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Fluid Dynamics by C. A. Sleicher, Jr., Cavendish Laboratory, University of Cambridge, Cambridge, England
R. A. Stern, University of California, Berkeley, Calif. 1. E. Scriven, University of Minnesota, Minneapolis, Minn.
A. K. Oppenheim, University of California, Berkeley, Calif. 1
The advent of space flight has emphasized the significance of chemical reactions and their kinetics in aerodynamics and other advanced fields of gas dynamics
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T H E prosperity of our times is so expansive that it encompasses even such an academic field as the fundamentals of fluid dynamics. Consequently, as time progresses the difficulties of keeping up with the literature mount at an amazing rate. This has been always recognized in these reviews, and instead of attempting comprehensive coverage of the literature the major effort is directed toward an account of progress in the field. Considering formulation of the subject matter, as expressed by equations of motion, some interesting examples are now available of the breakdown of the continuum approach to describe some specific nonsteady motions of a gas under conditions where the mean free path is still negligibly small. The basic subject of turbulence has been greatly enhanced by the publication of texts and proceedings of symposia. T h e most intriguing idea is the development of a revolutionary theory of isotropic turbulence based on the premise that statistical dependence among the Fourier amplitudes is induced wholly by the nonlinear terms in the Navier-Stokes equation. Understanding vortex motion as well as spray formation was aided greatly by some careful and penetrating experiments. Some of the most significant results are indicated by the illustrations shown. Most significant developments in the study of boundary layers are concerned with three-dimensional effects to account for the observed secondary flow phenomena. The classical subject of aerodynamics is today concerned primarily with the field of hypersonics. I n the advent of space flight the reason for such an interest is
quite obvious. Of particular significance is the fact that here, as in the other advanced fields of gas dynamics such as gas wave dynamics, dynamics of reactive fluids, and dynamics of conducting fluids (magnetohydrodynamics), the effects of chemical reactions and their kinetics play a predominant role.
Comprehensive Publications The growth of a subject is reflected best in the quality and quantity of textbooks. Last year there were fewer comprehensive books and more of a specialized nature. I n the first category there was an elegant mathematical treatise on inviscid flow ( 7 A ) . A well-known text (6A) has been throughly reorganized, and new material has been added, notably the compressible flow theory. A greatly improved version of the text of Knudsen and Katz ( 3 4 represents a well-conceived coverage of the classical fundamentals of fluid mechanics essential to ihe understanding of convection heat transfer. The second edition of a 1954 text ( 2 4 follows the traditional style of German textbooks. Two unique Russian monographs have been translated into English ( 5 4 73J). Specialized books (QC, SH, 72H,73J)are reviewed in their appropriate sections. I n addition to such well-known yearly events as the Heat Transfer and Fluid Mechanics Institute (7A) and the Midwestern Conference on Fluid Mechanics (#A),published last year were the seventh International Symposium on Combustion (79K), the third AGARD Colloquium in Palermo (4K), and the symposium on combustion of gaseous mix-
tures held at the 16th International Congress of Pure and Applied Chemistry
(74K). Equations of Motion and Stability Although the Navier-Stokes equations are regarded as the foundation of fluid dynamics, they are properly applied only to those motions in which a fluid closely adheres to the Newtonian continuum model. Much of rheology pertains to other continuum models of real fluid behavior, of which perhaps the next simplest is the Stokesian fluid recently reconsidered by Serrin (2.OB). An interesting breakdown of the continuum approximation occurs in a gas bounded by an infinitc plane set impulsively into motion parallel to itself. Rayleigh’s analysis by the Navier-Stokes equations predicted an infinite initial stress in the gas. From kinetic theory a result much closer to physical reality has been obtained (QB)which distinguishes between molecules which already have and those which have not yet interacted with the moving plane. The contrast between the microscopic relaxation process of attaining local equilibrium and the microscopic relaxation process of attaining spatial uniformity is central to Mori’s (27B) statisticalmechanical theory of transport in fluids. The latter process is represented by the hydrodynamic equations. T h e usual statement of Navier-Stokes equations rests on Stokes’ assumption that the second coefficient of viscosity of Newtonian fluid is such that bulk viscosity is nil. Though often immaterial, the assumption is inadequate in accounting for dissipation of energy in a VOL. 52, NO. 4
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sound wave. New evidence appeared in Nettleton’s (23B) considerations of bulk viscosity of nonassociated liquids from the standpoint of compressional relaxation effects. Related is the role of bulk viscosity in acoustic streaming, the steady flow that can accompany the propagation of sound waves through a fluid, which was studied by Kaugol’nykh (22B). An experimental search (73B) revealed the existence of transverse velocities generated by purely longitudinal shearing of a compressible fluid. This interesting effect is described by the Navier-Stokes and energy equations; heat generation by viscous dissipation is responsible. Flow at Low Reynolds Numbers. Interest continues in creeping flow past arrays of cylinders and spheres. A “free surface” modification of the curious unit cell model (see 1959 Fluid Dynamics Review) has been applied (70B) to the problem of flow parallel and perpendicular to an array of cylinders. T h e method yields average flow and total drag, in agreem nt with certain data, and seems promising so long as detailed knowledge of the whole velocity field is not required. Outstanding is the contribution of Jensen ( 7 5 B ) who calculated in great detail by relaxation methods the flow round a sphere at Reynolds numbers of 5, 10,20, and 40. Separation and formation of a circulating wake evidently occur at a Reynolds number of about 17. The classic Helmholtz-Korteweg theorem of minimum energy dissipation in every creeping flow in which velocity is specified on the system boundaries is not often applicable in practice. Christopherson and Dowson (5B) found it is a t least approximately verified by the actual motion of a heavy ball falling slowly down a tube slightly larger in diameter and filled with viscous liquid. As they point out, a more general principle of minimum dissipation, if it could be relied upon, would be very useful in the calculation of viscous flow. Something akin to this appears in the powerful variational method used (37B) to compute the laminar velocity distributions in square, rectangular, and circular sector ducts. An experimental technique for finding velocity distributions in two-dimensional laminar flows (2623)relies on the interference patterns in flowing, doubly refracting liquids when viewed by transmitted polarized light (see figure). Flow Stability. Experiments on the downstream decay of flow disturbances generated at a fixed station in a pipe are a noteworthy contribution (78B). At Reynolds numbers u p to a t least 13,000 small disturbances decay but at a decreasing rate with increasing Reynolds number. Certain large disturb-
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ances are amplified and finally result in transition to turbulent flow. In studying the calculation of hydrodynamic stability for plane Poiseuille flow, it was shown ( 3 5 B ) that the OrrSommerfeld equation can be converted to a Volterra integral equation well suited to iterative techniques of solution. Critical Reynolds number for instability was found ( 3 3 B ) to decrease with time in a developing plane Poiseuille flow. Taylor’s celebrated analysis of the stability of Couette flow was extended to the case of a large gap between cylinders (4B); theoretical predictions were again sustained by experiment (7B). The complication of a conducting fluid in the presence of a magnetic field was investigated by Niblett (24B), whose experiment supports his prediction that certain boundary conditions can grossly affect the mode of instability. The Rayleigh-Jeffreys stability analysis describing onset of cellular convection in a fluid layer hcated from below strictly applies only to layers without lateral boundaries. Yih ( 3 8 B , 39B) modified the analysis to describe thermal, or buoyancy-driven, instability of fluid confined between vertical planes or in a vertical tube. As in the unbounded layer, convection sets in with a fixed cellular pattern, but if the whole system is rotated-a feature of some importance in rocketry, for example-convection may set in with a moving cellular pattern, a mode known as overstability. The analysis should be compared with experimental results (30B) obtained when a cylindrical container of liquid was subjected to transient heating from below, with and without rotation. Meeting a challenge of several years’ standing. Fultz (8B) elucidated the mechanism of overstability in rotating fluid layers heated below, tracing it to the inertial stabilization associated with rotation. Another physical interpretation of overstability (34B) is overshadowed by highly significant results on finite amplitude, cellular convection in a rotating fluid layer. The analysis is pivoted about the more accessible solutions to the linearized stability problem. Another important contribution is a study (25B) on convection cells induced by surface tension. The stability analysis and the earlier experiments of Block illuminate a neglected source of fluid motion (sometimes called the Marangoni effect) : gradients of tension along a free liquid surface, which are in turn caused by differences in temperature or composition. Problems on the more familiar Taylor and Helmholtz types of hydrodynamic instability were treated by Carrier and Chang (3B). Although Helmholtz himself considered an inviscid fluid, current
INDUSTRIAL AND ENGINEERING CHEMISTRY
research favors the viscous case, as in the report on stability of a two-dimensional laminar jet in a viscous, incompressible fluid (72B). The jet is so highly unstable that there is great difficulty in studying it by the standard methods of small disturbances.
