Brard 
DISCUSSION 
Peter T.. Fink 
Untverstty of New South Wales 
Kenstngton, Australta 
The principal message of Admiral Brard's paper is that we 
should take bound vorticity more seriously, and although I like the vor- 
tex ribbons I cannot make up my mind about them before trying some 
examples. I want to make a contribution concerned with the problem 
of modelling shed vorticity, particularly when there are sharp edges 
fixing separation at known positions. Figure 1 is an example of a la- 
boratory simulation of a section of a ship heaving at its moorings, 
with no forward speed. A student, Mr. W. K. Soh, and myself are de- 
voting ourselves to potential flow modelling of (in this case) bilge keel 
separations and I think there is something of more general interest in 
this which I should like to speak about. 
In two-dimensional flow such problems can, of course, be 
transformed to the case of a flat plate moving normal to itself in ar- 
bitrary unsteady motion. For practical calculations it is usually ne- 
cessary to replace the vortex sheet by discrete vortices and one might 
expect the calculation to be straightforward, using Kutta conditions at 
the edges, the Kelvin Theorem to ensure that the total circulation re- 
mains constant and the condition of zero force on each vortex in the 
standard classical way. This certainly works quite satisfactorily for 
lifting aerofoils with unsteady motion in two dimensions. However, for 
a blade moving normal to itself the circulation is always zero, so that 
the Kelvin theorem does not help and a degree of arbitrariness has to 
be injected. The arbitrariness is required because the Kutta condition 
will not give both the strength and the position of the first of the vorti- 
ces one is going to putin. Figure 2 is an example which shows a plate 
of unit length moving towards the left, in this case five seconds 
after an implusive start, that is, when the motion is still largely iner- 
tial. Fresh discrete vortices have to be added to give the picture shown 
here and in one of the earlier attempts they were added at constant 
intervals of time ; and not long after a degree of zigzagging developed, 
as shown. My attention was drawn to the fact that Professor Birkhoff 
had issued a warning about this sort of procedure over ten years ago. 
Figure 3 shows the improvement when the time intervals are 
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