Category An Introduction to THE THEORY OF. AEROELASTICITY

Consider a single-degree-of-freedom system such as a mass-spring – dashpot model. An idealized equation of motion is (§ 1.8)

mx + fix + Kx = X (1)

where the constants m, /?, К are such that the kinetic energy, potential energy, and the dissipation function of the system are given, respectively, by

bmx2, Kx%, I fix2

The dissipation function has the dimensions of energy per unit time, f) is interpreted as viscous damping factor. A is a forcing function.

In § 2.4, p. 69, it is shown that Eq. 1 can be conveniently written as

x – f 2уш0х + a>02x — — X (2)

m

where a)02 = Kjm, and у is the ratio of actual damping /? to the critical damping factor (>„ = 2mco0.

If the forcing function is a harmonic function, the solution of Eqs. 1 and 2 has already been given in § 1.8.

It is instructive to compare the form of damping assumed in Eq. 1 with that commonly used in flutter analysis. Following § 6.9, we write

X + (1 + ig)u>fa = — X (3)

m

which is applied to harmonic oscillations only. But if a motion is har­monic so that x = x0eimi, Eq. 2 may be written as

x – r (1 + 2 iy — I a>02x = — X (4)

a>J m

This is identical in form with Eq. 3 if, and only if,

ш = 0,o. g = 2У (5)

It is therefore clear that the mechanisms of damping suggested by Eqs. 1 and 3 are entirely different if ш Ф ш0, and if the motion is not a harmonic one.

As is discussed in § 11.4, the detailed mechanism of damping in structures concerned with in aeroelasticity is yet unknown. Hence, a choice of the particular form of damping is open to question. For the same reason, the value of g is rarely accurately determined. It is known, however, that g is small for metal airplanes.

When g is small, a free oscillation of a system described in Eq. 2 follow­ing an initial disturbance will be almost sinusoidal with a frequency close to co0. Assuming that Eq. 3 is applicable also to this case, then we have approximately

g = 2У

as in Eq. 5. It is then easily derived from the solution given in § 1.8 that, when g<^l, y<^l, the logarithmic decrement 6 in a free oscillation following an initial disturbance is

<5 = ng = 2 ny

More generally, g can be deduced, approximately, from the rate of decay of free oscillations,

[1] Some authors define elastic axis as the locus of flexural centers or as a flexural line. This is not the sense to be used in this book.

[2] We shall assume tacitly that the structure is rigidly supported in a specific manner when the influence functions are measured. However, this does not prevent their usefulness in applications to an airplane in free flight, because all that is necessary is to measure the elastic displacements (or influence functions) of the airplane structures with respect to a set of rectangular coordinates attached to the airplane, with the origin located at a convenient point, e. g., a point on the center line of the main spar, or the air­plane center of gravity. The structure may be regarded as clamped with respect to the coordinate system at its origin. The change in direction of the coordinates and the motion of the origin can then be determined by the free-body motion of the airplane.

[3] The “local” lift coefficient at any point (x, y) on a lifting surface is defined as the limit of dL/qdS as dS -*■ 0, where dS is a surface element enclosing the point (x, y) and dL is the lift force acting on dS. The local lift coefficient Ct(y) used in Eq. 18 is defined over a chordwise section, so that dS = cdy. We shall use lower case subscript to indicate a local aerodynamic coefficient such as Cu whereas a capital letter subscript will be used for coefficients referred to the entire wing.

[4] A prime over the summation sign is introduced to indicate that the term j = і must be deleted from the sum.

[5] It is necessary only to satisfy the boundary conditions imposed by constraints against displacement. Such boundary conditions are called the “rigid” boundary conditions. The boundary conditions at an end where no constraint is imposed, such as those at the free end of the beam, are called the “natural” boundary conditions. The natural boundary conditions will be satisfied in the limit by the method of successive approximations.

[6] This type of oscillation, due to periodic shedding of vortices, may be called the oscillations of the Aeolian harp type. The principle of playing strings by the wind was recognized in ancient times. King David, according to Rabbinic records, used to hang his kinnor (Ki“ ra) over his bed at night, where it sounded in the midnight breeze.

[7] In some books, the Strouhal number is defined by replacing со by the number of cycles per second. For uniformity in notation, we shall not distinguish the reduced frequency and the Strouhal number in this book.

t There are some quantitative differences between Kovasznay’s and Relf and Simmons’ results, probably reflecting the difference between the wind tunnels they used.

[8] We shall assume that the Reynolds number is in the range of vortex shedding with a clearly defined predominant frequency, 40 < R < З X 106.

[9] We assume that the total angle of attack remain so small that Cj is linearly pro­portional to the angle of attack. This requires that the initial angle of attack do not approach the stalling angle. If the initial angle of attack is large, the additional elastic twist may cause the wing to stall. Divergence near stalling angle is an interesting nonlinear problem, which is closely related to stall flutter and buffeting.

[10] q&lv /1 _ 1 a qaiJ’

plotted against 1 lq will be a straight line. The divergence dynamic pressure can be found either from the slope of this line, ^div/“> or from its intercept on the Щ axis, l/?dtv – This relation can be used for an experimental determination of the divergence speed.

[11] See P. D. Grout, Marchant Calculating Machine Co. Bulletin ММ-182, MM-183 (Sept. 1941). Also Trans. AIEE, 60 (1941).

P. S. Dwyer, Psychometrika 6, 101-129 (1941).

W. E. Milne, Numerical Calculus, Princeton Univ. Press, Chapter 1 (1949).

Note that, according to Eq. 17, the &th column of A-1 can be obtained by solving Eqs. 16 for a column matrix В for which every element is zero except the Arth element which is 1. Thus the calculation of the inverse of a matrix is equivalent to the numerical solution of a system of linear equations.

[12] See for example, С. B. Millikan, Aerodynamics of the Airplane, Chapter 4, John Wiley & Sons (1941). C. D. Perkins, and R. E. Hage. Airplane Performance, Stability and Control, Chapters 5 and 6, John Wiley & Sons (1949).

t The characteristic wing-twisting angle 6W and the spanwise location of the wing reference section rjw will be defined by an equation analogous with Eq. 5 of § 4.6.

[13] In order to distinguish “flutter” from “stall flutter,” some authors use the term “classical flutter.”

f From the results of § 1.9, it is natural to expect that spanwise shift in phase angle exists at the critical flutter condition, because the aerodynamic force contributes a term proportional to the velocity of movement, but such spanwise shift in phase is relatively small.

[14] Flexural-stiffness criterion:

X

Fig. 6.1. Steady flow over a two-dimensional airfoil.