Turbulence
Of particular significance to students of fluid dynamics is a text (QC) on turbulence. With its realistic and practical approach and authoritative treatment of the subject, the book is indeed a true asset to modern engineering literature. A major, revolutionary theory of isotropic turbulence has been advanced (TIC, 72C) in a series of reports. Many aspects of the theory conflict \vith current ideas. ‘The basis of the theory is the assumption that the statistical dependence among the Fourier amplitudes is induced wholly by the nonlinear terms in the Navier-Stokes equations. This principle of “weak dependence” was exploited by a unique perturbation method, which leads to a solution in the form of infinite-series integrals. A number of interesting “physical” theories of turbulence were presented. Reid and Harris (74C) developed the consequences of KovBsznay’s theory to about the same point that Chandrasekhar and Proudman had developed Heisenberg’s theory. The two theories lead to similar conclusions, and an unambiguous comparison cannot be made without further experiment. I n an investigation (IOC) of isotropic turbulence, a differential equation was derived relating Q, the defining scalar of a second-order isotropic tensor, to X,the defining scalar of a particular third-order tensor. It was then assumed that Xis an arbitrary function of Q, and Kolmogoroff’s similarity principles were used to find the form of the function. A theory of decaying turbulence (75C) is based on mixing and interaction of viscous vortices with an initial probability distribution in space. A cascade process occurs-large vortices produce small ones. Results agree with experiment. A central problem in turbulence theory is the infinity of equations resulting from attempts to solve the Reynolds equations. Major differences among various theories lie in the assumptions made to limit the number of equations. Squire (76C) presented a radically different approach, attacking the foundations of the Reynolds equations. The discussion contains some misstatements but also has a number of interesting ideas. Theoretical work on temperature fluctuations in turbulent flow (3C) concerns the spectrum of temperature (or other scalar) fluctuations when thermal diffusivity is much less than or much
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Typical light interference patterns produced by flow double refraction in aqueous milling yellow solutions (266) a.
b. c.
d. e.
f.
Flow about a cylindrical obstacle at intermediate flow rate, circularly polarized light Flow about a cylindrical obstacle at low flow rate, plane polarized light Flow in diverging channel at high flow rate, circularly polarized light Flow in diverging channel at low flow rate, circularly polarized light Flow in converging channel at high flow rate, plane polarized light Flow in converging channel at low flow rate, plane polarized light
greater than kinematic viscosity. The authors show that the wave number marking the edge of the low-wave number region is a function of Prandtl number. The shape of the temperature spectrum was deduced for great and small Prandtl numbers. I n another theoretical study (4C), assumptions of the statistical theory of turbulence were given a rigorous mathematical examination. An interesting contribution would be the elucidation of this extremely intricate report and its comparison with the doubts expressed by Squire (76C). The behavior of weak, homogeneous turbulence subjected to three kinds of uniform distortion-rotation, shear, and irrotational distortionwas examined (73C). The latter proved to be the only one in which turbulent energy grew at the expense of mean motion until nonlinear effects became important. Turbulent measurements were made in a channel 0.18 X 2.4 X 12 meters (5C, 6C). The profile of 1/2i2/u* is in good agreement with Laufer’s near the wall but is 15% larger a t the center. Skewness, measured in the developing and fully developed boundary layer, was
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constant and negative in the center and increased to a positive value very close to the wall. Measurements (77C) of mean and fluctuating temperature above a heated, flat plate in a box confirm many predictions of the Malkus theory but few predictions of similarity theory. Other experiments (78C) involved measurements of turbulence in a plane jet of air by a hot wire anemometer and by diffusion measurements from a heated wire. Results of the two methods do not agree, and it was concluded that when turbulence level is high, a systematic error causes anemometer readings to be low. Reviews of several Russian studies on statistical theory of turbulence are available (7C). Gifford (8C) has urged caution in drawing physical interpretations from turbulence measurements. H e claims that estimates of physical eddy sizes from the one-dimensional energy spectrum may be large by an extent approximately explainable by the difference in shape between the one- and three-dimensional spectra. Data were gathered (2C) for testing Sutton’s hypothesis for diffusion from a
point source. Sulfur dioxide was released at 0.5 and 1.5 meters above flat ground, and concentrations were measured at 599 samplers. The parameters n, and n. (related to lateral and vertical diffusion rates, respectively) of Sutton’s hypothesis were invariant between 100 and 800 meters from the source, but were smaller clriser to the source.
