ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT (56B) Am. Inst. Chem. Engrs., presented at the 45th Annual Meeting, Cleveland, Ohio, December 1952. (57B) Am. Soc. Testing Materials, Philadelphia, Pa., Symposium on Statistical Methods for the Detergent Laboratories, 1953. (5SB) Chem. Eng. Neus, 31, 288 (1953).
(17C) Longwell, J. P., and Weiss, M. A., Ibid., 45,667 (1953). (18C) Melaak, Z. A., Quart. A p p l . Math., 11,231 (July 1953). (19C) Rleyer, P., Chem. Eng. Sci., 2, 53 (1953). (2OC) Opler, A., IND. EKG.CHEM.,45,2621 (1953). (21C) Perry, R.H., and Pigford, R. L., Ibid., 45,1247 (1953). (22C) Pfeiffer, P. W., Chem. Eng.Sci., 2,45 (1953). (23C) Reid, W.P., J . Phys. Chem., 57, 242 (1953). (24Ci Rose. A,. and Johnson. R. C.. Chem. Eno. Proor.. 49. 15 (1953). (25C) Rushton, J. H., and Oldshue, J. Y., Ibid., 49,161 (1953). (26C) Ibid., p. 267. (27C) Scheibel, E. B., Ibid., 49, 354 (1953). (28C) Schlinger, W. G., and Sage, B. H., IND.ENG.CHEM.,45, 657 (1953). (29C) Schwars-Bergkampf, E., Chem.-Ing.-Tech., 25, 177 (1953). (30C) Schwertz, F. A, IA-D.ENG.C m w , 45, 1592 (1953). (31C) Silberberg, I. H., and McKetta, J. J., Jr., Petroleum Refiner, 32, 179 (April 1953). (3%) Ibid., p. 147 (RIay 1953). (33C) Ibid., p. 101 (June 1953). (34C) Ibid., p. 129 (July 1953). (35C) Sofer, G. A, Diets, A. G. H., and Hauser, E. A,, IND.E m . CHEM.,45,2743 (1953). (36C) Stein, T. W., and Reid, R. C., Anal. Chem., 25, 1919 (1953). (37C) Van Krewlen, D. IT., and Hoftyeer, P. J., Chem. Eng. Scz., 2,145 (1953). (38C) Vermeulen, T.. IND. E m . CHmi., 45, 1664 (1953). (39C) Waugh, D. F.,and Yphantis, D. A , , J . Phys. Chem., 57, 312 (1953). (40C) Zeegers, J. A,, Chem. Eng. Sci., 2, 74 (1953).
M a t h e m a t i c a l Techniques
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Acrivos, -4., and Amundson, S . R., IND.ENG. CHEM.,45, 467 (1953). Barrer, R.M., J . Phys. Chem., 57, 35 (1953). Batchelder, H. R., Busche, R. M., and Armstrong, W. P., IND. ENG.CHEX.,45, 1856 (1953). Carley, J. F., and Strub, R. A,, Ibid., 45,970 (1953). Ceaglske, N. H., and Eckman, D. P., Ibid., 45, 1879 (1953). Danckwerts, P. V., Chem. Eng. Sci., 2, 1 (1953). J . Phys. Chem., 57, Gumprecht, R. O., and Sliepcevich, C. M., 90 (1953). Haruni, M. hI., and Storrow, J. A., Chem. Eng. Sci., 2, 164 (1953). Hiester, N. H., et al., IND. ENG.CHEM.,45, 2402 (1953). Hirschfelder, J. O., Curtiss, C. F., and Campbell, D. E., J . Phys. Cham., 57,403 (1953). Hudgens, C. R., and Ross, A. M., Anal. Chem., 25,734 (1953). Jost, W., Chem. Eng. Sci., 2, 199 (1953). ENG.CHCW., 45, 2634 (1953). Kayser, R. F., IND. Klinkenberg, A., Krajenbrink, J. J., and Lauwerier, H. d., Ibid., 45, 1202 (1953). Kramers, H., and Alberda, G., Chen. Eng. Sci., 2, 173 (1953). Licht, W.,and Pansing, W.F., IND.EKG. CHEX., 45, 1885 (1953).
F mg
A. M. OPPENHEIM University of California, Berkeley, Calif. T H O M A S BARON Shell Development Co., Emeryville, Calif.
The most significant feature of last year’s literature is the large number of books, essays, proceedings of symposia, and review papers providing a convenient and up-to-date summary of the present status of fluid mechanics.
HE: organization of this review is essentially the same as last year’s. The subject matter is scanned in a sequence corresponding to the degree of progress made up to date in the interpretation and analysis of each topic, starting from viscous flow and leading up to flows associated with chemical reactions. HOYever,we changed our method of approach; the survey contains this time only a few representative papers of each field selected primarily for their interest to the chemical and process industry. Consequently, the number of reported papers has been considerably reduced. The literature of 1953 is prolific in contributions to fluid mechanics in that an unusual number of books of fundamental importance appeared. Because of their significance, they are discussed in the first section.
T
Books and Proceedings of Symposia Classical hydrodynamics deals with ideal fluids which frequently bear little resemblance to real fluids. The development
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of classical hydrodynamics was culminated by Lamb’s well-knonrn treatise which characterizes the method of approach of the last century. The significant feature of the progress made since then is the emphasis of the effect of departures from idealizations of classical hydrodynamics. The advances made in this direction over the first half of this century are summarized most authoritatively by Prandtl, the founder of the mechanics of real fluids. The English-speaking world should welcome the translation of Prandtl’s last book entitled “Essentials of Fluid Dynamics” ( 1 0 A ) . I n words of Batchelor (Proc. Phys. Soc., 66, 518, 1953):
The book fills admirably the long-felt need for description of fluid floV it is in nature, It consists of four different and effectively self-containedchapters precededby a short introductory discussion of pressure relations applying t o fluids in equilibrium. The first of these chapters is concerned with the dynamics of frictionless fluids. The second describes the motion of viscous fluids and the associated phenomena of boundary layers and turbuIence. The third section deals with the high speed flow of gases not in great detail and, as elsewhere, avoiding mathematics if possible but nevertheless conveying the essential ideas involved in the flonr of a compressible fluid. $ ~ chapter is on miscellaneous topics. T o my knowledge, no other
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~ & ~
FUNDAMENTALS REVIEW book on fluid dynamics provides a description of cavitation, flow of solid fluid mixtures, bodies in accelerated flow, rotating fluids, and in particular the effect of the earth’s rotation on currents in the atmosphere and oceans, flow of stratified fluids with applications to meteorology, forces of free convection, and many other phenomena. The details and especially mathematical treatments which have been omitted from Prandtl’s book are presented in “Modern Developments of Fluid Mechanics,” a product of the Fluid Motion Panel of the Aeronautical Research Council in England. The first two volumes edited by Goldstein and published before the war dealt exclusively with incompressible flow. Their scope has been considerably enlarged by the recent addition of two new volumes edited by Howarth ( 6 A ) with the primary purpose of adding the effects of compressibility. Chapters deal with the salient features of compressibility, detailed derivations of the basic equations for continuum flow basic features of the method of characteristics, with the propagation of disturbances-a method which might be well suited for the analysis of unsteady behavior of process systems, shock and blast waves, exact solutions of the equations of steady isentropic flow, the hodograph transformation, approximate methods of solution used in the treatment of compressible flow phenomena, and unsteady motion; the short chapter on one-dimensional flow and a comprehensive chapter on boundary layers should be of direct interest. In the second volume considerable space is devoted t o the description of experimental methods. This section, as well as the last chapter on heat transfer, is essentially a revision of the corresponding parts of Goldstein’s volumes. Progress in the gasdynamic aspects of fluid mechanics was marked by the appearance of two fundamental texts, the “Gasdynamik” by Oswatitsch ( 9 A ) (in German) and the “Dynamics and Thermodynamics of Compressible Fluid Flow” by Shapiro (14A). For a chemical engineer !‘Gasdynamik” provides the best introduction to the conventional methods of gasdynamics. The most useful feature of the book is the exhaustive treatment of one-dimensional flow including unsteady flow phenomena which are discussed in more detail than in any other text. Only the first volume of Shapiro’s text has appeared. Half of this is devoted t o one-dimensional steady flow. The treatment is culminated by a detailed description of a general method developed by the author and based on the representation of the interaction between various flow parameters by means of influence coefficients which are only functions of the Mach number. This permits treatment of a great variety of problems including changes of chemical composition, change of phase, and addition of mass. One of the most important aspects of the behavior of real fluids is turbulence. In the books mentioned so far turbulence is treated only inasmuch as necessary to solve certain problems connected with boundary layers, The first book, dealing with turbulence alone, is Batchelor’s “The Theory of Homogeneous Turbulence” (8-4). Homogeneous turbulence is an idealization to which few practical situations conform. However, a proper understanding of the simpler problem of homogeneous turbulence is necessary before the more complicated problems involving shear can be attacked successfully. This is a particularly auspicious time for the appearance of such a text since the rapid progress during and after the war has brought the field to a point where further development is seriously hampered by the lack of understanding of the properties of the nonlinear partial differential equations governing turbulent flow phenomena. The work is complete (but not detailed) with the exception that all the analyses requiring a Lagrangian description of the flow are excluded. Thus, unfortunately for the chemical engineer, no mention is made of the important problem of turbulent diffusion. Free use is made of the tensor analysis and the theory of sto-
May 1954
