Category Aerodynamics for Engineering Students

Considering the jet flap (see also Section 8.4.2) as a high-velocity sheet of air issuing from the trailing edge of an aerofoil at some downward angle т to the chord line of the aerofoil, an analysis can be made by replacing the jet stream as well as the aerofoil by a vortex distribution.*

Fig. 4.17

The flap contributes to the lift on two accounts. Firstly, the downward deflection of the jet efflux produces a lifting component of reaction and secondly, the jet affects the pressure distribution on the aerofoil in a similar manner to that obtained by an addition to the circulation round the aerofoil.

The jet is shown to be equivalent to a band of spanwise vortex filaments which for small deflection angles r can be assumed to lie along the O. v axis (Fig. 4.17). In the analysis, which is not proceeded with here, both components of lift are considered in order to arrive at the expression for Cl’.

Cl — 4ттАот + 2тт( l + 250)n

where A о and Bo are the initial coefficients in the Fourier series associated with the deflection of the jet and the incidence of the aerofoil respectively and which can be obtained in terms of the momentum (coefficient) of the jet.

It is interesting to notice in the experimental work on jet flaps at National Gas Turbine Establishment, Pyestock, good agreement was obtained with the theoretical Cl even at large values of r.

A flapped-aerofoil characteristic that is of great importance in stability and control
calculations, is the aerodynamic moment about the hinge line, shown as Я in Fig. 4.16.
Taking moments of elementary pressures p, acting on the flap about the hinge,

where p — pUk and xf = x — (1 — F)c. Putting

c c c

xf = – (1 – cos 9) — – (1 — cos ф) = – (cos ф – cos 9)

and к from Eqn (4.51):

From Eqn (4.54):

1

b=–p; (1 + cos 0) (cos 0 – cos0)d#

J Ф

giving

b — — тто {2(7г – 0)(2cos0 – 1) + 4 sin ф – sin 2ф} 4 Fl

Similarly from Eqn (4.54)

bi = x соетсіеп1 °f V in Eqn (4.54)

This somewhat unwieldy expression reduces to[16]

The parameter a = dC^/da is 2тг and «2 = dCt/ch) from Eqn (4.52) becomes

«2 = 2(7г – ф + sin ф) (4-57)

Thus thin aerofoil theory provides an estimate of all the parameters of a flapped aerofoil.

Note that aspect-ratio corrections have not been included in this analysis which is essentially two-dimensional. Following the conclusions of the finite wing theory in Chapter 5, the parameters a, «2, b and 62 may be suitably corrected for end effects. In practice, however, they are always determined from computational studies and wind-tunnel tests and confirmed by flight tests.

Thin aerofoil theory lends itself very readily to aerofoils with variable camber such as flapped aerofoils. The distribution of circulation along the camber line for the general aerofoil has been found to consist of the sum of a component due to a flat plate at incidence and a component due to the camber-line shape. It is sufficient for the assumptions in the theory to consider the influence of a flap deflection as an addition to the two components above. Figure 4.14 shows how the three contribu­tions can be combined. In fact the deflection of the flap about a hinge in the camber line effectively alters the camber so that the contribution due to flap deflection is the effect of an additional camber-line shape.

The problem is thus reduced to the general case of finding a distribution to fit a camber line made up of the chord of the aerofoil and the Пар chord deflected through rj (see Fig. 4.15). The thin aerofoil theory does not require that the leading and/or trailing edges be on the x axis, only that the surface slope is small and the displacement from the x axis is small.

With the camber defined as he the slope of the part AB of the aerofoil is zero, and that of the flap – h/F. To find the coefficients of к for the Пар camber, substitute these values of slope in Eqns (4.41) and (4.42) but with the limits of integration confined to the parts of the aerofoil over which the slopes occur. Thus

(4.48)

where ф is the value of в at the hinge, i. e.

