KUNDU Fluid Mechanics 2 Episode 5 potx

For n 1 the irrotational flow velocity is infinitc at the origin, implying that thc boundary streamline + = 0 accelerates before rcaching this point and dccclcrslles alter it.. It will

1 54 lirututional How

which implies

(6.18)

It is easy to show that taking Sz parallel to the y-axis leads to an identical result The

dcrivativc d w l d z is therefore a complex quantity whose real and imaginary parts give Cartcsian components of the local velocity; d w / d z is therefore called the complex vebciry Ifthc local velocity vector has a magnitude y and an angle a! with the x-axis, then

(6.19)

It may be considered rcmarkable that any twice differentiable function w ( z ) , z =

x + iy is an identical solution to Laplace's equation in the plane ( x , y ) A general function of the two variables ( x , y) may be written as f ( z , z*) where z* = x - iy is the complex conjugate of z It is the very special case when f ( z , z*) = w ( z ) alone

that we consider here

As Laplace's equation is linear, solutions may be superposed That is, the sums

of clemental solutions are also solutions Thus, as we shall see, flows over specific shapes may be solved in this way

4 Flow a1 a Wall Angle

Consider the complex potential

qi = Ar" cos n8 = Ar" sin ne (6.21)

For a given n, lines of constant II can be plotted Equalion (6.21) shows that II = 0 for all values of r on lincs 8 = 0 and 8 = n / n As any streamline, including the $ = 0 line, can be regarded as a rigid boundary in the z-plane, it is apparent that Eq (6.20)

is the complcx potential for flow between two plane boundaries of included angle

a! = n / n Figure 6.4 shows the flow patterns for various values of n Flow within

a certain sector of the z-plane only is shown; that within other scctors can bc found

by symmetry It is clear hat thc walls form an angle larger than 180" for n e 1 and

an angle smaller than 180" lor n > 1 The complex velocity in terms of a! = n / n is

which shows that at thc origin d w l d z = 0 for a! e K , and diiildz = eo for a! > n

Thus, h e comer is a stagnation paint f o r f i w in a wall angle smaller than 180";

Figure 6.4 Irrotational flow at a wall anglc Equipotcntial lincr arc h h c d

F m 6.5 Stagnation flow itpresented by UI = AzZ

in contrust, it is a point of inJinile velocily for wull angles larger than 180“ In both cases the origin is a singular point

Thc pattcm for n = 1/2 corresponds to flow around a semi-infinite platc Whcn

la = 2, Ihe pattern represcnts flow in a region bounded by perpcndicular walls By

including the field within the second quadrant of the z-planc, ir is clear that n = 2 also represcnts thc flow impinging against a flat wall (Figure 6.5) Tbe streamlincs

and equipotential lines are all rectangular hyperbolas This is called a stagqnafionJluw

bccause it represents llow in thc ncighborhood of the slagnation point of a blunt body Real flows ncar a sharp change in wall slopc arc somewhat different than those shown in Figurc 6.4 For n 1 the irrotational flow velocity is infinitc at the origin, implying that thc boundary streamline (+ = 0) accelerates before rcaching this point and dccclcrslles alter it Bernoulli’s cquation implies that thc pressure force down-

stream of the corner is “adverse” or against the flow It will be shown in Chapter 10

that an adverse pressure gradient causes separation of flow and generation of station-

ary eddies A real flow in a corner with an included angle larger than 180” would

therefore separate at the comer (see the right panel of Figure 6.2)

The real and imaginary parts are

from which the velocity components arc found as

(6.28)

7 lloubbl

A doublet or dipole is obtained by allowing a sourcc and a sink of equal strcngth

to approach each othcr in such a way h a t their slrengths incrcase as thc separation

distance gocs to zero, and that h e product lends to afinite limit l h c complex potential