Vortex Flow a n d Rotation Under the condition of small Mach number, Burger’s solution for vortex flow in a viscous fluid was extended to account for compressibility (700). T h e solution shows that a small amount of cold gas is formed at the expense of a large amount of warm gas. However, the hot end of a Ranque-Hilsch tube is not explained by this model. Rietema and Krajenbrink ( 8 0 ) investigated theoretically the effect of turbulence and wall friction on the tangential velocity profile in a flat vortex chamber. T h e possible consequences of their assumption of an eddy diffusivity independent of position were not discussed. A study by Hide ( 7 0 ) revealed the fascinating flow pattern that develops in a liquid between rotating, concentric VOL. 52, NO. 4
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cylinders held at different temperatures, an experiment suggested by Thomson in 1892. At low speeds a spiral pattern develops. At higher speeds a thin “jet stream” of liquid moves in a wavelike pattern (shown). Thejet exists throughout the depth of liquid and travels in the direction of rotation. I n addition, the pattern drifts in a direction that depends on which cylinder is hotter. The number of modes of the jet depends upon speed and other conditions of operation. At sufficiently high speeds the pattern vacillates, and a t still higher speeds the motion becomes turbulent. A preliminary mathematical analysis of the situation was undertaken by Rogers ( g D ) , who treated the jet stream as a thermal boundary layer. A quite satisfactory theory of the jet stream that is couched in stability concepts has also been developed (5D). The fourth of a series of highly mathematical studies ( 7 0 ) presents a theory of oscillation-type viscometers concerning a thick disk, one for which the boundary layer thickness is small compared to both the radius and thickness of the disk. Using boundary layer techniques, an approximate solution for an edge correction was found. The solution was then improved by a variational technique. The thin disk was also reformulated. The influence of viscous dissipation on measurements in Couette viscometers was investigated theoretically by Zeibig ( 7 7 0 ) under the assumptions of in-
compressibility and viscosity proportional to temperature. Excellent agreement with experimental results was found.
Jets and Wakes Interesting experiments on the development of the wake behind a cylinder were made by Taneda (9E) usiug aluminum dust to make visible flow patterns in the water tank. The primary von KArmbn vortex streer broke down a t a certain distance behind the cylinder, but later the periodicity of the wake reappeared in a secondary vortex street of larger wave length. Taneda calls for a theoretical explanation of the phenomenon. Instability of the primary street to three-dimensional disturbances may be involved. Steady release of mass-momentum or buoyancy into a uniform or stably stratified fluid produces what Morton (GE) calls a forced plume. The jet, from a point source of momentum, and the plume, from a point source of buoyancy, are then special cases. Analysis of the general case of turbulent forced plumes showed that under certain conditions an increase in source strength has the effect a t first of reducing the total height of the plume. A mcthod has been devised (723) for estimating the heat required to ensure penetration of an atmospheric inversion by the buoyant plume from a stack. From measurements of velocity distributions in plane turbulent air jets the
turbulent shear stress, coefficient of momentum exchange, and mixing length were calculated (7023); they were only in partial agreement with existing theories. Also measured were the distribution of heat and matter in the jets, and the intensity of turbulent fluctuations. An extension (8E) of a theory of mixing in two-dimensional hot jets (see 1959 Fluid Dynamics Review) has produced asymptotic similarity solutions that allow for marked density variations. Prandtl’s hypothesis on the shear stress in free turbulence evidently applies despite the variable density. Another experimental study (2E)dealt with mixing in confined jets in jet ejector models. The ”degree of mixing” was characterized by the ratio of static pressure difference to stagnation pressure at a given distance from the nozzle. In continuation of an extensive research program at the University of Illinois, Chow (3E) demonstrated that the two-dimensional base pressure resulting from the interaction of two jets, of which one is supersonic, can be calculated by matching the Prandtl-Meyer solutions of appropriate flow fields. The results are verified by experimental data. As part of a program of research to attain stationary detonation, a detailed investigation was made ( I E ) of the structure of jets obtained from highly underexpanded nozzles. To rationalize the observations, an approximate method was developed for calculating initial jet boundary and estimating the position of the first normal shock or Mach disk.
Flow Near Solid Surfaces
Top-surface flow patterns of rotating annulus ( 7 D ) Aluminum powder used as indicator; sense of rotation i s clockwise. A corresponds to low rotation, in which case top-surface flow pattern i s a spiral; four remaining pictures are typical of wave-flow regime, with wave number going from 2 to 5
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Laminar Boundary Layers. In a supersonic wind tunnel the effect of bluntness on stagnation point velocity gradient was investigated (4F). Five axisymmetric models were used, ranging in bluntness from concave to hemispherical. It was found that truncated cylinder theories overestimate the gradient by at least 307,. Turbulent Boundary Layers. Temperature and velocity fluctuations in an adiabatic, turbulent boundary layer were measured ( 7 8 F ) for free stream Mach numbers u p to 4.76. The velocitytemperature correlation was negative and constant across the boundary layers a t all velocities, indicating that temperature fluctuations may be due to convection of the mean temperature field by the velocity fluctuations. An experimental study ( 7 0 F ) was made of various cylindrical shapes that experience a very sharp drop in drag coefficient as the Reynolds number is increased. I n some cases hysteresis of the drag coefficient us. Reynolds number relationship was found. Transition and Separation. In one
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study ( 3 7 F ) all previous data were carefully analyzed, and many new data were contributed on the effect of roughness on transition. The conclusions were that roughness shape is of major importance and for a given shape of roughness a critical roughness Reynolds number, R,, is substantially independent of both height and location of roughness, pressure distribution, and degree of freestream turbulence. Rhwas found also to be a characteristic parameter for supersonic flow. In the first of two consecutive reports Stratford (33F) extended his well-known analysis of separation of laminar boundary layers to the turbulent case. Mixing length theory and dimensional analysis were combined to yield a simple, accurate criterion for separation. The analysis leads to the concept of a boundary layer with no friction, and an experimental investigation of such a flow was described ( 3 4 F ) . Over one wall of a wind tunnel he produced a stable flow with no skin friction throughout the region of pressure rise, which was about 3 feet.
Boundary Layers with Variable Properties. The boundary layer equa-
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tions for flow along a vertical wall with heat transfer and variable density were solved ( 3 2 F ) using series expansion in which the first term is the solution for no buoyancy effects. I n this way a quantitative criterion was established for conditions under which the effects of free convection may be neglected in heat transfer and shear stress calculations for low-speed forced convection. In an investigation (727) of boundary conditions for which similar solutions exist for laminar boundary layer equations with and without heat transfer, the perfect gas law was used and viscosity was assumed to be proportional to temperature. All such solutions were enumerated, and it was shown that no others exist. A rapid method was reported ( 6 F ) for dealing with the effects of heat transfer from a uniform temperature wall to a laminar bqundary layer. A Prandtl number of unity and viscosity proportional to temperature were assumed. Agreement with the exact solution was good, but error in the prediction of separation increases with increasing Mach number and becomes quite poor at M = 4.
Three-Dimensional
Effects.