(I – F)c = ^(1 – cos</>)

Fig. 4.14 Subdivision of lift contributions to total lift of cambered flapped aerofoil

whence cos ф — 2F — 1. Evaluating the integral

£

-к) F

i. e. since all angles are small h/F = tan 77 ~ 77, so

‘lo =

Similarly from Eqn (4.42)

i4„=-|y 0 cos n(?d(? + J —^cosn6d0|

Thus

7Г 7Г

The distribution of chordwise circulation due to flap deflection becomes

and this for a constant incidence a is a linear function of 77, as is the lift coefficient, e. g. from Eqn (4.43)

Note that a positive flap deflection, i. e. a downwards deflection, decreases the moment coefficient, tending to pitch the main aerofoil nose down and vice versa.

In general, the camber line can be any function of x (or 9) provided that yc = 0 at x = 0 and c (i. e. at 9 = 0 and 7г). When trigonometric functions are involved a convenient way to express an arbitrary function is to use a Fourier series. Accord­ingly, the slope of the camber line appearing in Eqn (4.22) can be expressed in terms of a Fourier cosine series

Sine terms are not used here because practical camber lines must go to zero at the leading and trailing edges. Thus yc is an odd function which implies that its derivative is an even function.

Equation (4.22) now becomes

thin aerofoil is constant, depending on the camber parameters only, and the quarter chord point is therefore the aerodynamic centre.

It is apparent from this analysis that no matter what the camber-line shape, only the first three terms of the cosine series describing the camber-line slope have any influence on the usual aerodynamic characteristics. This is indeed the case, but the terms corresponding to n > 2 contribute to the pressure distribution over the chord.

Owing to the quality of the basic approximations used in the theory it is found that the theoretical chordwise pressure distribution p does not agree closely with

experimental data, especially near the leading edge and near stagnation points where the small perturbation theory, for example, breaks down. Any local inaccuracies tend to vanish in the overall integration processes, however, and the aerofoil coefficients are found to be reliable theoretical predictions.

In this simple case the camber Une is straight along Ox, and dycjdx = 0. Using Eqn (4.23) the general equation (4.22) becomes

к sin 0

(cose — cos 0i)

What value should к take on the right-hand side of Eqn (4.27) to give a left-hand side which does not vary with x or, equivalently, 0? To answer this question consider the result (4.25) with n = 1. From this it can be seen that

cos 0d0

(cos0 – cos 01)

Comparing this result with Eqn (4.27) it can be seen that if к = ki = 2Ua cos 0/sin 0 it will satisfy Eqn (4.27). The only problem is that far from satisfying the Kutta condition (4.24) this solution goes to infinity at the trailing edge. To overcome this problem it is necessary to recognize that if there exists a function k2 such that

then к = ki + кг will also satisfy Eqn (4.27).
Consider Eqn (4.25) with n = 0 so that

where C is an arbitrary constant.

Thus the complete (or general) solution for the flat plate is given by

The Kutta condition (4.24) will be satisfied if C = 2Ua giving a final solution of

(1 + cos 0)

sin0

Aerodynamic coefficients for a flat plate

The expression for к can now be put in the appropriate equations for Uft and moment by using the pressure:

and this shows a fixed centre of pressure coincident with the aerodynamic centre as is necessarily true for any symmetrical section.

In the general case Eqn (4.22) must be solved directly to determine the function A(.) that corresponds to a specified camber-line shape. Alternatively, the inverse design problem may be solved whereby the pressure distribution or, equivalently, the tangential velocity variation along the upper and lower surfaces of the aerofoil is given. The corresponding k(.) may then be simply found from Eqns (4.19) and

(4.20) . The problem then becomes one of finding the requisite camber line shape from Eqn (4.22). The present approach is to work up to the general case through the simple case of the flat plate at incidence, and then to consider some practical applications of the general case. To this end the integral in Eqn (4.22) will be considered and expressions for some useful definite integrals given.

In order to use certain trigonometric relationships it is convenient to change variables from. v to 9, through, v = (c/2)(l – cost)), and, V| to 0|, then the limits change as follows:

9 ~ 0 —> 7Г as. y ~ 0 —» c, and

d. v = – sin 9d9
2 [15]

The derivations of these results are given in Appendix 3. However, it is not necessary to be familiar with this derivation in order to use Eqns (4.25) and (4.26) in applica­tions of the thin-aerofoil theory.

