Theoretical lift and damping in roll at supersonic speeds of thin sweptback tapered wings with streamwise tips, subsonic leading edges, and supersonic trailing edges Page: 1 of 16
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THEORETICAL LIFT AND DAMPING IN ROLL AT SUPERSONIC SPEEDS OF THIN SWEPTBACE
TAPERED WINGS WITH STREAMWISE TIPS, SUBSONIC LEADING EDGES,
AND SUPERSONIC TRAILING EDGES
By FEANK S. MALVESTUTO, Jr., KENNETH MARGOLIS, and HERBERT S. RIBNER
On the basis of linearized supersonic-flow theory, generalized
equations were derived and calculations made for the lift and
damping in roll of a limited series of thin sweptback tapered
wings. Results are applicable to wings with strea-mwise tips
and for a range of supersonic speeds for which the wing is
wholly contained between the Mach cones spiinging from the
wing apex and from the trailing edge of the root section. A
further limitation is that the tip Mach lines may not intersect
on the wing.
For the portion of the wing external to the Mach cones spring-
ing from the leading edge of the wing tips, the pressure distri-
butions for lift and roll previously obtained for the triangular
wing are valid. For the portion of the wing contained within
the wing-tip Mach cones a satisfactory approximation to the
exact pressure distributions was obtained by application of a
point-source-distribution method developed in NACA TN 138.
A series of design curves are presented which permit rapid
estimation of the lift-curve slope C, and damping-in-roll
derivative C for given values of aspect ratio, taper ratio, Mach
number, and leading-edge sweep.
On the basis of linearized supersonic-flow theory the lift-
curve slope CLa and damping-in-roll derivative C , of thin
triangular wings with large vertex angles were treated in
references 1 and 2. In reference 3 stability derivatives in-
cluding CL and C, were presented for a series of sweptback
wings tapered to a point. The present analysis is an exten-
sion of the investigation reported in reference 3 in that, for
a similar range of Mach numbers, the derivatives C/L and
C,, are evaluated for a series of sweptback wings with finite
streamwise wing tips. The wings were derived by cutting
off the pointed tips of the sweptback wings reported in ref-
erence 3 along lines parallel to the free-stream direction.
The introduction of the finite wing tip causes an alteration
of the pressure distribution over the portion of the wing
contained within the Mach cone springing from the leading
edge of each wing tip. The wing-tip disturbances are con-
fined to these tip Mach cones and do not affect the remain-
ing portion of the wing.
For the case of a wing at an angle of attack the exact
solution for the pressure distribution in the tip region has
been reported in reference 4 by the method of superposition
of linearized conical flows. This solution (although inte-
grated therein for the tip loss in lift for some cases) does not
lend itself readily to the evaluation of the lift-curve slope for
families of wings. In the present report the pressure dis-
tributions in the tip region for lift, and for rolling as well,
are determined to a close approximation by the application
of the method used by Evvard in reference 5. The complete-
wing pressure distributions are integrated analytically to
obtain the lift-curve slope and damping in roll for general
families of wings.
The results of the analysis are given in the form of gen-
eralized equations for CzL and Crp together with a series of
design curves from which rapid estimations of CLa and Cr,
can be made for given values of aspect ratio, taper ratio,
Mach number, and leading-edge sweep. Some illustrative
variations of the derivatives with these parameters are also
presented. The derivatives are valid only for a range of
supersonic speeds for which. the wing is entirely contained
between the Mach cones springing from the wing apex and
from the trailing edge of the root chord. An added re-
striction (which, for practical configurations, materially
limits the range of Mach numbers for very small aspect
ratios only) is that the Mach lines emanating from the wing
tips may not intersect on the wing.
2, y, z Cartesian coordinates of an arbitrary point
x, -coordinate and y-coordinate of a point source
Uii, Vi, We induced flow velocities along X, Y, and Z
body axes (see fig. 2 (a))
i', v',w' . incremental flight velocities along X, Y, and Z
stability axes (see fig. 2 (b))
t, v oblique coordinates in plane of wing the axes
of which are parallel to Mach lines
M (-Bn); v (B (+By)
uw, Vw oblique coordinates of a particular point on
surface of wing (see fig. 13)
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Malvestuto, Frank S., Jr.; Margolis, Kenneth & Ribner, Herbert S. Theoretical lift and damping in roll at supersonic speeds of thin sweptback tapered wings with streamwise tips, subsonic leading edges, and supersonic trailing edges, report, February 15, 1949; (digital.library.unt.edu/ark:/67531/metadc65671/m1/1/: accessed September 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.