Category Aerodynamics of Wings and Bodies

Fundamentals of Fluid Mechanics

1- 1 General Assumptions and Basic Differential Equations

Four general assumptions regarding the properties of the liquids and gases that form the subject of this book are made and retained throughout except in one or two special developments:

(1) the fluid is a continuum;

(2) it is inviscid and adiabatic;

(3) it is either a perfect gas or a constant-density fluid;

(4) discontinuities, such as shocks, compression and expansion waves, or vortex sheets, may be present but will normally be treated as separate and serve as boundaries for continuous portions of the flow field.

The laws of motion of the fluid will be found derived in any fundamental text on hydrodynamics or gas dynamics. Lamb (1945), Milne-Thompson (1960), or Shapiro (1953) are good examples. The differential equations which apply the basic laws of physics to this situation are the following.[1]

1. Continuity Equation or Law of Conservation of Mass

image1(1-1)

where p, p, and T are static pressure, density, and absolute temperature.

Подпись: (1-2)Q = f/i + Vj + Wk

is the velocity vector of fluid particles. Here i, j, and к are unit vectors in the X-, y-, and г-directions of Cartesian coordinates. Naturally, com­ponents of any vector may be taken in the directions of whatever set of coordinates is most convenient for the problem at hand.

2. Newton’s Second Law of Motion or the Law of Conservation of Momentum

Подпись:DQ – _ Vp

Dt p

where F is the distant-acting or body force per unit mass. Often we can write

F = Vfi, (1-4)

where Я is the potential of the force field. For a gravity field near the surface of a locally plane planet with the «-coordinate taken upward, we have

Подпись: (1-5)F = – gk Я = — gz,

g being the gravitational acceleration constant.

3.

Подпись: _D Dt Fundamentals of Fluid Mechanics Подпись: (1-6)
image2

Law of Conservation of Thermodynamic Energy (Adiabatic Fluid)

Подпись: D P Dt Fundamentals of Fluid Mechanics Подпись: (1-7)
image3

Here e is the internal energy per unit mass, and Q represents the absolute magnitude of the velocity vector Q, a symbolism which will be adopted uniformly in what follows. By introducing the law of continuity and the definition of enthalpy, Л = e + р/р, we can modify (1-6) to read

Подпись: Ds Dt Подпись: 0, Подпись: (1-8)

Newton’s law can be used in combination with the second law of thermo­dynamics to reduce the conservation of energy to the very simple form

where s is the entropy per unit mass. It must be emphasized that none of the foregoing equations, (1-8) in particular, can be applied through a finite discontinuity in the flow field, such as a shock. It is an additional consequence of the second law that through an adiabatic shock s can only increase.

4. Equations of State

For a perfect gas,

p = RpT, thermally perfect gas ^

cP, c„ = constants, ealorieally perfect gas.

For a constant-density fluid, or incompressible liquid,

p = constant. (1-Ю)

In (1-9), cp and c„ are, of course, the specific heats at constant pressure and constant volume, respectively; in most classical gas-dynamic theory, they appear only in terms of their ratio У = cp/cv. The constant-density assumption is used in two distinct contexts. First, for flow of liquids, it is well-known to be an excellent approximation under any circumstances of practical importance, in the absence of cavitation. There are, moreover, many situations in a compressible gas where no serious errors result, such as at low subsonic flight speeds for the external flow over aircraft, in the high-density shock layer ahead of a blunt body in hypersonic flight, and in the crossflow field past a slender body performing longitudinal or lateral motions in a subsonic, transonic, or low supersonic airstream.

Transonic Small-Disturbance Flow

11- 1 Introduction

A transonic flow is one in which local particle speeds both greater and less than sonic speed are found mixed together. Thus in the lower transonic range (ambient M slightly less than unity) there are one or more super­sonic regions embedded in the subsonic flow and, similarly, in the upper transonic range the supersonic flow encloses one or more subsonic flow regions. Some typical transonic flow patterns are sketched in Fig. 12-1. Since in a transonic flow the body travels at nearly the same speed as the forward-going disturbances that it generates, one would expect that the flow perturbations are generally greater near M = 1 than in purely subsonic or supersonic flow. That this is indeed so is borne out by experi­mental results like those shown in Figs. 12-2 and 12-3, which show that the drag and lift coefficients are maximum in the transonic range. In the early days of high-speed flight, many doubted that supersonic aeroplanes could ever be built because of the “sonic barrier,” the sharp increase in drag experienced near M equal to unity.

Many of the special physical features, and the associated analytical difficulties, of a transonic flow may be qualitatively understood by con­sidering the simplest case of one-dimensional fluid motion in a stream tube. Combination of the Euler equation

and with the equation of continuity

і (pm = o,

where S(x) is the stream-tube area, yields after some manipulation

dp dS

pU2 S[1 – (C72/a2)]

This relation shows that for U/а — 1 the flow will resist with an infinite force any stream-tube area changes, i. e., it will effectively make the flow incompressible to gross changes in the stream-tube area (but not to curva­ture changes or lateral displacement of a stream-tube pattern). There­fore, the crossflow in planes normal to the free-stream direction will tend to be incompressible, as in the case of the flow near a slender body, so that much of the analysis of Chapter 6 applies in the transonic range to con­figurations that are not necessarily slender. This point will be discussed further below. It is evident that because of the stream-tube area constraint, there will be a tendency for a stronger cross flow within the stream tube and hence the effect of finite span will be maximum near M = 1. From (12-1) it also follows that in order to avoid large perturbation pressures and hence high drag one should avoid large (and sudden) cross-sectional area changes, which in essence is the statement of the transonic area rule discussed in Chapter 6. For the same reason one can see that the boundary layer can have a substantial influence on a transonic pressure distribution, since it provides a region of low-speed flow which is less "stiff” to area changes and hence can act as a “buffer” smoothing out area changes.

From such one-dimensional flow considerations, one practical difficulty also becomes apparent, namely that of wind tunnel testing at transonic speeds. Although a flow of M = 1 can be obtained in the minimum-area section of a nozzle with a moderate pressure ratio, the addition of a model, however small, will change the area distribution so that the flow no longer will correspond to an unbounded one of sonic free-stream speed. This problem was solved in the early 1950’s with the development of slotted – wall wind tunnels in which the wall effects are eliminated or minimized by using partially open walls.

