12 1 Statement of the Problem
Chapters 4 through 12 have dealt with aerodynamic loading due to uniform flight of wings and bodies. Obviously no air vehicle remains indefinitely bathed in steady flow, but this idealization is justified on the basis that the time constants of unsteady motion are often very long compared to the interval required for transients in the fluid to die down to imperceptible levels. There exist important phenomena, however, where unsteadiness cannot be overlooked; rapid maneuvers, response to atmospheric turbulence, and flutter are familiar instances. We therefore end this book with a short review of some significant results on time – dependent loading of wings. These examples merely typify the extensive research that has lately been devoted to unsteady flow theory, both linearized and more exact. We hope that the reader will be able to construct parallels with steadystate counterparts and thus prepare himself to read the literature on oscillating nonplanar configurations, slender bodies, etc.
Let irrotationality be assumed, under the limitations set forth in Sections 11 and 17. The kinematics of the unsteady field are then fully described by a velocity potential Ф, governed by the differential equation (174), from which the speed of sound is formally eliminated using (167). (We let Я = 0 here.) Pressure distributions and generalized aerodynamic forces follow from (164).
Flow disturbances in a uniform stream
Ф? = U„x (131)
are generated by a thin lifting surface (Fig. 131), which is performing rapid, small displacements in a direction generally normal to its x, yplane projection. Thus the wing might be vibrating elastically, undergoing sudden roll or pitch aerobatics, or an encounter with gusty air might give rise to a situation mathematically and physically analogous to vibrations.
With zu and zi as given functions of position and time, we have no difficulty in reasoning that the boundary conditions which generalize the steadystate requirement of flow tangency at the surface (cf. 55) read as follows:
Фг{х, у, z„, t) = Фх(х, у, zu, t) + Фу{х, у, zu, t) + Фг(ж, у, zi, t) = ^ Фх{х, у, zh t) + ~ Фy{x, y, zi, t) + ^
for (x, y) on S. There is the usual auxiliary condition of vanishing disturbances at points remote from the wing and its wake, but for compressible fluid this must be refined to ensure that such disturbances behave like outwardpropagating waves. The KuttaJoukowsky hypothesis of continuous pressure at subsonic trailing edges is also applied, although we should observe that recent evidence (Ransleben and Abramson, 1962) has cast some doubt on its validity for cases of highfrequency oscillation.
Provided that there are no timedependent variations of profile thickness, the upper and lower surface coordinates can be given by
Zu = eju(x, y, t) = т$(х, у) + el{x, y, t) (133)
Zi = eji(x, y, t) = —T$(x, y) + eh{x, y, t).
Here e is a dimensionless small parameter measuring the maximum crosswise extension of the wing, including the space occupied by its unsteady displacement. Angle of attack a (51) can be thought of as encompassed by the 0term; g and h are smooth functions as in steady motion; their x – and yderivatives are everywhere of order unity; the ^derivative of h will be discussed below.
Recognizing that in the limit € —» 0 the wing collapses to the x, yplane and the perturbation vanishes, we shall seek the leading terms by the method of matched asymptotic expansions. Let the inner and outer series be written
Ф0 = TJ«,[x + «Ф?(ж, у, z, t) + • • •], (134)
Ф* = Ux{x + еФгх(ж, y, z,t) + ■ ■ •], (135)
where
z = z/e, (136)
as in earlier developments. The presence of a uniform stream, which is clearly a solution of (174), has already been recognized in the zeroth – order terms in (134) and (135).
When we insert (133) into (132), a new question arises as to the size of dju/dt, that is of dh/dt. These derivatives may normally be expected to control the orders of magnitude of the time derivatives of Ф, hence of the terms that must be retained when (134)(135) are substituted into (174). Here we shall avoid the complexities of this issue by requiring time and space rates of change to be of comparable magnitudes. For example, within the framework of linearized theory a sinusoidal oscillation can be represented by[10]
l{x, y, t) = t{x, y)eiat. (137)
The combination of (137) with (132), followed by a nondimensionaliza – tion of Ф* and Л through division by JJX and typical length l, respectively, produces a term containing the factor
k = (138)
Here к is known as the reduced frequency and our present intention is to specify that к = 0(1). For the rich variety of further reductions, even within the linearized framework, that result from other specifications on the magnitudes of к, M, etc., we cite Table I, Chapter 1, of the book by Miles (1959).
With the foregoing limitation on sizes of time derivatives, we find that the development of smallperturbation unsteady flow theory parallels the steps (56) through (531) quite closely. Thus the condition that vertical velocity W must vanish as « —> 0 shows, as in (512), that Ф] must be independent of 2, say
ФІ = Vdx, У, t). (139)
By combining (133) with (132), we conclude that Ф is the first term to possess a nonzero boundary condition,
The differential equation
= 0 (1311)
requires a solution linear in z; thus