chastic integrals. This may make the book somewhat difficult to read for an average engineer. The remaining books are concerned with applications of fluid mechanics to special problems. The most comprehensive treatment u p to date of the aerodynamics of propulsion is presented by Kuchemann and Weber (“A). They present a unified and elementary treatment of all aeronautical propulsion devices from propellers to ramjets and oscillating wings. Another first in the literature is “The Atomization of Liquid Fuels” by Giffen and Muraszem (68). The book deals with the engineering aspects of spray formation, primarily with the correlation of available data, and with experimental methods used in assessing sprays. The book should be a useful source of information for the designer. Lyttleton’s book on the “Stability of Rotating Liquid Masses” (8A) is in an entirely different vein. The author investigates the fission theory of formation of binary systems in celestial mechanics. On the basis of the analysis of stability of rotating, gravitating liquids, he argues that the dynamical evidence does not support the fission theory. The importance of this book to the engineer lies in the extensive introduction of the mathematical subject matter, including a thorough discussion of stability conditions of a mechanical system, stability of a spherical form for small displacements, ellipsoidal harmonic analysis, properties of Lam6 functions, development of a function as a series of ellipsoidal surface harmonics, the secular stability of Maclaurin spheroids, and Jacobi ellipsoids, and the ordinary stability of the Jacobi ellipsoids and properties of Poincar6’s equations. Much of this information should be useful for the analysis of the disintegration of liquid masses. The third volume of the “Advances in Applied Mechanics” ( 1 A ) contains a number of authoritative and stimulating essays on various aspects of fluid mechanics, such as a generalization of the boundary layer concept, one-dimensional isentropic flow, turbulent diffusion, aerodynamics of blasts, shocks in mixed subsonic, supersonic flow, and vortex systems in wakes. Since these essays are otherwise unrelated to each other, they will be discussed separately under the proper headings of this review. The best indication of progress in a specific field is provided by proceedings of symposia. Last year was eEpecially prolific in this respect, in that five volumes of proceedings of recent symposia appeared dealing with various aspects of fluid mechanics: the Proceedings of the Second ( 1 2 A ) and Third ( 1 S A ) Midwestern Conference on Fluid Mechanics, the Proceedings of the First U. S. National Congress of Applied Mechanics (11A), Proceedings of the Fourth Symposium on Applied Mathematics ( S A ) ,and of the Fourth Symposium on Combustion (4A). These volumes alone contain a significant portion of the papers published on fluid mechanics last year, and they form a very useful compilation of references t o anyone interested in the subject. Although the majority of the papers are significant contributions to the field, even a brief mention of all of them is beyond the scope of this review. Only some papers of particular interest will be reviewed together with papers which appeared in journals.
Viscous Flow Fluid Properties. In the equations associated with the continuum treatment of the mechanics of real fluids certain coefficients enter by virtue of the fact that such a treatment deals only with the macroscopic behavior of the fluid. These coefficients represent properties of the fluid which, in principle, can be evaluated only by reference to the microscopic conditions. Even though the theoretical description of the latter is not always adequate due chiefly to mathematical difficulties, it is of importance since the theoretical concepts provide the proper basis for the correlation of experimental data.
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ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT The state of the art is described in an excellent review by Bird, Hirschfelder, and Curtiss (2%). The properties of matter are expressed in terms of the potential energy of interaction between a pair of molecules and the substance. The potential functions are empirical in nature. The most used potential function for nonpolar molecules is the Lennard-Jones potential and for polar molecules the Stockmayer potential. The former contains t v o parameters representing the maximum energy of attraction and the collision diameter; the latter includes, in addition, a third parameter representing the dipole moment of a single molecule. The authors present tables of these parameters for a large number of substances. The behavior of moderately dense gases is best described by the virial equation of state. The second and third virial coefficients can be evaluated using statistical mechanics. These coefficients are related to the parameters appearing in the potential function. In the case of dense gases and liquids, the virial treatment becomes too cumbersome and approximate treatments based on the lattice model or the radial distribution function have t o be used. Similarly, the transport coefficients-Le., the coefficient of viscosity, the coefficient of thermal conductivity, the coefficient of diffusion, and the thermal diffusion ratio-may be evaluated with high accuracy for dilute or moderately dense gases. Transport coefficients in dense gases and liquids may be calculated in terms of the nonequilibrium radial distribution function just as the equation of state of dense gases and liquids is given in terms of the equilibrium radial distribution function. However, a t the present time, only some rough calculations have been made, and accordingly the method does not yet provide means for practical computation. I n the meantime, the best molecular theories of transport phenomena are those of Eyring for liquids and of Enskog for dense gases. Eyring’s equations are based on the theory of absolute reaction rates and can be relied only to give the proper order of magnitude for the transport parameters. The usefulness of Enskog’s theory for dense gases is demonstrated by the fact that the pressure dependence of the viscosity of nitrogen is described quite accurately in a pressure range of 1000 atmospheres and that of carbon dioxide in a 100-atniosphcre range. Enskog’s theory has been extended t o binary mixtures but no numerical comparisons are available The authors conclude their review with comments on the applications of the principle of corresponding states. If molecules obey a two-constant potential function such as thc Lennard-Joncs potential then it can be demonstrated that they obey the principle of corresponding states based on critical properties. Similarly, if a set of substances having polar molecules associated with a Stockmeyer potential have the same reduced dipole moment, then the substances obey the same principle of corresponding states. To summarize, substances containing molecules mhich arc similar in structure tend to exhibit closely related behavior and obey one principle of corresponding states. I n an important paper RIcLellan (1JB) illustrates the use of the radial distribution function in deriving an expression for that part of the stress tensor which arises from intermolecular forces. This result is then used in the derivation of an expression for the tension of a spherical surface between liquid and vapor n-hich, in the limiting case of the plane surface, agrees with previous results of Kirk-rood and also of AIcLellan. Heath (IOB) measured the viscosity of binary mixtures over the complete range of concentration a t ordinary temperatures and atmospheric pressure. The mixtures investigated were helium-argon, helium-nitrogen, helium-carbon dioxide, hydrogennitrogen, and hydrogen-carbon dioxide. The results are in excellent agreement with the expression given by Chapman and Cowling and verify the curious fact that the addition of a limited amount of light and less viscous gas t o a heavy and more viscous gas may increase the viscosity. Similarly the addition of a small amount of heavy but less viscous gas t o a light but viscous gas may also increase the viscosity. 924
L a m i n a r Flow. Berman (SB)presents an exact solution of the Navier-Stokes equations giving a complete description of the fluid flow in a channel having rectangular cross section and two equally porous walls. The scope is limited to two-dimensional? incompressible, steady-state laminar flow. The solution describes the dependence of velocity and preasure on position, geometry, and fluid properties. This is of considerable interest t o chemical engineers since problems involving diffusion phenomena in flowing gas streams can be attacked only if the details of the flom are known. Synge ( 2 7 B ) describes the application of the hypercircle method amplified by the use of pyramid functions t o the numerical solution of Navier-Stokes equations for steady, incompressible laminar flow through channels with some unusual cross-section geometry and boundary conditions. Tifford (19B)describes an exact solution of the energy equation in the case of incompreseible flow near the forward stagnation region under the influence of extremely large surface temperature variations. He demonstrates that “solutions based upon Prandtl’s simplifying assumptions are found t o be erroneous near the line of stagnation when the fourth, sixth, etc., derivatives of the surface temperature are very large.” Wilson and Mitchell ( 2 2 B ) treat the problem of self-induced temperature effects on viscosity in laminar floiv of liquids. They assume an exponential variation of viscosity with temperature and present equations for the flow between parallel flat plates when both plates are stationary, and when one plate is in motion and for the flow in circular tubes. The variation of