Category Aerodynamics of Wings and Bodies

The pressure drag acting on a body in supersonic flow can be thought of as composed of two parts, the wave drag and the vortex drag (see further Chapter 9). If the body has a blunt base, there is, in addition, a base drag. The wave drag results from the momentum carried away by the pressure waves set up by the body as it travels at a speed greater than the speed of sound. In a subsonic flow there is, of course, no wave drag, since no stand­ing pressure waves are possible. The vortex drag arises from the momentum carried away by the vortices trailing from a lifting body and is governed by the same relations in both supersonic and subsonic flow.

The pressure drag of a slender body in a supersonic flow is most easily calculated by considering the flow of momentum through a control surface surrounding the body. We shall here follow essentially the approach taken by Ward (1949), which gives the total pressure drag but does not specify how the drag is split up into wave drag and vortex drag.

In Section 1-6 it was shown that, by considering the flow of momentum through a control surface S surrounding the body, the force on the body is given by

(1-51)

where n is the outward unit normal to the surface S, and Q is the velocity vector. It is convenient to introduce into (1-51) the perturbation velocity U„q = Q — it/*,, which gives

s

This may be simplified somewhat by use of the equation of continuity (cf. 1-45)

<£j>p(q + i) • n dS = 0. (6-68)

s

Hence, since S is a closed surface,

Fbody = —<0>l(p — Poo)n + pUlq(q « n + і * n)] dS, (6-69) s

which is the form we are going to use.

It is convenient in the present case to choose S in the manner shown in Fig. 6-6. Thus the surface S consists of three parts: Sx, S2, and S3, of which Si and S3 are circular disks and S2 a cylinder parallel to the main flow whose radius Rx will be chosen so that S2 is at the outer limit of the inner region. It is assumed that Sx lies ahead of the body so that the flow is undisturbed there, and S3 is located at the base section of the body. The body may have a blunt base, but the linearized theory is, of course, not valid for the calculation of the pressure on the base. We assume that the base pressure рв is known, so that the base drag contribution

Db = (p® — Pb)Sb (6-70)

to the total drag is given. In supersonic flow the effect of the blunt base will not be felt upstream of the base section. Hence the linearized theory may be used to calculate the flow ahead of the base section and thus the pressure drag on the remainder of the body.

Since S is located in the inner region, the flow is essentially incompres­sible so that, on S, p may be considered constant and equal to its free – stream value. The term q ■ n + і • n in (6-69) is simply the nondimensional velocity component normal to S. Thus, by taking the ж-component of (6-69) we obtain

This formula shows how the drag varies with Mach number. If S'(l) = 0, that is, if the body ends in a point or with a cylindrical portion, the drag becomes independent of the Mach number. Of particular interest is the drag of the equivalent body of revolution, for which

This is the equivalence rule for the pressure drag. It is a fairly easy matter to show that it must hold for all speed regimes whenever the equivalence rule for the flow is valid. Thus it will also hold for transonic speeds, and, as will be explained in Chapter 12, with less restrictions than for sub – or supersonic speeds.

In many cases the right-hand side of (6-87) is zero and hence the drag equal to that of the equivalent body of revolution. This occurs whenever:

(a) The body ends with an axisymmetric portion so that the two parts in (6-87) cancel.

(b) The body ends in a point.

(c) The body ends in a cylindrical portion parallel to the free stream so that dip/dn and S’ are zero.

Most practical slender missile or airplane configurations satisfy (a) or

(c) . For such a body one can thus experimentally test the validity of the equivalence rule simply by comparing the pressure drag with that of the equivalent body of revolution. Such measurements were made by Whitcomb (1956). Some of his results are reproduced in Fig. 6-7.

The agreement is good in the transonic region when the viscous drag has been separated out. From these results Whitcomb drew the conclusion that it should be possible to reduce the drag of a slender wing-body com­bination by indenting the body so that the equivalent body of revolution would have a smooth area distribution. This is the well-known transonic area rule, which has been used successfully to design low-drag configura­tions for transonic airplanes. The savings in drag that can be achieved are demonstrated in Fig. 6-7 (b).

6-7 Transverse Forces and Moments on a Slender Body

The transverse forces and moments (lift, side force, pitching moment, etc.) on a slender body can be obtained by considering the flow of mo­mentum through a control surface surrounding the body, as in the preced­ing section. However, we shall instead use a different method that makes use of results previously deduced for unsteady constant-density flow.

Let x, yi, Zi be a coordinate system fixed with the fluid so that

xi = x — UJ, yi = у, z = г. (6-88)

An observer in this coordinate system will see the body moving past with a velocity in the negative aq-direction. Consider now the fluid motion in a slab of width dxi perpendicular to the free stream as the body moves past. The crossflow in the neighborhood of the body will be governed by the equations for two-dimensional constant-density flow in the crossflow plane, but the flow will now be unsteady since consecutive cross sections of the body pass through the slab as the body travels by. The incompres­sible crossflow in the slab will thus be that around a two-dimensional body that changes shape and translates with time (and also rotates if rolling of the slender body is considered). This situation is illustrated in Fig. 6-8.

Since the flow has no circulation in the crossflow plane we can directly apply the methods developed in Section 2-4. [The Blasius’ equation (2-122) for unsteady flow is not applicable since it was derived under the assumption that the cross section does not change with time.] Thus, if £ is the crossflow momentum vector per unit body length, the force acting on the body cross section of width dx is, according to (2-61),

dF = jY + kL = – dxx ^ , (6-89)

where L is the lift and Y is the side force. The momentum vector £ is given by (2-54). Thus, in two-dimensional flow,

£ = —y>ia. ds,

where ipi is ip expressed in (x1: ylt zlt t) and the integral is to be taken around the (instantaneous) body cross section. The factor £/„ comes from the defini­tion of <p. It follows from Fig. 6-9 that

n ds = (j cos в + к sin в) ds

= j<fei — kdyi. (6-91)

Hence

£ = —f A^i(zi) dzi

— pxUJs. /Ду 1(2/1) dyu (6-92)

where Д<рі(уі) is the difference in <p between the upper and lower surfaces of the cross section and t<pi(zi) is the difference in <pi between the right and left surfaces of the cross section. Introducing (6-92) into (6-89) we thus obtain

By integrating over the body length, total lift, pitching moment, etc. can then be obtained. Particularly simple are the expressions for the total forces, which become

(6-97)

(6-98) where С в indicates that the integral is to be evaluated at the base cross

section. In order to calculate total forces, one thus only needs the cross flow at the base. Frequently, the flow is given in complex variables, in which case it is convenient to work with the complex force combination

X = у + iz.

