Periodic axisymmetric vortex breakdown in a cylinder with a rotating end wall

When the fluid inside a completely filled cylinder is set in motion by the rotation of the bottom end wall, steady and unsteady axisymmetric vortex breakdown is possible. The onset of unsteadiness is via a Hopf bifurcation.

Figure 1 is a perspective view of the flow inside the cylinder where marker particles have been released from an elliptic ring concentric with the axis of symmetry near the top end wall. This periodic flow corresponds to a Reynolds number Re=2765 and cylinder aspect ratio H/R=2.5. Neighboring particles have been grouped to define a sheet of marker fluid and the local transparency of the sheet has been made proportional to its local stretching. The resultant dye sheet takes on an asymmetric shape, even though the flow is axisymmetric, due to the unsteadiness and the asymmetric release of marker particles.When the release is symmetric, as in Fig. 2, the dye sheet is also symmetric. These two figures are snapshots of the dye sheet after three periods of the oscillation (a period is approximately 36.3 rotations of the end wall). Figure 3 is a cross section of the dye sheet in Fig. 2 after 26 periods of the oscillation. Here only the marker particles are shown. They are colored according to their time of release, the oldest being blue, through green and yellow, and the most recently released being red. Comparison with Escudier's experiment shows very close agreement.

The particle equations of motion correspond to a Hamiltonian dynamical system and an appropriate.

Complete and incomplete similarity

In Chapters 2 and 3 we considered two instructive and fundamentally different, albeit seemingly analogous, problems. In the problem of very intense, instantaneous and infinitely concentrated flooding considered in Chapter 2, following exactly the basic idea demonstrated in the Introduction for a very intense explosion, we arrived at an idealized statement of infinitely concentrated flooding. Applying to this idealized problem the standard procedure of dimensional analysis presented in Chapter 1 we were able to reveal the self-similarity of the solution, to find the self-similar variables and to obtain the solution in a simple closed form.

Deeper consideration showed, however, that this simplicity is illusory and that in making the assumption of an infinitely concentrated flooding we went, we might say, to the brink of an abyss.We demonstrated this when in Chapter 3 we modified the formulation of the problem, seemingly only slightly, by introducing fluid absorption. It would seem that in the modified formulation the same ideal problem statement would be possible and that all our dimensional reasoning would preserve its validity. However, in proceeding with the modified formulation we arrived at a contradiction. It turned out that in the modified formulation the solution to the ideal problem of very intense, instantaneous and infinitely concentrated flooding does not exist.

Turbulence at very large Reynolds numbers

Turbulence is the state of vortex fluid motion where the velocity, pressure and other properties of the flow field vary in time and space sharply and irregularly and, it can be assumed, randomly. Turbulent fluid flows surround us, in the atmosphere, in the oceans, in engineering and biological systems. First recognized and examined by Leonardo da Vinci, for the past century turbulence has been studied by engineers, mathematicians and physicists, including such giants as Kolmogorov, Heisenberg, Taylor, Prandtl and von Kármán. Every advance in a wide collection of subjects, from chaos and fractals to field theory, and every increase in the speed and parallelization of computers is heralded as ushering in the solution of the ‘turbulence problem’, yet turbulence remains the greatest challenge of applied mathematics as well as of classical physics.

It is very discouraging that in spite of hard work by an army of scientists and research engineers over more than a century, almost nothing became known about turbulence from first principles, i.e. from the continuity equation and the Navier–Stokes equations (Batchelor 1967; Germain 1986; Landau and Lifshitz 1987).

For the past seven years students and faculty at the University of California at Berkeley have had the privilege of attending lectures by Professor G.I. Barenblatt on mechanics and related topics; the present book, which grew out of some of these lectures, extends the privilege to a wider audience. Professor Barenblatt explains here how to construct and understand self-similar solutions of various physical problems, i.e. solutions whose structure recurs over differing length or time scales and different parameter ranges. Such solutions are often the key to understanding complex phenomena; there is no universal recipe for finding them, but the tools that can be useful, including dimensional analysis and nonlinear eigenvalue problems, are explained here with admirable conciseness and clarity, together with some of the multifarious uses of self-similarity in intermediate asymptotics and their connection with wave propagation and the renormalization group. Whenever possible, Professor Barenblatt shuns dry and distant abstraction in favor of the telling example from his incomparable stock of such examples; with the appearance of this book, there is no longer any excuse for any scientist not to master these simple, elegant, crucial and sometimes surprising ideas.