For the development of this theory it is assumed that the maximum aerofoil thickness is small compared to the chord length. It is also assumed that the camber-line shape only deviates slightly from the chord line. A corollary of the second assumption is that the theory should be restricted to low angles of incidence.

Consider a typical cambered aerofoil as shown in Fig. 4.10. The upper and lower curves of the aerofoil profile are denoted by yu and vi respectively. Let the velocities in the. v and г directions be denoted by и and v and write them in the form:

и = U cos a + и
v — U sin a + v’

u’ and v’ represent the departure of the local velocity from the undisturbed free stream, and are commonly known as the disturbance or perturbation velocities. In fact, thin-aerofoil theory is an example of a small perturbation theory.

The velocity component perpendicular to the aerofoil profile is zero. This constitutes the boundary condition for the potential flow and can be expressed mathematically as:

—и sin /3 + v cos /3 = 0 at у = yu and y

Dividing both sides by cos /3, this boundary condition can be rewritten as

—(t/cosa + m’)^+ t/sina + v’ = 0 at y = yu and yi (4.11)

By making the thin-aerofoil assumptions mentioned above, Eqn (4.11) may be simplified. Mathematically, these assumptions can be written in the form

Note that the additional assumption is made that the slope of the aerofoil profile is small. These thin-aerofoil assumptions imply that the disturbance velocities are small compared to the undisturbed free-steam speed, i. e.

u’ and v4t/

Given the above assumptions Eqn (4.11) can be simplified by replacing cos a and sin a by 1 and a respectively. Furthermore, products of small quantities can be neglected, thereby allowing the term u! dy/dx to be discarded so that Eqn (4.11) becomes

(4.12)

One further simplification can be made by recognizing that if _yu and yi c then to a sufficiently good approximation the boundary conditions Eqn (4.12) can be applied at у = 0 rather than at у = yu or y.

Since potential flow with Eqn (4.12) as a boundary condition is a linear system, the flow around a cambered aerofoil at incidence can be regarded as the superposition of two separate flows, one circulatory and the other non-circulatory. This is illustrated in Fig. 4.11. The circulatory flow is that around an infinitely thin cambered plate and the non-circulatory flow is that around a symmetric aerofoil at zero incidence. This superposition can be demonstrated formally as follows. Let

yu=yc+yt and yi=yc-yt

у = _yc(x) is the function describing the camber line and у = yt = (yu — yi)/2 is known as the thickness function. Now Eqn (4.12) can be rewritten in the form

where the plus sign applies for the upper surface and the minus sign for the lower surface.

Thus the non-circulatory flow is given by the solution of potential flow subject to the boundary condition v’ = ±Udyt/dx which is applied at у = 0 for 0 < x < c. The solution of this problem is discussed in Section 4.9. The lifting characteristics of the aerofoil are determined solely by the circulatory flow. Consequently, it is the solution of this problem that is of primary importance. Turn now to the formulation and solution of the mathematical problem for the circulatory flow.

It may be seen from Sections 4.1 and 4.2 that vortices can be used to represent lifting flow. In the present case, the lifting flow generated by an infinitely thin cambered plate at incidence is represented by a string of line vortices, each of infinitesimal strength, along the camber line as shown in Fig. 4.12. Thus the camber line is replaced by a line of variable vorticity so that the total circulation about the chord is the sum of the vortex elements. This can be written as

(4.13)

where к is the distribution of vorticity over the element of camber line 6s and circulation is taken as positive in the clockwise direction. The problem now becomes one of determining the function k{x) such that the boundary condition

is satisfied as well as the Kutta condition (see Section 4.1.1).

There should be no difficulty in accepting this idealized concept. A lifting wing may be replaced by, and produces forces and disturbances identical to, a vortex system, and Chapter 5 presents the classical theory of finite wings in which the idea of a bound vortex system is fully exploited. A wing replaced by a sheet of spanwise vortex elements (Fig. 5.21), say, will have a section that is essentially that of the replaced camber line above.