[This formula could, of course, also have been obtained by introducing the complex vector directly in (6-92).] The idea is then to introduce the complex potential W(X) = <p + іф, which would reduce the problem to that of evaluating a closed-contour complex integral. However, a direct replacement of <p by W will generally lead to an incorrect result unless the stream function ф happens to be zero, or constant, along the cross section contour. We therefore introduce, as in (6-59), the potential W’ for the related flow having zero normal velocity at the contour and velocity components at infinity proportional to the side-slip angle and angle of attack, respectively, at the base section. Thus

<p = Re {W'(X)} – aBz – РвУ, (6-100)

where а в and fis are the angle of attack and side-slip angle at the base (which would be different, from the overall angle of attack and side slip if the body were cambered). Now ф’ is zero along the contour, and we may therefore set

F = —ip„UxWV?’ dX — iPxUxSb(oib — іРв), (6-101)

where Sb is the base area. The last term follows from simple geometrical considerations that give, for example, that

In the first integral of (6-101) we may choose any path of integration that encloses the base contour, the most convenient one being a large circle at infinity. Assuming that W’ may be expressed by the following Laurent series for large |X|

w’ = «0x+x;^f, (6-юз)

and a_! being the residue at infinity, we obtain from (6-101)

and inserting the residue as given by the second term of (6-105), the fol­lowing result is obtained

L = irpaUla(s2 – R2 + R*/s2)B, (6-106)

where index В refers to the base section. (The side force is, of course, zero in this case.) This result contains as special cases those of the wing alone and body alone. In the latter case, setting sB = Rb in (6-106) we obtain

L = wpooU2R2Ba, (6-107)

that is, the lift coefficient based on the base area is simply

CL = 2a,

a result first derived by Munk (1924). An interesting conclusion from this is that on a body pointed at the rear no lift is exerted, only a pitching

moment. This is destabilizing, tending to increase the angle of attack (cf. ellipsoid example in Chapter 2 and Fig. 2-5). In reality, viscous effects will cause a small positive lift.

For the case of a wing alone (Rb = 0), (6-106) gives

L = 7гроо U&s%oc. (6-109)

Hence, any slender wing with a straight, unswept trailing edge will have a lift coefficient (R. T. Jones, 1946)

CL = ?Aa. (6-110)

Comparisons for delta wings with experiments and a numerical lifting – surface theory, presented in Fig. 6-10, show that this simple formula overestimates the lift by 10% and more for A > 1.0.

It should be pointed out that (6-106) and (6-109) hold only for wings having monotonically increasing span from the pointed apex to the base section, otherwise sections forward of the base section will produce a wake that will influence the flow at the base, so that it no longer becomes inde­pendent of the flow in other cross sections. For the case of an uncambered wing with swept-forward trailing edges, one can easily show that the lift on sections behind that of the maximum span is zero in the slender – wing approximation, and hence (6-109) will hold if Sb is replaced by Smai, the maximum semispan. In the case of swept-back trailing edges, as for an arrowhead or swallowtail wing, Mangier (1955) has shown that the determination of the flow requires the solution of an integral equation.

A practically useful formula to estimate the effect of a fuselage on total lift is obtained by dividing (6-106) by (6-109). Thus

^™g±body = 1 _ (RbY + (RbY t (6_ln)

which shows that the body interference tends to decrease the lift.

For later use in connection with the calculation of forces and moments on wings and bodies, we wish to be able to express these quantities in terms of the fluxes of linear and angular momentum through arbitrary control surfaces S. This approach will often be found to have a special convenience, because singularities which occur in velocities and pressures at the surface of a vehicle may not persist at great distances in the flow field around it, so that the integrations which must be carried out are facilitated. The basic tools for carrying out this task are irftegrated forms of Newton’s law of motion known as momentum theorems.

In connection with the presentation of the momentum theorems, we take the opportunity to discuss the question of conservation of other flow properties, as expressed in integral form.

Consider any quantity E which is characteristic of the fluid particles contained within a fixed control volume V. Let V be bounded on the inside by one or more impermeable bodies, whose collective surfaces are denoted by the symbol a, and bounded on the outside by a larger fixed surface S. See Fig. 1-5. The closed system under examination is that fixed mass of fluid that happens to be contained within V at a certain instant of time. To find the rate of change of the total quantity for this system at the instant it coincides with V, we observe that this change is made up of the sum of all the local changes at points within V plus changes which occur as a result of the motion of the system boundary. Supposing that E is referred to unit volume of the fluid, the former rate of change can be

where the interchange of the operations of differentiation and integration is permissible in view of the constancy of the volume V. There is also an increment to the total amount of the quantity E as a result of the fact that the fluid is moving across the bounding surfaces S and a with a normal velocity component (Q • n). At points where this scalar product is positive, the fluid adjacent to an area element dS of the boundary takes up new positions outside this boundary, the volume per unit time passing outside the boundary being given by (Q • n) dS. Thus the rate of change of E for the system due to passage of fluid across the boundary is given by

<$E(Q ■ n) dS.

S+<r

Combining these last two results, we find for the total rate of change of this generalized property for the system,

= fffd-~dV + <£f>E(Q-n)dS. (1-44)

v s+o-

This general result is now specialized to several cases of interest.

1. Conservation of Mass or Continuity. To derive an integral continuity equation we replace E by the mass per unit volume p and observe that, in the absence of sources and sinks, the total mass of the system must remain constant. Thus we are led to

/// Ы dV + ^P(Q ‘ n) dS = °- (b45)

V S+cr

For steady flow around an impermeable body of fixed position, of course, the first integral in (1-45) vanishes, and the contribution to the second integral from the inner surface must be zero because the quantity in parentheses vanishes.

We note incidentally how from Gauss’ theorem, (1-14),

jfip(Q-n)dS = IJjv-(pQ)dV. (1-46)

S+tr V

Substituting into (1-45), we obtain

Since the volume V is arbitrary, the only way that this integral can vanish is for its integrand to be everywhere zero. Thus, the differential form of the continuity equation, (1-1), is confirmed.

2. Linear Momentum. Let F,- represent the vector sum of all forces applied by the surroundings to the system. According to Newton’s second law, this sum is equal to the time rate of change of linear momentum of the system, which corresponds to replacing E with the quantity pQ in (1—44). Consequently, we obtain the following generalized version of the law of conservation of momentum:

+ <ffpQ(Q ■ n) (IS. (1-48)

* V S+a

We now examine the various contributions that might appear to the force system in (1-48). If there is a conservative body force field, the left- hand side will include a quantity

jfjpFdV = JJJpVQ dV. (1-49)

v v

This will be omitted from what follows because of its relative unimportance in aeronautical applications.