'Thc argument of transccndcntal functions such as thc logwithm must always he dimcnsionlcss Thus

a consttint must bc d d c d Lo @ in Fi (6.27) to put Ihc logarithm in proper form This is clonc cxplicitlp

when we arc solving a problcm as in Section 10 in what follows

(6.30)

x 2 + ( Y + & ) ’ = ( $ ) 2 -

The streamlines, reprcscntcd by * = const., arc thcrcforc circlcs whose centers lie

on thc y-axis and are tangent Lo the x-axis a1 the origin (Figure 6.8) Dircction of flow

at the origin is along the negative x-axis (pointing outward from the source of the

limiting source-sink pair), which is called the axis of the doublet It is easy to show that (Excrcisc 1) thc doublct flow Eq (6.29) can bc cquivalently defined by superposing

a clockwise vortex of strength -r on thc y-axis at y = E , and a counterclockwisc vortex of strcngth r at y = E

The complex potentials for concentrated source, vortex, and doublet are all sin- gular at the origin It will be shown in the following sections that several interesting flow patterns can be obtained by superposing a uniform flow on thcsc conccntrated singularities

8 Fk,w past a HuJJ-Body

An internsting flow rcsulls lorn superposition of a source and a uniform stream The complex potcntial for a uniform flow of strength U is u; = Ue, which follows from integrating the relation d w / d z = u - iv The complex potential for a source at the origin of strcngth in, immersed in a uniform flow, is

From Eqs (6.12) and (6.13) it is clear that there must be a stagnation point lo the

left ol the source (S in Figure 6.9), wherc thc uniform stream cancels the velocity of flow h m the source Tf thc polar coordinate or the stagnation point is (a, IC), then cancellation of velocity rcquircs

A plot of this smamline is shown in Figure 6.9 It is a semi-infinite body with a

smooth nosc, generally callcd a hay-body Thc stagnation s t r e d i n e divides thc field

The half-width or the body is found to be

m ( x - 6 )

h = r s i n Q =

2Ycu ’ where Eq (6.33) has been used The half-width tends to h,,, = m/2U as H + 0 (Figure 6.9) (This result can also be obtained by noting that mass flux from the source

is contained entirely within thc half-body, rcquiring the balance m = (2hmax)U at

a large downstream distancc where K = U.)

Thc pressure distribution can be found from Bernoulli’s equation

p + 4pq2 = p x + i p U 2

A convenient way of represcnting pressure is through the nondimensional excess

pressurc (called P~ESSKIZ coeflcient)

A plot of C , on the surface of the half-body is given in Figure 6.10, which shows

that there is pressure excess near the nose of the body and a pressure deficit beyond

it Tt is easy to show by integrating p over the surface that the net pressure force is zero (Exercise 2)

9 Flow pas1 a Cimular Cflinder wil/zout Cimulation

The combination of a uniform stream and a doublet with its axis directed against the stream gives the irrotational flow over a circular cylinder, for the doublet strength

I1 Figure 6.10

It is sccn h a t $ = 0 at r = u for all values of H , showing that the streamlinc

$ = 0 represents a circular cylindcr of radius N The streamlinc pattern is shown in Figurc 6.1 1 Flow inside the cuclc has no influcnce on that outsidc the circle Vclocity components are

from which thc flow s p e d on the surfacc of the cylinder is found as

41,- = l ~ e l , - - ~ = 2U sink): (6.36) where what is meant is the positivc value of sin 0 This shows that thcre are stagnation points on the surfxc, whose polar coordinates are (a, 0 ) and ( a , x ) The flow reaches

a maximum vclocity of 2 U at h e top and bottom or the cylindcr

Pressurc distribution on the surface of thc cylinder is given by

Surface distribution of prcssure is shown by thc continuous line in Figure 6.12 Thc symmetry of the distribution shows that therc is no net pressure drag In fact, a general

Figure 6.11 Irrotational flow past a circular cyhder without circulation

result of irrotational flow theory is that a steadily moving body experiences no drag

