Category THEORETICAL AERODYNAMICS

In Chapter 2, we saw that a perfect gas has to be thermally as well as calorically perfect, satisfying the thermal state equation and at least two calorical state equations. Thus for a perfect gas:

pv = RT

or

p = pRT,

where p is the pressure and R is the gas constant, given by:

Theoretical Aerodynamics, First Edition. Ethirajan Rathakrishnan.

© 2013 John Wiley & Sons Singapore Pte. Ltd. Published 2013 by John Wiley & Sons Singapore Pte. Ltd.

where Ru is the universal gas constant equal to 8314 J/(kg K) and M is the molecular weight of the gas. Thus, of the four quantities p, p, T, R, in Equation (9.1i>), only two are independent.

Taking log on both sides and differentiating Equation (9.1i>), we get:

Let us assume unit mass of a gas receiving a small quantity of heat q. By the first law of thermodynamics, we know that heat is a form of energy [2]. Thus the quantity of heat q is equivalent to q units of mechanical energy. Hence addition of heat q will supply energy to the gas, resulting in the increase of its specific volume from v to (v + dv). Thus, the heat q added does a mechanical work of pdv. Let us assume that the expansion is taking place very slowly, so that no kinetic energy is developed. For this process, we can write:

q = du + pdv, (9.3)

where du is the increase in the internal energy of the gas. It is essential to note from Equation (9.3) that only a part of the heat q supplied is converted to mechanical work pdv and the rest of the heat is dumped into the internal energy of the gas mass. This demonstrates that the energy conversion is 100% efficient. The work pdv is referred to as flow work. Thus, for doing, say, 1 unit of work (pdv) we need to supply q/n amount of heat, where n is the efficiency of the work producing cycle and n is always less than 1. For example, the work producing devices, such as spark ignition (SI) engine, compression ignition (CI) engine and gas turbine (jet) engines has efficiencies of 40%, 60% and 30%, respectively.

For a perfect gas, the internal energy u is a function of the absolute temperature T alone. This hypothesis is a generalization for experimental results. It is known as Joule ‘slaw. Thus:

du = kdT. (9.4)

Substituting this into Equation (9.3), we have:

q = kdT + pdv. (9.5)

For a constant volume (isochoric) process, dv = 0. Thus for a constant volume process, Equation (9.5) reduces to:

q = kdT.

We can express this as:

q = cvdT,

where cv is called the specific heat at constant volume. It is the quantity of heat required to raise the temperature of the system by one unit while keeping the volume constant. Thus from Equation (9.5), with dv = 0, we get:

k = cv.

Similarly the specific heat at constant pressure, cp, defined as the quantity of heat required to raise the temperature of the system by one unit while keeping the pressure constant. Now, for p = constant, Equation (9.2) simplifies to:

dp dT

Y’

V P•

Р

There are three unknowns; the pressure p, density p and velocity V in Equations (9.10) and (9.11). Therefore these two equations alone are insufficient to determine the solution. To solve this motion, we can make use of the process Equation (9.9) as the third equation, presuming that the motion is isentropic.

9.1 Introduction

Our discussions so far were on incompressible flow past lifting surfaces. That is, the effect of com­pressibility of the air has been ignored. But we know that the incompressible flow is that for which the Mach number is zero. This definition of incompressible flow is only of mathematical interest, since for Mach number equal to zero there is no flow and the state is essentially a stagnation state. Therefore, in engineering applications we treat the flow with density change less than 5% of the freestream density as incompressible[1]. This corresponds to M = 0.3 for air at standard sea level state. Thus, flow with Mach number greater than 0.3 is treated compressible. Compressible flows can be classified into subsonic, su­personic and hypersonic, based on the flow Mach number. Flows with Mach number from 0.3 to around 1 is termed compressible subsonic, flows with Mach number greater than 1 and less than 5 are referred to as supersonic and flows with Mach number in the range from 5 to 40 is termed hypersonic. In our discussions here, only subsonic and supersonic flows will be considered. In Chapter 2, we discussed some aspects of compressible flows only briefly. Therefore, it will be of great value to read books specializing on gas dynamics and its application aspects, such as Rathakrishnan (2010) [1], before getting into this chapter.

In our discussion in this chapter, the air will be treated as a perfect, compressible and inviscid fluid. In other words, the important consequence of viscosity, namely, the skin friction drag due to the viscous effects in the boundary layer will not be considered in our discussions.

