47
THE OSCILLATING RECTANGULAR AIRFOIL AT SUPERSONIC SPEEDS*
BY
JOHN W. MILES**
University of California, Los Angeles
Summary. The pressure distribution on a quarter infinite, zero thickness airfoil having
a prescribed distribution of down wash (on the wing only), which exhibits a harmonic
time dependence, is determined by a Fourier transform solution of the linearized,
potential equation for supersonic flow. The solution is effected with the aid of the
Wiener-Hopf technique and leads to a Green's function, which may be expressed either
as a finite, definite integral or as an expansion in powers of a dimensionless frequency
parameter. It is shown that the results are applicable to the calculation of the forces
and moments on rectangular airfoils of effective aspect ratio (A cot 6, where 6 is the
Mach angle) greater than unity. It appears that the force and moment coefficients of
practical interest may be expressed in terms of known functions, including certain
integrals which have been calculated for the two-dimensional, oscillating airfoil. The
extension of the two-dimensional results to rectangular wings for which the prescribed
down wash is constant along the span is particularly simple. The extension of the results
for harmonic time dependence to the step function (Heaviside) case is indicated.
1. Introduction. The linearized, two-dimensional problem of the oscillating airfoil at
supersonic speeds has been studied by a number of analysts, using a variety of ap-
proaches. This work has recently been collected and summarized in two reports prepared
by Biot et. al.,1'2 which give both the methods of analysis1 and the numerical results.3
(The reports also include the subsonic, compressible case.) These results are probably
adequate for the strip theory analysis of wings with supersonic leading and trailing edges
and aspect ratios sufficiently high to render tip effects small. (For an approximate treat-
ment of the swept wing, reference may be made to a recent paper by the writer.3) Un-
fortunately, those wings which meet the limitation of supersonic leading and trailing
edges are generally characterized by small aspect ratio; moreover, the most serious
supersonic flutter problems indicated by two dimensional analyses of such wings fre-
quently occur in the near sonic regime, where tip effects are by no means negligible.
Accordingly, it is of considerable practical importance to consider the three dimensional
problem of the oscillating airfoil.
The problem selected for study in the present paper is that of the rectangular wing
tip, since it is the simplest three dimensional configuration (excepting those wings with
no subsonic edges) of practical import. The results may be applied directly to the
rectangular airfoil of aspect ratio sufficiently large to prevent the Mach waves from the
leading edge wing tips from intersecting one another forward of the trailing edge and,
*Received November 21, 1949.
**Consultant, U.S. Naval Ordnance Test Station, Inyokern, Calif. The author wishes to thank
Dr. D. E. Zilmer for his careful criticism and helpful suggestions in the preparation of the manuscript.
'J. N. Karp, S. S. Shu, H. Weil, M. A. Biot, Aerodynamics of the oscillating airfoil in compressible flow,
F-TR-1167-ND, HQ, AMC, Wright Field, Dayton, Ohio (1947).
2J. N. Karp, H. Weil, M. A. Biot, The oscillating airfoil in a compressible fluid, F-TR-1195-ND (1948).
3J. W. Miles, Harmonic and transient motion of a swept wing in supersonic flow, J. Aero. Sci. IS,
343-347 (1948).
48 JOHN W. MILES [Vol. IX, No. 1
indirectly, to the case where these Mach waves intersect on the wing but do not intersect
the opposite side edges. It is also possible to solve the case of arbitrarily small aspect
ratio, but the results are so complex as to be of dubious practical value.
The method of solution to be used follows the Wiener-Hopf technique,4 which has
previously been applied to the rectangular wing in a steady flow.5
2. Statement of problem. A thin, rectangular airfoil is located in the vicinity of the
plane z = 0, and its projection there occupies the first quadrant of the xy plane. A flow
of supersonic velocity U is directed along the positive x axis, so that the leading edge
of the airfoil is projected on the positive y axis and the port side edge on the positive x
axis, as shown in Fig. 1. The boundary conditions are linearized in the usual manner,
so that they may be applied at the projection of the airfoil on the plane z = 0, rather
y
Fig. 1. x, y, z axes and projection of airfoil on xy plane.
than at the airfoil proper. The equations of flow will also be linearized,6 so that the
problem may be subdivided into antisymmetric and symmetric cases with respect to
the plane z = 0; only the former case is of interest here, since the latter situation does
not give rise to lift. Accordingly, it is sufficient to consider the half space z > 0 and to
apply the boundary conditions appropriate to the airfoil in the plane z = 0+. The
problem to be solved is then the specification of the perturbation pressure over the first
quadrant of the plane z — 0+ from a knowledge of the down wash there.
