Showing papers in "Annual Review of Fluid Mechanics in 1975"

The Structure and Dynamics of Vortex Filaments

Abstract: Recent concern over the hazard presented by the trailing vortices produced by large aircraft has stimulated research into one of the oldest subjects in fluid mechanics: the study of flows with concentrated vorticity in free motion. Since these vortices can be strong and persistent enough to pose a safety hazard to other aircraft, it is clearly desirable to be able to predict the structure, position, and persistence of such vortices as well as to understand the mechanisms by which vortex wakes are dissipated. Under most conditions, trailing vortices undergo a natural sinusoidal instability that eventually causes them to touch and break into a series of crude vortex rings. This process destroys the initial wake structure more rapidly than viscous or turbulent decay of the individual filaments. The character and potential hazard of the residual wake structure after the completion of the sinusoidal instability are still not well understood. It is apparent, however, that the motion, structure, and stability of flows with concentrated vorticity is of great practical interest and leads to an examination of many fundamental problems in fluid mechanics. The first organized attempt to compile what was known about the behavior of aircraft vortices, the process of generation and resulting structure, the hazard created by wake turbulence, and the possibilities for detection, avoidance, or destruction of wake vortices can be found in the proceedings of the conference on "Aircraft Wake Turbulence and Its Detection," edited by Olsen, GGldburg & Rogers (1971). Wc do not attempt a review of all of the technical activities in the field of aircraft wake turbulence; we arc here specifically interested in the fluid mechanics of trailing vortices and vortex rings as well as in the more general problem of the structure, motion, and stability of free vortices: compact regions of concentrated vorticity in free motion in a surrounding fluid that is either homogeneous and at rest or with weak background vorticity or stratification. In a very basic sense, the study of the motion of an incompressible homogeneous

New Trends in Experimental Turbulence Research

Abstract: In the past few years some novel statistical techniques and significant observations have been made in the search for a better understanding of the turbulence problem. These efforts have spearheaded what is believed to be a new trend in turbulence research. In this review an attempt is made to describe this new direction in a qualitative way, to place it in a proper historical perspective, and to assess its significance. Taking a brief, rather cursory glance at the manner in which experimental techniques have influenced prevailing ideas on treating turbulent flows over the past fifty years, one may recognize several fairly distinct periods of activity. The twenties and thirties were rich in ideas that tried to rationalize the concept of a turbulent viscosity coefficient. Mcan velocity measurement was the primary experi­ mental method available at the time and was extensively used to establish the advantage of one theory over another. While no conclusion could be reached on this point, the measurements gave some support to phenomenological approaches, including similarity arguments for helping to solve practical problems, but they served essentially as a passive tool. By the forties, the hot-wire technique for the measurement of velocity fluctuations was sufficiently developed that the various components of the Reynolds stress could be obtained with confidence. This development made it possible to test the assumptions of the various phenomenological theories more directly. The result was a complete rejection of these theories. On hindsight, this proved to be a rather ironic turn of events: Later results using more sophisticated statistical techniques developed in the early sixties gave some (although not well understood) support to the turbulent viscosity concept; furthermore, it turned many researchers' attention away from shear-flow turbulence to the "simpler" but more academic problem of homogeneous turbulence. With further improvement of the hot-wire technique and associated electronic instrumentation, subsequent experiments in the fifties concentrated on studying the spectral distribution of the turbulent kinetic energy with a particular emphasis

Experiments in Granular Flow

Abstract: The earliest example of a granular flow is the sand in an hourglass. Investigations into the question of how it really works began with a paper by Hagen (1852), but since the hourglass has been replaced by other chronometers, only a few specialists know the present state of its theory. Another old example is the art of ploughing, which our ancestors knew much more about than we do. Still, their knowledge was practical rather than theoretical. Hence, when draught animals were replaced by tractors, engineers were surprised to find that the drag of a plough is almost independent of speed. Although rich soil is not exactly a granular material, in this respect it behaves like dry sand, which nobody cares to plough. Nowadays only a few experts are concerned with the flow of powders and bulk solids (cf Brown & Richards 1970), whereas confrontation with aeroand hydro­ dynamic problems is commonplace. Hence, the dynamics of granular material has failed to attract general interest to the same degree as fluid dynamics, although first papers were written by well-known hydrodynamicists such as Hagen (1852) and Reynolds (1885). The flow of a discontinuum is a challenge to theoreticians, although it gives opportunities to experimenters too. Because we know so little of the behavior of granular matter from daily experience, we are often surprised by test results. By contrast, the statics of granular material has been the topic of soil mechanics for a long time. For example, on the recommendation of Simon Stevin, the first chair for an engineering science was established in 1600 for a professor of fortification at Leiden University. A theory of earth pressure and resistance that is still the backbone of practical calculations was given by Coulomb in 1776, about 50 years before Cauchy rigorously defined stress and strain. The statics of an idealized soil that is neither moist nor cohesive is, of course, the natural starting point for a review on granular flow.

