CIRCULAR-ARC AIRFOIL

IDEAL-FLUID FLOW

4.17 CIRCULAR-ARC AIRFOIL

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 ðzþmÞe^iaþ a² zþme^ia

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.

Thus the parametric equations of the airfoil pro¢le are x¼ Rþc²

R

cosn y¼ Rc²

R

sinn

The variableRmay be eliminated from these equations as follows:

x²sin²ny²cos²n¼ Rþc² R

2

sin²ncos²n Rc² R

2

sin²ncos²n

¼4c²sin²ncos²n

This is the equation of the airfoil surface in thezplane, but it still contains the variablen. This variable may be eliminated by applying the cosine rule to the FIGURE4.17 The circular-arc airfoil; (a) the mapping planes and (b) uniform flow past the airfoil.

triangle de¢ned by the radiusa, the coordinateR, and the imaginaryzaxis.

From this it follows that

a² ¼R²þm²2Rmcos p 2n

c²þm² ¼R²þm²2Rmsinn

where the fact thata²¼c²þm² has been used. Solving this equation for sinn, it follows that

sinn¼R²c² 2Rm

But it was shown above thaty¼ ½ðR²c²Þsinn=R; hence it follows that sinn¼ y

2msinn or

sin²n¼ y 2m and so

cos²n¼1 y 2m

Using these results to eliminate n, the equation of the airfoil surface becomes

x² y

2my² 1 y 2m

¼4c² y

2m 1 y 2m

Collecting like terms, this equation may be put in the form x²þy²þ2 c²

mm

y¼4c²

Completing the square inyshows that the equation of the airfoil surface is

x²þ yþc c mm

c

h i₂

¼c² 4þ c mm

c

₂

which is the equation of a circle. It should be noted that so far no approx- imations have been made. But to be consistent with the analysis in the

previous section and to permit superposition in the next section, the para- metere¼m=cwill again be assumed to be small compared with unity. Then, linearizing ine, the equation of the airfoil surface becomes

x²þ yþc² m

2

¼c² 4þ c² m²

That is, correct to the ¢rst order ine, the center of the circle in thezplane is located aty¼ c²=mand the radius of the circle isc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4þc²=m²

p .

The characteristic parameters of the airfoil are the chord l and the camber heighth, and these are shown in Fig. 4.17a. Since the equation of the airfoil has now been established, it is possible to relate these parameters to those in the zplane, namely,c and m. Since the ends of the circular- arc airfoil lie on the real axis y¼0, the foregoing equation of the airfoil shows that the corresponding values ofxare 2c. That is, the chord of the airfoil is

l¼4c

This is the same chord length as for the two previous airfoils.

The simplest way of establishing the camberhof the airfoil is to use the fact that, in view of the result that the center of the circular arc is atx¼0, the maximum value ofy will occur when x¼0. But the parametric equation x¼ ðRþc²=RÞcosnshows that this corresponds ton¼p=2. Then the other parametric equation, namely,y¼2msin²n, shows that the maximum value ofyis 2m. That is,

h¼2m

Using the foregoing results, thez-plane parametersc andmmay be replaced by the z-plane parameters l=4 and h=2, respectively. Then the equation of the airfoil surface in thezplane may be written in the form

x²þ yþl² 8h

2

¼l²

4 1þ l² 16h²

ð4:24aÞ

In order to satisfy the Kutta condition, the rear stagnation point must rotate through an angle greater thana, the angle of the free stream. By rotat- ing through the anglea, the rear stagnation point will be located on the sur- face of the circle in thezplane at a point which is in the same horizontal plane as the center of the circle. But the center of the circle is located a distancem

above the realzaxis. Thus, in order to be located at the pointz¼c, the rear stagnation point must rotate through a further angle given by

tan¹m

c ¼tan¹e

¼eþOðe²Þ

That is, in order to comply with the Kutta condition, the rear stagnation point must rotate through the angleaþm=c, to the ¢rst order ine.Then, from Eq. (4.16), the required circulation is

G¼4pUasin aþm c

buta¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c²þm²

p so that, to ¢rst order ine;a¼c. Hence G¼4pUcsin aþm

c

Then, from the Kutta-Joukowski law, the lift force is F_L¼4prU²csin aþm

c

and the corresponding lift coe⁄cient is C_L¼8pc

lsin aþm c

Using again the fact thatc¼l=4 andm¼h=2, the lift coe⁄cient becomes C_L¼2psin aþ2h

l

ð4:24bÞ Comparing this result with Eq. (4.22c), the corresponding result for the £at plate, shows that the e¡ect of positive camber in an airfoil is to increase its lift coe⁄cient. As a consequence of this increased lift coe⁄cient a nonzero lift exists at zero angle of attack.

Since the center of the circle in thezplane is atz¼imrather thanz¼0, the complex potential in thezplane may be obtained by replacingzbyzim in Eq. (4.29) and adding circulation. Thus the required complex potential is

FðzÞ ¼U ðzimÞe^iaþ a² zime^ia

þiG

2plog zim a

ð4:24cÞ

where

a¼ l 4 and

m¼h 2 The magnitude of the circulationGis given by

G¼pUlsin aþ2h l

and the parametercin the Joukowski transformation isl=4. The £ow ¢eld corresponding to this complex potential is shown in Fig. 4.17b. As was the case with the £at plate airfoil, this £ow ¢eld has a singularity at the leading edge. This singularity would not exist for airfoils of ¢nite nose radius and would not exist even for sharp leading edges because of separation of the £ow at the nose. In spite of this local inaccuracy, the results derived above are representative of the £ow around thin cambered airfoils.