The leading edge is taken as the origin of a pair of coordinate axes x and y; Ox along the chord, and Оу normal to it. The basic assumptions of the theory permit the variation of vorticity along the camber line to be assumed the same as the variation along the Ox axis, i. e. 6s differs negligibly from 6x, so that Eqn (4.13) becomes

Г = f kdx
Jo

Hence from Eqn (4.10) for unit span of this section the lift is given by

l = pUT = pU f kdx Jo

Alternatively Eqn (4.16) could be written with pUk = p:

1=1 pUk dx = pdx Jo Jo

Now considering unit spanwise length, p has the dimensions of force per unit area or pressure and the moment of these chordwise pressure forces about the leading edge or origin of the system is simply

Note that pitching ‘nose up’ is positive.

The thin wing section has thus been replaced for analytical purposes by a line discontinuity in the flow in the form of a vorticity distribution. This gives rise to an overall circulation, as does the aerofoil, and produces a chordwise pressure variation.

For the aerofoil in a flow of undisturbed velocity U and pressure po, the insert to Fig. 4.12 shows the static pressures pi and pj above and below the element 6s where the local velocities are U + щ and U + м2, respectively. The overall pressure difference p is p2 — p. By Bernoulli:

Pi +2^(^ + Mi)2 —Po + 2 pU2 Pi—^p(JJ + м2)2 = po —^pU2

1 m2 , /«12 ( Рг-P^jpU + -(■

and with the aerofoil thin and at small incidence the perturbation velocity ratios щ jU and uijU will be so small compared with unity that (щ/U)2 and (щ/U)2 are neglected compared with щ/U and щ! U, respectively. Then

P=P2~P =ри(щ – u2) (4.19)

The equivalent vorticity distribution indicates that the circulation due to element 6s is к 6x (6x because the camber line deviates only slightly from the Ox axis). Evaluating the circulation around 6s and taking clockwise as positive in this case, by taking the algebraic sum of the flow of fluid along the top and bottom of 6s, gives

k6x = +(U + щ)6х — (U + ui)6x = (mj — U2)6x (4-20)

Comparing (4.19) and (4.20) shows that p = pUk as introduced in Eqn (4.17).

For a trailing edge angle of zero the Kutta condition (see Section 4.1.1) requires mi = м2 at the trailing edge. It follows from Eqn (4.20) that the Kutta condition is satisfied if

к = 0 at x = c (4-21)

The induced velocity v in Eqn (4.14) can be expressed in terms of k, by considering the effect of the elementary circulation кбх at x, a distance x — x from the point considered (Fig. 4.13). Circulation кбх induces a velocity at the point x equal to

1 кбх 2ж x — x

from Eqn (4.5).

The effect of all such elements of circulation along the chord is the induced velocity v’ where

, _ J_ Ґ kdx 2nJo x-xi

k d. v

(4.22)

2tt

■V – – V і

The solution for к d. v that satisfies Eqn (4.22) for a given shape of camber line (defining dVc/d. v) and incidence can be introduced in Eqns (4.17) and (4.18) to obtain the lift and moment for the aerofoil shape. The characteristics С/. and C. v/Lt follow directly and hence kep, the centre of pressure coefficient, and the angle for zero lift.

The first successful aerofoil theory was developed by Zhukovsky.* This was based on a very elegant mathematical concept – the conformal transformation – that exploits the theory of complex variables. Any two-dimensional potential flow can be repre­sented by an analytical function of a complex variable. The basic idea behind Zhukovsky’s theory is to take a circle in the complex £ = (£ + і7;) plane (noting that here C does not denote vorticity) and map (or transform) it into an aerofoil-shaped contour. This is illustrated in Fig. 4.8.

