtain moisture content distributions a t various values of time in unsaturated semi-infinite horizontal porous media. Carman ( ? A )studies the adsorption of dichlorodifluoroniethane on plugs of Linde silica IT and shows that in capillary condensation the advorbed phase behaves as if it were a bulk liquid in viscous flow. Gas cooling of a porous heat source is described by Green (13A). The rangc of uniformly streamline flow is considered by Xielsen (27A). An important application of the theory of flow through porous media is to the flow of oil in wells (8.4,4 A , 1 2 8 , 16A, 36‘A). In a significant contribution, Morrison and Rogers (258 ) deal with the influence of flow patterni: on the initial convection in porous media. Xumerical solutions of the extant theory do not agree with experiment. Flow patterns are not of the cellular type observed for nonporous media. Experimental observations indicate that the flow is siniilar to the columnar flow described by Sutton [Proc. Roy. SOC.(London),A204, 297 (1950)l. Modifications of the classical theory are suggested.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
Rarefied Gas Flow Viscous flow is characterized by low Reynolds number. The physical characteristics of the flow, however, may be quite different, depending on whether the low Reynolds number is due to a combination of high viscosity and low velocity on the one hand or low density and small characteristic dimension on the other hand. I n the latter case, the ratio of the mean free path and the characteristic dimension, the Knudsen number, may become the governing parameter. It is easy to show that the Knudsen number is proportional to the ratio of the Mach number and the lteynolds number. According to the criterion suggested by Tsien, the continuum treatment fails when the Knudsen number becomes larger than 0.01, while for Knudsen numbers larger than 10 the flow can be treated as free molecular. Both the continuum and free molecular flow regions are subject to exact treatment, a t least in principle. The intermediate regime is treated a t present by modifying the boundary conditions of the limiting regions. This necessitates, for instance, the introduction of adjustable factors such as the slip coefficient. Accordingly, the major part of this regime is referred to as slip flow. The major effort of research on rarefied gas flow is concerned today with flight in upper atmosphere and is based on experiments performed in low pressure wind tunnels such as t h a t shown in Figure 1. The results, however, are of importance to chemical engineers not only because of their obvious application to vacuum techniques but also because they apply a t normal pressures if the body dimension is small as, for instance, in hot-wire anemometry, small total head tubes ( 6 B ) ,flow of gases through capillaries ( S R ) , and the evaporation of small droplets.
Figure 1.
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The basic difficulty in analyzing rarefied gas flow in the transition regime is due t o the inadequacy of the Navier-Stokes equations of motion and the corresponding energy equation, and the consequent necessity of introducing higher order terms in the stress tensor and the heat or mass flux vector. The second approximation involves the addition of the so-called Burnett terms which raise the order of the differential equations, thus requiring additional boundary conditions. Grad [Comm. Pure and A p p l . Math., 11, No. 4, 331 (1949)l presented a more general formulation of the required equations and their boundary conditions which is not restricted t o the Burnett assumption of small gradients. I n an effort to determine boundary conditions required for the solution of these equations, Chiang ( 1 B ) made measurements. of drag on a rotating cylinder in the slip flow regime. The experiments were performed a t Mach numbers lower than 0.5. He concludes that for this purpose the NavierStokes equations are sufficient, the difference between the results based on these equations and those of Grad’s thirteen-moment method amounting to less than 1%. The conclusions do not necessarily apply to larger Mach numbers. A more direct method of exploring the transition regime is provided by the study of the propagation of sound in gases. Wang Chang and Uhlenbeck ( 8 B ) attempt to utilize in this connection the experiments of Greenspan [Phys. Revs., 75, 197 (1949); J. Acoust. SOC.Amer., 22, 568 (1951)] who measured the velocity of sound at high frequencies and low pressures. The analysis based on the Burnett equations demonstrates the inadequacy of the second approximation for the rationalization of experiments performed under these conditions. The theory of drag under slip flow conditions, initiated by
Wind Tunnel for Rarefied Gas Flow Studies
Test chamber diameter, 4 feet; uniform flow iet diameter, P to 4 inches) free stream static pressure about 100 microns mercury; or calibration OF sages, ~ressure can be reduced to 0.1 micron mercury
1. A i r dryer 2. Flowmeter 3. Inlet valve
4. Baffler 5 Flow rtrai htening rcroens 6: Settling c l k b e r 7. Nozzle 8. Test chamber
9. Traversing mechanism 10. M o d e l
11. Instrument control panel 19 Transition section
13. Manifold and valve 14. M a i n control panel 15. PO-Inch oil diffusion and bscklns pumps
16. M a i n pump, ejector
five-stage steam
17. Third stage condenser 18. Fourth stage condenser 19. Cooling tower for condenser water
PO. Forced
recirculating steam seneratorr
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INDUSTRIAL AND ENGINEERING CHEMISTRY
Schaaf [Univ. Calif., Eng. Research Rept. HE-150-66(February 1950)] and based on the linearized boundary layer theory, is expanded by Mack (4B),who evaluates the aerodynamic properties for an inclined flat plate in slip flow and concludes that the aerodynamic formulas of linearized supersonic theory are sufficient except for the skin friction coefficient. The slip effect on compressible boundary layer skin friction is studied theoretically by Mirels ( 6 B )and experimentally by Sherman ( 7 B ) . Sherman’s data indicate that hlirels’ attempt to match the solutions of the slip flow and the continuum boundary layer theories is futile. Papers of immediate interest to chemical engineers are those of Folsom ( 2 B )on the nozzle characteristics in high vacuum flows, and of Sherman ( 6 B ) on the interpretation of impact pressure measurements in rarefied air streams.
