The fact that it is Reynolds number which is the decisive factor for the friction loss may be established by dimensional analysis. It is based upon the Buckingham's Theorem (Fig.D-9) which states that dependence between variables in a problem may be expressed by means of dimensionless parameter (described as parameters). Reynolds number may be easily shown to be one of the two parameters in the problem of evaluating the friction loss in fluid flow .

Fig.D-9

The parameters are sought as: ...(A)

where it is necessary do find the unknown exponents. A relation of the same form as (A), which is valid between the quantities, must be valid for their dimensions:



This may be re-written (addition of exponents in multiplication) as:

As a result, there are three equations for the five uknown exponents:


0 = -i + j + k + 3l + 2m ...(B)
0 = i - l ...(C)
0 = -2i - j - m ...(D)

Since number of equations is smaller than number of unknowns, the solution is not determined in any unique manner. Instead, there are two degrees of freedom. They are used for two independent choices:
1st choice: It is a good idea to have the first parameter directly proportional to the evaluated pressure drop across the investigated element. This means i=1. On the other hand, this parametr representing the results should not be dependent upon viscosity, as the property representing the input effect.
For the selected i = 1 and m = 0, the solution of the three equations (B) to (D) is:
j = -2, k = 0, l = 1
2nd choice: The second parametr should not be dependent upon the pressure drop (i=0) and it is desirable to have here in the first power the most easily and most often varied input variable, velocity w.
For this choice, i = 0 and j = 1, the three equations (B),(C), and (D) lead to:
k = 1, l = 0, m = -1
Inserting this into eq.(A) produces the Reynolds number.


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This is page Nr. D05 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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