the assumption t = const is introduced or whether the function t(x) as follows from 

 this relation is utilized. 



In spite of the imperfections of the existing theory of hull-propeller interaction 

 it has proven of great value in explaining experimental findings on the thrust deduction 

 coefficient which cannot be understood otherwise. In recent work on the scale effect of 

 propulsion coefficients by van Lammeren, van Manen and Lap [24] the result has been 

 obtained that the thrust deduction coefficient decreases both when the dimensions of 

 the model are decreased and when the roughness of the model surface is increased for 

 the same model. The explanation follows from the afore mentioned interplay of the 

 displacement thickness of the boundary layer with the singularities of the body which 

 may be axially distributed sources and sinks. To describe properly the displacement flow 

 in a viscous fluid, an "effective" hull should be introduced which is the hull of the 

 body augmented by the displacement thickness. The strength of the singularities 

 depends essentially on the curvature of the hull such that the singularities become 

 feebler if the curvature becomes smaller. This happens with the effective hull if the 

 displacement thickness becomes greater either because of a smaller Reynolds number 

 or because of increased roughness. Reducing the input of the singularities decreases 

 the interaction force from Lagally's law which gives the explanation for the afore 

 mentioned experimental results. 



III. Theory of the Wake Adapted Propeller 



The basis of the available theory is a steady relative flow, i.e., the wake is 

 assumed to depend only on the radius. The problem is to ascertain the bound circu- 

 lation such that the useful power of the system "ship and screw" becomes a maximum 

 value for given quantities of power input, advance ratio and wake distribution. As a 

 generalization of the optimum condition of a free-running propeller, the condition for 

 a wake adapted propeller may be written in the form 



tan |8i = F(x)/k (27) 



where k is independent of the radius. The optimum function F(x) is still to be deter- 

 mined. In the case of a free running propeller, F(x) = X/x. 



Combining the optimum condition with the geometrical relation 



tan |8 S = 



\„ = vJRc 



(28) 



which follows immediately from the diagram of the relative flow, and expressing the 

 components of the induced velocity by their respective induction factors, an integro- 

 differential equation for the bound circulation of the optimum wake propeller has 

 been deduced by Lerbs [6] : 



'-[*»« + F(x)i t ) —^— = 2 ] Fix) — - k[l - w(x)] ( (29) 



dxo x — x I \s ) 



G s = T/TrDv s 



With w — and with F(x) — X/x this equation passes over into the equation (10) 

 for a free-running optimum propeller. An approximate solution is obtained if both 

 G s and the induction factors are developed into Fourier series. This leads to a system 

 of linear algebraic equations for the coefficients of the bound circulation. For a prac- 

 tical application in propeller design work an assumption relative to the geometry of the 

 relative flow is introduced which simplifies the calculations considerably without losing 

 too much accuracy. This is explained in detail in a paper by Eckhard and Morgan [25]. 

 It remains to determine the optimum function F(x). This function follows 

 from a rule which has been expressed first by Helmbold [26], viz., that the propulsive 



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