Transition Regime -4complete design study should include the effects of small perturbations on the assumed design conditions, An inquiry of this character applied to laminar flow leads to the definition of the transition regime. The mathematical approach involves the solution of the problem dealing with the propagation of disturbances in viscous flow. Such a study is reported by Morawetz (4C), who investigates Couette flow, flow with symmetrical velocity profile between two walls, and flow in a boundary layer along a flat plate. The mathematical problem is involved and the introduction of simplifying assumptions is necessary. Squire [Proc. Roy. SOC. (London),A142,621 (1933)] has proved that the two-dimensional stability problem is equivalent to a three-dimensional one for the case of an incompressible flow. The generalization of this result is investigated by Lessen (WC, SC) and Dunn and Lin ( I C ) . They agree that up to, say, Mach number 2, Squire’s results apply, while for higher Mach numbers certain modifications are necessary. -4determination of the transition point from measurements of the static pressure along a surface is reported by Wijker (6C). Sackmann ( 5 C ) studies the transition region in pipe flow and shows that the probability of turbulence is a function of Reynolds number, the probability being approximately 0.4 a t Reynolds number of 890 and increases to about 0.9 a t Reynolds number of 1400. This represents a rational approach since it recognizes the statistical nature of transition limits. The essential difficulty in studying the stability of laminar f l o ~ is a mathematical one arising from the nonlinear nature of the equations of motion, -4s long as the disturbances are sufficiently small, a linearized treatment is sufficient. However, turbulence is associated with the region where the nonlinear terms take over, and consequently a linearized treatment cannot provide the basic conditions for the onset of turbulence. Vortex Flow I n leading up to turbulent flow, we shall first consider vortex flow and the closely allied subject of flow through turbomachinery in this section, and unsteady flow phenomena in the next section. These subjects concern the macroscopic features of flow characteristics which exhibit themselves microscopically in turbulent flow. A comprehensive study of steady rotational flow of ideal gases is presented by Prim (11D). Stewartson ( l 7 D ) analyzes the motion of a sphere along the axis of a rotating fluid. Various aspects of the theory of vortex sheets are treated by Birkhoff (ZD), Coddington ( 4 D ) , and Dolapchiev ( 6 D ) . The motions of atmospheric vortices are described by van Mieghem ( 9 D ) . For the purposes of this review, a turbomachine is considered as an apparatus, the operation of which is based on vortex flow. This subject is of such interest a t present that a complete survey of the papers cannot be made in a general review. The few
Vol. 45, NO. 5,
papers mentioned were selected to illustrate the type of problcmq on u-hich work is in progress. Methods and graphs for the evaluation of air induction systems are presented by Brajnikoff (SD).Theoretical and experimental analysis of one-dimensional compressible flow in a rotating radial inlet impeller channel is reported by Lieblein ( 7 D ) . Schlichting and Truckenbrodt (1ZD) treat the fundamental problem of flow in axial direction about a rotating disk. Although the problem is essentially a t least two-dimensional in nature, the analysis of the over-all characteristics can be based on a uni-dimensional treatment ( I D , ISD, 14D, 16D). The most important problem in axial flow machines is the flow through cascades treated in a great number of papera (10D, 15D. 19D, 2SD, 24D). Exact solutions must be based on a threcdimensional treatment. Examples of such attempts are the papers of Marble and Michelson (8D)and Wu et al. (200, 2 1 0 , a2D). A comprehensive and apparently satisfactory treatmcnt of the well-known Ranque-Hilsch cooling effect is given by van Deemter ( 5 D ) . Cyclonic flow is studied by Tarjan ( 1 8 0 ) .
Unsteady Flow The papers reviewed in this sec-tion dwi with effects of macroscopic temporal changes in boundary conditions on flow. Gerbes ( 1 0 E ) studies the incompressible laminar flow of a viscous fluid as affected by a pressure difference suddenly applied between two sections of a pipe. Ribaud (16E) treats the transient phenomena due to a compressor starting delivery a t constant rate to a gas pipeline of infinite length. A quasi onedimensional theory of unsteady liquid flow resulting in water hammer in long pipes is given by Grib (18E). Surging in compressors is treated by Stephenson (19E) and Bullock and Finger ( S E ) . The surge tank problem receives a considerable amount of attention ($E, 7E, 8E, 9E, 11E, 18E). The problem of pulsations in gas compressor systems is treated by Chilton and Handley (5E),who investigate experimentally the effect of various filtering devices and apply the electrical analogy to the rationalization of their results. This paper provides a useful technique for design. The measurement of pulsating flow by means of orifices is considered by Hall (14E), Baird and Bechtold ( l E ) , and Deschere ( 6 E ) . The chief value of the latter is the exhaustive list of references; the basic equations discussed in the paper, however, are questionable. h short report on an experimental study of the fundamentally important problem of wave distortion in a periodic gas flow is given by Truman and Lipstein (BOE). Carletti ( 4 E ) describes the effect of viscosity on transient waves in narrow fuel lines. This paper is of special importance in control problems. The remaining papers deal with the effects of transient conditions in practical applications, such as exhaust manifolds (17E), rocket nozzles ( 2 1 E ) ,impact on a plate ( l S E ) ,and bending vibrations of a pipeline (15E).
