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- W2323526431 abstract "For a given problem, the approximate solution given by boundary-layer theory depends on the system of coordinates used when the simplifying assumptions of boundary-layer theory are applied to the Navier-Stokes equations. In general, different systems of coordinates lead to boundary-layer equations which are not equivalent, that is, their solutions represent different flow fields. The object of this paper is to investigate in detail how these flow fields depend on the choice of coordinates, how different boundary-layer approximations to the same flow problem are related to each other, to the external flow and to the flow due to displacement thickness. The discussion is restricted to incompressible, steady, two-dimensional flow without separation. However, many of the results hold much more generally and will be discussed in a later paper. The main result of this paper is contained in Theorem 2. Normally, one uses boundary-layer theory in the following way in order to obtain a picture of the complete flow field of a viscous fluid (outside the wake) : The flow field is divided into two separate regions, that is, a boundary-layer region where the flow field is obtained from boundary-layer equations, and an outer region where the Euler equations are used to obtain an external flow, corrected for the displacement effect of the boundary layer. There has been considerable discussion about where and how to patch the two parts of the flow field, and about how to proceed to higher order approximations. However, Theorem 2 shows that one can find a system of coordinates, such that the boundary-layer sMution with respect to this system gives an approximation which is valid in the whole flow field. Both the external flow and the flow due to displacement thickness are included analytically in this approximation and, hence, the problem of patching is automatically eliminated. A coordinate system with these properties will be here referred to as optimal. In Section 6, the problem is discussed to what order such an optimal boundaryqayer solution is valid. In general it gives a better approximation to the exact flow field than does the composite flow field described above. It also appears to form a reasonable starting point for finding 1) Research prepared under Office of Naval Research Contract N 60NR-~44~ Task Order VIII," @default.
- W2323526431 created "2016-06-24" @default.
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- W2323526431 date "1954-01-01" @default.
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- W2323526431 title "The Role of Coordinate Systems in Boundary-Layer Theory 1)" @default.
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