Characteristics of Ship Boundary Layers 



VI. THE COORDINATE SYSTEM 



A set of mutually orthogonal lines on a surface S can be 

 selected In infinitely many ways. Such a net, together with the 

 distance along the normal to S form a system of space coordinates 

 which, in general, are triply orthogonal only on S. Although a non- 

 orthogonal systenn of space coordinates is usually an awkward choice 

 in formulating the Navier- Stokes equations, when these equations 

 are simplified in accordance with the usual assumptions of thin 

 boundary-layer theory. Squire [ 24] has shown that the boundary-layer 

 equations are identical in form with that for a fully orthogonal system. 



When the third coordinate is the distance C, along the normal 

 to S, the surfaces t, = const, are, for obvious reasons, said to be 

 parallel to S. It is shown in texts on differential geometry that the 

 lines of principal curvature, and only these lines, have the property 

 that the surface normals along them generate developable surfaces 

 I, = const, and r\ - const. , and that these, together with the parallel 

 surfaces t, = const. , form a mutually orthogonal family. For this 

 reason Howarth [ 25] and Landweber { 26] employed the lines of 

 principal curvature as surface coordinates in formulating the equations 

 of motion. Nevertheless, according to Crabtree, _et^_al. , [ 27] , "this 

 Is an undesirable restriction, " a feeling that seems to be shared by 

 most of the contributors to the subject of three-dimensional boundary 

 layers. Preferred is the streamline-coordinate system, although 

 geodesies and rectangular coordinates have also been used. Only 

 Howarth [ 28] has adopted the lines of principal curvature for the 

 coordinate system In his treatment of the three-dimensional boundary 

 layer near a stagnation point. 



There are two good reasons for using streamline coordinates. 

 One Is that. In the cases to which they have been applied, the Invlscld- 

 flow streamlines could be readily obtained; the other Is that practical, 

 approximate methods of solving the boundary-layer equations, em- 

 ploying techniques developed for two-dimensional boundary layers, 

 are available for the equations In streamline coordinates. The simplest 

 of these methods are based on the assumption of small cross-flow In 

 the boundary layer. According to Smith [ li] , however, who applied 

 five of these methods to compute the boundary layer on a yawed wing, 

 none of these was found to be completely satisfactory, as has already 

 been Indicated, 



For the case of present Interest, the boundary layer on a 

 ship form, the first of the aforementioned reasons does not apply. 

 Calculation of the velocity distribution and the streamlines on a ship 

 form at the particular Froude number Is a task of the same order 

 of difficulty as that of solving the three-dimensional boundary-layer 

 equations. For the zero-Froude-number case, methods are available 

 for computing the potential flow [ 8,9] ; at nonzero Froude numbers an 

 approximate method due to Gullloton [ 12,13] furnishes tables for 

 the calculation of three streamlines aJong a ship hull. 



457 



