Category Aerodynamics of Wings and Bodies

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


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)


r = xi + yj. (2-102)


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


Z = x — iy = re-‘9


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.


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).


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)


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



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


п туп

0 Ло J

Bn, і An • ,

— cos пф H—— sm пф



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*



Ф(Ф’) = ^ + ~fo е(ф) cot^



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


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


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


1 S'(xi) dxx /n nr4

<p = — 7T~ – — – -—:———- _ ’ (6-35)

6ТГ J a /(x — Xl)2 — B2r2


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


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.


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


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



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


djf -— фууй dll Фитй) dWj


which show that dx =

0 for

dw = {Фий/Фиъй) dll.


and that df = 0 for

du = —dw(fow/ifrwii).



– (a


m du

/ dx=0 [Фьт V’wwKV’wm/V’uuj)] dll

_ Фиу)

~ D ’



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


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)


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.


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)


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).


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.

General Slender Body

For a general slender body we assume that the body surface may be defined by an expression of the following form

B(x, y, z) = B(x, у/є, z/e) = 0, (6-41)

where e is the slenderness parameter (for example, the aspect ratio in the case of a slender wing and the thickness ratio in the case of a slender body of revolution). With the definition (6—41), a class of affine bodies with a given cross-sectional shape is studied for varying slenderness ratio e, and the purpose is to develop the solution for the flow in an asymptotic series in € with the lowest-order term constituting the slender-body approxima­tion. In the stretched coordinate system

V = 2//e, 2 = z/« (6-42)

the cross-sectional shape for a given x becomes independent of e. From the results of Section 6-4 it is plausible that the inner solution would be of the form

Ф* = Ux[x + Є2Ф2ІХ, У, z) + ■ • • ], (6-43)

that is, there will be no first-order term. The correctness of (6-43) will become evident later from the self-consistency of the final result. For a steady motion the condition of tangential flow at the body surface requires that the outward normal to the surface be perpendicular to the flow velocity vector:


Or, upon introducing (6-42) and (6-43) and dropping higher-order terms,

Bx + B&iy + ВІФІі = 0. (6-45)

This relation may be put into a physically more meaningful form in the following manner. Introduce, temporarily, for each point on the contour considered, a coordinate system n, s such that n is in the direction normal to, and 5 tangential to, the contour at the point, as shown in Fig. 6-2. Obviously, (6-45) then takes the form

Bx + ВкФІк = 0. (6-46)

Let dn denote the change, in the direction of n, of the location of the contour when going from the cross section at ж to the one at x + dx. Moving along the body surface with dS = 0 we then have

dB = Bx dx + Bn dn = 0. (6-47)

Upon combining (6-46) and (6-47) we obtain

*;, = g. (e-48)

a condition that simply states that the streamline slope must equal the surface slope in the plane normal to the surface.

By introducing (6-43) into the differential equation (1-74) for Ф we find that Ф2 must satisfy the Laplace equation in the 5, г-plane:

^2vv + Фггг = 0. (6-49)

A formal solution may be obtained by applying (2-124). (This solution was deduced by using Green’s theorem in two dimensions.) Thus

where index 1 denotes dummy integration variables as usual, and

As in the body-of-revolution case the function $2(x) must be obtained by matching. Note that (6-50) is in general not useful for evaluating Ф2, since only the first term in the integrand is known from the boundary condition on the body. Nevertheless, it can be used to determine Ф2 for large F, since then d/dnlnr1 may be neglected compared to In Fx and, furthermore, fi may be approximated by F. Hence the outer limit of the inner solution becomes


which is the same as the solution (6-11) for an axisymmetric body having the same cross-sectional area distribution as the actual slender body. We shall, following Oswatitsch and Keune (1955), term this body the equivalent body of revolution. By matching it will then follow that <?2(я) must be identical to that for the equivalent body of revolution. We have by this proved the following equivalence rule, which was first explicitly stated by Oswatitsch and Keune (1955) for transonic flow, but which was also implicit in an earlier paper by Ward (1949) on supersonic flow:

(a) Far away from a general slender body the flow becomes axisymmetric and equal to the flow around the equivalent body of revolution.

(b) Near the slender body, the flow differs from that around the equivalent body of revolution by a two-dimensional constant-density crossflow part that makes the tangency condition at the body surface satisfied.