Turbulent Flow The importance of turbulence in chemical engineering is due to its influence on the transport of momentum, mass, and heat. Early attempts a t the solution of turbulent flom were based on the various mixing length theories. They represent a frontal attack and provide valuable engineering techniques but fail t o penetrate into the core of the problem. In order to accomplish this, the retreat to the reformulation of the fundamentals is necessary. Far-reaching idealizations have t o be introduced t o such a degree that the practical importance of many fundamental papers becomes hard to appreciate. This reviev is presented with an attempt to emphasize the possible beadng of these papers on practical results. The comments are based largely on the excellent review papers of Batchelor ( 6 F ) and Liepmann ( 2 2 F ) . A characteristic feature of the equation of motion is the sensi-
May 1953
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INDUSTRIAL AND ENGINEERING CHEMISTRY
tivity of the solutions to small perturbations in boundary conditions. Were it possible to specify these perturbations as part of t h e boundary conditions, the consequent turbulent motion could be computed in principle. However, the disturbances are essentially uncontrollable, and, consequently, such calculations would be useless. The method of attack is based, instead, on the experimental observation of Batchelor, “the probability laws describing the turbulent motion are independent of the small extraneous disturbances from which the turbulence arises and depend only on the given data of the problem.” The problem thus becomes similar to that of classical, statistical mechanics. An attempt a t a statistical mechanical formulation of hydrodynamics is made by Hopf (18F). The basic difficulty is that the equations of motion are non-Hamiltonian and no analog of Liouville’s theorem exists. I n particular, the law of equal partition of energy is invalid. A refreshingly new approach is indicated by Burgers ( Y F ) , who points out that i t is preferable to average the solutions of the differential equations governing the motion rather than attempt to solve them for the averages. Turbulent motion consists of the superposition of a large nuniber of component motions differing from each other in their scale T h e instantaneous velocity distribution may be resolved into a linear expansion of orthogonal modes consistent with boundary conditions. The square of the coefficients in this expansion are proportional to the amount of kinetic energy associated with the component of a given wave number. The interaction between these components forms the characteristic feature of turbulent motion. Since the equations of motion cannot be solved, simplified approach is necessary. The turbulence process may be visualized as follows: The boundary conditions influence turbulent components having scales of the same order of magnitude as the apparatus; energy is then transferred from the largest eddies t o smaller and smaller ones of increasingly large wave numbers. Now a particularly useful hypothesis is that put forth by Kolmogoroff [ C o m p t . rend. acad. sci. U.R.S.S., 30, 301 (1941); 32, 16 (1941)], and Obukhoff [Compt. rend. acad. sci. U.R.S.S., 32, 19 (1941)], according to which the influence of boundary conditions is gradually lost in the process of transfer of energy to Components of higher wave number. The components belonging to large wave numbers become, thus, statistically independent of boundary conditions and are presumably determined by local flow characteristics. In particular, their motion may be assumed to be statistically isotropic. In this way the idealized notion of isotropic turbulence finds practical application. A number of papers deal with these and related topics (SF,9 F , ZOF, 23F,g5F, 26F, 36‘6F). The diffusion of scalar and vector quantities in turbulent fields is treated by Batchelor (BP), who shows that it is related to the behavior of material lines and surfaces. This as well as other papers dealing with turbulent diffusion ( 4 F , 15F, S5F, 41F, 4WF) are based on the Lagrangean point of view. The engineer, however, lives in a Eulerian world and the transition from one viewpoint to the other involves, a t present, unresolved difficulties. Corrsin (IOF),in an important contribution, treats the problem of heat transfer in a stationary isotropic turbulence with constant, small temperature gradients. By successfully making the transition between the Lagrangean and Eulerian formulations, he obtains an expression for the turbulent heat transfer coefficient which depends only on the velocity field. A preliminary result is given for the turbulent Prandtl number in a homogeneous shear flow in terms of the shear and heat transfer correlation coefficients. Heat transfer in turbulent flow is treated also by Chandrasekhar ( 8 F ) , Fleagle et al. (I@‘), and Schlinger et al. (37F,38F). Ogura ( 2 7 F ) studies the theory of turbulent diffusion in the atmosphere. The diffusion of small particles which follow the air motion as well as the diffusion of balloons which do not respond t o smaller motions is discussed. Bearing on atmospheric
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turbulence are the papers of Pelegrin ( S O F ) , Pillow ( $ I F ) , Theodorsen (&IF), and Yih ( 4 S F ) . An interesting practical application of turbulent theories is given by Lighthill (b4F),who considers the sound produced in a turbulent field and shows that the intensity depends on the frequency as well as on the wave length and, as a result, increases inproportion to approximately the eighth power of a typical velocity of flow. Lassiter (WlF) describes experiments on noise from subsonic jets in still air and shows that it is caused mostly by the jet mixing region. Using the technique of the mixing length theory, Pai (28F) analyzes an axially symmetrical jet mixing in a compressible fluid. This work is essentially an extension of the classical Prandtl and Tollmien theory. The mixing length theories have, a t best, a very uncertain physical basis. Consequently, they can be regarded only as definitions of various proportionality factors. Unfortunately, their usefulness is limited since they vary in an unpredictable manner and cannot be measured directly. Reichardt ( S b F ) points out that the technique of assuming a relationship which cannot be tested directly and drawing from i t conclusions which are then subject to experimental tests is essentially unsound. The fact, for instance, that the results of Tollmien’s jet theory are in reasonable agreement with experiment does not prove that the mixing length is constant along the lateral coordinate as assumed, but indicates only that the result is insensitive to the assumption. Reichardt investigates the assumptions to be made in the equations of motion in order t h a t the solutions agree with measurements on jets. The resulting hypothesis defines a new mixing length which has the virtue of being essentially constant along the lateral coordinate. While the physical basis of the Reichardt hypothesis may be seriously questioned, it represents probably the best attack on practical problems involving the mixing of two- and three-dimensional jets. An immediate consequence of the Reichardt hypothesis is the linearization of the equations of motion. This permits a direct treatment of the problem of intermixing. Baron and Bollinger ( 2 F ) report experiments conducted with interacting jets in order to find the limits of applicability of the Reichardt hypothesis. They find good agreement with theory provided the angle between the jet centerlines is not larger than 7“. The contours of free discharge jets and their technical applications are discussed by Hahnemann
(I7F). Alexander el aE. ( I F ) present a novel and evidently powerful extension of the Reichardt hypothesis by assuming that the free length is related not to the lateral rate of change of the momentum flow alone, but rather to the rate of change of the sum of momentum and pressure. The success of this theory is demonstrated by the analysis of the transfer of momentum in a jet of air issuing into a tube. Turbulent flow in convergent channels is treated theoretically by Szablewski ( S Q F ) , who derives velocity profiles and friction factor curves from mixing length theory and similarity assumptions. The results check Nikaradse’s measurements. An experimental study of the spectrum of turbulence in contracting streams is presented by Ribner and Tucker (338’). Donaldson ( I S F ) considers the turbulent skin friction laws; Deissler and Eian (IWF)describe analytical and experimental investigation of turbulent flow of air in a smooth tube with heat transfer; Johnson et al. ( 1 9 F ) report on measurements of heat transfer t o molten lead-bismuth eutectic in turbulent pipe flow. An ingenious approach to the problem of turbulent diffusion and erosion is made by Davies (11F). The dynamics of particulate matter in fluid suspensions is described with the aid of a new theory. I n .order that the statistical description be consistent with the force equation of sedimentation, a new diffusion equation is derived from the postulates of the past-future stochastic process. A steady state suspension of particles in a turbulent stream is treated like an atmosphere in analogy to the kinetic theory of gases. The treatment results in a qualitative theory of
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I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
sand rippling. The essential vieakness of the analytical treatment seems to be the assumption concerning equal partition of energy between the fluid and suspension. Richardson ( 3 4 F ) considers turbulence and silt load in channels. The article gives a brief summary of the theoretical aspects and measurements of turbulence parameters and describes a satisfactory apparatus for analyzing silt loads in channels. The last tn-o papers in this series ( 1 6 F , g 9 F ) are concerned with the measurement of turbulence paramcters.
