CIRCULAR CYLINDER WITH CIRCULATION

IDEAL-FLUID FLOW

4.9 CIRCULAR CYLINDER WITH CIRCULATION

FðzÞ ¼U zþa² z

þiG 2plogz

a ð4:14Þ

which describes a uniform rectilinear £ow of magnitudeU approaching a circular cylinder of radiusathat has a negative vortex of strengthGaround it.

As required, this result agrees with Eq. (4.13) whenG¼0.

In order to visualize the £ow ¢eld described by Eq. (4.14), the corre- sponding velocity components will be evaluated from the complex velocity.

WðzÞ ¼U 1a² z²

þiG 2p 1 z

¼U 1a² R²e^i2y

þ iG 2pRe^iy

¼ U e^iya² R²e^iy

þ iG 2pR

e^iy

¼ U 1a² R²

cosyþi U 1þa² R²

sinyþ G 2pR

e^iy Hence, by comparison with Eq. (4.6), the velocity components are

uR ¼U 1a² R²

cosy ð4:15aÞ

uy¼ U 1þa² R²

siny G

2pR ð4:15bÞ On the surface of the cylinder, whereR¼a, Eqs. (4.15) become

u_R¼0

uy¼ 2Usiny G 2pa

The fact thatu_R¼0 onR¼ais to be expected, since this is the boundary condition (II.3). A signi¢cant point in the £ow ¢eld is a point where the velocity components all vanishthat is, a stagnation point. For this £ow

¢eld the stagnation points are de¢ned by sinys¼ G

4pUa ð4:16Þ

where y_s is the value of y corresponding to the stagnation point. For G¼0;siny_s¼0, so thaty_s¼0 orp, which agrees with Fig. 4.8b for the cir- cular cylinder without circulation. For nonzero circulation, the value ofy_s

clearly depends upon the magnitude of the parameterG=ð4pUaÞ, and it is convenient to discuss Eq. (4.16) for di¡erent ranges of this parameter.

First, consider the range 0<G=ð4pUaÞ<1. Here sinys<0, so thatys

must lie in the third and fourth quadrants. There are two stagnation points, and clearly the one that was aty¼pis now located in the third quadrant while the one that was located aty¼0 is now located in the fourth quadrant.

The two stagnation points will be symmetrically located about theyaxis in order that sinys¼ constant may be satis¢ed.The resulting £ow situation is shown in Fig. 4.9a.

Physically, the location of the stagnation points may be explained as follows: The £ow due to the vortex and that due to the £ow around the cylin- der without circulation reinforce each other in the ¢rst and second quad- rants. On the other hand, these two £ow ¢elds oppose each other in the third and fourth quadrants, so that at some point in each of these regions the net velocity is zero. Thus the e¡ect of circulation around the cylinder is to make the front and rear stagnation points approach each other, and for a negative vortex they do so along the lower surface of the cylinder.

Consider next the case when the nondimensional circulation is unity, that is, whenG=ð4pUaÞ ¼1. Here sinys¼ 1, so thatys¼3p=2. The corre- sponding £ow con¢guration is shown in Fig. 4.9b. The two stagnation points have been brought together by the action of the bound vortex such that they coincide to form a single stagnation point at the bottom of the cylindrical sur- face. It is evident that if the circulation is increased above this value,the single stagnation point cannot remain on the surface of the cylinder. It will move o¡

into the £uid as either a single stagnation point or two stagnation points.

Finally, consider the case whereG=ð4pUaÞ>1. Since it seems likely that any stagnation points there may be will not lie on the surface of the

FIGURE4.9 Flow of approach velocityUaround a circular cylinder of radius a having a negative bound circulation of magnitudeGfor (a) 0<G=ð4pUaÞ<1, (b) G=ð4pUaÞ ¼1, and (c)G=ð4pUaÞ>1.

cylinder, the velocity components must be evaluated from Eqs. (4.15). Then ifRsandysare the cylindrical coordinates of the stagnation points, it follows from Eqs. (4.15) thatRsandysmust satisfy the equations

U 1a² R_s²

cosy_s¼0 U 1þa²

R²_s

siny_s¼ G 2pR_s

Since it is assumed that the stagnation points are not on the surface of the cylinder, it follows thatR_s6¼a, so that the ¢rst of these equations requires thaty_s¼p=2 or 3p=2. For these values ofy_s, the second of the above equa- tions becomes

U 1þa² R²_s

¼ G 2pRs

where the minus sign corresponds toy_s¼p=2 and the plus sign toy_s¼3p=2.

SinceU >0, the left-hand side of the above equation is positive, and since G>0, the minus sign must be rejected on the right-hand side. This might have been expected, since for G=ð4pUaÞ ¼1 the value of ys was 3p=2, whereas the minus sign corresponds toys¼p=2,which would require a large jump inysfor a small change inG. The equation forRsnow becomes

U 1þa² R²_s

¼ G 2pR_s or

R²_s G

2pUR_sþa²¼0 hence

R_s¼ G 4pU

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G

4pU

2

a² s

or

R_s a ¼ G

4pUa 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 4pUa

G

2

2 s 4

3 5

This result shows that as 4pUa=G!0;R_s! 1for the plus sign, but the corresponding limit is indeterminate for the minus sign. This di⁄culty may be overcome by expanding the square root for 4pUa=G1 as follows:

R_s a ¼ G

4pUa 1 11

2 4pUa

G

2

þ

" #

( )

where the dots indicate terms of orderð4pa=GÞ⁴or smaller. In this form it is evident that as 4pUa=G!0;Rs!0 for the minus sign. Since this stagnation point would be inside the surface of the cylinder, the minus sign may be rejected, so that the coordinates of the stagnation point in the £uid outside the cylinder are

ys¼3p 2 Rs

a ¼ G 4pUa 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 4pUa

G

2

2 s 4

3 5

This gives a single stagnation point below the surface of the cylinder. The corresponding £ow con¢guration is shown in Fig. 4.9c, from which it will be seen that there is a portion of the £uid that perpetually encircles the cylinder.

The £ow ¢elds for the circular cylinder with circulation, as shown in Fig. 4.9, exhibit symmetry about theyaxis. Then, following the arguments used in the previous section, it may be concluded that there will be no drag force acting on the cylinder. However, the existence of the circulation around the cylinder has destroyed the symmetry about thexaxis. so there will be some force acting on the cylinder in the vertical direction. For the negative circulation shown, the velocity on the top surface of the cylinder will be higher than that for no circulation, while the velocity on the bottom surface will be lower. Then, from Bernoulli’s equation, the pressure on the top surface will be lower than that on the bottom surface, so that the ver- tical force acting on the cylinder will be upward. That is, a positive lift will exist. In order to determine the magnitude of this lift, a quantitative ana- lysis must be performed, and this will be done in the next section.

The principal interest in the £ow around a circular cylinder with cir- culation is in the study of airfoil theory. By use of conformal transformations the £ow around certain airfoil shapes may be transformed into that of the

£ow around a circular cylinder with circulation.