64 J. A. Sparenberg 
W. H. Isay (Institute of Applied Mathematics of the German Academy of Science) 
In reply to Dr. Sparenberg’s discourse on my papers* dealing with the theory of the 
Voith-Schneider propeller I should like to remark the following: 
Equation (12) of my paper II mentioned by Dr. Sparenberg is obtained by the usual meth- 
ods of potential flow analysis if it is assumed that the free vortices move to infinity down- 
stream of the propeller without losing their intensity. In reality, however, these vortices 
will decay in the real turbulent flow downstream of the propeller and both their influence 
and their induced velocity will decrease accordingly. The use of Eq. (12) will therefore 
yield grossly erroneous results, particularly for the blades on the downstream half of the 
propeller which are hit by the free vortices produced by the blades on the upstream half of 
the propeller. This is because the influence of the free vortices is greatly exaggerated by 
Eq. (12) of potential theory. 
On the other hand, any attempt to arrive at an actually comprehensive theoretical de- 
scription of the turbulent mixing and decay of the free vortices would have little chance to 
succeed as the process is not only different at different points but also subject to random 
effects. It was therefore deemed important to evolve a physically reasonable equivalent rep- 
resentation of the velocity field of free vortices which may be used also for numerical calcu- 
lations without excessive difficulty. Equation (1) of my paper II, or Eq. (3) of my paper I, is 
satisfactory under this aspect though naturally different from the conventional formulae of 
potential theory. This formula yields useful results which are in satisfactory agreement with 
force measurements. A full discussion of the difference between Eqs. (1) and (12) along with 
numerical examples can be found in sections 2, 5, and 6 of my paper II. 
The problem thus encountered will be of interest with all types of propellers and turbo 
equipment in which part of the blades operate within the wake vortices produced by other 
blades. This, however, shall not imply that the method I have shown for Voith-Schneider 
propellers can be applied without change to other types of hydrodynamic machinery. 
Contrary to the opinion held by Dr. Sparenberg the induced velocity uy of the free vor- 
tices decreases with increasing incoming velocity u, also in the case of Eq. (1) of my paper 
II. This is caused by the resulting change in the distribution of circulation as can be read- 
ily seen from the numerical examples given in my papers. Furthermore, this reduction in uy, 
for increases in u,, is not more pronounced with Eq. (12) than it is with Eq. (1) as might 
erroneously be concluded when considering the initial factor w#R/u, without at the same time 
observing the behaviour of the circulation. The latter becomes apparent only from the bound- 
ary condition of flow past the profile, i.e., from the integral equation. 
Naturally it is possible also with my theory to determine — similarly as has been shown 
for the problem treated by Dr. Sparenberg in his paper — the distribution of the angle of inci- 
dence for a prescribed relationship for the change of blade circulation as the blade proceeds 
on the propeller circle. Also in this instance the difference between Eqs. (1) and (12) of my 
paper II becomes apparent. If, for example, a simple cosine law is prescribed for blade cir- 
culation (in which case the propulsive force produced by the upstream and downstream halves 
of the propeller would be about equal), then the incidence angle distribution obtained from 
*(1) “Zur Behandlung der Strémung durch einen Voith-Schneider-Propeller mit kleinem Fortschritts- 
grad,” Ing. Arch. 23:379 (1955); (II) “Zur Berechnung der Stromung durch Voith-Schneider-Propeller,” 
Ing. Arch. 24:148 (1956); (III) “Der Voith-Schneider-Propeller im Nachstrom eines Schiffsrumpfes,” 
Ing. a ge (1957); (IV) “Erganzungen zur Theorie des Voith-Schneider-Propellers,” Ing. Arch. 
26:220 (1958). 