Boundary Layer Theory Boundary layer theory is concerned with a special calculation technique necessitated by the fact that the general NavierStokes equations cannot be solved. This technique is bawd on the principle of neglecting the inertia terms and retaining the viscous terms for the region near the boundaries where the gradients are large and, conversely, retaining the inertia ternis and neglecting the viscous terms i n the regions far removed from the boundaries where the gradients are small. The conditions of matching these solutions define the thiclrness of the boundary layer. These problems concern the n-hole gamut of flow types and, accordingly, the presentation of this section follows roughly the outline of the whole review. A definite progress in this branch of fluid mechanics is marked by the appearance of Schlichtjng’s text (28G). Schlichting, an early student of Prandtl -the founder of boundary layer theorywas predestined for an authoritative presentation of the subject. The emphasis on physical insight into the phenomena makes this text especially valuable €or engineers. A concise summary of the principles of boundary layer theory is presented by Timman (31G). Truckenbrodt (SbG) describes an unconventional approach for the approximate analysis of the laminar and turbulent boundary layer by simple quadrature. The laminar boundary layer being most amenable to analysis is well presented in last year’s literature (iOG, IbG, i6G, i9G, 22G, 24G, 26G, 29G). Cooke ( 2 G ) considers the problem of matching solutions, indicating the error introduced by the assumption that all velocity components reach the value of the free stream velocities a t the same point. An additional complication in boundary layer analysis is introduced by mass transfer to the boundaries when a porous plate is utilized. This case is analyzed by Ringleb (d5G),while an experimental investigation is described by Libby et al. ( I 7 G ) . The problem of stability and transition in subsonic and supersonic boundary layer flow is treated by Gazley (9G). The behavior of boundary layers, as affected by certain transient phenomena, is considered by Bryson and Edwards ( I C ) and by Van Dyke (SSG). The topic of the former is the transient aerodynamic heating of thin-skinned bodies, while the latter considers the impulsive motion of an infinite flat plate in a viscous, incompressible fluid. The solution is improved in a systematic manner by iterating alternately upon the solution of the boundary layer in its regime and upon the acoustic solution in the outer flomfield. Treatments of the turbulent boundary layer are, of necessity, intuitive approximations rather than exact mathematical analyses (SG, 4G, 7G, i i G , IZG). The literature of recent years i- characterized by concentrated attempts to include the effects of compressibility, as attested by a large number of papers (5G, 6G, 8G, ISG, I J G , i8G, 20G, 2iG, 23G, WYG, SOG, 34G).
Multiphase Flow The importance of two-phase flow in chemical engineering can hardly be overemphasized. I n view of the mathematical difficulties of the problem, theoretical attacks are useful only in correlating and extrapolating experimental results. At low velocities stratified flow is usually encountered, there being a well-
Vol. 45, No. 5
defined boundary between phases. The main problem here is tllc instability of this boundary and the subsequent dispersion of the phases. Accordingly, the theories concerning the formation of waves dating back to Rayleigh have an important bearing and are treated in many recent references (7X, l i H , dOH, SdH, 25fI). Bearing on the process of breakup is the important mathematical investigation of Ericksen ( 8 H ) concerning the behavior of thin liquid jets. The work of Boussinesq is generalized to the asiallg asymmetric case. Capillary and body forces are taken into account and a description of the flow is given. The process OF atomization is treated by Monk ( 2 4 H ) and Stehling ( 2 9 H ) . Best ( S H ) considers the effect of turbulence and condensation on drop size distribution in clouds. Dorman ( 8 H ) describes an experimental study of the atomization of a liquid in a flat spray. He determines drop-size distribution from patterns made on filter paper by dyes added to the spray liquid. The Sauter 1110an diameter, as determined from the Nulriyama-Tanasawa equatioi~ is correlated against the volume of the liquid sprayed per unit, time, spray angle, surface temperature, liquid density, and pressure difference. Viscosity is neglected. I t is concluded that the drops form according to Castelman’s ligament theory. Atomization by means of two impinging jets of water is studied by IIeitlmann and Humphrey ( I d H ) . The velocity, jet diameter, and impinginent angle are varied. A sheet of liquid is formed perpendicular to t’heplane of jets which intermittently disintegrates, forming groups of drops which appear as waves propagating from the point of impingement. The frequency is approximately proportional to the velocity. This papcr is of special interest because it demonstrates that combustion instability may arise from the process of atomization. Of immediate practical interest is the flow of suspensions in pipes ( 6 H , 23H, 27H). In a particularly interesting study, Laird ( i 9 H ) invest’igatesthe stability of gas flow in a tube as related to vertical annular gas-liquid flow. By means of an ingenious expcriment, the author demonstrates that the waves forming on thc liquid surface may be responsible for an excessively largc prossurc drop in the gas phase. An important source of experirnontal data on heat transfer and pressure drop in turbulent flow of air and w t e r mixtures in a horizontal pipe is the work of Johnson and A4bou-Sabe( i 5 H ) . Additional papers bearing on the subject or gas-liquid flow are those of Sbramson ( 1 H ) and Linning (21fI). Experimental information of real practical value concerning the slip velocity of gases rising through liquid columns is offered by St,ein et al. ( 3 0 H ) , and a new method for the measurement of liquid film thickness is described by Pennie and Belanger ( 2 6 N ) . The work of Spells (28H) is important to the analysis