For the purpose of this paper the relation between interaction force and wake 

 is of predominant interest. It follows from the foregoing discussion that the wake and 

 correspondingly the interaction force consist of three parts, viz., a viscous, displacement 

 and wave part, the latter two being potential phenomena. These three parts are mutually 

 dependent, that is, writing the total wake or the interaction force as a sum of the 

 three components each one will be different from that which is obtained when the two 

 other ones are neglected. For instance, calculating the displacement wake from potential 

 flow, effects from viscosity will change the result since, so to speak, the hull is changed 

 by the displacement thickness of the boundary layer. 



It is to be expected that the relation between wake and interaction force is 

 represented by different laws for the different components. Our knowledge of these 

 laws is still fairly limited and is advanced best for the case of a deeply submerged body 

 of revolution in non-viscous flow. For this case, Dickmann has introduced singularity 

 methods. That is, the body as well as the propeller are replaced by proper singu- 

 larities from which follow both the perturbation velocity created by the hull, which is 

 the displacement wake, and the interaction force between hull and propeller, the latter 

 by an application of Lagally's law. This concept has proven very fruitful and has made 

 accessible the interaction problem to a detailed analytical treatment [14 to 20J. 



The singularities which replace the body with respect to its action on the flow 

 are known to be either axially distributed sources and sinks or surface distributions. 

 The absolute flow created by these singularities in the plane of the propeller is denoted 

 the "nominal" displacement wake. 



To replace the propeller by simple singularities infinitely many blades are 

 assumed. This reduces the vortex system of the propeller to cylindrical sheets of ring 

 vortices and straight line vortices. The latter are responsible for the tangential flow and 

 may be neglected since this flow component is of minor interest in these considerations. 

 To further simplify the problem the bound circulation is assumed independent of the 

 radius. What then remains of the vortex system is a cylindrical sheet of ring vortices 

 in the boundary of the slipstream. Relative to the inflow this sheet of ring vortices is 

 equivalent to an uniform distribution of sinks on the propeller disc which may be 

 replaced by a point sink in the propeller center if the action at large distances ahead 

 is considered. It should be mentioned that the equivalence of the propeller inflow 

 and the flow ahead of a sink disc, which has been obtained in the foregoing discussion 

 employing ideas of vortex theory, may also be directly deduced as a first approximation 

 from the fundamental equations of hydrodynamics as shown by Burgers [21]. 



On the basis of Lagally's law the following fundamental relation for the inter- 

 action force &R between the body singularities and the propeller point sink is obtained: 



t d = AR/T = P Epu d /T (13) 



From the potential of the propeller singularity and from considerations of momentum 

 it follows that 



Ep/Fp = 2w a (14) 



so that 



U = 2w d /(l +Vl + C T ) (15) 



The notation is as follows: 



t d — displacement thrust deduction coefficient =: AR/T, T — thrust, Ep — input 



of the propeller sink which is related to the loading coefficient of the propeller, viz., 



C T — T/ — - v s 2 F p , v s — speed of the body, F p = propeller disc area, u d = velocity 



induced from the body singularities at the propeller sink, w d = u d /v s — displacement 

 wake fraction. 



159 



