FLUID D NAMICS R. 8. HUGHES Shell Development Co., Emeryville, Calif. A. K. OPPENHEIM University o f California, Berkeley, Calif.
X T H E firsf Annual Revien of Fluid Dynamics [IsD. ENO. Cwm. 45, 941 (1953)l the subject matter mas organized in a sequence corresponding to a decreasing state of completion of the various topics into which it was divided. This principle has been maintained in subsequent years. If mch a premise were true, then one should expect that, as the years pass by, there ehould be less and less t o report on the sections treated a t the beginning. This indeed has been reflected in last year’s literature; the first section concerned with flow problems-namely, that on laminar flow-is conspicuously absent in the present survey. A few other changes werc also found necessary this year. Instead of a section on “Tranfiition Regime” a more general topic 011 “Flow Stability’ seemed more appropriate. Similarly, instrad of “Pipe Flow” there is now a more general section under thP heading “Ducted Flow.” It includes the consideration of conduits with variable cross-sectional area and channel flow. A new section on “Flow around Solids” has been added. Papers on unsteady-flow phenomena, which last year were treated in the section on gas dynamics, are now grouped together into a separate topic on “Transient and Periodic Flow.” I n this connertion, it became advisable to add a t least a rudimentary survey of acoustics, a subject not strictly within the scope of fluid dynamics but of fundamental importance to the treatment of transrient flow. Papers on shock interactions, also included in ‘‘Gas Dynamics” last year, have been distributed according t o the interacting phenomena: turbulence, boundary layer, transient flow, etc. Finally the title T y n a m i c s of Combustion” has been changed t o “Dynamics of Reactive Fluids” in order to convey the more general implication of the work done in this field. The increased activity on the fundamentals of fluid dynamics is reflected in the number of reviewed publications which exceed last year’s compilation by nearly 70%. The most significant progress hae been made in the theory of nonhomogeneous turbulence, notably that in shear flow; the basic understanding of the formation m d detailed mechanics of drops and bubbles; and the analysis and experinrental investigation of wave propagation and interaction. BOOKS OF GENERAL INTEREST
This section is concerned with recently published books dealing with the goneral field of fluid mechanics and with the compendia of articles which cover a wide field of application. Compilations of papers devoted t o special fields are discussed in the appropriate sections of the review. Books. The Englifih edition of Schlichting’s “Boundary Layer Theory” (IOA) provides a valuable up-to-date reference book on modern fluid mechanics. This is not merely a translation; it includes additions describing progress since thc Gcrnian edition appcarcd [(88G), 1953 Fluid Dynamics Review 1. As indicated by the title the approach i s through consideration of boundary layer flow; honrever, the book actually contains a thorough dis-
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cussion of all types of viscous motion, laminar-turbulent transition, and the nature of turbulent flow. It should therefore be considered as a fundamental text in fluid dynamics, fully recommended to anyone interested in this field of study. The only slight drawback is a natural concentration upon continental literature. Important English and American work is noted but the review of German literature is much more comprehensive. This, in itself, shows the value of the English edition, since workers in English-speaking nations are undoubtedly guilty of a similar neglect with respect to central European literature. A concise statement of the organization of thp book is given in the author’s preface: “The book is divided into four main parts. The first part contains two introductory chapters in which the fundamentals of boundary layer theory are expounded without the use of mathematics and then proceeds to compare tho mathematical and physical justification for the theory on the basis of the Xavier-Stokes equations. The second part contains the theories of laminar boundary layers and includes the theory of thermal boundary layers. The third part is concerned with the phenomenon of transition from laminar to turbulent flow (origin of turbulence), and the fourth part i s devoted to turbulent flows.” Space devoted to the four parts is, respectively, 20, 40, 10, and 30% of the total. Thus it can be seen that boundary layers are discussed in considerable detail. Problems of separa-. tion, secondary flow, shock wave interaction, frictional heating, etc., are thoroughly covered. The section on origin of turbulence includes both theoretical consideration of stability and discussion of experimental information on fluctuating edges of boundary layers and related subjects. The last part, on turbulent flow, includes discussion of flow through ~traight,curved, and noncircular pipes, skin friction on flat plates and rotating disks, and the behavior of jets and wakes. The text is clearly and concisely mitten throughout and well illustrated by numerous drawings, photographs, and numerical examples worked out in British engineering units, The first volume of the new text on chemical engineering by Coulson and Richardson ( $ A ) of the Imperial College of Science and Technology, London, is largely devoted to fundamentals; fluid mechanics naturally comprises the major share of the text. There has long been a need for an up-to-date, well founded, and reasonably elementary text for undergraduate study of chemiral engineering. I n this volume the authors seem to have made an excellent start which will probably be maintained in the second volume. After a short introduction on units and dimensions, the flow of fluids is considered in some detail. General enwgy relationships are followed by the information on friction in pipes and channels, using a well grounded theoretical approach. This section closes in a somewhat more practical vein with a chapter on flow measurement and another on pumps and compressors. The second section is concerned with heat transfer. The third considers the fundament& of mass t r a d e r , then discusses in some detail analogies among momentum, mass, and heat transfer, and finally describes the nature of the boundary layer The
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FUNDAMENTALS REVIEW latter is also used as means for an introduction t o more advanced topics in fluid mechanics. A final.short section is devoted t o humidification and water cooling, which, like heat transfer, is outside the scope of the present rcview. This method of organization s e e m very appropriate for a student of chemical engineering. He is allowed t o get some appreciation of the general features of flow and pressure drop nThere the behavior of the system can be visualiied directly; then he is presented the same sort of material in connection with heat transfer. Only after this has bren completed are the more abstruse theoretical details presented. As can be expected in any comprehensive book of this type, the expert can find objectionable points on which he differs from the authors. I n this case, however, such points are very few. T o the reviewers, only two criticisms scem worth mentioning. The first concerns the general presentation of the flow equations. They are derived only from the first and second law of thermodynamics. The corresponding relationship obtained from inomentum balance is not fully discussed until the chapter on boundary layer, and even there its importance is not sufficientlv emphasized. A brief statement of the momentum balance and the way it is derived from Newton’s law would considerably enhance the exposition of the fundamentals. A second, less important criticism concerns the units and dimensions. No recognition is given here of the (by now commonly approved) procedure of employing inconsistent English units Sith g. used as a unit conversion factor. Instead the discussion centers on using either the foot-pound-second systcm, involving use of a poundal as a unit of force, or the British engineering system involving use of a. slug as a unit of mass. These criticisms do not really detract from the book, which is certainly the best elementary chemical engineering text on the market today. New editions of two well-known elementary tests in fluid mechanics were published in the United States last year-namely, the third edition of Binder’s “Fluid R.lechanics” and the fifth edition of Daugherty and Ingersoll’s “Fluid Mechanics with Engineering Applications. ” Binder (1.4) reorganized his text, dividing it into two distinct parts: one on basic relations and the second on applications in fluid mechanics. XIe increased the scope of coverage by adding a few, mostly descriptive, sections in some chapters-e.g., on L‘galloping” structures in a fluid stream-aa well as including new chapters, notably one on propulsion and craft motion and another on unsteady flow and noise. Although intended primarily for civil engineers, the text of Daugherty and Ingersoll ( S A ) treats the subject in a general manner applicable t o all fluids. Conscquently, the title has been changed from “Hydraulics” t o “Fluid Mechanics. . and more emphasis placed on analyses derived from basic principles than on purely empirical relations The text is well organized and easv to read. The typical linking occurring in our times of the two historically distinct aspects of the subject-namely, hydro- and aeromechanics--is reflected in a German text of Kaufmann (5.4). The book contains detailed descriptions of most of the typical techniqucs of analysis used in hydro- , aero-, and gas dynamics. It represents a valuable reference text, especially with respcct t o the progress made in these fields in Germany over the past 20 years. Of particular interest to teachers is the text of Federhofer ( 4 A ) on “Problems in IIydromechanics,” also in German, which contains some 245 problems in statics and dynamics of incompressible flow with detailed solutions and explanations. Compendia. Of particular interest to the historian of the subject is the publication of two memorial volumes as tributes to two outstanding men in the field: one dcdicatcd to Richard von Mises, published in the United States t o commemorate his seventieth birthday (If A ) , and the other, published in France, honoring Dimitri Riabouchinsky on the occasion of the fiftieth anniversary of his scientific carrier in the field of aerodynamics,
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which started with the foundation of the Koutchino \Vind Tunnel in 1905 (7.4). Out of the 42 papers contained in the “Studies in Mathematics and PllechanicsJJpresented to ltichard von Mises, there are seven in the field of fluid mechanics-namely, that of J. M. Burgers on turbulent flow with shear; P. R. Garabedian on an axially synimetric flow with afree surface; G. Kuerti on a class of sphPrira11y symmetric flows; C. C. Lin on periodically oscillating wakes in the Oseen approximation; C. Loewner on some bounds for the critical free stream Mach number of a compressible flow around an obstacle; G. S. 8. Ludford on two topics in onc-dimensional gas dynamics; and W. Prager on slow viscoplastic flow. In contrast to the previous, there are some 44 papers out of a total of 49 in the Riabouchinsky volume which are concerned explicitlj with fluid mechanics. Some of these are discussed in specific sections of this review. Also published last year were the Proceedings of the 2nd U. 8. National Congress of Applied Mechanics ( 8 A ) with some 27 papers in the field of fluid dynamics; Volume 5 of the Proceedings of Symposia in Applied Mathematics ( I S A ) with seven papers bearing on the fundamental aspects of the subjcct; reprints of papers of the 1955 Heat Transfer and Fluid Mechanics Institute (6.4) with 17 (out of 20) papers of particular interest to this review; and Volume 1 of the Princeton series on high-sped aerodynamics and jet propulsion ( 9 A ) (Volumes 4 and 9 published previously were reviewed last year). Of the lattw the chapter on properties of liquids and liquid solutions by J. &I.Richardson and S. R. Brinkley and on relaxation phcnomena in gases by K. L. Herefeld are concerned with topics which are within the scope of this review. FLUID PROPERTIES
Gas Viscosity. Using a rigorous mathematical method, Iioga (6B)extends the kinetic theory of gases to cases where the gas departs from thermal equilibrium. The resulting equations of motion should be valid a t high shear stresses. I n the development, a second-order tensor appears in place of the usual viscosity and thermal conductivity; this is termed the “coefficient of conduction of strcss.” Two experimental studies of gas viscosity are worth noting Westmoreland ( I d B ) determined the viscosity of five samples of dry exhaust gas mixtures a t approxinlatcly atniospheric pressure and temperatures from 0” to l l O O o C. The correlation of the data by means of the Sutherland equation provides a convenient method of estimating exhaust gas viscosities. Lambert and others (623) measured the viscosity and thrrmal conductivity of 23 light hydrocarbons and also of carbon tetrafluoride a t atmospheric pressure and temperatures between 35’ and 100” C. For the lighter materials a pendulum visconieter was used; for the heavier, a capillary viscometer. -4comparison of viscosity and thermal conductivity data is given in terms of trnnslational, rotational, and vibrational energy of the various molecules. Carr, Parent, and Peck ( 1 B ) develop a correlation of g‘rs viscosity a t high pressures Plots of the ratio of the viscositT at a given pressure and temperature to that a t zero presswe and the same temperature are given as a funrtion of reduced pressure and temperature; for use of the correlation, a viscosity a t l o ~ vpressure is needed. A graph of thcsr is given for the usual permanent gases and hydrocarbons up to n-heptane. Comparisons of thc correlation Kith that of Uyehara and Watson [ S a l l . Pelroleurn News 36, 714 (1941)] are made for methane and for a 75-25 methane-ethane mixture. For thcsc data, the new correlation is much superior, probably because it depends on an evpeiiinentul value of low-pressure viscosity, rather than on a pret1it.t cd value of the critical viscosity. Liquid Viscosity, Recent work on liquid viscosity has definitely emphasized its non-Newtonian nature and the tramition from the viscosity of liquids t o the plasticity of solids. A
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FLUID DYNAMICS series of articles froni the laboratory of Schultz-Grunow gives a thorough discussion of the theoretical basis of liquid viscosity and the behavior of systems such as high polymer solutions. The first article, by Weymann ( I S B ) , extends Prandtl’s and Eyring’s theories of liquid viscosity, which lead to expressions of the form:
Here 7 is the shear stress along a plane normal to the y-axis, which is chosen in a direction parallel t o the gradient of the main streain velocity, V ; and C and A are constants characteristic of the fluid. Only for the case where T but the data are not converted t o show the true local shear stress vs. rate of shear relation. Anal. ses of non-Newtonian behavior show that viscosity determinations for nou-Newtonian fluids require an evaluation of eompletc curves of shearing stress us. rate of shear. “4recent report by Reltmann and Kuhns (21B) discusses a ncw, automatic, rotational-type viscometer ideally suited for the detrrmination of such curves. It is instrumented in such a way that thixotropic and rheopectic proper lies can be investignted. The revievers noted only one article on the behavior of Kewtonian liquids. Gruriberg ( 4 R )presents a thorough study of the viscosity of regular solutions involving carbon tetrarhlor icte, cyclohexane, and bcnzcnc. He sho.i\s that their behavior cnn only be explained in t e r m of Eyring’s theory if the interchange energy for viscosity is greater by a factor of 4 to 7 than the interchange energy derived from measurements of partial pressure. This is explained in terms of the constraint required on the motion of molecules in a solution of unlike molecules. TURBULENCE
Theory. Of general interest is the historical and rriticnl review by Munk (IBC). .4lthough entitled “Aerodynamirs,” the review concentrate8 on the necessity for the application of turbulence theory to aerodynamics. “Up to now,” the author says, ’‘our failure t o understand the mcchanism of turbulent fluid motion has prevented aero$ naniics from assuming the status of a logically unified science. This will soon be remedied as a consequence of recent advances. Aerodynamics will then be even more helpful than it is now.” Rlunk’s review is well written and of considerahle intcrest but rather belligerent in
A. K. OPPENHEIM studied aeronautical engineering at the Warsaw Institute of Technology and received the Dipl. Ing. in 1943. H e obtained his Ph.D. at the University of London and the D.I.C. (Imperial College) in 1945. Oppenheim served on the faculties of the City and Guilds College (London), 1945-1948, and of Stanford University, 1948-50. H e is now teaching mechanical engineering at Berkeley and serving Shell a s a consultant.
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FUNDAMENTALS REVIEW tone; the reader might best keep in mind that this is one man’s view of the field. Kamp6 de F6riBt’s ( 1 4 C ) serialized general discussion of turbulence theory which was mentioned in the last review [(UPE), 1955 Fluid Dynamics Review] has now been completed. The high standard of the first installment has been maintained throughout the entire publication. This makes the discussion an invaluable reference concerning the mathematics of turbulence. Most of the physical ideas stemming from the theory are concentrated in the last installment. Here the author suggests several new approaches to extend the statistical study of turbulence. In particular, he emphasizes that it may be preferable to return to the original l a m of fluid mechanics and put these on a statistical basis. His final conclusion is: “The use of iome methods of statistics in the theory of turbulence has certainly been fruitful. We cannot help admiring the really powerful physical intuitions which have illuminated the whole field, but it would be a dangerous illusion to think that almost everything has been solved by these intuitions. On the contrary it seems to us that we are still far from a completely rational theory of turbulence based on a statistical mechanics of fluids.” A new approach of considerable promise is that of Chandrasekhar ( 4 C ) . As is well-known, attempts to derive information concerning the turbulent velocity correlation from tho equations of motion involve a continuoua series of higher and higher order correlations. Thus the differential equation for the second-order correlation involves the third-order correlation, the equation for this third order involves the fourth order, etc. Chnndrasekhar suggests that a sufficiently valid approximation which would terminate this series is the assumption that the fourth-order or quadruple correlation i s related t o the second-order correlation as in a norrnal distribution. This does not limit the ease to a complete normal distribution, which mould imply that the thirdorder correlation is aero. The assumption leads to a rather formidable but atill useful differential equation for the secondorder correlation. In this paper, the author limits hirnself to finding some special solutions of this equation for infinite Reynolds number, at either very large or very small time lags between the measuring points. He suggests, however, that further studies are being made on more general solutions. The results obtained show the general properties expected of the correlation tensor. IIigher order correlations are also used by Proudman and Reid ( $ I C ) . Using both a t,hree-point correlation and the usual twopoint correlations, they consider the decay of normally distributed homogeneoucr turbulent velocity field. For large Reynolds numbers the exact solution of the equations for vorticity is obtained, and is found to be consistent with Kolmogoroff’s theorv Results concerning the energy spectrum and energy transier agree generally with previous knowledge of turbulence; however, energy transfer in large eddies does not agree with earlier work by Loitsiansky, Tin, and Batchelor. The .second-order correlation function alone is considered by Hyman (1JC) and applied to Burger’s well-known model of turbulence. Hynian has numerically integrated Burger’s diff erential equation for his second order correlation. Since the results follow the generally observed experimental variation of the correlation, some support is given to Burger’s model as a picture of real turbulence. A somewhat different approach is used by Peisen (2OC). He shows that von KQrniitn’s similarity theory can be made more general. Heretofore, the similarity constant, K , has always been considered universal. Persen suggests that it is only necessary that K be independent of the position and time in a given flow field. This still allows it to be a function of the main stream velocity, the initial intensity of turbulence, and the geometry of the turbulence gcncrator. Several recent papers are concerned with energy relationships in turbulence. In a short note Davies ( 7 C ) develops an equation for the energy spectrum function from existing limiting laws.
