Sec. 43.12 



DELINEATION OF SOURCE-SINK DIAGRAMS 



69 



to the uniform stream. At a large distance down- 

 stream, half-way to the sink, it has a velocity 

 approximating — U„ . Therefore the augment of 

 velocity AC/ imparted to it in the process of 

 flowing around inside the ovoid-shaped stream- 

 form boundary is — C/„ . Its momentum far 

 downstream is its mass times its velocity, or 

 — pQUo, , assuming that —U= —U„ . This is 

 also its increase in momentum which, by the laws 

 of mechanics, is a measure of the force necessary 

 to impart that increase. The force in question, 

 acting downstream, is balanced by an equal and 

 opposite force acting on the body surrounding 

 the source to shove the body upstream. The 

 latter force may be likened to a thrust which is 

 producing the relative motion depicted between 

 the source boundary and the stream. The different 

 signs for the two terms of the equality F = —pQUa, 

 indicate that the direction of F is opposite that of 

 U^ , indicated in diagram 1 of Fig. 43. P. 



At the left or trailing end of the body, an 

 "opposite" situation exists, depicted at the 

 left in diagram 3 of Fig. 43. P. As the velocity in 

 the uniform stream is every^vhere — U„ , the net 

 thrust or drag on the body is obviously zero. 

 When the forces on both source and sink are 

 taken into account, the expression 



f Bodv — / 



,oU^ + 



kU. 



is a simplified form of the Lagally Theorem, for 

 the special case considered, in which Fsody is zero. 



The practical and extremely useful applications 

 of the theorem embody situations in which the 

 flow is unsteady and non-uniform and in which 

 the closed body may be 2-diml or 3-diml, formed 

 by any desired number and combination of 

 sources and sinks. 



Leaving the schematic physical explanation 

 and turning to one that has hydrodynamic rigor, 

 a simplified form of the theorem is the expression 



F — 



J(Q.C/0 



where Qi is the output of the source or sink 

 within the body, expressed as a volume rate of 

 flow, positive for a source and negative for a 

 sink. The symbol C/; represents the velocity of 

 the stream, at the position of the source or sink 

 under consideration, which would occur if the 

 body, and hence all the sources and sinks forming 

 the body, were removed from the stream. For 

 each source or sink the line of action of the 

 individual force is through the singularity under 

 consideration and in the direction opposite to 



that of Ui . The net hydrodynamic force on the 

 body is the vector summation of the individual 

 forces, evaluated for each source and sink within 

 the body. 



If the body is placed in a uniform stream, 

 Ui at each singularity equals the uniform-stream 

 velocity U„ . And since, for any closed body 

 formed by a stream surface, the source outputs 

 must equal the sink inputs, and the mass density 

 p is constant, the net hydrodynamic force acting 

 on the body is zero, which simply confirms 

 d'Alembert's paradox. 



The complete theorem developed by M. Lagally 

 ["Berechnung der Krafte und Momente, die 

 stromende Fliissigkeiten auf ihre Begrenzung 

 ansiiben (Calculation of Forces and Moments 

 which Streaming Liquids Exert on Their Bound- 

 ary)," Zeit. fiir Ang. Math. Mech., Dec 1922, 

 Vol. 2, pp. 409-422 (in vector notation); MUne- 

 Thomson, L. M., TH, 1950, pp. 208-211] permits 

 the determination of both forces and moments 

 acting on a body in both steady and unsteady 

 flow [Cummins, W. E., "The Forces and Moments 

 Acting on a Body Moving in an Arbitrary 

 Potential Stream," TMB Rep. 780, Sep 1952]. 

 For example, if the uniform flow at the right of 

 diagram 1 of Fig. 43. P is replaced by a single 

 outside source on the x-axis, the flow from the 

 latter is radial rather than parallel and uniform. 

 The velocity C/„ in the momentum equation is 

 then replaced by an array of radial velocities, 

 varying with distance from the outside source. 

 If the outside source is offset from the a;-axis of 

 a source-sink body, there is a moment as well as a 

 force exerted on the ovoid-shaped boundary. 



The marine architect who wishes to delve 

 further into this subject may flnd a paper by 

 A. Betz of Gottingen somewhat more readable 

 than the references cited in the preceding para- 

 graph. This paper is entitled "The Method of 

 Singularities for the Determination of Forces and 

 Moments Acting on a Body in Potential Flow" 

 [TMB Transl. 241, Nov 1950]. 



In all applications of Lagally' s Theorem it is 

 important to remember that, although the line 

 of action of the force on a body, formed by 

 placing a source-sink system in a stream, passes 

 through the source or sink, the force is not to be 

 taken as acting on the source or sink proper but 

 on some type of surface or body boundary sur- 

 rounding each. However, in the conventionaUzed 

 diagrams published by Betz in TMB Translation 

 241, he apparently employed a schematic short- 