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classical methods the conditions under which similarity solutiyns exist for threedimensional boundary layers were investigated ( 7 5 F ) . A table of permissible solutions is given. I n the discussion following the report, it was pointed out that the conditions listed can be obtained from a general, less restrictive theory of obtaining similarity solutions. Apparently both techniques have their ad-
Chemical Engineering Fundamentals Review
vantages, and both the report and discussion are excellent contributions. Some numerical solutions for special cases of the equations werealso reported ( 4 7 F ) . An interesting contribution (30F) was the study of equations for laminar flow over a plane surface when the outerflow streamlines are spiral in shape. A similarity transformation reduced the independent variables from 3 to 2, and a polynomial fit to velocity profile in one direction permitted solution by the usual integral techniques. The work has applications to flow in turbines. The three-dimensional incompressible laminar boundary layer on a spinning cone was studied ( 3 Q F ) . Navier-Stokes equations were formulated in a coordinate system appropriate to the cone. Boundary layer approximations and a similarity transformation then reduce the equations and their boundary conditions to the equations for a spinning disk found by von K&rm&nand integrated by Cochran. Unsteady Boundary Layers. I n an interesting experimental study of the unsteady turbulent boundary (17F) the work was done in a wind tunnel in which free stream velocity was sinusoidally varied with frequencies up to 48 C.P.S. and amplitudes up to 34% of the mean. Measurements of mean velocity and amplitudes of components of the periodic and turbulent velocities were measured in the boundary layer over a flat plate. Remarkably, for all conditions studied the effect of nonlinear interaction on mean velocity was quite small. Thus linear methods may often be used to predict characteristics of transient boundary layers. Other investigations in this category are theoretical. One (ZQF) took into account the effect of the leading edge on the growth of a laminar boundary layer on a flat plate set impulsively in motion parallel to itself. For x / & t > 2, the influence of the leading edge was small. Another study (40F) gives a simple method of calculating the development of thermal and momentum boundary layers on a heated cylinder of arbitrary shape. T h e method is an extension of Schuh’s and gives good results except for inaccurate prediction of the separation point.
Miscellaneous Boundary Layer Effects. The case of laminar flow of a fluid with a single reactant over a catalytic plate was studied (28F) by an extension of a method of Liepman. Results are similar to those of ChambrC and Acrivos for a uniform temperature wall with first-order kinetics but may differ for other conditions. The instantaneous reaction A B +. C in a laminar boundary layer in which A is a t the plate and B is in the free stream was analyzed (26F) by the
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integral boundary layer method. Results differ from those of film or penetration theory in a direction that depends on the ratio of diffusivities of A and B. The drag of a flat plate in almost-freemolecule flow was investigated (22F). The ratio of molecular mean free path to characteristic size of the plate is assumed of order unity or larger, but collisions of molecules in the gas were considered. Flow in Ducts. Previous investigators of fluid flow in nonrigid tubes had linearized the equations, but Lambert ( 7 Q F ) has given an analysis that retains nonlinearities and has shown that they are important. The author assumed incompressible, inviscid flow and neglected mass, bending stress, and axial stress of the tube. The solution is given in a form suitable for machine computation, and the results have application to blood flow. The problem of flow distribution to and from simple manifolds has been analyzed ( I F ) , Calculations are based on one-dimensional flow equations for manifolds of constant cross-section with uniformly distributed side flow. Departure from one-dimensional flow is accounted for by a constant determined experimentally from data on a manifold with one side stream. Predicted flow distributions agree quite well with the calculations. Flow through gauzes has been studied ( 8 F ) ; a gauze is defined as any nearly regular distribution of obstructions that lie in or near a single surface. Parameters are velocity profiles upstream and downstream of the gauze, shape of the gauze, and gauze characteristics. Knowledge of any three of these can be used to compute the fourth. The change in friction factor and turbulent velocity distribution in the entrance region of a pipe were calculated ( 7 I F ) by assuming potential flow in the core and a logarithmic velocity profile in the developing boundary layer. With the aid of light transmission through solutions of bentonite, Lindgren (27F) has studied the transition process in pipes. I t was concluded that the transition process depends on some physical, perhaps structural, property of the liquid other than kinematic viscosity. Porous Walls. Previous analyses of laminar flow in a uniformly porous channel employed perturbation techniques that limited the validity of the solutions to low values of wall velocity. Now solutions of the equation for arbitrarily large wall velocity have been obtained ( 3 8 F ) by assuming the stream function to be a product of functions of the normal and axial directions. At high wall Reynolds numbers the velocity profile for suction differs greatly from that with injection, which is not much changed. VOL. 52, NO. 4
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The laminar, compressible boundary layer over bodies of revolution with a normal velocity a t the wall has been analyzed ( 7 3 F ) . Using the perfect gas law and assuming that viscosity is proportional to temperature, Prandtl number is unity, and wall temperature gradient is zero, the equations were reduced to those for incompressible flow over a flat plate. The separation of gas mixtures of unequal molecular weight by suction of the thermal diffusion boundary layer was analyzed (23F). An estimate of the rate of attaining an equilibrium concentration profile is included.
Multiphase and Free-Boundary Flow Liquid sloshing in a partially filled tank has acquired sufficient importance in space technoIogy to receive thorough theoretical analysis, for which the model of an incompressible inviscid fluid with a free surface is apt. The motion of an inviscid fluid is described by Hamilton’s principle of analytical mechanics, from which Laurence and others (24G) formulated a variational principle leading in turn to a very useful Rayleigh-Ritz procedure for calculating sloshing modes and frequencies in a tank subjected to transient accelerations. Oscillations in a rotating tank have been shown (27G) to be more complex than in the nonrotating case, the motion no longer being irrotational. Significant advances were made again this year in the attack on water waves generated by wind. In Phillips’ promising theory (34G) (see also 1959 Fluid Dynamics Review) the resonance between turbulent pressure fluctuations and the modes of wave propagation is of first importance. The analysis has been extended to the problems of whitecaps (32G) and scattering of gravity waves by turbulent pressure fluctuations over the surface ( 3 3 G ) . Although of immediate interest in oceanography, these studies may find more general application in transfer of turbulence across fluid interfaces. In a continuation of Benjamin’s and his own work on the formation of waves in liquid running down a surface (1959 Fluid Dynamics Review), Binnie ( 5 G ) reported experiments on the onset of turbulence, which appeared as curious, intermittent turbulent bores moving downstream more slowly than the main flow. When an airborne body is heated excessively, its surface melts and the resulting melt is swept downstream and ablated from the body. This complex free-boundary flow, in which even the location of the free boundary is wholly unknown at the outset, was partially analyzed by Goodman (76G). An interesting free-boundary flow is the splitting of a liquid film emerging from between
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rollers, a vitally important process in lubrication, adhesion, and the application of coatings, paints, and printing inks. Sequence of action in splitting seems to be cavitation in the highly stressed regions, bubble expansion, and finally elongation and rupture of liquid filaments ( 2 9 G ) . A masterly experimental and theoretical study was made (38G) of the forced penetration of one fluid into a porous region already occupied by another fluid, a two-phase, free-boundary flow situation common in oil fields. After examining the theory of fluid flow in a compressible, porous medium, Scheidegger concluded ( 4 0 G ) that a complex problem involving consolidation must be solved, as the motion of the fluid cannot be separated from that of the medium. Vaporous cavitation aside, the role of free nuclei dominates over that of dissolved gases so far as cavitation in pumps, propellers, and other hydraulic rnachinery is concerned ( 2 2 G ) . Experiments on supercavitating flow about simple bodies were performed (47G) in a twodimensional free jet water tunnel.