The main difficulty in the theoretical analysis of transonic flow is that the equations for small-disturbance flow are basically nonlinear, in con­trast to those for subsonic and supersonic flow. This again may be sur­mised from equations like (12-1), because even a small velocity change caused by a pressure change will have a large effect on the pressure-area relation. So far, no satisfactory general method exists for solving the transonic small-perturbation equations. In the case of two-dimensional flow it is possible, through the interchange of dependent and independent
variables, to transform the nonlinear equations into linear ones in the hodograph plane. However, solutions by the hodograph method have been obtained only for special simple airfoil shapes and, again, two-dimen­sional flow solutions are of rather limited practical usefulness for transonic speeds. For axisymmetrie and three-dimensional flow, various approxi­mate methods have been suggested, some of which will be discussed below.

11- 2 Small-Perturbation Flow Equations

That the small-perturbation theory for sub – and supersonic flow breaks down at transonic speeds becomes evident from the linearized differential equations (5-29) and (6-21) for the perturbation potential, which in the limit of M —> 1 become

<Pzz = 0, for two-dimensional flow, (12-2)

і (pr + tfrr = 0, for axisymmetrie flow. (12-3)

Thus, both the inner and outer flows will be described by the same differ­ential equation, and it will in general not be possible to satisfy the boundary condition of vanishing perturbation velocities at infinity. For transonic flow it will hence be necessary to consider a different expansion that retains at least one more term in the equation for the first-order outer flow.

In searching for such an expansion we may be guided by experiments. By testing airfoils, or bodies of revolution, of the same shape but different thickness ratios (affine bodies) in a sonic flow one will find that, as the thickness ratio is decreased, not only will the flow disturbances decrease, as would be expected, but also the disturbance pattern will persist to larger distances (see Fig. 12-4).

This would suggest that the significant portion of the outer flow will recede farther and farther away from the body as its thickness tends towards zero. In order to preserve, in the limit of vanishing body thick­ness, those portions of the outer flow field in which the condition of vanishing flow perturbations is to be applied we must therefore “compress ” this (in the mathematical sense). Taking first the case of a two-dimen­sional airfoil with thickness but no lift, we shall therefore consider an expansion of the following form:

where S(x) is the cross-sectional area. The outer flow must therefore be equal to that around the equivalent body of revolution as in the slender- body case, and we have thus demonstrated the validity of the transonic equivalence rule resulting from the form of the first-order term in an asymptotic series expansion as the disturbance level «, and M2 — 1 [, both tend to zero. The approach followed is essentially that taken by Messiter (1957). A similar derivation was given by Guderley (1957).

There is no requirement on aspect ratio except that it should be finite so that 8 A —> 0 as e —> 0, in order for the outer flow to be axisymmetric in the limit. A consequence of this is that slender-body theory should provide a valid first-order approximation to lifting transonic flows for wings of finite (and moderate) aspect ratios. In Fig. 12-3 the slender-body value for the lift coefficient is compared with experimental results for a delta wing of A = 2. It is seen that the agreement is indeed excellent at M = 1.

For a wing of high aspect ratio, the first-order theory will provide a poor approximation for thickness ratios of engineering interest. A different expansion is then called for, which does not lead to an axisymmetric outer flow. We therefore introduce in the outer expansion

V = 8y, f = 8z, (12-38)

with 8 chosen as before, (12-11). This then gives the following equation for the first-order outer term:

The matching will prescribe the normal velocity on the wing projection on f = 0, which in view of (12-38) will have all spanwise dimensions reduced by the factor S. Thus, if the limit of e —» 0, and hence 5 —> 0,

is taken with the aspect ratio A kept constant, the projection in the

x, y-plane will have a reduced aspect ratio AS that will shrink to zero in the limit, and the previous case with an axisymmetric outer flow is then recovered. In this case we therefore instead consider the limit of e —> 0 with A —» oo in such a manner that

AS = K2 (12-40)

approaches a constant. The reduced aspect ratio will then be finite and equal to K2. The matching procedure now parallels that for the two­dimensional flow and the choice (12-17) for e gives the boundary condition

Фн(*. V, 0±) = ± U (12-41)

to be satisfied on the wing projection of reduced aspect ratio

K2 = t1I3AM2,3( 7 + 1)1/3 (12-42)

Thus, the solution in this case depends on two transonic parameters K1 and K2. The previous case, for which the transonic equivalence rule holds, may be considered the limiting solution when

K2 -» 0.

As the approach to zero is made the solution defined by (12-39) and (12-41) becomes, in the limit, proportional to K2nK2. The two-dimensional case described by (12-12)-(12-14), or (12-19)-(12-21), is obtained as the limit of

K2 —» oo.

The most general transonic small-perturbation equation is thus

[1 — M2 — M2{ 7 + 1 )<pz]<pxx + <Pyy + <Ргг = 0, (12-43)

with the boundary condition in case of a thin wing

4>z(x, y, 0±) = ±r ~ on wing projection Sw. (12-44) ox

The pressure is given by (12-21) for a thin wing and by

Cp = —2ipx – <p2y – v2 (12-45)

for a slender configuration.

12- 3 Similarity Rules

The first-order terms in the series expansion considered above provide similarity rules to relate the flow around affine bodies.[9] Taking first the two-dimensional case, we see from (12-18) that the reduced pressure coefficient

я [My+ 1)]1/3

Ьр ~ т2/3 Ьр

must be a function of x/c (c = chord) and the parameter

M2 – 1 M2 – 1

fM2(T + 1) ~~ [М2т(У + l)]2/3

only. This conclusion follows because the solution must be independent of scale (see Section 1-4) and К i is the only parameter that enters the boundary value problem defined by (12-12) and (12-15).

The total drag is obtained by integrating the pressure times the airfoil slope, which leads in a similar way to the result that

[M2{ 7 + 1)]1/3„

——

must be a function of Kx only. The additional factor of r enters because the slope is proportional to r. Of course, (12—48) holds only for the wave drag, so that in order to use it to correlate measurements, the friction drag must be subtracted out. Such an application to biconvex-airfoil drag measurements by Michel, Marchaud, and LeGallo (1953) is illustrated in Fig. 12-5. As may be seen, the drag coefficients for the various airfoil thicknesses, when reduced this way, fall essentially on one single curve, thus confirming the validity of (12—48), and hence the small-perturbation equations.

Transonic similarity rules for two-dimensional flow were first derived by von Kdrmdn (1947b) and Oswatitsch (1947). These rules also included the lifting case.