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This, however, does not add much to our knowledge of energy spectra, since several well-known equations acconiplish practically the same thing. Of considerably greater importance is the thorough discussion of the equations of turbulent energy transfer presented by Lettau (I7C) and followed up with some comments by Rlackadar. Lettau considers Reynolds’ interpretation of the product of the Reynolds’ stresa tensor and the shear derivatives of mean velocity as the rate of transformation of energy of mean motion to energy of eddy motion. He suggests that this concept is overly Rimplified and rather arbitrary, and points out that alternative arrangements of the equations are possible which lead to dinerent expressions for the effect on the eddy motion. This objection is particularly valid in meteorological problems; in contrast to pipe and windtunnel turbulence, atmospheric turbulence very often generates mean motion. Lettau gives a few calculations for a typical atmospheric situation vhirh show rather interesting energy transformation relationships. Although Blackadar also speaks from a meteorological point of view, he defends the usual intcrpretation of the energy dissipation function. H e admits it is soinewhat arbitrary and states: “The energy eqnations should be written in such a way that each term represents a single phenomenon which is sufficiently regular in its behavior to be summarized by a rule or law.” In cases such as atmospheric turbulence, he suggests that confusion regarding the origin of the eddy motion might be avoided by segregating the eddies according to scale. The application of turbulence theory to atmospheric phenomena is considcred in more detail by IIutchings ( f 3 C ) . Among the various differences between atmospheric turbulence and normal sniall scale turbulence, he finds that the origin of the mean motion providrs the greatest dificulty, although the existence of Coriolis forces in large wale turbulence may also affect its statistical properties. Hutrhings feels that the most l i k l y application of turbulence theory in meteorology is to the large scnle motion of the upper atmosphere which is free from the loczl disturbances caused by the earth’s surface. Since theoretical reasoning cannot yet shorn whether turbu1mc-e theory can apply to this s j stem, he has studied correlation functions in the upper atmosphere obtained from atmosphcric velocity observations. Time lags are on thp order of several hours in contrast, to the time lags of minutes or even seconds usually observed in the lahoratory ; thus the measiirements refer to turbulence scales on the order of 500 to 1000 miles. The surprising thing is that even for thk. large a scale the second-order correlations follow the Bame general relationships previously observed in wind tunnel experiments. I n a somewhat similar vein, Blackadar (SC) presents a d ~ . tailed study of the thermodynamics of turbulent systems. Ire summarizes and organizes his earlier ideas concerning the nature of “eddy energy” and Richardson’s concept of eddy pressure. Then he gocs on and chows the value of also defining the eddy entropy and developing a “wcond law of eddy thermodynamics” which states: “Apart from the effects of molecular diffusion and the transformation of eddy energy into internal energy, the internal source of eddy entropv is positive.” He shorvs that the eddy entropv is proportional to the amount of infoxmation associated with the frequency distribution of eddy velocities. Thus it, is analogous to the usual thermodynamic cntrcpp which expresse8 the statistical statmeof random molecular motion. Two papers concern rather specialized aspects of turhulence theory. Krzywoblocki ( I S C ) analyzes the turbulence in rarefied gases using Grad’s equations of motion in a manner analogous t o Reynolds’ treatment of the usual Navier-Stokes oquatione. Kovasznay (16C) considers the interaction of a shock wave with a turbulent wake. His experiments show that the creation of turbulent eround-Le., pressure fluctuations -agrees with theory, but that the theoretically predicted augmentation of velocity fluctuations by some 30% is not attained. The turbulence was naturally decaying aa the wake moved downstrram; the
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author euggests tlmt this decay may have offset the increase predicted for the shock wave effect. Techniques of Turbulence Measurement. An unusual but highly effective method of obtaining turbulence correlations and energy spectra is that of Uberoi and Kovasznay (B7C). Starting with a shadow picture of the turbulent field, thcy obtain the correlation by observing the light transmittance of a negativepositive sandwich a8 a function of the displacement between the two plates. They show that such measurements can provide the velocity correlation even for three-dimensional flow fields. The rrsults arc applied t o measurements in the wake of a projectile and in a heated 10-7-speed jet. They find the method gives a satisfactory prediction for small distanccs-i.e., for the smaller eddies-but is difficult with respect to large distances. Accordingly, the microscale of turbulence is well determined, but the integral scale i s probably inaccurate. In fact, for the rases shown, the integral scale is about the same magnitudc as the microscale. This is obviously incorrect for the Bows studied. Workers in this field may recollect Kovmznay’s first discussion of the method at the 1949 Heat Transfer and Fluid Mechanics Institute in Berkeley. Another relatively uncommon method of measuring velocity and pressure fluctuations is discussed by Rouse (SdC). H e describes a special Pitot tube for the measurement of three turbulent velocity components and the turbulent pressure in a liquid. Considering the more usual hot-wire anemometer, Pearson (19C) shows that the true turbulent velocity correlation can be obtained from measurements affected by an imperfect velocity calibration and transient heat conduction along the wire. This had already been demonstrated for the more general case of any statistical homogeneous field by Uberoi and Kovasznay [ Q m r t . A p p l . Math. 10, 375 (1953)]. Shear Flow. Emmons (9C) offers a valuable contribution t o the atudy of turbulent flow with shear. The first portion of his paper actuslly constitutes a brief but inforniative historical review of turbulence theory. I n the main part he describes a meam of correlating turbulence information for several types of shear flow: free jets, boundary laycr turbulence, wake behind a cylinder, etc. From the energy equation, various terms are selected t o represent dissipation, growth, and convection of turbulent energy. Emmons then treats each of the terms separately, deriving expressions for them in terms of various assumcd universal constants, the magnitude of the turbulent energy, the velocity derivatives, and a unit of length. The latter is somewhat analogous to Prandtl’s mixing length, but is shown in this c a m t o correspond to the size of eddy a t the nia.;imum of the eneigy spectrum. Thc success of Ehmons’ correlations gives Borne support to the use of this “most energetic eddy size.” In a short paper, Burgers (32)considers the behavior in shear of a simple model of turbirle~ice,treating fluid particles a8 separate entities, subject to drag and rotation from Ihe main fluid. H e proves that his assumptions arc consistent, eince the final equations lead to reasonable relationships. However, these are not supported by any experimental evidence. A rather unique problem is described by Einstein and Li (SC). To explain the origin of turbulence in shear flow near a wall, they analyze the behavior of the laminar sublayer imniediately adjacent t o the wall. Instead of the usually assumed concept of a steady-state thin film, they propose a more dyriainic layer: this layer begins with zero thickness, gradually broadens ~tl! laminar flow is established, but finally reaches an unstable condition and b r c a h away, leaving again a layer of zero thickness. T o find experimental evidence for this layer, they measured the pressure fluctuations a t the bottom of a flume. Since measurement of low amplitude pressure fluctuation w o ~ l dhave been too difficult, oil was used as the fluid, permitting the attainment of high velocity, and therefore a relatively high amplitude pressure pulsation, Rith the relatively low Reynolds number, neceesary to keep the frequency low. They were able to show
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that pulsations did orcur of considerably greater magnitude io, turbulent than in laminar flow, and present a curve of the autocorlelation function. A s further support for their theory, the authors correlate the %,allvelocity data of Lauffer (NACA Tech. Note 2945, June 1953). They fiud that the resulting term provides a smooth transition through the data points between the region near the wall and the usual logarithmic velocity curve a t some distance from the wall. The problem of velocity distributions near a wall is also considpred by Van Driwt (28C). He follows the gencral line of von KBrnirin’s similarity treatment of Nikuradse’s data. However, instead of confining his equation t o the buffer layer with a sharp transition t o a laminar sublayer, he adds a damping factor t o the eddy viscosity term t o make it disappear gradually a s the wall is approached. For rough walls this damping factor is further modified so as to allow for vortex generation by roughness. Like Einstein and Li, he is able to obtain an equation which successfully predicts the Lauffer data over the entire range. In addition, the resulting friction factor curve is qualitatively in agreement with those reported by Nikuradse for roughened pipe. A third attempt a t the same problem is the direct empirical correlation of Rothfus and Monrad (33’2). They find that the scatter due to Reynolds number in the usual plot of reduced velocity, uf, us. reduced wall distance, y+, can be eliminated by multiplying the velocity coordinate by the rat>ioof the average velocity to the maximum velocity and dividing the distance coordinate by the aame factor. This factor is then plotted against Reynolds number to provide a working correlation. A similar method is found to apply t o Bow between flat plates. Szablemki ( S S C ) applies Prandtl’s mixing length t o turbulent flow in a plain diffuser with a half angle between 0” and 4”. IIe modifies the mixing length by adding a second degree dependence on the wall distance. The fit to Nikuradne’s data on diffusers is good, a t least up t o half angles of 3”. Eddy Diffusion. The general problem of point source diffusion in a turbulent flow field is considered by Beeton ( I C ) . Straightforward derivationa are presented using Taylor’s theory and the approximate correlation function, R = e-t/ta. Results are compared with previously developed approximate equations including those obtained for a constant eddy diffusivity. The more specific problem of turbulent diffusion from a point sonrce a t ground level in the atmosphere is studied by Davies (6C). He shows that 0. G. Sutton’s prediction formula (“Micrometeorology,” McGraw-Hill, New York, 1953) can be derived by a solution of the gencral three-dimensional equation of noniactropic atmospheric diffusion. The assumption made is that the three coefficients of diffusivity can be expressed as power-law functions of the time with the same exponent. The significance of this assumption is not fully explained; hovever, the time dependence is suggested by the fact that progressively large eddiez dominate the mean outward diffusion of the cloud as the time of diffusion increases. In a second article, Davies (6C) analyzes the problem of diffusion from a plane area source, using nonise. tropic diffusion coefficients, particularly applicable to atmospheric diffusion. Iticler (2dC) reports a n experimental study of the eddy diffusion of momentum, water vapor, and heat near the ground. His careful measurements show that eddy diffusivities for vapor and momentum are about equal over a large range of atmospheric stability characteristics. In grneral, the heat diffusivity is also approvimately the same, but there are scattered raplee where abnormally high thermal diff usivity occurs. Fleishman and Frenkiel(1OC) approach the problem of atniospheric diffusion in a somewhat different manner from the uaual development of 0. I. Taylor and 0. G. Sutton. Starting with the assumption that the mean concentration of fluid has a t all times a Gaussian distribution, they eliminate the need for partial differential equation. Then they derive approximate
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FUNDAMENTALS REVIEW relations for the mean square displacements, a t both very long times and very short times, from the definition and properties of the correlation function. Application of the method to the case of an arbitrary velocity correlation involves numerical solution. As an example, a comparison is made between the short-time formula and the exact numerical solution for the case where the horizontal displacement is onc fifth of the horizontal scale of turbulence Similar concepts are applied by Frenkiel (11C) t o the problem of atmospheric dilution in the I,os Angeles area. H e shows how various assumptions as to the source of pollution can yield various curves of pollution vs. the time of day, which can then be compared witshatmospheric data to locate the source. As a valuable preliminary to this application, Frenkiel discusses in some detail the differences between wind tunnel turbulence and atmospheric turbulence, especially that near the ground. A more practical approach to the problem of atmospheric pollution is given by Sherlock and Lesher (W5C). Their wind tunnel study of models of typical factory chimneys show quite clearly how the particular chimney design, the orientation of the building, and local atmospheric conditions can affect the dcgrce of pollution. FLOW STABILITY
Fundamental Analysis. In an inutructive paper, Lin (9D) formulates the problem of hydrodynamic stability in the following way: “The existence of two types of motions of a viscous fluid, laminar and turbulent, immediately raises the question: Which type of motion is the more likely to occur? I t has now been generally recognized that turbulent motion is the more natural state of fluid motion, and laminar motion occurs only when the Reynolds number is so low that deviations from it are liable to be damped out. For certain types of flows it has been found possible to keep the Bow laminar for higher and higher Reynolds numbers by keeping the disturbance smaller and smaller. The question then may be asked for a given flow: I s it stable relative to infinitesimally small disturbances? This is the problem of hydrodynamic staldity.” H e then goes on to a brief review of previous work in the field of stability and develops several critical points of the theory in some detail. The most important of these is the limiting behavior of the solutions as the Reynolds number appromhes infinity. In the Orr-Sommerfeld equations, calculations for the case of infinite Reynolds number lead to singularities a t the point where the mean velocity equals the propagation velocity of the disturbance; Lin shoms how to avoid their effects on the analysis. Di Prima ( S D ) also considers the singularity in the solution of the Orr-Sommerfeld equation. He calculates the error resulting from the infinite Reynolds number assumption by comparing the results with an improved solution vhich is valid in the neighborhood of the singularity, and shows that the difference is usually negligible. Pai presents two recent contributions to stability theory. The first short note (IWD)develops conditions for stability of any plain parallel flow from the general Orr-Sommerfeld equation. The second paper ( I I D ) deals with the same problem, but is not limited t o solutions of the Orr-Sommerfeld equation. Instead a complete, detailed analysis of the fluctuating flow field resulting from small disturbances i s developed. Both two-dimensional and thrce-dirnensional disturbances are considered. Two articles by Cbandrasekhar discuss unusual but interesting cases of instability. The first (2D) develops the general problem of instability of fluid motion in the presence of a magnetic field. The method of treatment follows the usual hydrodynamic procedure] but starts with the equations of iihydromagnetics.” A few examples show that the critical RayIeigh or Taylor numbers may be from t h e e t o twenty times the normal value if magnetic fields of 1000 t o 10,000 gauss are used. Chandrasekhar’s second paper (ID)describes the equilibrium of an incompressible fluid