Proofs similar to the one given here have been given by Harder and Klunker (1957) and by Guderley (1957). The equivalence rule allows great simplifications in the problem of calculating the perturbation velocity potential

First, the outer axisymmetric flow is immediately given by the results of the previous section. Secondly, the inner problem is reduced to one of two-dimensional constant-density flow for which the methods of Chapter 2 may be applied. The following composite solution valid for the whole flow field has been suggested by Oswatitsch and Keune [cf. statement (b) above]: Let <pe denote the solution for the equivalent body of revolution and <P2 the inner two-dimensional crossflow solution that in the outer limit becomes <p2 ~ (1/27Г)S'(x) In r. Then the composite solution

<pc = <Pe + <p2 — lnr (6-55)

holds in the whole flow field (to within the slender-body approximation).

As in the case of a body of revolution, quadratic terms in the crossflow velocity components must be retained in the expression for the pressure, so that (cf. Eq. 6-23)

Cp = — 2<(>x — ip% — ip. (6-56)

In view of the fact that the derivation given above did not require any specification of the range of the free-stream speed, as an examination of the expansion procedure for the inner flow will reveal, it should also be valid for transonic flow. As will be discussed in Chapter 12, the difference will appear in the outer flow which then, although still axisymmetric, must be obtained from a nonlinear equation rather than from the linearized (6-21) as in the sub – or supersonic case. The form of the differential equation for the outer flow does not affect the statements (a) and (b) above, how­ever, and it turns out that the validity of the equivalence rule is less restricted for transonic than for sub – or supersonic flow so that it can then also be used for configurations of moderate aspect ratio provided the flow perturbations are small.

Some Geometric or Kinematic Properties of the Velocity Field

Next we specialize for the velocity vector Q two integral theorems which are valid for any suitably continuous and differentiable vector field.

1. Gauss’ Divergence Theorem. Consider any volume V entirely within the field enclosed by a single closed bounding surface S as in Fig. 1-1.

j^Q-ndS = yyy (V ■ Q)dV. (1-14)

S ■ V

Here n is the outward-directed unit normal from any differential element of area dS. This result is derived and discussed, for instance, in Sections 2.60 and 2.61 of Milne-Thompson (1960). Several alternative forms and some interesting deductions from Gauss’ theorem are listed in Section 2-61. The theorem relates the tendency of the field lines to diverge, or spread out within the volume V, to the net efflux of these lines from the boundary of V. It might therefore be described as an equation of continuity of field lines.


Подпись: Here Подпись: JJn ■ (V X Q) dS. S Подпись: (1-15)

Stokes’ Theorem on Rotation. Now we consider a closed curve C of the sort employed in (1-12) and (1-13), except that the present result is instantaneous so that there is no question of a moving path composed of the same particles. Let S be any open surface which has the curve C as its boundary, as illustrated in Fig. 1-2. The theorem refers to the circulation around the curve C and reads


is called the vorticity and can be shown to be equal to twice the angular velocity of a fluid particle about an axis through its own centroid. The theorem connects the spinning tendency of the particles lying in surface S with the associated inclination of the fluid at the boundary of S to circulate in one direction or the other. Sections 2.50 and 2.51 of Milne-Thompson (1960) provide a derivation and a number of alternative forms.

Подпись: FIG. 1-2. Open surface S bounded by closed curve C. (Positive relationship of n- and ds-directions as indicated.)
Подпись: FIG. 1-І. Finite control volume V surrounded by closed surface S in flow field.

1-4 The Independence of Scale in Inviscid Flows

Подпись: FIG. 1-3. Two bodies of different sizes but identical shapes and motions.

Consider two bodies of identical shape, but different scales, characterized by the representative lengths h and /2, moving with the same velocity Qb through two unbounded masses of the same inviscid fluid. See Fig. 1-3. This motion is governed by the differential equations developed in Section 1-1 plus the following boundary conditions: (1) disturbances die out at infinity, (2) Qn = Qb • n at corresponding points on the two surfaces, n being the normal directed into the fluid.

We first treat the case of a steady flow which has gone on for a long period of time so that all derivatives with respect to t vanish. We can then apply the Newtonian transformation, giving the fluid at infinity a uniform motion minus Qb and simultaneously bringing the body to rest. Thus the boundary condition No. 2 becomes Qn = 0 all over the surfaces. We take the differential equations governing these two similar problems and make the following changes of variable: in the first case.