[16] It should be pointed out that the method used in these sections can be applied whenever the aerodynamic forces are expressed in the form of the so-cdlled “classical derivative coefficients” (see p. 228), and are not restricted to quasi-steady coefficients.

[17] The assumed functions f(y) and Ф(у) must satisfy the “rigid” boundary conditions at у = 0 (Eqs. 4, § 6.3). See footnote on p. 54.

[18] x, x • • • „ dR d*R d3R „ „

* Note that it is impossible to have R — — = —— — —; = 0 while —— ф 0.

‘ dU dUi dU3 dU1

t The validity of these relations is restricted by the quasi-steady assumption made in their derivations. The fluid is incompressible. The wing is unswept,, and the aileron is locked. For a more exact survey, see, for example, Ref. 6.23.

[19] The expression 8a gives somewhat better approximation for к < 0.5, whereas 86 is better for к > 0.5. The real part F is well approximated by both 8a and 86, the maximum percentage errors based on the exact values are + 2.6 per cent, — 2.1 per cent throughout the range (0, oo). The error of the absolute value of the imaginary part, — G, is much larger. The maximum percentage errors based on the exact values are as follows: For 8a (R. T. Jones), + 8.5 per cent, — 13.5 per cent. For 86 (W. P. Jones), + 10 per cent, — 11 per cent. A detailed numerical comparison can be found in Ref. 15.80.

[20] This is not true in all cases. For flutter involving ailerons, for instance, the effect of structural damping can be exceedingly large. See NACA Rept. 741.

[21] The (5 function can be defined as a limit of some well-defined functions. See Chapter V of van de Pol and Bremmer.8-2

[22] This curve is given by P. MacCready.8-52

[23] If the mean value does not vanish, we may first subtract from »(/) the mean value. In other words, we may consider only the deviation from the mean value.

[24] f®

p(co) = — І у>(т) cos сот dT rr Jo

[25] It must be observed that the possibility of such a separation into two independent problems depends on the assumption of small disturbances, so that the hydrodynamic equations may be linearized. Hence, the analysis is theoretically valid only for turbu­lences of small intensity.

[26] Within a certain region the total head of the flow is smaller than that of the un­disturbed stream. Although the transition to the normal value takes place gradually, the boundaries of this “total-head wake” are quite well defined. It is not clear how to interpret the total-head wake in a turbulent flow. However, the main indication of the experiment is that the buffeting intensity is related to the turbulences in the flow.

t A proper fillet, i. e., a smooth fairing at a wing-fuselage junction, is most effective in reducing flow separation caused by wing-fuselage interference. Von Karman, in Aerodynamics, Selected Topics in the Light of Their Historical Development, p. 151 (Cornell Univ. Press, 1954), relates an interesting story about his reporting of the effec­tiveness of a fillet in preventing tail buffeting in a lecture in Paris in 1932. Some French engineers were apparently troubled by the same difficulty and immediately tried K&rman’s fillets. Henceforth, the “fillet” was known in France as “karman,” and expressions like “large karman” and “small karman” are used. Von Kdrman himself attributes the development of fillet to the team work of himself, Clark Millikan, and Arthur Klein at the California Institute of Technology. See Karmdm, Th. von.: Quelques problemes actuels de 1’aerodynamique. ./. techniques internationales de Гaeronaut, 1-26 (Paris, 1933), Klein, A. L.: Effect of Fillets on Wing-Fuselage Inter­ference. Trans. A. S.M. E. 56, 1-7, AER-56-1 (1934).

[27] This is a Tauberian theorem. A sufficient condition is that F(s) be the Laplace transform of some function F(i) for all R1 у > 0, F (j) -* К as ^ -* + 0, and F(t) is of order oO/O as r-> oo. To apply this theorem to our example, we should consider the unit-step function 1(0 as the limit of e – A< 1(f) as Л —>■ 0.

[28] The argument of z, written as arg z, is the angle between the radius vector z = x + iy and the «axis; i. e., arg z = arc tan yjx. Thus arg/(z) = arc tan у>/ф. The argu­ment off(z) is increased by 2ir if the end point of the radius vector f(z) encircles the origin once in the positive direction; it is decreased by 2v if the radius vector encircles the origin once in the negative direction. Thus in Fig. 10.15, if Дг) traces C in the positive direction, arg/(z) is increased by 4- as г goes once around C in the positive direction.

harmonic excitations. Therefore the explicit form of the functions Д(г), ДиСг), etc., can be obtained from Eqs. 11 of § 6.9, by replacing ik by s.

f This conclusion is based on an important theorem by A. Erdelyi (Lecture Notes on Laplace Transformation, 1947, California Institute of Technology). Let S_д denote a region that consists of all points satisfying the condition |arg (s — j0)| <i (іг/2) + Д, where 0 < Д < ir/2. Then the theorem states that under the assumptions:

[30] The function /0) is analytic, and regular in S_A (but not necessarily at j0 or at 00).

[31] f(s) -> 0, uniformly in S„A, as j -> со.

[32] f(s) has the following asymptotic representation in the sense of Poincare:

N

fG) ~ 2 cn(s — ■*o)An> (— 1 < A0 < < • • ■ < A. v)

n== 0

uniformly in д, as s -> s0.

One concludes that the inverse Laplace transform of f(s) has the asymptotic repre­sentation

[33] There are numerous books and papers on automatic-control systems. See, for example, a book by H. M. James, N. B. Nichols, R. S. Phillips,*-81 and a paper by Bollay,10-10 from which further references can be found.

t For example, A. G. Webster, The Dynamics of Particles and of Rigid, Elastic, and Fluid Bodies, §§ 76-79, 91, and 102, reprinted by G. E. Stechert & Co., New York (1942). For the linearization of the inertia operator, see Refs. 10.10-10.13.

[34] The condition of tangency does not suffice to determine uniquely the flow around the airfoil. In addition, the Kutta-Joukowsky condition (§ 12.1) at the trailing edge must be satisfied. This condition determines the circulation around the airfoil.

The leading edge of an actual airfoil is rounded, and no special edge condition is needed. In a linearized theory, the airfoil being regarded as infinitely thin, the singu­larity at the leading edge may be regarded as the limiting form of such a rounded edge. In the small perturbation theory it suffices to require that the total integrated force be finite, and that the infinity in the pressure distribution at the leading edge be of a proper order, in analogy to the steady-flow theory.

[35] The following method is given by M. A. Biot.13,1

[36] 6 ~ v at the leading edge, б = 0 at the trailing edge, t Schwarz gives a second derivation of this equation in Ref. 13.29.

a + {тта + l)k + 277k2

[38] x0 is the point about which the moment is taken.

