Concerning Similarity Analysis Based on the Use of Governing Equations and Boundary Conditions and Long's Method of Generalized Dimensional Analysis

Long has illustrated the increased power for fluid mechanics of a generalized type of dimensional analysis based on consideration of the known mathematical form of the governing equations and boundary conditions. It is shown that Long's method is mathematically equivalent to an analysis of the transformational properties of the governing equations and boundary conditions under a particular class of simple linear transformations of the variables of the problem, and that similarity results obtained by his method may also be derived directly by a purely mathematical technique that avoids reference to the concepts of dimensional analysis and the Buckingham II-theorem. Application of this transformation technique is illustrated by several examples. Consideration of a recent attempt by Bernstein to apply Long's procedure to the problem of the wind distribution in the turbulent planetary boundary layer indicates that this application is not valid.