Rules for a slender body of revolution were formulated by Oswatitsch and Berndt (1950). There is an additional difficulty in this case associated with the logarithmic singularity at the axis. It follows from the formula­tion (12-26)-(12-30) for the outer flow and the matching to the inner flow
as given by (12-28) that, near the body,

$i = ^S'(aO lnp + £i(z).

The transonic parameter

M2 – 1 _ M2 – 1

Kl ~ eM*{7 + 1) M2r2(7 + 1)

thus enters into (ji only. By using (12-33) to calculate the pressure coeffi­cient we find that, on the body,

Cp = —2T2 {^S"(x) In [t2MVt~+7 R] + &'(*) + І(й’)2} * (12-51)

where R(x) = R(x)/t. Thus,

Cp = {— Cp + ^ S"(x) In [т2Мф + l]J (12-52)

is a function of K and x/l only. An application of (12-52) to correlate the measured pressures on two bodies of different thickness ratios, carried out by Drougge (1959), is shown in Fig. 12-6. As may be seen, the correla­tion is almost perfect, except at the rearmost portions of the bodies where boundary layer separation occurs.

From (12-52) one can also construct an expression for the drag, as shown in the original paper by Oswatitsch and Berndt (1950). They found that ^ 1

D = t D– + ~ [S'(l)]2 In [t-WyTT] (12-53)

2P« UQoT

must be a function of Ki only.

Fig. 12-6. Correlation of pressure measurements on two bodies of revolution using the transonic similarity law (12-52). P = Cp + (l/ir)S" In (7 + 1). (Adapted from Drougge, 1959. Courtesy of Aeronautical Research Institute of Sweden.)

It is a fairly straightforward matter to construct corresponding simi­larity rules for configurations of low-to-moderate aspect ratios. For wings of large aspect ratios one obtains results of the same form as for the two­dimensional case, except that now the reduced quantities depend on the second transonic parameter

K2 = t1/3AM2/3( у + 1)1/3

as well as on K. The rules for three-dimensional wings were derived by Berndt (1950) and by Spreiter (1953).

Examples of Constant-Density Flows Where Circulation May Be Generated

An elementary illustration of the ideas of the foregoing section is pro­vided by a two-dimensional vortex pair. We work here in terms of real variables rather than the complex variable, although it should be obvious to those familiar with two-dimensional flow theory that the results we obtain could be more conveniently derived by the latter approach. Con­sider a pair of vortices, which are equal and opposite and may be thought of as wrapped around very small circular cylindrical cores which constitute the boundaries S (Fig. 2-9). This motion can be generated by applying a downward force per unit area

ДР = —p ДФ = рГ (2-97)

across the barrier shown in the picture. The total Kelvin impulse per

image51

Fig. 2-9. Two equal and opposite line Fig. 2-10. Vortex pattern simulating vortices separated a distance d. flow around a wing of finite span.

unit distance normal to the page is directed downward and may be written

£ = – j(prd), (2-98)

where j is a unit upward vector and d is the instantaneous separation of the vortex cores.

If, for instance, one of the two vortices is bound to a wing moving to the left with velocity t/„ while the other remains at rest in the fluid in the manner of a starting vortex, the force exerted by the fluid on the supporting bodies is

Fbody = ~ ~it= ~ Jt

= )PTjt(d) = pU„Tj. (2-99)

This may be recognized as the two-dimensional lift called for by the theorem of Kutta and Joukowsky.

A more complicated system of vortices is used as an indirect means of representing the influence of viscosity on the flow around a lifting wing of finite span (Fig. 2-10). For any one of the infinite number of elongated vortex elements, the Kelvin impulse is directed downward and equals

£ = —jp ДФ X [area]. (2-100)

The area here changes at a rate dependent on the forward speed C7„. The reaction to the force producing the increased impulses of the various vortices adds up to the instantaneous lift on the wing. Moreover, from the spanwise distribution of vortex strengths the spanwise distribution of lift is obtainable, and the energy in the vortex system is connected with the induced drag of the wing. It is evident that these vortices could not be generated in the first place except through the action of viscosity in pro­ducing a boundary layer on the wing, yet we can obtain much useful information about the loading on the system without actually attempting a full solution of the equations of Navier and Stokes.

2-Q Two-Dimensional, Constant-Density Flow: Fundamental Ideas

We now turn to the subject of two-dimensional, irrotational, steady or unsteady motion of constant-density fluid. We begin by listing a number of results which are well-known and may be found developed, for instance, in Chapters 5 through 7 of Milne-Thompson (1960). The combined condi­tions of irrotationality and continuity assure the existence of a velocity potential Ф(г, t) and a stream function Ф(г, t), such that

Q = V4> = V X (кФ). (2-101)

Here

r = xi + yj. (2-102)

Подпись:

Подпись: ЭФ _ ЭФ . u ~ dx ~ ay ’ Подпись: ЭФ _ _ 5Ф dy dx

If (2-101) is written out in component form, we obtain.

The latter equalities will be recognized as the Cauchy-Riemann relations. For constant-density fluid,

V. Q = V2$> (2-104)

is the volume divergence, whereas for a rotational flow,

V X Q = – кУ2Ф (2-105)

is the vorticity vector. Therefore, in the case under consideration,

V4 = 0 = V24>. (2-106)

Among other ways of constructing solutions to the two-dimensional Laplace equation, a function of either

Z = x + iy = re’9

(2-107)

Z = x — iy = re-‘9

(2-108)

alone will be suitable.[3] To be more specific, the Cauchy-Riemann relations are necessary and sufficient conditions for Ф and Ф to be the real and imaginary parts, respectively, of the same analytic function of Z. This function we call the complex potential,

Подпись: (2-109)■w (Z) = Ф + г’Ф.

Examples of Constant-Density Flows Where Circulation May Be Generated Подпись: (2-110) (2-111)

The following formulas for particle velocity and speed are easily derived:

The lines Ф = const and Ф = const form orthogonal networks of equi – potentials and streamlines in the x, у-plane, which is usually referred to as the Z-plane.

An interesting parallelism between the imaginary unit г = %/—l and the vector operator kx is discussed in Milne-Thompson (1960), and some readers may find it helpful to study this more physical interpretation of a quantity which has unfortunately been given a rather formidable name.

The fact that the complex potential is a function of a single variable has many advantages. Differentiation is of the ordinary variety and can be conveniently cascaded or inverted. Also, it makes little difference whether we operate with the functional relationship W(Z) orZ(W); many flows are more conveniently described by the latter.