The remaining external force will then be broken into two parts: the reaction (—Fbody) to the force exerted by the fluid on the body, and the force exerted across the outer boundary S by the surroundings. Recalling that n is the outward-directed normal, this latter might take the form

ffi—pn + r] dS,

5

where p is the pressure across S, and r is the sum of shear stress and devi – atoric normal stress exerted by the surroundings, if these are significant. We may write r as the dot product of a dyadic or tensor of deviatoric stress by the unit normal n. Since we are dealing generally with a non­viscous fluid, however, the question will not be elaborated here. Leaving out effects of shear stress, (1-48) can be modified to read

Again we remark that if the body is fixed in our coordinate system, the contribution to the second integral on the right from a will vanish. Also in steady flow the last term on the right is zero, leaving

Fbody = — <jij>lpn + pQ(Q ■ n)] dS. (1-51)

s

Equation (1-51) is actually the most useful form for practical applications. The specialized versions of (1-51) which occur when the flow involves
small perturbations will be discussed in a later chapter. It usually proves convenient to use an integrated form of the equations of motion (Ber­noulli’s equation) to replace the pressure in terms of the velocity field.

3. Angular Momentum. Let r be a vector of position measured from the origin about which moments are to be taken. Then it is an easy matter to derive the following counterpart of the first form of the linear momentum theorem:

Er*x F* = /// Jt(pr XQ)# + ^p(r X QKQ • n) dS- (1-52)

* V S+<7

The summation of moments on the left here can once more be broken up into a body-force term, a reaction to the moment exerted by the fluid through a and a pressure or shear moment exerted on the system over the outer boundary. Substitutions of this sort, neglecting the deviatoric stress, lead to the working form of the theorem of angular momentum:

Afbody = — ^ p(r X n) dS

s

S+a V

The steady-flow simplification involves dropping the integral over V and over the inner boundary a.

4. Thermodynamic Energy. Integral forms of the laws of thermo­dynamics will be found developed in detail in Chapter 2 of Shapiro (1953). Since these will have little direct usefulness in later applications and since many new definitions are involved, none of these results are reproduced here. Shapiro’s equation (2.20), for instance, provides an excellent working form of the first law. It is of interest that, when the pressure work exerted on the boundaries is included, the quantity E in the second or boundary term on the right of (1-44) is found to be

Here z is the distance vertically upward in a parallel gravity field, and h is the enthalpy per unit mass, which proves to be the effective thermo­dynamic energy in steady flow.

As a sidelight on the question of energy conservation we note that, in a constant-density fluid without body forces, the only way that energy can be stored is in kinetic form. Hence, a very convenient procedure for calculating drag, or fluid resistance, is to find the rate of addition of kinetic
energy to the fluid per unit time and to equate this to the work done by the drag. This represents a balance of mechanical rather than thermo­dynamic energy. When the fluid is compressible and there are still no dissipative mechanisms present, energy can be radiated away by compres­sion waves in an acoustic fashion. Therefore, the problem of computing drag from energy balance becomes a good deal more complicated.

3- 1 Introduction

Viscous flows at high Reynolds numbers constitute the most obvious example of singular perturbation problems. The viscosity multiplies the highest-order derivative terms in the Navier-Stokes equations, and these will therefore in the limit of zero viscosity degenerate to a lower order. The boundary condition of zero tangential velocity on the body (or of continuous velocity in the stream) is therefore lost. This necessitates the introduction of a thin boundary layer next to the body constituting the inner region where the inviscid equations are not uniformly valid. Unfor­tunately, only a very restricted class of viscous-flow problems can be analyzed by the direct use of the method of matched asymptotic expansions. First, only for a very limited class of bodies will the boundary layer remain attached to the body surface. When separation occurs, the location of the region where viscosity is important is no longer known a priori. The second difficulty is that for very high Reynolds numbers the flow in the boundary layer becomes unstable and transition to turbulence occurs. As yet, no complete theory for predicting turbulent flow exists.

Before considering some of the model problems that may be analyzed, we will give a short description of the qualitative effects of viscosity.

7- 1 Compressibility Corrections for Wings

This chapter deals with the application to finite, almost-plane wings of the linearized, small-perturbation techniques introduced in Chapter 5. By way of introduction, we first review the similarity relations which govern variations in the parameter M, the flight Mach number.

In the light of the asymptotic expansion procedure, the principal un­known, from which all other needed information can be calculated, is the first-order term* Ф? in the outer expansion for the velocity potential. The term Ф)’ is connected to the more familiar perturbation velocity potential <p(x, y, z) by (5-28). The latter is governed by the differential equations and boundary conditions (5-29)-(5-30), which we reproduce here (see also Fig. 5-1):

(1 M )<Pxx “b <Pyy “f" <Pzz ——– 0,

for (x, y) on S. (5-30)

The pressure coefficient at any point in the field, including the upper and lower wing surfaces z = 0±, is found from

Cp — 2 <px.

Extending a procedure devised by Prandtl and Glauert for two-dimen­sional airfoils (see Fig. 7-1), Gothert (1940) introduced a transformation of independent and dependent variables which is equivalent to

* The zeroth-order term is, of course, the free stream Фо = x.

124

Where /3 = /l — M2, as in Chapter 5. Equation (7-1) converts (5-29) into the constant-density perturbation equation

(<Po)x0x0 + (‘Po)y0Vo + (<Po)z0zo = 0. (7-2)

Some care must be observed when interpreting the transformed boundary condition at the wing surface. Thus, for example, the first of (5-30) states that just above the wing’s projection on the x, у-plane the vertical velocity component produced by the sheet of singularities representing the wing’s disturbance must have certain values, say Fu(x, y). After transformation, we obtain

(<Po)z0 = Fu(0x о, у о) s FUo(x0, Vo)

at z0 = 0—f—, for (x0, y0) on »S’o, where <S0 is an area of the x0, уо-ріапе whose lateral dimensions are the same as the original planform projection S, but which is stretched chordwise by a factor l/13. (See Fig. 7-2.)

Equation (7-3) and the equiva­lent form for the lower surface state, however, that the “equivalent ” wing in zero-M, constant-density flow has (at corresponding stations) the same thickness ratio t, fractional camber

в, and angle of attack a as the origi – ——–

nal wing in the compressible stream. _____

Fig. 7-2. Equivalent wing planform in zero-Mach-number flow. If sweep is present, tanAo = (1 //3) tan A. The aspect ratio is А о = /ЗА.

The similarity law might be abbreviated

where the semicolon is used to separate the independent variables from the parameters.

By way of physical explanation,[6] Gothert’s extended Prandtl-Glauert law states that to every subsonic, compressible flow over a thin wing there exists an equivalent flow of constant density liquid (at the same flight speed and free-stream ambient conditions) over a second wing, obtained from the first by a chordwise stretching 1//3 without change of surface slope distribution. It is obvious from (5-31) that pressure coefficients at corresponding points in the two flows are related by

(7-5)

Since they are all calculated from similar dimensionless chordwise and spanwise integrations of the Cp-distribution, quantities like the sectional lift and moment coefficients Ci(y), Cm{y), the total lift and moment co­efficients Cl, Cm, and the lift-curve slope дСь/да are found from their constant-density counterparts by the same factor 1//3 as in (7-5). It is of interest in connection with spanwise load distribution, however, that the total lift forces and running lifts per unit //-distance are equal on the two wings, because of the increased chordwise dimensions at M = 0.