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sphere, subject to radial acccleration. This is a problem of considerable importance in geophysics. The author s h o w that instability occurs if the viscosity is less than a certain critical value, determined by gravitational attraction and the size of the sphere. For the case of the earth, the critical value is about 3 X 10’3 sq. em. per second. Since the earth’s mantle is now generally considercd to have a viscosity of about 10% to lO*3 sq. c m . per second, its present condition is definitely stable. A number of papers discuss problems of stability of boundary layers. D u m and T,in ( 4 0 ) present a general theory for compressible three-dimensional boundary layers. Gotler’s ( 6 0 ) fundamental work on three-dimensional instability of boundary layers on concave walls has recently been translated. This work ifi extended in a rather complete study by Hammerlin ( 7 ) ) . Finally, Low (1OD)analyzes the stability effects of heat sinks with the boundary layer, which can be practically attained by a spray of liquid droplets He shows that stability is augmented by the existence of sinks and reduced by sources. Laminar-Turbulent Transition. Schubauer and Klebanoff ( 2 4 0 ) present an extremely valuable picture of turbulence characteristics in the neighborhood of the transition between laminar and turbulcnt flow. They made their observations in the boundary layer of a flat plate, placed in a wind tunnel operated a t velocities of around 80 to 100 feet pcr second. One study concerned the generation of turbulence by artificial roughness. Figure 1 (p. 632) describes the various features of the wake inside the boundary layer, immediately behind a 1/8-inch sphere. The separation into fully turbulent, intermittent, and laminar zones is evident from characteristic hot-wire anemometer traces. Figure 2 (p. 632) shows the wake from an intermittent disturbance caused by firing a spark at the surface. The similarity to the turbulence spots considered by Mitchner in the paper discussed last year [ ( 4 0 ) ,1955 Fluid Dynamirs Review] is striking. Moreover, the similarity of the oscillogram records for spark-excited and natural-transition turbulence is notable. Transition phenomena in pipes and annuli were studied by Prengle and Rothfus ( I S D ) , using Reynolds’ dye filament technique. Both the Reynolds number range of the disturbances and the exact nature of the waves and spirals are described. In pipes, a t Re m 1200, the first observable stable deviation appears as an irregular wave a t the ccuterline of the pipe. As the Reynolds nuinher increases (between 1225 and 2100), the fluid in the central portion of the pipe takes up a sinuous motion while the fluid near the wall remains laminar. A t a Reynolds number of 2100 the first true disturbance eddy appears. Further incenses in Rejnolds number result in an increased shedding frequency of these eddies. Finally, a t Reynolds nurnbci fi grcatcr than 3000, the disturbance eddies appear to be stable. Similar results occur in annular sections, except that the first appearance of a slight wave is a t Re M 700, and the first disturbance eddy a t Re m 2200 to 2300. I n both pipes and annuli, the velocity a t the edge of the laminar film remains constant for any given geometry. This was observed up t o Reynolds numbers in the neighborhood of 25,000, but the authors warn that, extrapolation to higher Reynolds numbers may be dubious. On the whole, this study is a valuable one and should contribute to our undcrstanding of transition phenomena. In a short note Eckert and Irvin (60) present observations showing the simultaneous existence of turbulent and lamina] flow in ducts with noncircular cross sections. For shapes of the form of an isosceles triangle with a 12’ apex angle, turbulenrc begins near the base a t low Reynolds numbers and gradually propagates further and further toward the apex as the Reynolds number is increased. They plan to study this phenomenon for ducts of other cross sections to see whether thc apex angle is crucial. Liquid Films. The flow of thin films of liquid is of considerable interest in chemical engineering. The problem of wave f o m tion in such films has received a great deal of attention in recent
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FLUID DYNAMICS years. The latest example is the thorough analysis by Tih (16D).IIe first derives the Orr-Sommerfeld equation, then obtains a solution using a series expansion in terms of the Reynolds number. The solution applies to any inclination of the film, but numerical results are only obtained for the vortical case. Instability occurs for Reynolds numbers larger than about 6 (on Yih’s curves this value is actually 1.5, but he uses a definition of Reynolds uuinber whirh is one fourth as large aa that usually used for film flow); it first appears with wave lengths about 10 times the film thickness. In this case the wave velocity is about 6 times the average film velocity. If disturbances of other nave lengths occur a t high Reynolds numbers where they are unstable, their wave velocity is higher. Yih’s results arc given approximate experimental confirination by Jackson (SD). Film thicknesses of sir liquids in a 1.4-inch vertical tube were nieasureti by means of a Geiger counter, meamring beta radiation emitted from trace amounts of yttrium dissolved in the liquid. Jachson sho\+sthat this experimental technique gives a very reliable figure for the averagc film thickness. Film flow rates of 0.26 to 0.44 square feet per hour were used with liquids varying in liinematic viscosity from 0.02 to 0 45 square foot per hour; the resulting Reynolds numbers ranged from 4 to 4600. Observations show that wave motioii begins a t Reynolds numbers in the neighborhood of 12 to 20. This is slightly higher than the value predicted by Yih, but the agreement in order of magnitude is gratifying. The film thickness compares favorably n-ith Nusselt’s mll-known film equation, especially for the low viscosity materials. For high viscosity materials, film thicknesses tend to be up to 20% lower than predicted; deviations seem to depend on the extent of nave formation. The mechanism of the film flow is explained in terms of turbulent waves appearing intermittently on an undisturbed laminar fhn. The postulated profile of the wave is shown in A; Jackson suggests B as an analogous model---Le., solid rods rolling over the surlace of a thin film. Further experimental study of falling liquid films is reported by Stirba and Hurt (16U). Most of their observations concerns mass transfer from the solid wall t o the film. Hoaever, they point out that this is a very good means of identifying turbulence in the film, since molecular diffusivities for mass transfer are so much lower than for momentum transfer-i.e., the Schmidt numbers for liquids are usually large They report visual obseivations of wave motion in the film and also some studies by nieans of dye injection. Both mass transfer results and dye injection show that turbulence exists, a t least as far as gross effects are concerned, even a t Reynolds number on the order of 300 t o 1000. This is apparently tied up with wave motion on the surface. Stirba and Hurt as well as Jackson mahe some observations on the dfects of adding wetting agents to the liquid. One experiment of Stirba and Hurt showed that addition of agent not only eliminated the ripples appearing on the film, but also suppresscd turbulence, as indicated by a dye filament. On the other hand, wetting agents were used in most of their mass transfer experiments and did not affect rippling. Similarly Jackson observed no reduction in rippling from the addition of a wetting agent. Moreover, there was no change in film thickness for runs with wetting agents added. These experimental studies are very interesting and provide considerable information on the behavior of the liquid films; they also reveal many questions for further study.
VORTEX FLOW AND ROTATION
Theory of Vortex Flow. I n spite of the substantial progress made in the direct application of fundamental principles, heuristic arguments based solely on intuition still play an important role in the solution of problems in fluid dynamics. An excellent example of this is the solution presented by Barua (123) of the flow
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due t o a source placed on the avis of rotation in a uniformly rotating incompressible fluid. In the steady state, the irrotational flow due to the source is confined to a region inside a cylindrical discontinuity surface which has a bulge in the neighborhood of the sourec. The surface is symmetric about the axis of rotation and approaches a t infinity a circular cylinder of diameter 2a; the maximum diameter of the bulge a t the cross section of the aource is 2b. The differencc between the pressure a t the point of maximum deforniation a t the bulge and that a t infinity is evaluated, on one side, from the simple free-vortex relation in terms of the angu1a.r velocity of flow in the region unaffected by the source and, on the other side, from Bernoulli’s equation by noting that, from symmetry of flow, the point of maximum deformation of the bulge must be a stagnation point. By cqiiating the tn-0, a relation is obtained betwecn the angular velocity and the total output of the source with the unknown paramcters a and b. Since the problem is indeterminate a t this point the author inti oduces, what he calls, “a variational principle” by simply hypothesizing that of all the values of h only that for nhich (dbldu) = 0 need be adopted, BO that corrcsponding to a small variation in the value of a, there will be no change in the value of b. This yields a unique solution of b/a,-Le., 1.670--for which the whole flow field is then mapped in detail. The case of a plane vortex in a confined space is analyzed by Rahnberg (1123)’ on the basis of, again, a heuristic assumption which admits the solution of the problem by direct application of conformal mapping. The assumption is based on the postulate that any particle not a t rest moves with a speed exceeding a critical constant value characteristic of the fluid. Then, close to the vortex, the flow is assiimcd nonviscous, incompressible, and irrotational, this region being separated by a surface of discontinuity from outer parts which do not flow. On this basis some special cases are solved such a8 that of a vortex between two parallel planes, inside a wedge, and inside a triangle. Naylor (10E) analyzes adiabatic rotational flows of perfect gases by the application of the Legendre translorination to the Crocco stream function Squire (16E) studies the gro4th of a vortex in turbulent flow on the basis of Navier-Stokes’ equations with an eddy viscosity assumed proportional t o circulation, used in place of the kinematic viscosity. This permits hiin to trace the growth of a line vortex with tinre or the spread of a trailing vortex v.+,h downstream distance. Secondary Flow. The problem of secondary flow caused by axial rotation of solid boundaries provides an interesting field of study. Barua ( 2 E ) presents a solution to such a problem in the case of laminar flow a t low Reynolds numbers in a rotating straight pipe, and derives an exprcssion for the resistance to flow along the pipe. Eiranier and Stanitz (BE) describe the basic theory for the analysis of steady, nonviscous, and incompressible flow in a rotating syetem. Concerned with direct applications to turbo-machines are the papers of Ileraig and others on secondary flows in turbine nomles (12E) and on the visualization studies of such flows in rotating cascades of blades ( 7 E ) . Rotating Machinery. Worth noting are the German contributions to cascade flow problems, as exemplified by the products of Schlichting’s Institute of Fluid Mechanics at the Technical University of Braunsrhweig-e.g., Srhaffcr’s (I.@) investigation of thc three-dimensional flow through axial cascades with CJ lindrical blades, and Speidel’s (1523) study of the effect of n:ill roughness on flow losses in a two-dimensional cascade. Flow conditions in rotating turbine blades with transpiration cooled, porous walls are analyzed by Eekert, Livingood, and Prasse (5E). On the basis of a one-dimensional analysis, they determine the local wall pernieability necessary to obtain a prescribed local distribution of ejected gas, and also solve the reverse problem of the local distribution of ejected gas resulting from a given local wall permeability. The problem of stall in turbo-machines continues to attract
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wide attention, as exemplified by the paper of Cornell (SIC), t h e experimental report of Costilow and Huppert (4E), and the fundamental treatise of Ernmons, Peareon, and Grant ( 6 E ) on compressor surge and stall propagation. Of particular interest t o chemical engineering practice is the comprehensive and concise paper of Iversen (8E) on the performance of the periphery pump-that is, a machine which pumps by the direct action of shear due t o wall friction of a surface moving in a channel containing fluid. Also of practical value is the experimental technique for observing discharge flow from turbine-type mixing inipellers described by Sachs and Rushton (fJE). BOUNDARY LAYERS
Of major importance in this field is the appearance of the English edition of Schlichting’s “Boundary Layer Theory” (10A) discussed in the first section of this review. Although this book covers the entire field of fluid mechanics, its view is that obtained through the study of boundary layers. Also of general intercst is a discussion of Kaplun ( 1 1 F ) about the role of coordinate systems in boundary layer theory. Laminar. Only a few papers have been published on inronipressible laminar boundary layers. Wiegliardt (31F)presents a simple method for calculating laminar boundary layers based on momentum and energy equations. Shapiro, Siegcl, and Kline ( B S F ) compare their measurements of friction in the entry region of a smooth tube with predictions from laminar boundary layer theory. Toong and Kaye ( d 6 F ) analyze the same problem with a compressible flow; numerical results are obtained from laminar boundary layer theory. For secondary flow within laminar boundary layers, Wilkinson ( S B F ) and Loos (I@‘) present analyses, which differ only in thc assumed curvature of streamlines in the main flow field. Crabtree (6F)extends this type of analysis to the compressible laminar boundary layer. He shows that a sniall interaction term must be taken into account and, conscquently, separation into cordwise and spanwise components is not strictly justified in the case of compressible flow. .4s in previous years, the major effort in this field of study has been devoted to the case of compressible flow, with allowance for temperature gradients associated with supersonic flows Meksyn (178’)integrates the boundary layer equation8 for a plane wall with specified temperature distribution a t its surface. Flugge-Lotz and Johnson ( 8 F ) study the layer along a curved insulated surface. Finally, Morduchow and Grape ( 1 8 F ) present a detailed analysis, including numerical examplcs, of the Beparation, stability, and other properties of a layer involving both pressure gradient and heat transfer. Turbulent. Several valuable contributions have been made t o our knowledge of turbulence in a boundary layer. Townsend (27F) discusses the nature of the turbulent boundary layer as an intermediate between channel flow and free turbulence. He considers that in the thin layer, close to the wall, conditions of universal similarity exist, The outer part is intermittently turbulent and bears close similarity to wake flow. Interaction between the two portions is limited to the energy exchange; the inner layer must absorb the energy arising from the entrainment processes of the outer layer. Consequences of this assumption of relative independence are investigated and compared succe8.sfully with experimental shear stress distributions. Klebanoff ( 1 9 F ) presents hot-wire anemometer data for the boundary layer on a smooth flat plate. I n general agreement with Townsend’s suggestion, these data show that approximately 85% of the total energy dissipation occurs in the laminar sublayer near the wall. The turbulence data indicate quite c l e d y the intermittency of the edge of the boundary layer; in fact, they provided some of the background for the discussion of such fluctuation presented by Schubauer in a paper reviewed last year [(16E), 1955 Fluid Dynamics Review].
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In two reports Sandborn and Slogar present even more detailed data concerning the turbulence structure in a boundary layer. Their studies refer to an adverse pressure gradient caused by an expanding cross section. However, all measurements are made a t pointa upstream of the separation of the boundary layer. The first report ( B I F ) is primarily concerned with a comparison of terms in the turbulent stress tensor and in the momentum equation. Final results generally conform closely to the usual hypotheses of boundary layer theory. The second report ( 8 g F ) presents detailed discussion of the spectrum of turbulence and its variation along the turbulent boundary layer. A somewhat similar case is treated by Uram (&gF)--naniely, the axi-symmetric boundary layer in the expanding aection of a flow nozzle. However, only static pressure diatributions and velocity profiles are obtained. This ia sufficient to demonstrate the validity of applying two-dimensional boundary layer theory to the axi-symmetric case. Tults ( 2 8 F ) reports experiments with two-dimensional flow in an expanding channel of rectangular cross section. Data on pressure profiles provide information on the mechanism of flow separation. The author also presents a correlation for optimum divergence at a given rate of expansion. H. U. Eckert ( 6 F ) presents a simplified analysis of turbulent boundary layer development on a long cylindrical surface placed with the axis parallel to the stream; this represcxits an extension of the work of Batchelor [(f G ) , 1955 Fluid Dynamics Review J to the case of turbulent boundary layer. Sincc compressible flow i.s considered, an extension to high blnch numbers is also made. At several instances empirical relationships are used to permit application of a simple two-dimensional incompressible annlysk to the case under considcration. Unsteady Motion. Two papers desrribe the growth of 8 boundary layer on a surface starting a t rest and accelerating. TlTatson (SUF) considers the growth on a cylinder whose velocity has a simple power-law dependence on time. The method use8 the momentum equation to provide burcessive approximations to the true solution. Stewartson ( 2 5 F ) discusses the motion of a flat plate a t high speed in a tiscous compressible fluid when the plate is given an impulsive start. Related publications dealing more directly with the transient nature of flow are discmeed in the section on “Transient and Pulsating Flow.” Shock Interactions. A second paper by Stewartson (&F) analyzes the interaction betwem the bow shock wave, the mean motion behind it, and the boundary layer on a flat plate in steady motion a t high speed. He finds that the boundary layer equations may be reduced to those for an inconipressible fluid. The nature of flow in the inviscid layer between the boundary layer and the shock wave is discussed in some detail. A somewhat different interaction between shock waves and boundary layers is comidered in three other papers. Morkovin (19F) studies the behavior of a typical boundary layer upstream and downstream of a disturbance caused by a shock wave generated by a wedge in the main stream. Velocity profiles, temperature, and velocity fluctuations are reported. Morkovin and Bradfield (BOP) also describe the probe interference observed in these measurement8 in a supersonic boundary layer. Gadd and others ( 9 F , IOF) publish the results of their experimental work concerned with boundary layer-shock wave interaction. FinaIIy, Bogdonoff ( S F ) studies, by means of wall pressure rneasurementfl, both the effect of an impinging shock upon a turbulent boundary layer and the development of a shock due to a sharp deflection of the turbulent boundary layer by the wall. Injection, Suction, and Mass Transfer. The effect of injection for transpiration cooling on the boundary layer is considered by Low (16F)for a compressible laminar case and by Clarke, Menkes, and Libby ( 4 F ) for the incompressible turbulent cme. The same problem is studied experimentally by Berger ( I F ) who injected carbon dioxide through a sintered bronze plate into a turbulent boundary layer.
