It was observed in Sec. 4.6 that the potential £ow solution for £ow around a sharp edge contained a singularity at the edge itself.This singularity required an in¢nite velocity at the point in question, which, of course, is physically impossible.The question arises, then, as to what the real £ow situation would be in a physical experiment. Depending upon the actual physical con¢gura- tion, one of two remedial situations will prevail. One possibility is that the £uid will separate from the solid surface at the knife-edge. The resulting free streamline con¢guration would be such that the radius of curvature at the edge becomes ¢nite rather than being zero. As a consequence, the velocities there will remain ¢nite. Examples of this type of solution will be discussed later in this chapter.
A second possibility is that a stagnation point exists at the sharp edge. For the £ow around ¢nite bodies, stagnation points exist, and it seems pos- sible that a stagnation point could be induced by the £ow ¢eld to move to the location of the sharp edge. This possibility leads to the so-called Kutta con- dition, and it will be discussed below in the context of the £at-plate airfoil that is, a £at plate which is at some angle of attack to the free stream.
In the previous section, the £ow around an ellipse was obtained from the Joukowski transformation [Eq. (4.20)] by considering the £ow around a circular cylinder of radius a> c in the z plane. Now, if the constant c is allowed to approach the magnitude of the radius a, the resulting ellipse in the z plane degenerates to a £at plate de¢ned by the strip2a x 2a. The resulting £ow ¢eld, as de¢ned by Eq. (4.21a), is shown in Fig. 4.15a. Because of the angle of attack, the stagnation points do not coincide with the leading and trailing edges of the £at plate. Rather, the upstream stagnation point is located on the lower surface and the downstream stagnation point is located on the upper surface at the points x¼ 2a cos a. Then, around both the leading and trailing edges, the £ow will be that associated with a sharp edge, which was discussed in Sec. 4.6. In that section it was observed that in¢nite velocity components existed at the edge itself, a situation that is physically impossible to realize.
The di⁄culty encountered above with the £at-plate airfoil does not occur at the leading edge of real airfoils because real airfoils have a ¢nite thickness and so have a ¢nite radius of curvature at the leading edge. How- ever, the trailing edge of airfoils is usually quite sharp, so that the di⁄culty of in¢nite velocity components still exists there. However, this remaining di⁄- culty would also be overcome if the stagnation point which is near the trailing edge was actually at the trailing edge. This would be accomplished if a circu- lation existed around the £at plate and the magnitude of this circulation was
just the amount required to rotate the rear stagnation point so that its location coincides with the trailing edge. This condition is called the Kutta condition, and it may be restated as follows: For bodies with sharp trailing edges that are at small angles of attack to the free stream, the £ow will adjust itself in such a way that the rear stagnation point coincides with the trailing edge.
The amount of circulation required to comply with the Kutta condition may be determined as follows: In the z plane of Fig. 4.15a, the rear stagnation point is located at the point z¼ aeia. But, according to the Kutta condition,
the rear stagnation point should be located at the point z¼ 2a, which corre- sponds to the point z¼ a. That is, the stagnation point on the downstream face of the circular cylinder in the z plane should be rotated clockwise through an angle a. But from Eq. (4.16), the magnitude of the circulation that will do this is
G ¼ 4pUa sin a ð4:22aÞ
FIGURE4.15 Flow around a flat plate at shallow angle of attack (a) without circu- lation and (b) satisfying the Kutta condition.
in the clockwise direction (that is, negative circulation). Then the complex potential for the required £ow in the z plane is, from Eqs. (4.14) and (4.29),
FðzÞ ¼ U zeiaþa2
z e
ia
þ i2Ua sin a logz a But the equation of the mapping is
z¼ z þa
2
z and the inverse, which gives z! z as z ! 1, is
z¼z 2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2 2 a2 r
Then the complex potential in the z plane is
FðzÞ ¼ U z 2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2 2 a2 r " # eiaþ a 2eia z=2 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz=2Þ2 a2 q 8 > < > :
þi2a sin a log 1 a z 2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2 2 a2 r ! " #
)
ð4:22bÞThe £ow ¢eld corresponding to this complex potential is shown in Fig. 4.15b. Although the £ow at the trailing edge is now regular, the singularity at the leading edge still exists. In an actual £ow con¢guration the £uid would separate at the leading edge and reattach again on the top side of the airfoil. The streamline c¼ 0 would then correspond to a ¢nite curvature, and the velocity components would remain ¢nite at the leading edge.
