Sec. 14.3 



FORCE AND FLOW DATA FOR HYDROFOILS 



73 



L per unit span = pU^T 



(44.i) 



where 



p (rho) is the mass density of the liquid in the 

 stream 

 [/„ is the velocity of the uniform stream relative 

 to the hydrofoil 

 r (capital gamma) is the strength of the circu- 

 lation in a plane parallel to the stream 

 motion and normal to the body axis. 



For a hydrofoil of span or breadth h, measured 

 normal to the stream direction and to the plane 

 of circulation, the total lift force 



L = bpU^T 



(44. ii) 



The circulation required to insure that the 

 liquid leaves the hydrofoil tangentially at its 

 trailing edge is approximated by [Rouse, H., 

 EMF, 1946, Eq. (192), p. 279] 



r = xct/oo sin 



(44.iii) 



where a(alpha) is the geometric angle of attack 

 of an equivalent flat plate and c is the chord. 



Expressed in terms of the Uft coefficient Cl 

 and the drag coefficient Cd , the overall Uft and 

 drag are 



Lift L = Ci^^ Ulhc = CA UlA„ (44.iva; 



Drag D = CoT) Ulhc = Co^ UlA„ (44.ivb) 



For the special case of an infinitely thin flat 

 plate lying at an angle of attack a in a uniform 

 stream, when the plate is also of infinite span, 

 with infinite aspect ratio, the 



Lift coefficient Cl = 2x sin a 



(44.V) 



[Durand, W. F., "Aerodynamic Theory," 1935, 

 Vol. I, Div. B]. 



By making an arbitrary assumption as to the 

 distribution of the intensity of lift over a hydrofoil 

 of finite span, an approximate induced-drag co- 

 efficient is Cdi = Cl/[T{b''/A„)], where 67 A „ is 

 the aspect ratio [Rouse, H., EMF, 1946, p. 285]. 

 By the same reasoning the downwash angle 

 e(epsilon) is approximately C/,/[T(?>V^ff)l. 



The magnitude of the lift by the Magnus Effect 

 on a rotating cylinder in a stream is given by the 

 simple expression L = p[/„r for unit length of the 

 cyhnder along its axis. 



44.3 Test Data from Typical Simple Airfoils 

 and Hydrofoils. There is, in the technical litera- 



ture, a great wealth of published data on the lift, 

 drag, and moment coefficients and other factors 

 of a great variety of airfoil shapes, when tested 

 in air. Some of these data are in tabular form but 

 most are in graphic form, corresponding somewhat 

 to the information in Figs. 44. A and 44. B. Here, 



Moment Coefficient C^ 

 0.1 O.a 0.3 04 



Fig. 44.A Lift-Coefficient, Drag-Coefficient, 



AND Moment-Coefficient Graphs for Gottingen 



Profile 409 op Aspect Ratio 1.00 



as in many cases in the literature, the geometric 

 or the nominal angle of attack is the independent 

 variable. For the data in Figs. 44.A and 44.B, 

 adapted from the work of H. Winter at Danzig 

 on Gottingen section 409 ["Flow Phenomena on 

 Plates and Airfoils of Short Span," NACA Tech. 

 Memo 798, Jul 1936, Figs. 16 and 17, respectively], 

 the values of the angle of attack a are spotted 

 along the Cl and Co curves. To use these graphs, 

 follow along the proper curve until a mark is 

 found for the particular value of the angle of 

 attack that has been selected. Then the abscissa 

 and the ordinate of this mark are the values of 

 drag and lift coefficient, respectively. For ex- 

 ample, for the full-line graph of Fig. 44.A, take 

 the point where the angle of attack a is 17.8 deg, 

 at which partial breakdown or preUminary stalling 

 occurs. The corresponding value of Cl is about 

 0.592 and of Co about 0.12. The lift-drag ratio 

 at this hydrofoil position is 0.592/0.12 or about 

 4.93. 