Spheres, Bubbles, and Drops in Liquids. In addition to Jensen’s analysis of flow around a sphere (15B), Kynch’s analysis (23G) of the slow motion of two or more spheres through a viscous liquid is noteworthy. He solved for velocities as a function of separation and radii of the two spheres, commenting then on the three particle problem and on the applicability of his results to hindered settling. These results were not available to Eveson and others (72G), who found that experimental observations on two settling spheres did not bear out the predictions of theoreticians. Data were again reported (45G)on terminal velocities of liquid drops; this time the effects of high continuous phase viscosity and of low interfacial tension were investigated. The proposed correlation suffers from scant systematic attention to details of motion and dynamic interfacial properties. In an impressive investigation of the motion of rigid and fluid spheres in stationary and moving fluids inside tubes (79G), a comprehensive theoretical analysis backed by experiments was carried out. Detailed calculations of velocity fields are included, as is a thorough review of pertinent literature. After continuing valuable researches on drop mechanics, Bartok and Mason reported (7G) equations describing fluid motion inside and outside a fluid sphere suspended in plane hyperbolic and parallel shear flow. Even more interesting are experiments on shear-induced circulation, rotation, deformation, and twobody interactions of liquid droplets (ZG). I n these experiments, it was not too surprisingly found that age of the suspension affected coalescence of colliding
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fluid spheres. I n this connection, it was concluded ( 3 0 G ) that coalescence involves nonhydrodynamic factors. Sprays and Mists. Break-up of lowspeed liquid jets depends chiefly on surface tension forces, the surrounding gas having little effect. At higher speeds atomization changes owing to various tractions exerted by the gas; at first the jet acquires an irregular wavy profile leading to quicker complete disintegration, but a t still higher speeds multitudes of tiny droplets are torn off the whole surface before the main column has broken up. The process has been investigated by means of high-speed photographic (43G), conductivity (44G). and other techniques. In a related theoretical analysis the breakdown of a film issuing from a narrow slit was treated (25G), considering in particular a mode of deformation which leads to separation of an unstable filament from the surface. On theoretical grounds and in agreement with experimental results, it was predicted ( 8 G ) that periodic variations of liquid velocity and of gas density both cause a decrease in drop sizes produced in jet break-up. The importance of the conrinuous phase in spray formation was also emphasized by Ranz (35G), who approached the problem in anothei way; he injected liquid jets into another liquid. The velocities were high enough that the jets were largely broken up by inertial forces arising in the continuous phase, and the process was slow enough that it could relatively easily be photographed in detail (see figure). The results were analyzed in terms of characteristic stresses, drop size, induced motion of the continuous phase, and development of the spray zone. Particle impaction on spheres a t high Mach numbers was studied ( 7 5 G ) , setting u p equations of motion of a small particle traversing the detached shockwave (bow wave) upStream of the forward stagnation point. Flow of Suspensions. By considering in the creeping motion approximation the dissipation of energy due to solid particles suspended in a viscous liquid, Brenner (6G) computed the permanent pressure drop in flow through a fluidized bed and also arrived at Einstein’s formula for apparent viscosity of a dilute suspension of spherical particles. The equivalence of the different viewpoints of Einstein, Jeffrey, and Burghers was shown. The viscosity of dilute suspensions was examined experimentally (ZDG) ; in the capillary viscometer a wall effect was found that curiously causes apparent viscosity to decrease with decreasing tube size. The Einstein relation was confirmed. Drawing on his theory of isolated bubble motion (48G), Zwick (49G) devised a statistical model of a liquidbubble mixture which permits treatment as a continuum [an approach to be com-
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Atomization a t surface of liquid-intoliquid jet (35G)
4 Interferogram of detonation wave in 2Hz O2 a t 65 mm. of mercury ( I 4J)
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Density increase moves fringes downward. Note that front consists of crumpled shock followed b y reactive rarefaction zone
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pared with Oldroyd's (37G)J and applied this in a mathematical description of bubble swarm behavior in a tank undergoing forced vertical vibrations. H e predicted formation of permanent clusters of bubbles at certain intermediate depths, in agreement with experimental observations. Four different regimes were noted (47G) in the flow of fluidized solids through horizontal pipes, each with a counterpart in gas-liquid flow. The analogy is not so close in considering flow of granular solids through orifices (74G). The mechanism by which a dense bed of sand passes through a fairly small hole is rather complicated. Gas- and Liquid-Liquid Flow in Pipes. Dimensional reasoning was used (78G, 28G) to arrive at an odd representation of various flow regimes a6 regions in a tetrahedron constructed on the Reynolds, Weber, and Froude numbers. Data on pressure drop and hold-up in gas-liquid flow in horizontal pipes were a significant contribution (27G). I t was found that although the LockhartMartinelli correlation is verified for plug, slug, and froth flow at atmospheric pressure, it is otherwise inadequate. New correlations were proposed. In continuation of earlier work, the effect of diameter was determined (77G) on flow pattern, hold-up, and pressure drop in upward flow in vertical pipes. Data were obtained a t 36 p.s.i.a. and may be useful in finding the effect of gas density. Similar experiments were made by Ueda (42G), but he also measured the static pressure along the pipe and looked into the effects of obstructions, too.
Under some conditions it is evidently possible to reduce appreciably the pressure drop in oil pipelines by injecting water into the flow, and this has excited interest in liquid-liquid flows in horizontal pipes. O n the assumption that water forms a continuous annulus around the oil, Russell and Charles (36G) predicted the reduction in pressure drop attendant upon injecting water into a pipeline. Their prediction greatly exceeds the reduction usually observed; it was concluded that the actual twophase flow is usually somewhere between the annular and stratified regimes. The possibility of coarse oil-in-water emulsions, which have relatively low effective viscosities, was not mentioned. A mathematical problem was solved ( 7 7G) representing laminar, one-dimensional flow of two viscous fluids of different densities through a circular or rectangular conduit. Numerical examples considered flow of water and benzene and of water and air.