The lift force generated by the £at-plate airfoil may be calculated from the Kutta-Joukowski law. Then, denoting the lift force by Y and using the value of the circulation given by Eq. (4.22a),
Y ¼ 4prU2a sin a
It is usual to express lift forces in terms of the dimensionless lift coe⁄cient CL,
which is de¢ned as follows:
CL¼
Y
1 2rU2l
where l is 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
CL¼ 2p sin a ð4:22cÞ
This result shows that the lift coe⁄cient for the £at-plate airfoil increases with angle of attack, and for small values of a, for which sin a a, 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.
4.16
SYMMETRICAL JOUKOWSKI AIRFOIL
A family of airfoils may be obtained in the z plane by considering the Jou- kowski transformation in conjunction with a series of circles in the z plane whose centers are slightly displaced from the origin. These airfoils are known as the Joukowski family of airfoils.
Consider, ¢rst, the case where the center of the circle in the z plane is displaced from the origin along the real axis. It must then be decided in which direction the center should be moved and what radius should be employed, relative to the Joukowski constant c. From previous sections it is known that if the circumference of the circle passes through either of the two critical points of the Joukowski transformation, z¼ c, then a sharp edge or cusp is obtained in the z plane. Then, if the leading edge of the airfoil is to have a ¢nite radius of curvature and if there should be no singularities in the £ow ¢eld itself, it follows that the point z¼ c should be inside the circle in the z plane. Also, since the trailing edge of the airfoil should be sharp as opposed to being blunt, the circumference of the circle should pass through the point z¼ c.These conditions will be satis¢ed by taking the center of the circle to be at z¼ m, where m is real, and by choosing the radius of the circle to be cþ m. Such a con¢guration is shown in Fig. 4.16a.The radius a is given by
a¼ c þ m ¼ cð1 þ eÞ
where the parameter e¼ m=c will be assumed to be small compared with unity. When e¼ 0, the £at-plate airfoil is recovered, so that for e 1 it may be anticipated that a thin airfoil will be obtained. The signi¢cance of the restriction e 1 will be that all the equations may be linearized in e, which will permit a closed-form solution for the equation of the airfoil surface in
the z plane. Also shown in Fig. 4.16a is the airfoil that is obtained in the z plane and its principal parameters, the chord l and the maximum thickness t. It is now required to relate these parameters to the free parameters a and m and to establish the equation of the airfoil surface in the z plane.
To establish the chord of the airfoil in terms of the chosen radius a and o¡set m, it is only necessary to ¢nd the mapping of the points z¼ c and z¼ ðc þ 2mÞ, since these points correspond to the trailing and leading edges, respectively. Using the Joukowski transformation, the mapping of the point z¼ c is z ¼ 2c. Also, the mapping of the point z ¼ ðc þ 2mÞ ¼ cð1 þ 2eÞ is
z¼ cð1 þ 2eÞ c 1þ 2e
Since it was decided to linearize all quantities in e, the value of z will be obtained to the ¢rst order in e only.
z¼ cð1 þ 2eÞ c½1 2e þ Oðe2Þ ¼ 2c þ Oðe2Þ
FIGURE4.16 The symmetrical Joukowski airfoil: (a) the mapping planes and (b) uniform flow past the airfoil.
That is, to the ¢rst order in e the lending edge of the airfoil is located at z¼ 2c, so that the chord length is
l¼ 4c
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 the z plane.
