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National Advisory Committee for Aeronautics, Technical Notes - Pitching Moment Derivatives at Supersonic Speeds for a Slender Delta Wing and Slender Body Combination and Approximate Solutions for Broad Delta Wing and Slender Body Combinations

The pitching-moment deriVatives Cmq 'ana Cmd at supersonic speeds
are developed for a slender— —delta—wing and slender-body combination
having no afterbody. By drawing an analogy between the aerodynamics of
the wing—body section of the combination and the aerodynamics of a delta
wing alone, the results for the slender- -deltaéwing and slenderebody com—
bination are modified to the extent that approximate solutions for C
and Cmd for broad- -delta-wing and slender4body combinations can.be
obtained.

Various methods, based on linear theory, for obtaining solutions
for the flow about wing—body-combinations have been developed for the
determination of the lift and moment due to angle of attack. Refer-
ences l to 7 comprise a fairly'comprehensive list of most of the signif-
icant of these methods, which include both approximate and exact solu-
tions. All the exact solutions to the linearized differential equation
of steady supersonic flow, however, employ iteration processes, infinite
series, or both, and their practical application results in approximate
solutions although the error is often negligible, depending upon the
particular prdblem, rate of convergence, number of iterations, and so
forth. Spreiter (reference 7) has presented solutions in closed form
to the two-dimensional Laplace .equation of potential flow for the lift
and moment of winngody combinations. These sblutions apply to the super-
sonic range for the limiting case of a slender wing-body configuration.

For the stability derivatives of wing—body combinations, there are
a few papers on the damping-in—roll characteristics (see, for example,
references 8 and 9) but none for the damping in pitch.

The purpose of the present paper is to extend the method used by
Spreiter in reference 7 to the calculation of the pitching-moment deriva-
tives due to constant rate of pitch Cmq and due to constant accel-
erated motion in the vertical direction CD361.) for a slender-delta-wing
and slender-body combination. In addition, an approximate solution to
these derivatives is developed for a broad-delta—wing and slender-body
combination in supersonic flow by introducing certain modifying factors
into the slender-delta—wing and slender—body results.

Certain conditions are placed upon the configuration. The body
ahead of the wing is slender, has a circular cross section, and is
pointed at the nose, and the slope of the body meridian section is
continuous. For the wing-body section, the Wing semiapex angle is small;
along the wing-body juncture, the body radius is a maximum and is con-
stant; and finally, the configuration has no afterbody (see fig. 1).