= x/h,



Zi = z/h;


in the second case, *2

= x/h,


z2 = z/l2-


In the dimensionless coordinate systems the two bodies are congruent with each other. Moreover, the differential equations become identical. That is, for instance, continuity reads, in the first case,

Подпись:Подпись: (1-20)BjpU) , d{pVl, аШ1 = о

dxi dyi dz і

and in the second case,

d(pU) , dJpV) d(pW) =

dx2 dy2 dz2 ‘

We are led to conclude, assuming only uniqueness, that the two flows are identical except for scale. Velocities, gas properties, and all other dependent variables are equal at corresponding points in the two physical fields. This is a result which certainly seems reasonable on grounds of experience.

The same sort of reasoning can be extended to unsteady flows by also scaling time in proportion to length:

Xt = x/h, Ух = у/lx, zx = z/l 1, tx = t/lx, (1-21)

x2 = x/l2, 2/2 = y/h, z2 = z/l2, t2 = t/l2, (1-22)

in the two cases. With regard to boundary conditions, it must also be specified that over all time Qb(h) in the first case and Qbih) are identical functions.

Evidently, it makes no difference what the linear scale of an ideal fluid flow is. In what follows, therefore, we shall move back and forth occasion­ally from dimensional to dimensionless space and time coordinates, even using the same symbols for both. A related simplification, which is often encountered in the literature, consists of stating that the wing chord, body length, etc., will be taken as unity throughout a theoretical develop­ment; the time scale is then established by equating the free-stream to unity as well.

One important warning: the introduction of viscosity and heat conduc­tion causes terms containing second derivatives, the viscosity coefficient, and coefficient of heat conductivity to appear in some of the governing differential equations. When dimensionless variables are introduced, the Reynolds number and possibly the Prandtl number[2] appear as additional
parameters in these equations, spoiling the above-described similarity. The two flows will no longer scale because the Reynolds numbers are differ­ent in the two cases. Even in inviscid flow, the appearance of relaxation or of finite reaction rates introduces an additional length scale that destroys the similarity. It is satisfactory, however, for real gas which remains either in chemical equilibrium or in the “frozen” condition thermody­namically.

If shock waves and/or vortex wakes are present in – the field, it can be reasoned without difficulty that they also scale in the same manner as a flow without discontinuities. An excellent discussion of this whole subject of invariance to scale changes will be found in Section 2.2 of Hayes and Probstein (1959). Finally, it might be observed that differences between the two flows in the ambient state of the fluid at infinity, such as the pres­sure, density, and temperature there, may also be included in the scaling by referring the appropriate state variables to these reference values. The Mach numbers in the two flows must then be the same.

Unsteady Flow

12- 1 Statement of the Problem

Chapters 4 through 12 have dealt with aerodynamic loading due to uniform flight of wings and bodies. Obviously no air vehicle remains indefinitely bathed in steady flow, but this idealization is justified on the basis that the time constants of unsteady motion are often very long compared to the interval required for transients in the fluid to die down to imperceptible levels. There exist important phenomena, however, where unsteadiness cannot be overlooked; rapid maneuvers, response to atmospheric turbulence, and flutter are familiar instances. We therefore end this book with a short review of some significant results on time – dependent loading of wings. These examples merely typify the extensive research that has lately been devoted to unsteady flow theory, both linearized and more exact. We hope that the reader will be able to construct parallels with steady-state counterparts and thus prepare himself to read the literature on oscillating nonplanar configurations, slender bodies, etc.

Let irrotationality be assumed, under the limitations set forth in Sec­tions 1-1 and 1-7. The kinematics of the unsteady field are then fully described by a velocity potential Ф, governed by the differential equation (1-74), from which the speed of sound is formally eliminated using (1-67). (We let Я = 0 here.) Pressure distributions and generalized aerodynamic forces follow from (1-64).

Flow disturbances in a uniform stream

Ф? = U„x (13-1)

are generated by a thin lifting surface (Fig. 13-1), which is performing rapid, small displacements in a direction generally normal to its x, y-plane projection. Thus the wing might be vibrating elastically, undergoing sud­den roll or pitch aerobatics, or an encounter with gusty air might give rise to a situation mathematically and physically analogous to vibrations.