viscosity in the direction of flow is determined and the ratio of tho discharge with the assumptions made to that with constant viscosity is also evaluated. Carley el al. (BB) present a simplified analysis of the theory of screw extruders which is of direct interest to chemical engineers. Assuming Newtonian behavior, the rate of extrusion and the die pressure are related t o the screw and die geometry and to the operating variables. Butler ( 5 B ) describrs a method of analyzing flow induced by spherical surfaces; his theory may prove useful for problems of settling. Man-Newtonian Flow. The flow of high polymer solutions betnreen rotating cylinders exhibits phenomena which the NavierStokes equations do not describe. For instance, the liquid climbs the walls of the inner cylinder. The rather extensive controversy on this subject resulted in various theories such as the elastic theory of Weissenberg and Garner’s elaborate explanations of a chcmicol nature. The simplest explanation is b a v d on a purely hydrodynamical analysis with viscous stresses assumed as linear functions of the rate of deformation only. Greensmith and Rivlin ( Q R )report new experiments on the torsional motion of the fluid confined between parallel plates, one of which rotates and the other is fixed. While experiments of similar nature have been reportcd before, the authors’ merit is that they analyzed, eliminated, and corrected for sources of experimental error vith extraordinary care. They show that the Reiner-Rivlin hydrodynamical theory is consistent with experiment and evaluate the LLnormalstress coefficient” which governs the departures from predictions of the theory of linear viscosity. Thornton (18B)reports measurements of the time dependence of viscosity of thixotropic materials-Le., substances whose viscosity depends not only on shear stress but also on previous history of motion. Flow of Suspensions. The effect of concentration on the rate of settling of dispersed particles can be evaluated in esscntially two different ways. The effect of neighboring particles on a test particle may be assumed t o be the same as that of a suitably placed solid boundary, or, the effect of any one particle may be calculated on the test particle and the total effect evaluated by summing over-all particles in the field. McNown and Lin (14B) follow the latter approach in an attempt to improve on the work
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FUNDAMENTALS REVIEW of Smoluchowski and Burgers. They differ from the latter authors in that their summation is carried out slightly differently and that they use the Oseen modification of the stream function and are thus able to take into account the effect of Reynolds number on the rate of settling. The results seem to be in excellent agreement w-ith authors’ experiments. Rosenberg (16B) analyzes the motion of small slender particles in a viscous incompressible fluid. He finds that the results obtained for Couette flow are not sufficiently general to be applicable to all types of two-dimensional flows. Instead, for any but parallel flow, there are two equilibrium positions. Wilhelm ( 2 l B ) reviews the ideas proposed in recent years to explain the behavior of fluidized beds. This is an excellent source of information with valuable references. Reviewers remark that the analogy with the kinetic theory of gases, proposed by Bowman, should not be carried too far and that such concepts as vapor pressure, boiling, and melting as applied to fluidized beds, while superficially conveying a feel for the behavior of a fluidized bed, are probably not analogous to the corresponding concepts in the kinetic and statistical theories of liquids and gases. Flow through Porous Media. The stwdy-state flow of gases through porous media presents no new problems, since the pressure field is determined simply by the Laplace equation in the square of the pressure. In the unsteady-state case the flow is governed by the nonlinear equation:
In last year’s literature considerable attention has been paid to the solutions of this equation. Aronofsky and Jenkins ( l B , IBB), as well as Bruce et al. (4B),present numerical solutions obtained by digital computers for the cases of one-dimensional and radial flow systems. Green and Wilts ( 8 B ) use analogous electrical networks for the same purpose. Roberts ( M B )solves the problem analytically by breaking up the time coordinate into small increments for each of which the equation can be linearized. The problem is thus reduced to the solution of the linear differential equation of heat diffusion with variable thermal conductivity. Typical papers concerning s l eady-flow are those of Cornel1 and Xatz ( 7 B )on procedures for predicting and correlating turbulent flow through consolidated porous media, of Tollenaar ($OB)on the exchange of liquid between paper pores, and of Hudson and Roberts (11B) on the transition from laminar to turbulent flow through granular media.
Transition Regime Dryden (1C) reviews and reanalyzes published data on the effect of roughness on transition from laminar to turbulent flow. He shows that the transition Reynolds number of a flat plate with zero pressure gradient depends on the ratio of the height of the roughness to the displacement thickness of the boundary layer a t the element. Thesc effects are similar in streams of different initial turbulence. A good correlation is obtained by expressing the effect, of the ratio of the height of the roughness element to displacement thickness on the ratio of transition Reynolds number of the rough plate to that for the smooth plate. This correlation is valid when transition occurs downstream from the roughness element. The transition position reaches the roughness element and remains there as the stream speed is further increased a t a certain value of the height to thickness ratio. The effect of distributed roughness on transition on a flat plate is also discussed. Lessen (3C) considers the stability problem from the point of view of locally incompressible disturbances in a compressible flow field in the large and obtains a tentative check of existing compressible theory. Seneca1 and Rothfus (4C) present friction
May 1954
factors and velocity profiles for transition flow of fluids in smooth tubes. In a most interesting paper Lauer (2C) considers steady flow in the vicinity of the critical Reynolds number in a slightly conical tube. He describes experiments with air and carbon dioxide and reports the effective Reynolds number for various configurations. Using a thermodynamic argument, he shows that for subcritical Reynolds number the flow should be laminar while for supercritical Reynolds number turbulent flow must take place.
Vortex Flow A subject allied to transition from laminar to turbulent flow E ia the formation and stability of vortex systems and wakes. This field was summarized in an excellent critical review by Rosenhead (60). He points out that the problem of two-dimensional KArm&n street of vortices has been investigated exhaustively. All that is needed a t present is a quantitative, theoretical treatment which will consolidate the facts already available. However, the experimental investigation of wakes behind three-dimensional bodies is still quite fragmentary and a systematic study of this problem is recommended. I n general one may note that the available stability considerations are restricted to special kinds of disturbances while the results concerning more generalized cases, such as three-dimensional disturbances and disturbances due to the diffusion of vorticity, are inconclusive. The stability of vortex streets in a nonviscous fluid is analyzed by Birkhoff ( 2 0 )who demonstrates that it is the periodicity of a vortex trail which is unstable and not the ratio of the mean longitudinal and transverse spacing. In a viscous fluid the Iongitudinal spacing is invariant while the transverse spacing increases in the direction of flow. The analysis yields a rough explanation for the order of magnitude of the Strouhal number representing the nondimensional frequency of the shedding of vortices. The more practical aspects of vortex flow are illustrated in the following papers concerned with direct engineering applications. Bader ( I D ) considers frictionless flow in a cyclone following an isentropic expansion of the gas a t the entrance. Expressions are derived for the flow rate and rate of spin of the gas core in terms of cyclone geometry. Ter Linden ( 7 D ) describes a laboratory investigation of cyclone dust collectors. The paper includes a short summary of cyclone theory, data on pressure drops in cyclones, and information concerning the influence of cyclone geometry on efficiency. An example of a method used for the analysis of vortex flow generated in turbomachinery which recently attracted a good deal of attention is provided by the paper of Torda, Hilton, and Hall (8D). The method was introduced by Lorenz at the beginning of the century and is based on the simulation of the effects of blades by an axially symmetric force field. Since it represents in effect an infinitesimally spaced blade system, the resulting analysis is particularly useful for the calculation of the pressure and velocity distribution in modern turbomachines with their closely spaced blades. The paper is limited to viscous, laminar, incompressible flow. Smith, Traugott, and Wislicenris ( 6 0 ) present some practical solutions of three-dimensional flow problems occurring in axial flow machines. Huppert and Benser ( 4 0 ) describe some stall and mrge phenomena in axial flow compressors. In particular, they discuss the phenomenon of rotating stall which added considerably to our understanding of the stability limits of rotating compressors.