[39] The pressure perturbation p is continuous outside the airfoil, is antisymmetric in y, and hence vanishes behind the trailing edge when у — 0. Therefore, the lift, which is the difference of pressure across the airfoil, vanishes at the trailing edge. It does not vanish at the leading edge because the leading edge is a singular point for L.

t A shorter and more general derivation is given by Kiissner16,12 on the basis of Lorentz transformation of the wave equation. Kiissner’s integral equation holds for three-dimensional flow in both supersonic and subsonic cases. Specialization – into Possio’s integral equation in the subsonic case, into Birnbaum’s equation in the incom­pressible case, and into Prandtl’s equation in the steady-state case are demonstrated. But the form of Kussner’s integral equation is simple only if it is stated in terms of divergent integrals. In practice, an involved limiting process is required to trans­form such divergent integrals into those for which Cauchy’s principal value can be formed.

t See M. Muskhlishvili: Singular Integral Equations, in Russian, translated by Radok and Woolnough, Ministry of Supply, Australia. Published by P. Noordhoff, Groningen, Holland (1953).

[40] The notation is as follows: If a table is given for a range of the argument from 0.10 to 0.50 at intervals of 0.05; this is indicated by writing 0.10 (0.05) 0.50. The notation (5 fig.), (6 dec.), etc., implies that most of the figures in the table in question are given to the apparent accuracy of 5 figures or 6 decimals, respectively.

[41] Most of the experimental investigations are restricted to the determination of the forces normal to the plane of the airfoil, i. e., lift, moment, and pressure distribution. Little is known about the unsteady drag force, which plays only a minor role in aeroelastic problems. Only the normal forces will be considered here.

[42] Timoshenko, Theory of Elasticity, p. 301, McGraw-Hill (1934). t Trefftz, op. cit.

The problem of the stability of a dynamic system is often reduced to determining whether the real parts of all the roots of a polynomial are negative.

For a polynomial of fourth degree with real-valued coefficients,

alxi + a3x3 + a2x2 + axx + a0 = 0 (an > 0) (1)

Routh* shows that the real parts of all the roots are negative if and only if all the coefficients a0, alt ■ • •, a4, and the discriminant

R = ахаф з — афі — axax2 (2)

are of the same sign. For a polynomial of nth degree, Hurwitz established a criterion in terms of a series of determinants. Hurwitz’s determinants for the fourth-degree equation 1, with a0 > 0, are

These must all be positive if the real parts of all the roots are negative, and vice versa. For a fourth-degree polynomial, determinants of order 1 to 4 are involved. The diagonal from the upper left to lower right contains coefficients with increasing indices beginning with 1. The indices of the coefficients decrease from left to right in each row. Negative indices and indices greater than the degree of the polynomial involved are replaced by zero. These rules are sufficient to establish the determinants for a polynomial of any degree n.

It is easy to show that Routh and Hurwitz conditions are equivalent. The proof for Routh’s conditions for polynomials of third and fourth degrees can be found in Kdrman and Biot’s book;f that for Hurwitz can be found in Uspensky. J The literature related to this algebraic problem is very extensive. In Bateman’s review§ on this subject, more than 100 papers are quoted.

* Routh, Advanced Rigid Dynamics, Vol. II, Macmillan Co., London (1930).

f Mathematical Principles in Engineering, McGraw-Hill, New York (1940).

J Uspensky, Theory of Equations, McGraw-Hill (1948).

§ H. Bateman, The Control of Elastic Fluids, Bull. Am. Math. Soc. 51, 601-646 (1945).

There exists several interpretations for the term “torsion-free bending” on which the definition of shear center (§ 1.2) is based.

Consider a uniform cantilever beam (Fig. A 1.1) of length /, built-in at the right end (z = /), and loaded at the left end by a force 8 which acts in the negative jy-axis direction. The origin is taken at the centroid of

.y

Z

the cross section at the free end. The horizontal axis x and the vertical axis у are the principal axes of the cross section.

First Analytical Definition. The stresses and deflection of this beam can be found by Saint-Venant’s theory of bending. Let u, v represent the components of displacement in the x, у axes’ directions, respectively. It is well known that the “rotation” ш of an element in the cross section of the beam is expressed by the equation

(i)

The rate of change of w in the axial direction is

where exz and eyz are the shearing-strain components. By Hooke’s law, one obtains

^ = J_ дтхг

dz 2G l 3x dy

where rxz and ryz are shearing-stress components.

It turns out that the right-hand side of Eq. 2 can be expressed very simply in terms of the stress function used in Saint-Venant’s theory. The result indicates that the “local twist” 3a>/3z at different points in the cross section has different values. It is impossible to have this zero for all elements of the cross section.

It seems natural to define the torsion.-free bending by the condition that the average value of the local twist over the whole section vanishes.* Hence, the first analytic definition of a torsion-free bending is

(3>

By using Eq. 3, the problem of “torsion-free” bending can be solved in a classical way. The shear center is then determined from the fact that the resultant of the shearing stresses in the section and the load S must be equal and opposite and have the same moment arm about the z axis. The distance І of the resultant from the у axis is therefore the abscissa of the shear center. Equating the moment about the z axis of the shearing stresses rxy and ryz with that of S, one obtains

f = £ J J(z V – yrj dx dy (4)

In this formulation, the position of the shear center depends on the Poisson’s ratio y.

Second Analytical Definition. Trefftz proposes-}- another definition of torsion-free bending on the basis of energy considerations. If a beam is twisted by a couple M at the free end, so that that end rotates through an angle a, the elastic strain energy stored in the beam is equal to the work done by the couple during deformation

A1 = IMat (5)

* This is the view taken by J. N. Goodier, J. Aeronaut. Sci. 11, 272-280 (1944). R. D. Specht, in a note to J. Applied Mech. 10, A-235-236 (1943), attributed this definition to A. C. Stevenson {Phil. Trans. Roy. Soc. London, A. 237, 161-229 (1938- 1939).) This definition agrees with the works of Timoshenko.

t E. Trefftz, Z. angew. Math. u. Mech. 15, 220-225 (1935). For a different point of view, which leads to results agreeing with Trefftz’s definition, see P. Cicala, Atti. R. Acc. Sci. Torino 70, 356-371 (1935), and A. Weinstein, Quart. Applied Math, 5, 97-99 (1947).

On the other hand, if a single force S is applied at the free end where the deflection is d, then the elastic strain energy is

A2 = iSd (6)

In combined action of the couple M and the force 8, the elastic strain energy stored in the beam is, in general, different from the sum Ax + A2. Let us apply the torsional moment first, so that it does the work Av Then apply the bending load 8, keeping M fixed. The load S does the work A2, while the moment M must do additional work corresponding to the angle of rotation of the section induced by the action of 8. Trefftz defines the torsion-free bending by the condition that this energy of “interaction” be zero. Alternately, if a shear acts through the “shear center,” so that the beam is in “pure” bending, and then a torque is added, the elastic energy in the beam is simply the sum of elastic energies due to the torsion and the “pure” bending alone. The order of applica­tion of S and M is evidently immaterial to this definition.

