Potential flow in engine valves Page: 12 of 29
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F.A.CA. Technical Memorandum No. 343 11
toffel integral, but the purely formal difficulties are consid-
erably greater than by the following method. The problem can
here be more conveniently solved by three auxiliary diagrams,
which successively show the log w strips on a slotted half-
plane and on an unslotted half-plane.
The strip of the log w plane on a half-plane is repre-
sented by z = 1 + n which is out out along the imagi-
nary axis from to o (Fig. 14). A reflection on the
unit circle z = 1 shifts the slot from m to 0 and from
i to -i tanc (Fig. 15). The half-plane slotted from 0
to -i tana is finally depicted on the half-plane (Fig. 16) by
the third diagram (2 = tan2a (f2 - 1). On eliminating the in-
dividual steps, we find, for the diagram of log w plane on
the half plane, the function
- cos2 - cos2 (log vy)
cos 2 - cos2 cos2 (log vV) - cos2a
The potential planes depicted on the half-plans by
s = exI or s = eX2 are covered by a linear transformation,
so that every three corresponding points coincide. The ex-
f + a a
pressions 1 =... and 2 "= .. fC +a1 and
1 fc1 +b
f + a.
f - a2 lead (after determining the constants and eliminating
the individual auxiliary variables) to the equations
cos (log w) + 1 - K2
eX2= 4/cos (log w)--.cosh .cx / (1l- K2-) + 4 cos2a 5)
cos (log w) 1
J cosa (log w) - cos2a sina
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Eck, Bruno. Potential flow in engine valves, report, December 1, 1925; (digital.library.unt.edu/ark:/67531/metadc65148/m1/12/: accessed September 24, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.