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FLUID DYNAMICS The opposite problem, that of suction used to stabilize laminar boundary layers and prevent separation, is considered by Eujen ("9 by Lachmann ( I S F ) , and by Maeder and Carroll ( 1 6 F ) . The latter is of practical interest since the authors describc the process in terms of finite perforations instead of uniform sniall scale porosity. The problem of injection or suction is closely related to that of mass transfer, differing only in the n a t i m of the boundary condition and its dependence on the distance downstream. Bcrnian ( 2 F ) presents a review of laminar diffusion boundary layer thcorg. The discussion is generally correct. Hoverer, no modern references from chemical engineering literature are reported; in fact, detailed discussions of most of the material can be found in several chemical engineering textbooks. JETS
Details of Jet Mixing. Of major importance is the publication by Alexander and coworkers ( d G ) of a thorough report on the thcory and experiments concerning a ducted jet. Experimentally the work is an extension of that reported last year [ ( I J ) , 1955 Fluid Dynamics Revie%*]. I n the present work a heated jet was used to permit study of heat transfer as well as momenturn transfer. Theoretical equations are developed for momentum and energy transfer, assuming Reichardt'a hypothesis. For the case of a ducted jet, the Illinois norkers suggest the following form for the Rcichardt coefficient:
ITere
r
and x are the radial and axial coordinates, and u and
u the corresponding velocity coniponents; p the fluid density;
D the duct diameter; and b and c are empirical coefficicnts to be evaluated from the data; c is the usual spreading coefficient for a free jet, while b expresses the suppression of jet tmhulence by the flow through the duct. Unfortunately, observable data are not very strongly influenced by the value of b; in fact, the authors find that their data are satisfactorily correlated by assuming 6 is equal t o zero, thus reducing Equation 2 to that of a free jet. More precise measurements made further downstream might have demonstrated a more pronounced effect. An experimental study of ducted jets is reported by Tani and Koharhi ( I I G ) . Distributions of mean velocity and flurtuations are ohon- for several values of the ratio of jet velocity to secondary stream velocity. Torda, Thompson, and Gcnetti ( I S G ) consider the effect of nonuniform velocity profiles in the jet and the Furrounding fluid on the mixing of the two fluids. The solution is derived for the laminar case alonc, and also U S C . ~simplified equations of motion which neglert pressure variations entirely. The analysis is based upon assumed velocity profiles for varioiis regions of the jets. The constants for the assumed profiles are then evaluated from momentum balances for various regions. Chapman and Korst ( 4 G ) study the same problem wibh respect to a turbulent jet,. Their treatment is more thorough; the sohition is obtained from the cquations of motion and mass transfer containing eddy viscosity and eddy diff usivity terms. Following I'ai's suggestion, they assume that eddy viscosity is a power function of the distance, x, downstream from the jet. From their experimental data, they find that 2 9 . 7 should be used. The experimental apparatus permitted testa vith a variety of different velocity profiles in a jet. The exponent on a power-law dependenre between velocity and the distanre from the wall varied from 0.11 to 0.4. Using the same general treatment, whirh he had previously suggested, ia€' (DG) now studies the case of mass transfer in a turbulent jet. I n this particular case he aasiimes, a priori, that the eddy viscosity is proportional to the distance downstream. Thus a complete solution is obtained without any exMarch 1956
perimental data. Unfortunately, no experimental cxmfirmation of his results is given. An experimental study of momentum and mass transfer in a submerged water jet is reported by Forstall and Gaylord (SG). Both velocity profiles and salt concentration data are well correlated by error curves. Results for various runs are reported in terms of a turbulent Schmidt number shown to be the square of the ratio of thc spreading radii for momentum and mass transfer. These Schmidt numbers are somewhat higher than corresponding Schmidt and Prandtl numbers previously reported for gas systems. Howcver, all of the values are less than unity, thufi indicating a more rapid transfer of mass than of momentnm, Knon-ledge of free jet expansion has been extended by the study of Anderson and Johns (SC), who determined velocity, pressure, and temperature profiles downstream of a supersonic jet exhausting into quiet air. At points downstream of the supersonic cone, they find that the velocity profiles and spreading charzcteristics follow those of subsonic jets. Data on the length of the suprrsonic cone are well correlated in terms of dirnensionless length us. Mach number. Ejector Design. A greatly detailed discussion of onc-diniensional ejector theory is given by Szczcnionski (IOG). Although the subject is not too complex, the author prolongs and complirates discussion by exccwive attention to algebra and insistence on deriving everything from first principles. I n fact, the first 4 or .5 pagcs present derivations of the equations of motion of a fluid and consider certain examples which havc nothing to do nith the problem a t hand. Even more serious is the fact that essentially nothing has been added to the previous analyses of Flugel [Z. Ver. deut. Ing. 83, 1065 -9(1939)l and of Keenan, Neumann, and 1,ustwerk [J. A p p l . Mechanics 17, 299 (1950)]. In fact, Szrzeniowski's analysis is more limited, since it assumes inronipressihle flow. A second paper on the same subject also ignores the earlier mork by Flugel and Keenan and coworkers. I-IOwever, since it reports experimental work, Acharya's paper ( I G ) on the design of a cylindrical ejector is of some value. velocity profiles, static pressure variations, and total momenta are reported along a constant area mixing duct. The author finds that the data are well correlated by a simple momentum balance with some allowance for skin friction along the duct. Nonsymmetric and Baffled Jets. The use of D vertical air jet as a mind screen is considercd theoretically by Taylor (IgG). The shape of the axis of the jet is computed m d found to provide a sheltered area to the rear. Possible practical deviations from the theory are discussed. A more detailed experimental study of a cross-flow jet is reported by D c m y and Vick (5G). They report flow characteristics, pressure distributions, and thrust coefficicnts for rectangular and ckcular outlets a t various angles n ith respect to the main stream flow. Tests were made a t main stream Mach numbers varying from 0.7 to 1.3. Gough ( 7 G ) reports a series of experiments intended to improve basic knowledge for the dcsign of gas burners. Particular attention is devoted to the impingement of the jet on a flat plate. The sharp break to turbulent flow in the boundary layer is shovn quite clearly by the concentration profiles of the entrained gas. A notc by Horn and Thring (8G) presents a t,echnique which should be useful in model studies irivolving jets of different density in the surrounding fluid. These authors were particularly interested in the case of flame jets in the surrounding combustion air. An inverted picture of this was obtained by observing a jet of magnetite slurry in water. A brief discussion of the validity of the model is given. DUCTED
FLOW
Pipe Flow. The fundamental problem of longitudinal dispersion of matter introduced as a pulse into fluid flowing down a pipe waa discussed by Taylor in a series of papers reported in last
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FUNDAMENTALS REVIEW year’s Review of Fluid Dynamics (@I-611). In a subscquent paper ( 2 2 H ) , he suniniarizes thc results, discusses their experimental validity, and describes the problem of stability of solutions in vertical tubes when the density increases upnard. On this baRis, he suggests a simple method for measuring diffusion coefficients. Sandborn (2011) reports on experimental evaluation of the mean- and turbulent-momentum terms in fully developed turbulent pipe flow. For this purpose both total- and staticpressure data as well as constant current and constant temperature hot-wire anemometer measurements were made. Mutual agreement between rcsults obtained by these varioiis techniques is demonstrated. The author concludes that the terms appearing in the radial direction momentinn equation are as large as or larger than the terms of the equation for longitudinal direction. Two other experimental papers in the general field of pipe flow characteristics are worth noting. One, by Ferrell, Richardson, and Beatty ( S H ) , describes a dye displacement technique for measurement of velocity distribution; and the other, by K a y , Brown, and others (1Sli),reports experimental velocity profiles for supersonic flow of air in a tube with and without heat transfer Concerned with pressure drop are the papers of Knapp and Metzger (14 H ) on graphical representation of the frictional losses, in commercial pipe. of air and steam flowing turbulently a t low pressure, and of Bohnet and Stinson ( 4 H ) on Fanning friction factors for air flow a t low absolute pressures in cylindrical pipcs. Of particular significance is the report of Weltniann (25EI) on an analytical evaluation of pressure losscs for non-Newtonian flow in pipelines, derived from the rate of shear us. shearing st rcss curves. The results are summarized in form of a generahed friction diagram, describing the flow characteristics of Ne\+toriian and non-Newtonian materials in pipelines. Flow in ducks with noncircular but constant area cross section is treated in a number of publications, such as that of Nuttall ( I 7 H ) on the evaluation of the flow rate of a viscous incompressible fluid in uniform triangular ducts; those of Deissler and Taylor ( B H ) , Woods ( 2 7 H ) ,and Rothfus and others ( 1 9 H ) on various aspects of annular flow; and that of Eckert and Irvine (7’81) on the flow in corners of passages nith noneircular cross sections. The secondary flow due to pipe cwvature is analyzed by Cuniing ( 5 H ) on the basis of Navier-Stokes equations. Flow through ducts with porous walls is treated by Eckert, Diaguila, and Donoughe ( 8 H ) in a report on “Experiments on Turbulent Flow through Channels Having Porous Rough Surfaces with or without Air Injection”; by Yuan and Finkelstein (28H) in an analysis of laminar pipe flow with injection and suction through a porous wall; and by Wcissberg and Bernian (24ZI)in an analytical and experimental study on velocity and pressure distributions in turbulent pipe flow M ith uniform wall suction. Channel Flow. Fundamental contributions in this field Rere on the flow of nratcr offered by Rinnie, Davies, and Orkney (3”) from a reservoir through an open horizontal channel; by Ananian (111) on equations of turbulent flov a t the bend of a water d11ct; and by Tong ( 2 S H ) on the analysis of tno-dimensional potential flow in a gravitational field n ith a known free streamline. Ducts with Variable Cross Section. A fascinating problem of an analytical evaluation of flowmeter coefficients is treated in two publications. Siinnions (21H ) determines anal) tically the discharge coefficient2 of flow nozzles by integrating the laminar incompressible momentum equation for an assunicd velocity profile in the boundary layer (a hyperbolic tangent of an angle proportional to distance from the wall). He then notes that the rcsulting ratio of the thickness of the bouiidary layer to the nozzle radius is virtually independent of the assumed velocity profile and of the point of start of the boundary layer. The results agree with experimental data nith an accuracy better than 0.5% for Reynolds numbers between 104 and 106. Hutton ( I O H ) 642
describes the evaluation of coefficients for Venturi mcters and their dependence on wall roughness. He considers two types of effccts: ( 1) “external”-caused by upstream flow conditions, and installation, and (2) “internal”-depending on the geometry and nall surface condition of the meter. The results of analysis concerned with effects of pipe roughness are shown to be in good agreement with values recommended by British code, demonstrating that the latter correspond t o roughness characteristic of cast iron. The author points out that the theory may bc useful for extrapolating code values t o other wall roughness conditions and pipe sizes. Flow in clastic tubes, of fundamental interest to biophysics, attracted a lot of attention. Papers noted by the reviewers range from considerations of steady flow by Rlarchetti ( I681) t o wave propagation by Jacobs ( 1 1 H ) and Morgan and Kiely ( 1 6 H ) , oscillatory motion by Womersley (ZBH), and the effcct of Burrounding fluid on pressure waves propagated through a liquid contained in an elastic tube by Junger ( I Z H ) . Texts. Worth noting is the publication of the sccond edition (after 20 years) of Richter’s handbook on “Hydraulics of Pipe Flow” (in German) ( 1 8 H ) . After a thorough introduction of basic concepts concerned with fluid properties, fundamental principles, and similitude conditions, the salient theoretical as well as experimental aspects of ducted flow are systematically described. The book contains a wealth of numerical examples and critical discussions of experimental pipe flow formulas. Published last year in the United States under the intriguing title “Flow and Fan” is the text of Berry (ZEI) on the design and perforrriance of ventilation equipment. I t contains a thorough discussion of fluid dynamic aspects of the subject starting from basic principles and leading to problems of control and matching of fanswith the duct system. The analysis is restricted to steadyflow characteristics but, within this scope, covers practically all the problems of interest to the designer. FLOW AROUND SOLIDS
Single Bodies. The exact equations of motion can be solved numerically in any given geometry for laminar flow of an incompressible viscous fluid. Allen and Southwell ( 1 J ) obtained this solution by relaxation method for flow past a fixed circular cylinder a t Reynolds numbers of 0, 1, 10, 100, and 1000. Of special interest is the manner in n hich the trailing eddy appears a t the higher Heynolds numbers. Since the solutions are all steady state, the effects of turbulence of course do not appear. However, the behavior of the numerical operations showed that the flow was becoming more and more unstable a t higher Iteynolds numbers. Calculated drag coefficients arc in good agreement with experimental values for Re = 0, 1, 10, and 100. The value a t Re = 1OOOwas about 1.5- to 2-fold high, In other nords, the characteristic dip in the curve due to the start of turbulence does not appear. The relaxation techniques are well described; the authors point out that solution for cylinders of other cross section and other Beynolds numbers would be relatively simple Dean ( S J )obtains an approvimate solution for the motion of a viscous liquid past a parabolic cylinder in two-dimerisiorial flow. In the limit, this case corresponds to flow past a flat plate. The rcsulting viscous layer ai ound the cylinder s h o w similarities to the usual boundary layer picture. Experimental drag coefficients for skin friction on a parabolic body of revolution are reported by Mottard and Loposer ( 9 J ) . A photograph of the model is shown on p. ,561. The skin friction is obtaincd through a momentum balance on the boundary layer, using three fins a i t h pressure taps to determine the velocity profile near the tail of the parabolic shape. The data were found to be in good agreement with skin friction coefficients obtained from flat-plate skin friction data. Drag on a cone in rarefied supersonic air flow is studied experimentally by Ipsen (62). The results were in poor agreement
