Chapter 5 attempts to extend the basic knowledge gained and (homework) problem solution skills acquired from studying Chapters 1 through 4 and using Appendices A to F. Section 5.1 deals with mathematical modeling aspects important for the development of computer simulation models, which are becoming more and more acceptable in solving complex fluid mechanics and convection heat transfer problems. The selected case studies begin with turbulent shear-layer flows applied to external curved surfaces (Sect. 5.2.1) and internal walls (Sect. 5.2.2). The basic idea of boundary-layer theory is then extended in Section 5.3 to non-Newtonian fluid flow with exothermal chemical reaction (Sect. 5.3.1) and with wall heat transfer (Sect. 5.3.2). The fifth and sixth case studies are more complicated, focusing on two-phase flows with moving gas–liquid interface due to droplet vaporization (Sect. 5.4.1) and moving liquid–solid interface due to ice formation (Sect. 5.4.2). The last two case studies represent the state of the art in (bio-) fluid dynamics applied to laminar pulsatile flow in branching blood vessels (Sect. 5.5.1) and temperature-driven flow in a local hyperthermia treatment device that is a concentric heated cylinder in a water-filled balloon (Sect. 5.5.2). Both biofluid flow systems are transient three-dimensional and hence require the numerical solution of the complete Navier–Stokes equations.

The open literature provides many additional case studies that encapsulate both basic knowledge in engineering fluid dynamics and computer solution steps for real-world (thermal) flow problems.

Scaling laws for the atmospheric surface layer

Geophysical fluid dynamics has become in the last few decades a broad subject (see Pedlosky, 1979) with many applications in earth sciences and in engineering practice. In all branches of geophysical fluid dynamics using similarity considerations, scaling laws and self-similar solutions play an important, often decisive role. We have chosen in this chapter for demonstration's sake some topics from geophysical fluid mechanics related mainly to geophysical turbulence.

The surface layer of the atmosphere is usually modelled (see, e.g., Monin and Yaglom, 1971) by a turbulent flow that is statistically horizontally-homogeneous and stationary, and is bounded below by a horizontal plane. The shear stress τ in the surface layer is also assumed to be constant. The essential difference from the flow in the wall region considered in section 10.2 consists in the presence in the surface layer of thermal stratification – temperature inhomogeneity over the height of the layer. The stratification is stable if the temperature increases with height and unstable in the opposite case. Owing to the thermal inhomogeneity, a vertical displacement of fluid particles, produced by a vertical velocity fluctuation, is accompanied by work done against the force of gravity (or extracted, depending on whether the stratification is stable or not). This work is either taken from the turbulent energy or added to it, thus influencing the turbulence level, i.e. the transfer of heat, mass and momentum, and consequently also influencing the vertical distribution of the mean longitudinal velocity across the flow. The effectiveness of the influence of thermal stratification on the balance of turbulent energy is governed by the product of the coefficient of thermal expansion of the air and the acceleration of gravity, the so-called buoyancy parameter. The air in the atmospheric surface layer is usually considered to be a thermodynamically ideal gas, for which the coefficient of thermal expansion is equal to 1/T, where T is the absolute temperature.

Solutions of travelling-wave type

In various problems in mathematical physics an important role is played by invariant solutions of the travelling-wave type. These are solutions for which the distributions of the properties of the motion at different times can be obtained from one another by a translation rather than by a similarity transformation as in the case of self-similar solutions. In other words, one can always choose a moving Cartesian coordinate system such that the distribution of properties of a motion of travelling-wave type is stationary in that system. One can reduce to a consideration of travelling waves the study of the structure of shock-wave fronts in gas dynamics (see, e.g. Kochin, Kibel' and Roze, 1964; Zeldovich and Raizer, 1966, 1967) and in magneto-hydrodynamics (Kulikovsky and Lyubimov, 1965), the structure of flame fronts (Zeldovich, 1948; Zeldovich, Barenblatt, Librovich and Makhviladze, 1985), the investigation of solitary and periodic waves in a plasma and on the surface of a heavy fluid (Jeffrey and Kakutani, 1972; Whitham, 1974; Karpman, 1975; Lighthill, 1978; Eilenberger, 1981; Drazin and Johnson, 1989; Fordy, 1990), and many other problems. In recent years many processes have been studied involving the effects of the propagation of plasma fronts in electrical, electromagnetic, and light (laser) fields, the so-called waves of discharge propagation. These processes also lead to the consideration of solutions of travelling-wave type (Raizer, 1968, 1977).