A potential flow can be represented by a complex potential defined by Ф = ф + it’ where, as previously, ф and </> are the velocity potential and stream function respect­ively. The same Zhukovsky mapping (or transformation), expressed mathematically as

C2

(where C is a parameter), would then map the complex potential flow around the circle in the f-plane to the corresponding flow around the aerofoil in the z-plane. This makes it possible to use the results for the cylinder with circulation (see Section 3.3.10) to calculate the flow around an aerofoil. The magnitude of the circulation is chosen so as to satisfy the Kutta condition in the r-plane.

From a practical point of view Zhukovsky’s theory suffered an important draw­back. It only applied to a particular family of aerofoil shapes. Moreover, all the

members of this family of shapes have a cusped trailing edge whereas the aerofoils used in practical aerodynamics have trailing edges with finite angles. Karman and Trefftz[13] later devised a more general conformal transformation that gave a family of aerofoils with trailing edges of finite angle. Aerofoil theory based on conformal transformation became a practical tool for aerodynamic design in 1931 when the American engineer TheodorsenT developed a method for aerofoils of arbitrary shape. The method has continued to be developed well into the second half of the twentieth century. Advanced versions of the method exploited modern computing techniques like the Fast Fourier Transform.[14]

If aerodynamic design were to involve only two-dimensional flows at low speeds, design methods based on conformal transformation would be a good choice. How­ever, the technique cannot be extended to three-dimensional or high-speed flows. For this reason it is no longer widely used in aerodynamic design. Methods based on conformal transformation are not discussed further here. Instead two approaches, namely thin aerofoil theory and computational boundary element (or panel) methods, which can be extended to three-dimensional flows will be described.

The Zhukovsky theory was of little or no direct use in practical aerofoil design. Nevertheless it introduced some features that are basic to any aerofoil theory. Firstly, the overall lift is proportional to the circulation generated, and secondly, the magni­tude of the circulation must be such as to keep the velocity finite at the trailing edge, in accordance with the Kutta condition.

It is not necessary to suppose the vorticity that gives rise to the circulation be due to a single vortex. Instead the vorticity can be distributed throughout the region enclosed by the aerofoil profile or even on the aerofoil surface. But the magnitude of circulation generated by all this vorticity must still be such as to satisfy the Kutta condition. A simple version of this concept is to concentrate the vortex distribution on the camber line as suggested by Fig. 4.9. In this form, it becomes the basis of the classic thin aerofoil theory developed by Munk* and Glauert.8

Glauert’s version of the theory was based on a sort of reverse Zhukovsky trans­formation that exploited the not unreasonable assumption that practical aerofoils are

Fig. 4.9 thin. He was thereby able to determine the aerofoil shape required for specified aerofoil characteristics. This made the theory a practical tool for aerodynamic design. However, as remarked above, the use of conformal transformation is restricted to two dimensions. Fortunately, it is not necessary to use Glauerl’s approach to obtain his final results. In Section 4.3, later developments are followed using a method that does not depend on conformal transformation in any way and. accordingly, in principle at least, can be extended to three dimensions.

Thin aerofoil theory and its applications are described in Sections 4.3 to 4.9. As the name suggests the method is restricted to thin aerofoils with small camber at small angles of attack. This is not a major drawback since most practical wings are fairly thin. A modern computational method that is not restricted to thin aerofoils is described in Section 4.10. This is based on the extension of the panel method of Section 3.5 to lifting flows. It was developed in the late 1950s and early 1960s by Hess and Smith at Douglas Aircraft Company.

In Eqn (3.52) it was shown that the lift / per unit span and the circulation Г of a spinning circular cylinder are simply related by

l = pVT

where p is the fluid density and V is the speed of the flow approaching the cylinder. In fact, as demonstrated independently by Kutta[12] and Zhukovsky*, the Russian physi­cist, at the beginning of the twentieth century, this result applies equally well to a cylinder of any shape and, in particular, applies to aerofoils. This powerful and useful result is accordingly usually known as the Kutta-Zhukovsky Theorem. Its validity is demonstrated below.