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with the first-order boundary laler analysis but in fairly good agreement Kith an analysis including the influence of transversc curvature. The effect due t o slip associated with the relatively large mean free path in the gas was unexpectedly small. Reichardt ( 1 1 J ) describcs a uniquc experimental setup lor obtaining straight-line Couettc flow a t small Reynolds numbers between two parallel walls moving in opposite directions. Within this flow, he studied effects of the insertion of onc or t n o cylinders at different distances from the moving walls. The flow about blunt-nosed bodies is successfully pictured by Breslin ( 2 J )in terms of the superposed steady f l o ~and a plane L‘sourcelamina” perpeiidirular to the flow. Both an infinite flow field and that betneen t\\o parallel walls is considered. The wall effcct is found to be quite signifirant; the authois narn that experimental facilities should be designcd u ith this in mind. Wakes. Three papers by Roshko concern the flow around a blunt body and the resulting wakes. The first ( 1 S J ) presents a new hodograph for free streamline calculations. For flow past a flat plate normal to the stream, this allon s arbitrary separation velocity and base pressure. Thus the calculations can bc fitted more closely t o evpcriinental results. The second paper ( 1 4 J ) describes esperinicntal studies of drag and shedding frequency of two-dimensional bodies. Correlation is obtained betn cen a universal Strouhal number m d a universal Reynolds number. When combined 5%ith the frce streaniline theory discuwed above, this allo\Ts drag to be calculated from the measured shedding frequency. The third report ( f 2 J ) considers the nature of the turbulent wake generated by a vortex street. Hot-wire anemometer measurements R ere made in the wakes behind ciicular cylinders a t Reynolds numbers of 40 t o 10,000. The manner i n m hich the regular vortex pattern digintegrates into a random turbulence is clearly presented. Woods ( S O J ) also considers t u o-dimensional flou past a curved obstacle. In a somewhat diffeient manner from Itoshko, he modific,s Helmholtz’ thcory to eliminate the arbitrary nature of the free streamline IIe shons that this permits a very closc aqeement u i t h Fage and E’alkner’s data on the flow about a circular cylinder. Both the pressure profile and the separation point arc very closely predicted. T1700ds’ throry also permits an evaluntion of compressibility effects, Increase in Mach numbcr rednces the pressure at the equator soinm hat more severely than that, predicted by incompressible flow theory. A related problem is cavitation, since it often results i n flow separation and thr starl of the nake. Rerniecn, hlcGran-, and Parkin ( 8 J )report a study of the mechanism of cavitation inception. They find that it only occurs after a significant tension is achieved in the liquid. Thns the usual procedure of designing t o ensure no pressure points below the vapor pressure of the liquid is a conservative one. Kaniimoto and Matsuoka ( 7 J ) report specific cavitation studies a t Reynolds numbers of 80,000 to 200,000 using a 25-nim. cylinder. At small values of the cavitation number, cavitation occurred only in a locali7ed region, but a t larger values the flow changed rather abruptly with cavitation extending right up to thc rear end of the cylinder. They also describe the effect of the increase in free stream turbulence produced by a grid upstream of the cylinder. Rotating Bodies. Tifford and Chu (18J)report on an exact solution of the Navier-Stokes equations for the flow and temperature field in the neighborhood of a rotating disk. Because of its evact nature it is an improvement on the approximate analysis of Schlichting and Truckenbrodt [(ZZG), 1955 Fluid Dynamics Review]. Plots of the three velocity components in
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the neighborhood of the surface are given for the various ratios of the approach velocity to peripheral speed. Nigam ( I N ) and Fadnis ( 4 J ) discuss the flow of a fluid layer on a rotating sphcrical and spheroidal surface. These notes bring in improvements and extensions to the original ~ o r kof Howarth [PhzZ.Mag., Scr. 7 , 42, 1308 (1951)]. Solid Arrays. The form drag of plane surfaces of varying solidity made up of combinations of disks, rings, and plates provides an interesting and significant objective. Rouse and Siao ( 1 5 J )present a brief discussion of the free streamline thcory of flow past multiple plates, and then report on several experiments dealing with various combinations of rings, disks, and rectangular plates. The drag coefficient (defined in terms of the solid area) is a monotonic function of the solidity (defined as the ratio of solid area t o the total area of the zrrayj. More specific studies of grids of parallel rods are reported by Sato ( 1 6 J )and by Tamaki and Oshima ( 1 7 J ) . The first is concerned with unheated grids and the second, with heated rods. In both cases hot-nire anemometer studies are made of the wakes behind the wire. The obvious purpose of this study is to consider the generation of turbulence as normally produced in modern wind tunnels Grootenhuis ( 5 4 has measured the pressure drop through various assemblies of wire gauzes: a single laycr, multiple packed layers, and layers sintered into a solid mass. These data were successfully correlated with those obtained from litrrature in form of a resistance coefficient-Reynolds number curve. In the discussion Roylance claims that Grootenhuis has used an incorrect expression t o calculate the total surface area. iZlthough Grootenhuis does not agree with this suggestion, Roylance shows the data plotted on his basis give a resistance curve nioie nearly comparable with that for a solid cylinder. Tn o articles discufis the problem of f l o ~through packed beds of solid particles. Wyllie and Gregory ( Q I J )report a thorough study of liquid flow through packed beds of spheres, tubes, cylinders, triangular prisms, and disks iangiiig in Biz? from 28 microns t o 6 mm. Data are correl.zted i n terms of the Ko7en)ever, they find the “constant” in the Carman equation 110~1 equation is really a function of porosity and shape of the particles. Curves are given which should be valuable for using the Kozeny-Caiman equation in predi t i n g the pressure drop for other packed beds. I n the second article Wagstaff :tiid Kirmaier (19J)present data on pressure drop for air flow through bcds of packcd wooden and plastic particles in the range of 1/10 to l/, inch. Thry find that Erguu’s equation yields the best correlation when the particle sui-facc area and density are cvaluated from pressure drop data MULTIPHASE FLOW
Liquid Dispersion. The monograph by Marshall (26Kj on “Atomization and Spray Drying” is a comprehensive and thorough revien. of the science and art of producing liquid dispersions, and of using them in spray dryers t o produce powdered solids. Excellmt illustrations are provided in the form of line drawings, plots, still photographs, and motion picture sequences. The monograph begins with an exposition of the principles of jet breakup. Then, in turn, the various types of atomizers are considered: first, the centrifugal or swirl type, then the spinning disk and pneumatic nozzles, and, finally, a variety of miscellaneous atomizers. Following this, the drop s h e distribution is considered, various distribution functions are described, and actual distributions are given for several types of atomizers. The
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FUNDAMENTALS REVIEW drop size. The distribution plots were correlated by means of the uppcr limit equation of ENG.CHEM.43, 13-17 Mugele and Evans [IND. (1951)I. Even though the Nukiyama-Tanasawa equation provides a good empirical fit to the data, the agreement is probably fortuitous. The equation is derived for a region where breakup occurs by distortion of drops or filaments, while in Bitron’s experiments it results from the stripping off of liquid layers from the surface of the large drops as they reach the supersonic jet. These data provide a valuable guide for theoretical studies of such “boundary-layer stripping.” TWO articles discuss the performance of sperific types of atomizers. Radcliffe ($OK) discusses the normal t a n g e n t i a l - i n l c t s w i r l atomizer. Most of his data concern flowpressure relationships and spray shape, but a few drop size data and an empirical equation for Sauter mean diameter are also included. Valuable contributions are made by Carlisle, Knight, and others in a rather lengthy discussion of Radcliffe’s work. Herring and Marshall ( 1 8 K ) present an excellent discussion of a vaned disk atomieer. The emphasis i s on size distribution and average diameter of the drops produced. Extensive data, correlations, and photographs of the atomization mechanism are included. Two publications of the Imperial Collcge of Science and Technology in London review the use of photographic methods in fluid mechanics and related research fields. Much of the work reportcd has t o do with atomization. Fraser ( I O K ) presents a number of photographs of Figure 3. High-speed photographs of disruption of iet of viscous high-speed flow and atomization. Some of them fluid in flight ( 7 OK) were previously published, and one was presented in last year’s review (Figure 4, 1955 Fluid Dynamics Review). A particularly interesting section on spray drying (about one third of the monograph) series (Figure 3) illustrates the disruption of a jet of viscoua contains: the general principles of evaporation from drops and liquid in fiight. The top photograph shows the head of the jet with its characteristic knotty form; in the remaining sprays, 1he air flow in spray drier vessels, and general design and pictures, the general trajectory is maintained even though a performance considerations. Marshall has performed an important task in produring this monograph, but a few drawbacks large numbcr of lumps, twists, and knots are formed. Coulson are noticeable. First, review of work from other sources is and Dombrowslci ( 6 K ) present photographic work morc closely not so complete as that of Marshall and his coworkers. This is related to chemical engiueering. Figure 6 of last year’s review is again included in t,his article. Figure 4 is a good example of probably natural and is not serious since Marshall and his their use of a relatively long duration flash t o show not only the fellow workers have contributed a great deal in the recent progsize of drops but also their trajectory. ress in this field; however, in some instances, other articles are After a long period of neglect, several scientific studies of the not clearly explained. A reader particularly concerned with some mechanism of entrainment of droplets from an aerated liquid special subject would do well to look up the original references have been puhlishcd. Garner, Ellis, and Lacey ( I d k ‘ ) report cited by Marshall. Some typographical errors were noted in the equations. On the whole this is an extremely worth-while both pilot plant studies of evaporator operation and laboratory project. It is hoped that Marshall will be able to extend his studies of the droplets formed by bursting of individual bubbles. I n the former, both a double effect evaporator with 12-inch vessels discussions, correct a few of thew minor difficulties, and produce and a 4inch single effect evaporator were studied. Data oba second edition in a more permanent form. I n a very interesting note, Baatard (1R)considers the breakup tained include size distributions and ent,rainmcnt rates for the of a turbulent jet. He shows that the spreading of the jet prior spray leaving the evaporating surface and the miat which is still entrained after passing through the exit vapor line. The addition to breakup provides an experimental verification of Heisenberg’s of potassium nitrate, which resulted in a layer of foam several equation for the spectrum of turbulence in the flow within the nozzle. inches deep upon the liquid surface, suppressed the entrainment A few important experiments are dcscribed by 13itron ( S K ) . of spray. The layer of foam apparently acted as a viscous blanHe has measured the fine drops produced when large drops of ket, eliminating some of the splash. dibutylphthalate are allowed t o fall into a supersonic air jet. The authors’ study of droplets formed by individual bubbles Experiments were mado a t five velocities between 460 and 680 involved air bubblcs from 1t o 12 mm. in diameter rising through water, methanol, and benzene. For most bubbles the drop size meters per second. Gauter mean diameters for the drop distribudistribution is binodal. A large number of fine droplets 10 to tions ranged from 5.7 to 7.2 microns. These wcre reasonably well predicted by the Nukiyama-Tanasawa equation for mean 20 microns in size are formed when the bubble dome ruptures 644
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at the surface and the gas in the bubble escapes A smaller number of much larger drops is formed from the jet which rises from the surface as the bubble crater collapses after the burst. As the bubbles increase in size, the sine of these jets and the drops formed from them also increase. At the mme time, the frequency of formation of drops from the jets decreases until evcntually, for bubbles larger than about 6 or 7 mm. in diameter, practically no large drops are formed. As might be expected, the total entrainment, expressed as a fraction of the gas flow, is largest for small bubbles and decreases sharply in the range of 4 to 6 nim., leveling off a t about 2 to 5% for the larger bubbles. This phenomenon of bubble breakup was studied in even more detail by Kcwitt, Dombrowslci, and Knelman ( 8 7 K ) . -4 preliminary note describing their results was reviewed last year [ ( M K ) , 1965 Flmd Dynamics Review]. Although made independently, their observations are in cxccllent agrcernent with those of Garner and coworkers. The same binodal distribution and the same effects of‘ bubble diameter are observed. These authors also develop a correlation for the characteristics of the rising jet and show that existing theories of jet breakup prcdict well the observed drop sizes. The large drops formed from the jet often fall back into the liquid; from available drag coefficients, Newitt and his coworkers predict a maximum height of fall and show that reasonably good agreement with observation is obtained. A second part of Newitt’s article conccrns the formation of drops from liquid collected on baffles. The experiments were made with vertically hanging baflles having eitber square edges or beveled edges with 90’ or 45” apex angles. Drop diameters formed on these baffles were evaluated as a function of baffle thickness. The drop diameter decreases toward a sharp niinimuin a t about 0.2-em. thickness, then increases more gradually, approaching an mymptotic value at thicknesses of about 1.5 to 2 em. For water, the minimum drop diameter is in the neighborhood of 4 mm. and the large. asymptotic value is about 8 mm. Drop formation from baffles was also observed wii h air flow past the baffle. Drop diameters, angular deflections from the vertical, and average velocities were determined for sir velocities ranging from 800 t o 2340 cm. per second. The third article by Gleim ( 1 6 K ) presents similar data on the breakup of individual bubbles. Heights of rise of large dropa formed from the jet following the bubble burst are reported for bubbles ranging from 2 to 7 mm. in diameter. The height decreascs steadily irom about 18 cm. for the 2-mm. bubbles to practically eero for the 7-mm. bubbles. Gleim also gives extrnsive data on the wetncss of steam produced in boiling various solutions and suspenwions in an ordinary lahorntory flask. He tiiida that the maximum entrainment occurs at an i n k m e d i a t e value of boil-up rate. Like Garner and others (12li) he observes that foam tends to suppress the entrainment by blanketing the bubble-bursting effects. Drop Motion in a Gas. Scvcral inqlortant contributions have been made to our knowledge of the detailed motion of drops in a gas. In a short note, Pllagono ( 2 5 K ) presents data on the fall of water drops in air. Verv dear photographs of the shape
Figure 4.
March 1956
of the droplcte are given. In the discussion of this note, Blanchard points out that it is very possible that Magono’s drops did not achieve terminal velocity. This is important for the evaluation of drag coefficients from the data, but is not particularly significant with regard t o the shape of the drop. Using Magono’s data, McDonald ( 2 J K ) prewnts a n excellent analysis of the distortion of drops. From the profile of the drops as prescnted in Magono’s pictures, McDonald calculates thr pressure profile on the outside of tho drop. Not only are tE.? resulting curves in good qualitative agrecment with aerodynamic theory, but they also can be integr.bted to give the total drag. For the four cases considered, the drag is from 15% low to 2% high compared with the weight of the drop. This agrees well with Blanchard’s cxpectation that Magono’s drop8 had not achieved terminal velocity. The four drops studied range in size from 0.8 t o 6.5 nun.,with Reynolds numbers from 1000 t o 4000. A few more observations on drop motion and size distribution are reported by MacPhail (84K)in connection with research on aeronautics in Arctic regions. TIis concern with drops is, of course, associated with the problem of aircraft icing. Figure 5 taken from this article is a rather good illustration of the internal circulation observed for drops moving a t low Reynolds numbers without excessive surface rigidity. Two National Advimry Committee for Aeronautics reports discuss drop dynamics in particular situations. Reed ( 3 1 K ) has made a very interesting analytical study of the effect of airplane wake and especially of trailing vortex oii lateral diepersion of aerial sprays. The obvious application is t o agricultural spraying. Sernfini (S4K) studies the impingement of water droplets on wedges at supersonic speeds. The usual collection efficiencies me plotted but, in addition t o the normal variables, the free stream Mach number is considered. Both of the above papers are fundamentally sound with one exception: no mention is made of the fact that drag coefficients are assumed to be the same as those for steady-state flow. In other words, the effect of accelcration on drag is neglected. This probably has a minor influence on the calculated results, but its complete omission leaves the point in doubt. Another article by Ryley (S2K) also dcals with drop motion in a particular Bituntion: the behavior of wet steam. His observations are interesting, but the article is mainly descriptive and does not lead to quantitative conclusions. Two articles by Gillcspie and Rideal ( 1 6 K ) consider the adhesion of drops and particles upon impact a t solid surfaces. .4lthough this has a n obvious importance for any type of impingement separator, it has not been studied thoroughly bcfore. The first article presents results for impingenwnt on various wires and other cylinders The second concerns impingement of drops on surfaces a t vaiious angles of contact. Both articles present valuable experimental data and analyses of results. Another article by Gillespie ( 1 4 K ) considers specifically tho role of electric forces in the filtration of acro.sols.