We recall that many fundamental flow patterns are associated with simple singular forms of the complex potential. Thus In (Z) implies a point source or point vortex, 1/Z is a doublet, and Z“ corresponds to various fluid motions with linear boundaries meeting at angles related to a. A failure of one or more of the underlying physical assumptions occurs at the singular point location. Nevertheless, the singular solutions are useful in constructing flows of practical interest in regions away from their cen­ters. Forces and moments can be expressed in terms of contour integrals around the singularities and are therefore connected with residues at poles.

Examples of Constant-Density Flows Where Circulation May Be Generated Подпись: (2-112)

The complex potential is itself a kinematical concept. To find pressures and resultant forces in steady and unsteady flows, further information is required. Thus Bernoulli’s equation, (1-63), is our tool for pressure calcu­lation. The necessary quantities are taken from (2-111) and

where the operator on the right means to take the real part of the quantity in braces.

Examples of Constant-Density Flows Where Circulation May Be Generated Examples of Constant-Density Flows Where Circulation May Be Generated

For forces and moments on a single closed figure in steady flow, we have available the classical Blasius equations

where Mo is the counterclockwise moment exerted by the fluid on the profile about an axis through the origin, and C is a contour that surrounds the body but no other singularities of the flow field, if such exist. In the absence of external singularities, the contour may be enlarged indefinitely. Then if ‘W(Z) can be expanded into an inverse power series in Z, which is nearly always the case, we identify all forces as coming from the l/Z term and all moments as coming from the l/Z2 term. We conclude that an effective source or vortex, plus a uniform stream, will lead to a resultant force. Moreover, a doublet may give rise to a moment, as can certain other combinations of source-like and vortex-like singular solutions.

image52Fig. 2-11. Pressure force acting on a short segment of body surface in two­dimensional flow.

Подпись: However, Подпись: dFx — idFy = —pdy — ipdx = —ipdZ. Q2 . дФ nonessential p = —p 2 + dt_ + increment Подпись: (2-115)

In a limited way, (2-113) and (2-114) can be extended to apply to unsteady flows. The development follows Section 6.41 of Milne-Thompson (1960), but a restriction is required which is not carefully stated there. Let us consider a two-dimensional body whose position is fixed and whose contour does not change with time but which is in an accelerated stream U„(t) or otherwise unsteady regime. See Fig. 2-11. We derive the force equation and simply write down its analog for the moment. Let Cb be a contour coinciding with the fixed body surface. We note that

Подпись:p<W<№_ дФ 2 dZ dZ P dt ‘

The quantity labeled “nonessential increment” is dropped from (2-116) since even a time function will contribute nothing to the total force or moment. We substitute into (2-115)

dFx – і dFy = » I % (2-H7)

On the body surface,

Ф = Фв(<) (2-118)

independent of the space coordinates because of the assumed fixed position (i. e., the body is always an instantaneous streamline). Noting that the integration of force is carried out for a particular instant of time, we may write, following the contour Cb,

dw = <M> = dW = dZ, (2-119)

Подпись:Подпись: (2-121)ЭФ _ dW. d*B dt ~ dt +l dt ‘

The latter holds true because

W = Ф — г’Ф.

dZ + ip f *£dZ

CB dt

Подпись: F, Examples of Constant-Density Flows Where Circulation May Be Generated Подпись: (2-122)

Finally, we integrate around the contour С в and observe that the integral of the quantity d^fe/dt must vanish.

In this latter form the integrals are carried out around the contour C because each integrand is recognized as an analytic function of the variable of integration only, and contour deformation is permitted in the usual fashion. Of course, no pole singularities may be crossed during this de­formation, and branch points must be handled by putting suitable cuts into the field. The extended Blasius equation for moments in unsteady flow reads

M0 = – Re j| £ Z (^f )2 dZ + p I £ fw + itB(t)]Z dzj • (2-123)

Here the second contour may not be deformed from Cb since Z is not an analytic function of Z.

Подпись: Ф(ж, у; t) Подпись: 2ir J cB Examples of Constant-Density Flows Where Circulation May Be Generated Подпись: (2-124)

We close this section by setting down, without proof, the two-dimen­sional counterpart of (2-28). For an arbitrary field point x, y, this theorem expresses the velocity potential as follows:

Here the fluid must be at rest at infinity; line integration around the body contour is carried out with respect to dummy variables x, y\ and

The natural logarithm of r is the potential of a two-dimensional line source centered at point X, y. When differentiated with respect to out­ward normal n, it is changed into a line doublet with its axis parallel to n.

2-10 Two-Dimensional, Constant-Density Flow:

Conformal Transformations and Their Uses

A consequence of the mapping theorem of Riemann is that the exterior of any given single closed figure, such as an airfoil, in the complex Z-plane can be mapped into the exterior of any other closed figure in the f-plane by an analytic relation of the form

£ = /(f)- (2-126)

See Fig. 2-12. It is frequently convenient to choose a circle for the f-figure. The angle between any two intersecting lines is preserved by the trans­formation; for example, a set of orthogonal trajectories in one plane also turns out to be a set in the other. If the point at infinity is to remain unchanged, the transformation can always be expanded at large distances into something of the form

2 = f + y + p + — -> (2-127)

where the a„ are complex constants.

image53

Fig. 2-12. An illustration of the Z – and f-planes connected through conformal transformation.

It is occasionally pointed out that a special case of conformal trans­formation is the complex potential itself, (2-109), which can be regarded as a mapping of the streamlines and equipotentials into а (Ф + id’j-plane, where they become equidistant, horizontal and vertical straight lines, respectively.

The practical significance of the mapping theorem is that it can be used to transform one irrotational, constant-density flow’Wi(f) with elementary

boundaries into a second flow

W(Z) = ‘Wi[f1(Z)]l (2-128)

which has a more complicated boundary shape under the control of the transformer. Contours of engineering interest, such as a prescribed wing section, are readily obtained by properly choosing the function in (2-126). Conditions far away from the two figures can be kept the same, thus allowing for a prescribed flight condition.

Velocities, and consequently pressures, can be transformed through the relation

£ and – q being the real and imaginary parts of f, respectively. The equa­tions of Blasius can be employed to determine resultant force and moment in either of the two planes, and the transformation of variable itself is helpful when determining this information for the Z-figure. To provide starting points, many elementary complex potentials are known which characterize useful flows with circular boundaries.

Several transformations have proved either historically or currently valuable for constructing families of airfoil shapes and other two-dimen­sional figures with aeronautical applications. The reader is presumed to be familiar with the Joukowsky transformation, and much can be done with very minor refinements to the original investigations of Kutta and Joukowsky. No effort is made to expose in detail the various steps that have been carried out by different investigators, but we do list below a number of the more important transformations and something about their consequences.