Turbulent Flow Shear Flow. I n the discussion of Batchelor’s book on homogeneous turbulence, it waR pointed out that most of the engineering problems involve the additional dificulties introduced by the presence of shear. The general problem of transport processes in turbulent shear flows is reviewed by Kuethe ( 7 E ) . In dis-
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ENGINEERING. DESIGN. AND PROCESS DEVELOPMENT cussing the mecha,nism of turbulence, he calls attention t o a possible mechanism suggested by Theodorsen (16E) which involves the formation of horseshoe vortices in shear flow. If in a small region of a boundary layer there is temporarily little or no turbulent shear stress, then that region will slow down relative to the surrounding fluid. The faster moving fluid around this region creates then a horseshoe vortex with the feet toward the wall. The feet are anchored to slower moving fluid than the top of the horseshoe. As the vortex tips downstream, the flow through the opening carries momentum and fluid properties from near t o the mall to the outer regions of the boundary layer. Kuethe uses wind maps obtained by Sherloclc and Staut during winter storms to show evidence of the horseshoe structure. A similar mechanism was also postulated by Townsend [Proc. Cambridge Phil. SOC.,47, Pt. 2, 375-95 (1951); Phil. Mag., 41, 890-907 (19jO)I. The three most important featurcs of turbulent shear flow are the concept of local isotropy, according to which the large eddies are responsible for the transport of fluid and flow properties; the phenomenon of intermittency, which 30 far has been observed only in unconfined turbulent flo~vsand which has been described by Corrsin ( 4 E ) ; and the flow of turbulent energy toward the wall where it is consumed in the region of high velocity gradients. The Reynolds equations for turbulence in incompressible flow are obt,ained by introducing fluctuations in the Navier-Stokes and continuity equations and taking mean values. The characteristic difficulty is that, there are ten dependent variables and only four equations, the three equations of motion and the equation of continuity. The first successful attempt t o integrate Reynolds equations is due to Pai (11E) for the case of fully developed turbulent flow in a channel. I n this case the problem reduces to t,he solution of two equations with four unknowns. Accordingly, the distribution of mean velocity and turbulent shear stress are given in expressions containing two empirical constants. The empirical constants are evaluated from Laufer’s experiments. The results are in remarkable agreement with both the velocity and shearing stress distributions. The laminar su blayer a-hich has been excluded from all theories to date is automatically included in Pai’s analysis. In a brief and excellent discussion of the mixing length theories, Kuethe points out that they assume that the transport, processes can be described by local values of the gradient of the transportable quantities. The mixing length is defined as the distance normal to the main stream where the fluid particle will carry the propert’ies to a given location. However, one finds experiment,ally that the mixing length away from the immediate vicinity of thc wall is relatively large in comparison to the boundary layer thickness. It is, therefore, questionable whether a mixing length can be described by conditions a t a point. Reviewers remark that perhaps a more fundamental objection to t.he use of the mixing length concept is that the mixing length is not directly measurable and is not the property of a physical entity since the particles continuously mix during t,heir motion. Xuethe closes his paper by discussing heat transfer in force fields, an interesting application being the theory of the Ranque-Hilsch tube. Burgers (1E) presents an intuitive attempt to express turbulent shear stress in terms of funct’ions resembling turbulent energy spectra and the Lagrangian autocorrelation function; using a number of postulates, he obtains experimental orders of magnitude. Laufer (823)describes measurements taken principally with a hot-wire anemometer in fully developed turbulent flow in pipes at speeds ranging from 10 to 100 feet per second. The results include Reynolds stresses, triple correlations, turbulent dissipation, and energy epectra. The turbulence field may be divided into three regions: 1. Kear the wall where the production of turbulence, diffusion, and viscous action are all of approximately equal importance. 2. The central region where the diffusion of energy plays the predominant role.
926
3 . A region between the first and second regions where the local rate of change of turbulent energy production dominates the energy received by diffusive action. This paper supplements the author’s two-dimensional channel study and is a valuable addition to our knowledge of turbulrnt shear flo~vs. Turbulent Diffusion. Frenkiel’s essay ( 5 E ) on turbulent diffusion concerning mean concentration distributions in a field of homogeneous turbulence is based on Taylor’s theory of diffusion by continuous movement. I n their contributions of considerable engineering interest, Sage and coworkers ($E, 14E, 15E) report on a series of investigations of eddy diffusion in turbulent shear flows. Longwell and Weiss (10E) consider the mixing and distribution of liquids in high velocity air streams. In a brief theoretical investigation they replace the random motion of the air by a sinusoidal motion and assume Stokes law. They conclude that the eddy diffusivity of particles is considerably less than the eddy diffusivity of molecules at the same conditions of turbulence. Their derivation, however: is not conclusive. The replacement of a random field by a sinusoidal field may affect not only the quantitative but also the qualitative behavior of the droplets; nor can their own experimental results be dismissed which show that the eddy diffusivity of drops may be larger than the corresponding eddy diffusivity of molecules. Statistical Theory. Chandrasekhar ( S E ) gives an excellent review of the statistical theory of turbulence. He considers the invariant theory of isotropic and axisymmetric tensors, extensions t o the formal equations of turbulence that are necessary when magnetic fields are present, and density fluctuations in isotropic turbulence. In a critical discussion of similarity concepts in isotropic turbulence Lin ( $ E )discusses the problem of the extent of self-preservation of the energy spectrum function during the decay of homogeneous isotropic turbulence. Author adopts Heisenberg’s formula for the transfer function and analyzes the stability of the quasi-equilibrium spectrum to a small perturbation. Results indicate that the large eddies are slow to adjust to the condition of quasi-equilibrium. I n his paper entitled “Similarity Theory of Isotropic Turbulence” Rotta ( 1 S E )argues that an arbitrary homogeneous and isotropic turbulent motion tends in a course of time to a universal state. A law of similarity for isotropic turbulence is deduced from dimensional considerations. The equation for the corresponding spectrum has a one parametric solution, but only one of these solutions satisfies the continuity condition in the region of smallest wave numbers, Rotta considers this solution as the one corresponding to the universal state. Comparison with measurements indicates that the universal atate is correctly described by such a similarity solution. A sophisticated insight into the properties of isotropic turbulence is given by Uberoi (17E)who shows that the correlation of fluctuating static pressure with any fluctuating quantity in the flow field can be expressed in terms of the rorrclation of the same quantity with two or more components of the velocity. With this as a premise, pressure correlations are related to correlations between two velocity components at one point and bvo velocity components a t another point. A postulated relationship between fourth-order and second-order velocity correlations is supported by experimental evidence. Miscellaneous. Ribner and Tucker (IRE) consider the problem of turbulence in a contracting stream. They extend the Prandtl and Taylor theory by the introduction of the spectrum concept. The selective effect of contraction on the components of turbulent intensity and the induced changes inthespectrumand correlation tensors are treated for the first time. While of lesser importance to chemical engineers, Kovasznaq ‘s ( 6 E ) paper on turbulence in supersonic flow cannot be omitted from this review. By means of a first-order perturbation theory, he shows from the linearized equations of flow that a compressible, viscous, and heat conductive gas can have thrce distinc-
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FUNDAMENTALS REVIEW tively different types of disturbance fields. The three modes are the vorticity mode, entropy mode, and sound wave mode. They are independent when the fluctuations are small, but interact a t intensities where linearization is not permissible. Hot-wire measurements are presented exploring the entropy mode and pure sound-wave fields.