This result is at variancc with observations and is sometimes known as d’ Alembert’s pcrrdox The existence of tangential stress, or “skin friction,” is not the only reason for the discrepancy For blunt bodies, the major part of the drag comes from separation of the flow from sides and the resulting generation of eddies The surface pressure in the wake is smaller than that predicted by irrotational flow theory (Figure 6.12), resulting

in a pressure drag These facts will be discussed in further detail in Chapter 10 The flow due to a cylinder moving steadily through a fluid appears unsteady to

an observer at rest with respect to the fluid a1 infinity This flow can be obtained by

- + ” +

8 +- c u =

Figure 6.13 Decomposition of irmtational flow pattcm duc to a moving cylindcr

supcrposing a uniform strcam along the negative x direction to the flow shown in Figurc 6.1 1 The resulting instantaneous flow pattcm is simply that of a doublet, as

is clear from thc dccornposition shown in Figure 6.13

It was seen in thc last section that there is no net form on a circular cylindcr in steady

irrotational flow without circulation It will now bc shown that a lateral force, akin

to a lift force on an airfoil, rcsults when circulation is introduccd into the flow Tf

a clockwise line vortex of circulation -r is added to the irrotational flow around

a circular cylinder, the complex potential becomes

ui = U ( :) z + - + -ln(z/u)! 1:

whose imaginary part is

(6.37)

(6.38)

where we have added to 111 the term - ( i r / 2 x ) l n a so that the argumcnl of the logd-

rithm is dimcnsionless, as it must be always

Figurc 6.14 shows thc resulting streamline pattern for \w-ious valucs of r The close sl.reamline spacing and higher velocity on top of thc cylinder is due to the addition of velocity fields of the clockwise vortcx and the uni€orm stream In contrast, the smallcr velocities at the bottom of the cylinder are a result of the vortex field

countcracling the uniform stream Bernoulli’s cquation consequently implics a higher pressurc below thc cylinder and an upward ‘‘lift” lorce

Thc tangential vclocity component at any point in the flow is

At the surface of the cylinder, velocity is entirely tangential and is givcn by

(6.39)

r

ug I r a = -2U sin8 - -,

2rra

Figure6.14 Irrotational flow past a circular cylinder lor differcnl values of circulation Point S reprcscnts

the stagnation point

which vanishes if

r

sine =

For r < 47caU, two values of 0 satisfy Eq (6.40), implying that there are two stag-

nation points on the surface The stagnation points progrcssively move down as

r inmases (Figure 6.14) and coalesce at r = 47caU For r > 4naU, thc stag-

nation point moves out into the flow along the y-axis The radial distance of the

stagnation point in this case is found from

r

ueIs=-rjz = u ( 1 + - - - ::) 2nr = 0

This gives

r = - [r f Jr* - (415au)q 47c u

one root of which is r > a; the other root corresponds to a stagnation point inside the

cylinder

Prcssure is found from the Bernoulli equation

P + P92/2 = poc + p u 2 / 2

Using Eq (6.39), the surface pressure is found to be

p , , = p o o + ~ p - 2 ~ s i n e - - 2Yra )'I (6.41) The symmetry of Row about the y-axis implies that the pressure force on the cylinder has no component along the x-axis The pressure force along the y-axis, called the

"lift" force in aerodynamics, is (Figure 6.15)

L = - 12" pr=" sin e de

Substituting Eq (6.41), and carrying out the integral, we finally obtain

L = pur, (6.42) where we havc used

sin e de = 6" sin3 e de = 0

It is shown in the following section that Eq (6.42) holds for irrotational flows around

m y two-dimensional shape, not just circular cylinders The rcsult that lift force is proportional to circulation is of fundamental importance in aerodynamics Rela- tion Eq (6.42) wa% proved independently by the German mathematician, Wilhelm Kuttsl(1902), and the Russian aerodynamist, Nikolai Zhukhovsky ( 1 906); it is called thc Kufiu-Zhukhovsky lift theorem (Older western texts translitcrated Zhukhovsky's name as Joukowsky.) The intcmsting question of how certain two-dimcnsional shapes, such as an aidoil, develop circulation when placed in a stream is discussed in Chap- ter 15 It will be shown then: that fluid viscosity is responsible for the development of circulation The magnitude of circulation, however, is independent of viscosity, and depends on flow speed U and the shape and "attitude" of the body

