tex sheets built up from radial filaments become necessary to account for the change 

 of bound circulation in a circumferentially non-uniform inflow. These additional sheets 

 generate additional perturbation velocities on the body singularities which affect the 

 interaction force. Studies are indicated by Martinek and Yeh on a general ellipsoid 

 and by Korvin-Kroukowsky on 3 -dimensional bodies. It is to hope that these studies 

 will shed some light on the question whether the increase of the interaction force in 

 a non-homogeneous wake may be explained from non-viscous flow or whether the 

 frictional wake is responsible. The first opinion has been expressed by Dickmann [14] 

 on the basis of first order estimates; the latter by van Manen [23] on the basis of tests 

 on ship models conducted by van Lammeren. In these tests it has been found that 

 the potential nominal wake is fairly evenly distributed over the propeller disc. From 

 this is concluded that a circumferentially non-uniform wake arises essentially from 

 frictional wake which would then be responsible for the increase of the thrust deduc- 

 tion coefficient. 



We will now briefly consider a viscous fluid. Unfortunately, the methods to 

 establish expressions for the frictional parts of both the wake and the interaction force 

 cannot yet be regarded as sufficiently complete. What has been done essentially in 

 the afore mentioned papers by Dickmann [14] and Martinek and Yeh [17] is to utilize 

 the laws of the boundary layer of a flat plate. That is, the increment of the frictional 

 force is determined from the increment of the free stream velocity which takes place 

 from the action of the propeller race. The modifications of the boundary layer of a 

 flat plate which arise from the pressure fields of both the body and the propeller and 

 the modification of the displacement flow from the displacement thickness are left out 

 of consideration. With these assumptions the frictional part of the interaction force 

 is found to be a small quantity as compared to the displacement part. Somewhat more 

 detailed are the calculations made by Korvin-Kroukowsky [20]. In this paper velocity 

 profiles as measured experimentally on a body of revolution are introduced into the law 

 of momentum. Further, the effect of the propeller flow on the boundary layer is esti- 

 mated. Again, the result is obtained that the frictional thrust deduction is small and is 

 essentially independent of the frictional wake fraction. However, these results refer to 

 axi-symmetrical flow. Trying to estimate the influence of a non-homogeneous frictional 

 wake on the interaction force on a basis of boundary layer theory great difficulties 

 are encountered and it has not yet been possible to establish an order of magnitude 

 of this effect. In this situation one is compelled for the time being when designing a 

 wake-adapted propeller to use semi-empirical relations for the dependence of the local 

 thrust deduction coefficient on the wake fraction, the latter being known as a function 

 of the radius from experiment. Either, the assumption is made that the local thrust 

 deduction coefficient is independent of the wake fraction, or the following relation is 

 employed which has been given by van Manen [23]: 



[1 - t(x)]/[l - U] = {1 - w(x)]/[l - wo]} - 25 (25) 



The quantities t and w are average values as determined by analysis of a model 

 propulsion test. The relation is an interpolation formula obtained from numerical 

 results for t(x). Denoting the thrust deduction coefficients which are related to the 

 elements of an annual ring of the propeller disc by t(x, <p), t{x) is considered the aver- 

 age value of t(x, cp). To calculate t(x, <p) two essential assumptions are made. Starting 

 from the axi-symmetrical case, it is assumed that the local frictional thrust deduction 

 is proportional to the local frictional wake fraction. Secondly, when the loading changes 

 because of a circumferentially non-uniform wake, that the frictional thrust deduction is 

 inversely proportional to the frictional wake. This means that 



t f {x,tp) « tf(x)/w f (x,<p) 

 tt Wf(x)/w f (x,<p) 



is written. Whether or not these assumptions are physically sound is unproven. It 

 should be mentioned that it makes only little difference for a propeller design whether 



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