1. The Joukowsky-Kutta Transformation

Z=[ + j – (2-131)

Here l is a positive real constant, and the so-called singular points of the transformation where the dZ/d{ — 0 are located at f = ±1, correspond­ing to Z = ±21. When applied to suitably located circles in the f-plane, (2-131) is well known to produce ellipses, flat plates, circular arc profiles of zero thickness, symmetrical and cambered profiles with their maximum thickness far forward and with approximately circular-arc camber lines.

The shape obtained actually depends on the location of the circle relative to the aforementioned singular points. A cusped trailing edge is produced by passing the circle through the singular point on the downstream side of the circle.

2. The von Mises Transformations. These transformations are special cases of the series, (2-127), in which it is truncated to a finite number of terms:

Z=f + ^ + — -+ £- (2-132)

S f”

When an airfoil is being designed, the series is constructed by starting from the singular point locations at

and subsequently integrated in closed form. By the rather laborious process of trial-and-error location of singular points, many practical air­foils were developed during the 1920’s. It is possible to adjust the thick­ness and camber distributions in a very general way. Interesting examples of von Mises and other airfoils will be found discussed in a recent book by Riegels (1961).

3. The von Karman-Trefftz Transformation. This method derives from a scheme for getting rid of the cusp at the trailing edge, produced by the foregoing classes of transformations, and replacing it by a corner with a finite angle t. To see how it accomplishes this, consider a trans­formation with a singular point at f = fo, corresponding to a point Z = Z0. In the vicinity of this particular singular point, it is easy to show that the transformation can be approximated by

Z-Z0= Alt – ГоГ, n > 1, (2-135)

where A is some complex constant. Evidently, the quantity

S = – ~Ai’ – f°r" (2“136)

vanishes at the point for n > 1, as it is expected to do. Let r0 and в0 be the modulus and argument of the complex vector emanating from the point f0- Equation (2-136) can be written

Подпись: (2-137)d(Z – Z0) = nAr3-V("-1)*0d(r – fo).

image55

This is to say, the element of arc d((" — f0) is rotated through an angle (n — 1)0O in passing from the ("-plane to the Z-plane, if we overlook the effect of the constant A which rotates any line through ("0 by the same amount. Figure 2-13 demonstrates what this transformation does to a continuous curve passing through the point (" = ("0 in the ("-plane. By proper choice of n, the break in the curve which is produced on the Z-plane can be given any desired value between ir and 27t.

Подпись: Z - 21 Z A 21 Подпись: 2—(r/ir) Подпись: (2-139)

Yon K&rm&n and Trefftz (1918) suggested replacing (2-138) as follows:

As in Fig. 2-14, the outer angle between the upper and lower surfaces of the airfoil at its trailing edge (T. E.) is now (2тг — т), so a finite interior angle has been introduced and can be selected at will.

In a similar fashion a factor

can be included in the von Mises equation (2-134). Then if the circle in the f-plane is passed through the point f = f,,T E, an adjustable trailing edge is provided for the resulting profile.

4. The Theodor sen Transformation[4] Theodorsen’s method and the several extensions which have been suggested for it are capable of con­structing the flow around an airfoil or other single object of completely arbitrary shape. All that is needed is some sort of table or equation pro­viding the ordinates of the desired figure. The present brief discussion will emphasize the application to the airfoil.

The transformation is actually carried out in two steps. First the airfoil is located in the Z-plane as close as possible to where a similarly shaped Joukowsky airfoil would fall. It can be proved that this will involve locating the Joukowsky singular points Z = ±2Z halfway between the nose and the trailing edge and their respective centers of curvature. (Of course, if the trailing edge is pointed, the singularity Z — —21 falls right on it.) By applying Joukowsky’s transformation in reverse, the airfoil is transformed into a “pseudocircle” in the Z’-plane

Z = Z> + D • (2-140)

z

Подпись:The second procedure is the conver­sion of the pseudocircle to an exact circle centered at the origin in the Z’-plane by iterated determination of the coefficients in the transforma­tion series

Z’= fexpfe^). (2-141)

i=i 1 7

Here the Cn are complex constants. The two steps of the process are illustrated in Fig. 2-15.

Let us consider the two steps in a little more detail. We write

Z = x + iy, (2-142)

Z> = іеФ+іч, (2-143)

That is, the argument of Z’ is

denoted by в and its modulus is le*. It is not difficult to derive the direct and inverse relationships between the coordinates of the airfoil and pseudocircle:

Подпись:(2-144)

Подпись: (2-145) (2-146) 2 sin2 в = p + л/p2 + (y/l)2
2 sinh2 ф = — p – f – Vp2 + (y/l) 2>

where

image58

Note that ф will be quite a small number for profiles of normal thickness and camber.

The function ф(в) for the pseudocircle may be regarded as known at as many points as desired. We then write

J – = Д0еіф = и*°еіф. (2-147)

For points on the two contours only, the transformation, (2-141), can be manipulated as follows:

Z’ le^+i8

J = = exP к* – *o) + W – Ф)}, (2-148)

Z’ = f exp [{ф — фо) + i(8 — ф)] = f exp ^ (2-149)

Theodorsen (1931) adopted the symbol e to denote the shift in argument going from the Z’- to the ("-plane,

Подпись:€ ~ ф — в ОГ ф = в + €.

Подпись:Cn ~ “1“

An. Bn.

—– COS пф І———- sm пф

Д

п туп

0 Ло J

Bn, і An • ,

— cos пф H—— sm пф

Д

п ТУП

о Ло J

These two equations imply that (ф — Фа) and e are conjugate quantities. They are expressed as Fourier series in the variable Ф, so that the standard formulas for individual Fourier coefficients could be employed if these quantities were known in terms of ф. Moreover, the conjugate property

can be used to relate the functions e and ф directly,

as follows:

■<*’>- Li *<+>“*(* 2*

^ёф,

(2-154)

Ф(Ф’) = ^ + ~fo е(ф) cot^

ф’)ёф.

(2-155)

In actuality, only ф(в) is available to begin with; 0 can be regarded as a first approximation to ф, however, and (2-154) employed to get a first estimate of e. Equation (2-150) then yields an improved approximation to ф and to ф(ф), so that (2-154) may be used in an iterative fashion to obtain converged formulas for the desired quantities. For typical airfoils it is found that this process converges very rapidly, and the numerical integration of (2-154) need be iterated only once or twice. Of course, it is necessary to be careful about the pole singularity at ф = ф’.