Trefftz derives an expression for the coordinates of the shear center with the aid of the “warping” function in St.-Venant’s torsion theory.

The shear center so determined is independent of the Poisson’s ratio ц and the average value of the local twist over the cross section, given by the integral in Eq. 3, does not always vanish.

When the section is symmetrical about the у axis, the two definitions yield the same location of the shear center. It can also be shown that, for a single-cell closed thin-walled section, the two definitions agree, whereas, for a multicell thin-walled section, the two definitions in general disagree.

Example. For a semicircular cylinder the abscissa of the shear center is

(a) By the first definition,[42]

8 3 + 4fi * ~ Ш 1 + p, °

(b) By the second definition, f

For a Poisson’s ratio /л — 0.3; the locations of the shear center P are shown in Fig. A 1.2.

Thin-Walled Sections—Third Definition. For a single-cell closed thin – walled tube, it is shown in § 1.2 that the shear flow q at a point s is

S f*

4 = 4a + 7 У* ds (7)

і Js0

where q0 is the value of q at s0, and s is the distance measured along the wall. In order to find the shear flow, it is necessary to determine q0. In aeronautical literature it is customary to define a “pure” bending of the tube as one in which the value of q0 is so chosen that the strain energy

Fig. A 1.2. Locations of shear center according to
different definitions.

stored in the tube is a minimum. Now, the elastic strain energy per unit length of the tube, due to the shearing strain, is

where the integral is taken over the entire section, pure bending,

But гq/Эq0 = 1 according to Eq. 7; hence, the condition for pure bending is

The last result can be generalized to multicelled tubes. For an и-celled tube it is necessary to determine и integration constants of the nature of

qQ above. It can be shown that an equation like 10 holds for every possible closed circuit drawn along the walls. Since there are n inde­pendent circuits for an и-celled tube, the и integration constants can be uniquely determined.

Having determined the shear flow, the abscissa of the shear center is obtained from Eq. 18, § 1.2.

Measurements of responses to sudden start of motion, and to gusts, reveal many interesting phenomena. Due to experimental difficulty, most of the results available so far are qualitative in nature. In the following, whenever comparison with theoretical results is mentioned, it is meant
that the results given by the two-dimensional linearized theory for infinitely thin wings, as exposed in the preceding chapters, are being compared. All the experiments quoted below are performed at low air speeds; hence, only the theory of incompressible flow are checked.

Change of Circulation due to Sudden Start of Motion. Figure 15.1

Asymptotic

t, no. of semichord lengths traveled after start of motion Fig. 15.1. Growth of circulation and lift after a sudden start of motion. The theoretical values and Cj№ are for two-dimensional flat plate of zero thickness. The steady-state circulation and lift of the experimental airfoils are smaller than rmTh and CLm, respectively. If the ratio of the experi­mental instantaneous circulation to the corresponding experimental steady – state value (shown as asymptote) were plotted as a function of time, the experimental curves will appear to be in better agreement with the theory. (From data given by Walker, Ref. 15.119, and Francis, Ref. 15.113. Cam­bridge Aeronautical Laboratory. Wing chord 4 in. Reynolds number

1.4 x 105 (water). Span 6 in. between plane walls.)

shows the results obtained by Walker15119 and Francis15113 by photo­graphing the flow patterns in a water tunnel. The airfoil was suddenly moved with constant velocity in still water. In one case the angle of attack was so small (a = 7.5°) that the flow remained unseparated In the other case, however, the angle of attack was so large (a — 27.5° measured from the zero-lift line) that the flow began to separate after a

In Fig. 15.1 the measured values

Fig. 15.2. Effect of rate of change of angle of attack on the maximum lift
coefficient. (Courtesy of Dr.-Ing. H. Drescher of Max-Planck-Institut fiir
Strommungsforschung.)

of circulation Г and lift coefficient CL are compared with Wagner’s theoretical values (§ 15.1), Гтї11 and CLaa being the theoretical limiting value of Г and CL, respectively, as time / -»■ oo. The limiting values of the experimental curves were determined from steady lift measurements.

It is seen that the experimental values of Г are always lower than the theoretical ones. In the small-angle-of-attack case the shape of the experimental curve resembles closely that of the theory. In the large – angle-of-attack case the shape differs considerably.

Increase of the Maximum Lift Coefficient—Kramer’s Effect. The
maximum lift coefficient CLmax of a moving airfoil is different from that
of a stationary airfoil in a steady flow. Kramer15’115 first showed that

CL max increases when the angle of attack increases with time. In Fig. 15.2 are shown the results of Wieselsberger16,120 for a stationary airfoil situated in a flow the angle of which increases with time, and those of Farren15’111 when the angle of attack is first increased and then decreased. The rate of change of angle of attack dafdt is made nondimensional by multiplying with the wing chord c and dividing by the speed of flow V.

The increment of the maximum lift coefficient, ACimax over the stationary c da.

values is plotted against — — in Fig. 15.3, where the results of other

authors are also presented. The values found by Kramer are represented by the straight line

AC£raax = 21.7^ (1)

which is seen to be invalid for small values of angular speed parameter. c dcx.

For — — <0.21 x 10”3, the NACA data obtained by Silverstein,

/T Л* 7 J

Katzhoff, and Hootman15118 gives a straight line whose slope is about 17 times higher than that of Kramer’s Eq. 1. In the intermediate range, Ehrhardt’s15117 test results show a curious transition from the NACA curve to Kramer’s curve.

Motion of a Wing Encountering a Gust. By dropping a wing model through the jet of an open-section wind tunnel, and photographing the trajectories of two little lamps attached to the. wing leading and trailing edges, Kiissner15116 obtained a good qualitative agreement with his theory of gust response. The theory predicts that the lift created by a gust is due to circulation and that the resultant force acts through the forward aerodynamic center. In Kilssner’s tests a rectangular wing (chord 0.204 meter, span 0.415 meter), with and without tip plates, fell (at zero angle of attack) into the horizontal free jet of the wind tunnel. When it reached the geometric jet boundary, its velocity was of order 6.13 meters per second. The ratio of the wind velocity w to the falling velocity of the airfoil U could amount to w/U — 0.359 without separation of the flow. A ratio w/U = 0.359 corresponds to an angle of attack a = 19.5°, at which separation will occur in a steady flow. This delay of separation is another revelation of Kramer’s effect. In experiments with higher wind speed, the wing turned (pitched) to the wind, which means that the center of pressure moved backward because of the separation of flow.

