IDEAL-FLUID FLOW
4.16 SYMMETRICAL JOUKOWSKI AIRFOIL
wherelis the length or chord of the airfoil, which, for the £at plate under consideration, equals 4a. Then the value of the lift coe⁄cient for the £at- plate airfoil is
CL¼2psina ð4:22cÞ This result shows that the lift coe⁄cient for the £at-plate airfoil increases with angle of attack, and for small values ofa, for which sinaa, the lift coe⁄cient is proportional to the angle of attack with a constant of pro- portionality of 2p. This result is very close to experimental observations, and so the Kutta condition appears to be well justi¢ed. If the Kutta condition were not valid, there would be no circulation around the £at plate, and con- sequently no lift would be generated. This would mean that kites would not be able to £y.
thezplane. Also shown in Fig. 4.16a is the airfoil that is obtained in thez plane and its principal parameters, the chordland the maximum thicknesst.
It is now required to relate these parameters to the free parametersaandm and to establish the equation of the airfoil surface in thezplane.
To establish the chord of the airfoil in terms of the chosen radiusaand o¡set m, it is only necessary to ¢nd the mapping of the points z¼c and z¼ ðcþ2mÞ, since these points correspond to the trailing and leading edges, respectively. Using the Joukowski transformation, the mapping of the point z¼c is z¼2c. Also, the mapping of the point z¼ ðcþ2mÞ ¼ cð1þ2eÞis
z¼ cð1þ2eÞ c 1þ2e
Since it was decided to linearize all quantities ine, the value ofz will be obtained to the ¢rst order ineonly.
z¼ cð1þ2eÞ c½12eþOðe2Þ
¼ 2cþOðe2Þ
FIGURE4.16 The symmetrical Joukowski airfoil: (a) the mapping planes and (b) uniform flow past the airfoil.
That is, to the ¢rst order inethe lending edge of the airfoil is located at z¼ 2c, so that the chord length is
l¼4c
This means that, correct to the ¢rst order in e, the length of the airfoil is unchanged by the shifting of the center of the circle in thezplane.
In order to determine the maximum thickness t, the equation of the airfoil surface must be obtained. This may be done by inserting the equation of the surface in thezplane into the Joukowski transformation. But in thez plane the polar radiusRto the circumference of the circle is a function of the anglen. In order to establish this dependence, the cosine rule will be applied to the triangle de¢ned by the radiusa, the coordinateR, and the realzaxis, as shown in Fig. 4.16a. Thus
a2 ¼R2þm22RmcosðpnÞ
¼R2þm2þ2Rmcosn
Buta¼cþm, so that the equation above may be written in the form ðcþmÞ2 ¼R2 1þm2
R2þ2m Rcosn
Now sinceRc, it follows thatm=Rm=cso that, to ¢rst order ine¼m=c, the termm2=R2may be neglected. The equation forRthen becomes
cþm¼R 1þ2m Rcosn
1=2
¼R 1þm
RcosnþOðe2Þ
h i
Thus to the ¢rst order ine, this relation becomes cð1þeÞ ¼Rþmcosn
::: R¼c½1þeð1cosnÞ
This is the required equation that gives the variation of the radiusRwith the anglenfor points on the circumference of the circle in thezplane. Then, in order to determine the equation of the corresponding pro¢le in thezplane, this result should be substituted into the Joukowski transformation [Eq. (4.20)]. Thus points on the surface of the airfoil will be de¢ned by
z¼c½1þeð1cosnÞeinþ cein 1þeð1cosnÞ
This equation may also be linearized ineas follows:
z¼c½1þeð1cosnÞeinþc½1eð1cosnÞ þOðe2Þein
¼c½2 cosnþi2eð1cosnÞsinnþOðe2Þ
Then, by equating real and imaginary parts of this equation, the parametric equations of the airfoil are, to ¢rst order ine,
x¼2ccosn
y¼2ceð1cosnÞsinn
Using the ¢rst of these equations to eliminatenfrom the second equation gives the following equation for the airfoil pro¢le:
y¼ 2ce 1 x 2c
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 x
2c r 2
The location of the maximum thickness may now be obtained, and this is most readily done by using the parametric equation for the coordinateyas derived above.Thus settingdy=dn¼0 for a maximum inygives the following equation for the value ofnat the maximum thickness:
sin2nþ ð1cosnÞcosn¼0 This relation reduces to
cos 2n¼cosn
which is satis¢ed byn¼0;n¼2p=3, andn¼4p=3. This solutionn¼0 cor- responds to the trailing edge and so is the minimum thickness. The solu- tions n¼2p=3 and n¼4p=3 give the maximum thickness, and for these values ofnthe coordinates of the airfoil surface are
x¼ c y¼ 3 ffiffiffi
p3 2 ce
The maximum thicknesst will be twice the positive value ofy, so that the thickness ratiot=lof the airfoil will be
t l¼3 ffiffiffi
p3 4 e
That is, the thickness-to-chord ratio of the airfoil is proportional toe, which is the ratio of the o¡set of the center of the circle in thezplane to the radiusc of the critical points of the transformation. Since the thickness ratio of the airfoil is a parameter that may be thought of as being speci¢ed, it is useful to eliminateein terms of this parameter. Hence
e¼ 4 3 ffiffiffi
p3t
l¼0:77t l
Then the equation of the airfoil surface may be written in the form y
t ¼ 0:385 12x l
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2x
l r 2
ð4:23aÞ where the maximum value ofy=twill be 0.5 and the minimum value will be 0:5, both of which occur atx¼ c.
The magnitude of the circulation required to satisfy the Kutta condi- tion is, from Eq. (4.16), 4pUasina, wherea¼cþmandm¼ce¼0:77tc=l.
Thus the required amount of circulation is G¼pUl 1þ0:77t
l
sina ð4:23bÞ
wherechas been replaced byl=4. In this form the required circulation may be evaluated for the given free-stream velocity, angle of attack, and the chord and thickness of the airfoil. Using the Kutta-Joukowski law [Eq. (4.18)], the lift force acting on the airfoil may be evaluated as
FL¼prU2l 1þ0:77t l
sina
Then the lift coe⁄cient for the symmetrical Joukowski airfoil is CL¼2p 1þ0:77t
l
sina ð4:23cÞ It will be noticed that this result reduces to Eq. (4.22c) for the £at-plate air- foil ast !0. This indicates that the e¡ect of thickness of an airfoil is to increase the lift coe⁄cient. However, this fact cannot be used to produce high lift coe⁄cients through thick airfoils, since the £ow tends to separate from blu¡ bodies much more readily than it does from streamlined bodies.
This separation of the £ow is a viscous e¡ect, and it will be discussed in the next part of the book. In the meantime, it is su⁄cient to say that separation of the £ow results in a low-pressure wake that destroys the lift. The same result may occur for slender bodies, such as airfoils, that are at large angles of attack. In this context the separation is usually referred to asstall.
The center of the circle in thezplane is located atz¼ mrather than z¼0. Thus the complex potential for the £ow in thezplane may be obtained from Eq. (4.29) by replacingzbyzþmand adding circulation. The required complex potential then becomes
FðzÞ ¼U ðzþmÞeiaþ a2 zþmeia
þiG
2plog zþm a
ð4:23dÞ where
a¼l
4þ0:77tc l and
m¼0:77tc l
The magnitude of the circulation G is given by Eq. (4.23b), and in the Joukowski transformation the parametercequalsl=4. The £ow ¢eld corres- ponding to this complex potential is shown in Fig. 4.16b.