Unfortunately, when one is treating a given three-dimensional configura­tion, the foregoing transformation requires that a different planform be analyzed (or a different low-speed model be tested) for each flight Mach number at which loading data are needed. This is not true for two-dimen­sional airfoils, since then the chordwise distortion at fixed a, etc., is no more than a change of scale on an otherwise identical profile; we have already seen (Section 1-4) that such a change has no effect on the physical flow quantities at fixed M.

Measurements like those of Feldman (1948) correlate with the Gothert – Prandtl-Glauert law rather well up to the vicinity of critical Mach number, where sonic flow first appears at the wing surface. They also verify what we shall see later theoretically, that the coefficient of induced drag should be unaffected by M-changes below Afcrit. There exist, of course, more accurate compressibility corrections based on nonlinear considerations which are successful up to somewhat higher subsonic M.

Inasmuch as (5-29) applies also to small-perturbation supersonic flow, M > 1, one might suspect that the foregoing considerations could be extended directly into that range. This is an oversimplification, however, since the boundary conditions at infinity undergo an essential change— disturbances are not permitted to proceed upstream but may propagate only downstream and laterally in the manner of an outward-going sound wave. (The behavior is connected with a mathematical alteration in the nature of the partial differential equation, from elliptical to hyperbolic or “wavelike.”) What one does discover is the existence of a convenient reference Mach number, M = /2, which plays a role similar to M = 0 in the subsonic case. When M = /2, the quantity В = y/M2 – 1 becomes unity and all flow Mach lines are inclined at 45° to the flight direction. Repetition of the previous reasoning leads to a supersonic similarity law

<p(x, V, 2; M, А, г, 0, a) = <P (jj ‘ V, 2; M’ = л/2, BA, T, 0, a) . (7-6)

Pressure coefficients at corresponding points, lift coefficients, etc., are related by

CP = (Cp)*_* (7-7)

Once more the equivalent planform at M = /2 is obtained from the original by chordwise distortion, but now this involves a stretching if the original M < /2 and a shrinking if M > y/2. The process has been likened to taking hold of all Mach lines and rotating them to 45°, while chordwise dimensions vary in affine proportion.

Clearly, Eqs. (7-5), (7-7), and the associated transformation techniques fail in the transonic range where M ^ 1. It has been speculated, because the equivalent aspect ratio approaches zero as M —> 1 and slender-body – theory results for lift are independent of Mach number (Chapter 6), that linearized results for three-dimensional wings might be extended into this range. This is, unfortunately, an oversimplification. Starting from the proper, nonlinear formulation of transonic small-disturbance theory, Chapter 12 derives the actual circumstances under which linearization is permissible and gives various similarity rules. It is found, for instance, that loading may be estimated on a linearized basis whenever the param­eter At1/3 is small compared to unity.

Enough has been said about the subject of vorticity, its conservation and generation, that it should be obvious that an initially irrotational, uniform, inviscid flow will remain irrotational in the absence of heat transfer and of strong curved shocks. One important consequence of per­manent irrotationality is the existence of a velocity potential. That is, the equation

f = VxQ = 0 (1-55)

is a necessary and sufficient condition for the existence of a potential Ф such that

Q = V<f>, (1-56)

where Ф(х, у, г, t) or Ф(г, t) is the potential for the velocity in the entire flow. Its existence permits the replacement of a three-component vector by a single scalar as the principle dependent variable or unknown in theo­retical investigations.

Given the existence of Ф, we proceed to derive two important conse­quences, which will be used repeatedly throughout the work which follows.

1. The Bernoulli Equation for Irrotational Flow (Kelvin’s Equation).

This integral of the equations of fluid motion is derived by combining (1-3) and (1-28), and assuming a distant acting force potential:

The vanishing of the gradient implies that, at most, the quantity involved will be a function of time throughout the entire field. Hence the least restricted form of this Bernoulli equation is

f + T + ly-11-™- <1Hi0)

In all generality, the undetermined time function here can be eliminated by replacing Ф with

Ф’ = Ф — jF(t) dt. (1-61)

This artificiality is usually unnecessary, however, because conditions are commonly known for all time at some reference point in the flow. For instance, suppose there is a uniform stream I7„ at remote points. There Ф will be constant and the pressure may be set equal to p„ and the force potential to SI» at some reference level.

F(t) = + j ~ — Q, * = const. (1-62)

The simplified version of (1-60) reads

ҐР

~ + m2 – Vl]+ ^ + [0. – Q] = 0. (1-63)

dt JPtx p

In isentropic flow with constant specific heat ratio 7, (1-63) is easily reorganized into a formula for the local pressure coefficient

Here a is the speed of sound and M = Ux/ax is Mach number. For certain other purposes, it is convenient to recognize that

(Note that, here and below, the particular form chosen for the barotropic relation is isentropic. Under this restriction, dp/dp, (dp/Sp),, and a2 all have the same meaning.)

This substitution in (1-63) provides a convenient means of computing the local value of a or of the absolute temperature T,

a2 – al = -(7 – 1) + i(Q2 – ul) + (Q. – Q)j • (1-67)

Finally, we remind the reader that the term containing the body-force potential is usually negligible in aeronautics.

2. The Partial Differential Equation for Ф. By substituting for p and Q in the equation of continuity, the differential equation satisfied by the velocity potential can be developed:

lf + V. Q_0. (Ь68)

The second term here is written directly in terms of Ф, as follows:

V ■ Q = V • (УФ) = V4,

which we identify as the familiar Laplacian operator,

To modify the first term of (1-68), take the form of Bernoulli’s equation appropriate to uniform conditions at infinity, for which, of course, a special case would be that of fluid at rest, U«, = 0. The body-force term is dropped for convenience, leaving

[ ~ ~ – UQ2 – ul). (1-70)

j Poo P 01

By the Leibnitz rule for differentiation of a definite integral,

d fP dp __ 1 dP h* P ~ P

We then apply the substantial derivative operator to (1-71) and make use of the first three members of (1-66),

ГР г – ГР ■

dp _ d I dp

Jpc p ~ Up Jp« p.

In view of (1-67) and of the simple relationship between the velocity vector and Ф, this is essentially the desired differential equation. If it is multiplied through by a2, one sees that it is of third degree in the unknown dependent variable and its derivatives. It reduces to an ordinary wave equation in a situation where the speed of sound does not vary signifi­cantly from its ambient values, and where the squares of the velocity components can be neglected by comparison with a2.