of trnnsfer in extraction. The circulation patterns within liquid drops moving through another liquid are studied. Flash photographs show that the circulation patterns in slowly falling drops corrcspond closely to the theoretical patterns predicted by Hadamard. For the case of drops falling with higher velocities, and this is the usual case, the drops are not spherical and the cent’er of circulation is displaced below the equatorial plane. Trilling ( S i H ) studies the collapse and rebound of a gas bubble, a subject of considerable int,erest in connection with an eventiial theory of heat transfer to boiling liquids. This problem T ~ Sfirst studied by Rayleigh, who neglected compressibility of the liquid and identified the content o€ the bubble as a gas compresscd a t constant temperature. Trilling computes the velocity and pressure fields in a slightly compressible liquid resulting from the collapse of a spherical bubble. Thc calculations are within thc acoustic approximation. The results are accurate as long as the liquid Mach numbers are small, and agree with those previously obtained by Herring [Ofice of Scientific Research and Development,, Rept. 236 (October 1931)]. The bubble is supposed to he filled v i t h inviscid, nonconducting gas. Special emphasis is given to the gas motion, which involves a series of shock waves. A fraction of the energy of compression is degraded so that the radius of the bubble after rebound is decreased. The pressure ~
INDUSTRIAL AND ENGINEERING CHEMISTRY
May 1953
variation on the bubble wall is virtually the same as if the gas were compressed uniformly and isentropically. Trilling's work is accurate to the first power of the liquid Mach number. Gilmore (13H) generalizes the solution to include higher order compressibility terms and considers also the effects of viscosity and surface tension. The problem of gas bubbles is closely associated with cavitation phenomena (2H, 4H, 6H, 10H, 12H), which are of special importance since equipment failure can be traced frequently t o their occurrence. A comprehensive review of the present status of this problem is presented by Knapp (18H).
t
\ i W , I
-X __
I
RAREFACTlON WAVE\
/ SHOCK
WAVE
HIGH PRESSURE L L O W PRESSURE
CELLOPHANE D I A P H R h
(t=o)
COURTESY INSTITUTE OF AEROPHYSICS, UNIVERSITY OF TORONTO, CANADA
Figure 2.
Wave System Generated in a Shock Tube
Bubble formation in supersaturated hydrocarbon mixtures is described by Kennedy and Olson ( 1 7 H ) , and a novel technique based on the utilization of radioactive tracers for improved multiphase flow studies is presented by Josendal et al. (16H).
Compressible Flow Compressible flow problems play an outstanding role in today's fluid mechanics. The foundations are laid on well-established mathematical techniques (61,111,l7I, 301, 331,361). The problems are suitable for purely academic studies as well-for instance, that of ShXman (461) on the esiEtetice of subsonic flows of a compressible fluid, The combined effects of viscosity and compressibility are particularly difficult to treat. Some progress in this class of problems is reported by Kariin and Rosenhead (251) and Hess (%I), the latter, moreover, including the effect of heat conduction. While the fundamental studies of compressible flow are restricted to isentropic flows, the limits of their applicability and their interaction with dissipative flows are brought to focus by von Mises (541) and Crocco and Lees (ISI), respectively. The problem of heat transfer in a compressible flow is studied in its fundamental aspects by Cole and Wu (81,5211)and as applied to pipe flow by Weinlich (491).
947
The most basic property of compressible flow is exhibited by the propagation of elastic waves treated by Biot (SI),Lawley (g81),and Glass (19Z),the latter being concerned with an interesting experimental technique of determining the speed of sound by the measurement of the velocity of a rarefaction wave in a shock tube (Figure 2, reproduced from J . Aeronaut. Sci.). A paper deserving special mention in connection with the reception of sound is that of Fletcher (161),dealing with an analysis of the dynamics of the cochlea. A detailed consideration of the many papers concerned with compressible flow is beyond the scope of this review. In order to satisfy the reader with specialized interest, typical papers are listed; they are grouped in the usual classes of subsonic flow (11, 161, 421, 601),transonic flow (,$I, 181,871,431), and supersonic flow (61, $11, 591, 4OI, 411, 611). Sauer (44I) presents a concise review of supersonic flow theories. While the major effort is directed toward the analysis of perfect gases, two simplified extensions to real gases are presented by Durham(1dI) and Traupel(481). The following papers are concerned with experimental technique: Kaye et al. (241,261) report on the measurement of recovery factors and friction coefficients for supersonic flow of air in a tube; Coles (91) presents the results of direct measurement of supersonic skin friction; Cooley and Stever (101)describe a novel method of air velocity determination by ion transit-time measurements; and Clark and Rohsenow (7Z)introduce a new method for the determination of static temperature of high velocity gas streams. The dependence of density on pressure in compressible flow is the essence of the nonlinearity resulting in the generation of shock waves. To a chemioal engineer they are of importance not only because they have to be taken into account in design but also because of their possible utilization for the measurement of fast reactions. In the literature they attract a great deal of attention because of their importance in modern aerodynamics. Legras (291) and Ting (461) treat the two-dimensional aspects of shock waves. The profile and structure of plain shock waves are analyzed by Grad (ZOZ),Bernard (ZZ),and Ludford (311). The problem of the interaction of shock waves is treated by Kofink (261), Moeckel (%I), and Morkovin et al. (371). An interesting method of determining the pressure time curves of shock waves is presented by Criborn (121). McFadden ( S t Y ) , Newton (SSZ),and Ting and Ludloff (471)treat the theory of spherical blasts.