4N. Wiener and R. Paley, The Fourier transform in the complex domain, Amer. Ma. Soc. Colloq.
Publ. 19, Ch. IV (1934); E. C. Titohmarsh, Theory of Fourier integrals, Oxford Press (1937) 339-349;
E. Reissner, J. Ma. & Ph. 20, 219-223 (1941).
6J. W. Miles, On the rectangular airfoil at supersonic speeds, No. Amer. Avia. Report AL 866 (1948).
6A more complete discussion of the linearizing process and its various aspects is given by P. A.
Lagerstrom, Linearized supersonic theory of conical wings, J.P.L. Progress Report 4-36. California Insti-
tute of Technology (1947); NACA T.N. 1685 (1948).
1951] OSCILLATING RECTANGULAR AIRFOIL AT SUPERSONIC SPEEDS 49
The vector, perturbation velocity (q) due to the presence of the airfoil in the flow is
specified as the gradient of a velocity potential Ub<j>, viz:
q(z, y, z, t) = UbV<l>(x, y, z, t) (2.1)
(b is a characteristic length, to be chosen in any convenient manner.) The gage pressure
follows from Newton's law as:
p(x, y, z, t) = -pUb jft<t>(x, y, z, t) (2.2)
where D/Dt is the time differentiation operator in a fixed reference frame and, in linearized
form, is given by:
it-ui + it ™
The condition of continuity, after linearization, leads to the scalar Helmholtz equation
in the fixed reference frame, viz.:
v2</> = ? W2 * (2-4)
where c is the sonic velocity for the ambient stream conditions.
At this point, it is convenient to introduce the harmonic time dependence exp (iwt),
the Mach angle 6, the frequency parameter k, the dimensionless coordinates (x', y', z'),
and the modified potential, pressure, and downwash functions t(x', y', z'), y(x', y'), and
a(x', y'), in accordance with the relations
6 = sin-1 (c/U) (2.5)
k = (aib/c) tan 6 (2.6)
x = (6 cot 6) x' (2.7a)
y = by' (2.7b)
z = bz' (2.7c)
4>(x, y, z, t) = exp [i(ut — kx' csc 6)]\p(x', y', z') (2.8)
<t>(x, y, 0 +, <) = (U tan d/b) exp [i{ut — kx' csc 6)]y(x', y') (2.9a)
y(x', y') = - in sin dji(x', y', 0+) (2.9b)
a(x', y') = -^7 t(,x', y', 0+) (2.10)
Substitution of Eq. (2.8) in Eq. (2.4) yields the reduced equation
tv'v- + t.-z- = tz'x' + Ki (2.11)
50 JOHN W. MILES [Vol. IX, No. 1
while substituting Eq. (2.9a) in Eq. (2.2) yields
p(x, y, 0+, t) = — pU2 tan 9 exp — kx' csc 0)]y(x', y') (2.12)
The boundary value problem to be solved now may be posed as: find a solution to
Eq. (2.11) which satisfies the boundary conditions
y', 0+) = -«+(*', y'), y' > o (2.13)
y(x', y') =0, y' < 0 (2.14)
t(x', y', z') = 0, x' < z' (2.15)
Eq. (2.13) states that the (modified) downwash a+(x', ?/), is prescribed on the airfoil;
Eq. (2.14) states that the pressure must vanish off the airfoil, since it is presumed asym-
metric with respect to z, and only the airfoil is capable of supporting a discontinuity in
pressure; and Eq. (2.15) states that the disturbance is propagated downstream and must
vanish forward of the Mach waves originating at the leading edge of the airfoil.
