Effect of Compressibility at High Subsonic Velocities on the Moment Acting on an Elliptic Cylinder Page: 4 of 45
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NACA TN No. 1218
MCME FORUMIA
Specifically, the problem treated herein is to obtain an
improvement of the Prand-tl-Glauert approximation of the effect of
compressibility on the moment acting an an elliptic cylinder set
at a small angle of attack in a uniform stream. Let Z denote
the physical flo~w plane, z the affinely transformed plane,
and z' the plane of the circle into which the affinely distorted
profile is mapped by a conformal transformation. (See fig. 1.)
As in the calculation of the resultant lifting force given in
reference 1, it is a great labor saving device to choose a large
circle in the z'-plane to correspond to the control contour in the
physical Z-plane during the calculation of the moment and also to
choose as independent variables the polar coordinates Rel,-
of the z'-plane, with
. -.... 2' = Re (2)
where ( + i and R is the radius of the conformal circle.
(See fig. l( ).)
Since the large circle in the z '-plane corresponds to a large
control ellipse in the physical flow plane Z, the expression or
the moment must contain, in addition to the usual momentum integral,
a term involving the integration of the pressures around the control
ellipse. This additional term is necessary because the normal
vector to an ellipse does not pass through its center. The general
vector expression for the moment in a compressible fluid M. with
respect to the origin, obtained from reference 2, is
Mc= pI1(i) de + s (3)
where brackets andparentheses denote vector and scalar products,
respectively, and
r radius vector from origin
n unit normal vebtorq velocity vector of fluid
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Kaplan, Carl. Effect of Compressibility at High Subsonic Velocities on the Moment Acting on an Elliptic Cylinder, report, March 1947; (https://digital.library.unt.edu/ark:/67531/metadc54275/m1/4/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.