Sec. 41.S 



GENERAL LIQUID-FLOW FORMULAS 



17 



they are retained here to insure that the same 

 answer is obtained by different methods of 

 calculation. 



In view of the dimensionless character of the 

 parameters described in this section they have 

 the same numerical values when derived by 

 consistent units in the metric or any other system 

 of measurement. 



41.8 Derivation of Stream-Function and 

 Velocity-Potential Formulas for Typical Two- 

 Dimensional Flows. The combination of the 

 stream functions '/'(psi) and the velocity potentials 

 <^(phi), respectively, of two liquid flows is dis- 

 cussed briefly in Sees. 2.11 and 2.14. To illustrate 

 how this procedure is employed in analytic 

 hydrodynamics a brief outline is given of the 

 formation of the equations for the resultant 

 stream functions and velocity potentials for flow 

 about the following simple forms: 



(a) Single-ended 2-diinl body with a single source 

 in the nose; body axis parallel to the flow. A 

 partial longitudinal section through this body is 

 shown by the heavy line in Fig. 43. B. 



(b) Two-dimensional Rankine stream form de- 

 veloped around a 2-dinil source-sink pair whose 

 axis is parallel to the flow. A similar section 

 through such a body is drawn in Fig. 43. D. 



(c) Two-dimensional circular cyhnder or rod, 

 with its axis normal to the flow, illustrated in 

 diagram 1 of Fig. 41.G 



(d) Two-dimensional circular cylinder with its 

 axis normal to the flow, and about which circula- 

 tion is taking place, corresponding to diagram 3 

 of Fig. 14.E in Volume I. 



The formulas given for the yp- and ^-values of 

 these five classes of bodies enable the coordinates 

 for their potential-flow streamline patterns to be 

 determined and the resultant velocities and 

 pressures at selected points to be calculated. The 

 methods of developing the formulas in the text 

 may be followed for formulas applying to bodies 

 of varied shapes. For all of these bodies, the 

 mathematical expressions characterizing the con- 

 tours and the flow can be manipulated with 

 mathematical operators to derive other useful 

 data, practical as well as analytical. 



The purely mathematic steps in the derivations 

 of this section and the one following are omitted 

 from the text. 



I. It is convenient to take first the case of a 

 single 2-diml source of strength m in a uniform 



l^u for Uniform Flo 



2"-Diml Source f-^ '^SO f<"- Smijle E-DimI 



q\ V-q for Combination of these two 

 'i --Uoo V <-m9°-U o oRsir 



^■~~-Reference 

 Axis 

 0s= -U„xtmlo(5gR= -UooRco5e*mloijeR 



Fig. 41. E Definition Sketch and Formulas for 



Stream Function and Velocity Potential of 

 Combination of Uniform Flow and Single Source 



stream of velocity — U„ , listed under (a) of the 

 preceding tabulation. The liquid in the stream is 

 moving from right to left, as in Fig. 4 I.E. Here 

 the 2-diml stream function fso for the single 

 source is m9 (theta.) , from Eq. (3.xi). The 2-diml 

 stream function ^pu for the uniform flow is — U^y. 

 Adding the two gives the stream function for the 

 combined flow 



^s = - U^y + me = - U„R sin 6 + mB (41.ivx 



The 2-diml stream function for ^,5 = outlines 

 a single-ended body, of which a portion of one half 

 is pictured in Fig. 43. B. The values of ^.9 = — 1, 

 — 2, —3, and so on, produce resultant streamlines 

 around the body. 



The 2-diml velocity potential <j)s o for the single 

 source is m loge R, from Eq. (3.xii). That for the 

 uniform flow is — U^x. Adding the two gives the 

 velocity potential for the combined flow 



4>s = — U^x + m log, R 



= - Ua.R COS e + m log, R (41. v) 



II. Take next the case of the 2-diml Rankine 

 stream form, produced by a source-sink pair, 

 lying in a uniform stream having a direction of 

 flow parallel to the stream-form axis, as listed 

 under (b) preceding. It is necessary first to derive 

 the stream function \f/ (and the velocity potential 

 <l>) for the source-sink combination, diagrammed 

 in Fig. 41.F. The stream function is obtained by 

 adding the stream functions of the source and 

 sink. For the source, ypso = fn tan"' [y/{x — s)]; 

 for the sink, ypsK = —in tan~' [y/(x -(- s)], where 

 the origin of the cartesian coordinates is at 

 the midlength of the source-sink axis and s is 

 the half-distance between the two. Combining 

 the stream function for the source and sink gives: 



= m tan 



y 



— tan 



= m tan 



■ L-.^^] 