With zu and zi as given functions of position and time, we have no diffi­culty in reasoning that the boundary conditions which generalize the steady-state requirement of flow tangency at the surface (cf. 5-5) read as follows:

Фг{х, у, z„, t) = Фх(х, у, zu, t) + Фу{х, у, zu, t) + Фг(ж, у, zi, t) = ^ Фх{х, у, zh t) + ~ Фy{x, y, zi, t) + ^

for (x, y) on S. There is the usual auxiliary condition of vanishing dis­turbances at points remote from the wing and its wake, but for compressible fluid this must be refined to ensure that such disturbances behave like outward-propagating waves. The Kutta-Joukowsky hypothesis of con­tinuous pressure at subsonic trailing edges is also applied, although we should observe that recent evidence (Ransleben and Abramson, 1962) has cast some doubt on its validity for cases of high-frequency oscillation.

Provided that there are no time-dependent variations of profile thick­ness, the upper and lower surface coordinates can be given by

Zu = eju(x, y, t) = т$(х, у) + el{x, y, t) (13-3)

Zi = eji(x, y, t) = —T$(x, y) + eh{x, y, t).

Here e is a dimensionless small parameter measuring the maximum cross­wise extension of the wing, including the space occupied by its unsteady displacement. Angle of attack a (5-1) can be thought of as encompassed by the 0-term; g and h are smooth functions as in steady motion; their x – and y-derivatives are everywhere of order unity; the ^-derivative of h will be discussed below.

Recognizing that in the limit € —» 0 the wing collapses to the x, y-plane and the perturbation vanishes, we shall seek the leading terms by the method of matched asymptotic expansions. Let the inner and outer series be written

Ф0 = TJ«,[x + «Ф?(ж, у, z, t) + • • •], (13-4)

Ф* = Ux{x + еФгх(ж, y, z,t) + ■ ■ •], (13-5)


z = z/e, (13-6)

as in earlier developments. The presence of a uniform stream, which is clearly a solution of (1-74), has already been recognized in the zeroth – order terms in (13-4) and (13-5).

When we insert (13-3) into (13-2), a new question arises as to the size of dju/dt, that is of dh/dt. These derivatives may normally be expected to control the orders of magnitude of the time derivatives of Ф, hence of the terms that must be retained when (13-4)-(13-5) are substituted into (1-74). Here we shall avoid the complexities of this issue by requiring time and space rates of change to be of comparable magnitudes. For example, within the framework of linearized theory a sinusoidal oscillation can be represented by[10]

l{x, y, t) = t{x, y)eiat. (13-7)

The combination of (13-7) with (13-2), followed by a nondimensionaliza – tion of Ф* and Л through division by JJX and typical length l, respectively, produces a term containing the factor

k = (13-8)

Here к is known as the reduced frequency and our present intention is to specify that к = 0(1). For the rich variety of further reductions, even within the linearized framework, that result from other specifications on the magnitudes of к, M, etc., we cite Table I, Chapter 1, of the book by Miles (1959).

With the foregoing limitation on sizes of time derivatives, we find that the development of small-perturbation unsteady flow theory parallels the steps (5-6) through (5-31) quite closely. Thus the condition that vertical velocity W must vanish as « —> 0 shows, as in (5-12), that Ф] must be independent of 2, say

ФІ = Vdx, У, t). (13-9)

By combining (13-3) with (13-2), we conclude that Ф| is the first term to possess a nonzero boundary condition,

The differential equation

= 0 (13-11)

% + l*l dx Ux dt J

requires a solution linear in z; thus

for z > Ju, with a similar form below the lower surface. As in steady- flow, the z-velocities are seen to remain unchanged along vertical lines through the inner field, and it will be shown to serve as a “cushion” that transmits both W and pressure directly from the outer field to the wing.

It is an easy matter to extract the linear terms from (1-74) and derive the first-order outer differential equation

By matching with W derived from (13-12), we obtain indirectly the fol­lowing boundary conditions:

*;.<«, o+,« = § + £#

dfi. 1 dji

(7« dt

Moreover, matching Ф itself identifies with the potential Ф[ at the inner limits z = 0±.

The linear dependence of Ф( on /„ and fi, evident from (13-14), suggests that, in a small-perturbation solution which does not proceed beyond first order in e, we should deal separately with those portions of the flow that are symmetrical and antisymmetrical in г [cf. (5-32) or Sections 7-2 through 7-3]. Indeed, one may even isolate that part of h(x, y, t) from (13-3) that is both antisymmetrical and time-dependent. This we do, realizing that we may afterward superimpose both the thickness and lifting contribu­tions of the steady field, but that neither has any first-order influence on the unsteady loading.