Boundary Layer Laminar. Characteristic of the modern trend in boundary layer theory is the emphasis on the extension of the scope to include arbitrary variations in velocity, pressure, and surface temperature profiles and to allow for variations in fluid properties. Morris and Smith ( 7 F ) extend the Karman-Polhausen method to the case of compressible boundary layer with arbitrary pressure and surface temperature gradients. Brown (fF)presents some exact solutions of the boundary layer equations for a porous plate with variable fluid properties and pressure gradient in the main stream. Levy and Seban ( 6 F ) obtain numerical solutions of the momentum and energy equations including the effects of variation of viscosity and thermal conductivity; their results reveal that, for variable free-stream velocity, the temperature difference between the wall and the free stream exerts a considerable influence on the velocity distribution in the boundary layer and also affects the skin friction coefficient. Drake ( 4 F ) describes a method for the calculation of three-dimensional rotationally symmetrical boundary layers with arbitrary free-stream velocity and arbitrary wall temperature profiles which, by the use of Mangler’s transformation, makes it possible to translate the results obtained for two-dimensional cases to three-dimensional flow. Turbulent. -4 survey of the present status of the turbulent boundary layer with pressure rise is given by Tetervin ( I I F ) . An example of the engineering application of boundary layer theory is given by Schola ( 1 0 F ) , who extends the known procedure for calculating the turbulent boundary layer without pressure gradient to the case where prc’ssure gradients exist and on this basis determines the drag of some slender bodies having surfaces of arbitrary roughness. Rubesin (QP)re-examines the relationship between heat transfer and skin friction including the effects of compressibility which causes a thickening of the laminar sublayer and the buffer layer. The Reynolds analogy is modified with the result that the ratio of the Stanton number to half the local skin friction coefficient becomes approximately 1.2 instead of unity as before. Chapman and Xester (SF)report measurements of skin friction on cylinders in axial flows for Reynolds numbers of order 107 over a large variation of Mach numbers. Friction was determined by measuring force directly rather than deducing the frictional force from the conventional boundary layer Pitot-pressure surveys. Results are compared to a large number of available theories and previous data, showing best agreement with theories of Cope (British ARC 7364, 1943), Wilson [ J . Aeronaut. Sci., 17, No. 9, 585 (1950)], and Tucker (Natl. Bdvisory Comm. Aeronaut., Tech. Note 2337, 1951). Jet Mixing. Closely related to turbulent boundary layer theory is the problem of jet mixing. Torda ( 1 d F ) extends the Von K&rm&nintegral concept t o the analysis of wakes behind flat plates. Pai ( 8 F ) analyzes the laminar jet mixing of two compressible streams. Torda et al. ( 1 S F ) consider the symmetric turbulent mixing of two parallel streams. FVilder and Hindersinn (15F) report experiments on spreading of supersonic jets in supersonic streams. General. Lighthill’s original contribution ( B F ) is concerned with a fundamental study of the transmission of the influenee of a disturbance upstream through the boundary Iayer. Although the problem is of particular importance in the ease of supersonic flow, the analysis of the boundary layer separation along a flat plate induced by a step is of general interest. This case is worked May 1954
out in detail. The location of a separation point is determined as a function of geometry for laminar and turbulent flow. The analysis successfully rationalizes a flow phenomenon which was only qualitatively described heretofore (Goldstein, S., “Modern Developments in Fluid Dynamics,” Vol. 1, p. 88, Plate 19b, London, Oxford University Press, 1938). The practical significance of the paper is that it deals essentially with the basic mechanism of flow in front of a sudden constriction in ducts. Another unconventional aspect of boundary layer theory is investigated by Tulin (148’). He obtains solutions of the linearized boundary layer equations which are of particular value for the qualitative description of unsteady two-dimensional boundary layer flows, such as the flow involving an impulsive start from rest. In an essay entitled “Boundary Layer Problems in Applied Mechanics” Carrier ( d F ) calls attention to the fact that the techniques used in solving boundary layer problems in fluid mechanics are more generally applicable to a large class of problems in applied mechanics. If a physical situation can be described by a differential equation, it may be recognized as a boundary layer problem provided: “(1) the coefficient of the most highly differentiated terms is small compared to unity; (2) the other important terms have coefficients of order unity; (3) the size of the domain is characterized by lengths of order unity in the coordinate system chosen.” The boundary layer in the generalized sense is simply that region near the boundary in which the most highly differentiated term has an appreciable effect. The most important engineering aspect of this method is that if a physical problem can be formulated in this sense the thickness of a region involving steepest gradients becomes immediately evident without any further calculation. The solution of the problem umally provides only details which are rarely of engineering importance. Of Carrier’s examples, the two most interesting to chemical engineering practice concern the problem associated with heat penetration occurring during the manufacture of linoleum and a problem connected with heat transfer across cellular materials.
Multiphase Flow Surface Instability. Pork, Stubbs, and Tek ( I d G ) give an excellent example of applying the fundamental methods of fluid mechanics to engineering problems. They investigate the disintcgration of a plain sheet of liquid of finite thickness moving relative to a surrounding fluid by the classical methods of instability analysis. They investigate the conditions of maximum instability which are then supposed to be the conditions of disintegration of the sheet. Conical sheets formed by swirl chamber spray nozzles are supposed to break up into rings, the dimensions of which are determined by the most unstable wave length. The rings are then assumed to break into drops under the influence of surface tension as described by Rayleigh. Beautiful photographs are included which support the assumed mechanism. Eckart’s paper (JG) on the generation of wind waves on a water surface is of first importance to the theory of instability of liquid surfaces. Statistical methods are applied to the generation of surface gravity waves. The normal wind stress is assumed to be caused by a known ensemble of gusts. The main departure from the previous work of Jeffries is that while the latter supposes that the variable component of the wind pressure is caused by the waves already present, the author assumes that wind even on a flat surface has a variable component. Before applying the statistical equations, the effect of a single gust is evaluated and then calculations are made for an idealized long-continuing storm. Bubbles. In a paper inspired by chemical engineering problems, Hughes, Handlos, Evans, and Mayeock (SG) discuss the formation of bubbles a t simple orifices. Their major contribution is an excellent analysis based on a simplified model of the effect of a surge chamber introduced in the supply train. The analysis,
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aided by reference to an analog electrical circuit (Figure l), is carried sufficiently far to demonstrate the reasons for the discrepancy among various investigators. Rayleigh (“Scientific Papcrs,” Vol. 6, pp. 504-7, Cambridge, Cambridge Univ. Press, 1911-13) shomd that the collapse of a spherical cavity in an incompressible nonviscous liquid occurs in a finite time. Poritsky (8G) considers the effect of viscosity which curiously enough enters only through the boundary condition. Numerical integration of the basic equations show that the time of collapse may be infinite in some cases. Shu (10G) establishes this result rigorously, showing that the time of collapse is infinite if a certain parameter of viscosity is greater than a critical value and is finite otherwise. When sur-
I
R
Figure 1. Typical Gas Train and Its Electrical Analog
face tension is taken into account, the time of collapse can always be shown to be finite. In a remarkable experiment, Rouse (YG) demonstrates the case for which incipient cavitation does not Q C C U ~on the solid boundaries but within the free stream. Turbulence in the mixing zone of a free jet may lower the instantaneous pressure sufficiently to cause cavitation. Miscellaneous. Baron, Sternling, and Schueler (SG) suggest that certain incompressible multiphaee dispersed flows can be treated with the techniques for single phase flow. They suggest that a multiphase flow may be treated as a single phase flow with suitable effective properties if a single set of Navier-Stokes equations written for the mixture suffice. Conditions derived from this criterion indicate that the single phase flow technique is applicable to most liquid-liquid and liquid-solid dispersions but is not applicable to dispersions of liquid in vapor except when the dispersion is extremely fine. Application of these ideas is demonstrated by a successful correlation of experimental results obtained on a water-carbon tetrachloride system. They indicate that the dispersion may have been due t o high, localized shear stresses in the turbulent field. The flow of air-water mixtures is treatcd by Straub et al. (21G) and Einstein and Sibul ( 4 G ) while a theory of pressure drop and heat transfer for annular steady state two-phase, two-component flow in pipes is presented by Levy ( 7 G ) . As a result of a straightforward derivation, he obtains results consistcnt with Martinelli’s correlations. Droplet trajectories are of primary import,ance in the theory of filtration. A method of calculation of trajectories using an electronic analog computer is described by ilbramson and Torgeson
(IC). 928
Hughes, Evans, and Sternling (5G)present the most stimulating account of problems encountered in flash vaporization. Much of the discussion concerns the phase diagram of tn-0-phase flow and conditions of atomization. They might be the first t o call attention to thc effect of turbulence on the separation in cyclones.