Dynamics of Combustion Problems of this class should be of particular interest to chemists because they are associated with an inherent interaction between chemical processes and flow. Since in the treatment of these problems all possible types of flow are involved, this discussion will again follow the over-all pattern of review. Lessen (8J)presents the problem of the stability of laminar flame fronts; Manton et al. (9J)describe the essential features of the nonisotropic propagation of combustion waves in explosive gas mixtures, and the development of cellular flames. Kippenhan and Croft ( 6 J ) describe experiments investigating the effect of sound waves on a flame. The waves do not affect the magnitude of the normal flame velocity, but change their shape and enhance the stability of both laminar and turbulent flames. An application of a classical aerodynamic technique to the analysis of the steady state two-dimensional flow field in a duct where combustion of a homogeneous gaseous mixture is stabilized by a small obstacle located on the axis of symmetry is presented by Fabri el al. (4J). The nove1,featureof this analysis, in contrast to the original one presented by Tsien, is the assumption of constant ratio of stagnation enthalpies instead of stagnation temperatures. The study indicates that the heat release by combustion imposes an upper limit on the axial velocity, which in
948
INDUSTRIAL AND ENGINEERING CHEMISTRY
a given system adjusts itseli sporilarieously t o the conditions of the boundary. Contributions to the analysis of shock waves and combustion are presented b y Roy ( I a J , 1 S J ) ; Oppenheim (1OJ) describes a uni-dimensional gas dynamic analysis of the development and stability of gaseous detonation emphasizing the conbequences of the relative motion of the shock and combustion front during the development of the process. He demonstrates, as a consequence, that the locus of states behind such a combustion wave is different from the classical Hugoniot cuive. Kistiakowsky ( 6 J ) adds one more paper to the series of reports on his experiments on detonation describing an ingenious method of density measurement b y means of x-ray absorption. n’aturally, there is a group of papers concerned with applications to rocket propulsion ( I J , I I J , 1 4 J ) . 01 particular interest are the papers of Crocco ( S J ) on the subject of sweat or film cooling with reactive fluids, and ( R J ) on combustion stability in liquid propellent rockets. Ledinegg’s paper ( 7 J ) is a design study of vortex combustion chambers based on the elementary aspects of combined effects of combustion and flow.
Aerodynamics As mentioned in the introduction, the papers were selected here only for the purpose of indicating the present trends in aerodynamics. Considerable attention in recent years was centered on the BOcalled reverse problem-Le., the problem of calculating the airfoil geometry for prescribed pressure distribution ( 8 K ) . Flax ( I K ) and Manwell ( 5 K ) discuss certain applications of the calculus of variations to lifting surface theory. Mohr ( 6 K ) , Longhorn ( 4 K ) ,and Keumarlr ( 7 K ) are concerned with nonuniform motion. Germain ( 2 K )describes a general method of linearized supersonic aerodynamics. A coniprchensive book on aerodynamic drag, dealing with all possible aspects of the subject, was published by Hoerner (bK).
General Recently published books are those b y Kuznetsov (6L)(in Russian) on general hydrodynamics of special interest to hydrometeorology, Taylor (91,) on detonation; Redding (7L), consisting of an extensive bibliography with abstracts on the subject of flow through orifices and nozzles; and Carrier ( I L )presenting a well-selected compilation of original papers forming the foundations of modern aerodynamics. Review papers of general interest ale those of Rouse ( 8 L ) on present-day trends in hydiaulics; Truesdell (IOL) on the mechanical foundations of elasticity and fluid dynamics; ahd Glad ( 4 L ) on the generalized theory of statistical mechanic¶, thermodynamics, and fluid dynamics. The following reports on recent meetings are worth noting: on the Eighth International Congress on Theoretical and Applied Mechanics ( S L ) , on the Berlin conference on the progress of research in fluid mechanics (%), and on the Wiesbaden meeting on the movement of dust particles in gases (5L).
