A Solution of the Direct and Inverse Potential Problems for Arbitrary Cascades of Airfoils Page: 4 of 51
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NACA ARR No. L4K22b
transformed to a circle, p-plane, by a modification of
the Theodorsen-Garrick procedure (.reference ..3) contesting
in the use of log dp'/dp instead of log p'/p as
the mapping function. The transformation of the given
cascade to the near circle appears to be much more
laborious and the near circle much more different from
a circle than is the case for the transformation of
isolated airfoils. This general procedure has recently
been treated in reference 4. A similar method has also
been used in a British paper by A. R. Howell. The
basic transformation that reduces the given cascade to
a single shape is taken as L = tan' z, where z is the
physical plane. Inasmuch as the resulting c-plane shape
is not of near-circle type, several Joukowski transforma-
tions are applied to produce a near circle. The transfor-
mation from near circle to circle is then accomplished by
the Theodorsen-Garrick method. This method, too, involves
an excessive amount of numerical labor.
In the present paper, the Cartesian mapping function
method of reference 5 is extended to the solution of the
direct and inverse potential problems for arbitrary cas-
cades of airfoils. After an exposition of some basic
properties of the Cartesian mapping function for cascades,
procedures are given for the solution of the direct and
inverse problems. The procedures are then illustrated
by numerical examples.
Acknowledgment is made to Mrs. Lois Evans Loran :f the
computing staff of the lan ley full-rcale tunnel for her
assistance in making the calculations.
z = x + iy plane of cascade of airfoils
S= g + in plane of cascade of straight lines
p plane of unit circle
( central angle of circle
dx component of Cartesian mapping function (CaF)
parallel to chord
Ay component of Cartesian mapping function per-
pendicular to chord
displacement constant for locating airfoil
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Mutterperl, William. A Solution of the Direct and Inverse Potential Problems for Arbitrary Cascades of Airfoils, report, December 1944; (https://digital.library.unt.edu/ark:/67531/metadc61263/m1/4/: accessed April 22, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.