We again adopt a perturbation potential, given by

еФ? = <p(x, y, z, i)

and satisfying, together with proper conditions at infinity, the following system:

for (x, y) on S. Corresponding to the pressure difference, has a discon­tinuity through S. As we shall see below, the Kutta-Joukowsky hypothesis also leads to unsteady discontinuities on the wake surface, which is ap­proximated here by the part of the x, y-plane between the downstream wing-tip extensions.

Finally, a reduction of (1-64) and matching, to order e, of Ф or Фх shows that

Cp = —2<px – JLVI + 0(e2) (13-18)

throughout the entire flow. (From Chapter 6, the reader will be able to reason that the small-perturbation Bernoulli equation again contains non­linear terms when used in connection with unsteady motion of slender bodies rather than wings.)

13- 2 Two-Dimensional, Constant-Density Flow

The best-known of the classical solutions for unsteady loading is the one, found almost simultaneously by five or six authors in the mid-1930’s, for the oscillating thin airfoil at M = 0. In this case of nearly constant density, a key distinction disappears between the steady and unsteady problems because the flow must satisfy a two-dimensional Laplace equation

<Pxx + <Pzz = 0. (13-19)

We may accordingly rely quite heavily on the results for a steadily lifting airfoil from Section 5-3, particularly on (5-58) and (5-73), which supply the needed inversion for the oscillatory integral equation while simul­taneously enforcing Kutta’s condition at the trailing edge.

With the lifting surface paralleling the ж, г/-р1апе between x = 0 and c, it can be assumed from (13-17) that <pz(x, 0, t) is known over that area and given by (dimensionless)

w0(x, t) = ib0(x)e’at for 0 < x < c. (13-20)

The perturbation field has <p and и antisymmetric in z, and allowance must be made for ^-discontinuities through the x, y-plane for x > 0. Hence, (13-19) and all other conditions can be satisfied by a vortex sheet similar to the one described by (5-58) but extended downstream by replacing the upper limit with infinity. Equations (13-17) and (13-20) are intro­duced through

For later convenience, we define an integrated vortex strength

Г(ж, t) = ( У(хі, t) dxі = 2 f 4>x(xi, 0+, t) dxi Jo Jo

= 2ф, 0+, t), (13-23)

and note that Г(с, t) is the instantaneous circulation bound to the airfoil.

From (13-18) and the antisymmetry of Cp in z, we deduce that a (physi­cally impossible) discontinuity of pressure through the wake is avoided only if

<Px + <Pt = 0, (13-24)

for x > c along г = 0+. Equation (13-24) is a partial differential expres­sion for <p(x, 0+, t), which is solved subject to continuity of <p at the trailing edge by

ф, 0+, t) = * (с, 0+, t – ■ (13-25)

From (13-23) and (13-25) are derived the further relations for the wake: Г(х, t) = T(c, t – > (13-26)

У(х, t) = — Vt (c, t – . (13-27)

Equation (13-27) has the obvious interpretation that wake vortex ele­ments are convected downstream approximately at the flight speed U„, after being shed as countervortices from the trailing edge at a rate equal to the variation of bound circulation.


We next introduce (13-27) into (13-21) and use the assumption that a linear, simple harmonic process has been going on indefinitely to replace all dependent variables with sinusoidal counterparts and cancel the com­mon 4 Indicial Motion in a Compressible Fluid

In the analysis of linear systems there exists a well-known duality between phenomena involving simple harmonic response and “indicial” phenomena—situations where an input or boundary undergoes a sudden step or impulsive change. Fourier’s theorem enables problems of one type to be treated in terms of solutions of the other, and this is frequently the

most useful avenue to follow in unsteady wing theory. There are cases, however, when a direct attack on the indicial motion is feasible.