Of the whole field of compressible flow theory which has been advanced mostly in connection with aeronautical applications only the gas dynamic aspects of, primarily, the unidimensional flow are of interest to chemical engineering. Continuum Flow. Znldastani (92OH) reviews the analytical methods of treating transient one-dimensional isentropic fluid flow. He obtains continuous analytical solutions of the hypcrbolic differential equations governing such motions by generalizing the step-by-step procedure of the conventional method of characteristics. McVittie ( 1 4 H )introduces a mrthod of deriving solutions of the unidimensional Euler equation of motion without involving any thermodynamic restrictions. The results are given in terms of an arbitrary function which can be determined a posteriori from thermodynamic considerationn. The method is applied to spherically symmetric gas motions. Ludford ( 1 2 H ) presents an analysis of compresiible flow involving shock phenomena baaed on the Navier-Stokes equations. The ensuing mathematical difficulties are avoided by thc use of a particle model of continuum which results in difference equations. Examples are worked out for gas motions in closed tubes. A most interesting paper with engineering implications is that of Goldsworthy ( 7 H ) on the theory of the initial flow accaoinpanying the opening of a high pressure valve. Hearth and Perchonolr ( 3 H ) analyze the effect of heat addition in the divergent section of a de Lava1 nozzle on flow parameters a t the exit. They demonstrate that increasing heat addition results in an increase in thrust and in exit static pressure and in a decrease in exit Xach numhcr and total presaure. The attenuation of shock waves in shock tubes due t o interaction rvith the wall should be an important consideration in possible process applications. Emrich and Curtis ( 6 H ) present measurements and conclude that
where z is the relative pressure change across the shock and X is the hydraulic radius. However, the exact nature of the interaction of the shock and wall is not determined. The general problem of nonuniform propagation of shock waves is treated by Burgers ( 5 H ) . The interaction between disturbances and shock waves attracted a considerable amount of attention. Bogdonoff’s paper ( 4 H ) represents an example of the study of shock wave, turbulent boundary layer interaction; Ribner and Moore ( 1 6 H ) analyze the interaction betwecn shock waves with weak turbulence and noise; a comprehensive account of the interaction between shock wavea and obstacles is given by Ludloff ( I S H ) in his essay on “Aerodynamics of Blauts.” Kline and Shapiro ( 1 1 H )investigate the influence of fluid properties represented by the shape of the isentrope on the pressure volume diagram on the formation of shock wavrs. Their paper provides in effect the generalized theory of the Rayleigh Line. Neice ( 1 5 H ) descrihes a method for stabilizing shoclc waves by means of surge chambers. Green ( 8 H ) derives exprmions for velocity and thiclcnpss of shock waves in liquids. An engineering application of the progressive wave theory is the gas dynamic investigation of a valveless pulsr jet tube by Yen, Korst, and 1lcCloy ( I S H ) . Rarefied Gas Flow. Thc analysis of free molrcule flow is usually restricted to the case in vhich the gas is in Maxwellian
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FUNDAMENTALS REVIEW equilibrium. Although this restriction is satisfactory in an unbounded flow field, it introduces a severe limitation in problems concerned with the analysis of flow near local boundaries. Such boundaries are invariably introduced by any measuring instrument. Therefore, the theory of free molecular flow with a nonMaxwellian distribution is of particular interest in the interpretation of experimental data. A significant contribution in this respect is offered by Bell and Shaaf ( B H ) who obtain the expressions for aerodynamic forces on a cylinder under such flow conditions based on a distribution function, including the Grad’s terms representing the effects of viscosity and heat flux. The convective heat transfer coefficient is evaluated on the same basis ( 1 H ) . Bernard and Siestrunck ( S H ) describe a modification of the Maxwellian distribution function introduced by a weak shock wave. Data of interest in connection with the interpretation of measurements in rarefied gas flow are presented by Sherman ( 1 7 ” ) with respect to impact pressures in supersonic and subsonic air streams, by Ipsen (IOH) in relation with drag on cones and by Talbot ( 1 8 H )on viscosity corrections to cone probes.
Dynamics of Combustion This subject, which combines the interests of fluid mechanics, thermodynamics, and chemistry, waR advanced considerably over the last few years. The attention devoted to this topic is demonstrated by the large number of papers on the dynamic aspects of combustion contributed to the last Combustion Symposium. The proceedings of this symposium are conveniently available in a single volume ( S J ) . The progress made to date is described in a large number of excellent survey papers: “Limits of Inflammability” by Egerton, “The Problem of Ignition” by von Elbe, “Methods of Measuring Burning Velocities” b y Linnett, “The Structure of Laminar Flames” by Gerstein, “Instability Phenomena in Combustion Waves” by Markstein, “Open Turbulent Flames” by Karlovitz, “Quenching, Flash Back, Blowoff” by Wohl, ‘(Flame Stabilization by Bluff Bodies and Turbulent Flames in Ducts” by Longwell, and [‘Burning in Laminar and Turbulent Fuel Jets” by Hottel. For details of the hundred-odd original contributions, the reader is referred to this volume. Of particular significance to fluid mechanics are the fundamental papers on the structure of stationary flame fronts and detonation waves, on cellular flames and oscillatory combustion, on the relationship between laminar and turbulent flames, and, especially, on turbulent flame theory which has been advanced considerably. Besides theoretical analyses, the volume rontains a wealth of experimental data. Of other publications worth noting is the paper of Von KBrmBn ( 7 J ) in which the author presents a fundamental formulation of the aerothermodpnamic analysis of combustion, based on onedimensional flow equations which include viscosity and a parameter characterizing the progress of chemical reaction. This approach provides a convenient viewpoint for the evaluation and interpretation of progress made up to date in the Rubject. A simple application of the one-dimensional gas dynamic analysis t o a reactor is presented by Price (QJ)in his paper on the theory of steady flow with mass addition applied to solid propellant rocket motors. The most widely studied topic concerned with the engineering aspects of combustion is the problem of combustion instability. Although it is believed that the fundamental mechanism of flame instability is associated with transverse oscillation, the majority of the papers deal with the somewhat simpler problems of unidimensional longitudinal vibrations. Crocco and Cheng (WJ) discuss such a high frequency combustion instability in rocket motors assuming concentrated combustion as contrasted to the case of distributed combustion, the analysis of which was presented by the same authors a t the combustion Symposium (SJ). Putnam and Dennis (5J) study the organ-type oscillation of May 1954
flames. Tishler and Bellman (64 analyze the combustion instability in a monopropellant rocket with a pressurized gas propellent pumping system. An experimental investigation of combustion instability in rocket motors is reported by Berman and Cheney ( 1 J ) who provide detailed information on wave motions in the combustion chamber.
General In addition to the large number of books, an unusual number
of excellent review papers was published last year. Some of these were already described, since they were concerned with specific aspects of the topics discussed. The following papers are of more general character. De Groot ( 1 K )develops the equations of hydrodynamics from the point of view of the thermodynamics of irreversible processes. The laws of conservation of mass, momentum, and energy and the second law of thermodynamics are used to derive an entropy balance equation which has the usual form associated with equations of continuity. The rate of accumulation of entropy is the sum of the accumulation due to the divergence of the entropy flow and the accumulation due to the production of entropy. The latter characterizes the irreversible processes taking place inside the system and has the form of a sum of products of fluxes and forces. Following the scheme of the thermodynamics or irreversible processes, certain linear relationships are established between these fluxes and forces. The fluxes expressed in terms of the forces are reinserted into the fundamental laws and result in completely general equations describing the flow. The primary virtue of this procedure is demonstrated in the derivation of the usual equations of hydrodynamics as the Navier-Stokes equation. The scheme of assumptions of the thermodynamics of irreversible processes is general enough to include the idealizations necessary for the derivation of the flow equations. I t must be emphasized that the number of assump tions is thereby not lessened. For instance, the premise that linear relationships exist between the so-called forces and fluxes is simply a generalization of the usual assumption that the pressure tensor is proportional to the velocity gradient tensor. However, the generality of the procedure has its practical importance in calling attention to the possibility that certain forces may cause certain fluxes not generally associated with them; a wellknonrn example of this is mass diffusion under a temperature gradient. The author describes a more unusual example concerned with the behavior of liquid helium. Because of a restricted interchange of momentum between two species of liquid helium, the equations of motion are quite different from those ordinarily used. For example, in the case of mechanical equilibrium, the pressure gradient is balanced by the gradient of temperature. This could never occur in an ordinary fluid. Von KBrmBn ( Q K )gives a refreshing survey of the foundations of high speed aerodynamics. I n addition to the conventional aspects of the subject he includes a section on the aerothermcdynamics of combustion, summarizing in essence his views on this topic as described in the previous section ( 7 4 . Recent progress made in the field of aerodynamics is concisely described by Dryden (WK).The analytical methods used in the description of secondary flows are summarized by Kronauer (9X). These flows are of three-dimensional character arising in otherwise two-dimensional flows due to the action of t,he boundary layer and should not be neglected when the displacement thickness of the boundary layer is greater than 1/20 of the duct width.
References Books and Proceedings of Symposia (1A) “Advances in Applied Mechanics,” Vol. 3, New York, Academic Press Inc., 1953.