For thin aerofoils, which can be approximated by replacing them by their plan areas in the xy-plane, the acceleration potential can be applied comfortably. Let us consider such an aerofoil, shown in Figure 8.26(a), and replace it by its plan area represented by its section AB, shown in Figure 8.26(b).

If pu is the pressure at a point on the upper surface of AB and pi is the pressure at a corresponding point at the lower surface, then it can be shown that:

Pi — Pu = P (Фі — Фи) , (8.91)

where Фі and Фи are the corresponding values of the acceleration potential. Thus we have the lift and pitching moment as:

(Фі — Ф^ dS	(8.92)
S
x (Фі — Фи) dS,	(8.93)

L = р

M = р

S

(b)

where S is the surface area of the aerofoil planform. The center of pressure is at a distance xp = M/L from the origin.

The downwash velocity w is obtained by equating the values of the z-component of the acceleration, Equations (8.86) and (8.87). Thus:

dw дФ dx dz

But the downwash w vanishes at x = ж, therefore:

1 Г ЭФ

w = – —dx. (8.94)

V Л» dz

The induced drag is given by:

For a given y, the profile z = z(x, y) is determined by:

Example 8.5

A wing with elliptical loading, with span 15 m, planform area 45 m2 is in level flight at 750 km/h, at an altitude where density is 0.66 kg/m3. If the induced drag on the wing is 3222 N, (a) determine the lift coefficient, (b) the downwash velocity, and (c) the wing loading.

Solution

Given, 2b = 15 m, S = 45 m2, p = 0.66 kg/m3, V = 750/3.6 = 208.33 m/s.

W 180462.98

~S

8.12 Summary

The vortex theory of a lifting aerofoil proposed by Lancaster and the subsequent development by Prandtl made use of for calculating the forces and moment about finite aerofoils. The vortex system around a finite aerofoil consists of the starting vortex, the trailing vortex system and the bound vortex system.

From Helmholtz’s second theorem, the strength of the circulation round any section of a bundle of vortex tubes is the sum of the strength of the vortex filaments cut by the section plane.

If the circulation curve can be described as some function of y, say f (y), then the strength of the circulation shed by the aerofoil becomes:

Sk = – f ‘(y) dy.

At a section of the aerofoil the lift per unit span is given by:

l = pUk.

The induced velocity at yi, in general, is in the downward direction and is called downwash.

The downwash has the following two important consequences which modify the flow about the aerofoil and alter its aerodynamic characteristics.

• The downwash at y1 is felt to a lesser extent ahead of y1 and to a greater extent behind, and has the effect of tilting the resultant wind at the aerofoil through an angle,

	Zw	w
e = tan 1	Ы	^~U

The downwash reduces the effective incidence so that for the same lift as the equivalent infinite or two-dimensional aerofoil at incidence a, an incidence of a = a^ + e is required at that section of the aerofoil.

• In addition to this motion of the air stream, a finite aerofoil spins the air flow near the tips into what eventually becomes two trailing vortices of considerable core size. The generation of these vortices requires a quantity of kinetic energy. This constant expenditure of energy appears to the aerofoil as the trailing vortex drag.

The forward wind velocity generates lift and the downwash generates the vortex drag Dv.

This shows that there is no vortex drag if there is no trailing vorticity.

The expression k = f (y) which can be substituted in expression for L, w and Dv is:

The lift of an aerofoil of span 2b is:

The circulation for elliptical distribution is:

The downwash becomes:

This is an important result, which implies that the downwash is constant along the wing span. The drag caused by the downwash is:

Therefore, the drag coefficient becomes:

For modified elliptical loading:

The lift coefficient becomes:

The downwash for modified elliptic loading at any point y along the span is:

The vortex drag for modified loading is:

The drag coefficient becomes:

This drag coefficient for the modified loading is more than that for elliptical loading by an amount S, which is always positive since it contains X2 terms only.

If the lift for the aerofoils with elliptical and non-elliptical distribution is the same under given con­ditions, the rate of change of vertical momentum in the flow is the same for both. Thus, for elliptical distribution the lift becomes: ■b

m wady.

b

For non-elliptic distribution, the lift is:

L (X m (wo + Л(у))dy,

J-b

where m is a representative mass flow meeting unit span. But lift L is the same on each wing, therefore:

b

mfi(y) dy = 0.

b

Now the energy transfer or rate of change of the kinetic energy of the representative mass flow is the vortex drag (or induced drag). Thus, for elliptical distribution the vortex drag is:

1 / 2

Dva X 2 m J Wody.