Formation and distribution of drops from a spinning bowl (6K)
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Figure 5. Circulation inside a water drop falling through castor oil, as shown by suspended solid particles (24K) Bubbles. A thorough comprehensive treatment of the whole problem of bubble formation and motion is given by Siemes (35K). The article includes excellent analyses of the process, correlation of new data and of data from the lit'erature, and comprehensive discussion of previous articles in the field. In this regard, the only noticeable omissions ar? two recent publications which, admittedly, did not receive wide distribution [(6G), 1954 Fluid Dynamics Review; ( I 7 K ) , 1955 Fluid Dynamics Itevien.]. In part one of his paper, Siemes describes bubble formation a t upward-facing circular orifices. He first constructs a very sound theory of bubble formation a t very low flow rates, nhere inertial and viscous forces are unimportant. IIc then reports extensive measurements, both with and without resonating chambers upstream. A successful correlation is developed for the case of 110 resonating chamber and the effects shown by the rcsonating chambers are clearly esplained. There is some discussion of bubble formation at higher rates than those in wlrich individual bubbles are formed and a consideration of the effects of multiple-tube systems. The second part of Siemes' article is concerned with the mechanics of bubble rise. Here he shows that the shapes of niedium-sized bubbles can be satisfactorily predicted by assiiming an oblate spheroid n-hose distortion is predicted by potential floncquations. The drag on such bubbles can also be predicted directly from the potential f l o ~theory, except for certain data where a von I(&rman vortex trail is observed. IIaberman and Morton [ ( I Y K ) , 1956 Fluid Dynamics Review] show t,hat the effects of surface viscosity can supprcss the circulation normally occurring within a bubble by supporting the tangential stress a t the surface. I n such cases lower velocities and higher drag coeflicients with their attendant vortex shedding are observed. Siemes' suggestion of pobential flow, which is well supported by experimental bubble velocities in pure systems, coincides very well with this picture; whenever the surface is circulating, no opportunity is provided for the build-up of normal boundary layer. Thus potential flow streamlines should provide a good approximation. For large bubbles Siemes presents a theory which is essentially the same as that of Davies and Taylor [Proc. Roy. SOC.200A, 375 (1950)l. I n the region of large bubbles, Ladyzhensky ( 2 2 K ) reports ob646
servations and measurements of air bubbles in water. Bubbles with volumes from 0.15 to 20 cc. were observed. As they became very large, disturbance waves appearcd a t their edges. I n fact, for the largest bubbles, the disturbances became so large that a toroidal bubble separated from the edge of the main bubble. I t is possible that Borne of these effects might have been due to the fact' that the observations were made in a relatively sniitll cylinder 25 cm. in diameter. Spclls ( 3 6 K ) contributes a valuable discussion and analysis of previously reported experiments on bubble formation a t high air rates through vertical slots. His postulated mechanism Eatislactorily explains many detailed experimental observations. In turn, Spells discusses: the resonating cffect in the reservoir I,ehind the slot, the preperiod during which the bubble grows \Ihile actually in contact with the slot, and t.he later period nhere additional gas flows through the neck connecting the slot with the liubble. I n a short note describing his photographic arrangcment, Tuson (S8K)offers a few comments on bubble formation a t a capillary orifice. He finds that even though the bubble period is about 0.1 sccond, the actual bre:tk-ofT occurs in about 50 micruseconds. As a. preliminary in their study of mass transfer to gas bubbles, Quiylty, Johnson, and Harris ( 2 Q K )measured the size of bubbles formed a t single square-edged orifices from 1 tu 10 mm. iri tliameter. Ruhbles ranging in diameter from 8 to 50 mm. were observed i n glycerol-water solutions and carbon tetrachloride. The authors also studied holdup of air bubbles in the same liquids. Although bubble velocities are not reported, sufficient data are provided so that they could be calculated. Since the experiments were made in a square column 2.5 inches on the side, the value of these velocities for general correlation is probably doubtful. Gas-Liquid Flow. Only one new article was noted by the reviewers t,his year -namely, Calvert and W'illiams' ( 4 K ) study of upward cocurrent annular air-water flow. One- to 2-inch tu1)c.s carried water a t rates up to 2000 lb. per hour and air a t rates from 10 to 112 cu. feet per minute, The results are rat,ionalized on the basis of the Prandtl-Nikuradase inking length theory. This yields a plot of film thickness against water rate and interfacial shear. By using both pressure drop and holdup data, it is possible to obtain an experimental curve relating a drag coefficient group with the film thickness. Deviations between experiment and theory are mostly confined to the largc water rates, arid probably are caused by the fact that formation of wl'itvcs eventually leads t o partial dispersion of liquid from the annular film. Liquid-Liquid Systems. I n liquid-liquid systems, the vai.ious dist,ortion and circulation effects eonimon to all drops and bubbles can be very conveniently studied; both interfacial tension and density difference can be reduced to such an extent t,hat these motions can be considered in large, sIowly moving drops. Important contributions have been made to this field by Garner and his colleagues [(16K),1955 Fluid Dynamics Itevien-] . Garner and Skelland ( I I K ) now present a more detailed discussion of various flow transitions: from laminar to eddy flow in the droplet wake, and from stagnancy to circulation within the droplets. The circulation transition constitutes t,he major portion of the article. Interesting experiments are quoted showing the effects of continuous phase viscosity and interracial teiision on the minimum velocity neccssary for circulation. Progressive surface adsorption of impurities and surface active materials can suppress the circulation in a given drop; desorption of these impurities by diffusion of a solute to or from the drop will preserve the circulation. I n a final short section, the authors present a few experiments showing how circulation, distortion, and oscillation vary for different liquids. Hu and Kintner ( 1 9 K ) present experiments and a correlation for velocity of droplets ranging from 1 to 12 nim. in diameter, in a variety of liquid-liquid systems, all using water as a continuous phase. Nearly all data correlate well as drag coefficient us. Reynolds number with deviations from the solid sphere line
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due t o distortion. The single exception is the aniline-water system. This is the only one in which drag coefficients lower than those for solid spheres with the same Heynolds number are observed. Although the authors feel that this can be explained largely by lack of chemical stability of their aniline, it is very possible that the lower interfacial viscosity of the system permits internal circulation. Formation of droplets from jets of organic liquids in water is studied by Keith and Hixson (BOK). The twelve systems studied range in interfacial tenbion from 2 to 40 dynes pcr cm. and in viscosity of the jet liquid from 0.2 to 79 centipoises. Nozzles ranging from 0.7 t o 6.7 mm. were used. Studies concentrated on jet length, drop size, and drop size distribution, but also included some observatious on the effect of solutes, motion of the continuous phase, rise velocities of the drops, and their eccentricity. In all, this article provides a storehouse of valuable data for the study of liquid-liquid systems. Although slightly apart from the normal Geld of fluid mechanics, one more article i s cited because of its importance in the study of liquid-liquid systems. Criddle and Meader ( 7 K ) study the viscosity and elasticity of oil surfaces and oil-water interfaces. A nuniber of careful experiments are described. This sort of sui face effect may control the circulation within drops or bubbles, dcscribed above. Gas-Solid Systems. Kith the usual assumption that steadystate drag coefficients apply to accelerating particles, Gilbert, Davis, and Altnian ( 1 3 K ) analyze the behavior of particles in linearly accelerated combustion gases. i l u application is made to : the floiv of particles through a laminar flame, behavior of particles within the gas inside a rocket chamber, and flow of particles through a rocket nozzle. An interesting contribution to gas-solid fluidization is made by Iihudiakov (21K). IIe reports photographic studies of particle motion in a downflow system, involving air-solid flow in 14- , 20-, and 32-mm. tubes. Average particle sizes ranged from 70 to 845 microns. From the data, drag coefficients for the slip between solids and gas are computed. The authors point out the errors inherent in this computation, due t o the possible nonuniformity of the gas flow. Dussourd arid Shapiro ( 8 K )reviea- their research into the problem of measuring stagnation pressure and velocity of a particleladen gas stream and present a successful design of a probe to accomplish this purpow. Theoretical analysis confirmed by experiments shows the nature of the errors in stagnation pressure and the corrections necessary to obtain a true value. The authors also report some study on the modification of the instrument and experimental technique t o yield particle sizes and ve1ocitie.y as well as the gas stagnation pressure. A less usual problem is reported by Franklin and Johanson ( 9 K ) . From studies on a variety of different solids, in orifices ranging from 0.25 to 2.25 inches, they develop a correlation for the weight rate in terms of orifice and particle diameters, particle density, and angle of repose. A detailed discussion is given of the angle of respose and its dependence on the method of measurement. A n apparatus suitable for standardizcd nieasurcmente is proposed. Liquid-Solid Systems. Threr articles hich recently appeared are concerned with sedimentation. Khitniore ( S 9 K )studied the sedimentation of 100-micron plastic spheres in aqueous solutions of lead nitrate. Settling rates were measured for plastics of two different densities a t volume concentrations as high as 25%. In another experiment, spheres of two differcnt densities were suspended simultaneously in a fluid with a density equal t o that of the lighter plastic. The effects of the suspended (lighter) spheres, in small amounts, reduce the settling vclocity of the falling (heavier) spheres. When the total volume concentration of solids increased, instability occurred, m ith resultant vertical streaming. Gurel, Ward, and Whitmore ( I 7 K ) describe the sedimentation of nonspherical particles about 1.5 to 10 mm. in
March 1956
size. For a variety of different compact shapcs, they find the resistance is approximately proportional to the square root of the surface area of the body. Finally, Cheng and Schachman ( 6 K ) report some centrifuge measurements on very sinall polystyrene latex particles. These studies arc chiefly of interest in that Stokes’ law for settling of spheres was validated for particles of the order of 0.25 micron in size. Three articles deal with the viscosity of liquid-solid suspensions. Orr and I3locker ( d 8 K )present a simple correlation of literature data in the form:
Here 7 = the viscosity of a suspension, 70 the viscosity of the pure liquid, c the volume fraction of the particles, and u and k are characteristic functions of the suspension. The authors show from the data that u is closely related to the reciprocal of the maximum volume concentration possible for the given solid, while k is related to the reciprocal of the geometric standard deviation of the particle size distribution. Starkey (37R) in a similar treatment, develops an equation using a parabolic dependence on the volume concentratioii rather than the power law given in the above equation. Scheraga ( S S K ) calculates the intrinsic viscosity of solutions of ellipsoidal particles b3- machine evaluation of series solutions. The turbulent flow of solid-liquid suspensions is treated by Bhattacharya and Roy (BK). They report flow of suspensions of various solids in water and kerosene a t Reynolds numbers of 2400 to 7600 in glass tubes with internal diameters of 10 and 19 mm. The solids werc designed t o represent tvpical FischerTropsch catalysts. The increase in pressure drop due to the \\ere presence of the solid a t aeight ratios approaching ’Z.i70 successfully correlated by an equ‘ition soniev hat qirnllxr to that of Voyt and White for gas-solid sksterne [INUEYG.C ~ X . 40, 1731 (1948)l. TRANSIENT A N D PULSATING FLOW
Perhaps the most significant feature in modern developnienl of fluid mechanics i s the widespread attention given t o tranrient flow problems. Consequently, publications in this field of study are here grouped in a separate section. Wave Propagation. The study of nave propagation reprcsents, in turn, the most important and interesting aspect of the progress made in the field of transient f l o ~ . Wave phenomena arc describcd mathematically in terms of hyperbolic partial differential equations. These arc uwnlly solved by the method of characteristics which leads essentially to a graphical solution. Comprehensive treatment of such an analysis applied t o unidimensional wave propagation through gases is prcreiitcd by Rudinger in a text entitled “Wave h a grams for Non-steady Flow in Ducts” ( S S L ) . The wave diagram is a description of the process in the tinic-space plane. In the first chapter, construction of such diagrams and the characteristic relations due to Riemann are derived from basic principles. This is followed by discussion of some general flow problems such as isentropic flon s in ducts of constant as ~vellas variable crossscctional area, flow Rith heat addition, and flows with body forces due to gravitational and centrifugal fields. The consideration of boundary conditions is then describcd in detail. The method of characteristics is generalized t o include finif e discontinuities such as shock w&ves, contact surfaces, discontinuous change of the duct cross scction, and flame fronts. Finally, problems concerned with thc interact ion between discontinuities are systematically treated. The book is very well organized, clearly written, and contains an exhaustive list of references. Related to the above is the text of Parmakian on “Waterhammer Analysis” (2QL). The book covers virtually all the practical aspects of the subject and contains a large number of
INDUSTRIAL AND ENGINEERING CHEMISTRY
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FUNDAMENTALS REVIEW detailed solutions of authentic problems. I t will provide, undoubtedly, invaluable help to anyone concerned with the problem of water hammer in any kind of liquid pump-line operation. Some unconventional aspects of wave propagation are described by Lighthill and Whitham (ML), who presrnt a theory of “kinematic waves”-that is, waves whose propagation is governed only by the equation of continuity. The theory is restricted t o a unidimensional treatment of problems v hich can be described in terms of two parametem: the f l o ~ rate passing through a given point, q, and the quantity per unit diiitance or flow concentration, L. The flow rate, q, is a constant for cach kinematic wave, and the wave velocity is given by the partial derivative of q with respect to k. The most interesting feature of the phenomenon is the fact that successive waves inay coalesce, forming “kinematic shock waves.” Thc authors apply these notions to the analysis of flood movement in long rivers !+here hinematic waves play the leading role, although the long, gravity, dynamic waves also have to be considered. In the second part they then derive a theory of traffic flow on long, crowded roads, a classical example of a problem concerned only with kincmstic waves. MirelR (2515) studies the laminar compressible boundary Iaycr induced by a shock wave propagating into a stationary fluid bounded by a wall. He notes that, for w a k naves, the boundary layer is identical with that obtained when an inGnitc wall is impulsively set into uniform motion, known asthe13 ayleigh problem [see (2z?F)];the effects of strong shocks are analyzed by means of a numerical solution. A linearized analysis of pressure qaves generated by addition of heat in a gaseous medium is presented by Chu (BL). Worth noting is his description of the analogies between prcssure waves generated by heat release &nd those obtained by mass release or piston motion, or established in front of a two-dimensional body in a steady supersonic stream. Concerned with the problem of free surface waves is Proudman’s (SOL) treatment of the propagation ol tide and surge in an estuary, and the recent publication of Havelock (151)) who, in pursuance of his studies of free surface wave motion caused by obstacles, presents an analysis of waves due t o a floating sphere making periodic heaving oscillations. Of interest here also are recent publications on the hydraulic analogy to gas dynamics, by Laitone ( d l L )who points out the important limitation on mater depth imposed by the phenomenon of fluctuating group diStiirbance, and by IIarlcinan and Ippen (14.5) who describe the use of the analogy for studies of shock wave diffraction. Spherical Waves. Kumerical analysis of the proriagation and decay of spherical shocks generated by a very strong point ~ o u r c eexplosion &as made recently by Goldstine and von Neumann ( I J L ) by the use of the digital computer in the Princeton Institute of Advanced Study. The medium was considered BB a perfect gas with constant specific heat and the change of state across the nave front was assumed t o obey the KmhineHugoniot relation. The considcration of the latter n as handled by an iterative procedure. The results are given in graphical and tabular form describing in detail the history of the process. As part of their studies on the gcometry of acoiibtics, Friedrichs and Keller (9.5) present a n analysis of the diffraction, reflection, and refraction of a weak spherical or cylindrical shock a t a plane interface. The early development of a spherical blast from a “particular’, explosive (for which the ratio of specific heats equals 3) is analyzed by Berry, Butler, and Kolt (ZL)using numerical integration of equations of motion along characteristics. Time-distance description of the early stages of the process is traced in detail. Two shock wavcs are considered, one propagating in air and the other following it in the gas released by explosion. The results of computation confirm a very interesting behvior of the second blast wave, known from experimental observations namely, that, after the initial outward rnovemcnt,
648
it turns to move back toward the center of the explosion. Latter ( W L )presents an interesting treatment of the similarity solution for a spherical shock wave, allowing for the effects of eddy viscosity ahead of the expanding shock. Shock Tubes. Especially significant in the field is the fast growing popularity of the shock tube technique. The gencral theoretical as well &s e.xperimenta1 aspects of this technique are described by Glass and Patterson (1%). The problem of shock attenuation due to friction and heat transfer is treated by Trimpi and Cohen (361,). They demonstrate that the analysis based on skin friction coefficients of steady-flow turbulent boundary layer yields results in good agreement with experimental data in the range of diaphragm pressure ratios from 4 to 18. A similar conclusion is reached by Mucklow and Wilson (86~5)who report a n experimental investigation of wave attenuation and rationalize their results, obtained in the range of diaphragm pressure ratio 1.35 t o 10.74, by a simple analysis based on the consideration of the Blasius skin friction coefficient. They also report pressure measurements after wave reflection from a closed- and a n open-end of the tube, dernonstrating an excellent agreement with the theory. Dealing with wellknown practical application of wave reflection phenonienanamely, that of scavenging two-stroke engines-Wallace ( S T L ) demonstrates that the use of diffusers is superior t o that of constant area rxhaiist ducts. Dressler ( 7 L ) descrities turbulent flow in shock t u b s of varying cross section, and Jones (18L) the propagation of ahock wave8 in regions of nonuniform density. Freemen ( 8 L )presents a theory of the stability of plane shock waves which are obtained when a corrugated piston is moved impulsively from rest a t R constant velocity. ITis paper has a definite bearing on the shock tube technique, where the rupture of a diaphragm generates a perturbed wave which becomes plane only after it has traveled several shock tube diameters down the tube. The shock tube technique is especially useful for the study of high velocity, high temperature flow, Recent progress maclc in such an application to hypersonic investigation a t the California Institute of Technology is summarized by Nagamatsu ( d T L ) , who describes in particular the increased siipersaturation assoriated with air condensation and the evtreme stability of the laminar boundary layer a t high Mach numbers. Other significant applirations of the technique are for the study of rchxation phenomena in gases as mentioned by I-Ierzfrld in his chapter of Volume 1 of “High Speed Aerodynamics and Jet Propulsion” (9.4 p. 686). Such measurements of vibration relaxation in carbon dioxide and chlorine in the temperature range from 1400’ to 1500” K. are reported by Smilry and Winkler (S4L). Important to chemistry arr the recent shock tube applications to the study of kinetics of chemical reactions reported here in the section on Dynamics of Reactive Fluids. Pulsating Flow. The problem of pulsating flow mrasurement is rcviewed by Oppcnheim and Chilton (28L). They emphaqiae the fundamental aspects of the probleni and the applications to pressure-differential meters. The suhject matter ia classified into that pertaining t o flow through the meter test Aection, the effect of the flow system on the meter, and the transmiwion of signals from the sensing element to the recorder. Application to pulsating flow measurement of other meters such as the turbinc type, the electromagnetic, and the hot-wire anemometer, as well as the so-called “true masvrate flovmeter” is described and a summary of information available iii codes and manuals is included. Kastner and Williams (19L) present a eomprehensive report on the study of viscous flowmeters used for the measurement of pulsating flow with a particular referencc to the air supply of internal combustion engines. Kronauer and Grant (gUL) investigate the response of a pressure probe in fluctuating flow and demonstrate that the equilibrium pressure recorded by conventional probes may deviate from the time average of the imposed fluctuation by as much as 15% of the fluctuation ampli-
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
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FLUID DYNAMICS tude when the probe hole i s short aiid the frequency low. The results of the analysis, although developed for incompressible flow, are confirmed by experiment. An extensive bibliography of some 850 items on dynamic pressure measurement has been compiled by Brombacher and Lashof (4L). Performance analysis of mrge-bottle pulsation absorbers used in reciprocating pump installations is described by Chilton and Handley (61,). The application of direct electrical analogs to such problems is summarized by Isakoff (1%). Problems of surge tank stability are considered by Geza Data (If L ) and Gerber (IOL)and the transmission of water-hammer pressures through surge tanks is investigated experimentally by Zienkiewicz and Hawkins (391,). Concerned with recently developed application of pulsating 9ow to extraction columns is the paper of Williams and Little They describe on the hydrodynamics of pulsed columns (3%). a practical method for the evaluation of pressures developed in a simple system and a determination of the point of cavitation. Bearing on the latter is the paper of Truesdell (36L) on the hydraulic pendulum, where some fundamental theorems are derived for occurrence of cavitation in a liquid-filled column moving under the influence of a conservative force field, inside some generally shaped tube of constant cross-sectional area. Acoustics. The most significant event in this field is the publication of three texts. The books of Reranek (ZL)and of Hueter and Bolt (16L), published iii the United States, stress direct engineering applications, the first dealing with the general aspects of the subject and the second devoted primarily to techniques for the use of sound and ultrasonirs in engineering and science. Both make extensive use of electric analogs. The book of Skudrzyk ( S S L ) , published in Gerniany, is a much more complete treatise covering virtually the whole field. It will undoubtedly become a classical reference on the subject. Of interest is the use of ultrasonic techniques for the study of relaxation phenomena in gases [IIerzfeld (9A)] and liquids [Litovitz and others (%$L)]and the problems of noise generation by jets [Bouyoucos and Hyborg (SL)]and shocks [Ribner
WLN.
