Sec. 41.12 



GENERAL LIQUID-FLOW FORMULAS 



25 



X, y, and z, respectively. Substituting selected 

 values for the coordinates in the region being 

 investigated, the component velocity values are 

 readily calculated. These comments apply equally 

 to the stream function f, and to 3-diml as well 

 as 2-diml forms. 



Data on velocity and pressure fields, already 

 calculated or otherwise available for a considerable 

 number of typical body shapes, are referenced in 

 several of the sections of Chap. 42. 



G. S. Baker and J. L. Kent give an example of 

 D. W. Taylor's method of using line sources and 

 sinks to delineate a 2-diml ship-shaped body and 

 to determine the magnitude and distribution of 

 pressure around it in an ideal liquid [IN A, 1913, 

 Vol. 55, Part II, pp. 50-54]. They describe an 

 adaptation of this method to a determination of 

 the same features around a ship-shaped form in a 

 restricted channel with straight walls parallel to 

 and equidistant from the ship axis. 



A. F. Zahm, in NACA Report 253 of 1926, 

 entitled "Flow and Drag Formulas for Simple 

 Quadrics," gives calculated and obsei-ved pres- 

 sures for a series of geometric forms, including a 

 sphere, a circular cylinder, an elliptic cylinder, 

 prolate and oblate spheroids, and a circular disc. 

 He also gives diagrams of isobars and isotachyls 

 about some of these forms, and discusses velocity 

 and pressure in oblique flow. 



4L11 Conformal Transformation. An ingeni- 

 ous mathematical process, involving complex 

 variables, was utilized by W. Kutta in the early 

 1900's to determine the flow characteristics 

 around typical or schematic airfoils [Kutta, W., 

 111. Aeronaut. Mitt., 1902; AHA, 1934, p. 173]. 

 Knowing the flow characteristics in the region 

 surrounding some simple geometric form such as 

 a circular rod, by the doublet construction illus- 

 trated in Figs. 3.M and 43. J, the circular form 

 and the flow pattern are transformed simul- 

 taneously into the form and pattern desired. 

 However, the nature of Kutta's method restricts 

 its use to 2-dunl problems. 



The transformation is effected by retaining the 

 essential shape of the "curvilinear squares" in 

 the flow net around the typical body as the size 

 of these squares is reduced from visible to infini- 

 tesimal dimensions. Taking its name from the 

 presei'vation of angles in each small area as the 

 "mesh" of the flow net is reduced, the process is 

 known as conformal transformation. Put in another 

 way, conformal transformation or conformal 

 representation is defined "as a distortion of a 



geometric figure that preserves geometric simi- 

 larity of infinitesimal parts of the figure. This 

 means in particular that all angles between cor- 

 responding lines are the same in the original 

 figure and in its conformal representation" 

 [Wislicenus, G. F., FMTM, 1947, p. 582]. 



L. M. Milne-Thomson gives an excellent illus- 

 tration of conformal transformation, or conformal 

 mapping, as it is sometimes called [TH, 1950, 

 p. 140]. This: 



". . . is afforded by an ordinary map on Mercator's 

 projection. It is well known that the angle between two 

 lines as measured on the map is equal to the angle at 

 which the two corresponding lines intersect on the earth's 

 surface; in fact, it is this property which renders the map 

 useful in navigation. 



"In particular the lines on the map which represent the 

 meridians and parallels of latitude are at right angles. If 

 we confine our attention to a small portion of the map, we 

 also know that distances measured on the map will 

 represent to scale the corresponding distances on the 

 globe, but that the scale changes as the latitude increases." 



By this method of mapping it is possible, 

 starting with a circular rod and the accompan}dng 

 2-diml flow pattern of Fig. 3.M, depicting a field 

 in which the velocities and pressures are known, 

 to flatten the rod into a hydrofoil section with a 

 blunt nose and a somewhat pointed tail. The 

 flow pattern is flattened or transformed with 

 the rod so that the velocity and pressure charac- 

 teristics for the transformed pattern are fully 

 determined. 



Stated mathematically, the modified flow 

 picture obeys the same general laws for continuity 

 and for irrotational flow, namely 



du , dv „ , dv du 



— + — = and T- = 



dx dij dx dy 



as the original flow picture from which it was 

 derived [Wishcenus, G. F., FMTM, 1947, p. 193]. 

 Wislicenus gives a brief discussion of conformal 

 transformation in Sec. 44 of the reference, pages 

 211-218. H. Rouse gives a more complete treat- 

 ment in Chap. V of "Fluid Mechanics for Hy- 

 draulic Engineers" [McGraw-Hill, New York, 

 19.38, pp. 96-124]. 



An excellent presentation of this subject is the 

 one by A. Betz, entitled "Konforme Abbildung 

 (Conformal Representation)," pubhshed by Juhus 

 Springer in Germany. 



41.12 Quantitative Relationship Between Ve- 

 locity and Pressure in Irrotational Potential Flow. 

 Having determined the magnitudes and directions 

 of the velocity at selected (or at all) points in the 