It is of interest that Garrick (1957) has pointed out that (1-74) can be reorganized into

v2*=Ml+о -v) (f+«• – v#) – h Ш *■ (i-re>

where the subscript conQc and on the substantial derivative is intended to indicate that this velocity is treated as a constant during the second application of the operators d/dt and (Q • V). Equation (1-75) is just a wave equation (with the propagation speed equal to the local value of a) when the process is observed relative to a coordinate system moving at the local fluid velocity Q.

The question of boundary conditions and the specialization of (1-74) and (1-75) for small-disturbance flows will be deferred to the point where the subject of linearized theory is first taken up.

1-8 The Acceleration Potential

It is of interest that when the equations of fluid motion can be simplified to the form

In a manner paralleling the treatment of irrotational flow, we can conclude that

a = V¥,

where ¥ (r, t) is a scalar function called the acceleration potential. Clearly,

¥ = П – ^ + G(t),

J P

G(t) being a function of time that is usually nonessential.

The acceleration potential becomes practically useful when disturbances are small, so that

In the absence of significant body forces, we then have

– V

poo

This differs only by a constant from the local pressure, and doublets of ¥ prove a very useful tool for representing lifting surfaces. The authors have not been able to construct a suitable partial differential equation for the acceleration potential in the general case, but it satisfies the same equation as the disturbance velocity potential in linearized theory.

It is a common feature of most flows of engineering interest that the viscosity of the fluid is extremely small. The Reynolds number, Re = TJJ,/v, which gives an overall measure of the ratio of inertia forces to viscous forces, is in typical aeronautical applications of the order 10® or more. For large ships, Reynolds numbers of the order 109 are common. Viscosity can then only produce significant forces in regions of extremely high shear, i. e., in extremely thin shear layers where there is a substantial variation of velocity across the streamlines. The thickness of the laminar boundary layer on a flat plate of length l is approximately ~ o///Re. For Re = 108, 8/1 ~ 0.005, which is so thin that it cannot even be illustrated in a figure without expanding the scale normal to the plate. A turbulent boundary layer has a considerably greater thickness, reflecting its higher drag and therefore larger momentum loss; an approximate formula given in Schlichting (1960), p. 38, is 8/1 ~ 0.37/Re1/s. For

71

Re = 10® this gives 6/1 = 0.023, which is still rather small. Viscosity is only important in a very small portion of the turbulent boundary layer next to the surface, in the “viscous sublayer. ”

The transition from a laminar to a turbulent boundary layer is a very complicated process that depends on so many factors that precise figures for transition Reynolds numbers cannot be given. For a flat plate in a very quiet free stream (i. e., one having a rms turbulent velocity fluctua­tion intensity of 0.1% or less) transition occurs at approximately a distance from the leading edge corresponding to Re ~ З X 10®. With a turbulent intensity of only 0.3% in the oncoming free stream the transition Reynolds number decreases to about 1.5 X 10®. These distances are far beyond that for which the boundary layer first goes unstable. Stability calculations show that on a flat plate this occurs at Re ~ 10s. The complicated series of events between the point where instability first sets in until transition occurs has only recently been clarified (see Klebanoff, Tidstrom, and Sargent, 1962). Transition is strongly influenced by the pressure gradient in the flow; a negative (“favorable”) pressure gradient tends to delay it and a positive (“adverse ”) one tends to make it occur sooner. As a practical rule of thumb one can state that the laminar boundary layer can only be maintained up to the point of minimum pressure on the airfoil. On a so-called laminar-flow airfoil one therefore places this point as far back on the airfoil as possible in order to try to achieve as large a laminar region as possible. Laminar-flow airfoils work successfully for moderately high Reynolds numbers (<107) and low lift coefficients but require ex­tremely smooth surfaces in order to avoid premature transition. At very high Reynolds numbers transition starts occurring in the region of favor­able pressure gradient.

Both laminar and turbulent boundary layers will separate if they have to go through extensive regions of adverse pressure gradients. Separation will always occur for a subsonic flow at sharp corners, because there the pressure gradient would become infinite in the absence of a boundary layer. Typical examples of unseparated and separated flows are shown in Fig. 4-1. In the separated flow there will always be a turbulent wake

behind the body. In principle one could have instead a region of fairly quiescent flow in the wake, separated from the outer flow by a thin laminar shear layer attached to the laminar boundary layer on a body. However, a free shear layer, lacking the restraining effect of a wall, will be highly unstable and will therefore turn turbulent almost immediately. Because of the momentum loss due to the turbulent mixing in the wake the drag of the body will be quite large. On a thin airfoil at a small angle of attack the boundary layer will separate at the sharp trailing edge but there will be a very small wake so that a good model for the flow is the attached flow with the Kutta condition for the inviscid outer flow determining the circulation.

Having shown how steady, constant-density flow results are useful at all subcritical M, we now elaborate them for the finite wing pictured in Fig. 7-1. As discussed in Section 5-2 and elsewhere above, it is convenient to identify and separate portions of the field which are symmetrical and
antisymmetrical in z, later adding the disturbance velocities and pressures in accordance with the superposition principle. The separation process involves rewriting the boundary conditions (5-30) as

«* = Tfx + etx~a at г = 0+ ,

do. „dh –

<pz — — Гу + в——– a at z = 0—

dx dx

where h(x, y) is proportional to the ordinate of the mean camber surface, while 2i? (ж, у) is proportional to the thickness distribution. The differential equation is, of course, the three-dimensional Laplace equation

V2<p = 0. (7-9)

As a starting point for the construction of the desired solutions, we adapt (2-28) to express the perturbation velocity potential at an arbitrary field point (ж, у, z),

Here n is the normal directed into the field, and the integrals must be carried out over the upper and lower surfaces of S. Dummy variables (xi, 2/i, zi) will be employed for the integration process, so the scalar distance is properly written

r = V(x — xi)2 + (y — yi)2 + (z — zi)2. (7-11)

In wing problems Zj = 0 generally.

Considering the thickness alone, we have

ip{x, y, z) = <p(X, y, —z)

for all z and the boundary condition

Moreover, no discontinuities of <p or its derivatives are expected anywhere else on or off the ж, y-plane. In (7-10), dS = dzj dyb the values of <p(xi, //і, 0+) and <p(xi, yi,0—) appearing in the integrals over the upper and lower surfaces are equal, while the values of д/дп(1/4ят) are equal and opposite. Hence the contributions from the first term in brackets cancel,

where (7-13) and (7-11) have been – employed. Physically, (7-14) states that the flow due to thickness can be represented by a source sheet over the planform projection, with the source strength per unit area being pro­portional to twice the thickness slope dg/dx. [Compare the two-dimensional counterpart, (5-50).]