For a circular cylinder, however, the only way to develop circulation is by rotating

it in a flow stream Although viscous effects arc important in this case, the observed

'f

Figure 6.15 Calculation ofprerrurc force on a circular cylindcr

166 Inwlalinud Hou:

pattern for large values of cylinder rotation displays a striking similarity to the ideal flow pattern for r > 47ruU; see Figurc 3.25 in thc book by Prdndtl (1952) For

lower rates of cylinder rotation, the retarded flow in the boundary layer is not able

to ovcrcorne the adverse pressure gradicnt behind the cylinder, leading to scparation; the rcal flow is therefore rather unlike the irrotational pattern However, even in the

presence of separation, observed speeds are higher on the upper surface of thc cylinder, implying a lift force

A second reason for generating lift on a rotating cylinder is the asymmewy gen- erated due to delay of scparation on the upper surface of the cylinder The resulting asymmetry generates a lift force The contribution of this mechanism is small for two-dimensional objects such as the circular cylinder, but it is the only mechanism for side forces experienced by spinning the-dimensional objects such as soccer, tcnnis and golf balls The interesting question of why spinning balls follow curved paths is discussed in Chapter 10, Scction 9 Thc lateral lorcc experienced by rotating bodics is called the Mugnus efect

The nonuniqueness of solution for two-dimensional potential flows should be noted in the example we havc considered in this section It is apparent that solutions for various values of r all satisfy the same boundary condition on the solid surfacc (namely, no normal flow) and at infinity (namely, u = U), and there is no way to detcrmine the solution simply from the boundary conditions A general result is that solutions of the Laplace equation in a multiply connected wgion are nonunique This

is explaincd further in Swtion 15

In the precedmg section we demonstrated that the drdg on a circular cylinder is zero

and the lift equals L = pur We shall now demonstrate that these results are valid for cylindrical shapes of arhifrtrry cross section (The word “cylidcr” refers to any

planc two-dimensional body, not just to those with circular cross sections.)

B l d u s Theorem

Considcr a general cylindrical body, and let D and L be thc x and y components of

thc force excrted on it by the surrounding fluid; we rcfer to D as “drdg” and L as

“lift.” Because only normal pressures are exerted in inviscid flows, the forces on a

surfacc elemenl dz are (Figure 6.16)

P h

Figure 6.16 Forcer exerted on an clcmcnl of a body

where C denotes a counterclockwise contour coinciding with the body surface Neglecting gravity, the pressurc is given by the Bernoulli equation

I

~ 3+ o T P ~ ’ = p + $p(u* + v2) = p + $ p ( u + i v ) ( u - i v )

Substituting for p in Eq (6.43), we obtain

D - i L = -i k [pm + 4pU’ - i p ( ~ + i~)(u - i v ) ] d z * , (6.44) Now the integral of the constant term ( p m + i p U 2 ) around a closed contour is zero

Also, on the body surface the velocity vector and the surface element d z are parallel (Figure 6.16), so that

(u + iu) de* = (u - iu) d z

Equation (6.44) then becomes

(6.45)

where we have introduced the complex velocity d w l d z = u - i v Equation (6.45)

is called the Blcrsius theorem, and applies to any plane steady irrotational flow The integral need not be camed out along the contour of the body because the theory

of complex variables shows that any contour surmunding the body cun be chosen,

providcd that there are no singularities between thc body and the contour chosen

Kutta-Zhukhovsky Lift Theorem

We now apply the Blasius theorem to a skady flow around an arbitrary cylindrical body, around which there is a clockwise circulation r The velocity at inhity has

a magnitudc U and is directcd along the x-axis The flow can be considered a supcr-

position of a uniform stream and a set of singularities such as vortcx, doublet, source, and sink