In accordance with the definition given above, solutions of travelling-wave type can be expressed in the form

v = V(x − X) + V0(t). (7.1)

Here v is the property of the phenomenon being considered; x is the spatial Cartesian coordinate, an independent variable of the problem; t is another independent variable, for simplicity identified with time; and X(t) and V0(t) are time-dependent translations along the x- and v-axes. In particular, if the properties of the process do not depend directly on time, so that the equations governing the process do not contain time explicitly, the travelling-wave propagates uniformly:

v = V(x − λt + c) + μt. (7.2)

Homogeneous and isotropic turbulence

10.1.1 The problem of turbulence

This chapter differs from the previous ones in that scaling laws and self-similar solutions of the first and second kind will be established by making essential use of experimental data and without turning to a mathematical formulation of the problem, which, for turbulence, is lacking at the present time.

The problem of turbulence, to which this chapter is devoted, is considered with good reason to be the number-one problem of contemporary classical physics. Discovered by Leonardo and baptized by Lord Kelvin, it has attracted the greatest minds of the century, including such giants as W. Heisenberg, A.N. Kolmogorov, G.I. Taylor, L. Prandtl, and Th. von Kármán. Nevertheless, it remains an open problem: none of the results available has been obtained from first principles. They are based essentially on strong additional assumptions, which may or may not be correct.

The phenomenon of turbulence, as is well known, consists in the following. As we have seen in chapter 1, the basic similarity parameter that governs the global properties of the flow of an incompressible viscous fluid is the Reynolds number ρUl/μ (ρ being the density and μ the viscosity of the fluid, U a characteristic speed, and l a characteristic length scale of the flow). When the Reynolds number reaches a certain critical value Recr, different for different flows (for example, for flow in a smooth cylindrical pipe of circular cross section, Recr ∼ 103, for flow in a boundary layer, Recr ∼ 105), the character of the flow changes suddenly and sharply. A stream that at subcritical values of the Reynolds number was regular and ordered – laminar – becomes essentially irregular both in time and in space. The flow properties for supercritical values of the Reynolds number undergo sharp and disorderly variations in space and time, and the fields of flow properties, – pressure, velocity, etc. – can to a good approximation be considered random. Such a regime of flow is called turbulent.

Professor Grigorii Isaakovich Barenblatt has written an outstanding book that contains an attempt to answer the very important question of how to understand complex physical processes and how to interpret results obtained by numerical calculations.

Progress in numerical calculation brings not only great good but also notoriously awkward questions about the role of the human mind. The human partner in the interaction of a man and a computer often turns out to be the weak spot in the relationship. The problem of formulating rules and extracting ideas from vast masses of computational or experimental results remains a matter for our brains, our minds.

This problem is closely connected with the recognition of patterns.lt is not just a coincidence that in both the Russian and English languages the word ‘obvious’ has two meanings – not only something easily and clearly understood, but also something immediately evident to our eyes. The identification of forms and the search for invariant relations constitute the foundation of pattern recognition; thus, we identify the similarity of large and small triangles, and so on.

Let us assume now that we are studying a certain process, for example a chemical reaction in which heat is released and whose rate depends on temperature. For a wide range of parameters and initial conditions, a completely definite type of solution is obtained – flame propagation. The chemical reaction occurs in a relatively narrow region separating the cold combustible substance from the hot combustion products; this region moves relative to the combustible substance with a velocity that is independent of the initial conditions. (Of course, the very occurrence of combustion depends on the initial conditions.)

