Although problems of turbulence are in this text discussed only in introductory manner, it is impossible to avoid special notation for turbulent quantities which vary stochastically - in particular in chap.[I] we shall need to characterise basic statistical concepts of such fluctuations, e.g. by introducing notation for ensemble averaged and time-averaged quantities and for fluctuations as deviations from such mean. Of particular importance are time-mean values, which are denoted by a dash above the particular symbol: is time-mean value of fluctuating pressure, while e.g. is time-mean value of velocity component parallel to axis , etc..

While in deriving differential equation it is essential to work with specific values of extensity variables, when the equations are integrated, we shall receive total magnitudes of such quantities and we cannot escape the necessity to have notations for them. In the case of extensity variables, the total value depends upon the amount of fluid taking part in the particular process. In flowing fluid, delimitation of a certain amount (mass) of fluid is not straightforward. It is useful to define a reference amount of a quantity passing through an inlet - and similarly also outlet - cross section of an element during infinitesimal time period. This definition leads to flow or flowrate values of extensity variables. It may be noted that matter existing in space and time is the basic object of our investigation in mechanics and the variables describing them (mass, time, distance) are neither extensive nor intensive quantities, but form a special basic group of protensive quantities. While specific values are defined by differentiation with respect to mass, the flowrate values are defined by differentiation with respect to time.
Flowrate values are here denoted by adding a time derivative operator to the (upper-case) total value symbol cf. Fig.A-13. In this version of the textbook, the operator is a dot above the symbol.

There is a number of disadvantages to this the operator and the multiplicative used in earlier editions of the present textbook is better in several respects - apart from its being not traditional. It is the force of tradition which forced the author to use "dot" form here. The problem with the dot is that it does not print well, and especially on lower quality paper used in earlier editions it was easily misread for accidental faults. The dot also interferes with the dash above the symbol of time-mean values. Symbol has also the advantage of multiplicative character: it may be e.g. it may be extracted in front of parentheses. In modern fluid mechanics it is common to work with nondimensional quantities. Because of their large number used here, it became useful to apply several different notations, depending upon their character:

1) Simple ratios of two quantities of the same kind are denoted by lower-case greek letters. E.g. ... ratio of flows, ... ratio of cross sections, often called contraction ratio.
2) Coefficients, often used for empirical correction of values obtained by some simplified calculation, are denoted by the letter with an index determining it mode closely, e.g.: ... pressure coefficent, ... drag coefficient, ... coefficient of friction.
3) Quite a number of more complex non-dimensional expression is named after someone who discovered or introduced them or explained their meaning (or sometimes was simply honoured this way). They are denoted by a two-letter symbol, the first an upper-case letter and the secon one lower-case, e.g.: ... Mach number, ... Knudsen number. In chap. [D] we shall learn the method of deriving these quantites.
It should be said, unfortunately, that various reasons - such as an old tradition or different views, prevent the notation in some cases being truely systematic.


Although im this textbook we shall learn a large number of relations and formulas of fluid mechanics, there are, among them, two relations of exceptional position, deserving a name of a "law". In fact, the whole present textbook will be in its first half subdivided into chapters according to terms gadually inserted into one of this laws. The laws are called by old names derived from their originators (who, however, introduced and used them in quite different form). Both laws are nothing else but special cases of conservation of mass and energy. They were, however, discovered a long time before the concepts of mass an energy became known (and that is why they have been - and sometim,es still are - used in forms in which the energy and mass are not recognisable at all). We shall use them in a form of differenial equantions for an infinitesimal length element of one-dimensional flow (the were, however, introduced and used in a form which has nothing to do with a differential equation). Perhaps the most instructive derivation is from energetic balance on a pipe element.


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This is page Nr. A08 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS"
Any comments and suggestions concerning this text may be mailed to the author to his address tesar@fsid.cvut.cz

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