As described in the references, Theodorsen’s so-called e-method has many uses in the theory of low-speed airfoils. For instance, one can gener­ate families of profiles from assumed forms of the function е(ф). Approxi­mate means have been developed, starting from an airfoil of known shape and pressure distribution, for adjusting this pressure distribution in a desired fashion. This scheme formed the basis for the laminar flow profiles

image59

Fig. 2-16. Comparison between predicted and measured pressure distribution over the upper and lower surfaces of a Clark Y airfoil at an effective geometrical incidence of —1 deg 16 min. Theoretical incidence chosen to give approximately the measured value of total lift. [Adapted from Theodorsen (1931).]

which played such an important role in the early 1940’s. Their shapes sustain a carefully adjusted, favorable pressure distribution to assure the longest possible laminar run prior to transition in the boundary layer. In wind tunnel tests, they achieve remarkable reductions in friction drag; unfortunately, these same reductions cannot usually be obtained in engi­neering practice, and some of the profiles have undesirable characteristics above the critical flight Mach number.

Figure 2-16 shows a particularly successful example of the comparison between pressure distribution measured on an airfoil and predicted by this so-called “e-method.”

5. The Schwarz-Christoffel Transformations. These transformations are described in detail in any advanced text on functions of a complex variable. They furnish a useful general technique for constructing flows with boundaries which are made up of straight-line segments.

Solutions for Subsonic and Supersonic Flows

The outer flow is easily built up from a continuous distribution of sources along the ж-axis. The solution for a source in a subsonic flow is given by (5-37). Thus, for a distribution of strength f(x) per unit axial distance,

Подпись: (6-25)J_ f f(xi) dxі

4-7Г v/(x — Жі)2 + /32r2

The source strength must be determined such that the boundary condition (6-22) is satisfied. It follows directly from (6-22) that the volumetric outflow per unit length should be equal to the streamwise rate of change of cross-sectional area (multiplied by Ux). Hence we have

Подпись: (6-26) (6-27) / = S'(x).

The result thus becomes

_ 1 £ <S'(xi) dx і

4:ir Jо л/(ж — xi)2 + /32r2

We need to expand the solution for small r in order to determine an inner solution of the form (6-24). This can be done in a number of ways, for example by Fourier transform techniques (Adams and Sears, 1953) or integration by parts. Here we shall select a method used by Oswatitsch
and Keune (1955) for its physical perspicuity. It is seen that the kernel in the integral (6-27) for small r is approximately

Solutions for Subsonic and Supersonic Flows Подпись: 1 S'(Xl) - S'(x) dx i, (6-32) X) AirJo |x — X! | 4ж. Ґ S'M - S'(x) 10 x - Х!І dXl■ (6-33)

In the second term of (6-29) we may use the approximation (6-28) because the numerator tends to zero for xx —> x (the error actually turns out to be of order r2 In r). Thus, collecting all the terms in (6-29), we find that for small r

The last term gives the effect due to variation of source strength at body stations fairly far away from station x. This form of the integral is par­ticularly convenient when the cross-sectional area distribution is given as a polynomial, since then the integrand will become a polynomial in xx. We may obtain an alternate form by performing an integration by parts in (6-33). This gives

cx

g(x) = ^~lnf — 4~ Jo s"(xi)ln (x — xi) dxx

+ Il-f S"(x0 ln (Xi — x) dxx, (6-34)

where we have assumed that S'(0) = S'(l) = 0, that is, the body has a pointed nose and ends in a point or in a cylindrical portion.

The solution for supersonic flow can be found in the same manner. Using (5-38) we obtain

rx—Br

1 S'(xi) dxx /n nr4

<p = — 7T~ – — – -—:———- _ ’ (6-35)

6ТГ J a /(x — Xl)2 — B2r2

where

В = VM2 – 1.

The upper integration limit follows because each source can only be felt inside its downstream Mach cone; hence the rearmost source that can influence the flow in the point x, r is located at xx = x — Br. Rewriting of (6-35) in a similar manner as (6-30) gives

(6-36)

For the first term we obtain

Подпись: x-Brdx 1

Подпись: = —ln Br + ln [x + /x2 — B2r2 ] Solutions for Subsonic and Supersonic Flows Подпись: (6-37)

уДх — xx)2 — B2r2

In the second integral we may replace the square root by x — xx as before. In addition, the upper integration limit may be replaced by x for

Подпись: , V S’(x) В g{x) = ^ In- Solutions for Subsonic and Supersonic Flows Solutions for Subsonic and Supersonic Flows

That the correct factor 1/27Г (cf. 6-24) was obtained for the first term confirms the constant for the supersonic source solution (5-38) selected by intuitive reasoning. For the supersonic case we thus have

Подпись: g(x) Подпись: S' (ж), в FFln2 Подпись: s. Подпись: S"(xi) In (x — xi) dxi, (6-40)

As with subsonic flow, an alternate form can be obtained by integrating the last term by parts. This yields

where we have assumed that <8′(0) = 0. This form will be used later for the calculation of drag.

Подпись: 47Г Подпись: <S"(xi) In (xi Подпись: x) dxi,

It is interesting to note how g(x) changes from subsonic to supersonic flow, as seen by comparing (6-34) with (6-40). First, /3 is replaced by B. Secondly, the integral

which represents the upstream influence in subsonic flow, changes to

— Jo s"(xi) ln (x — xi) dxb

that is, becomes equal to half the total downstream influence. To under­stand this behavior, consider the disturbance caused by a source in one cross section x as it is felt on the body at other cross sections. The dis­turbance will spread along two wave fronts, one wave moving downstream with a velocity of (approximately) ax + Ux and the other either upstream or downstream with a velocity of aw — Ux, depending on whether the flow is subsonic or supersonic. The effect of fast-moving waves is given by the first integral in (6-34), whereas that of slow-moving waves is given by the second integral. In the supersonic case, the fast and slow waves each contribute half of the integral in (6-40). Because of the small cross­wise dimensions of the body, the curvature of the waves may be neglected in the present approximation. Hence their fronts may be treated as plane, the total effect being given by a function of x only.

image86

Fig. 6-і. Pressures on the forward portion on a body of revolution. [Adapted from Drougge (1959). Courtesy of Aeronautical Research Institute of Sweden.]