This result can be obtained by direct numerical integration of the partial differential equations that describe the heat transfer, diffusion, chemical reaction, and (in some cases) hydrodynamics. Such a computational approach is difficult; the result is obtained in the form of a listing of quantities such as temperature and concentration as functions of temporal and spatial coordinates.

Mandelbrot fractals and incomplete similarity

12.1.1 The concept of fractals. Fractal curves

In the scientific and even popular literature of recent time fractals have been widely discussed. By fractals are meant those geometric objects, curves, surfaces, volumes and higher-dimensional bodies having a rugged form and possessing certain special properties of homogeneity and self-similarity. Such geometric objects were studied intensively by mathematicians at the end of the last and the beginning of present century, in particular in connection with the construction of examples of continuous nowhere-differentiable functions. To most physicists and engineers they seemed mathematical monsters having no applications in the problems of natural science and technology. In fact, it is not so.

The revival of interest in such objects and the recognition of their fundamental role in natural science and engineering is due to a series of papers by B. Mandelbrot and especially to his remarkable monographs (1975, 1977, 1982). Mandelbrot coined the very term ‘fractal’ and introduced the general concept of fractality. In the monographs and subsequent papers Mandelbrot and his followers showed that, contrary to what was expected, this concept, enclosing many known special examples, appeared to be exceptionally fruitful in such diverse and important applications as polymer physics, geomorphology, the theory of Brownian motion, turbulence theory, astrophysics, fracture theory and many others. In the monographs of Mandelbrot are referenced from a unified viewpoint the preceding works of other authors that relate to these topics.

In this section we will demonstrate the concept of Mandelbrot fractals for the simplest example, fractal curves. We will discuss the properties of homogeneity and self-similarity that make a continuous curve fractal, and we will show that the very idea of fractals is closely related to the incomplete similarity concept. A non-trivial example related to the fractality of respiratory organs will be presented in conclusion.

Dimensions

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

We say without thinking that the mass of water in a glass is 200 grams, the length of a ruler is 0.25 metres, the half-life of radium is 1 600 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 the water is compared with a standard – a unit for the measurement of mass – the gram; the length of the ruler is compared with a unit for the measurement of length – the metre; the half-life of radium is compared with a unit for the measurement of time – the year; and the velocity of the car is compared with a unit for the measurement 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 for these quantities – either natural or artificial – 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 force, length, and time standards. However, these standards are insufficient for the description of, for example, heat transfer. Additional standards also become necessary when studying electromagnetic phenomena, economic phenomena etc.

Complete and incomplete similarity

In chapters 2–4 several instructive and fundamentally different self-similar problems were considered. In the problem of an instantaneous concentrated heat source and the problems of the propagation of very intense thermal and blast waves the situation turned out to be relatively simple. Namely, for them there exists some completely schematized idealized statement of the problem (energy release at a point, initial temperature and pressure equal to zero). Considering this statement of the idealized problem and applying the procedures of dimensional analysis to it in the standard way, we can reveal the self-similarity of the solution, construct the self-similar variables, and obtain the solution in finite form owing to the existence of certain integrals.

Deeper consideration shows, however, that this simplicity is illusory and that, for example, in making the assumption of pointwise release of energy we have, as it is said, gone to the brink of an abyss. In fact, by changing the formulation of the problem, apparently only slightly, in such a way that it would seem that all similarity considerations must preserve their validity, we arrived at a contradiction; it turned out that in the modified problems the required solutions simply do not exist. More detailed analysis showed that in trying to find solutions of the modified problems by the same standard procedure, and starting from the formulation of an idealized problem, it turned out that the very statement of the question was improper. What we actually needed was not exact solutions of the simply formulated idealized, degenerate problem, corresponding to the instantaneous removal at a plane of a finite mass of fluid or the instantaneous release at a point of a finite amount of energy. We were interested, rather, in the asymptotics of solutions of the corresponding non-idealized problems, the existence and uniqueness of whose solutions are either strictly proved or evoke no doubt; the non-idealized problems naturally turned out to be non-self-similar.