The lift on any aerofoil moving relative to a bulk of fluid can be derived by direct analysis. Consider the aerofoil in Fig. 4.7 generating a circulation of Г when in a stream of velocity V, density p, and static pressure po. The lift produced by the aerofoil must be sustained by any boundary (imaginary or real) surrounding the aerofoil.

For a circuit of radius r, that is very large compared to the aerofoil, the lift of the aerofoil upwards must be equal to the sum of the pressure force on the whole periphery of the circuit and the reaction to the rate of change of downward momen­tum of the air through the periphery. At this distance the effects of the aerofoil thickness distribution may be ignored, and the aerofoil represented only by the circulation it generates.

Fig. 4.7

The vertical static pressure force or buoyancy 4 on the circular boundary is the sum of the vertical pressure components acting on elements of the periphery. At the element subtending 66 at the centre of the aerofoil the static pressure is p and the local velocity is the resultant of V and the velocity v induced by the circulation. By Bernoulli’s equation

Po+jpV2 =p + jp[V2 + v2 + 2Fvsin 6]

giving

p = po – pVvsin#

if v2 may be neglected compared with V2, which is permissible since r is large.

The vertical component of pressure force on this element is

—pr sin 6 66

and, on substituting for p and integrating, the contribution to lift due to the force acting on the boundary is

л2тг

k = ~ (po – pFvsin6)rsin 6d6

Jo (4.7)

+pVvnr

with po and r constant.

The mass How through the elemental area of the boundary is given by pVr cos ObO. This mass flow has a vertical velocity increase of v cos 9, and therefore the rate of change of downward momentum through the element is —pVvr cos2 9b9 therefore by integrating round the boundary, the inertial contribution to the lift, /j, is

/і = + / /э Kit cos2 9d9

J о

= pVvrTT

Thus the total lift is:

I = 2pVvrir

From Eqn (4.5):

2nr

giving, finally, for the lift per unit span, /:

I = pVT

This expression can be obtained without consideration of the behaviour of air in a boundary circuit, by integrating pressures on the surface of the aerofoil directly.

It can be shown that this lift force is theoretically independent of the shape of the aerofoil section, the main effect of which is to produce a pitching moment in potential flow, plus a drag in the practical case of motion in a real viscous fluid.

From the discussion above it is evident that circulation and vorticity, introduced in Section 2.7, are key concepts in understanding the generation of lift. These concepts are now explored further, and the precise relationship between the lift force and circulation is derived.

Consider an imaginary open curve AB drawn in a purely potential flow as in Fig. 4.3a. The difference in the velocity potential ф evaluated at A and В is given by the line integral of the tangential velocity component of flow along the curve, i. e. if the flow velocity across AB at the point P is q, inclined at angle a to the local tangent, then

Фа — Фв = Я cos a ds (4.1)

J AB

which could also be written in the form

Фа~Фв= {udx+vdy)

J AB

Equation (4.1) could be regarded as an alternative definition of velocity potential.

Consider next a closed curve or circuit in a circulatory flow (Fig. 4.3b) (remember that the circuit is imaginary and does not influence the flow in any way, i. e. it is not a boundary). The circulation is defined in Eqn (2.83) as the line integral taken around the circuit and is denoted by Г, i. e.

Г = cos ads or T = j>(udx + vdy) (4.2)

It is evident from Eqns (4.1) and (4.2) that in a purely potential flow, for which Фа must equal фв when the two points coincide, the circulation must be zero.

Circulation implies a component of rotation of flow in the system. This is not to say that there are circular streamlines, or that elements of fluid are actually moving around some closed loop although this is a possible flow system. Circulation in a flow means that the flow system could be resolved into a uniform irrotational portion and a circulating portion. Figure 4.4 shows an idealized concept. The implication is that if circulation is present in a fluid motion, then vorticity must be present, even though it may be confined to a restricted space, e. g. as in the core of a point vortex. Alternatively, as in the case of the circular cylinder with circulation, the vorticity at the centre of the cylinder may actually be excluded from the region of flow con­sidered, namely that outside the cylinder.

