Studies on the Motion of Viscous Flows— III 



POTENTIAL FLOW AS A BASE FLOW FOR 

 AN ITERATIVE PROCESS IN ACTUAL FLOWS 



We shall consider the significance of the potential flow solution given by 

 Eqs. (1.17), (1.18), (1.19), and (1.20) from both the physical and mathematical 

 points of view. . ; 



Let us write the Navier-Stokes Eqs. (1.12) and (1.13) as follows: -' - ' 



Here, the Reynolds number is Re = u„ a/v . .. _., , . _ . , :, 



There are three ways by which the Eqs. (1.12a) and (1.13a) can be reduced 

 to the Euler equations: (a) We may assume, a priori, the fluid to be nonvis- 

 cous, in which case 1/Re = /x//x^a/3 is identically equal to zero. Then the gov- 

 erning equations /or all such flow conditions are the Euler equations, (b) We 

 may consider the fluid to be viscous, but assume l Re to be so small that in 

 some domain the right-hand sides of Eqs. (1.12a) and (1.13a) can be regarded as 

 negligible and hence put equal to zero. In this case the Euler equations are the 

 governing equations only when the specific conditions are met. (c) We may con- 

 sider the fluid to be viscous, but assume the velocity field to be irrotational, in 

 which case the terms in parentheses of Eqs. (1.12a) and (1.13a) are identically 

 equal to zero. Here the Euler equations are the governing dynamic equations 

 for a viscous fluid when the flow is assumed to he irrotational. 



Possibilities (a) and (b), which are quite distinct, impose restrictions on 

 the Reynolds number Re. Possibility (c) imposes a restriction on the nature of 

 the flow. Historically, the Euler equations are derived through the first possi- 

 bility (a). ...^ vv :, 



The boundary layer theory assumes that, when the Reynolds number is very 

 large, i.e., 1/Re is very small, a real flow is a potential flow except near a solid 

 boundary; accordingly, it asserts that near a solid boundary the terms in paren- 

 theses of the right-hand sides of Eqs. (1.12a) and (1.13a) are significant, even 

 though 1/Re may be made arbitrarily small. This theory, then, makes use of the 

 second possibility (b). It has led to a widely held view that the potential flow is 

 an approximation to the corresponding real flow only at very large Reynolds 

 number. However, an alternative view, which is very significant in the present 

 work, can be taken as shown by the following considerations. 



First, we note that the possibilities (a), (b), and (c) show that when I/Re is 

 assumed to be zero the velocity field need not be assumed irrotational and, con- 

 versely, when the velocity field is assumed irrotational Re may have any finite 

 value so that 1/Re differs significantly from zero. As we have already seen, the 

 potential flow solution given by Eqs. (1.17), (1.18), (1.19), and (1.20) satisfies 

 Eqs. (1.22) and thus satisfies the requirements of the third possibility (c). 



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