A comparison of calculated and measured pressure distributions given by Drougge (1959) is shown in Fig. 6-1. The excellent agreement despite the fairly large thickness ratio (r = £) demonstrates the higher accuracy of slender-body theory than thin-airfoil theory. In the former theory the error term is of order e4 (or, rather, e4 In e), whereas in the latter it is of order e2. In assessing the accuracy of slender-body theory for practical cases, however, one must remember that the body considered in Fig. 6-1 is very smooth, with small second derivative of the cross-sectional area distribution, and should therefore be ideally suited for the theory.

The weak Mach number dependence of In |1 — Л/21 as compared to |1 — A/2|~1/2 in the thin-airfoil case, with the associated weaker singu­larity at M = 1, is significant. It indicates that the linearized slender-body theory generally holds closer to M = 1 than does the thin-airfoil theory for the same thickness ratio, i. e., the true transonic region should be much

smaller. For the body in Fig. 6-1, the linearized theory gives accurate pressure distributions for M < 0.90 and M > 1.10. In the transonic region the slow-moving waves will have time to interact and accumulate on the body, thus creating nonlinear effects that cannot be treated with the present linearized theory. The transonic case will be further discussed in Chapter 12.

Implicit in the derivation of the theory was the assumption that S’ is continuous everywhere, as is also evident from the results which show that <p becomes logarithmically singular at discontinuities of S’ and the pressure thus singular as the inverse of the distance. However, the slender-body theory may be considered as the correct “outer” solution away from the discontinuity with a separate “inner” solution required in its immediate neighborhood. Such a theory has in effect been developed by Lighthill (1948).

Conservation Laws for a Barotropic Fluid. in a Conservative Body Force Field

Conservation Laws for a Barotropic Fluid. in a Conservative Body Force Field Подпись: S2 — Подпись: dp P. Подпись: (1-11)

Under the limitations of the present section, it is easily seen that the law of conservation of momentum, (1-3), can be written

The term “barotropic ” implies a unique pressure-density relation through­out the entire flow field; adiabatic-reversible or isentropic flow is the most important special case. As we shall see, (1-11) can often be integrated to yield a useful relation among the quantities pressure, velocity, density, etc., that holds throughout the entire flow.

Подпись: where Подпись: DT Dt Подпись: (, dp c P Подпись: c T ds’ Подпись: (1-12)

Another consequence of barotropy is a simplification of Kelvin’s theorem of the rate of change of circulation around a path C always composed of the same set of fluid particles. As shown in elementary textbooks, it is a consequence of the equations of motion for inviscid fluid in a conservative body force field that

image4(1-13)

is the circulation or closed line integral of the tangential component of the velocity vector. Under the present limitations, we see that the middle member of (1-12) is the integral of a single-valued perfect differential and therefore must vanish. Hence we have the result DT/Dt = 0 for all
such fluid paths, which means that the circulation is preserved. In par­ticular, if the circulation around a path is initially zero, it will always remain so. The same result holds in a constant-density fluid where the quantity p in the denominator can be taken outside, leaving once more a perfect differential; this is true regardless of what assumptions are made about the thermodynamic behavior of the fluid.

Methods of Solution

Here we shall make a short review of some of the methods that have been proposed and used for solving the transonic nonlinear small-disturb­ance equations. The only case, so far, that has been found amenable to a mathematically exact treatment is the two-dimensional, for which it is possible to transform the nonlinear problem to a linear one by going to the hodograph plane. Let

0

n>


where Kx is given by (12-47). The quantities її and w represent (to first order), with appropriate constants, the components of the difference between the local velocity and the speed of sound. Thus, on the sonic line

її = 0, (12-55)

with w taking on any value, and

и = К i, w = 0

at infinity.

We may then write (12-12) with the aid of the condition of irrotation – ality as follows:

— йїїх + Wf = 0, (12-57)

щ = wx. (12-58)

We shall now transform this system so that и and w appear as the depend­ent variables. For this purpose we introduce a function ф(й, w) such that

x = fc, (12-59)

Ґ = Фіг – (12-60)

It will be shown subsequently that the irrotationality condition (12-58) is thereby automatically satisfied. For from (12-59) and (12-60) it follows that

dx — фгіи dll ~J – ^pu w dv)j

(12-61)

djf -— фууй dll Фитй) dWj

(12-62)

which show that dx =

0 for

dw = {Фий/Фиъй) dll.

(12-63)

and that df = 0 for

du = —dw(fow/ifrwii).

(12-64)

Thus

– (a

U^p

m du

/ dx=0 [Фьт V’wwKV’wm/V’uuj)] dll

_ Фиу)

~ D ’

(12-65)

where

D — фиифигш (фиш)

is the functional determinant (Jacobian).

Similarly, it is found by use of (12-64) that

which confirms (12-58).

Proceeding in this manner we find that

^x — fiww/D

and

Щ = Фш/D-

Hence (12-57) transforms to

— Www + Фш = 0,

which is known as Tricomi’s equation after Tricomi (1923), who first investigated its properties. It is seen that the linear equation (12-69) preserves the mixed subsonic-supersonic character of the original equation because it is hyperbolic for її > 0 and elliptic for м < 0. However, the linearization of the equation has not been bought without considerable sacrifice to the simplicity in the application of the boundary conditions. In fact, solutions have so far been obtained only for some very simple shapes.

To illustrate the difficulties involved, the boundary conditions for a simple wedge are formulated in Fig. 12-7. The case of a subsonic free stream for which Кi < 0 is illustrated. In view of (12-54) the point й = Ki, = 0 represents infinity in the physical flow field so that, following

Fig. 12-7. The transonic flow around a nonlifting single wedge.

the definitions (12-59) and (12-60), the derivatives of ф must be infinite at this point, i. e., the solution must have a singularity at (Кb 0). The form of this may be determined from the linearized subsonic solution, except for M = 1 which requires special treatment.

The transonic wedge solution has been worked out for M less than unity by Cole (1951) and by Yoshihara (1956), for M = 1 by Guderley and Yoshihara (1950), and for M > 1 by Yincenti and Wagoner (1952). The results for the drag coefficient are plotted in Fig. 12-8 together with experimental data obtained by Bryson (1952) and Liepmann and Bryson (1950). As seen, the agreement is excellent.

Fig. 12-8. Theoretical and experimental results for the drag of single-wedge airfoils. (From Spreiter and Alksne, 1958. Courtesy of the National Aero­nautics and Space Administration.)