Fluid Mechanics of Waste-Water Disposal in the Ocean

Abstract: Outfall pipes into the ocean are analogous to chimneys in the atmosphere: they are each intended for returning contaminated fluids to the environment in a way that promotes adequate transport and dispersion of the waste fluids. A waste-water treatment plant and an adjoining outfall constitute a system for environmental control; it is practically never feasible to provide such complete treatment that an outfall is not necessary, nor is it common to depend entirely on an outfall with no treatment. Although outfalls and chimneys are functionally similar, there are important differences in their relationships to the coastal waters and atmosphere respectively. Urban and industrial areas, generating waste water, are located along the shallow edge of the ocean, with often tens or even hundreds of kilometers of continental shelf between the shoreline and the deep ocean. The bottom slope on the shelf is typically less than one percent. Thus outfalls extending several kilometers offshore discharge into a body of water of large lateral extent compared to the depth, and are still remote from the main body of ocean water. In contrast, most atmospheric contaminants are introduced at the base of the atmosphere and circulate throughout the whole atmosphere much more readily. Vertical convection mixes the troposphere rapidly in most places and the wind systems circulate the air around the globe in a matter of weeks. Outfalls and chimneys are useful in reducing pollutant concentrations only locally. Far away from the sources, it makes little difference how the pollutants are discharged. The decay times of the pollutants are important in the choice of effective discharge strategies. For example, the problems of very persistent contaminants such as DDT cannot be alleviated by dispersion from an outfall; such pollutants must be intercepted at the source and prevented from entering the environment. On the other hand, degradable organic wastes, as in domestic sewage, may be effectively disposed of through a good ocean outfall. Since the decay time is only a few days, potential problems are only local, and not regional or global.

Hydrodynamics of Large Lakes

Abstract: difficult nonlinear terms in the equations of motion could be neglected. The resulting linearized equations are akin to those of acoustics, and solutions could be found with a variety of realistic boundary conditions. The same equations adequately describe some other large-scale atmospheric and oceanic motions, and it is therefore no surprise that much can be learned with their aid about large-scale motions in lake s. The equations of tidal theory were in fact used with great success in elucidating the properties of seiches and internal seiches in small and moderate size lakes. A seiche is a back-and-forth oscillation of lake level, much like the sloshing of water in· a bathtub. There is a large literature on seiches (for a recent review see Wilson 1972) and its dynamics is well understood. An internal seicpe is a similar motion of the cold, bottom layer of a stratified lake, the interface between cold and warm layers (the thermocline) playing the role of a free surface, although the interface displacements are very much larger in an internal seiche than surface dis­ plac ements in an ordinary seiche. Internal seiches in small lakes have also been understood for three quarters of a century (see e.g. Mortimer 1953). During the last decade or so considerable further advances have been made in the understanding of motions in lakes so large that the effects of the earth's rotation play an important role. These advances were also achieved with the aid of linearized equations (containing Coriolis force terms), with some assist from the conservation of potential vorticity. Platzman (1963) ha s treated in detail the effects of Coriolis force on seiches in large homogeneous lakes. Here we discuss some other in teres ting hydrodynam ic phenomena involving density stratification, which occur in large lakes and for which we have recently acqu ired a reasonable degree of dynamical insight. Originally this review started out with the tentative title "circulation and

Nonlinear Thermal Convection

Abstract: The topic of this. review article is thermal convection in thin horizontal fluid layers, uniformly heated from below and/or cooled from above. If the temperature difference between the two horizontal boundaries is sufficiently small, the heat will be transferred through the fluid by conduction alone. For greater temperature differences the conduction state becomes unstable and a convective motion is set up. The first experimental investigations on thermal convection date back to Thomson (1881) and Benard ( 1900). The experiments by Benard in particular have attracted great attention and are today considered classical in fluid mechanics. This is essentially due to the surprising and fascinating pattern of very regular hexagonal cells obtained for large values of time in his experiments. Benard studied convective motion in a very shallow layer of viscous fluid (molten spermaceti) and made the motion visible by graphite or aluminum powder. The theory of thermal convection was initiated by Rayleigh ( 1916). He assumed that the amplitude of the motion was infinitesimal such that the equations could be linearized. He thus derived the critical temperature gradient (or in modern language, the critical Rayleigh number) for the onset of convection together with the wavenumber for the marginal stable mode. The resemblance of the cell pattern in the Rayleigh theory to certain cloud forms was early noticed by meteorologists. For example, it was pointed out that often the observed ratio between the vertical height and the lateral extent in cumulus clouds is the same as found in the theory for hexagonal cells. The theory on cloud formation is perhaps best applied to the convective motion set up in an altostratus layer when it breaks up into altocumulus clouds because of radiative cooling at the top of the layer. This application is, however, obscured by the effect of the release of latent heat, which is not accounted for in the theory. The theory on thermal convection has become very important in the study of motion in the earth's interior (see Turcotte & Oxburgh 1972) and also in some branches of astrophysics (Spiegel 1971 , 1972). The great interest shown to the theory for the last 10--15 years is partly due to these and other applications. The main