3. Fourier integral formulation. A general solution to Eq. (2.11) may be conveniently
formulated in terms of Fourier integrals. The Fourier transformation of a function of
the space coordinates (x, y) into its representative in the (p, v) spectrum will be denoted
by a transition from lower to upper case letters in the functional notation, and the
conjugate transform operators T and T are defined by
fix, y) = T{F(n, »>)} = ^ J dn J dvF(n, v) exp [iipx + vy)] (3.1a)
F(ji, v) = T{f{x, y)} = ~ J dx f dyf(x, y) exp [~i(jxx + vy)} (3.1b)
In general, the parameters m and v may be allowed complex values, but the paths of
integration in the m and v planes must be suitably restricted in order to comply with
both physical and mathematical requirements. Frequent reference will be made to
Titchmarsh7 and Campbell and Foster8, simply by using the letters T and CF, followed
by the appropriate equation number in the original source, although the notation used
herein is not entirely consistent with either of these references.9
In addition to the entire transforms of Eq. (3.1), it is expedient to introduce the
notation
F+(ji, v) = T{f(x, y)l(x, y)} (3.2a)
F-(v, v) = T{f(x, y)l{x, -y)} (3.2b)
7E. C. Titchmarsh, Theory of Fourier integrals, Oxford Univ. Press (1937).
SG. A. Campbell and R. M. Foster, Fourier integrals for practical applications, Bell Tel. Syst. Mono.
B-584 (1942); also published by D. Van Nostrand and Co., New York, N. Y. (1948).
9In the ease of transformation with respect to a single variable, we find it convenient, however, to
use the notation of ref. 8, such that, e.g./(x) = T^Fip.) J = 1 /2tt /"_«> dpi F(n) exp (ifix).
Thus, the inversions of section 5 correspond to ref. 8.
1951] OSCILLATING RECTANGULAR AIRFOIL AT SUPERSONIC SPEEDS 51
l(x, y) = 1, x > 0 and y > 0
(3.3)
0, x < 0 or y < 0
Eq. (3.3) defines the Heaviside step function in two variables, but if only one variable
is indicated, e. g\, l(x), the step is in that variable alone. F+(n, v) then represents the
transform of a function which vanishes off the wing, while F- (p., v) represents the trans-
form of a function which vanishes on the wing, recalling, cf. Eq. (2.15), that the solution
vanishes identically forward of the wing.
It is readily shown that an elementary solution to Eq. (2.11) is given by
i0{x, y, z) = exp [i(p.x + vy) — \z] (3.4)
X = [„2 - (M2 - k2)]1/2 (3.5)
the sign of the exponent Az being chosen to represent a disturbance which is bounded
for large z. The primes have been dropped from the coordinates, but they are assumed
to be the dimensionless (primed) coordinates of Eq. (2.7). By virtue of the linearization
of the problem, these elementary solutions may be synthesized with the aid of the
Fourier integral to form solutions capable of satisfying prescribed boundary conditions.
In particular, the most general solution to Eq. (2.11) reducing to y(x, y), cf. Eqs. (2.9b)
and (2.14), is given by
(3.6)
provided that the paths of integration in the n and v planes are suitably chosen.
In order to determine the appropriate paths in the complex transform planes, it is
necessary to establish a domain of regularity for r+(/j, v). If it is simply assumed that
y(x, y) is bounded for large x (the behavior of y in y affects the behavior of T in v only),
it follows from Eq. (3.1b) that T+(ti, v) will be regular in Im (p.) < 0, so that the yu inte-
gration may be carried out along a path in the lower half of the complex n plane. More
specifically, T+(n, v) will be found to have a simple pole at n — +(k csc d + it) if the
pressure on the wing, cf. Eq. (2.9a) behaves as exp (—ex) for large x, where e is a positive
real constant, and will have a zero at n = k sin 6, as may be verified a posteriori. (Due
to the zero in T at ju = k sin 6, ^ does not have a pole there.) In the present analysis,
it suffices to take e = 0, insofar as Im (p.) < 0. Accordingly, the integrand in Eq. (3.6)
is regular in n except for a simple pole on the real axis and the branch points in X, the
latter being designated as ±moM, where, cf. Eq. (3.5),
X = 04 - m2)1/2 (3.7)
MoM = (k + vy/2 (3.8)
The location of these branch points of course depends on v, but if the path in the v plane
is chosen such that
| Im (y) | < k (3.9)
they will always possess real parts, and their imaginary parts will always be less (in
magnitude) than | Im (v) |. It follows that, if the branch cuts from ±ju0 are both extended