As there are no singularities outside the body, we shall take the contour C in the Blasius theorem at a very large distance from the body From large distances, all

singularities appear to be located near the origin z = 0 The complex potential is then

should set m = 0 The Blasius theorem, Eq (6.45), Lhcn becomes

(6.46)

To carry out the contour integral in Eq (6.46), wc simply have to find the coeffi-

cient of the term proportional to 1 /L in the integrand The coefficient of 1 /z in a power series expansion for f (z) is called the residue of f(z) at z = 0 It is shown in complex variable theory that the contour integral of a function f ( z ) around the contour C is 2ni times the sum of the residues at the singularities within C:

f ( z ) d z = 2rri[sum of residues]

The residue of the intcgrand in Eq (6.46) is easy to find Clearly the term p / z 2 does

not contribute to the residue Completing the square (U + i r / 2 n z ) ' , we see that the coefficient of 1 / z is i r U/rr This gives

which shows that

D = 0,

L = pur (6.47)

I

The first of these equations states that there is no drag experienced by a body in

steady two-dimensional irrotational flow The second equation shows that there is a

lift force L = pur perpendicular to the stream, experienced by a two-dimensional

body of arbitrary cross section This result is called the Kutba-Zhuwlovsky lzft the- orem, which was demonstrated in the preceding scction for flow around a circular

cylinder The result will play a fundamental role in our study of flow around airfoil shapes (Chaptcr 15) We shall sec that the circulation dcveloped by an airfoil is ncarly proportiofid to U , so that thc lift is nearly proportional to U 2

Thc following points can also be dernonstratcd First, irrotational flow over a finite three-dimensional object has no circulation, and there can be no nct force on the body in steady statc Second, in an unsteady flow a force is required to push a body,

essentially because a mass of fluid has to be accclerated from rest

Let us redrive the Kutta-Zhukhovsky lift theorem from considerations of vector calculus without referencc to complex variablcs From Eqs (4.28) and (4.33), for

steady flow with no body forccs, and with I the dyadic equivalent of the Kronecker delta Sij

With r = xi, + yiyr dr = dxi, +dyiy = ds, dA1 = ds x i, 1 = -iy dx +ix dy

Now let u = Ui, + u', where u' + 0 a, r 4 30 at least as fast as 1 / r Substituting for uu and u2 in the intcgral for Fg, wc find

Fu =-.ll (VUi,i, + Uix(ufix + diY) + (u'ix + diy)ixU

+ ufuf + (ixix + iyiy)[po/p - u2/2 - UU'

- (ua + vf2)/21 (-ir d x + i, d y ) }

F i g m 6.17 Domain or integration for the Kulltl-Zhukhovsky theorem

Let r += 00 so that the contour C is far from the body The constant terms U2,

p o / p , -U2/2 integrate to zero around the closcd path Thc quadratic terms u’u’:

(uR + vR)/2 5 I / r 2 as r + oc and thc perimeter of the contour increases only

as r Thus the quadratic terms + 0 as r + o Separating the force into x and y components,

FB = -i,pU [(u’dy - v’dx) + (u’dy - u’dy)] - iypU (u’dy + u’dx)

We note that the first intcgrand is u’ - ds x i,, and that we m a y add the constant

Vi, to each of the integrands because thc integration of a constant velocity over a closed contour or surface will result in zero force The integrals for the force then become

F g = -ixpU (Vi, + u’) - dAl - iypU (Vi, + u’) - ds

The first integral is zero by Eq (4.29) (as a consequence of mass conservation for constant density flow) and the second is the circulation r by definition Thus,

F g = -iYpUr (force/unit depth),

where r is positive in the counterclockwise sense We see that there is no force component in the dircction af motion (drag) undcr the assumptions necessary for the derivation (steady, inviscid, no body forces, constant density, two-dimensional, uniform at infinity) that were bclieved to be valid to a reasonable approximation for

a wide varicty of flows Thus it was labeled a paradox-d’ Alembert’s paradox (Jean

Lc Rond d’Alembert, 16 November 1717-29 October 1783)