For non-elliptic distribution the vortex drag is:

Lancaster-Prandtl lifting line theory is a representation to improve on the accuracy of the horseshoe vortex system. In lifting line theory, the bound vortex is assumed to lie on a straight line joining the wing tips (known as lifting line). Now the vorticity is allowed to vary along the line. The lifting line is generally taken to lie along the line joining the section quarter-chord points. The results obtained using this representation is generally good provided that the aspect ratio of the wing is moderate or large – generally not less than 4.

The integral equation from which the bound vorticity distribution may be determined is:

The lift generated by the wing is:

The lift coefficient is:

Thus, the lift coefficient Cl depends on A1, which in turn depends on the values and distribution of a and g.

The induced drag is:

where:

and is usually very small. Also, A1 = CL/n/R, so that:

where (1 + S) > 1, is the induced drag factor, and hence Г depends on the values of the Fourier coeffi­cients, and hence on the wing geometry, especially on the planform.

For an aerofoil:

• Geometrical incidence is the angle between the chord of the profile and the direction of motion of the aerofoil.

• Absolute incidence is the angle between the axis of zero lift of the profile and the direction of motion of the aerofoil.

When the axes of zero lift of all the profiles of the aerofoil are parallel, each profile meets the freestream wind at the same absolute incidence, the incidence is the same at every point on the span of the aerofoil, and the aerofoil is said to be aerodynamically untwisted.

An aerofoil is said to have aerodynamic twist when the axes of zero lift of its individual profiles are not parallel. The incidence is then variable across the span of the aerofoil.

The drag and lift ratio can be expressed as:

Cdv ‘	w t
Cl’	V (

For an actual aerofoil in a subsonic flow the main components of the drag are the profile drag and the skin friction drag. The induced drag caused by the downwash is an additional component of drag. Therefore, the total drag coefficient of the strip (profile), using Equation (8.24), is:

Cd’ = cd0 ‘ + (Cl ‘

where CDo’ is the coefficient of profile drag for the profile.

It may be noted that the profile drag is largely independent ofincidence in the working range. Profile drag is the sum of the skin friction due to viscosity and form drag due to the shape.

The form drag due to the shape is owing to the high pressure at the leading edge and low pressure at the trailing edge (that is the low pressure in the wake).

The following are the two problems associated with aerofoils:

• For a given circulation k(y), the form of the aerofoil and the induced drag are to be determined.

• For a given form of aerofoil, the distribution of circulation and the induced drag are to be determined.

In practice, in addition to induced drag there is profile drag due to skin friction and wake. The coefficient of profile drag is indicated by CD0. This the complete drag coefficient is:

Cd = Cd0 + Cdv.

The lift curve slope for an aerofoil of finite aspect ratio Ж with elliptical loading is:

If the aspect ratio is reduced to Ж and if the ‘primes’ refer to the new aerofoil with the same incidence, we have:

In problem I the aerofoil shape is found for a given circulation k(y). Problem II is an inverse problem in which, for a given aerofoil geometry the circulation is determined.

At the point of the trailing edge of an aerofoil, whose eccentric angle is в, we have:

For elliptic loading this becomes:

we = U Aj.

By Equation (8.7), we have:

We = —. 4b

Therefore:

which is constant across the wing span.

For wings with loading other than elliptic, the drag polar becomes:

where e is known as the Oswald wing efficiency and for elliptic loading e = 1.
For elliptic loading, S = 0 and e = 1, therefore, the drag polar becomes:

For drag minimum:

Rectangular aerofoil is an aerofoil whose planform is a rectangle. An aerofoil whose shape is that of a cylinder erected on an aerofoil profile satisfies this requirement.

Cylindrical rectangular aerofoil is the simplest type, of span 2b and chord c, which is constant at all sections. All the sections are similar and similarly situated.

In the general case, where the loading or lift distribution is not symmetrical about mid-span section, even terms appear in the distribution, and as a consequence of the asymmetry other characteristics of aerofoil appear.

When the lift distribution is not symmetrical about the centerline, one wing will have higher lift than the other and a net rolling moment about the longitudinal axis through the mid-span will result.

