The Aerodynamics of Sails 



relates the skin friction coefficient to the Reynolds number based on momentum 

 thickness and H , the ratio of displacement thickness to momentum thickness, as 



Cf = 0.246 R0-'-'^' 10-0. 678H _ (38) 



This law has been used in connection with a number of existing semi-empirical 

 theories. Moses (1964) has supplied a computer program for his semi- 

 empirical theory. He uses a skin friction law where the skin friction coeffi- 

 cient is dependent only on r^ in order to compensate for some approximations 

 in his theory. 



In almost all cases the presentations of the semi-empirical theories in- 

 clude a comparison with experiment, and the semi-empirical results are shown 

 to be in excellent agreement. However, when a number of the theories are ap- 

 plied to a given experimental situation, there is often a significant discrepancy 

 between their various predictions. For example. Fig. 4 shows results from the 

 four theories investigated for a normalized chordwise velocity distribution 

 given by 



(l-e *''') - ( 1-e '')x 



(39) 



where the chord is taken as the line < x < 1. K is chosen to locate the point 

 of maximum u' 40% of the chord length aft of the leading edge. A is chosen to 

 correspond to a lift coefficient of 1.8. The predominating influence on separa- 

 tion is the velocity gradient. The velocity distribution is also shown in Fig. 4. 

 The semi-empirical theories predict separation points between 71 and 92 per- 

 cent of the chord. The normalized velocities at these two points are 1.45 and 

 1.06, respectively. This range is too large to accept the accuracy of all of the 

 theories, and accordingly an examination of them has been carried out to deter- 

 mine which one, if any, is likely to be accurate. The experimental comparisons 

 considered by Von Doenhoff and Tetervin (1943), Truckenbrodt (1955), and 

 Spence (1956) were for airfoils on which it is quite difficult to make accurate 

 pressure measurements. Furthermore, there are three-dimensional effects 

 affecting the entire flow field, and there is no way to determine the results of 

 these effects. The experiments of Moses (1964) were carried out in an annular 

 chamber with axial flow in which the axial pressure distribution could be varied 

 by varying the leakoff on the outer wall. Boundary layer growth was studied 

 on the inner wall. It is less difficult to make accurate pressure measurements 

 on such a device than on an airfoil. Three-dimensional effects are minimized, 

 since the purely axisymmetric effects can be accounted for. 



Almost all section data (Abbott and Von Doenhoff, 1959) indicates that rais- 

 ing the Reynolds number results in an increase in lift coefficient and a decrease 

 in drag coefficient, indicating that the separation point moves aft when the Reyn- 

 olds number is increased. The semi-empirical theories of Spence and of Von 

 Doenhoff and Tetervin (Fig. 4) indicate the reverse of this. This is always the 

 case with the theory of Von Doenhoff and Tetervin and occurs on some pressure 

 distributions with the theory of Spence. Spence and Truckenbrodt present 



1417 



