Showing papers in "Annual Review of Fluid Mechanics in 1972"

One-Dimensional Flow of Liquids Containing Small Gas Bubbles

Abstract: Early papers on the subject matter of this review, such as Mallock's (1910) "The Damping of Sound in Frothy Liquids" or Minnaert's (1933) "Musical Air Bubbles and the Sound of Running Water," were written out of interest in physical phenomena. In papers written during and after World War II this interest was mixed with motives of a more grim nature because it was sought to take advantage of the acoustical properties of bubbly liquids to control the sound produced by propellers, both of surface ships and sub­ merged ships. The literature on the subject deals therefore for a large part with theoretical and experimental studies of the various aspects of propaga­ tion of pressure waves of small amplitude in bubbly liquids. A major part of this review will be devoted to this topic, Sections 2-4. Apart from this, various schemes have been envisaged in which the dynamic properties of bubbly liquids could be profitably used in the design of devices to propel high-speed waterborne craft, such as hydrofoil and aircushion boats. This made it valuable to investigate the flow of bubbly liquids through Laval nozzles and similar configurations. This type of flow will be discussed in Section 5. In recent years research not directed by the need for data on acoustical properties has been carried out on waves in bubbly fluids beyond the acoustical regime, where disturbances to a given quiescent state are no longer of infinitesimal smallness. The effects of nonlinearity, dispersion, and dissipation, discussed in Section 6, lead as in other areas, such as those of plasma waves and water waves, to mathematical description by Burgers' equation and that of Korteweg and de Vries. Nonlinearity, dispersion, and dissipation cooperate in the formation of shock waves in bubbly liquids, the structure of which has been investigated to some extent, as reviewed in Section 6. The present review is restricted to flows described along the above lines. This means that bubbly flow forming the coolant of nuclear reactors is not included. There the bubbles are mostly large vapor bubbles and the interest is primarily in heat transfer.

Oil Spreading on the Sea

Abstract: winds and currents, and the second of the increase in area of the oil due to the tendency of the oil to spread in calm water. The drift due to wind may be estimated by arg uin g that the turbulent­ shear-stress law at the water interface is approximately the same in both the air and the water. If the wind velocity some distance (usually 10 meters) above the water surface is U, then the turbulent stress is

Self-Similar Solutions as Intermediate Asymptotics

Abstract: Hydrodynamics, like other branches of continuum mechanics, has long been dealing with particular solutions of the special type m(t)F[x/l(t)], where x and t are independent variables that may]someUmes.}be interpreted as a spatial coordinate and time, and re(t) and l(¢) are time-dependent scales of the unknown variable and the spatial coordinate. The name of these solutions, self-similar, comes from the fact that the spatial distribution of the characteristics of motion, i.e., of the dependent variables, remains similar to itself at all times during the motion. Self-similar solutions always represent solutions of degenerate problems for which all constant parameters entering the in]’,tlal and boundary conditions and having the dimensions of the independent variables vanish or become infinite. Interest in self-similar solutions was originally due to the fact that they are representable in terms of functions of one variable. Hence the solution of partial[ differential equations for the corresponding problems is reduced to the solution of ordinary differential equations, and this was thought to be simpler according to the hierarchy of complexity existing at that time. By this means it was possible,to obtain a numberof exact special solutions, sometimes even in analytic form. F.or linear equations, the principle of superposition is valid and these exact solutions can be used directly in constructing solutions of more general problems, such as the self-similar fundamental solution of the inltial-value problem for the heat-conductlon equation. But what is the place of self-slmilar solutions in the case of the general nonlinear problem? Much later it was realized that self-slmilar solutions do not represent merely specific examples. In actual fact they describe the \"intermediateasymptotic\" behavior of solutions of wider classes of initial, boundary, and mixed problems, i.e., they describe the behavior of these solutions away from the boundaries of the region of independent variables or, alternatively, in the region where in a sense the solution is no longer dependent on the details of the initial and/or boundary conditions but the system is still far from being in a state of equilibrium.

Mantle Convection and the New Global Tectonics

Abstract: Within the last five years new unifying concepts in geology and geo­ physics have won general acceptance. These concepts explain the origin of earthquakes, volcanism, and mountain building as well as the evolution of oceanic and continental crust. It is postulated that the surface of the earth is divided into a series of plates that are in relative motion with respect to each other. Earthquakes, volcanism, and mountain building occur on plate margins. Most geophysicists accept that some form of thermal convection drives the motion of the plates. The hot crystalline upper mantle behaves like a highly viscous fluid, owing to solid-state creep processes. Heating from the decay of radioactive elements drives thermal convection cells. Observations of surface heat flux, surface topography, and variations in the surface gravitational field may be compared with predictions of convection theories. Seismic observations and surface measurements of variations of magnetic and electric fields yield information on the properties of the earth's interior that must be consistent with the predicted convective flows.

