IDEAL-FLUID FLOW
4.14 FLOW AROUND ELLIPSES
point is passed. That is, the value ofn1n2changes fromp=2 top=2, giving a di¡erence of p. From the resulty1y2 ¼2ðn1n2Þ, it follows that the corresponding di¡erence in the value of y1y2 will be 2p. This yields a knife-edge or cusp in thezplane as shown in Fig. 4.13b. That is, if a smooth curve passes through either of the critical pointsz¼ c, the corresponding curve in thezplane will contain a knife-edge at the corresponding critical pointz¼ 2c.
An example of a smooth curve that passes through both critical points is a circle centered at the origin of thezplane and whose radius isc, the con- stant that appears in the Joukowski transformation. Then, on this circle z¼cein, and the value ofzwill be given by
z¼ceinþcein
¼2ccosn
That is, the circle in thezplane maps into the stripy¼0,x¼2ccosnin thez plane. It is readily veri¢ed that all points that lie outside the circlejzj ¼c cover the entire zplane. However, the points inside the circlejzj ¼calso cover the entirezplane, so that the transformation is double-valued. This is readily veri¢ed by observing that for any value ofz, Eq. (4.20) yields the same value of zfor that value ofzand also forc2=z. It will be noted thatc2=zis simply the image of the pointzinside the circle of radiusc.
This double-valued property of the Joukowski transformation is trea- ted by connecting the two pointsz¼ 2cby a branch cut along thexaxis and creating two Riemann sheets. Then the mapping is single-valued if all the points outside the circlejzj ¼care taken to fall on one of these sheets and all the points inside the circle to fall on the other sheet. In £uid mechanics, dif-
¢culties due to the double-valued behavior do not usually arise because the pointsjzj<cusually lie inside some body about which the £ow is being stu- died, so that these points are not in the £ow ¢eld in thezplane.
z¼aeinþc2 aein
¼ aþc2 a
cosnþi ac2 a
sinn
Equating real and imaginary parts of this equation gives x¼ aþc2
a
cosn y¼ ac2
a
sinn
These are the parametric equations of the required curve in thezplane. The equation of the curve may be obtained by eliminatingnby use of the identity cos2nþsin2n¼1. This gives
x aþc2=a
2
þ y ac2=a
2
¼1
which is the equation of an ellipse whose major semiaxis is of lengthaþc2=a, aligned along the xaxis, and whose minor semiaxis is of lengthac2=a.
Then, in order to obtain the complex potential for a uniform £ow of magni- tudeUapproaching this ellipse at an angle of attacka, the same £ow should be considered to approach the circular cylinder in thezplane. But it is shown in Prob. 4.5, Eq. (4.29), that the complex potential for a uniform £ow of magnitudeUapproaching a circular cylinder of radiusaat an angleato the reference axis is
FðzÞ ¼U zeiaþa2 z eia
Then, by solving Eq. (4.20) forzin terms ofz, the complex potential in thez plane may be obtained. From Eq. (4.20),
z2zzþc2¼0 ::: z¼z 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2
2c2 r
Since it is known thatz!zfor large values ofz, the positive root must be chosen. Then the complex potential in thezplane becomes
FðzÞ ¼U z 2þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2
2c2
" r #
eiaþ a2eia z=2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðz=2Þ2c2 q
8>
<
>:
9>
=
>;
¼U zz 2þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2
2c2
" r #
eiaþa2 c2
z 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2
2c2
" r # eia
( )
where the last term has been rationalized. By writingz=2 aszz=2 in the
¢rst term, two of the terms may be combined as follows:
FðzÞ ¼U zeiaþ a2
c2eiaeia
z 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2
2c2
r !
" #
ð4:21aÞ
Equation (4.21a) is the complex potential for a uniform £ow of magnitudeU approaching an ellipse whose major semiaxis isaþc2=aand whose minor semiaxis isac2=aat an angle of attackato the major axis. In this form it may be seen that the complex potential consists of that for a uniform £ow at an angleato the reference axis plus a perturbation which is large near the body but vanishes for large values ofz. The £ow ¢eld generated by the com- plex potential (4.21a) is shown in Fig. 4.14a together with that for the circular cylinder in thezplane.
The stagnation points in the z plane are located at z¼aeia and z¼aeiðaþpÞ, that is, atz¼ aeia. Then, from Eq. (4.20), the corresponding points in thezplane are
z¼ aeia c2 aeia
¼ aþc2 a
cosa i ac2 a
sina
This gives the coordinates of the stagnation points as x¼ aþc2
a
cosa y¼ ac2
a
sina
Equation (4.21a) includes two special cases within its range of validity.
Fora¼0 it describes a uniform rectilinear £ow approaching a horizontally oriented ellipse, and for a¼p=2 it describes a uniform vertical £ow approaching the same horizontally oriented ellipse. However, it is of interest to note that the solution for a uniform rectilinear £ow approaching a verti- cally oriented ellipse may be obtained directly from the Joukowski transfor- mation with a slight modi¢cation. Substitute c¼ib, where b is real and positive, into Eq. (4.20)
z¼zb2 z
Then, as with the horizontal ellipse, examining the mapping of the circle z¼aeingives the parametric equations of the mapped boundary.
FIGURE4.14 (a) Uniform flow approaching a horizontal ellipse at an angle of attack, and (b) uniform parallel flow approaching a vertical ellipse.
x¼ ab2 a
cosn y¼ aþb2
a
sinn
Thus the equation of the contour in thezplane is x
ab2=a
2
þ y aþb2=a
2
¼1
which is the equation of an ellipse whose major semiaxis isaþb2=awhich is aligned along theyaxis.Then to obtain uniform rectilinear £ow approaching such an ellipse the same £ow should approach the circle in thezplane. Thus the required complex potential, from Eq. (4.13), is
FðzÞ ¼U zþa2 z
But the inverted equation of the mapping for whichz!zasz! 1is
z¼z 2þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2
2þb2 r
Hence the complex potential in thezplane is
FðzÞ ¼U z 2þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2
2þb2 r
þ a2
z=2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz=2Þ2þb2 q
2 64
3 75
FðzÞ ¼U z 1þa2 b2
z 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2
2þb2
r !
" # ð4:21bÞ
in which the same rationalization and simpli¢cation has been carried out as before. Again the complex potential is in the form of that for a uniform £ow plus a perturbation which is large near the body and which vanishes at large distances from the body. Equation (4.21b) describes a uniform rectilinear
£ow of magnitudeUapproaching a vertically oriented ellipse. The £ow ¢eld for this situation is shown in Fig. 4.14b.
4.15 KUTTA CONDITION AND THE FLAT-PLATE