For tests in which wjU < 0.359 the wing with its center of gravity at 0.236 chord length behind the leading edge did not turn (pitch) when passing the jet boundary. Assuming the aerodynamic center of this rather short wing to be located at 0.236 chord (the theoretical value is 0.25 for a two-dimensional wing without thickness), the experiments may be taken as confirming the theoretical location of the center of pressure.

From measurements of the curvature of the trajectories of the airfoil, the lift force acting on the wing can be calculated. However, the accuracy of such measurements was rather low. Within the experimental error, no significant difference between the theory and experiment has been found with respect to the transient lift force, provided that separation did not occur.

Flap Motion—Adherent Flow. An example of the measurements of transient pressure distribution over an airfoil due to a sudden deflection of a flap is shown in Fig. 15.4, which is given by Drescher15110 from experiments in a water tunnel. The pressure distribution was measured by a multiple manometer. At the top of this figure is a curve of the normal force coefficient Cn vs. time, Cn being the force (normal to wing chord) divided by q x (wing area). Next is a curve of the flap force coefficient CF vs. time; CF, like Cn, is referred to the main-wing area (including flap) and is the component of force acting on the flap in the

direction normal to the main wing chord. In the lower part of Fig. 15.4 are plots of pressure distribution over the airfoil and the flap. The pressure coefficient Cv is defined as the actual pressure jump across the airfoil divided by the dynamic pressure q — pU2. The angular speed of flap £2 was a constant in each experiment, and was expressed in non – dimensional parameter Q. c/2U. The wing model had a symmetric profile Go-409; it spanned wall to wall in the tunnel, so that in the mid-span section, where the measuring holes were arranged, the flow

Fig. 15.5. Transient normal force and flap force coefficients for a higher
angular speed of flap deflection. Reynolds number 6 x 106. a = — 5°.
ft = 0° -> 15°. Qcj2U = 0.356. (Courtesy of Dr.-Ing. H. Drescher.)

approximated well a two-dimensional one. The flap was mounted to the main wing without a slot.

In the experiment of Fig. 15.4, the angle of attack was a = — 5°, and the flap was deflected from /3 = 0° to /3 = 15° at constant angular speed Q. In general, Cn(t), CF(t), and Cv{x) curves all agree fairly well with theoretical values (dotted curves) based on infinitely thin two-dimensional plate. The airfoil profile shape causes a discrepancy in the pressure distribution, particularly at the leading edge, where a sharp edge is assumed in the theory.

In Fig. 15.5 is plotted another result by Drescher15110 for a higher angular speed of the flap motion. A periodic oscillation in Cn(t), which gradually dies out, appears after the flap has ceased to move. This is connected with the flow picture of Fig. 15.6. The vortex surface deflected by the flap motion is followed by periodic vortices of decreasing intensity.

Such periodic vortices had been observed after sudden starting or sudden stopping of the flap motion. One may conclude that the circulation about the wing cannot instantaneously assume the value which is given by the kinematic conditions, but that it oscillates about the prescribed value with appreciable amplitude.

Flap Motion—Separated Flow. Experimental pressure distribution on an airfoil with flow separation is also reported by Drescher.15110 The beginning of the separation process depends on the Reynolds number. Low Reynolds number will favor the separation. But the most important factor influencing the separation process is the angular velocity with which the flap moves. If the main-wing angle of attack is small, and the flap

is deflected to a large angle, the flow separates in the flap region, and in later stages a Karman vortex street is developed. For Go-409 airfoil, a = 0, /3 moves from 0 to 60°, separation occurs at /3 = 12° when Oc/2£/ is 0.014, but it occurs after /3 = 60° when Qc/2U is Г.16. In the latter case very high suction is obtained at the flap nose, and the CL and CF values may exceed the corresponding theoretical values at the instant before separation occurs. Very large oscillations in CL(t) and CF(t) curves are often observed after separation.

If the main-wing angle of attack a is small and /3 decreases from 60° to 0, the separated flow becomes adherent again, but often after an ap­preciable time delay, which is required to scavenge away the dead water accumulated behind the wing. More complicated motions of the flap and the main wing induce more complicated responses; but such responses can generally be understood on the basis of the facts mentioned above: oscillation in circulation before it reaches a steady value, and time delay required for the scavenging process.

Harmonic Oscillations. Results of oscillating-wing experiments have been published by many authors. (See bibliography.) For a wing without a flap, the experimental results in general agree with the theo­retical. There are, however, some small quantitative difference between theory and experiment, and among various authors. Typical results obtained by Halfman15129 are shown in Figs. 15.7 and 15.8. The airfoil (NACA 0012 section, chord 1 ft, span 2 ft) was tested in a 5 X 7-ft wind tunnel with plates shielding the wing tips so that two-dimensional flow condition was assured. Oscillations in two degrees of freedom were imparted to the airfoil: h and a. The expressions of aerodynamic force and moment corresponding to the translational motion h = h0eimt (h0 real) are, respectively,

Ь – = + IL,/ e«»*+*4 <f, LT = tan-1 Ь*

Aqb Rlt

Мж. — /~r a _i_ r 2 жм+ф„.) j ~ Imil

Aqb KMT

Similarly, those corresponding to pure pitch a — a0eto* (a0 real) are given by the same formulas except that the subscript T is replaced by P. The force L, as well as the displacement h, is taken positive downward in these figures. Both the magnitude and phase angle are plotted in Figs. 15.7 and 15.8. The solid curves are theoretical values.

More serious differences between theoretical lift and moment coefficients and experimental ones are found for oscillating flaps. But test results are meager. See papers by Drescher15126 and Walter.15-135,15136 Experimental results for oscillating wings having a mean angle of attack near or greater than the static stalling angle have been reviewed in § 9.4.

Many methods have been used in measuring the unsteady aerodynamic forces (lift, moment, or pressure distribution) acting on an airfoil which is moving or is situated in a nonuniform flow. One may speak of “direct” and “indirect” measurements. If the forces are measured directly by dynamometers, manometers or strain gages, the measurement is said to be direct. If they are determined from their effect on the motion of the airfoil, air density, or other quantities, it is said to be indirect.

The requirements imposed on direct-measuring instruments are not easy to meet. The influence of the measuring instruments on the phenomena under examination must be kept as small as possible. The installation must not affect the flow to any appreciable degree. The variation of forces with time must be recorded with sufficient accuracy. In particular,

the time lag in the recording instrument due to the mass inertia of the sensing elements must be kept small.