Examination of (7-14) leads to the conclusion that the thickness prob­lem is a relatively easy one. In the most common situation when the shape of the wing is known and the flow field constitutes the desired information, one is faced with a fairly straightforward double integration. For certain elementary functions g(x, y) this can be done in closed form; otherwise it is a matter of numerical quadrature, with careful attention to the pole singularity at aq = x, у = y, when one is analyzing points on the wing z = 0. The pressure can be found from (5-31) and (7-14) as

There is no net loading, since Cp has equal values above and below the wing. Also the thickness drag works out to be zero, in accordance with d’Alem­bert’s paradox (Section 2-5). Finally, it should be mentioned that, for any wing with closed leading and trailing edges,

avydaq = Рте — !7le — 0. (7-16)

J chord OX

This means that the total strength of the source sheet in (7-14) is zero. As a consequence, the disturbance at long distances from the wing ap­proaches that due to a doublet with its axis oriented in the flight direction, rather than that due to a point source.

1- 1 Introduction

For the present chapter we adopt all the limitations listed in Section 1-1, plus the following:

(1) p = constant everywhere; and

(2) The fluid was initially irrotational.

The former assumption implies essentially an infinite speed of sound, while the latter guarantees the existence of a velocity potential. Turning to (1-74), we see that the flow field is now governed simply by Laplace’s equation

V4 = 0. (2-1)

Associated with this differential equation, the boundary conditions prescribe the values of the velocity potential or its normal derivative over the surfaces of a series of inner and outer boundaries. These conditions may be given in one of the following forms:

(1) The Neumann problem, in which vn = дФ/дп is given.

(2) The Dirichlet problem, in which the value of Ф itself is given.

(3) The mixed (Poincare) problem, in which Ф is given over certain portions of the boundary and дФ/дп is given over the remainder.

A great deal is known about the solution of classical boundary-value problems of this type and more particularly about fluid dynamic applica­tions. Innumerable examples can be found in books like Lamb (1945) and Milne-Thompson (1960). The subject is by no means closed, how­ever, as will become apparent in the light of some of the applications presented in later sections and chapters. A useful recent book, which combines many results of viscous flow theory with old and new develop­ments on the constant-density inviscid problem, is the one edited by Thwaites (1960).

Among the very many concepts and practical solutions that might be considered worthy of presentation, we single out here a few which are especially fundamental and which will prove useful for subsequent work.

21

2- 2 The Three-Dimensional Rigid Solid Moving Through a Liquid

Let us consider a single, finite solid S moving through a large mass of constant-density fluid with an outer boundary 2, which may for many purposes be regarded as displaced indefinitely toward infinity. For much of what follows, S may consist of several three-dimensional solids rather than a single one. The direction of the normal vector n will now be re­garded as from the boundary, either the inner boundary S or the outer boundary 2, toward the fluid volume.

In the early sections of Chapter 3, Lamb (1945) proves the following important but relatively straightforward results for a noncirculatory flow, which are stated here without complete demonstrations:

(1) The flow pattern is determined uniquely at any instant if the boundary values of Ф or дФ/дп are given at all points of S and 2. One important special case is that of the fluid at rest remote from <S; then Ф can be equated to zero at infinity.

(2) The value of Ф cannot have a maximum or minimum at any interior point but only on the boundaries. To be more specific, the mean value of Ф over any spherical surface containing only fluid is equal to the value at the center of this sphere. This result is connected with the interpreta­tion of the Laplacian operator itself, which may be regarded as a measure of the “lumpiness” of the scalar field; Laplace’s equation simply states that this “lumpiness ” has the smallest possible value in any region.

(3) The magnitude of the velocity vector Q = |Q| cannot have a max­imum in the interior of the flow field but only on the boundary. It can have a minimum value zero at an interior stagnation point.

(4) If Ф = 0 or дФ/дп = 0 over all of S and 2, the fluid will be at rest everywhere. That is, no boundary motion corresponds to no motion in the interior.

Next we proceed to derive some less straightforward results.

1. Green’s Theorem. For the moment, let the outer boundary of the flow field remain at a finite distance. We proceed from Gauss’ theorem for any vector field A, (1-14),

j^A-ndS = – jjjv ■ AdV. (2-2)

S+S V

(The minus sign here results from the reversal of direction of the normal vector.) Let Ф and Ф’ be two continuous functions with finite, single­valued first and second derivatives throughout the volume V. We do not yet specify that these functions represent velocity potentials of a fluid flow. Let

Equations (2-4) and (2-5) are now substituted into Gauss’ theorem. After writing the result, we interchange the functions Ф and Ф’, obtaining two alternative forms of the theorem:

2. Kinetic Energy. As a first illustration of the application of Green’s theorem, let Ф in (2-6) be the velocity potential of some flow at a certain instant of time and let Ф’ = Ф. Of course, it follows that

у2Ф = VV = 0.

We thus obtain a formula for the integral of the square of the fluid particle speed throughout the field

“ -///lvt>’dr – "///«*"• 0«>

S+2 V V

Moreover, if we multiply the last member of (2-9) by one-half the fluid density p and change its sign, we recognize the total kinetic energy T of the fluid within V. There results

S+2

Such an integral over the boundary is often much easier to evaluate than a triple integral throughout the interior of the volume. In particular, ЭФ/дп is usually known from the boundary conditions. If the solid S is moving through an unlimited mass of fluid, with Q = 0 at infinity, it is a simple matter to prove that the integral over S vanishes. It follows that (2-10) need be integrated over only the inner boundary at the surface of the solid itself,

s

3. A Reciprocal Theorem. Another interesting consequence of Green’s theorem is obtained by letting Ф and Ф’ be the velocity potentials of two different constant-density flows having the same inner and outer bounding surfaces. Then, of course, the two Laplacians in (2-6) and (2-7) vanish, and the right-hand sides of these two relations are found to be equal. Equating the left-hand sides, we deduce

<2-,2)

S+S S+2

4. The Physical Interpretation of Ф. To assist in understanding the significance of the last two results and to give a meaning to the velocity potential itself, we next demonstrate an artificial but nevertheless meaning­ful interpretation of Ф. We begin with Bernoulli’s equation in the form (1-63), assuming the fluid at infinity to be at rest and evaluating the pres­sure integral in consequence of the constancy of p,

Imagine a process in which a system of very large impulsive pressures P = Ґ p dr (2-14)

is applied to the fluid, starting from rest, to produce the actual motion existing at a certain time t. In (2-14), т is a dummy variable of integration. We can make the interval of application of this impulse arbitrarily short, and integrate Bernoulli’s equation over it. [Incidentally, the same result is obtainable from the basic equations of fluid motion, (1-3) and (1-4), by a similar integration over the interval dt.] In the limit, the integrals of the pressure p„, pQ2/2, and p(Sl„ — fi) become negligible relative to that of the very large p, and we derive

pf = “p – (2-15)

J u-bt) от

Hence, P = — рф constitutes precisely the system of impulsive pressures required to generate the actual motion swiftly from rest. This process might be carried out, for example, by applying impulsive force and torque to the solid body and simultaneously a suitable distribution of impulsive pressures over the outer boundary 2. The total impulse thus applied will equal the total momentum in the instantaneous flow described by Ф. Unfortunately, both this momentum and the impulse applied at the outer boundary become indeterminate as 2 spreads outward toward infinity, so

that there are certain problems of physical interpretation when dealing with an externally unbounded mass of liquid.