Further, as the lift is not symmetric nor is the spanwise distribution of circulation, the downwash will vary across the span without being symmetrical about the centerline and so will be the vortex drag grading. Hence, more drag will be experienced on one wing (the one with more lift) than on the other and a net yawing moment will result about the vertical (normal) axis through the mid-span section. In

addition to these there will be the overall lift and vortex drag force normal and parallel to the plane of the aerofoil in the plane of symmetry.

The lift acting on any section of spanwise length Sy at a distance y from the centerline (ox-axis) will produce a negative increment of rolling moment equal to:

ALr = —l y dy,

where l is the lift grading given by l = pVk. The total moment becomes:

b r-b

ly dy = — I pVk y dy.

b J—b

The asymmetrical drag grading across the span, gives rise to yawing moment N. The yawing moment can be expressed as:

N = -2 pV 2SbCN

where CN is the yawing moment coefficient.

Lifting surface theory is a method which treats the aerofoil as a vortex sheet over which vorticity is spread at a given rate. In other words, the aerofoil is regarded as a surface composed of lifting elements. This is different from the lifting line theory. The essential difference between the lifting surface theory and lifting line theory is that in the former the aerofoil is treated as a vortex sheet, whereas in the latter, the aerofoil is represented by a straight line joining the wing tips, over which the vorticity is distributed.

Munk’s theorem of stagger states that “the total drag of a multiplane system does not change when the elements are translated parallel to the direction of the wind, provided that the circulations are left unchanged.” Thus the total induced drag depends only on the frontal aspect.

The total drag mutually induced on the pair of lifting elements becomes:

, j2n pTxTzdsxdsz cos^! + ф2)

d Di2 + d D21 = —————— ~———————– 2——— ,

2n n2

which is independent of the angle of stagger. This yields Munk’s theorem of stagger, that is:

“the total drag of a multiplane system does not change when the elements are translated parallel to the direction of the wind, provided that the circulations are left unchanged."

When the system is unstaggered (that is, when в = 0):

d2Di2 = d2D21

and thus if the lifting systems are in the same plane normal to the wind, the drag induced in the first by the second is equal to the drag induced in the second by the first. This result constitutes Munk’s reciprocal theorem.

The total mutual induced drag is:

p Гi Г2 cos (фі + ф2) 2n n2

Blenk’s method is meant for wings of finite aspect ratio and is based on the lifting line theory of Prandtl, hence limited to aerofoils moving in the plane of symmetry and with a trailing edge which could be regarded as approximately straight. This method considers the wing as a lifting surface, that is to say the wing is replaced by a system of bound vortices distributed over its surface rather than along a straight line coinciding with the span. However, this method has the limitation that the wing is assumed to be thin and practically plane.

The following are the two main approaches employed in Blenk’s method:

1. Given the load distribution and the plan, find the profiles of the sections.

2. Given the plan and the profiles, find the load distribution (that is the vorticity distribution).

For aerofoils with aspect ratio less than about unity, the agreement between theoretical and experimental lift distribution breaks down. The reason for this break down is found to be the consequence of Prandtl’s hypothesis that the free vortex lines leave the trailing edge in the same line as the main stream. This assumption leads to a linear integral equation for the circulation.

For aerofoils with aspect ratio less than about unity, the agreement between theoretical and experimental lift distribution breaks down. The reason for this breakdown is found to be the consequence of Prandtl’s hypothesis that the free vortex lines leave the trailing edge in the same line as the main stream. This assumption leads to a linear integral equation for the circulation. Let us consider a portion of a flat rectangular aerofoil whose chord c is large compared to the span 2b, as shown in Figure 8.23. Let us take the chord axes with the origin at the center of the rectangle.

The bound vorticity у (x) is assumed to be independent of y, that is to say is constant across a span such as PQ but is variable along the chord. The downwash is also assumed to be independent of y and may therefore be calculated at the center of each span. The main point of the theory developed for aerofoils of small aspect ratio is that the trailing vortices, which leave the tips of each span such as PQ, make an angle в with the chord which is different from a, the incidence, since the trailing vortices follow the fluid

Figure 8.23 A portion of a flat rectangular aerofoil of small aspect ratio.

particles which leave the edges of the aerofoil at angle в will presumably be a function of x. To a first approximation it is assumed to be constant.

8.13.1 The Integral Equation

Let us begin with the calculation of the velocity induced at the center C(x, 0, 0) of the span PQ, shown in Figure 8.24.