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ENGINEERING, DESIGN, A N D PROCESS DEVELOPMENT Batchelor, G. X., “Theory of Homogeneous Turbulence,” New York, London, Cambridge University Press, 1953. “Fluid Dynamics,” Proceedings of Symposia in Applied Mathematics,” 1-01. 4, New York, McGraw-Hill Book Co., Inc., 1953. “Fourth Symposium (International) on Combustion (Combustion and Det,onation Waves) ,” Baltimore, Williams and Wilkins Co., 1953. Giffen, E., and Murasaew, A., “Atomization of Liquid Fuels,” Kew York, John Wiley & Sons, Inc., 1953. Howarth, L., “Modern Developments in Fluid DynamicsHigh Speed Flow,” Vols. 1 and 2, London, Oxford University Press, 1953. Kiichemann, D., and Weber, J., “Aerodynamics of Propulsion,’’ New York, McGraw-Hill Book Co., Inc., 1953. Lyttleton, R. A.. “Stability of Rotating Liquid Masses,” New York, London, Cambridge University Press, 1953. Oswatitsch, K., “Gasdynamik,” Wien, Springer-Verlag, 1952. Prandtl, L., “Essentials of Fluid Dynamics,” New York, Hafner Publishing Co., 1952. Proc. First U. S. Xatl. Congr. Appl. Mech., Am. Soc. Mech. Engrs., New York (1952). Proc. Second Midwestern Conference on Fluid Mechanics, Ohio State University, Columbus (1952). Proc. Third hlidwestern Conf. Fluid Mech., University of Minnesota Institute of Technology, Minneapolis (June 1953). Shapiro, A. H., “Dynamics and Thermodynamics of Ccmpressible Fluid Flow,” Yol. 1, Xew York, Ronald Press Co., 1953.
(22B) n7ilson, W. E., and bIitchel1, m7. I., Proc. First U. S. Natl. Cong. Appl. RIech., Am. Soc. Mech. Engrs., New York, PP. 789-95 (1952). Transition Regime (1C) Dryden, H, L., J . Aeronaut. Sci., 20, KO.7,477-82 (1953). (2C) Lauer, O., 2. angew. Phys., 5, 3,Sl-90 (1953). (3C) Lessen, M., Proc. Second Midwestern Conf. Fluid Mech., Ohio State University, Columbus. pp. 195-9 (1952). (4C) Senecal, V. E,, and Rothfus, R. R., Cham. Eng. Progr., 49, No. 10, 553-8 (1953).
Vortex Flow (1D) Bader, W., Z A M M , 33, 1/2, 66-7 (1953). (2D) Birkhoff, G., J . A p p l . Phys., 24, 1, 98-103 (1953). (3D) Brown, F. N. hl., Proc. Second Midwestern Conf. Fluid hlech.,
Ohio State University, Columbus, p. 123 (1952). (4D) Huppert, 31. C., and Benser, W. A., J . Aeronaut. Sci., 20, No. 12, 835-45 (1953). (5D) Rosenhead, L., “Advances in Applied Mechanics,” Vol. 3, 185-95 (1953). (6D) Smith, L. H., Traugott, S.C., and Wislicenus, G. F., Trans. Am. Soc. Mech. Engrs., 7 5 , KO.5,789-804 (1952). (7D) Ter Linden, -4.J., Ihid., 75, No. 3, 433-40 (1953). (8D) Torda, T. P., Hilton, H. H , and Hall, F. C., J . Appl. Mech., 20, KO.3, 401-6 (1953). T u r b u l e n t Flow
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Arcnotsky, J. S., and Jenkins, R., Proc. First U. 9. liatl. Congr. iippl. Nech., Am. SOC.Mech. Engrs., New York, pp, 763-71 (1952). Bird, R. B., Hirschfelder, J. O., and Curtiss, C. F., Proc. Third Midwestern Conf. Fluid Mech., University of Rlinnesota Institute of Technology, Minneapolis, pp. 3-84 (1953). Berman, -4. S., J . A p p l . Phys., 24, S o . 9, 1232-5 (1953). Bruce, G. H., Peaceman, D. TV., Raohford, H. H., Jr., and Rice, J. D., J . Petroleum Technol., 5 , No. 3,79-92 (1953). Butler, S. F. J., Proc. Cumbridge Phil. SOC.,49, Pt. 1, 169-74 (1953). Carley, J. F., Mallouk, R. S.,and McKelvey, J. PI.,IXD. EXG.CHEM., 45,974-8 (1953). Cornell, D., and Katz, D. L., Ibid., 45, 2145-52 (1953). Green, L., Jr., and Wilts, C. H., Proc. First U. S.Katl. Congr. Appl. Mech., Am. Soc. Mech. Engrs., New York, pp. 777-81 (1952). Greensmith, H. W.,and Rivlin, R. S.,P h i l . Tmns. B o g . SOC. London, A245, No. 899,399-428 (1953). Heath, H. R., PTOC.Phys. SOC.,6 6 , Pt. 5, SO.401B, 362-7 (1953). Hudson, H. E., and Roberts, R. E., Proc. Second Midwestern Conf. Fluid Mech., Ohio State University, Columbus, pp. 105-17 (1952). Jenkins, R., and Aronofsky, 3. S., J . Appl. Meeh., 20, No. 2, 212-14 (1953). McLellan, A. G., Proc. R o g . Soc., A217, No. 1128, 92-6 (1953). McNown, J. S., and Lin, P. N., Proc. Second Midwestern Conf. Fluid hlech., Ohio State University, Columbus, pp. 401-11 (1952). Roberts, R. C., Proc. First C. S. Natl. Congr. Appl. Mech., Am. Sac. Mech. Engrs., New York, pp. 773-6 (1952). Rosenberg, B., Ibid., pp. 807-11. Synge, J. L., in “Fluid Dynamics,” Vol. 4, Kew York, McGraw-Hill Book Co., Inc., pp. 141-65, 1953. Thornton, S., Proe. Phus. Soc. (London), B66, Pt. 2, No. 398B, 115-28 (1953). Tifford, A. K.,Proc. First U. S. Natl. Congr. Appl Mech., Am. SOC.Mech. Engrs., New York, pp. 783-8 (1952). Tollenaar, D., A p p l . Sci. Research, A3, S o . 6 , 451-61 (1953). Wilhelm, R. H., Proc. Second Midwestern Conf. Fluid Mech., Ohio State University, Columbus, pp. 379--88 (1952).
(1E) Burgers, J. bI., Koninkl. Ned. Ahad. Wetenschap., Proc. B56,SO,2,125-47 (1953). (2E) Cavers, S . D., Hsu, N. T., Schlinger, W. G., and Sage, B. €I., IND.EXG.CHEN.,45,2139-45 (1953). (3E) Chandrasekhar, S., in “Fluid Dynamics,” Vol. 4, pp. 1-17, New York, LIcGraw-Hill Book Co., Ino., 1953. Corrsin, S., Proc. Third hlidwestern Conf. Fluid Mech.,
University of Minnesota, Minneapolis, PD. 435-8 (1953). Frenkiel, F. N., “Advances in dpplied Mechanics,” Vol. 3, 61-107, New York, Academic Press, Inc., 1953; Proc. First U. S. Katl. Cong. Appl. Meoh., Am. SOC.Mech., Engrs., New York, pp. 837-41 (1952). Kovasanay, L. 9. G., J . Aemnaut. Sci., 20, So. 10, 657-74 (1953). Kuethe, A. M., Proc. Third Midwestern Conf. Fluid Rlech., University of Minnesota, Minneapolis, pp. 85-101 (1953). Laufer, J., Nat. Advisory Comm. Aeronaut., Tech. Note 2954 (1953). Lin, 6. C., in “Fluid Dynamics,” Vol. 4, pp. 19--27, New York, McGraw-Hill Book Co., Inc., 1953. Longwell, J. P., and Weiss, &I. A , IND.E m . CHEX,,45, 667-77 (1953). Pai, S.I., J . Appl. Mech., 20, No. 1, 109-14 (1953). Ribner, H. S., and Tucker, M., Proc. Second Midweutern Conf. Fluid nlech., Ohio State University, Columbus, pp. 57-66 (1952). Rotta, J. C., J . Aeronaut. Sci., 20, No. 11, 769-73 (1953). Schlinger, W. G., and Sage, B. H., ISD. ENQ. CHEM.,45, 657-61 (1953). Ibid., pp. 2636-9. Theodorsen, T., Proc. Second Midwestern Conf. Fluid Mech., Ohio State University, Columbus, pp. 1-18 (1952). Uberoi, h l . S.,J . Aeronaut. Sei., 20, No. 3, 197-204 (1953). Boundary Flow
(1F) Brown, W. B., Proc. First U. S. Natl. Cong. Appl. Mech., Am. SOC.AIech. Engrs., New York, pp. 843-52 (1952). (2F) Carrier, G. F., “Advances in applied hleohanics,” Vol. 3, pp. 1-19, New York, Academic Press, Inc., 1953. (3F) Chapman, D. R., and Xester, R. H., J . Aeronaut. Sci., 20, NO.7, 441-8 (1953). (4F) Drake, R. hl., Jr., Ibid., 20, 309-16 (1953). (5F) Levy, S., and Seban, R. A., J . A p p l . Mech., 20, KO.3, 418-21 (1953).