The hodograph method is not very useful for axisymmetric flow, since the factor 1/p in the second term of (12-26) makes the equation still non­linear after transformation to the hodograph plane. Because of such limitations, a considerable effort has been expended in finding methods that work directly in the physical plane. The solutions developed so far are all based on one or more approximations. In the method for two­dimensional flow proposed by Oswatitsch (1950) and developed in detail by Gullstrand (1951), the differential equation is rewritten as an integral equation by the aid of Green’s theorem, and the nonlinear term is approxi­mated under the implicit assumption that the value of an integral is less sensitive to errors in the approximations than is a derivative. Further improvements in this method have been introduced by Spreiter and Alksne (1955).

An approximation of a radically different kind was suggested by Os- watitsch and Keune (1955a) for treating the flow on the forward portion of a body of revolution at M = 1. In the differential equation (12-31) for M = 1,

— (7 + l)<Px<Pxx + – Vr + <Prr = 0, (12-70)

the nonlinear term was approximated by

(T – j – 1)iPx(Pxx = ^p*Pxj (12“71)

where the constant Xp is to be suitably chosen. The justification of this approximation is that on the forward portion of the body the flow is found to be everywhere accelerating at a fairly constant rate. Also, the resulting differential equation is parabolic, which intuitively is satisfying as an intermediate type between the elliptic and hyperbolic ones. The constant Xp was chosen arbitrarily (but in a way consistent with the similarity law) so as to give good agreement with the measured pressure distribution in one case, and it was proposed to use this as a universal value in other cases.

Fig. 12-9. Pressure distribution on a cone cylinder at iff = 1. (From Spreiter and Alksne, 1959. Courtesy of National Aeronautics and Space Administration.)

Maeder and Thommen (1956) also used the approximation (12-71) for flows with M slightly different from unity and suggested a new, but still arbitrary, rule for determining Xp.

An interesting extension of Oswatitsch’s method, which removes the arbitrariness in selecting Xp, has been presented by Spreiter and Alksne (1959). In this the parabolic equation resulting from the approximation (12-71) is first solved assuming Xp constant, and the value of и = <ря is calculated on the body. Now (7 + 1 )ux is restored in place of Xp and a nonlinear differential equation of first order is obtained for u, which may be solved numerically. As an example, the pressure distribution on a slender cone-cylinder calculated this way is shown in Fig. 12-9 together with values obtained from the theory by Oswatitsch and Keune (1955) and measured values. As seen, the agreement with the improved theory is excellent and considerably better than with the original one.

Spreiter and Alksne (1958) also employed this technique with consider­able success for two-dimensional flow, and for flows that have a Mach number slightly different from unity. In the latter case they replaced the nonlinear term

[1-М2 – M2(7 + l)Vx]Vtx (12-72)

by

X<p„, (12-73)

and proceeded similarly to solve the resulting linear equation with X con­stant. Thereupon (12-72) was resubstituted into the answer, producing a nonlinear first-order differential equation for и = <px as before. They were able to show that in the two-dimensional supersonic case, this gave an answer that was identical to that given by simple-wave theory.

The Kutta Condition and Lift

As is familiar to every student of aerodynamics, Joukowsky and Kutta discovered independently the need for circulation to render the two­dimensional, constant-density flow around a figure with a pointed trailing edge physically reasonable. This is a simple example of a scheme for fixing the otherwise indeterminate circulation around an irreducible path in a doubly connected region. It is one of a number of ways in which viscosity can be introduced at least indirectly into aerodynamic theory without actually solving the equations of Navier and Stokes. The circulation gives rise to a lift, which is connected with the continually increasing Kelvin impulse of the vortex pair, one of the vortices being the circulation bound to the airfoil, while the other is the “starting vortex ” that was generated at the instant the motion began.

image60

With respect to the lifting airfoil, we reproduce a few important results from Sections 7.40-7.53 of Milne-Thompson (1960). Let the profile and the circle which is being transformed into it be related as shown in Fig. 2-17.

It is assumed that the transformation

£=f + ^ + p + — -> (2-156)

which takes the circle into the airfoil, is known. It can be proved that the resulting force is normal to the oncoming stream and equal to

L = pt/„r. (2-157)

Here Г is the circulation bound to the airfoil, which incidentally may or may not satisfy the full Kutta condition of smooth flow off from the trailing edge. If this condition is entirely met, which is equivalent to neglecting the effect of displacement thickness of the boundary layer and’ the wake thickness at the trailing edge, then the circulation is given by

Г = ітгі/^а sin <*z. l.- (2-158)

All the quantities here are defined in the figure. In particular, az. L. is the angle of attack between the actual stream direction and the zero-lift (Z. L.) direction, determined as a line parallel to one between the center of the circle in the f-plane and the point Hi which transforms into the airfoil trailing edge. Combining (2-157) and (2-158), we compute the lift

L = 4wpl7^a sin «z. l.- (2-159)

From this we find that the lift-curve slope, according to the standard aeronautical definition, is slightly in excess of 2ir, reducing precisely to 2ж when the airfoil becomes a flat plate of zero thickness, i. e., when the radius a of the circle becomes equal to a quarter of the chord.

The airfoil is found to possess an aerodynamic center (A. C.), a moment axis about which the pitching moment is independent of angle of attack. This point is located on the Z-plane as shown in Figs. 7.52 and 7.53 of Milne-Thompson (1960). The moment about the aerodynamic center is

MA. C. = — 2жрІІ212 sin 27, (2-160)

where

ai = (le~iy)2. (2-161)

Figure 2-18 gives some indication of the accuracy with which lift and moment can be predicted. The theoretical value of zero for drag in two dimensions is the most prominent failure of inviscid flow methods. It represents a nearly achievable ideal, however, as evidenced by the lift-to – drag ratio of almost 300 from a carefully arranged experiment, which is reported on page 8 of Jones and Cohen (1960).

image62

Fig. 2-18. Comparisons between predicted and measured lift coefficients and quarter-chord moment coefficients for an NACA 4412 airfoil. “Usual theory” refers to Theodorsen’s procedure, whereas the modifications involve changing the function е(ф) so as to make circulation agree with the measured lift at a given angle of attack. [Adapted from Fig. 9 of Pinkerton (1936).]

More information will be found in Chapter 4 of Thwaites (1960) on refined ways of calculating constant-density flow around two-dimensional airfoils. In particular, these include reference to a modern theory by Spence and others which makes allowance for the boundary layer thickness and thus is able to carry the calculation of loading up to much higher angles of attack, even approaching the stall.