# Two-dimensional unsteady lift problems in supersonic flight Page: 10 of 13

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REPORT 945-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

The load distributions which have been developed were

obtained for flight velocities in the supersonic regime. It is

apparent, however, that the basic differential equation is not

restricted to the case where M> 1 and that the method of

analysis affords a means whereby transient load effects can

be studied for subsonic speeds. The essential difference

between the latter problem and the results derived here lies

in the relative position of leading- and trailing-edge traces

in the x,t plane and the trace of the characteristic cone.

Thus, for subsonic flight, the trace x= -t does not cut

across the region occupied by the airfoil; whereas the cone

stemming from the trailing-edge point c,O does. A qualita-

tive picture of the problem is obtained if the analogy between

the nonsteady two-dimensional case and three-dimensional

wing theory is used. The loading functions given in equa-

tion (11) are equivalent to loading existing on a swept-for-

ward tip of a three-dimensional wing. Thus, in figure 1 (a),

x can represent distance measured spanwise, t can represent

distance measured chordwise, and the shaded area can

represent a portion of the plan form of the wing. Using this

analogy, the loading which has been determined is merely

load distribution over the swept-forward tip of a wing with

constant chord and supersonic leading edge. When the

case of the airfoil section traveling at subsonic speeds is to

be considered, the problem becomes one of determining the

loading over the swept-forward tip of a wing with constant

chord and subsonic leading edge.

DEVELOPMENT OF INDICIAL LIFT FUNCTIONS

Since section lift coefficient c, is given by the expression

cz- 1 dx

c q

the relations presented in equations (11) and (12) are suffi-

cient for the determination of cl,, indicial lift coefficient for

change in angle of attack, and c, indicial lift coefficient for

an airfoil entering a gust. As a result of direct integration,

the following results are obtained:First time interval

c

O<t<l+M

4a

c l, _ m ol

4wo0t

c c-Vo(13a)

(14a)c c

Second time interval +M[t < Ali

cz -=4 - 2+arc sin c +

M7-I1 arc cos t+Mct--M + - /t2 (ctl) (13b)

VM_1 C I

cig= 4 (Vc + arc sin -Mt)+

4wo Alc +t2-M't

4w- arc cos Act2M2t(14b)

rVo/AfP- 1 cThird time interval c c t

M-l14a

C _ 4wo

l- Vo M1'-1(13c)

(14c)Values of the lift functions are plotted in figure 6 as func-

tions of s, the distance traveled by the airfoil measured in

2Mt

half-chord lengths where s=- . The curves shown were

calculated for values of M equal to 1.2, 1.31, and 1.46, since

calculated for values of M equal to 1.2, 1.31, and 1.46, sinceO 2 4 6 8 0 /12 /4

Half-chords traveled, s

FIGURE 6.-Indicial lift functions Cj,(s) and ci(s) for various free-stream Mach numbers.

the asymptotic values of c: and c1, for the three cases agree

with the values given in reference 1 for the subsonic wing at

aspect ratios of m, 6, and 3, respectively. No direct analogy,

of course, can be made between the two cases. It is, how-

ever, worthy of note that the variations in the indicial func-

tions for the supersonic case are of the same order of magni-

tude as those found in the finite-span incompressible case.

From a knowledge of the lift function resulting from a

sudden unit angle of attack, it is possible to express the lift

corresponding to a given variable motion by considering

the given motion as being composed of infinitesimal steps

and summing the lifts corresponding to each step. Math-

ematically, the problem corresponds to the use of the

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Heaslet, Max A. & Lomax, Harvard. Two-dimensional unsteady lift problems in supersonic flight, report, December 5, 1947; (https://digital.library.unt.edu/ark:/67531/metadc60268/m1/10/: accessed May 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.