shock waves, compressible flow with chemical reaction, deflagration and detonation waves, gas turbine and ejector design. I n the field of transonic flow there is an interesting note of Laitone ( I s M ) , who points out an analogy between the normal shock wave terminating the local supersonic flow on an airfoil and the hydraulic jump above a hydrofoil moving in the vicinity of the free water surface. Both are governed by the maximum pressure (depth) available from the wave front. Most of the papers are concerned, of course, with supersonic flow. Of general interest here is Birkhoff’s and Walsh’s ( 4 M ) comprehensive treatment of the possible conical, axially symmetric, irrotational flows of a compressible, nonviscous fluid; and Giese’s (8M)report on calculations of steady, inviscid, axisymmetric supersonic flows. The latter made u ~ eof digieal computers, whereby he dispensed with the conventional linearizing assumptions. As a consequence, flows about cones with semi-apex-angles between 5” and 50” for Mach numbers of the incident stream of 1.3 to 7.0 were evaluated. Kavanau (It M )reports mrasurements of base pressures, that is, pressures exerted on the back wall of a cylindrical body, in a supersonic stream flowing parallcl to its axis. Pai ( I S M ) analyzes the twodimensional supersonic shear flow around a corner. Heaslet and Fuller ( 1 O M ) describe a method for determining axially symmetric shapes with minimum wove drag. nggers, Savin, and Syvertson ( 7 M ) present the generalized shock-expansion method and its application to bodies traveling a t high supersonic speeds. Concerned with nozzle flow is Beckwith’s and Moore’s “Accurate and Rapid Method for the Design of Supersonic Nozzles” ( S M ) , Wegener’s ( 1 6 M ) description of water vapor condensation processes in supersonic nozzles, and Dailey’s ( 6 M ) analysis of supersonic diffuser instability. In relation to shock wave phenomena, worth noting is Griffith’s ( 9 M ) analysis of the structure of shock waves in polyatomic gases and Broer’s ( 6 M ) recent addition to his theory of shock structure (third in the sequence). Publications dealing with specific aspects of shock wave theory are reviewed in the sections on Turbulence, Boundary Layer, and Transient and Pulsating Flow.
GAS DYNAMICS
DYNAMICS
This section contains a cursory survey of only a few publications selected from the field of compressible aerodynamics to give the reader some idea of recent progress and direct his attention a t those which bear directly on the analysis of problems occurring in chemical engineering practice. Worth noting is the publication of the second edition of the text on applied gas dynamics by Abramovich ( 1 M ) in Russia. ‘The scope of the book is perhaps best described by the following list of contents: gas dynamic equations for a streamline; some information on hydrodynamics; shock waves; accelrration of the gas flow; single state flow; boundary layer and turbulent flow; pressure losses in nozzles, diffusers, and gas ejectors; elements of airfoil and airfoil cascade theory; elements of compressor and turbine gas dynamirs; and reaction force and propulsion. Of direct practical interest is Rosenhead’s and others’ ( 1 4 M ) selection of graphs used in calculations of compressible air flow, published by the Oxford University Press, and the “Equations, Tables, and Charts for Compressible Flow” compiled by the research staff of the NACA Ames Laboratory (biM). Spalding ( 1 6 M ) presents graphs for computation of flow of a perfect gas with variable specific heats. Temperature and velocity, in logarithmic scales, are used as coordinates and lines of constant encrgy, constant entropy, constant ratio of flow rate per unit area to absolute pressure, constant momentum, and unit Mach number are plotted. The application of the graph for the solution of a number of engineering problems i s described; these include: nozzle design, heat addition t o gas flow in a pipe,
March 1956
OF
REACTIVE FLUIDS
The title of this section has been changed from “The Dynamics of Combustion” in order t o emphasize the more general scope of this field of study. Nevertheless, the majority of the papers reviewed here are still concerned primarily n-ith combustion phenomena. General. The most fundamental contribution in the general field is the text of Penner on “Chemical Reactions in Flow Systems” (19N). The book is concerned primarily with the theory of chemical reactions in flow systems of perfect gas mixtures, It starts with a summary of chemical kinetics developed from basic principles and definitioiis, written clearly for a reader without previous familiarity with the subject; this is followed by a discussion of conservation laws for reacting media and the derivation of various transport coefficients; a chapter is devoted to thermodynamics and chemical kinetics of flow through a Lava1 nozzle; the last of the four chapters is concerned with the problems of heterogeneous chemical reactions and includes a discussion of the salient features of heterogeneous diffusion flames. A welcome addition to the engineer’s library is the text of Spalding on “Some Fundamentals of Combustion” (96iV). The book gives a clear summary of the subject, in particular relation to gas turbine application. Emphasis is placed on the fundamental aspects. This is reflected by the organization of the subject matter: Chapter 1 gives a resume of thermodynamics with emphasis on mas8 and energy balances; Chapter 2 suinmarizea the most important aspects of fluid flow; Chapter 3 is on the fundamental principles of heat and mass transfer; Chapter 4 dealer
INDUSTRIAL AND ENGINEERING CHEMISTRY
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FUNDAMENTALS REVIEW with the additional consideration of chemical reaction; chemical effects of combustion are described in Chapter 5 ; and Chapter 6 brings in some remarks on engineering implications. Other current publications of the author demonstrate the extent of his present activities in the field, ranging from an experimental investigation of laminar flame speed (with heat extraction from the flame, obtained by the use of a burner fitted with a porous disk) ( S N ) , and a theory of the extinction of diffusion flames ( M N ) , to the investigation of flame stability in high velocity gas streams (97“). Of interest also are his critical remarks ( S 6 N ) on the L‘excessenthalpy hypothesis” of Imvis and von Elbe.
Figure 6. Schlieren picture of inverted stoichiometric pentane-air flames (IN) Jet velocity 28 fi./sec., H P igniter (Left) undisturbed free jet (Right) 25-mesh screen 1/2 inch below ignition point
Last year marked the publication of two compcndin : first of papers presented at the sixth and seventh Advisory Group for Aeronautical Research and Development (AGARD) Combustion Panel Meeting (6N)held in The Netherlands and in France, and the second of papers presented a t the fifth International Syniposium on Combustion, at the University of Pittsburgh in 1954 ( ?‘Ar). Although the latter R as devoted primarily to combustion kinetics, it included a number of papers concerned with dynamic aspects. Noteworthy among these are the review articles on combustion problems in internal combustion engines, gas turbines, and rockets. Related t o fluid dynamics is a group of six papers on combustion of fuel droplets, some scven papers on propellent burning, four papers on djffusion flames, and a couple of papers on comhustioii in engines. Interesting to note is thc emphasis, reflected in many papers, on the relative importance of mixing as the rate-controlling process. Flame Propagation. A revealing study of flames in turbulent streams is reported by Berl, Rice, and llosen ( I N ) , who consider especially the effect of turbulence on the increase of the laminar flame propagation rate. They point out again that “the rate of the chemical process and it? temperature dependence are profoundly influenced by the mixing process.” For instance, too vigorous mixing may produce a lower heat relcase in spite of the simultaneous increase in the turbulent diffusion coefficient. The details of flame structures are illustrated by a numbrr of interesting Schlieren photographs, exemplified by Figure 6. Mickelsen and Ernstein ( I S N ) report an expcrimrntal investigation of flame propagation in a turbulent gas stream, made in a carefully designed wind tunnel, fitted with turbulencc-generating grids. Tucker ( S O N ) presents a detailed analysis of the interaction of a free flame front with a turbulent field. Summerfield, Reiter and others ( S S N ) publish the results of an exhaustive investigation designed to test the validity of the 650
popular fluctuating-laminar-front model of turbulent flames. They conclude that ‘I Ref. ., (sa),pp. 687-95. Watson, E J , Proc. Roy. SOC.(London) 231A, 101-16 (1955). m’ieghardt, K. E. G , Aero Quart. 5, part 1, 25-38 (May 1954) Wilkinson, J., Aero. Quart. 5, part 1, 73-84 (1954).
Acharya, Y . V. G., A p p l . Sci. Research 5A, 296-308 (1955). Alexander, L. G., Kirnick, A., others, A m . Inat. Chena. Rngrs. 1, 55-73 (1955). Anderaon, A. R., Johns, F. R., Jet Propulsion 25, 1, 13-15, 25 (1955). Chapman, A. J., Korst, H. H., Ref. (SA). pp. 723-31. Dewey, P. E., Vick, A. R., Natl. Advisory Comm. Aeronaut., Tech. Notes, 3466 (July 1955). Forstall, W., Gaylord, E. W., J . A p p l . Mechanics 22, 2, 161-4 (1955).
INDUSTRIAL AND ENGINEERING CHEMISTRY
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FLUID DYNAMICS (7G) Gough. R. II., Proc. P h y s . Soc. (London) 68B, 4,424B, 234-40
(8G) (W) (10G)
(11G) (12G) (13G)
(1955). Horn, G., Thring, M. Wr.,Nature 175, 1081-2 (1955). Pai, 5. I., J . A p p l . Mechunks 22, 1, 41-7 (1955). Szczeniowski, B., Natl. Advisory Comm. Aeronaut., Tech. Notes, 3385 (May 1955). Tani, I., Kobashi, Y., &‘roc. 1st Japan iVatZ. Congr. A p p l . Mechanics 1951; Natl. Comm. Theoret. A p p l . M e c h u n k , 4 6 6 8 (May 1952). Taylor. Sir Geoffrey I., Ref. ( 7 A ) , pp. 313-17. . Torda, T. P., Thompson, Vi’. J., Genetti, B. K., Ref. ( 8 4 , pp. 715-21.
(135) Roshko, A., Natl. Advisory Comm. Aeronaut., Tach. Notes, 3168 (July 1954). (145) Roshko, A., Ibid., 3169 (July 1954). (15J) Rouse, H., Siao, T. T., Kef. (7A),pp. 743-9. (16J) Sato, H., Proc. 1st J a p a n ;Vat2. Congr. Appl. Mechanics (1951); Natl. Comm. Theoret. A p p l . &fechanics,469-73 (May 1952). (175) Tamaki, II., Oshima, K., I b i d . , 459-64. (185) Tifford, A. N., Chu, S. T., Kef. ( 8 A ) , pp. 793-800. (19J) Wagstaff, J . B., Sirmaier, E. A., IND.ENG.CHEM.47, 1129 -35 (1955). (205) Woods, 1,. C., Proc. Rou. SOC.(London) 227A, 367-86 (1955). (215) Wyllie, M . R. J., Gregory, h.R., IND. EXG.CHEM.47, 1379-88 (1955).
DUCTfD FLOW
(1H) Ananian, A. K., Doklady -4kud. iVaulc S.S.S.R. 93, 633--6
(1953). (211) Be&, 6.TI., “Flow and Fan,” Industrial Press, Sew York, VI 226 pp., 1954. (3H) Binnie. A. hI., Davies, P. 0. A. L., Orkney, J. C., Proc. Rou. Soc. London 230A, 225-45 (1955). (4H) Bohnet, J., Stinson, 1,. S., Trans. A S M E 77, 5, 683-92 (1955). (511) Cuming, H. G., Aeronaut. Research Council (Great Britain), Rcpt. and hlerno, 2880 (1955). (613) Deissler, R. G., Taylor, 31. F., Nail. Advisory Comm. Aeronaut., Tech Aretes, 3451 (May 1955). (71%)Eckert, E. 17. G., Irvine, T. F., Jr., Ref. ( 6 A ) ,Paper XI. (8R) Eckcrt, E. R. G., Diagiiila, A. J., Donoughe. P. L., Y a t l . Adv:7:sory Cornm. Aermaut., Tech. N o t e s , 3339 (February 1955). (9H) Ferrell, J. K., Richardson, F. LI., Beatty, K. O., Jr., IND. E N G . C H E M . 47, 29-33 (1955). (10H) Hutton, S. P., Proc. I m t . Civil Engrs. 3, part 3, 1, 216.~35 (1954). (1 1H) Jacobs, R. H . , B d l . Math. Biophys. 15, 395-409 (1953). (12I-I) Junger, hf. C., J . AppZ. Mechanics 22, 2, 227-31 (1955). (13H) Kaye, J., Bron-n, G. A., others, Ref. ( 7 A ) ,pp. 787-92. (14H) Knapp, W. C., Aletzger. J. W.,Trans. A S M E 77, 5, 675-82 (1955). (15H) iUarchett,i, h l . , Energia eletf. 30, 2, 65--79 (1953). (1611) Morgan, G. IT., Kiely, J . P., J . Acoust. Soc. Anier. 26, 3, 323-8 (1954). (171%) A-lItcall, H., E ~ ~ i ~ ~ 178, p d 4623, n g 298-300 (1951). (18II) Richter, H., “IIydraulics of Pipe Flow. Ilandbook for Design Flow Calculat,ions (Hohrhvdraulik. Ein IIantibuch zur Praktischen St rtimungsherPrhnung) Springer-Verlag, 328 pp., 1954. Berlin, X I (1911) Rothfus, R . R., llonrad, C. C., others, IND.ENG.CHEW.47, 913-18 (1955). :2OII) Sandhorn, V. h , .?‘ail. Advisory Comm. Aeronaut., Tech. 11-ofes, 3266 (Jan. 1955). (211%) Sirnmons. F. 9.. Ibid.. 3447 (AIDril1955’3. { Z I T ) Taylor, Sir Geoffrey I , Proc P h y s Soc ( L o n d o n ) 67B, 420R, part 12, 857-69 (1954). (2311) Tong, K. N., Ref. (Sil), pp. 603-6. (24H) Weissbera, 1%. I,., Herman, -1 9 , Ref ( 6 A ) , Paper XIV. (2511) Weltmann. It. N., S a t l . Adnisory Comm. Aeronout., Tech. Notes, 3397 (February 1955). (2611) Womrrsley, J. R., P h i l . Mag., ( i ) , 46, 373, 189-221 (1955). (27H) Woods, L. C., Ptoc. R o y . SOC.L o d o n 2298, 63-85 (1955). (281%)Yuan, S. W.,Finkelstrin, -A. B., Ref. (6A), Paper XVII.