As a particularly simple indicial problem, let us consider the initial development of flow near the upper surface of a wing (e. g., Fig. 7-1) when a step change occurs in the normal velocity of the surface. Such a specifica­tion demands that we reexamine the fundamental development of Section 13-1. Essentially what we are saying is that z„ in (13-3) is given by

= eWo4t)’


(x, y) on S,



1(0 =


t < 0



t > 0

and w0 = Wq/Uv is a constant of order unity. Clearly, in the vicinity of the time origin, there is now some interval where rates of change of flow properties are very large. It is useful to study this zone by defining

* = f (13-71)

and replacing the inner series (13-5) by

Ф* = U*[x + €ФІ(ж, у, г, t) + • • •]. (13-72)

Once again we are led to the conclusion that Ф2 carries the first signifi­cant disturbances, but now its differential equation and upper-surface boundary condition are

= -7 *25 (13-73)



Ф‘2г = Jj~ % = wol(0, at 2 = Ju, for (x, у) on S. (13-74)

(This statement is actually unchanged if w0 depends on x and y.) Equa­tions (13-73)-(13-74) describe the linearized field due to a one-dimen­sional piston moving impulsively into a gas at rest. The solution reads

Фгг = :r = w°l 6 – ’ (13~75)

dec doo/

and it is easily shown that the overpressure on the wing surface (or piston face) is


Pu Poo ———— P«Ctoe дї

All of these solutions are quite independent of flight Mach number M, so long as the disturbance velocities remain small compared with a„.

After a short time interval, the foregoing results make a continuous transition to solutions determined from (13—16)—(13—17). Moreover, for t » c/17x, the indicial solution must settle down to the steady-state result for a wing at angle of attack (—ew0). This behavior can be demon­strated using the method of matched asymptotic expansions, but the details are much too complicated to deserve elaboration here.

Perhaps the most interesting aspect of (13-75) and (13-76) is their general applicability, when M >5> 1, for any small unsteady motion. At high Mach number, fluid particles pass the wing surface so rapidly that all of the disturbed fluid near this surface remains both in the inner z – and f-fields; except for large values of x far behind the trailing edge the outer field experiences no disturbance at all. Hence the piston formula, (13-76), yields for any instant the pressure distribution over the entire wing, a result which can also be extended into the nonlinear range (Lighthill, 1953).

Singular Perturbatio Problems

3- 1 Introduction

The treatment given in the preceding chapter is of fairly limited practical use for engineering problems. Constant-density inviscid and irrotational flow is there rather considered as a physical model for subsonic flows in general from which interesting qualitative information can be extracted but not always accurate quantitative results. Thus, despite d’Alembert’s paradox, the drag in a two-dimensional flow is certainly not zero, but the proper interpretation of the theoretical result is that, in a steady (attached) flow, drag forces are generally much smaller than either lift or forces due to unsteady motion. The practical conclusion one can draw from the inviscid model is that flow separation should be avoided at all costs. This calls for rather blunt-nosed shapes with no abrupt slope or curvature changes or protuberances, and with gently sloping rear portions, i. e., “streamlined ” bodies.

Apart from the drag which is dominated by viscosity—the very thing that was neglected in the simplified model—the constant-density theory is able, in many practical cases, to produce remarkably good approxima­tions to pressure distributions for speeds less than, say, half the speed of sound. Unfortunately, the calculation of inviscid flow for shapes of en­gineering interest is usually so difficult that one is forced to make some further approximation in order to obtain a result. For all the simple shapes considered in Chapter 2 (with the exception of certain airfoils and the ellipsoids with large fineness ratio) the nonviscous solutions happen to be almost completely useless, since in reality the flow will separate and the simplified model then loses its validity. The example emphasizes the fact that extreme caution must be exercised when using the physical model to obtain approximate engineering results rather than just to gain a general qualitative understanding of the physical situation and the mathematical structure of the problem. Considerable insight is generally required to judge when a simplified model will provide a useful first approximation to an actual physical situation.

The discussion may suffice to emphasize the basic difference between physical models and approximate solutions; for the former, one seeks an exact solution to a simplified and very often an artificial problem whereas,

loosely speaking, in the latter, one seeks a simplified solution to a real problem. The distinction should always be kept in mind although it is not always clear-cut. For example, the continuum model of gas is also a very good approximation for all the problems that we will consider.

There are a variety of methods to obtain useful approximations. We shall discuss two different methods which are the ones mostly used in aerodynamic problems. One is the expansion in powers of a small parameter. Often the first term in this expansion may by itself be con­sidered as a physical model. Some of the expansions that will be discussed and the physical models derived in this manner are illustrated in Fig. 3-1.

A great majority of problems in fluid mechanics have been successfully attacked by series-expansion methods. The series obtained are usually only semiconvergent, i. e., asymptotic, and also frequently not uniformly valid. These features, which are closely associated with the so-called singu­lar perturbation nature of such problems, will be thoroughly discussed in the following section.