Mathematical Analysis of Navier-Stokes Equations for Incompressible Liquids

Abstract: Let viscous incompressible liquid occupy a volume ~ of three-dimensional Euclidean space E3. Introducing a Cartesian orthogonal frame xl, x2, x3 in E3, we denote a point (xl, x2, x3) by x. Let us denote the velocity and pressure of the liquid at the point x and at the instant t by v(x, t) and p(x, t), respectively, v~(x, t), i-1, 2, 3, being the projections of the velocity on axes x~. The scale may be chosen in such a way that the density is equal to unity. Then according to the Navier-Stokes theory the motion of the liquid caused by external forces f(x, t) will be described by the following system of four equations : 3 v~-vAv+ ~ vkv~,~ = -gradp+f, (1) k=l

Fluid Mechanics of Heat Pipes

Abstract: Many specialists in fluid mechanics probably have not heard of heat pipes or fully realized the potential of this new technology. The heat pipe is a thermal device recently discovered and developed for the efficient transport of thermal energy (Grover et al 1964, Cotter 1965). It is a closed structure (Figure 1) containing a working fluid that transports thermal energy from one part (evaporator) to another (condenser) by means of liquid vaporization in the evaporator, vapor flow in the core region, vapor condensation in the condenser, and condensate return to the

Relaxation Methods in Fluid Mechanics

Abstract: The present work considers the iterative solution of a coupled set of difference equations and examines methods that carry successive approximates to a state that is invariant with further iteration and independent of the initial guess. Methods are studied with regard to their efficiency and economy of computer resources. The basic principles of classical relaxation are set forth, with attention confined to linear elliptic equations. This discussion involves the evaluation of the spectral radius that is the magnitude of the eigenvalue with largest modulus. The subject of relaxation is then related to the study of ordinary differential equations and hyperbolic partial differential equations. Problems that occur when linearly dependent eigenvectors appear in the relaxation matrix are discussed, leading to multiply connected eigenvalues in the Jordan canonical form. Finally, a brief survey of relaxation methods used in aerodynamics is given.

The Effect of Waves on Rubble-Mound Structures

Abstract: For thousands of years breakwaters have been built at or near the coast to protect harbors or coastlines from wave attack. One of the earliest known harbor protection schemes was devised in about 2000 B.C. for the Port of Pharos on the open coast of Egypt; it had a rubble-mound breakwater approximately 8500 ft long composed of large blocks of stone with smaller stone filling the spaces between blocks (Savile 1940). Until the development of experimental laboratory techniques to investigate the effect of waves on breakwaters, these structures were designed primarily from experience gained from other similar structures. It is the purpose of this review to discuss various aspects of the hydrodynamics of wave attack on such structures and the relation of certain analytic considerations and experimental results to the design of a rubble-mound. A breakwater built as a rubble-mound is constructed by placing material of various sizes layer by layer (or unit by unit) until the desired cross-section shape is achieved. Generally, the units are not structurally connected, so that the integrity of the rubble-mound depends on features such as the weight of the material, the interlocking nature of the material, and the cross section of the structure. Usually the structure is built with material graded from smaller sizes in the core to larger material armoring the face against wave attack. The armor layer may be composed of quarry-stone, if it is available in the required sizes and is economically feasible to use. When these conditions are not met, specially designed concrete units for armoring the face of the rubble-mound have been developed that tend to interlock better than rock when properly placed; hence, it may be possible to use armor units lighter than the required quarry-stone. Over the years numerous geometric shapes have been developed for such armor units, with each shape generally introduced in an attempt to improve on the interlocking characteristics of its predecessors. To mention only a few, various names used for different units are: tribars, tetrapods, quadripods, and dolosse. A brief description of two of these is presented; for a more detailed discussion of shape along with drawings of the units the interested reader is directed to CERC (1966) and Hudson(1974). Tribars, which consist basically of three circular cylinders connected by a yoke of three cylinders, are usually placed in a uniform geometric pattern on the face of the rubble-mound. Dolosse are shaped like the letter "H" with the vertical legs rotated 90° to each other, and are generally placed randomly on a rubble-mound face. It is the effective interlocking of dolosse that leads to the use of random placement techniques. Obviously an important aspect in the design of a rubble-mound is its stability under wave attack. This subject is discussed in detail, along with descriptions of the basis for certain design approaches currently used. The support of these design criteria as well as their limitations are discussed with reference to available experimental data. Three other aspects of the effect of waves on rubble-mounds are treated in this review: wave run-up, transmission, and overtopping. Run-up is defined as the vertical height above still water level to which waves incident upon a structure can be expected to travel up the face of the structure. Wave run-up is important in defining both the amount of wave energy transmitted over and through permeable rubble-mounds and also the quantity of water that may be expected to overtop the structure. In each of the following sections the discussion is directed toward understanding the fluid-mechanic aspects of the various problems and the features and the shortcomings of analytical and experimental models used in connection with the design of breakwaters constructed as rubble-mounds.