Consider then the span RS, center D(§, 0, 0). The bound vortex associated with RS is of circulation Y(§) d§ and induces at C a velocity, in the z-direction:

Y (§) d§ .

dwi =————— (cos ZCRS + cos ZCSR) .

4n • CDK +

This gives the downwash due to the whole set of bound vortices as:

This is the velocity induced in the z-direction. If ui is the velocity induced in the x-direction, it can be shown that:

ui = -2 Y(x) or + 2 y(x). (8.77)

To find the velocity induced at C by the vortices trailing from R and S, let T, M, U be the projections of P, C, Q on the plane of these vortices. Then the vortex trailing from R induces at C a velocity of magnitude dq perpendicular to the plane RCT, and the vortex trailing from S induces at C a velocity of the same magnitude perpendicular to the plane SUC. Let dqn be the resultant induced velocity and its

direction is along CM. Then, if the angle ZTCM = S, we have:

, , 2y(§) d§ x b/2 Іл

dqn = 2dq sin S = ——— 2————– (1 — cos ZCR!).

Therefore, for all the trailing vortices, the resultant induced velocity becomes:

If u2, w2 are the components of qn in the x – and z-directions:

w2 = qn cos в, u2 = —qn sin в = —w2 tan в.

The boundary condition is that there shall be no flow through the aerofoil, this implies that the normal induced velocity just cancels the normal velocity due to the stream. Therefore:

wj + w2 = V sin a. (8.80)

The required integral equation can be obtained by substituting the values from Equation (8.76) to Equation (8.79). At this stage it will be useful to employ “dimensionless” coordinates. For this problem, the dimensionless coordinates are 2x/c and 2§/c. Also, b/c = JR. In terms of the dimensionless coordinates we can cast Equation (8.78) as:

This is a nonlinear equation since в itself is a function of y (§).

The method proposed for the tentative solution of Equation (8.81) is to put:

which is valid for large aspect ratios and then to use Equation (8.80) to determine Y0 in terms of в and then approximate to a suitable mean value for в. This is a laborious exercise, so without venturing into this let us examine the variation of the normal force coefficient on an aerofoil CN, defined as:

Cn =

where N is the normal force. Variation of CN for JR = 1/30 with incidence angle a is shown in Figure 8.25.

For the same profile, the variation of CN, calculated with lifting line theory, with a will be as shown by the line of dashes. From Figure 8.25, it is seen that for aerofoils with very small aspect ratio the stalling incidence is very high and hence they can fly at high values of a, without stalling.

8.13.2 Zero Aspect Ratio

For the limiting case of a profile with zero aspect ratio (c ^ to) , it has been found that:

CN = 2 sin2 a.

This is the same as the behavior predicted by Isaac Newton for a flat plate which experiences a normal force proportional to the time rate of change of momentum in inelastic (incompressible) fluid particles impinging on it. In fact here we should have the normal force:

N = pV sin a • V sin a • S,

which gives the above CN.

8.13.3 The Acceleration Potential

Let us consider an aerofoil placed in a uniform flow of velocity — V in the negative direction of x-axis. We can as usual consider the aerofoil replaced by bound vortices at its surface enclosing air at rest and accompanied by a wake of free trailing vortices. Outside the region consisting of the bound vortices and the wake the flow is irrotational, and hence the flow field can be represented by a velocity potential ф such that the air velocity is given by:

q = — уф.

If the velocity induced by the vortex system is v, then:

q =—V + v.

The motion is steady, therefore the acceleration is:

If we assume[16] that the magnitude of the induced velocity v is small compared to the main flow velocity V, Equation (8.85) can be expressed, using Equation (8.83), as:

dq d

a = -(V у) q = – V-q = V— (у ф).

dx dx

Thus we have:

a = —у Ф, Ф = — V —. (8.87)

dx

where Ф is the acceleration potential. Since the velocity potential ф satisfies Laplace’s equation у2ф, it follows from Equation (8.87) that:

у2 Ф = 0. (8.88)

Assuming the flow to be incompressible and neglecting external forces, the acceleration can be expressed as:

This shows that an acceleration potential always exists. However, only with our assumption of small magnitude of induced velocity v this satisfies Laplace’s equation. Comparing Equations (8.87) and (8.89) we see that Ф and р/р can differ only by a constant, and we can take:

Ф = P—. (8.90)

P

where I is the pressure at infinity.