In the light of this interpretation of the velocity potential, we reexamine the kinetic energy in (2-11), rewriting this result

(2-16)

The work done by an impulse acting on a system which starts from rest is known to be the integral over the boundary of the product of the impulse by one-half the final normal velocity at the boundary. Thus, the starting impulses do a total amount of work given by exactly the last member of (2-16). Since the system is a conservative one, this integral would be expected to equal the change of kinetic energy which, of course, is the final kinetic energy T in the present case. The difficulty in connection with carrying 2 to infinity disappears here, since the work contribution at the outer boundary can be shown to approach zero uniformly. Hence, the kinetic energy of an unbounded mass of constant-density fluid without circulation can be determined entirely from conditions at the inner bound­ary, and it will always be finite if the fluid is at rest at infinity.

The reciprocal theorem, (2-12), can be manipulated, by multiplication with the density, into the form

(2-17)

As such, it becomes a special case of a fairly familiar theorem of dynamics which states that, for any two possible motions of the same system, the sum over all the degrees of freedom of the impulse required to generate one motion multiplied by the velocity in the second motion equals the same summed product taken with the impulses and velocities interchanged.

One final important result is stated without proof: it can be shown that for a given set of boundary conditions the kinetic energy Г of a liquid in a finite or infinite region is a minimum when the flow is acyclical and irrotational, relative to all other possible motions.

We shall consider the viscous laminar high Reynolds number flow over a semi-infinite flat plate at zero angle of attack. This is the simplest case of a boundary layer that may serve as a model for the calculation of bound­ary layer effects on a thin airfoil. For incompressible flow the Navier – Stokes equations and the equation of continuity read

The boundary conditions are that the velocity vanishes on the plate and becomes equal to XJx far away from the plate surface. Thus for two­dimensional flow

Q(x, 0) = 0 for x > 0, (4-2)

Q иЛ £ог г ^ . (4-3)

We have assumed that the leading edge of the plate is located at the origin (see Fig. 4-2). By assuming that the plate is semi-infinite one avoids the problem of considering upstream effects from the trailing edge. These are actually quite small and do not show up in the boundary layer approximations to be derived.

In the process of introducing nondimensional coordinates a minor difficulty is encountered because there is no natural length in the problem to which spatial coordinates could be referred. We will circumvent this by selecting an arbitrary reference length and thereby implicitly assume that the behavior of the solution for x — 0(1) and z = 0(1) will be studied.

For the purely antisymmetrical case we have

<p{x, y, z) = —<p{x, y, —z) (7-17)

with corresponding behavior in the velocity and pressure fields. The
principal boundary condition now reads

<Pz = — oc at z = 0± for (ж, y) on S. (7-18)

ox

We must make allowance for a discontinuity in <p not only on the planform projection S but over the entire wake surface, extending from the trailing edge and between the wingtips all the way to x = +oo on the x, y-plane. For this reason, the simplest approach proves to be the use of (7-10) not as a means of expressing <p itself but the dimensionless ж-component of the perturbation velocity

и = <pX! (7-19)

where и is essentially the pressure coefficient, in view of (5-31). Equation (1-81) shows us that we are also working with a quantity proportional to the small-disturbance acceleration potential, and this is the starting point adopted by some authors for the development of subsonic lifting wing theory.

Obviously, и is a solution of Laplace’s equation, since the operation of differentiation with respect to x can be interchanged with V2 in (7-9). We may therefore write

(7-20)

We specify for the moment that S’ encompasses both wing and wake, since the derivation of (2-28) called for integration over all surfaces that are sources of disturbance and made no allowance for circulation around any closed curve in the flow external to the boundary. It is an easy matter to show, however, that the choice of и as dependent variable causes the first term in the (7-20) brackets to vanish except on S and the second term to vanish altogether. Because there can be no pressure jump except through a solid surface, и is continuous through z = 0 on the wake. But

are equal and opposite on top and bottom everywhere over S, so the first-term contributions remain uncanceled only on S. There и jumps by an amount

У(хі, V) = uu — m. (7-22)

(By antisymmetry, щ = —uu.)

As regards the second bracketed term in (7-20), the condition of irrota – tionality reveals that

du _ du _ dw dn dz і dxi

on top and one finds that

on the bottom of S’. Both w and its derivatives are continuous through all of S’, and therefore the upper and lower integrations cancel throughout. One is left with

s

We have inserted (7-21) and (7-22) here, along with the fact that

A=+_*_____

dz d(z1 — 2)

d_ dz

when applied to a quantity which is a function of these two variables only in the combination (2 — 21).

The modification of (7-24) into a form suitable for solving lifting-wing problems can be carried out in several ways. Perhaps the most direct is to observe that nearly always w(x, y, 0) is known over S, and (7-24) should therefore be manipulated into an expression for this quantity. This we do by noting that и = рх and w = so that

w(x, y, z) = u(x0, y, 2) dx0, (7-26)

where account has been taken that <p(— ж, у, z) — 0. Inserting (7-26) into (7-24) and interchanging orders of differentiation and integration, we get

w(x, y, z) = — ^.JjУ(хь Vi) s

_____________ dx0____________

V (x0 — Zl)2 + (у — У1)2 + 22′

When deriving (7-28b), an integration by parts* on yx is carried out at finite z. The singularity encountered as z —> 0 is then similar to the one in upwash calculation at a two-dimensional vortex sheet and can be handled by the well-known Cauchy principal value. The integral in (7-28a) containing 7(xj, уi) itself is, however, the more direct and directly useful form. In the process of arriving at it, we find ourselves confronted with the following steps:

dx0

V (x0 — Xl)2 + (y — Vi)2 + Z2

For г = 0 all terms here will vanish formally except the one arising from the 2-derivative of the numerator which will give a nonintegrable singu­larity of the form 1 /{y — уi)2. It is precisely with such limits, however, that Mangler’s study of improper integrals [Mangier (1951)] is concerned. Indeed, if we examine Eqs. (33) and (34) of his paper, replacing his £ with our yi, we observe that our ^-integral should be evaluated in accordance
with Mangler’s principal-value technique and thereupon assumes a per­fectly reasonable, finite value. The result implies, of course, that the self­induced normal velocity on a vortex sheet should not be infinite if it is calculated properly.

