31
SOURCE-SUPERPOSITION METHOD OF SOLUTION OF
A PERIODICALLY OSCILLATING WING AT SUPERSONIC SPEEDS*
BT
H. J. STEWART and TING-YI LI
California Institute of Technology
Introduction and summary. In a recent paper Evvard (Ref. 1) discussed the linearized
theory of the non-steady motion of three dimensional wings by methods which he had
previously developed for the treatment of the corresponding steady flow problems
(Refs. 2 and 3). Evvard represented the wing by a distribution of sources, and the
important result of his steady state theory concerned the determination of the flow in
a region influenced by a subsonic leading edge or wing tip. He showed that the influence
of the flow around this subsonic edge of a flat lifting wing on the velocity potential at
X
Fig. 1. Geometry of Wing.
a point within the region of influence of this edge is exactly equal to and of opposite
sign to the contribution to the potential from the sources distributed over a simply
determined region of the wing. In his paper on non-steady motion, he was able by similar
methods to determine an explicit formula for the velocity potential; however he could
not express the results in a similar, "equivalent area", form.
The present paper is concerned with the same problem of the non-steady lift of
finite wings at supersonic speeds, particularly in regions which are influenced by subsonic
leading edges or wing tips. It is shown that the simple "equivalent area" theorem de-
veloped by Evvard for the steady state case is also valid for oscillating wings. The
*Received December 30, 1949.
32 H. J. STEWART AND TING-YI LI [Vol. IX, No. 1
theorem is not extended to arbitrary non-steady motions, and an example where the
theorem in this simple form is apparently not valid is demonstrated.
The basic differential equation and boundary conditions. Consider a wing in a steady
supersonic flow of velocity U and Mach number M in the direction of the x-axis. Then
the velocity potential <p which governs any small, possibly non-steady, disturbance pro-
duced by the wing satisfies the linearized differential equation
i ,2U&p_ ,(&_ i^_3V m
a2 df + a2 dx dt + \ a2 / dx2 dy2 + dz2' ^ '
where (x, y, z) are cartesian coordinates, t is the time variable and a is the speed of sound
in the undisturbed flow so that U = Ma. The wing is assumed to be near the plane
z = 0.
With the approximation of the linear equation, it is permissible to replace the
boundary conditions at the point (x, y, z) on the actual wing surface by the same boundary
conditions applied on the plane z = 0 at the point (x, y, 0). In order to express the
boundary conditions it is necessary in general to divide the wing surface into two different
types of regions (see Fig. 1). The origin of the coordinate system is taken at the point
0 where the Mach line Ov is tangent to the leading edge. The leading edge is thus divided
into two segments, the segment OA which is defined by x = (y) or y = y, (x) and is
a supersonic leading edge and the segment OS which is defined by x = x2(y) or y — y2(x)
and is a subsonic leading edge. As a matter of convenience it is assumed that the trailing
edge, x = x3 (y), is a supersonic trailing edge where the Kutta condition need not apply.
The Mach line Ou then divides the wing into two regions. Region I, which is bounded
by a; = x^y), x = x3(y) and Ou, may be referred to as a purely supersonic region. Region
II, which is bounded by a: = x2(y), x = x3(y) and Ou may be referred to as a mixed
supersonic region (Ref. 4).
At any point on the surface of the wing the flow must be tangential to the surface
at any instant. This boundary condition, linearized, and applied to an oscillating condi-
tion is
•\
-J1 = wT(x, y, + 0) exp {ivt) = UkT{x, y, + 0) exp (ivt), (2)
dz
dz
= wB(x, y, — 0) exp (ivt) — —UAB(x, y, — 0) exp (ivt), (2a)
where, except for the time factor, w(x, y, z) is the z component of the velocity and A
is the effective slope of the streamline and v is the frequency of oscillation. The subscript
T refers to the top of the wing and the subscript B refers to the bottom of the wing. In
general wT and wB (or AT and AB) need not be related. A sign convention for Ar and As ,
adopted in Ref. 1 is also used here and is shown in Fig. 2.
From the definition of a purely supersonic region, there can be no disturbance in the
flow ahead of the line x = x^y). For any point P in Region I the velocity w is thus
known at every point on the plane z = 0 in the forward Mach cone from the point P.
On the wing w is given by Eq. 2 or Eq. 2a, ahead of the wing w = 0.
For a point Q in Region II conditions are more complex. As before, the velocity w
is given for that portion of the plane z = 0 in the forward Mach cone from Q which is
1951] PERIODICALLY OSCILLATING WING AT SUPERSONIC SPEEDS 33
covered by the wing by Eq. 2 or 2a. Also w = 0 and <p = 0 ahead of the line segments
x = x1(y) and Ov. Since x = x2(y) is a subsonic leading edge, there is, in general, an in-
teraction between the upper and lower surfaces which produces an upwash across the
plane z = 0 in Region III which is bounded by x = x2(y) and (to. This upwash cannot,
in general, be specified in advance. For this region the pressure must be continuous
across the plane z = 0 so the linearized boundary condition for this region is thus
(3)
at ox dt dx
The boundary conditions on the plane z = 0 for a point Q in Region II are thus of a
mixed type, involving w over the wing, pressure over Region III and no disturbance
ahead of the lines Ov and x = a\(y).
u
Fig. 2. Sign convention of A's.