+
+
,I’
FLOW AROUND SOLIDS
(1J) Allen, I). N. de G., Southwell, 11. V., Quart. J . Mechanics and A p p l . Math. 8, 129-45 (1955). (25) Breslin, J. P., J . Appl. Mechanics 22, 1, 35-40 (1955). (35) Dean, W. R., Proc. Cambridge Phil. Soc. 50, part 1, 125-30 (1964). (4.1) Fadnis, 13. S., ZAMP 5, 2, 156-63 (1954). (55) GrootenLuis, P., Proc. I n s t . Mech. Engrs. 168, 34, 837-46 (1954). (FJ) IDscn. D. C.. Ref. (6A1. PaDer X. (7Jj kamimoto, G., Maisuoka, ?., T r a n s . Japan Soc. X e c h . &.pa. 19., 32-7 (19581. -, ~
~~
(85) KeIrmcen, R. W., McGraw, J. T., Parkin, B. H., Trans. A S M E 77, 4 , 53342 (1955). (9J) Mottard, E. J., Loposer. J. D., N a t l . Advisory Comna. Aeronaut., Rcpts., 1161 (1954). (1OJ) Kigsm, S. D., Z A M P 5 , 151-5 (1954). (11J) Reichardt,, 11.. “Straightline Couette Flow around a Cylindrical Body,” (in Gerinan) , M i t t . Max-Planck-Inst. Stromungsforschung, KO.9 (1954). (125) Roshko, A., Nutl. Adczsoru Cornrta. Aeronaut., Repta., 1191 (1954).
March 1956
MULTIPHASE FLOW
(1K) Raatard, P., C‘ornpt. rend. 239, 7, 531-3 (1954). (PK)Bhattrtcharya, A., ltoy, A. N.. IND.ENG.CHEW.47, 268-74 (1955). (3K) Ritron, .\I. D., Ibid., 47, 1, 23-8 (1955). (4K) Calvert, S., Williams, B., Ana. I n s t . Chem. Engrs. 1, 78--86 (1955). (5K) Cheng, P.Y., Schachman, H. K.. J . PoEumer Sci. 16, 19-30 (1955). (6Kj Coulson, J. M., Dombrowski, N.,J . Photographic Sci. 3, 88-94 (1955). (710 Criddle, D. W., hfeader, A. L., Jr., J . Appl. I‘hys. 26, 838-42 (1955). (8K) Dussourd, J. I,., Shapiro, -2.IT., Kef. (GA), Paper XX. (9K) Franklin, F. C., Johanson, I,. N., Chem. Eng. Sci. 4, 119-29 (1955). (10K) Fraser, R. P., J . Photographic Sci. 3, 21-32 (1955). (11K) Garner, F. H., Skelland, A. H . P., Chem. Eng. Sci. 4, 149-58 (1.955). (12Kj Garner, F. H., Ellis.,Brit. J . A p p l . Phys. 6, 83-7 (1955). (18K) IIerrinn, W. M.. Jr., hIarshal1. W. R., ,Jr., -4m.Inst. Chem. Engri. 1, 200-9 (1955). (19K) IIu, Shengen, Kintner, R . C. Ibid., 1, 42-8 (1955). (2010 Keith, F. Pi.,Jr., IIixRon, A. N , IND, 1 ’ : ~ . CHEM.47, 258-67 (195.5). (21K) Khucliakov, G. N., Akad. .Vozlk S S.S.R., Otdel. Tekh. Nauk, Izvest. 1953, 1022-34. (22K) Ladyzhensky, K. AT., J . B p p l . Chem. ( U 8.R.R.) 27, 22 -32 (1954). (23K) McDonald. J . E . J . MeteoroE. 11. 478 04 (1954). ( 2 4 ~ llacPhai1, j D. c., proroc. 4th AGARD ani. ~ s x w n ~73-83, y NATO Memo .4G 14,/P5(May 1954). (25K) l\Iagono. C., J . Meteorol. 11, 77-9 (1954); [comment,$ by D. C. Blanchard, Ibid., 12, 91-2 (1955)]. (26K) 3Iarshal1, W. R., Jr., “Atomization and Spray Drying,” Chem. Eng. Progr. Mono. Scr., No. 2, VI11 + 122 pp., 1954. (27K) Newitt, D. M., Dombrowski, N., Knelman, F. IL, Tmna. Inst. Chem. Engrs. 32, 24441 (1954). (28K) Orr, C., Jr., Blocker, 1%.G., J . Colloid Sci. 10, 24-8 (1955). (29K) Quigley, C. J., Johnson, A. I., Harris, B. I,., “Mass TransferTransport Properties,” Ch,em. Eng. Progr., Symposium Ser. KO. 16, 31-45 (1955). (30K) Hadcliffe, A., “The Perforinance of a Type of Swirl Atomizer,” Inst. .Mech. Engrs. 1954. (31K) Rced, W. H., 111, Nutl. Advisory Comm. Aeronaut., Repts., 1196 (1954). (32K) Ryley. D. J., Engineer 198, 7,4-8 (1954). (33K) Scheraga, H. A , , J . Chem. P h y s . 23, 1526-32 (1955). (34K) Serafini, J. S., Natl. Advisor$/ Comm. Aeronaut.. Repts., 1159 (1954). (35K) Sicmes, W., Chem.-Ing.-Tech. 26, 479-96, 614-30 (1954). (36K) Spells, K. E., Trans. Inst. Chem. Engrs. 32, 167-73 (1954). (37K) Starkey, T . V., Brit. J . AppE. P h y e . 6,34-7 (1955). (38K) Tuson, K. K., Ibid., 6, 99-100 (1955). (39K) Whitmore, R. Id.,Ibid., 6, 239-44 (1955). TRANSIENT AND PULSATING FLOW
(1L) Beranek, 1 2 . L., ”Acoustics,” McGraw-Hill, New York, Toronto. London. X 481 DD.. 1954. (2L) Berry, F. J., Butler, D. S.,Hoi< &I., Proc. Roy. Soc. London 2271). 258-70 (1955).
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I N D U S T R I A L AND E N G I N E E R I N G CHEMISTRY
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FUNDAMENTALS REVIEW Bouyoucos, J. V., Hyborg, TV. L., J . Acoust. SOC.Amer. 26, 4, 511-14 (1954). Brombacher, W. G., Lashof, T. M.,N a L B u r . Standards ( U . 9.) Circ. 558 (February 1955). Chilton, E. G., Handley, 1,. R., T r a n s . A S M E 77, 225-30 (1955). Chu, Boa-Teh, Natl. Adrisory Comm. Aeronaut., Tech. Sotes, 3411 (June 1955). Dressler, R. F., J . Research Xatl. B u r . Standards 53, 253-60 (1954). Freeman, N. C., Proc. R o y . SOC. (London) 228A, 341-63 (1955). Friedrichs, K. O., Keller, J. B., J . A p p l . Phys. 26, 961-6 (1955). Gerber, S.,Comp. rend. 239, 5, 391-3 (1954). Geza Bata, AI., Ibid.,239, 6, 476--8 (1954). Glass, I. I., Patterson, G. X., J . Aeronaut. Sci. 22, 2, 73-100 (1955). Goldstine, H. H., von Seuinnn, J., Comm. Pure and -4ppZ. Math. 8 , 2, 327-53 (1955). Harleman, R. F., Ippen, A. T., Ref. ( T A ) ,pp. 91-112. IIavelock, T., Proc. Eou. SOC.(London) 231A, 1-7 (1955). Hueter, T. F., Bolt, R. II., “Sonics,” John Wiley, New York, X I f 456 pp., 1955. Isakoff, S. E., IND. ENG.CHEM.47, 413- 21 (1955). Jones, C. W., Proc. Roy. SOC.(London) 228A, 82-100 (1955). Kastner, L. J., Williams, T. J., I n s t . Mechanical Engrs., Proc., 3-16 (1955). Kronauer, R. E., Grant, I€. P., Ref. ( B A ) ,pp. 763-70. Laitone, E. V., Ref. (7A),pp. 203-17. Latter, R., J. A p p l . P h y s . 26, 954--60 (1955). Lighthill, M. J., Whitham, G. B., Proc. R o y . SOC. (Loiadon) 229A, 281-345 (1955). Litovitz, T. A., Lyon, T., Peselnick, L., J . Acoust. SOC.Amer. 26, 4,566-76 (1954); Litovitz, T. A., Lyon, T., Ibid., pp. 577-80. RIirels, H., Natl. Advisory Comni. Aeronauf., Tech. ,Votes, 3401 (March 1955). Mucklow, G. F., Wilson, A. J., “Wave-action in Gases: The Attenuation and Reflection of Compression Waves Propagated in Pipcs. Parts I, 11,” Inst. Mechanical Engrs. Proc., 12 pp. (London) 1954. Nagamatsu, 11. T., J . Aeronaut. Sci. 22, 3, 165-72 (1955). Oppenheim, A. K., Chilton, E. G., Trans. A S M E 77,2,231-48 (1955). 161 pp., Parmakian, J., “Waterhammer Analysis,” XIV Prentice-Hall, New York, 1955. Proudman, J., €‘roc. R o y . SOC.(London) 231A, 8-24 (1955). Ribner, H. S.,Natl. Advisory Comm. Aeronaut., Tech. ,Votes, 3255 (July 1954). Rudinger, G., “Wave IXagrams for Non-steady Flow in Ducts,” X I 278 pp., Van Nostrand. New York, 1955. Skudrzyk, E., “Foundations of Acoustics,” (Die Grundlagen der Akustik), Wien, Springer-Verlag, XXII 1084 pp., 1954. Smilcy, E;. F., Winkler, E. I€., J . Chem. Phys. 22, 12, 2018-22 (1954). Trimpi, R. I,., Cohen, N. B., Natl. Advisory Conim. Aeronaut., Tech. Notes, 3375 (March 1955). Truesdell, C., Ref. ( 7 A ) ,pp. 383.46. Wallace, E’. J., Wngineering 178, 4630, 524-8 (1954). Williams, J. A., Little, D. J., Trans. Inst. Chem. Eng1.s. 32, 174-80 (1954). Zienkiewicz, 0. C., Hawkins, P., Proc. I n s f . Mechanical Xngra. 168, 23, 629-42 (1954).
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GAS DYNAMICS
(lhl) Abramovich, G. N., “.ipplied Gas Dynamics,” 2nd cd., (in Russian), 732 pp , lloscow, Gos, Izd. Tech.-Teor. Lit., 1953. (2hI) Ames Research Staff, Nail. AdLisoru Conzm. Aeronaut., Repts., 1135 (1953). (3M) Beckwith, I. E., Moore, J. .4.,Natl. Advisory Conzm. Aeronaut., Tech. Notes, 3322 (February 1955). (411)Birkhoff, G., Walsh, J. IT.,Ref. ( 7 A ) , pp. 1-12.
654
(5M) Broer, L. J. F., A p y l . Sci. Research, Sec. A, 5 , 76-80 (1954). (6M) Dailey, C. L., Ref. ( 6 A ) ,Paper IX. (711) Eggers, A. J., Jr., Savin, R. C., Syvertson, C. A., J . Aeronaut. S C ~22, . 4, 231-8, 248 (1955). (8M) Giese, J. H., Coinm. Pure A p p i . Math. 7, 65-77 (1954). (9M) Griffith, W., Ref. ( 6 A ) ,Paper 11. (10M) Heaslet, It.A., Fuller, E’. B., S a t l . Aduisory Conzrra. Aeronaut., Tech. .Votes, 3389 (E’ebruary 1955). (1111) Kavanau, I,.L.,J . Aeronaut. Sci. 21, 4, 257-60, 274 (1954). (12hI) Laitone, E. V., Ref. ( 8 A ) ,pp. 751-3. (13hf) Pai, S.I , Ibid.,pp. 637-42. (14h2) Iioaenhead, L.. Bickley, W. G., others, “A Selection of Graphs for Use in Calculations of Compressible Airflow,” Clarendon Press, Oxford, X 115 pp., 1954. (15M) Spalding, D. B., Engineering 177, 4612, 777-81 (1954). (l6M) Wegencr, P. P., J . Appl. P h ~ s25, 1485-91 (1954)
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DYNAMICS OF REACTIVE FLUIDS
(1N) Berl, W. G., Rice, J. L., Rosen, P., Jet Propulsion 25, 7 , 3416 (1955). (2N) Bolz, R. E., Burlage, II., Jr., Ibid., 25, 6, 265-75, 283 (1955). (3N) Botha, J . P., Spalding, .E).B., Proc. Rou. SOC.(London) 225A, 1160, 71-06 (1954). (4N) Cheng, Sin-I, J e t Propulsion 25, 4, 163-7 (1955). (5N) Combustion Researches and Reviews, invited papers presented a t the 6th and 7th AGARD Combustion Panel Meet,ings held respectivoly at Scheveningen, The Netherlands, May 1954, and Paris, Prance, Nov. 1954, XV -i- 188 pp., Buttcrmorths Scientific Publications, London, 1955. (6K) Dugger, G. L., IND.ESG.CHEM.47, 1, 109-14 (1955). (7N) Fifth Symposium (International) on Combustion (Combustion in Engines and Combustion Kinetics) at the University of Pittsburgh, Fittsburgh, Pa., 4ug. 30-Sept. 3,1954; Reinhold, Kew York, XXVI 802 pp., 1955. (8N) Gaydon, A. G., Wolfhard, H. G., Fuel 33, 3, 286-90 (1954). (9N) Glick, H. S., Squire, W., IIertzberg, A,, Ref. (7A7, 393-402. (lOK) Gross, R. A., J. Aeronaut. Sci. 21, 2, 139-41 (1954). (11s) Gross, R. A., Jet Propulsion 25, 6 , 288---go,293 (1955). (12K) Grunier, J., Harris, hI. E., IXD. ENG.CIIEM. 46, 11, 2424-30 (1954). (13N) Kaskan, W. E., Soiccn, A . E., Trans. A S M E 77, 6, 885-96 (1955). (14N) Markstein, G. II., Schwartz, D., Jet Propulsion 25, 4, 174-6 (1955). (15N) Riickelsen, W. R., Ernstein, N. E., Natl. Advisory Cornni. Aeronaut., Tech. .Votes, 3456 (July 1955). (16X) Moore, F. K., Maslen, S. I€. Ibid., 3152 (October 1954). (17N) Office of Naval Research, “Second ONR Symposium on Detonation, Feb. 9-11, 1955,” V + 502 pp., Department of the Navy, Washington, D. C., 1955. (18K) Olsen, R.I,., Gayhart, E. L., Jet Propulsion 25, 6, 276-83 (1955). (19K) Penner, S.S., “Chemical Reactions in Flow Systems,” V I 1 1 86 pp., Buttemorths Scientific Publications, London, 1955. (20K) Putnam, 8. A , Dennis, W.It., Trans. A S M E 77, 6. 875-84 (1955). (21ru’) Rogers, D. E., JIarble, F.E., Ref. ( 6 A ) , Paper VIII. (22K) Schafler, A., Cambel, -4.B., Jet Propulsion 25, 6 , 284-7 (1955). (23N) Schultz-Grunow, F.,Z. Elektiochem. 59, 9, 804-8 (1954). (24N) Spalding, D. B., Fuel 33, 3, 255-73 (1954). (25K) Ibid.,34, Supplement 100-6 (1955). (ZGK) Spalding, D. H., ”Somc Fundamentals of Combustion,” X t250 pp., Academic Press, S e w York; Butterrvorths Scientific Publications, London, 1955. (27N) Spalding, D. B., Tall, B. 9.. Aeronaut. Quart. 5, 195-217 (1954). (28s) Summcrlield, RI., Reitcr, S. H., others, J e t Propulswiz 25, 8, 377-84 (1955). (29N) Thorpe, h1. I,., Browning, J . A., IXD.ENG. CHEAf. 46, 10, 2203-5 (1954). (305) Tucker, AI., S a f l . ddUiS01.y Coinm. Aeroimut., Tech. .Votes, 3407 March 1955). (315) Van Wonterghem, J.; Van Tiggelen, A., Bull. Soc. chim. Kelges 63, 235-60 (1954).
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INDUSTRIAL AND ENGINEERING CHEMISTRY
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Vol. 48, No. 3