, if an indefinite integral can be found for the integrand, the answer is obtained simply by inserting the limits y = a and yx = b, provided any logarithm of (y — yi) that appears is interpreted as In |г/ — yx. The validity of Mangler’s principal value depends on the condition that the integrand, prior to letting z —* 0, be a solution of the two-dimensional Laplace equation. It is clear that this is true, in the present case, of the function that causes the singu­larity in the ^-integration of (7-29), since

Equation (7-32) provides confirmation for (5-35).

The question of exact or approximate solution of (7-28) is deferred to later sections. We note here that, when the angle of attack a and camber ordinates h(x, y) of the wing are given and the load distribution is required, (7-28) is a singular double integral equation for the unknown 7. Thus the problem is much more difficult mathematically than the corresponding thickness problem embodied in (7-14)-(7-15). On the other hand, when the loading is given and the shape of the wing to support it desired, the potential and upwash distributions are available by fairly straightforward integrations from (7-31) and (7-28), respectively. Finally, the thickness
shape g(x, y) to generate a desired symmetrical pressure distribution must be determined by solving the rather complicated integral equation which results from г-differentiation of (7-14).

We finish this section with some further discussion of the lift, drag, and nature of the wake. From (5-31), (7-19), and (7-22), we see that the difference in pressure coefficient across the wing is

. Cvi – CPu = 27. (7-33)

Using the definition of Cp,

Vi — Vu = PmUiy. (7-34)

Because the surface slopes are everywhere small, this is also essentially the load per unit plan area exerted on the wing in the positive z-direction. Since (7-34) is reminiscent of the Kutta formula (2-157), we note that 7 can be interpreted as a circulation. As shown in Fig. 7-3, let the circula­tion about the positive ^/-direction be computed around a small rectangular box C of length dx in a chordwise cross section of the wing. Since the con­tributions of the vertical sides cancel, except for terms of higher order in dx, we find that the circulation around C is

UK[ 1 + uu] dx — f7„[l + ui] dx = U«,(uu — щ) dx = f7„7 dx. (7-35)

Hence U„У(х, y) is the spanwise component, per unit chordwise distance, of the circulation bound to the wing in the vicinity of point (x, y).

The lift per unit span at station у on the wing is l(y) = f [Pi — Pu] dx = pxUl ( У(х, y) dx = pxU^T(y), (7-36)

./chord./chord

Г being the total bound circulation. If the wing tips are placed at у — ±5/2, the total lift becomes

L = Ґ12 l(y) dy = pxt/o, fbl2 Г(у) dy. (7-37)

J-Ы 2 J-bl 2

The total pitching moment, pitching moment per unit span about an
arbitrary axis, rolling moment, or any other desired quantity related to the loading may be constructed by an appropriate integration of (7-34).

Unlike a two-dimensional airfoil, the finite lifting wing does experience a downstream force (drag due to lift or “induced drag,” Zh, sometimes also called “vortex drag, ” see Chapter 9) in a subsonic inviscid flow. An easy way to compute this resistance is by examining the wake at points remote behind the trailing edge. In fact, if we observe the wing moving at speed Ux through the fluid at rest, we note that an amount of mechanical work DiUx is done on the fluid per unit time. Since the fluid is nondissi­pative and can store energy in kinetic form only, this work must ultimately show up as the value of T (cf. 2-11) contained in a length Ux of the distant wake. The nature of this wake we determine by finding its disturbance velocity potential. Over the wing region S, the ^discontinuity is calcu­lated, as in (7-26), to be

A<p(x, y) = <p(x, y, 0+)

(7-38)

The last line here follows from the definition of 7, (7-22), the lower limit — oo being replaced by the coordinate zle of the local leading edge since there is no w-discontinuity ahead of this point. Beyond the trailing edge on the x, ?/-plane, [uu — щ] = 0 in view of the condition of continuity of pressure. Hence

A<£>wake(2/) ~ / 7(^0, y) dx0 — ‘ (7“39)

J chord ^ qo

Since A^wake is independent of x, the wake must consist of a sheet of trailing vortices parallel to x and having a circulation per unit spanwise distance

A j-p

Ux b(y) = Ux ± (AVwake) = J • (7-40)

The complete vortex sheet simulating the lifting surface, as seen from above, is sketched in Fig. 7-4. It is not difficult to show that the Kutta – Joukowsky condition of smooth flowoff is equivalent to the requirement that the vortex lines turn smoothly into the stream direction as they pass across the trailing edge. At points far downstream the motion produced by the trailing vortices becomes two-dimensional in x, z-planes (the so – called “Trefftz plane”). Although the wake is assumed to remain flat in accordance with the small-perturbation hypothesis, some rolling up and downward displacement in fact occurs [Spreiter and Sacks (1951)]. This rolling up can be shown to have influences on the loading that are only of

third order in в and a. For the plane wake, we use (2-11) and the equality of work and kinetic-energy increment to obtain

Di = — <p dS, (7-41)

*^wake

where Swake comprises the upper and lower surfaces of unit length of the wake, as seen by an observer at rest in the fluid. Clearly, dS = dy, while

rbl 2

д<Р = = 1 і jTP dyi

дП <Рг 2tU„ J-b/2 dy! (y — yi)

A more symmetrical form of (7-43) can be constructed by partial integration with respect to y. In the process, we make use of the Cauchy principal value operation at у = y and note the fact that Г(±Ь/2) = 0 in view of the continuous dropping of load to zero at the wing tips.

It is interesting that (7-37) and (7-44) imply the well-known result that minimum induced drag for a given lift is achieved, independently of the

7- 4 Lifting-Line Theory

The first rational attempt at predicting loads on subsonic, three-dimen­sional wings was a method due to Prandtl and his collaborators, which was especially adapted in an approximate way to the large aspect-ratio, unswept planforms prevalent during the early twentieth century. Although our approach is not the classical one, we wish to demonstrate here how the lifting-line approximation follows naturally from an application of the method of matched asymptotic expansions. Let us consider a wing of the sort pictured in Fig. 7-4, and introduce a second small parameter e.4, which is inversely proportional to the aspect ratio. Then if we write the local chord as

c(y) = eAc(y) (7-49)

and examine the matched inner and outer solutions associated with the process tA —> 0 at fixed span b, we shall be generating a consistent high aspect-ratio theory. As far as the thickness ratio, angle of attack, and camber are concerned, we assert that they are small at the outset and
remain unchanged as eA —> 0. Our starting point is therefore the problem embodied in (5-29)-(5-30), with M = 0; we study the consequences of superimposing a second limit on the situation which they describe.

Placing the wing as close as possible to the у-axis in the x, у-plane, we define new independent variables

Fig. 7-5. Coefficients of lift plotted vs. angle of attack a and total drag coeffi­cient Cd for several rectangular wings (values of A indicated on the figures). In parts (b) and (d), the same data are adjusted to a reference A = 5 by formulas based on lifting-line theory and elliptic loading. [Adapted from Prandtl, Wiesels – berger, and Betz (1921).]