In some of the indirect methods of measurement, the mass-inertia time-lag problem is completely eliminated. These methods are listed as follows:

1. Indirect Methods, (a) Determination of circulation from photo­graphs of the flow. In Walker15-119 and Farren’s experiments,15-112 a wing model is drawn through a glass vessel filled with water. The flow is made visible by small drops of olive oil and ethylene dibromide, and can be photographed. The velocity of the fluid particles can be deter­mined from the time of exposure and from the streak lines of the oil drops. The circulation about the wing is then obtained from the velocity field by a numerical integration.

(b) Determination of pressure from the variation of air density. Since the index of refraction of light varies with the density of a fluid, which in turn is related to the pressure field, the pressure can be determined optically. The variable index of refraction is measured by light inter­ference. Zender-Mach interferometer can be used for this purpose. This method is effective for high-speed flow, in the transonic and super­sonic ranges.

(c) Determination of unsteady forces from the motion influenced by these forces. The lift acting on an airfoil in entering a gust can be calcu­lated from the motion of the airfoil. The gust response may be verified by measuring the trajectory of an airfoil in passing through a jet stream (e. g., by dropping an airfoil through the test section of an open-jet wind tunnel). This method has been applied with certain degree of success by Kiissner.15110

A more commonly used method is based on forced oscillations. The wing model is excited by harmonic external forces, and the aerodynamic forces are determined from the kinematic quantities involved. Generally, the aerodynamic forces are not large compared to the inertia and elastic forces. Hence, the model must be built as light as possible (yet sufficiently rigid to prevent appreciable distortions) or a water tunnel must be used. See the works of Cicala,15-125 Dresher,15128 Greidanus,15,128 etc.

Flutter experiments may be considered as another approach. But, owing to the large number of parameters involved, it is unsuitable for an exact determination of the aerodynamic forces.

2. Direct Methods, (a) Spring balances. The transient force to be measured is opposed by a spring whose deflection is converted, for instance, into a rotation of a mirror, which is recorded by a pencil of reflected light. Various versions of spring balances are used by Farren,15111 Silverstein,15131 Scheubel,15117 Reid and Vincenti,15130 etc.

(b) Electric measuring elements. Forces can be measured by a number of electromagnetic devices, such as: (i) Piezoelectric gages (Kramer15115), (ii) Wire-resistance strain gages, (iii) Inductance transducers (Wiesels – berger15,120).

A piezoelectric gage measures the electric charges (or their voltages) produced on the end surfaces of certain crystal (e. g., quartz) when it is subject to pressure. It has been used for stationary models in a flow whose direction is changed at constant velocity.

A wire-resistance strain gage measures the change of resistance of a fine wire (e. g., tungsten) due to elastic strain. It can be used to measure the elastic deflections of springs. If the wire is attached to a thin metal membrane which deflects under pressure, it can be used as a pressure gage. Wire-resistance gages are used extensively both in wind-tunnel and in flight testing.

There are many types of inductance transducers. Either the air gap or the position of an iron core may be varied, and the corresponding change in inductance is measured. The displacement of the air gap or the iron core can be made proportional to the deflection of a spring, thus measuring a force. Very high accuracy can be achieved in certain designs.

Use of electrical means for measuring forces often involves a compli­cated electronic system. To improve the accuracy and ease the analysis, various ingenious schemes have been invented. As an example, one may name the “wattmeter” harmonic analyzer of Bratt, Wight, and Tilly,15,121 in measuring the aerodynamic damping for pitching oscillations. The modulated output from the stress indicator is first rectified and then analyzed electrically by means of an electronic wattmeter; the damping coefficient is obtained directly from meter readings.

The electrical measurements have the important advantage of min­imizing the mass-inertia effect of the sensing elements. With proper electronic equipment and circuits, it is probably the most convenient, accurate, and versatile of all methods.

(c) Manometers. Pressure distribution over the model surface can be measured by manometers. Drescher15,110 describes a successful multiple manometer used in measuring unsteady pressure distribution over an airfoil in a water tunnel.

The problem of estimating the spanwise distribution of lift and moment on an unswept lifting surface of finite span executing simple-harmonic oscillations in an incompressible fluid has been studied by many authors, among them the most notable being Cicala, Lyon, W. P. Jones, Skan, Sears, R. T. Jones, Kiissner, Biot, Boehnlein, Wasserman, Reissner, Zartarian, Hsu, Ashley, Dengler, Goland, and Shen (see bibliography). Unfortunately, the problem is so complicated that, even after the standard linearization, there exists no practicable, exact solution. Moreover, the independent lines of approach initiated by various authors have led to answers that cannot be shown to be entirely equivalent. Experimental results available at present, due to their scatter, cannot discriminate definitely which one of these theories approximates best the physical reality, although generally the methods of Reissner and that of Biot and Boehnlein are favored.

For a compressible fluid, the linearized theory in a supersonic flow is quite advanced. A number of exact solutions have been obtained for some special wing planforms and modes of motion. A few solutions are known also in the high subsonic and transonic speed range, but, generally speaking, the three-dimensional oscillating-airfoil theory is still a subject for future research.

The aerodynamics of unsteady motions of airfoils with arbitrary time history is described in § 15.1 as a simple generalization, involving one integration, of the harmonic-oscillation case. In particular, the indicia! admittance (response to a unit-step function) is derived from the oscillating case. This does not imply that there are no shorter methods of deriving the indicial admittance, but rather emphasizes the reciprocal relation between the admittance and the indicial admittance. The effects of finite span is briefly mentioned in § 15.2.

The ultimate test of a theory lies in experiment. Some general con­siderations in experiments are given in § 15.3. Some of the established experimental results are outlined in § 15.4. Owing to the limitation of space, the experiments cannot be described in detail. Moreover, since the numerical uncertainty of the experimental results available at present is such that a scheme of empirical correction of the theoretical results is not yet generally acceptable, the aim of our discussion will be to point out the conditions under which the linearized theory is inapplicable and to describe some of the phenomena yet unexplained by theory.

15.1 UNSTEADY MOTIONS OF A TWO-DIMENSIONAL AIRFOIL

The aerodynamic response of an airfoil performing an arbitrary un­steady motion about a mean uniform rectilinear translation can be cal­culated from that of an oscillating airfoil by means of a Fourier analysis. In § 8.1 it is shown that, if the response?/ to a forcing function F= F„eimt is

(1) then the response to a periodic function F(t)

CO

F(t) = ^ Cn einmt

n~~ CO

IS

These results can be applied directly to the airfoil problem if the linearized theory is accepted. The airfoil displacements (change of angle of attack, velocity of translation, aileron angle, etc.) may be considered as the forcing function and the induced lift, moment, or pressure distribution as the response. The admittance l/Z(iw) is given by the theory of har­monically oscillating airfoils, as applied to у (x, t) =f (x) ewt The procedure can be expressed also in Laplace transformation. In Eqs. 5 and 6, put

im = s (7)

and assume F(t) — 0 for t < 0; then

(9)

Hence, formally, V2n^( — is) is the Laplace transformation of F(t), and Eq. 8 shows that y(t) is the inverse Laplace transformation of JSf{F}/Z(s), i. e.:

These formal steps can be mathematically justified for suitable classes of forcing functions F(t). The Laplace transformation of the response is equal to the Laplace transformation of the forcing function multiplied by 1 IZ(s), which is obtained by replacing /со by s in the admittance to harmonic oscillation.

