Free Surface Effects tn Hull Propeller Interaction 
Here , Equation (C 1) represents the predetermined two - 
dimensional foil characteristics , i.e. the lift coefficient C, asa 
function of angle of attack a . Equations (C2) to (C4) establish the 
local angle of attack o@ as the difference between the predetermined 
geometric pitch angle @ and the unknown hydrodynamic pitch angle f; . 
The velocities ug and uy induced by the free vortex trail of the 
propeller may be obtained from Biot-Savart's Law and are expressed 
in Equations (C5) and (C6) as integrals involving the slope of the 
bound circulation [(R) and two special functions in and i- (of 
three variables ) , the so-called induction factors , see Lerbs (1952) . 
Equation (C7), finally , is the relation between lift and circulation in 
accordance with Kutta-Joukowsky's theorem . Mathematically , the 
problem is an integral equation for the unknown function 8; (R/Rp) ; 
which can be solved by iteration if efficient algorithms are available 
for computing the induced velocities ug and u-+ 
The solution of the above problem yields the distribution of 
circulation , and hence lift , over the radius 
dL = p {(1-w )V tug} aR/(sin 6; ) (C 8) 
Now the drag can also be estimated from the known foil charac- 
teristics 
Cy = on (a ) (C 9) 
dD Sa R {(1-w) v tug }?c ar /(singi)” (C10) 
Hence , by resolving lift and drag along the axial and circum- 
ferential directions and integrating over the radius , one can calcula- 
te the thrust and torque produced by the propeller . 
C.2. Method of solution 
Our method of solution was dictated by the computational tools 
and the information on propeller characteristics available tous . The 
principal computational tools at our disposal were two computer pro- 
grams for propeller design , both based on the lifting line theory and 
incorporating efficient algorithms for numerical evaluation of induced 
velocities . One was obtained from the Naval Ship Research and Deve- 
lopment Center and the other from the Technical University of Berlin 
by courtesy of Dr. O stergaard . They are well documented in the lit- 
terature , cf. Haskins (1967) and Ostergaard (1970) , and therefore 
1907 