Mandelbrot fractals and incomplete similarity

The concept of fractals. Fractal curves

In the scientific and even popular literature of recent time fractals have been widely used and discussed. By fractals are meant those geometric objects, curves, surfaces and three- and higher-dimensional bodies, having a rugged form and possessing certain special properties of homogeneity and selfsimilarity. Such geometric objects were studied intensively by mathematicians at the end of the nineteenth century and the beginning of the twentieth century, euphony particularly in connection with the construction of examples of continuous nowhere-differentiable functions. To many pure mathematicians (starting with Hermite) and most physicists and engineers they seemed for a long time mathematical monsters having no applications in the problems of natural science and technology. In fact, it is not so and in clarifying this point the concept of intermediate asymptotics plays a decisive role.

The revival of interest in such objects and the recognition of their fundamental role in natural science and engineering is due primarily to a series of papers by Mandelbrot and, especially, to his monographs (1975, 1977, 1982). Mandelbrot coined the very term ‘fractal’ and introduced the general concept of fractality.

Applied mathematics is the art of constructing mathematical models of phenomena in nature, engineering and society. In constructing models it is impossible to take into account all the factors which influence the phenomenon; therefore some of the factors should be neglected, and only those factors which are of crucial importance should be left. So we say that every model is based on a certain idealization of the phenomenon. In constructing the idealizations the phenomena under study should be considered at ‘intermediate’ times and distances (think of the impressionists!). These distances and times should be sufficiently large for details and features which are of secondary importance to the phenomenon to disappear. At the same time they should be sufficiently small to reveal features of the phenomena which are of basic value.We say therefore that every mathematical model is based on ‘intermediate asymptotics’.

Dimensions

Measurement of physical quantities, units of measurement. Systems of units

We say without any particular thought that the mass of water in a glass is 200 grams, the length of a ruler is 0.30 meters (12 inches), the half-life of radium is 1600 years, the speed of a car is 60 miles per hour. In general, we express all physical quantities in terms of numbers; these numbers are obtained by measuring the physical quantities. Measurement is the direct or indirect comparison of a certain quantity with an appropriate standard, or, to put it another way, with an appropriate unit of measurement. Thus, in the examples discussed above, the mass of water is compared with a standard – a unit of mass, the gram; the length of the ruler is compared with a unit of length, the meter; the half-lifetime of radium is compared with a unit of time, the year; and the velocity of the car is compared with a unit of velocity, the velocity of uniform motion in which a distance of one mile is traversed in a time equal to one hour.

The units for measuring physical quantities are divided into two categories: fundamental units and derived units. This means the following.

A class of phenomena (for example, mechanics, i.e. the motion and equilibrium of bodies) is singled out for study. Certain quantities are listed, and standard reference values – either natural or artificial – for these quantities are adopted as fundamental units; there is a certain amount of arbitrariness here. For example, when describing mechanical phenomena we may adopt mass, length and time standards as the fundamental units, though it is also possible to adopt other sets, such as force, length and time.

General properties

From the class of flows that are termed geophysical, there are three that are distinct and these more than illustrate the salient properties that such flows possess when viewed from the basis of perturbations. First, there is stratified flow. In this case there is a mean density variation and it plays a dominant role in the physics because there is a body force due to gravity. At the same time, the fluid velocity, to a large degree of approximation, remains solenoidal and therefore the motion is incompressible. The net result leads to the production of anisotropic waves, known as internal gravity waves, and such motions exist in both the atmosphere and the ocean.

Second, because of the spatial scales involved, motion at many locations of the earth, such as the northern or southern latitudes, are present in an environment where the effects of the earth's rotation cannot be taken as constant. On the contrary, rotation plays a dominant role. Again, this combination of circumstances leads to the generation of waves.

Viscous effects can be neglected in the analysis for both the stratified flow and the problem with rotation but the presence of a mean shear in either flow – as we have already seen so often – does lead to important consequences for the dynamics when determining the stability of the system.

Third, there is the modeled geophysical boundary layer where the rotation is present but taken as constant and the surface is flat.