Gas Dynamics This section is concerned primarily with compressibility effects in steady flow. Although this subject occupies prime position in fluid mechanics, it is related mostly to problems of high-speed flight; consequently, as in previous years, only a representative sample is included to give the reader an impression of the advances made in those branches of most interest to chemical engineering. General. As a reflection of current interests, recent gas dynamics progress has been made mostly in the field of
hypersonics. Present status has been discussed comprehensively in textbook form ( 6 H ) . The potential theory of unsteady, supersonic flow has been put together (72H) in a book containing 12 chapters on linearized, unsteady flow problems and one on nonlinear problems. Internal Flow. T o obviate difficulties associated with the appearance of weak shocks, unadmissible by conventional linearized theory, Mahony ( Q H ) has presented a scheme of approximate solution for supersonic flow in a circular duct of slowly varying cross section. It is essentially an extension of Whitham's modification of linearized theory to allow for convergence of Mach lines. External Flow. A detailed review of the perturbation problem for steady, plane transonic flow has been given by Manwell (70H). H e discusses in particular the conditions under which the perturbation solution will break down in the vicinity of a body. Supersonic Flow. A comprehensive report on the extensive experimental program at the Jet Propulsion Laboratory has been presented ( 4 H ) . I t concerns the effect of uniformly distributed sand-grain roughness on skin friction drag and boundary layer velocity profile at supersonic speeds. The results show that as long as the quadratic resistanco law holds up to speeds corresponding te Mach 5, skin-friction drag is exactly the same as in the incompressible case, while compressibility effects occur only in a reduction of the fluid density at the surface. Results of a similar program of investigation carried out at the NASA Ames Laboratory have been reported (77H). Local skin friction in turbulent boundary layers is measured from Mach 0.2 to 9.9 and Reynolds numbers up to 1 0 8 by means of a floating element in a smooth, flat surface using air and helium as working fluid. Other experimental results have been obtained (73") in the course of exploring the flow field in the vicinity of a rectangular depression on a flat plate in a supersonic stream. The study reveals some details of shock formation and the generation of local vorticity. For obvious reasons, the blunt-body problem attracted a good deal of attention. A comprehensive survey of existing analytical treatments was presented (77H),in which it was concluded that none is adequate for predicting details of the flow field. Then a numerical procedure was described suitable for solution of the full inviscid equations with a medium-size electronic computer. Experimental study (5H)of hypersonic viscous effects on a flat plate with finite leading edge was performed in the helium wind tunnel at Princeton. Previous measurements made at Mach 11 to 15 indicated that surface pressures are conVOL. 52, NO. 4
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siderably higher than those predicted by available analytical methods. This was confirmed again by the present results, which extended operating conditions u p to Mach 20 and Reynolds numbers between 0.5 X 106 and 2.0 X 106 per inch.
Gas Wave Dynamics
Basic Aspects. The most significant event has been the translation of the remarkable text of Stanyukovich ( 7 3 4 . The volume represents a comprehensive coverage of the whole subject matter starting right from the introduction of mathematical techniques and basic thermodynamic concepts needed for the development of the analysis of unsteady flow phenomena. The wave system has been treated (72J)as the solution of quasi-linear equations satisfying certain functional relations between dependent variables imposed by the nature of the medium and the process. The report is concerned primarily with isentropic simple waves in a polytropic gas. I n continuation of his work on the approximate theory of “shock dynamics,” Whitham ( 7 5 J ) ,has extended the scope of his treatment to three-dimensional problems. The theory is based upon the premise that there exists a functional relation between the strength of the shock wave a t any point and the area of the “ray tube” or stream tube. The main applications discussed are the diffraction by a slender axisymmetrical body and the stability of the plane shock. Shock Thermodynamics. The application of shock tube techniques for studying of hypersonic flow has been described in a comprehensive report ( 7 7 4 . The facility described permits the attainment of shock waves in air moving with a velocity of u p to 55,000 feet per second, corresponding to a calculated equilibrium temperature of 16,000” K. produced in the shock tube. Observations of detached shock wave on blunt body indicates the attainment of flow Mach number of 19.6. Concerned with a similar application of shock tubes is a report (76J) in which an increased nominal testing time is obtained by the use of a two-diaphragm shock tube, where the propagation of the primary contact surface is delayed by the reflected shock wave from the diaphragm a t the inlet to the tunnel nozzle. I n continuation of studies on radiation from hot air, Keck and others (8J) present results of absolute intensity measurement of radiation emitted by shockheated oxygen, nitrogen, and air a t temperatures between 4000’ and 9000° K., by means of both spectroscopic and photometric techniques. Waves with Central Symmetry.
Ex-
istence of the Cauchy problem for non-
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steady motion of the gas in front of a spherically expanding piston has been discussed ( 6 J ) in terms of particular self-similar solutions. Sedov’s similarity solutions of the point-blast problem have been generalized (9J)by considering three new forms of the equation of state. The details of the motion induced by a blast wave from a spherical charge have been explored by Brode ( 7 J ) using a numerical solution of the equations of motion based on the von NeumannRichtmyer viscosity approximation.
Dynamics of Reactive Fluids T h e main topic in this branch of fluid dynamics is, of course, combustion. Progress made over the last two years has been reviewed ( S K ) . Consequently, only publications of more fundamental character are included here. T h e trend of “progress by symposia,” so characteristic in this field of fluid mechanics, has produced two significant publications: proceedings of the Seventh Symposium on Combustion (79K) and of the Third AGARD Colloquium on Combustion and Propulsion (4K). The first volume contains 124 technical reports with records of discussions. The most significant advances were made in the field of kinetics and in wave dynamics, especially in connection with the development of detonation. The AGARD Colloquium presented a cross-section of recent work on propulsion systems and related problems, such as component interaction, noise, and flame stabilization. Modern topics in high speed aerodynamics, such as heat transfer, shock kinetics, and magnetic effects were also discussed. Proceedings of the session on combustion of gaseous mixtures a t the 16th International Congress of Pure and Applied Chemistry have been published (74K). There are a number of significant contributions on flame kinetics, propagation, and structure. I n a critical review of the current status of combustion studies, Emmons (GK) discussed the nature of problems posed by combustion processes and their applications. H e showed that the large number of variables introduced by chemical effects, heat and mass transfer, and combustion chamber geometry constitute limiting factors which must be tackled if this science is to progress. Steady Flow Phenomena. The flow field of flame stabilized in a two-dimensional tunnel has been studied by Uberoi (27K). This work, together with his previous publication on Bunsen flames (1959 Fluid Dynamics Review) are of fundamental importance to the aerodynamics of laminar flames. T h e study of non-
INDUSTRIAL A N D ENGINEERING CHEMISTRY
equilibrium flow of an ideal dissociating gas has been further advanced ( 8 K ) in continuation of a research problem on the effects of irreversible chemical reactions on flow through shock wave and past spheres. Most significant is the decrease in stand-off distance of the shock from the blunt body because of reaction effects. A thorough experimental and analytical study of supersonic nozzle flow with a reacting gas mixture was reported (23K). Flows were produced either in chemical equilibrium or in states between this and frozen flow. Measured flow parameters agreed well with those calculated on the basis of independent optical absorption measurements. Optical studies of flame structure were performed (75K)by means of the “inclined slit” method permitting direct observation of temperature distribution. This was combined with burning velocity measurements made by the particletrack technique to determine distribution of heat release rates. S t a b i l i t y Considerations. Energy transfer through a dissociated diatomic gas in Couette flow has been analyzed ( 3 K ) . Particular attention was directed to the interrelation between heat transfer to the wall and reaction rate for two idealized cases : chemical equilibrium and chemically frozen flow. I n continuation of his analysis of heatdriven oscillations of gas tube flow, Merk (77K) considered the influence of heat transfer on combustion stability. The characteristic speed defined by equations of motion for a chemically reacting gas has been shown (2K) to be the high frequency limit of sound wave phase velocity if the effects of viscosity and heat conduction are neglected and the reaction rate is finite. At infinite reaction rates-chemical equilibriumthe characteristic speed changes suddenly to the low frequency velocity of sound. An experimental study (26K) of highfrequency combustion oscillations in rocket engines has led to an interesting controversy with Crocco on the utility of his concept of time lags for analysis of combustion stability.