Elementary oscillating source potential. Elementary solutions of Eq. 1 which can be
superimposed to obtain more complex solutions can easily be obtained by the method
of separation of variables. For this purpose it is convenient to introduce the following
coordinate transformation:
r=[x2- fiV + z2)]1/2
W+i!)"1"
I -
]-l/2
'
03 = tan (z/y), r = tfa[t -
(4)
where /32 = M2 — 1.
These space variables were found useful in the treatment of steady conical flows (Ref. 5).
The time transformation is similar to a combined Lorentz and Galilean transformation
and has been used by Miles (Ref. 6). In these coordinates Eq. 1 is
n2
=r2^ I 9,^ + 1
dr2 dr2 + dr + dfj.
(1 — 2\ ^1 j_ 1 d2y> . .
" M ) a.. J + i _ „2 3..2- (5)
dy.J 1 — yu da>
This is identical with the form of the classical wave equation in spherical coordinates.
34 H. J. STEWART AND TING-YI LI [Vol. IX, No. 1
The solutions of Eq. 5, obtained by the method of separation of variables are
(cos mu) (P:GO) (r-1/2J_n_1/2(Zr)j
¥> = I] AlmA X (\ ? exp (±Ut), (6)
(sin mw) (0:W) ir'U2Jn+lJlr) )
where Z, m and n are the separation parameters. Pn(jx) and QTAp) are Associated Legendre
functions and J,(n+U2)(lr) is a Bessel function of order ±(n + 1/2).
For m = n = 0, a simple solution of Eq. 6 is
Vl = Ar~1/2J-i/t(lr) exp (ilr). (7)
If r is replaced by the physical time variable from Eq. 4, the Bessel function is written
in the trigonometric form and I is eliminated by the relation v = Iffa, Eq. 7 becomes
Ai (vr \ X. / Ux\
•" = - coswexp A'-w?)
(8)
where At is a new arbitrary constant. Equation 8 may be considered as defining a super-
sonic oscillating source. This basic solution has been used in this form by Miles (Ref. 7).
If the basic solutions used by Garrick and Rubinow (Ref. 4) or by Evvard (Ref. 1) are
applied to oscillating problems, they can be reduced to this same form. It may be noted
that for v — 0 Eq. 8 reduces to the usual steady state source potential. The complete
velocity potential field for an oscillating source is defined by Eq. 8 in the downstream
Mach cone and as zero outside the Mach cone.
Velocity potential of an oscillating wing. For a point in the purely supersonic region
the velocity potential due to the wing can readily be obtained by replacing the wing
z
(?,y,z)
y i, - (x-pz,y,o)
Fig. 3. Singularities or sources in the x, y plane, that affect conditions at (x, y, z) at instant t.
by a distribution of sources over the wing surface. If the region of dependence of a given
point includes only that portion of the wing which is purely supersonic, the velocity
potential for z > 0 due to the source distribution is thus
Hx, y, z, t) = /J Ar(V) exp ?)]} cos (^) ^
(9)
where
n = [(* - f)2 - fi2(y - v)2 - fcT2. (10)
Here AT(£, rj) is the source strength per unit area at the coordinate (£, on the wing
surface. The region of dependence, which determines the region of integration on the
wing surface, is bounded on the downstream side by the line r, = 0 (Fig. 3).
1951] PERIODICALLY OSCILLATING WING AT SUPERSONIC SPEEDS 35
If the source strength AT(ij, 17) can be chosen so the boundary conditions for the
purely supersonic region are satisfied, Eq. 9 is the proper solution. In order to do this
it is convenient to replace the integration variable 17 by
Ky ~ v) = ~r3 sin 6, (11)
where
r2 = [(* - £)2 - W2. (12)
With this notation Eq. 9 becomes
H%, V, z, t) = 1 exp (ivt) exp 0 ~ £>] $
X J y + ^ sin ej cos cos 0^ dd,
(13)
where & is the least value of £ on the leading edge. Since
dr2 ffz
dz r,'
3$
= —ttAt(x — fiz, y) exp (ivt)
az
(14)
l»x—pz r 'ivjj
- pz exp (ivt) exp [-^2 (a: - £)
x y +1 Bi° °)cos (fa cos').
i« (15)
I 2
dd.
If the function A r(£, 17) is continuous in the neighborhood of the point (x, y), the magni-
tude of the double integral in Eq. 15 is finite for sufficiently small values of z; so
f d$\
lim I— j = — tAt(x, y) exp (ivt). (16)
e-* + 0 \uZ /
By comparison of Eqs. 16 and 2, it is seen that
At(x, y) = -- wT(x, y, + 0) = AT(x, y, + 0). (17)
7T 7T
For a point below the wing, z < 0, a similar analysis shows that
AB(x, y) = - wB(x, y, - 0) = —— \B(x, y, - 0) (17a)
7T IT
The required source strength, A (x, y) in the plane z = 0 is thus completely determined
for any point in the purely supersonic region. On the wing A(x, y) is given by Eq. 17 or
Eq. 17a. Ahead of the wing the disturbance (d$/dz)2_0 is zero; so A(x, y) = 0 in this
region. With these values of A(x, y), Eq. 9 defines the velocity potential and thus the
velocity components and the pressure on the wing. This analysis was given in a similar
form by Miles in Ref. 6 (some errors in his presentation were corrected in Ref. 7).