for z > Ju, with a similar form below the lower surface. As in steady flow, the zvelocities are seen to remain unchanged along vertical lines through the inner field, and it will be shown to serve as a “cushion” that transmits both W and pressure directly from the outer field to the wing.
It is an easy matter to extract the linear terms from (174) and derive the firstorder outer differential equation
By matching with W derived from (1312), we obtain indirectly the following boundary conditions:
*;.<«, o+,« = § + £#
dfi. 1 dji
(7« dt
Moreover, matching Ф itself identifies with the potential Ф[ at the inner limits z = 0±.
The linear dependence of Ф( on /„ and fi, evident from (1314), suggests that, in a smallperturbation solution which does not proceed beyond first order in e, we should deal separately with those portions of the flow that are symmetrical and antisymmetrical in г [cf. (532) or Sections 72 through 73]. Indeed, one may even isolate that part of h(x, y, t) from (133) that is both antisymmetrical and timedependent. This we do, realizing that we may afterward superimpose both the thickness and lifting contributions of the steady field, but that neither has any firstorder influence on the unsteady loading.
We again adopt a perturbation potential, given by
еФ? = <p(x, y, z, i)
and satisfying, together with proper conditions at infinity, the following system:
for (x, y) on S. Corresponding to the pressure difference, has a discontinuity through S. As we shall see below, the KuttaJoukowsky hypothesis also leads to unsteady discontinuities on the wake surface, which is approximated here by the part of the x, yplane between the downstream wingtip extensions.
Finally, a reduction of (164) and matching, to order e, of Ф or Фх shows that
Cp = —2<px – JLVI + 0(e2) (1318)
throughout the entire flow. (From Chapter 6, the reader will be able to reason that the smallperturbation Bernoulli equation again contains nonlinear terms when used in connection with unsteady motion of slender bodies rather than wings.)
13 2 TwoDimensional, ConstantDensity Flow
The bestknown of the classical solutions for unsteady loading is the one, found almost simultaneously by five or six authors in the mid1930’s, for the oscillating thin airfoil at M = 0. In this case of nearly constant density, a key distinction disappears between the steady and unsteady problems because the flow must satisfy a twodimensional Laplace equation
<Pxx + <Pzz = 0. (1319)
We may accordingly rely quite heavily on the results for a steadily lifting airfoil from Section 53, particularly on (558) and (573), which supply the needed inversion for the oscillatory integral equation while simultaneously enforcing Kutta’s condition at the trailing edge.
With the lifting surface paralleling the ж, г/р1апе between x = 0 and c, it can be assumed from (1317) that <pz(x, 0, t) is known over that area and given by (dimensionless)
w0(x, t) = ib0(x)e’at for 0 < x < c. (1320)
The perturbation field has <p and и antisymmetric in z, and allowance must be made for ^discontinuities through the x, yplane for x > 0. Hence, (1319) and all other conditions can be satisfied by a vortex sheet similar to the one described by (558) but extended downstream by replacing the upper limit with infinity. Equations (1317) and (1320) are introduced through
For later convenience, we define an integrated vortex strength
Г(ж, t) = ( У(хі, t) dxі = 2 f 4>x(xi, 0+, t) dxi Jo Jo
= 2ф, 0+, t), (1323)
and note that Г(с, t) is the instantaneous circulation bound to the airfoil.
From (1318) and the antisymmetry of Cp in z, we deduce that a (physically impossible) discontinuity of pressure through the wake is avoided only if
<Px + <Pt = 0, (1324)
for x > c along г = 0+. Equation (1324) is a partial differential expression for <p(x, 0+, t), which is solved subject to continuity of <p at the trailing edge by
ф, 0+, t) = * (с, 0+, t – ■ (1325)
From (1323) and (1325) are derived the further relations for the wake: Г(х, t) = T(c, t – > (1326)
У(х, t) = — Vt (c, t – . (1327)
Equation (1327) has the obvious interpretation that wake vortex elements are convected downstream approximately at the flight speed U„, after being shed as countervortices from the trailing edge at a rate equal to the variation of bound circulation.
We next introduce (1327) into (1321) and use the assumption that a linear, simple harmonic process has been going on indefinitely to replace all dependent variables with sinusoidal counterparts and cancel the common 4 Indicial Motion in a Compressible Fluid
In the analysis of linear systems there exists a wellknown duality between phenomena involving simple harmonic response and “indicial” phenomena—situations where an input or boundary undergoes a sudden step or impulsive change. Fourier’s theorem enables problems of one type to be treated in terms of solutions of the other, and this is frequently the
most useful avenue to follow in unsteady wing theory. There are cases, however, when a direct attack on the indicial motion is feasible.
As a particularly simple indicial problem, let us consider the initial development of flow near the upper surface of a wing (e. g., Fig. 71) when a step change occurs in the normal velocity of the surface. Such a specification demands that we reexamine the fundamental development of Section 131. Essentially what we are saying is that z„ in (133) is given by

