The approach based upon nondimensional parameters is of basic importance even for other branches of mechanics and sciences in general. Fig.D-9 shows how it makes possible, through decrease in the number of variables in aproblem, simplifying the graphical presentation of relations between quantities: a single curve sufficed in a situation where the original dependence between five variables required several bound volumes of complicated diagrams. Even more important is the obtained simplification of experimental investigations (remember that for turbulent flows - the ones almost exclusively encountered by an engineer in practical applications - where no exact solution exists, experiments are still of essential importance): measurement of a frictional loss by systematic investigation of all five variables in the problem would mean unacceptably expensive and time-consuming experimental program. After the reduction of the number of variables, the simple measurement of the relation between two variables may be performed perhaps during an afternoon.
Fig.D-10
The dimensionless parameters generated by dimensional analysis have an intereting significance: they may be interpreted as the ratio of an external and an internal characteristic dimension of the particular problem or phenomenon. The outer characteristic dimension the usually the one a designer designing a device based upon the phenomenon is free to choose. The inner characteristic dimension is usually determined by the mechanism of the processes. It is typical for processes in Nature that as a certain critical ratio of the two characteristic dimensions the phenomenon in question changes its character. At a critical value of the Abbe number (where is the diameter of the aperture which limits the transverse dimension of a light beam, is the wavelenght of the light) the laws of geometrical optics cease to be valid and different laws of wave optics become relevant. Already mentioned
Fig.D-11
was a similar meaning of the Knudsen number , the critical value of which delimits the region of validity for laws of superaerodynamics. Another already mentioned nondimensional parameter is the Mach number , above a critical value of which the different laws of supersonic aerodynamics. Event though in the present case of the Reynolds number a meaning of the ratio of two characteristic dimensions is not so immediately apparent, Figs D-10 and D-11 present a document that this number as a dimensionless parameter has actually quite similar meaning and the importance of its critical value (Fig.12), delimiting validity of different laws of turbulent flow is also very similar to what was mentioned in the case of Ab, Kn, and Ma.


Practical calculation procedure for a frictional loss problem

1) Reynolds number is evaluated. This determines wheter laminar or tubulent laws apply.
2) Suitable formula is chosen in Fig.D-12 and the coefficient of friction loss is evaluated.
3) The Weisbach formula, Fig. D-4 is used to compute the loss (drag) coefficient .
4) The amount of dissipated ("lost") energy is calculated according to Fig.D-2.
5) This result is inserted to the right-hand side of integrated Bernoullis' Theorem equation.


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This is page Nr. D06 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS"
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