12 Soume neur n Wall: MeChod oJlmages

The melhod of imagcs is a way of determining a flow field due to one or more

singularides near a wall It was introduced in Chapter 5 , Section 7, where vortices

near a wall were examined We found that the flow due to a line vortex near a wall can

be found by omitting the wall and introducing instead a vortex of opposite strength

at the “image point.” The combination gencrates a straight streamline at the location

of the wall, thereby satisfying the boundary condition

Another example of this technique is given here, namely, the flow due to a line source at a distancc u from a straight wall This flow can be simulated by introducing

an imagc source of the same strength and sign, so that thc complcx potential is

Wc know that the logarithm of any complex quantity C = I< I exp ( i Q ) can be written

as In 5‘ = In 15 I + io The imaginary part of Eq (6.48) is lhereforc

@ = - tan-’

2?c x2 - y 2 - u2

Y

Figorc 6.18 Irrotational flow due to two equal souzccs

from which the equation of streamlines is found a,

The streamline pattern is shown in Figure 6.1 8 The x and y axes form part of the

streamline pattern, with the origin as a stagnation point It is clear that the complex potential Eq (6.48) represents three interesting flow situations:

( I ) flow due to two equal sourCes (entire Figurc 6.18);

(2) Bow due to a source near a plane wall (right half of Figure 6.18); and (3) flow through a narrow slit in a right-angled wall (first quadrant of Figure 6.18)

13 Conformal Mqping

We s h d now introduce a mcthod by which complex flow patterns can be transformed into simple ones using a technique known as conjormal mcrpping i n complex variable theory Consider the functional relationship w = f(z), which maps a point in the w-plane to a point in the z-plane, and vice versa We shall prove that infinitesimal figures in thc two planes preserve their geometric similarity if UJ = f ( z ) is analytic

Let lines C, and Ci in the z-planc be transformations of the curves C and CL in the

w-plane, respectively (Figure 6.19) Let Sz, S’z, Sw, and S’u; be infinitesimal elements

along thc curves as shown The four elements are related by

d ti)

Sw = -Sz,

d z du: ,

b‘lU = -8 z

d z

(6.49)

(6.50)

6.20 Flow pattans in thc wplane and the z-planc

If w = f(z) is analytic, thcn d w / d z is independent of orientation ofthe elements, and therefore has the same valuc in Eq (6.49) and (6.50) These two equations ihcn imply

that the elcments Sz and S'z are rolatcd by h e sume arnounl (cqual to the argument

of d w / d e ) to obtain the elements S w and S'UJ It follows that

a = B ,

which demonstrales that infinitesimal figures in the two planes are geometrically similar Thc demonstration fails at singular point at which d w / d z is either zero or infinite Because d w / d z is a function of z, the amount of magnification and rotation Lhdt an element Sz undergoes during transformation from the z-planc to thc w-plane varies Consequently, luQe figures become distorted during the transformation

In application of conformal mapping, wc always choosc a rectangular grid in the w-plane consisting of constant Q, and 9 lines (Figure 6.20) In other words, wc define

I$ and @ to be the real and imaginary parts of w:

IL' = Q, + i@

The rtctangular net in thc w-plane represents a uniform flow in this plane Thc con- stant 4 and $ lines are transformed into ccrtain curves in the z-plane through the transformation w = J'(z) The parfern in the z-plane is the physical pattern under investigation, and the images of constant 4 and @ lines in the z-plane form the equipo- tential lines and streamlines, respectivcly, of the desired flow We say that UI = f(z)

transforms a uniform flow in the w-plane into the desired flow in the z-plane In fact, all h e preceding flow patterns studied through the transformation UI = f(z) can bc interpreted this way

If the physical pattern under investigation is too complicated, we m a y introduce intermediate transformations in going from the w-plane to the z-plane For example,

the transformation u; = In (sin z) can be broken into

w = In( J' = sinz

Velocity components in the z-plane are given by

An example of conformal mapping is shown in the next section Additional applica- tions are discussed in Chapter 15

We shall briefly illustrate the method of conformal mapping by considering a trans- formation that has important applications in airfoil theory Consider the following Iransformation:

(6.51)

b2

z = J ' + -

J ' :

relating z and J' planes We shall now show h a t a circle of radius b centered at the

origin of the <-plane transforms into a straight line on the real axis or the z-plane To

Figure 6.21

z = 5 -k I>'/(

Transformulion of a circle into an cllipse by means of thc Zhukhovsky transl-ormation

prove this, consider a point < = b exp (io) on the circle (Figure 6.21), for which the

corresponding point in the z-plane is

z = bei9 + be-io = 2b cos 8

As 8 varies from 0 to A , z goes along the x-axis from 2h to -2h As 8 varics from 5c

to k, z goes from -2h to 2b The circle of radius b in the <-plane is thus mnsformcd into a straight h e of length 4b in the z-plane It is clear that the region outside the circle in <-plane is mapped into the entire z-plane It can be shown that the region

inside the circle is also transformed into the entire z-plane This, howevcr, is a1 no concern to us because we shall not consider the interior of the circle in the <-plane Now consider a circle of radius a > b in the <-plane (Figure 6.21) Points 3' =

a exp (io) on this circle are transformed to

For various values of a =- b, Eq (6.53) represents a family of ellipses in the z-plane, with loci at x = f 2b

The flow around one of these cllipses (in the z-plane) can be determined by first h d i n g the flow around a circle of radius a in the <-plane, and then using the transformation Eq (6.5 1 ) to go to the z-plane To be specific, suppose the desired flow

in the z-plane is that offlow around an elliptic cylinder with clockwise circulation r,

which is placed in a stream moving at U The corresponding flow in the <-plane is that of flow with the same circulation around a circular cylinder of radius u placed in

a stream of the same strength U for which the complex potential is (see Eq (6.37))

(6.54)

The complex potential w ( z ) in the z-plane can be found by substituting the inverse

of Eq (6.5 I), namcly,

< = i z + ;(z2 - 4h2)'/2, (6.55)

into Eq (6.54) (Notc that the negative root, which falls inside h e cylinder, has bcen

excluded h m Eq (6.55).) Instead of finding the complex velocity in thc z-plane by directly differentiating I U ( Z ) , it is easier to find it as

dw d w d <

u - - I v = - =

d z d < d z '

The resulting flow around an elliptic cylinder w i h circulation is qualitatively quite

similar to that around a circular cylinder as shown in Figure 6.14

1.5 ihiquencws oJlrrO~ationul Flown

In Section 10 we saw that plane irrotational flow over a cylindrical object is nonunique

Tn particular, flows with m y amount of circulation satisfy the same boundary

conditions on the body and at infinity With such an example in mind, wc are ready

to make certain general statements concerning solutions of the Laplace equation We shall see that the topology of the region of flow has a great influence on the uniqueness

multiply connected because certain circuits (such as C1 in Figure 6.22b) are reducible

while others (such as C2) are not reducible

To see why solutions are nonunique in a multiply connectcd region, consider the

two circuits CI and Cz in Figure 6.22b The vorticity everywhere within C1 is zero, thus Stokes’ theorem requires that the circulation around it must vanish Tn contrast,

the circulalion around C2 can have any strength r That is,

(6.56)

where the loop around the integral sign has been introduced to emphasize that the

circuit C2 is closed As the right-hand side of Eq (6.56) is nonzero, it follows that

u 9 d x is not a “perfect differential,” which means that the line integral between any

two paints depends on the path followed (u dx is called a p e ~ e c t diflerentiul if it

can be expressed as the diffcrential of a function, say as u dx = d f In that case the

line intcgral around a closed circuit must vanish) In Figure 6.22b, the line inlegals

between P and Q are the same for paths 1 and 2, but not the same for paths 1 and 3

Thc solution is therefore nonunique, as was physically evident from the whole family

of irrotational flows shown in Figure 6.14

i /c

0

Figure 6.22 Singly connccld and multiply conncctcd regions: (a) singly connected, (h) multiply con-

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