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FUNDAMENTALS REVIEW (6F) Lighthill, M. J., PTOC.Roy. Soc., A217, No. 1130, 344-57 (1953); NO. 1131, 478-507 (1953). (7F) Morris, D. E.,and Smith, J. W., J . Aeronaut. Sei., 20, No. 12, 805-18 (1953). (8F) Pai, S. I., Proc. Second Midwestern Conf. Fluid Mech., University of Minnesota, Minneapolis, pp. 297-307 (1952). (9F) Rubesin, M. W., Natl. Advisory Comm. Aeronaut., Tech. Note 2917 (1953). (10F) Scholz, N., Jahrb. Schiflbautech n. Gesellschuft, 45, 244-63 (1951). (11F) Tetervin, N., First U. S.Natl. Cong. Appl. Mech., Am. Soo. hfech. Engrs., Iiew York, pp. 853-8 (1952). (12F) Torda, T. P., Proc. Third Midwestern Conf. Fluid Mech., University of Minnesota, Minneapolis, pp. 613-29 (1953). (13F) Torda, T. P., Ackermann, W. O., and Burnett, H. R., J . Appl. Mech., 20, KO.1, 63-71 (1953). (14F) Tulin, M. P., Proc. Second Midwestern Conf. Fluid Mech., Ohio State University, Columbus, pp. 155-69 (1952). (15F) Wilder, John G., Jr., and Hindersinn, K., Aeronaut. Eng. Rev., 12, NO.10,54-68 (1953).
Multiphase Flow (1G) Abramson, A. E., and Torgeson, W. L., Proc. Third Midwestern Conf. Fluid Mech., University of Minnesota, Minneapolis, pp. 353-66 (1953). (2G) Baron, T., Sternling, C. V , and Schueler, A. P., Ibid., pp. 103-28 ‘1953). (3G) Eckart, Carl, J . A p p l . Phys., 24, No. 12, 1485-94 (1953). (4G) Einstein, H. A., and Sibul, O., “Heat Transfer and Fluid .Mechanics Institute,” Stanford, Calif., Stanford University Press, 211-20 (1953). (5G) Hughes, R. R., Evans, H. D., and Sternling, C. V., Chem. Eng. Progr., 49, No. 2, 78-86 (1953). (6G) Hughes, R. R., Handlos, A. E., Evans, H. D., and iMaycock, R. L., “Heat Transfer and Fluid Mechanics Institute, Stanford, Calif,, Stanford University Press, 143-64 (1953). (7G) Levy, S., Proc. Second Midwestern Conf. Fluid Mech., Ohio State University, Columbus, pp. 337-48 (1952). (8G) Poritsky, H., Proc. First U. S. Natl. Cong. Appl. Meoh., Am. Soc. Mech. Engrs., New York, pp. 813-21 (1952). (9G) Rouse, H., Houzlle blanche, 8 , No. 1, 9-19 (1953). (10G) Shu, S. S., Proc. First U. S. Natl. Cong. Appl. Mech., Am. SOC. Mech. Engrs., New York, pp. 823-5 (1952). (1lG) Straub, L. G., Killen, J. M., and Lamb, 0. P , Proc. Am. Soc. Civ. Engr., 79, Separate No. 193 (1953). (12G) York, J. L.. Stubbs, H. E., and Tek, M. R., Trans. -4m. Soc. Mech. Engrs., 75, No. 7, 1279-88 (1953).
Dynamics of Combustion (1J) Berman, K., and Cheney, S. H., Jr., J . A m . Rocket Soc., 23, KO’ 2, 89-96 (1953). (25) Crocoo, L., and Cheng, S. I., Ibzd., 23, No. 5, 301-13 (1953). (35) “Fourth Symposium (International) on Combustion (Combustion and Detonation Waves) ,” Baltimore, Williams and Wilkins Co., 1953. (45) Price, E. W., J . Am. Rocket Soc., 23, NO. 4, 237-41 (1953). (55) Putnam, A. A,, and Dennis, W. R., Trans. Am. Soc. Mech. Engrs., 75, No. 1, 15-28 (1953). (6J) Tischler, A. O., and Bellman, D. R., Natl. Advisory Comm. Aeronaut., Tech. Note 2936 (May 1953). (75) Von Kiirrnh, T., Aerotecnica, 33, No. 1, 80-6 (February 1953). General
Gas Dynamics (1H) Bell, S., Univ. Calif. Tech. Rept., HE-150-115 (September 1953). (2H) Bell, S., and Schaaf, S. A., J . Am. Rocket sot., 23, No. 5, 314-17, 322 (1953). (3H) Bernard, T. T., and Siestrunck, R., Recherche adronaut., No. 31, 45-8 (1953).
May 1954
(4H) Bogdonoff, S. &I., “Heat Transfer and Fluid IIechanics Institute,” pp. 71-82, Stanford, Calif., Stanford University Press, 1953. (5H) Burgers, J. M., in “Fluid Dynamics,” Vol. 4, pp, 101-8, New York, fiTcGraw-Hill Book Co., Inc., 1953. (6H) Emrich, R. V., and Curtis, C. W., J . Appl. Phys., 24, No. 3, 300-3 (1953). (7H) Goldsworthy, F. A., Proc. Roy. Soc. (London),A217, No. 1132, 69-87 (1953). (8H)Green, R. B., Proc. First U. S. Natl. Cong. Appl. hlech., Am. Soc. Mech. Engrs., Iiew York, pp. 883-6 (1952). (9H) Hearth, D. P., and Perchonok, E., Natl. Advisory Comm. Aeronaut., Tech. Note 2938 (1953). (10H) Ipsen, D. C., Univ. Calif. Tech. Rept. HE-150-114 (A4ugust 1953). (11H) Kline, S. J., and Shapiro, A. H., “Heat Transfer and Fluid Mechanics Institute,” pp. 193-210, Stanford, Calif., Stanford University Press (1953). (12H) Ludford, G., J . Appl. Phys., 24, No. 4,490-5 (1953). (13H) Ludloff, H. F., “Advances in Applied Mechanics,” Vol. 3, pp. 109-14, New York. Academic Press, Inc., 1963. (14H) McVittie, G. C., Quart. Appl. Math., 11, 327-41 (1953); Proc. Roy. SOC.( L o n d o n ) , A220, No. 1142, 339-55 (1953). (15H) Neice, S. E., Natl. Advisory Comm. Aeronaut., Tech. Note 2694 (June 1953). (1BH) Ribner, H. S.,and Moore, F. X., “Heat Transfer and Fluid Mechanics Institute,” pp. 46-56, Stanford, Calif., Stanford University Press (1953). (17H) Sherman, F. S., Natl. Advisory Comm. Aeronaut, Tech. Note 2995 (September 1953). (18H) Talbot, L., Univ. Calif. Tech. Rept. HE-150-113 (June 1953). (19H) Yen, S. M., Korst, H. H., and McCloy, R. W., Proc. Second Midwestern Conf. Fluid Mech., Ohio State University, Columbus, pp. 507-20 (1952). @OH) Zaldastani, O., “Advances in Applied Mechanics,” Vol. 3, pp. 21-59, New York, Academic Press, Inc., 1953.
(IK) De Groot, S. R., “Fluid Dynamics,” Proc. Symp. Appl. Math., Vol. 4, pp. 87-99, New York, McGraw-Hill Book Co., Inc.,
1953. (2K) Dryden, H. L., Aeronaut. Eng. Rev., 12, KO.12, 88-95 (1953). (3K) Kronauer, R. E., PTOC.Am. SOC.Mech. Engrs., pp. 747-56 (1952). (4K) Von Kbrmiin, T., Ibid., pp. 673-85.
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