Potential flow in engine valves Page: 5 of 29
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N.A.C.A. Technical M rorandurr ro. 313 4
scribes the quadrant from C to E. From E to A, u = 0 and
v falls from 1 to a certain value K which we will provision-
ally assume to be less than 1. From A to B, u = 0, so that
the whole flow in' the w plane is portrayed by a quadrant.
The connection between x = cp it and w = u - iv fol-
lows from dx/dz = wv. This is a complex differential equation,
the solution of which is decisive for our problem. In our so-
lution, we find x = fz and utilize the differential equation
for finding this expression, by first obtaining x = g w and
then finding z = J d by an integration. An expression for
the function x, similar to the one for w, is indeed known,
since its limits can also be given. In the theory of func-
tions, the conformable diagram affords a convenient method for
finding x = g w.
For the further calculation, it is more convenient to work
with the natural logarithm of w instead of with w itself,
a simplification also employed in the cases subsequently treated
The further calculation is as follows. The strip BCEAB
is plotted on a half-plane, so that the boundary falls on the
real axis. The same thing happens with the potential.plane x.
These half-planes are then so superposed that the correspond-
ing points coincide, by which operation the function x = g w
The plotting of BCEA on the half-plane (Fig. 4) is ac-
complished by means of Christoffel's integral
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Eck, Bruno. Potential flow in engine valves, report, December 1, 1925; (digital.library.unt.edu/ark:/67531/metadc65148/m1/5/: accessed December 13, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.