When F(t) is a unit-step function,

J?{1(0} = – (11)

s

the response, called indicial admittance and denoted by A(t), is given by

m <12»>

or

"«-•НтЫ <12‘>

in agreement with Eq. 32 of § 8.1.

Note that the downwash distribution over the airfoil must be the same in the unsteady motion as in the harmonic-oscillation case in order that the above formulas be applicable.

There are other methods of deriving the indicial admittance. For special problems special methods may be devised that are much shorter than the Fourier-transformation method mentioned above. Recently, great advance has been made in the calculation of indicial admittance with respect to a sudden motion of the airfoil or to a sharp-edged gust, for both the two-dimensional case and the finite-aspect ratio case. The success is remarkable, particularly at high subsonic Mach numbers and at M = 1 (linearized theory). On the other hand, the second – and higher-order theories are investigated in the supersonic case, and significant corrections to the first-order linearized theory (which is presented in the last chapter) are revealed for Mach numbers near and below Vl. See bibliography, and in particular, Lomax,15 84 and van Dyke.14-45

Example. Wagner’s Problem. Consider an airfoil moving recti – linearly in a fluid with a relative speed U which is so small that the fluid may be considered incompressible. Let the angle of attack be suddenly increased by an amount a. Owing to this change the relative velocity of the fluid will have a component normal to the airfoil. This normal velocity is uniformly distributed along the chord and is a unit-step function of time:

v(x, t)= – U*l(t) (- 1 < я < 1) (13)

The lift and moment induced by this upwash can be found by the Laplace transformation. Note that the same problem arises if an airfoil suddenly starts to move in a stationary fluid with a constant velocity U and angle of attack a.

In § 13.4 it is shown that, if

v(x, t) = v0 eimt = iwy0 eimt

and the stalling moment about the mid-chord point is

M/a = – 7TPU*iy0k C(k) 4*

C(k) is Theodorsen’s function.

Replacing tk by s, we obtain, for the lift force,

■ TrpbUv0 8(t) — 2-rrpbUv0ЛЄ

The first term gives an impulse function. The total impulse is obtained by an integration over an infinitesimal time interval. If we remember the scale factor b/U (Eq. 15) in changing from т to t, the magnitude of the total impulse is seen to be – прЬЪ0. This is the total impulse that is needed to move abruptly a mass тгрЬ2 to a velocity v0. The quantity npb’2 is the apparent mass of the fluid associated with the vertical motion (cf. § 6.7, Eq. 9).

The second term gives the lift due to circulation. Define a special function Ф(т):

Ф(т) = &-1 {^“7^} (R1 * > 0) (19)

Then the circulatory lift can be written as

L,(r) = – 2ттрЬ Шй Ф(т) (20)

The function Ф(т) is the Wagner’s function defined in § 6.7, and is shown graphically in Fig. 6.6.

In the case of supersonic flow, many authors give numerical tables of aerodynamic coefficients for oscillating airfoils. For flutter-calculation purpose the following reference, prepared by E. C. Kennedy, is the most comprehensive:

1. Handbook of Supersonic Aerodynamics. liM M 1.1 (0.1) 2.0 (0.2)

4.0 (0.5) 5.0 (1.0) 12. Q: 0.01 (0.01) 0.04 (0.02) 0.10 (0.05) 0.40 (0.10)

1.0 (0.20) 3.0 (0.50) 5.0, 7.5, 10, 15, 20. (8 fig.)

The coefficients CLh, CL(t, CMh, CMa listed in this Handbook are, respectively, the Lh, Le, Mh, Ma defined in Chapter 6. The independent entry in the Handbook is the frequency parameter Cl, which is related to the reduced frequency к and Mach number M by the relation

In the following reference, compiled by Y. L. Luke on the basis of tables published by Garrick and Rubinow and Jordan, the independent entries are the reduced velocity Ujbco = l/к, and the coefficients Lh,

Mh, Ma:

2. Tables of Coefficients for Compressible Flutter Calculations M: if, , if, I, 2, f, if, 5. 1/k: approx. 0.1 (irreg.) 100. (5 dec.)

The scope of these tables is indicated above in the ranges of M and Cl[40] Note that in the supersonic case it suffices to tabulate the four fundamental aerodynamic coefficients named above. The lift and moment coefficients

involving the motion of a flap and a tab can be expressed in terms of these four fundamental coefficients. Explicit formulas expressing these relations involving a flap are given in Ref. 14.44. A complete list of formulas including both the flap and the tab can be found in Ref. 14.1.

For subsonic flow, the published data are meager. The principal sources are the papers by Possio, Frazer, Frazer and Skan, Dietze, Schade, Schwarz, Turner and Rabinowitz, Timman, Van de Vooren and Greidanus, and Fettis (see bibliography). The numerical results of the first eight authors named above have been compiled and converted into the Lh, Mh, • • ■ coefficients by Luke in Ref. 14.8. A summary of the published tables is given in Table 14.1. In Table 14.2, the notations used in some of the most important references, for both the supersonic and the subsonic cases, are listed.

In Table 14.1, the symbols Lh, Mh> etc., are defined in § 6.10. These coefficients are referred to the 1/4-chord axis for both the rotation a and the moment Mx. In Timman, van de Vooren and Greidanus’s papers, the rotation and moment are referred to the mid-chord axis; hence, a transformation is needed when comparison of the data is to be made. This is indicated in Table 14.1. Under each column the factors in paren­theses are the tabulated quantity expressed in the author’s notation. The adjacent coefficients are the factors necessary to convert to the corres­ponding Lh, La, Mh, or Ma, which is listed at the left on the same horizontal

line. Thus, under Timman and opposite Lh, we find the entry ~ (ka).

This indicates that (ka) is given by Timman and that Lh = (ka)/k2. With the exception of the symbols for the quantities actually tabulated in the references, the notation of this book is used throughout. The notations of the original authors may be found in Table 14.2.