= eWo4t)’

for

(x, y) on S,

(1369)

where

1(0 =

f

t < 0

(1370)



u

t > 0


and w0 = Wq/Uv is a constant of order unity. Clearly, in the vicinity of the time origin, there is now some interval where rates of change of flow properties are very large. It is useful to study this zone by defining
* = f (1371)
and replacing the inner series (135) by
Ф* = U*[x + €ФІ(ж, у, г, t) + • • •]. (1372)
Once again we are led to the conclusion that Ф2 carries the first significant disturbances, but now its differential equation and uppersurface boundary condition are
= 7 *25 (1373)
a„
and
Ф‘2г = Jj~ % = wol(0, at 2 = Ju, for (x, у) on S. (1374)
(This statement is actually unchanged if w0 depends on x and y.) Equations (1373)(1374) describe the linearized field due to a onedimensional piston moving impulsively into a gas at rest. The solution reads
Фгг = :r = w°l 6 – ’ (13~75)
dec doo/
and it is easily shown that the overpressure on the wing surface (or piston face) is
dz,
Pu Poo ———— P«Ctoe дї
All of these solutions are quite independent of flight Mach number M, so long as the disturbance velocities remain small compared with a„.
After a short time interval, the foregoing results make a continuous transition to solutions determined from (13—16)—(13—17). Moreover, for t » c/17x, the indicial solution must settle down to the steadystate result for a wing at angle of attack (—ew0). This behavior can be demonstrated using the method of matched asymptotic expansions, but the details are much too complicated to deserve elaboration here.
Perhaps the most interesting aspect of (1375) and (1376) is their general applicability, when M >5> 1, for any small unsteady motion. At high Mach number, fluid particles pass the wing surface so rapidly that all of the disturbed fluid near this surface remains both in the inner z – and ffields; except for large values of x far behind the trailing edge the outer field experiences no disturbance at all. Hence the piston formula, (1376), yields for any instant the pressure distribution over the entire wing, a result which can also be extended into the nonlinear range (Lighthill, 1953).