Wave Propagation and Detonation. I n continuation of an analysis of the structure of detonation waves ( 5 K ) the effects of viscosity, diffusion, and heat conductivity were discussed. The current analysis is based on a simplified model of the reaction wave, emphasizing the effect of ignition temperature. The concept is that below ignition temperature reaction rates are zero, while a t temperatures above this threshold reaction rates are only functions of chemical composition and are independent of temperature. An interesting result is the proof that, if ignition temperature is above that of the Von Neumann spike, detonation wave thickness is of the order
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of one mean free path. An experimental study of detonation wave structure ( ? 3 K ) , 12th of a series, deals with spectroscopic measurements of the ultraviolet absorption of hydroxyl from radicals in hydrogen-oxygen detonations. The equipment permits a time resolution of a few microseconds. T h e transition from deflagration to detonation has been analyzed (7ZIC) for the simplified model of a flame front preceded by a pressure fan under the assumption that absolute flame velocity is proportional to the mass velocity of the gas into which the flame propagates. Induction distance is then computed for two cases: constant flame acceleration and acceleration varying linearly with time. T h e development of detonation was also observed (76K) by means of streak pictures and flash photography. Main purpose of the study was to check the “hot gas piston” model of Zeldovich. While, according to Zeldovich, the accelerating flame front resembles the velocity profile in the tube, ample evidence was obtained that the flame propagates primarily because of turbulent action in the boundary layer, giving it a scoop shape. Strong warning has been given (7K, 78K) against the too-literal, one-dimensional interpretation of data on detonations in tubes. T h e two-dimensional flow effects can be apparently of a sufficient order of magnitude to render invalid conclusions based on exact onedimensional evaluation of propagation velocity. An apparent evidence of detonation overpressures was presented ( I K ) on the basis of records obtained for hydrogen-oxygen mixtures with a pressure transducer mounted at the end of a detonation tube.
Dynamics of Conducting Fluids
Comprehensive Publications. After i
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the initial phase of rapid growth the stage of stock-taking has been reached. This is reflected in the program of the last Lockheed Symposium (7L), which was made up of a number of authoritative reports summarizing the present status of the art. I t is also significant that almost two full issues of Physics of Fluids were devoted to the report of advances made by Project Matterhorn. They start with the general description of the Stellarator concept given by its originator, Spitzer (77L). Some general features of problem formulation in magneto-gas dynamics have been considered (6E, 7L),with proper account taken of finite conductivity. For nonsteady one-dimensional flow the problem can be solved by approximating its description in terms of the hyperbolic set of equations, while for steady two-
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dimensional flow, conditions for potential motion are shown to be sufficient for investigating its most interesting characteristics. Steady Flow. Calculations of magneto-gas dynamics effects in hypersonic Couette flow have been reported (215). By comparison with an exact solution obtained in the absence of magnetic field, it is shown that relatively weak fields produce large increases in total drag accompanied with a large reduction in skin friction. This implies that total drag in the present case is due primarily to magnetic effects. A fundamental experimental and analytical study on magnetohydrodynamic shock-tube flow has been carried out (74.L). It was demonstrated that magnetic fields produce effects similar to viscous friction, and evidence was given of the reduction in electrical conductivity of the gas because of the presence of the magnetic field. The latter was ascribed
either to Hall currents generated by charged particle drift across magnetic field lines or to ion slip whereby ions and neutral particles travel through the field at different speeds. Wave Dynamics. An interesting analysis of the structure of oblique shocks in plane hydromagnetic flow has been performed (9L,I O L ) . The basic equations in the absence of heat conduction and viscosity are satisfied by relations between dependent variables whose topological properties are such that the shock condition does not ensure uniqueness either of final state nor of transition. T o describe the propagation of small disturbances the roots of solutions of the linear equations were examined, and the existence of various modes of propagation was demonstrated. Magnetohydrodynamic flow in a shock tube has been studied by Mitchner (72L) who generalized Riemann invariances to a form applicable to plane flows with
Miscellaneous Publications in Fluid Dynamics
Subject Equations of Motion and Stability Solutions of boundary-value problems in flow of inviscid, compressible fluid past a wedge Solutions for inviscid rotational flow with corner eddies which describe effect of upstream vorticity on eddies preceding sudden duct contraction Inviscid model for elucidating inhibitory influences of rotational, gravitational, and electromagnetic fields on fluid motion Exact solutions for incompressible fluids of high conductivity Review of hydrodynamic impact Analysis of impact of liquid on solid walls Analysis of creeping flow past cubic array of spheres Drag and lift coefficients for lattice of elliptic cylinders Drag and lift coefficients for randomly distributed spheres or parallel circular cylinders Drag coefficient of lattice of parallel glass fibers at low Reynolds numbers Stability of axially symmetric disturbances in Poiseuille flow Stability of viscous flow between horizontal concentric cylinders Problems of hydrodynamic instability Analysis of stability in wakes, including wake behind cylinder Turbulence Review of basic theory, turbulent boundary layers, and recent theories Vortex Flow and Rotation Measurements of circulation and vorticity in several types of vortices Experimental and theoretical study of flow over enclosed rotating disk Calculation of drag on rough rotating disk by assuming logarithmic velocity profile Calculation of drag and heat transfer in turbulent boundary layer of rapidly rotating disk by Reynolds’ analogy and von Ktrrmln velocity profile Jets and Wakes Transverse velocity component in plane, compressible jet Graphical method for computing gas dynamic jet mixing ; special consideration of thrust augmentation Flow Near Solid Surfaces Four alternative treatments of displacement thickness of boundary layers and wakes for two-dimensional, laminar, or turbulent, incompressible flow; extension of three-dimensional boundary layers Auxiliary equation for approximate calculation of growth of turbulent boundary layers with adverse pressure gradient Incompressible laminar flow near stagnation point of cylinder having arbitrary transverse motion Unsteady flow over inclined porous flat plate Mass transfer in laminar boundary layers Theory and application of fluid meters (revision of book) VOL. 52.