Vol. 45, No. 5
Cumper, C . W. ii.,and Ilexander, A. E., Australiicn .J. Sei. Research, A5, S o . 1, 146-59 (1952). Dean, W.R., Proc. Cambridge Phil.SOC.( L o n d o n ) , 48, S o . 1, 149-67 (1952). Doolittle, A. K., J . App1. P h y s . , 23, No. 2, 236-9 (196’). Gadd, G. E., P h i l . M a g . ( 7 ) ,43, KO.341, 663-70 (1952). Geffen, T.M., Parrish. D. R., Haynes, Cr. TV., and Vorse, R. A , , J . Petroleum Tech?iol., I V , No. 2 (Petroleum Yiiins., 195),29-38 (1952). Green, Leon, Jr., J . Appl. Mechanics, 19, No. 2, 173-7 (19521. Rahnemann, H. IT,,Forsch. Gebiete Ingenieurw., B18, S o . 1, 25-6 (1952). Heinrich, G., ;Ilaschznenhau m’arnzewirtech, 6, No. 4, 57 60: S O .5,78-82 (1951). Kern, 1,. R., J . Petroleum Technol., IV, KO.2 (Petroliacirn T r a n s . , 195),39-46 (1952). Klnte, A,, Soil Sci., 73, N o . 2, 105-16 (1952). Kramer, J. J., and Stanits, J. D., Natl. Advisory Conzm. Aeronaut., Tech. N o f c 2736 (June 1952). Krieger, J. M , , and Maron, S. K., J . A p p l . P h y s . , 23, 147-50 (1952). Lighthill, M. J., Conmzun. P u r e a n d A p p L M a t h . , 5, No. 2, 10918. Magnusson, K., I n g . T’etensk. A k a d . T i d s k . T e k n . Forsk., 2 3 , NO. 2, 86-99 (1952). hlillsaps, K., and Pohlhausen, K., J . Aeronaut. Sci., 19, S o . Z?$ 120-6 (1952). Mochisuki, S., Mem, Fac. Technol., Tokyo Metropol. Unio., 1952,p. 41-7. Morgan, G. W., B u l l . M a t h . Biophys., 14, No. 1, 19-26 (1952). Morrison, H. L., and Rogers, F. T., Jr., J . A p p l . Ph?/s., 23, 1058 (1952). Xielsen, R. F., Oil Gas J., 51, 151 (Augu8t 1952) . I b i d . , p. 159. Ohji, ICI. I., R e p t . Research I n s t . A p p l . Mechanics, ICuiisiizi linin., 1, No. 2, 23-8 (1952). Roscoe, R., B r i t . J . A p p l . Phys., 3, No. 8, 267 (1952). Rosenberg, B., David W. Taylor Model Basin, Rrpt. 617 (March 1952). Rumer, Yu. B., Prikl.. M a t . Mekh., 16, No. 2, 255-6 (1952). Simha, R., J . A p p l . P h y s . , 23, 1020 (1952). Squire, K. B., Phil. M a g . ( L o n d o n ) , 43, No. 344, 942-5 11952).
Taylor,’Sir Geoffrey, Proe. R o y . SOC.( L o n d o n ) , A211, S o . 1105,225-39 (1952). Truesdell, C., 2.Physik, 131, KO.3, 273-89 (1952). Welge, H. J., J . Petroleunz Technol., IV, KO,4 (Petroieztm Trans., 195), 91-8 (1952). Whitfield, H. B., and Baron, T., J . Colloid Sci., 7, So. 8, 268-71 (1952). Ybrahim, A. A. K., and Kabiel, A. M., J . Appl. Phijs., 23, 754 (1952). Rarefied Gas Flow
(1B) Chiang, S.F., Univ. Calif. (Berkeley), Tech. Rept. HE-150-100 (May 1952). (2B) Folsom, R. G., T r a n s . Am. Soc. Mech. Engrs., 74, No. 0. 915-18 (1952). (3B) Huggill, A. W. J., Pioc. R o y . SOC.(London),A212, 123 (1952). (4B) Mack, S.F , Univ. Calif. (Berkeley), Tech. Rept. HE-150-98 (March 1952). (5B) hiirels, H., Natl. Advisory Comm. Aeronaut., Tech. 2609 (January 1952). (6B) Sherman, F. S.,Univ. Calif. (Berkeley), Tech. Rept. HE-150-99 (December 1951). (7B) Ibid., HE-150-105 (July 1952). (8B) Wang Chang, C. S., and Uhlenbeck, G. E., Univ. Michigan, Ofice IYanal Reseawh Rept. Project M999 (October 1952). Transition Regime
Literature Cited Viscous Flow Ackeret, J., 2.angew. M a t h . u. P h y s . , 3, No. 4 , 259-70 (1952). Andrade, E. N. da C., Proc. R o y . SOC. ( L o n d o n ) , A215, S o . 1120,36-43 (1952). Aronofsky, J. S., J . Petroleum Techriol., IV, Xu. 1 (Petroleum Trans., 195),15-24 (1952). Barenblatt, G. I., P r i k l . , M a t . M e k h . , 16, N o . 1, 67-78 (1952). Beyer, C. E., and Towsley, F. E., J , Colloid Sci., 7, S o . 3, 23643 (1952). Brinkman, H. C., J . Chem. Phys., 20, S o . 4, 571 (1952). Carman, P. r., Proc. R o y . Soc, (I,ondon), A211, KO. 1107, 526-35.
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(1952). (2C) Lessen, M., Ibid., 19, KO. 6 , 431-2 (1952). (3C) Ibid., 19, No. 7,492 (1952). (4C) Morawetz, C. S., J . Rational Mechanics and Analyszs, 1, Y o . 4, 579-604 (1952). (5C) Sackmann, L. A,, Publ. sci. tech, ministhre a i r ( F m n e t ) , Actes Coll. inter. M b c a n I I I , No. 251, 221-9 (1951). (6C) TVijker, H., Natl. Lzceht Lab. (Amstepdam), Rept. A1210 (1952). Vortex Flow
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Birkhoff, G., Proc. Natl. Acad. Sei. H. S., 38, No. 5, 409-10 (1952). Brajnikoff, G. B., Natl. Advisory C o m m . Aeronaut., Tech. Note 2697 (May 1952). Coddington, E. A., J . M a t h . and Phys., 30, No. 4, 171-99 (1952). Deemter, J. J. van, A p p l . Sci. Research, A3, No. 3, 174-96 (1952). Dolapchiev, B. I., Doklady A k a d . N a u k S.S.S.R., 78, No. 2, 225-8 (1951). Lieblein, S., Natl. Advisory C o m m . Aeronaut., Tech. Note 2691 (April 1952). Marble, F. E., and Michelson, I., Ibid., 2614 (March 1952). Mieghem, J . van, Tellus, 3, N o . 2, 75-7 (1951). Polyakhov, N. N., Doklady A k a d . Nazkk S.S.S.R., 84, N o . 2, 233-6 (1952). Prim, R. C., 111, J . Rational Mechanics and A n a l y s i s , 1, No. 3, 425,497 (1952). Schlichting, H., and Truckenbrodt, E., Z . angew. M a t h . u. Mech., 32, No. 4/5,97-111 (1952). Stanitz, J . D., N a t l . Advisory C o m m . Aeronaut., Tech. Note 2610 (February 1952). Stanite, .J. D., T r a n s . Am. Sei. Mech. Engrs., 74, No. 4, 473-97 (1952). Stanitz, J. D., and Ellis, G. O., ,Vatl. Advisory C o m m . Aeronaut., Tech. Note 2654 (March 1952). Stanitz, J. D., and Sheldrake, L. J., Ibid., 2652. Stewartson, K., Proc. Cambridge P h i l . Soc., 48, Pt. 1, 168-77 (1952). Tarjan, G., Acta Tech. Acad. Sci. Hung., 1, No. 1, 22-32 (1950). Vallander, S. V., Doklady A k a d . Nauk S.S.S.R., 82, No. 3, 345-8 (1952). Wu, C. €I., Natl. Advisory Comm. Aeronaut., Tech. Note 2604 (1952). \Vu, C. H., Trans. Am. Sac. Mech. Engrs., 74, No. 8 , 1363-80 (1952). Wu, C. H., and Brown, C. A,, J . Aeronaut. Sci., 19, No. 3, 183-96 (1952). Yeh, H., Ibid., 19, No. 4, 279-80 (1952). Ibid., 19, NO. 9,630-8 (1952).