The problem of an oscillating airfoil in a two-dimensional supersonic flow can be solved in several different ways. In this section the Laplace – transformation method, in a form due to Stewartson,14 40 will be used. The theory will be limited to the linearized case, so that the airfoil must be infinitesimally thin, and executing harmonic oscillations of small ampli­tudes. The principle of superposition holds. It is sufficient to consider airfoils of zero thickness and zero camber, with stationary mean position. The fluid moves over it with an undisturbed velocity U at infinity. The x axis is taken in the direction of the free stream, and the origin of co­ordinates is taken at the leading edge of the airfoil. In the first-order theory the wing may be assumed to lie in the plane у = 0. The coordinate z, in the spanwise direction, does not appear in the problem. The flow is assumed to be irrotational, with a velocity potential Ux + Ф, and deviations in velocity components, pressure, and density are so small that squares and products of these deviations may be neglected in comparison with the first-order terms.

The equation of the velocity potential, referred to a frame of reference at rest relative to the fluid at infinity, is (§ 12.5):

УФ Э*Ф__1Э*Ф_

За;2 + dy2 a2 dt2 ~ ° ^

where a is the velocity of sound, which, in our order of approximation, is a constant. Transforming to axes moving with speed U in the negative x direction, so that x is replaced by x + Ut, Eq. 1 becomes (Eq. 2, p. 418):

The increment of pressure at any point due to the disturbance is given by the Eulerian equations of motion

3 u, 3tii 1 3

37 + Щ 5— = — – -— p

at oXj p dx{

and linearizing the result, one obtains

where M is the Mach number Ufa and

p. = M2 – (7)

The boundary conditions must be formulated according to the following considerations:

1. The disturbances created at the leading edge propagate along a wedge which is called the Mach wedge. In front of the Mach wedge the disturbances cannot be felt, and the flow is uniform relative to the wing. Hence, one may put Ф = 0 for x < 0.

2. Inside the Mach wedge, the velocity of flow normal to the airfoil must conform to the actual motion of the airfoil. If the equation of the airfoil surface is specified by

У = Y(x, t) (8)

On the rest of the plane у = 0, the pressure must be continuous.

Equation 6 with the boundary condition 11 may be solved by Laplace transformation. Define

V = JSf’f’F} = JV8*¥(*) dx (12)

Then f satisfies

since Y and Э’Г/Эж vanish when x = becomes

where g(s) is the Laplace transform of the right-hand side of Eq. 11. Thus we require a solution of Eq. 13 so that -> 0 as y -> oo and such that Eq. 14 is satisfied. Let

y2 = /SV + 2Ms — + (—)2 (15)

a a }

The general solution of Eq. 13, if we take y. to be the branch on the right half plane, i. e., with Щу > 0, is

f = Ae-m + Bem

The constants A and В are determined by the boundary conditions. It is necessary to distinguish the solution on the upper and lower half-space. For the upper half-space (y > 0), the condition f -> 0 as у -> oo requires 5 = 0; and the condition 14 requires A — — g(s)/y. Similarly the solution for у < 0 can be determined. Hence,

where the symbol sgn у indicates a sign to be taken as positive on the positive side (y > 0) of у — 0 plane, and as negative on the negative side.

It is now necessary to find the inverse transform of Eq. 18. The inverse transform of g(s) is (Э’Г/Эу)г/=0 and is given by Eq. 11. From Table 10.1, we find

Пи™)} = – t=L= (19)

Now

The pressure change on the airfoil is, according to Eq. 3,

Since the pressure changes on the upper and lower surfaces of the airfoil are equal and opposite

Note that the integral is a function of the parameters M and Cl, or, alternatively, M and k.

The total lift on the airfoils is

L = 2bj^(x’) dx’ (32)

The moment (positive nose-up) on the airfoil about the leading edge is

The integration of Eqs. 32 and 33 has been discussed by von Borbely,14-29 Schwarz,14 39 Garrick and Rubinow.14-32

It is a general feature of Carleman’s integral equation that the solution admits a singularity. For an airfoil, such a singularity is located at the leading edge where the acceleration potential should tend to infinity like V(l —x)j{ 1 + x) when — 1. This particular form of the singu­larity can be derived from Carleman’s general theory, or from analogy with the incompressible-flow case. The intensity of the singularity can

* Jahnke and Emde, Tables of Functions, p. 3. Dover Publications.

t See § 14.7 and Bibliography. Summaries of numerical tables are given in Refs. 14.1 and 14.8.

be determined by the Kutta-Joukowski condition that the lift be zero at the trailing edge.

When we write L(x, i) — L(x)ela, i, the function L(x) consists of a term const V(1 — x)j([ + x) and a nonsingular part. Possio14,12 and Frazer14-6 write the nonsingular part as a series involving a number of constant coefficients and determine these constants by the collocation method. Schade14,15 writes the nonsingular part of L(x) as a series in Legendre polynomials. The nonsingular part of the kernel Ky(M, z) as well as the upwash distribution v(x, 0) are also expressed as series in Legendre polynomials. The undetermined coefficients are then obtained on the basis of the orthogonality of Legendre polynomials in the interval – 1 < 1.

The best-known method is probably Dietz’s iteration procedure,14-2 which will be outlined below. Let us introduce the notation of the composition product

P K(M, х-І) Щ) dH = K-L (1)

p0U2J-i

— P m X-S) m dS =– Ka ■ L (2)

PqU* J-i

Then Possio’s equation can be written as

K-L = v (3)

where v{x) is the given vertical velocity component on the airfoil. If the fluid is incompressible, the corresponding lift distribution is given by L0, and Eq. 3 is reduced to

K0-L0 = v (4)

The solution of this equation is known explicitly. (See § 13.5.) Let us write L = L0+ AL0 which defines AL0; we have

K0- L0 = v — К ■ L — К ■ L0- K – AL0 (5)

or

К • AL„ = % (6)

if

v^{K0 ~K)-L, (7)

The function vy being known, Eq. 6 is completely analogous to Eq. 3. Hence, the same procedure can be applied. Let

AL0 i- – f~ A /.}

where Ly is the solution of the incompressible-fluid problem

Vy = KQ- Ly

SEC. 14.6 OSCILLATING AIRFOILS IN SUPERSONIC FLOW and ALj is governed by the equation

Continuing this process, we obtain the approximate solution L = L0 + + L2 + • ‘ ‘ + Ln + A Ln

where Lm is the solution of the incompressible-fluid problem K0 ‘ Lm = vm (m < ri)

while

Vm — (Ко К) ■ Lm_J

v0 = v being specified, and

K-ALn= (K0 – K)Ln

The convergence of this procedure has not been proved, but is indicated by Dietz’s numerical examples. The rapidity of convergence is found to deteriorate for increasing M and k.

Recently, Fettis14-4 introduced a method that avoids the iterative procedure and gives a relatively simple solution on the basis of an ap­proximate kernel, in which the nonsingular part of Possio’s kernel is replaced by a polynomial.