Fig. 3-і. Use of series expansions for obtaining approximate solutions to aero­dynamic problems.

The second method of approximation that will be considered is the purely numerical one. In the future this will undoubtedly become of increasing importance as the full potentialities of modem computing devices are realized among aerodynamicists. There are many difficulties associated with numerical solutions. First of all, the equations for fluid flow are so complicated that no one has as yet succeeded in a step-by-step integra­tion of the full gas dynamic equations even assuming a perfect, inviscid gas. Therefore, the examples given will concern the numerical solution of linearized problems. Second, it is very hard to estimate the error induced by the approximation scheme employed. In contrast, this could in principle always be done in the analytical series solution by estimating the first neglected term. For this purpose, and for checking out the computational program, the analytical solutions for the limiting cases are extremely useful. Thus, far from making them obsolete, the new possi­bilities for numerical solutions give the analytical solutions extended practical usefulness.

Examples of Lifting Slender-Body Flow

A particularly fortunate consequence of the equivalence rule is that the outer flow is needed only for the calculation of g(x), which is a symmetric term that only influences the drag but not transverse forces and moments. For the calculation of lifting flows one therefore seeks the inner constant – density two-dimensional crossflow which is independent of the Mach number and which may be obtained, for example, by using complex variables. Some of the results of the classical two-dimensional theory may be directly applied. Thus, the flow around a circular cylinder applies to the lifting slender body of revolution, and the solution for a flat plate normal to the stream can be used for the flow around a thin, slender wing. A simple, yet practically useful configuration that incorporates these as special cases is that of a mid-winged body of revolution (see Fig. 6-3).


Fig. 6-3. Wing-body configuration.



To determine the perturbation velocity potential (</> = е2Фг2 in this case) it is, in this problem, convenient to align the x-axis with the body axis and let the free-stream vector be inclined by the angle a to the x-axis. That we then consider the flow in cross sections normal to the body axis instead of normal to the free stream will only introduce differences of order a2, which will be negligible in the present approximation; they will only be of importance for the calculation of higher-order terms. The prob­lem becomes that of finding a two-dimensional constant-density flow having a nondimensional vertical velocity of


= sin a ~ a (6-57)

U со

at infinity and zero normal velocity component at the body contour. Let the velocity potential corresponding to this flow be <p’:

tp’ = ip + az. (6-58)

We may obtain <p’ from a complex potential:

W'(X) = V'(y, z) + гф'(у, z), (6-59)


X = у + iz. (6-60)

The complex potential W’ will be constructed in steps using conformal transformation. First, the Joukowsky transformation

Хг = X + R2/X (6-61)

maps the outside of the contour onto the outside of a slit along the у i-axis (see Fig. 6-4) of width 2s i, where



Fig. 6-4. Mapping of wing-body cross section onto a slit.


Corresponding points are marked in the figure. A second transformation

X2 = (Xf – sf)112 (6-63)

transforms the horizontal slit to a vertical slit of width 2s i. Since in both transformations the plane remains undistorted at infinity, the flow in the X2-plane is simply

V?'(X2) = – iaX2, (6-64)

that is, an undisturbed vertical flow of (nondimensional) velocity a. By substituting (6-61)-(6-63) into (6-64) we obtain

Подпись:Подпись: 1/2Подпись: (6-66)image92"V?'(X)

Thus for the complex velocity perturbation potential,

W(X) = V?'(X) + iaX = – ia

This solution was given by Spreiter (1950). It is a straightforward process to derive from it the cases of body alone (R — s) and wing along (R = 0).

It is interesting to note that the crossflow considered above has no physical significance in a truly two-dimensional case, since then the flow will separate and the flow becomes rotational and nonpotential. In the slender-body case the axial flow keeps the crossflow from separating so that the potential-flow solution gives a realistic result. However, for large angles of attack the flow will separate, particularly when the aspect ratio is very small. The type of flow that then will be encountered is illustrated in Fig. 6-5. The flow separates at the leading edges and forms two station­ary, more-or-less concentrated vortices above the wing. Separation gives rise to an additional “drag” in the crossflow plane, which is equivalent to


Fig. 6-5. Leading-edge separation on | j

image94a slender wing. T 1

increased lift and drag on the vehicle. Simplified models of this type of flow have been considered by, among others, Bollay (1937), Legendre (1952), and Mangier and Smith (1956).

The calculation of total lift and moments on slender bodies will be considered in Section 6-7.