Cavity and Wake Flows

Abstract: The phenomenon of wake formation behind a body moving through a fluid, and the associated resistance of fluids, must have been one of the oldest experiences of man. From an analytical point of view, it is also one of the most difficult problems in fluid mechanics. Rayleigh, in his 1876 paper, observed that "there is no part of hydrodynamics more perplexing to the student than that which treats of the resistance of fluids." This insight of Rayleigh is so penetrating that the march of time has virtually left no mark on its validity even today, and likely still for some time to come. The first major step concerning the resistance of fluids was made over a century ago when Kirchhoff (1869) introduced an idealized inviscid-flow model with free streamlines (or surfaces of discontinuity) and employed (for steady, plane flows) the ingenious conformal-mapping technique that had been invented a short time earlier by Helmholtz (1868) for treating two-dimensional jets formed by free streamlines. This pioneering work offered an alternative to the classical paradox of D’Alembert (or the absence of resistance) and laid the foundation of the free-streamline theory. We appreciate the profound insight of these celebrated works even more when we consider that their basic idea about wakes and jets, based on a construction with surfaces of discontinuity, was formed decades before laminar and turbulent flows were distinguished by Reynolds (1883), and long before the fundamental concepts of boundary-layer theory and flow separation were established by Prandtl (1904a). However, there have been some questions raised in the past, and still today, about the validity of the Kirchhoff flow for the approximate calculation of resistance. Historically there is little doubt that in constructing the flow model Kirchhoff was thinking of the wake in a single-phase fluid, and not at all of the vapor-gas cavity in a liquid; hence the arguments, both for and against the Kirchhoff flow, should be viewed in this light. On this basis, an important observation was made by Sir William Thomson, later Lord Kelvin (see Rayleigh 1876) "that motions involving a surface of separation are unstable" (we infer that instability here includes the viscous effect). Regarding this comment Rayleigh asked "whether the calculations of resistance are materially affected by this circumstance as the pressures experienced must be nearly independent of what happens at some distance in the rear of the obstacle, where the instability would first begin to manifest itself." This discussion undoubtedly widened the original scope, brought the wake analysis closer to reality, and hence should influence the course of further developments. An expanded discussion essentially along these lines was given by Levi-Civita (1907) and was included in the survey by Goldstein (1969). Another point of fundamental importance is whether the Kirchhoff flow is the only correct Euler (or outer) limit of the Navier-Stokes solution to steady flow at high Reynolds numbers. If so, then a second difficulty arises, a consequence of the following argument: We know that the width of the Kirchhoff wake grows parabolically with the downstream distance x, at a rate independent of the (kinematic) viscosity u. If Prandtl’s boundary-layer theory is then applied to smooth out the discontinuity (i.e. the vortex sheet) between the wake and the potential flow, one obtains a laminar shear layer whose thickness grows like (ux/U)^-1/2 in a free stream of velocity U. Hence, for sufficiently small u/U the shear layers do not meet, so that the wake bubble remains infinitely long at a finite Reynolds number, a result not supported by experience. (For more details see Lagerstrom 1964, before p. 106, 131; Kaplun 1967, Part II.) The weaknesses in the above argument appear to lie in the two primary suppositions that, first, the free shear layer enveloping the wake would remain stable indefinitely, and second (perhaps a less serious one), the boundary-layer approximation would be valid along the infinitely long wake boundary. Reattachment of two turbulent shear layers, for instance, is possible since their thickness grows linearly with x. By and large, various criticisms, of the Kirchhoff flow model have led to constructive refinements of the free-streamline theory rather than to a weakening of the foundation of the theory as a valuable idealization. The major development in this direction has been based on the observation that the wake bubble is finite in size at high Reynolds numbers. (The wake bubble, or the near-wake, means, in the ordinary physical sense, the region of closed streamlines behind the body as characterized by a constant or nearly constant pressure.) To facilitate the mathematical analysis of flows with a finite wake bubble, a number of potential-flow models have been introduced to give the near-wake a definite configuration as an approximation to the inviscid outer flow. These theoretical models will be discussed explicitly later. It suffices to note here that all these models, even though artificial to various degrees, are aimed at admitting the near-wake pressure coefficient as a single free parameter of the flow, thus providing a satisfactory solution to the state of motion in the near part of the wake attached to the body. On the whole, their utility is established by their capability of bringing the results of potential theory of inviscid flows into better agreement with experimental measurements in fluids of small viscosity. The cavity flow also has a long, active history. Already in 1754, Euler, in connection with his study of turbines, realized that vapor cavitation may likely occur in a water stream at high speeds. In investigating the cause of the racing of a ship propeller, Reynolds (1873) observed the phenomenon of cavitation at the propeller blades. After the turn of this century, numerous investigations of cavitation and cavity flows were stimulated by studies of ship propellers, turbomachinery, hydrofoils, and other engineering developments. Important concepts in this subject began to appear about fifty years ago. In an extensive study of the cavitation of water turbines, Thoma (1926) introduced the cavitation number (the underpressure coefficient of the vapor phase) as the principal similarity parameter, which has ever since played a central role in small-bubble cavitation as well as in well-developed cavity flows. Applications of free-streamline theory to finite-cavity flows have attracted much mathematical interest and also provided valuable information for engineering purposes. Although the wake interpretation of the flow models used to be standard, experimental verifications generally indicate that the theoretical predictions by these finite-wake models are satisfactory to the same degree for both wake and cavity flows. This fact, however, has not been widely recognized and some confusion still exists. As a possible explanation, it is quite plausible that even for the wake in a single-phase flow, the kinetic energy of the viscous flow within the wake bubble is small, thus keeping the pressure almost unchanged throughout. Although this review gives more emphasis to cavity flows, several basic aspects of cavity and wake flows can be effectively discussed together since they are found to have many important features in common, or in close analogy. This is in spite of relatively minor differences that arise from new physical effects, such as gravity, surface tension, thermodynamics of phase transition, density ratio and viscosity ratio of the two phases, etc., that are intrinsic only to cavity flows. Based on this approach, attempts will be made to give a brief survey of the physical background, a general discussion of the free-streamline theory, some comments on the problems and issues of current interest, and to point out some basic problems yet to be resolved. In view of the vast scope of this subject and the voluminous literature, efforts will not be aimed at completeness, but rather on selective interests. Extensive review of the literature up to the 1960s may be found in recent expositions by Birkhoff & Zarantonello (1957), Gilbarg (1960), Gurevich (1961), Wehausen (1965), Sedov (1966), Wu (1968), Robertson & Wislicenus (1969), and (1961).