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Miscellaneous Publications in Fluid Dynamics Subject Flow Near Solid Surfaces Analysis of dispersion of light, heavy, and neutral density particles in channel flow Charts for eliminating calculations of compressible flow in pipes when upstream conditions are calculated from downstream conditions Suppression of turbulence in pipe flow of mercury by axial magnetic field Influence of pipe curvature on friction and transition in turbulent flow Mean velocity profiles for laminar, transition, and low turbulent flow between concentric tubes Including lubricant inertia in basic equations for hydrodynamic theory of lubrication increases calculated capacity slightly Effect of turbulence on lubrication Multiphase and Free-Boundary Flow Methods of observing wave motion with light polarization technique Effects of shape and Reynolds number on drag for single, freely oriented solid body Measurement and correlation of size distributions of drops obtained by injecting molten wax into flow Quick method for determining drop-size distribution in certain sprays Method for calculating efficiency of particle collection by falling drops, assuming particles having negligible inertia and subject to solenoidal force field Turbulent flow of suspensions, with results as friction factor us. Reynolds number for various volume concentrations Review of gas-liquid flow, especially flow regimes and pressure drop in horizontal and vertical conduits Gas-liquid flow in annular regime Pressure drop and hold-up in stratified and bubble flow regimes in I-inch diameter tube Gas Dynamics Theory of Lava1 nozzle Fundamental aspects of transonic flows ; degenerate solutions Approximate similarity solution for transonic flows Approximate procedure for transonic flow past biconvex, circular arc airfoil; determination of locations of sonic line and shock wave limiting supersonic regime Similarity solution for transonic flow past finite wedges by application of Weber-Orr transform to linearized perturbation equations Analysis of flow at two-dimensional air intake in sonic stream by application of transonic small-disturbance theory Analysis of flow between detached shock wave and symmetrical profile Gas Wave Dynamics Thermal radiation front penetrating into cold fluid Determination of motion of plane shock reflected from conducting wall and interacting with wave system generated by decaying contact region Detonation waves ; shock tube techniques for determining equilibrium equation of state data Influence of diaphragm opening time on shock-tube flow Thermal decomposition of NO2 Study of performance of shock tubes at low initial pressures by electron beam densitometer Interferometric measurement of dissociation and recombination rates of 0 2 Dynamics of Reactive Fluids Experimental studies of turbulent flames with two-dimensional open burner Optical studies of turbulent flames Optical observations and ionization probe measurements of turbulent flames Statistical formalism for describing behavior of sprays and spray combustion Simplified analysis of processes in liquid bipropellant rocket motors Review of recent advances in gaseous detonation Dynamics of Conducting Fluids Introduction to magnetohydrodynamics Survey of literature on magnetohydrodynamics Steady, plane flow of incompressible fluid around thin cylindrical obstacles with uniform magnetic fields either parallel or perpendicular to undisturbed uniform stream Two-dimensional flow of incompressible, constant-conductivity fluid through elliptically shaped solenoid producing constant magnetic field normal to flow field Effect of magnetic field on flow of conducting fluid with free surface Analysis of flow of electricallv conducting- fluid through duct with transverse magnetic field Propagation of one-dimensional flow produced by strong shock in infinitely conductive gas subject to forces of gravity and transverse magnetic field
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transverse magnetic fields. A study was made (3L) of the structure of a steady magnetohydrodynamic switch-on shock wave-a shock where velocity and magnetic field are normal to its plane on the upstream side. They become oblique to it on the downstream side. Such turning, which is not possible in conventional gas dynamics, is accomplished by Maxwell stresses parallel to the shock. Thus the shock “switches on” the transverse components of velocity and magnetic field. T h e investigation of wave propagation and structure in magnetohydrodynamics led (IgL) to the examination of some general properties of wave propagation. It was demonstrated that different levels of approximation to the governing equations may lead to completely different types of wave motions. For complex sets of waves, propagation speeds, defined by the highest derivatives in the equations, may be quite different from speeds at which the main disturbance travels, making the unraveling of experimental observations quite difficult.
Literature Cited Comprehensive Publications (1A) “1959 Heat Transfer and Fluid Mechanics Institute,” Stanford University Press, Stanford, Calif., 1959. (2A) Kaufmann, Mi., “Technical Hydroand Aeromechanics (Technische Hydround Aeromechanik),” 2nd ed., Springer, Berlin, 1958. (3A) Knudsen, J. G., Katz, D. L., “Fluid Dynamics and Heat Transfer,” McGrawHill, New York, 1958. (4A) “Proc. Sixth Midwestern Conf. Fluid Mechanics,” University of Texas, Austin, Tex., 1959. (5A) Sedov, L. I., “Methods of Similarity and Dimensional Analysis in Mechanics,” 4th ed., Moscow, 1957; transl. by B. H. Cleaver-Hume, Blackwell’s, Oxford, 1959. (6A) Streeter, V. L., “Fluid Mechanics,” 2nd ed., McGraw-Hill, New York, 1958. (7A) Temple, G., “An Introduction to Fluid Dynamics,” Clarendon Press, Oxford, 1958. Equations of Motion and Stability (1B) Arkhipov, V. N., Soviet Phys. Doklady 3, 1117-19 (1958). (2B) Brewster, D. B., Grosberg, P., Nissan, A. H., Proc. Roy. Soc. (London) 251A, 76-91 (1959). (3B) Carrier, G. F., Chang, C. T., Quart. App1. Math. 16,436-9 (1959). (4B) Chandrasekhar, S., Proc. Roy. Sac. (London) 246A, 301-11 (1958). (5B) Christopherson, D. G., Dowson, D., Zbid., 251A, 550-64 (1959). (6B) Corcos, G. M., Sellars, J. R., J. Fluid Mech. 5 , 97-112 (1959). (7B) Donnelly, R. J., Proc. Roy. Soc. (London) 246A, 312-25 (1958). (8B) Fultz, D.. J. Meteorol. 16, 199-208 (1959). (9B) Gross, E. P., Jackson, E. A., Phys. of Fluids 1, 318-28 (1958). (10B) Happel, J., A.I.Ch.E.Journal5,174-7 11959’1. - - ,\ - -
(11B) Hasimoto, H., J. Fluid Mech. 5 , 317-28 (1959).
an (12B) Howard, L. N., J. Math. Phys. 37, 283-98 (1959). (13B) Hromas, L., Degroff, H., J . Fluid Mech. 5, 140-50 (1959). (14B) Ivanova, L. S., J. Appl. Math. and Mech. (U.S.S.R.) 22, No. 2, 344-8 (1958). (15B) Jensen, V. G., Proc. Roy. Soc. (Lon. don) 249A. 346-66 (1959). (16B)’ Kuwabara, S.,~ J . Phys. SOC.Japan 14, 522-6 (1959). (17B) Ibid., pp. 527-32. (18B) Leite, R. J., J . Fluid Mech. 5, 81-96 (1959). (l