Batchelor, G. K., Proc. Roy. Sac. ( L o n d o n ) , A213, 349 (1952). Batchelor, G. K., Repts. Progr. in Phys., 15 101-41 (1952). Burgers, J. M., Publ. sci. et tech. ministdre air (France), Actes Call. inter. MBcan 111, No. 251, 261-6 (1951). Chandrasekhar, S., P h i l . T r a n s . R o y . Sac. ( L o n d o n ) , A244, NO.884,357-84 (1952). Corrsin, S.,J . Aeronaut. Sci., 18, No. 11, 773-4 (1951). Corrsin, S.,J. A p p l . Phys., 23, No. 1, 113-18 (1952). Davies, R. W., Ibid., 23, No. 1, 941-8 (1952). Deissler, R. G., and Eian, C. S.,Natl. Advisory Comm. Aeronaut., Tech. Note 2629 (February 1952). Donaldson, C. du P., Ibid., 2692 (May 1952). Fleagle, R. G., Parrott, W. H., and Barad, M. L., J . Meteorol., 9, NO.1,53-60 (1952). Frenkiel, F. N., Proc. N a t l . Acad. Sci. U . S., 38, No. 6,509-15 (1952). Glaser, A. H., J . Scz. Inst., 29, No. 7, 219-21 (1952). Hahnemann, H. W., Forsch. Gebiete Ingenieurw., B18, No. 2, 45-55 (1952). Hopf, E., J. Rational Mechanics, 1, 87-125 (1952). Johnson, H. A., Hartnett, J. P., and Clabaugh, W. J., Proc., “Heat Transfer and Fluid Mechanics Inst.,” California, Stanford Univ. Press, 1952. Krzywoblocki, M. Z. E., Ibid., pp. 65-72. Lassiter, L. W., and Hubbard, H. H., Natl. Advisory C o m m . Aeronaut., Tech. Note 2757 (August 1952). Liepmann, H. W., 2. angew. M a t h . u. Phys., 3,321-43 (1952). Liepmann, H. W., Laufer, J., and Liepmann, Kate, N a t l . Advisory Comm. Aeronaut., Tech. Note 2473 (November 1951). Lighthill, M. J., Proc. R o y . SOC.( L o n d o n ) , A211, 564 (1952). Mattioli, E., A t t i accad. naz. Lincei R e n d . Classe sci. fis. mat. e nat. ( 8 ) , 11,N o . 5,260-4 (1951). Moyal, J. E., Proc. Cambridge P h i l . Sac., 48, Pt. 2, 329-44 (1952). Ogura, Y. I., J . Meteorol. Sac. ( J a p a n ) , 30, No. 1, 23-8 (1952). Pai, S. I., Quart. A p p l . Math., 10, No. 2, 141-8 (1952). Pearson, C. E., J . Aeronaut. Sci., 19, No. 2, 73-82 (1952). Pelegrin, M., Compt. rend., 235, 19 (1952). Pillow, A. F., Aeronaut. Research Lab., Melbourne, Rept. A79 (May 1952). Reichardt. H.. Forsch. Gebiete Inoenieurw.. BDI-Forschunasheft. 1951. ~.’414. (33F) Ribner,H. S , and Tucker, M., Natl. Advisory C o m m . Aeronaut., Tech. Note 2606 (1952). (34F) Richardson, E. G., &err. Ing.-Arch., 6, No. 2, 86-92 (1952). (35F) Richardson, L. F., Proc. Roy. Soc. ( L o n d o n ) , A214, 1-19 (1952). Rotta, J., 2.P h y s i k , 131, No. 1, 51-77 (1951). Schlinger, W. G., Berry, V. J., Mason, J. L., and Sage, B. H., Calif. Tech., Chem. Eng. Lab., Rept. 5016 (August 1952). Schlinger, W. G., Hsu, N. T., Cavers, S.D., and Sage, B. H., Ibid., 5017 (August 1952). Szablewski, W., Ing.-Arch., 20, No. 1, 37-45 (1952). Theodorsen, T., J. Aeronaut. Sci., 19, No. 9, 645 (1952). Townsend, A. A., Proc. R o y . SOC.( L o n d o n ) , A209, No. 1098, 418-30 (1951). Uberoi, M. S., and Corrsin, S., “22. Advisory C o m m . Aeronaut., Tech. Note2710 (June 1952). Yih, C . S., Trans. Am. Geophys. U n i o n , 33, No. 1,8-12 (1952). 1
U n s t e a d y Flow
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