Wing-Body Aerodynamic Interaction

Abstract: that has proved most interesting to the theoretical aerodynamicist. Clea rly the former is concerned with the flow over, and the pressures, forces, and moments experienced by, generally shaped streamlined bodies and multiple lifting surfaces (wings, tails, con­ trols) in intimate combination. Until recently he has had to place primary reliance for such informa tion on measurements by wind tunnel and other experimental means. Except in particular cases, he has obtained only a rather piecemeal and preliminary sort of guidance from the produc ts of rational theory, whose focus has been sing le, isolated, planar wings and slender bodies of revolution. Interference was the word chosen to describe situations where flow over one lifting or body element was significantly affected by the presence of an­ other-as if, somehow, this interaction interfered with the purist's more appealing efforts to analyze each aerodynamic entity standing alone. It is no accident that 32 yea rs intervened between the attainment of powered air­ craft flight and the first appearance of even a short , descriptive chapter on flow over the "Airplane as a W hole " (Durand 1935). Many additional years preceded the pub lication of two truly comprehensive survey articles (Law­ rence & Flax 19 54, Fe rrar i 1957). Most of the basic tools of interference or in­ teraction theory, as it exists today, were understood by the latter date. With regard to the complete description of fluid motion past an assemblage of streamlined elements, notable subseque nt progress has been made toward realizing the full potentialities of this theory by means of successive gen­ erations of high-speed data-proces sing equipment. The limited objective of the present authors will be to review and

Fluid Mechanics of Heat Disposal from Power Generation

Abstract: The problem of disposing of the waste heat produced as a result of the generation of electrical energy has its roots in the growth of demand for electrical power and in the economics of power production. Electrical gen­ erating capacity in the United States is expected to continue doubling each decade. As hydroelectric sites are developed to their full capacity, the de­ mand will be met increasingly by fossil-fuel and nuclear steam plants. Regional economics and economics of scale have resulted in increases in plant size and concentrations of waste heat. The efficiency of steam-electric power production by either fossil or nu­ clear fuels is governed by the thermodynamics of the heat cycle. The ideal or Camot efficiency is determined by the temperature of the heat source and by the temperature of the surrounding air or water, which acts as a heat sink. The ideal efficiency is given by

As Luck Would Have It-A Few Mathematical Reflections

Abstract: The kind invitation of the Editors has encouraged me to set down one or two personal reminiscences concerning some of those scientific discoveries with which I have been more or less directly connected. It may be worth­ while , I think, to examine a few circumstances that are the result of con­ temporary scientific thought. One phenomenon that has always struck me very forcibly is the impor­ tance of the role played by chance, not only in the actual discovery of certain scientific facts, for that is well known, but in the awarding of merit to those who originally make these discoveries-the attribution, for example, to such and such a scholar, of a result later proven to be essential. For example, it is most certainly not widely known that the seeds of a great number of the main findings of modern mathematics are to be found in the complete works of Cauchy. This illustrious mathematician has been the victim of his own vast output, a mass of work contained in an unbeliev­ able number of volumes, an entire library in themselves. The consequence of this enormous output is that few mathematicians have been able to acquire a detailed knowledge of the whole of Cauchy's thought. To give one instance, it is generally not known that the essential fact of Fredholm's theory of equations (1902) was familiar to Cauchy as far back as 1848. But all that is, however, of very little importance-the vital thing is that a useful discovery should be made, no matter by whom, and that the world should be informed of it and able to make use of it to push ahead even further. The question of priority remains ever secondary in the face of reality. I t so happens that I have known just such a case for having closely fol­ lowed it, a case that clearly demonstrates how much the true story of mathematical discovery can differ from the story commonly accepted. Allow me to relate this adventure. It is understood that I do so with no intent of claiming precedence of discovery for anyone, and, moreover, the mathema­ tician concerned will be nameless. At the turn of this century a young and unknown mathematician, toiling