Fig. 4.4

Consider this by the reverse argiunent. Look again at Fig. 4.3b. By definition the velocity potential of C relative to A (^ca) must be equal to the velocity potential of C relative to В (фсъ) in a potential flow. The integration continued around ACB gives

Г = Фса ± Фсъ = 0

This is for a potential flow only. Thus, if Г is finite the definition of the velocity potential breaks down and the curve ACB must contain a region of rotational flow. If the flow is not potential then Eqn (ii) in Section 3.2 must give a non-zero value for vorticity.

An alternative equation for Г is found by considering the circuit of integration to consist of a large number of rectangular elements of side 6x 6y (e. g. see Section 2.7.7 and Example 2.2). Applying the integral Г = J (u dx + v dy) round abed, say, which is the element at P(x, y) where the velocity is и and v, gives (Fig. 4.5).

The sum of the circulations of all the areas is clearly the circulation of the circuit as a whole because, as the ДГ of each element is added to the ДГ of the neighbouring element, the contributions of the common sides disappear.

Applying this argument from element to neighbouring element throughout the area, the only sides contributing to the circulation when the ATs of all areas are summed together are those sides which actually form the circuit itself. This means that for the circuit as a whole

This shows explicitly that the circulation is given by the integral of the vorticity contained in the region enclosed by the circuit.

Fig. 4.5

If the strength of the circulation Г remains constant whilst the circuit shrinks to encompass an ever smaller area, i. e. until it shrinks to an area the size of a rectangular element, then:

Г = £ x 6x 6y = £ x area of element

Therefore,

Here the (potential) line vortex introduced in Section 3.3.2 will be re-visited and the definition (4.2) of circulation will now be applied to two particular circuits around a point (Fig. 4.6). One of these is a circle, of radius r, centred at the centre of the vortex. The second circuit is ABCD, composed of two circular arcs of radii r and r2 and two radial lines subtending the angle (3 at the centre of the vortex. For the concentric circuit, the velocity is constant at the value

C

where C is the constant value of qr.

Fig. 4.6 Two circuits in the flow around a point vortex

Since the flow is, by the definition of a vortex, along the circle, a is everywhere zero and therefore cos a = 1. Then, from Eqn (4.2)

Now suppose an angle в to be measured in the anti-clockwise sense from some arbitrary axis, such as OAB. Then

dj = rid#

whence

f2irC

Г = – nd# = 2TrC

Jo r і

Since C is a constant, it follows that Г is also a constant, independent of the radius. It can be shown that, provided the circuit encloses the centre of the vortex, the circulation round it is equal to Г, whatever the shape of the circuit. The circulation Г round a circuit enclosing the centre of a vortex is called the strength of the vortex. The dimensions pf circulation and vortex strength are, from Eqn (4.2), velocity times length, i. e. L2T- , the units being m2 s-1. Now Г = 27гС, and C was defined as equal to qr hence

Г = 2-irqr and

Г

2жг

Taking now the second circuit ABCD, the contribution towards the circulation from each part of the circuit is calculated as follows:

(i) Radial line AB Since the flow around a vortex is in concentric circles, the velocity vector is everywhere perpendicular to the radial line, i. e. a = 90°, cos a = 0. Thus the tangential velocity component is zero along AB, and there is therefore no contribution to the circulation.

(ii) Circular arc BC Here a = 0, cos a — 1. Therefore

(iii)

Radial line CD As for AB, there is no contribution to the circulation from this part of the circuit.

(iv)

Circular arc DA Here the path of integration is from D to A, while the direction of velocity is from A to D. Therefore a = 180°, cos a = — 1. Then

Therefore the total circulation round the complete circuit ABCD is

(4.6)

Thus the total circulation round this circuit, that does not enclose the core of the vortex, is zero. Now any circuit can be split into infinitely short circular arcs joined by infinitely short radial lines. Applying the above process to such a circuit would lead to the result that the circulation round a circuit of any shape that does not enclose the core of a vortex is zero. This is in accordance with the notion that potential flow is irrotational (see Section 3.1).