Studies on the Motion of Viscous Flows— III 



III— Theoretical Aspects and Application 

 of the Linear Substructure Underlying 

 the Complete Navier-Stokes Equations 



S. Desai and P. Lieber 



: University of California •■ • ' ■ 



: ■ Berkeley, California 



" ' ABSTRACT ' ; ,' 



In this paper the potential flows are shown to be fundamental as a basis 

 leading to the construction of analytical representations of viscous in- 

 compressible flows by a process of iteration. This concept reveals the 

 linear substructure underlying the Navier-Stokes equations as applied 

 to the problem of a two-dimensional flow of a viscous incompressible 

 fluid past a circular cylinder. This linear substructure is here under- 

 stood as characteristic everywhere in the domain and for all Reynolds 

 nuinbers. A viscous flow is regarded as a deviation, not necessarily 

 small, for the basic potential flow. A theorem (Theorem I) is estab- 

 lished on the basis of a principle of minimum dissipation, to the effect 

 that for a large class of real flows the velocity field tends to become 

 irrotational and hence derivable from a potential. Iterative equations 

 representing the linear substructure are obtained, and it is shown that 

 at least two iterations are necessary, and are to a large extent suffi- 

 cient, to obtain with good approximation an analytical solution which 

 corresponds to the flow field around a circular cylinder as observed in 

 nature 



On the basis of the linear substructure equations, an intimate relation 

 between asymmetry and the time dependence of the flow field around 

 the cylinder is shown to exist, and a symmetry theorem concerning 

 thenn is proved. Experimental results are shown to be in accord with 

 this theorem. 



An idea of a physically infinite distance is introduced and applied to 

 obtain solutions to the sets of equations governing the first two itera- 

 tions for a steady flow. These solutions are obtained in power series 

 expansions of 1/c log^ r as well as 1/c log^ (log^ r+ 1), where r is 

 the radial distance in polar coordinates and c is a suitable scale fac- 

 tor, both having an infinite radius of convergence. It is shown that these 

 two transformations, viz., s = 1/c logg r and s = 1/c log^ ( log^ r+ 1), 

 belong to a group of transformations. However, only the analysis using 

 the second transformation is presented here. The analysis using the 

 first transformation is presented in Refs. 1 and Z. Information about 

 the structure of vortices and the wake is implicit in these analytical 

 solutions insofar as they give the complete streamiline field around the 

 circular cylinder. 



Computer programs using double-precision arithmetic are developed 

 and presented to evaluate these analytical solutions for any Re in the 



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