Sec. 42.8 



POTENTIAL-FLOW PATTERNS 



43 



Flow About Elongated Bodies of Revolution," 

 TMB Rep. 761, Aug 1951. On pp. 59-61 the author 

 lists 28 references, some of which are given here. 

 (14) Campbell, I. J., and Lewis, R. G., "Pressure Distri- 

 butions About Axially Symmetric Bodies in 

 Oblique Flow," ARL (Admiralty Research Labora- 

 tory) Report (ACSIL/ADM/54/254) of Apr 1954. 



42.7 Velocity and Pressure Diagrams for 

 Various Two- and Three-Dimensional Bodies. 



When the body shape, form, and contour can not 

 be expressed by some type of mathematical 

 equation, or when the liquid flow around a body 

 can not be expressed in the form of a given stream 

 function \}/ or velocity potential <^, analytic expres- 

 sions for the velocity and pressure can rarely be 

 established or derived. It becomes necessary, as 

 a rule, to fall back upon experimental observa- 

 tions, or upon data previously derived, assembled, 

 and pubhshed by other workers. 



Some references containing these data for 

 various 2-diml and 3-diml bodies, usually with 

 graphs of both velocity and pressure, are given 

 under the appropriate categories in Table 42.e. 

 Could this list be made complete, for all published 

 works, the marine architect might find readily 

 at hand a great amount of data that would be 

 directly useful and occasionally most valuable. 



42.8 The Distribution of Velocity and Pressure 

 About an As3mimetric Body. Many underwater 

 craft, such as submersibles and submarines, have 

 shapes resembUng roughly a body of revolution 

 but they almost invariably possess transverse 

 asymmetry, at least above and below the principal 

 longitudinal axis. Submerged bodies, and craft 

 of the type mentioned, usually possess asymmetry 

 in a fore-and-aft direction as well, reckoned about 

 the midlength. 



However, considering first the geometric forms 

 having asymmetry about the principal axis, it so 

 happens that formulas are available for computing 

 the velocity and pressure distribution about what 

 may be termed elUptic elhpsoids. For these bodies, 

 the profiles, planforms, and sections are all of 

 elliptic shape, as in diagram 2 of Fig. 42. D. Indeed, 

 this particular form is special in two respects; 

 first, in that the flow of an ideal fluid around it 

 lends itself to computation, and second, in that, 

 when placed with one axis in the direction of 

 uniform flow, the velocity at the surface is 

 constant everywhere around the girth of the 

 midsection. The latter feature appears to be 

 inherent in these elliptic shapes only. 



The flow around an elhptic ellipsoid having 



axes in the ratio of 6 : 2 : 1, with a uniform flow of 

 velocity C/„ taking place parallel to the longest 

 axis, has been investigated by H. Chu, P. C. Chu, 

 and V. L. Streeter [Illinois Inst. Tech., Report 

 on Project 4955, sponsored by ONR Contract 

 47onr-32905, dated 15 Mar 1950]. They found 

 that around the elUptic midsection periphery the 

 surface velocity parallel to the stream axis had 

 a constant value of 1.0714f7» , At the quarter- 

 lengths the velocity around the girth of the body 

 varied by only about one per cent from the mean. 

 The uniformity of tangential surface velocity 

 at the midlength, parallel to the undisturbed 

 stream direction, is a sign that the surface pres- 

 sure around the midsection girth is hkewise 

 everywhere the same. However, at a constant 

 given distance from the body surface, in the plane 

 of the midsection, the velocity and pressure do 

 vary around the girth. This variation has been 

 investigated by R. K. Reber for the elhptic 

 ellipsoid having axes in the ratio of 6 : 2 : 1 [unpubl. 

 memo of 13 Apr 1950 to HES], lying with its 



Circular A,'^ 



stream-tube boundarvj 



Port streamline 



Fig. 42.D Typical Schematic Flow Abound 



AxiSYMMETRIC AND ELLIPTIC ELLIPSOIDS 



longest axis parallel to the uniform stream. The 

 results are shown in Fig. 42.E. Here it is noted 

 that, in the plane of the midsection, the isotachyls 

 or loci of constant velocity (parallel to the body 

 axis and to the stream direction) he closer than 

 the average distance to those portions of the 

 transverse midsection having the sharpest curva- 

 ture. Opposite the portions of least curvature 

 they are farther from the body. The transverse 

 velocity gradient is therefore greater in way of 