Consider first the velocity induced at P(x, y, 0) by a vortex MM’, shown in Figure 8.22, parallel to the span (lifting line) and the trailing vortices which spring from it. Let Q(x’, y’, 0) be a point on the vortex MM’. The circulation at Q is then у1 (x’, y’) dx’ and from Q there trails a vortex of circulation:

дуіС! У) , ,

————– dy dx.

dy

The downwash velocity induced by the trailing vortex caused by MM’ (see Chapter 5) at point P is:

Thus from Equation (8.69) we get:

,2 (§ §)2 , ( /) 2

к = ——— — + (n — n) .

AT

Note that, if §’ = § Equation (8.72) reduces to its first term and if we put y W, n) = Yo (§0 /(1 — n’2)- the elliptic distribution across the span, we get:

Yo (§’) + c(§ — §,)Yo(§,) f+l_________ n’ dn’______

2b 2nb2 J—i k(n — n’) /(1 — n’2)

Substituting Equation (8.74) into Equation (8.7i) we get the downwash velocity. The induced drag is given by:

which includes the suction force at the leading edge and hence the leading edge should not be rounded.

The integrals in Equation (8.74) are elliptic type and cannot be evaluated in terms of elementary functions. Blenk therefore adopted an ingenious method of approximation, even though it is lengthy. However, this approximation is valid only to the middle part of the wing so that the end effects are uncertain. The approximation is better suited for larger aspect ratios. The method leads to replacing Equation (8.74) by:

where the coefficients Ai, Bi, Ci, Di and Ei are functions of n which depend on the particular case among the four wing shapes considered. For the rectangular wing moving in the plane of symmetry:

The downwash may be calculated from Equation (8.7i) with the aid of Equation (8.75)

To determine the profile of the section at distance y from the center, let us assume the relative wind to blow along the x-axis. Since we consider the perturbations in the freestream to be small and the air must flow along the profile, we have:

dz w

дХ = V’

Therefore:

і г. .

z = — / w(x, y) dx. V

Comparison of this result with the theory of the lifting line gives the following mean additions to the incidence and curvature for the rectangular wing:

C і C

Aa = 0.059 A – = 0.056

M R b

In the case of sweep-back wing, the mean increase of incidence, according to Blenk, should be 1’6 в (4 — М) percent of the absolute incidence without sweep-back.

Let us assume that the load distribution is given for the rectangular planform of Figure 8.21(a). Now the task is to find the profiles at different sections.

Let us assume the profile to be thin so that the whole aerofoil may be considered to lie in the xy plane, as shown in Figure 8.22.

Let Yi (x, y) be the circulation per unit length of chord at the point (x, y, 0) so that the circulation round the profile at distance y from the plane of symmetry is:

[where c is the chord and b is the span (note that here b is taken as the span, instead of 2b, for convenience)] we get the circulation as:

nfa) = J Y(§, n) d§. (8.66)

For y(§, n), let us choose the following elliptic distribution over the span:

Y(§, n) = У0Й) V7! – П2

and for у0(§), let us consider the following three different functions:

where a0, b0, c0 are arbitrary constants. Note that Equation (8.67a) is the distribution for a thin flat aerofoil in two-dimensional motion. The most general distribution considered here will then be of the form:

This method is meant for wings of finite aspect ratio and is based on the lifting line theory of Prandtl, discussed in Section 8.7, hence limited to aerofoils moving in the plane of symmetry and with a trailing edge which could be regarded as approximately straight. This method considers the wing as a lifting surface, that is to say the wing is replaced by a system of bound vortices distributed over its surface rather than along a straight line coinciding with the span. However, this method has the limitation that the wing is assumed to be thin and practically plane. The shapes considered are shown in Figure 8.21

In all cases the arrow indicates the direction of motion. The angle в, which is considered to be small, is the angle between a leading edge and the normal, in the plane of the wing, to the direction of motion.

In all the cases in Figure 8.21 it is assumed that the bound vortices are parallel to the leading edge, so that in particular for wing shape (c) the bound vortex lines are also arrow-shaped.

The following are the two main approaches employed in Blenk’s method:

Figure 8.21 Some wing shapes meant for Blenk’s method; (a) rectangular wing moving in the plane of symmetry, (b) a skew wing in the shape of a parallelogram moving parallel to its shorter sides, (c) a symmetrical arrow-shaped wing (sweep-back), (d) rectangular wing side-slipping.