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 the z plane into the Joukowski transformation. But in the z plane the polar radius R to the circumference of the circle is a function of the angle n. In order to establish this dependence, the cosine rule will be applied to the triangle de¢ned by the radius a, the coordinate R, and the real z axis, as shown in Fig. 4.16a. Thus
a2 ¼ R2þ m2 2Rm cosðp nÞ ¼ R2þ m2þ 2Rm cos n
But a¼ c þ m, so that the equation above may be written in the form ðc þ mÞ2 ¼ R2 1þm2
R2þ 2
m Rcos n
Now since R c, it follows that m=R m=c so that, to ¢rst order in e ¼ m=c, the term m2=R2may be neglected. The equation for R then becomes
cþ m ¼ R 1 þ 2m Rcos n 1=2 ¼ R 1 þm Rcos nþ Oðe 2Þ h i
Thus to the ¢rst order in e, this relation becomes cð1 þ eÞ ¼ R þ m cos n
::: R ¼ c½1 þ eð1 cos nÞ
This is the required equation that gives the variation of the radius R with the angle n for points on the circumference of the circle in the z plane. Then, in order to determine the equation of the corresponding pro¢le in the z plane, 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
z¼ c½1 þ eð1 cos nÞeinþ ce
in
This equation may also be linearized in e as follows:
z¼ c½1 þ eð1 cos nÞeinþ c½1 eð1 cos nÞ þ Oðe2Þein ¼ c½2 cos n þ i2eð1 cos nÞ sin n þ Oðe2Þ
Then, by equating real and imaginary parts of this equation, the parametric equations of the airfoil are, to ¢rst order in e,
x¼ 2c cos n
y¼ 2ceð1 cos nÞ sin n
Using the ¢rst of these equations to eliminate n from the second equation gives the following equation for the airfoil pro¢le:
y¼ 2ce 1 x 2c ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x 2c 2 r
The location of the maximum thickness may now be obtained, and this is most readily done by using the parametric equation for the coordinate y as derived above.Thus setting dy=dn ¼ 0 for a maximum in y gives the following equation for the value of n at the maximum thickness:
sin2nþ ð1 cos nÞ cos n ¼ 0 This relation reduces to
cos 2n¼ cos n
which is satis¢ed by n¼ 0; n ¼ 2p=3, and n ¼ 4p=3. This solution n ¼ 0 cor- responds to the trailing edge and so is the minimum thickness. The solu- tions n¼ 2p=3 and n ¼ 4p=3 give the maximum thickness, and for these values of n the coordinates of the airfoil surface are
x¼ c y¼ 3 ffiffiffi 3 p 2 ce
The maximum thickness t will be twice the positive value of y, so that the thickness ratio t=l of the airfoil will be
t l¼
3pffiffiffi3
That is, the thickness-to-chord ratio of the airfoil is proportional to e, which is the ratio of the o¡set of the center of the circle in the z plane to the radius c 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 eliminate e in terms of this parameter. Hence
e¼ 4 3pffiffiffi3 t l¼ 0:77 t l
Then the equation of the airfoil surface may be written in the form y t ¼ 0:385 1 2 x l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2x l 2 r ð4:23aÞ
where the maximum value of y=t will be 0.5 and the minimum value will be 0:5, both of which occur at x ¼ c.
The magnitude of the circulation required to satisfy the Kutta condi- tion is, from Eq. (4.16), 4pUa sin a, where a¼ c þ m and m ¼ ce ¼ 0:77tc=l. Thus the required amount of circulation is
G ¼ pUl 1 þ 0:77t l
sin a ð4:23bÞ
where c has been replaced by l=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
FL¼ prU2l 1þ 0:77
t l
sin a
Then the lift coe⁄cient for the symmetrical Joukowski airfoil is
CL¼ 2p 1 þ 0:77
t l
sin a ð4:23cÞ
It will be noticed that this result reduces to Eq. (4.22c) for the £at-plate air- foil as t ! 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 as stall.
The center of the circle in the z plane is located at z¼ m rather than z¼ 0.Thus the complex potential for the £ow in the z plane may be obtained from Eq. (4.29) by replacing z by zþ m and adding circulation. The required complex potential then becomes
FðzÞ ¼ U ðz þ mÞeiaþ a2 zþ me ia þiG 2plog zþ m a ð4:23dÞ where a¼l 4þ 0:77 tc l and m¼ 0:77tc l
The magnitude of the circulation G is given by Eq. (4.23b), and in the Joukowski transformation the parameter c equals l=4. The £ow ¢eld corres- ponding to this complex potential is shown in Fig. 4.16b.
4.17
CIRCULAR-ARC AIRFOIL
It was shown in the two previous sections that, using the Joukowski trans- formation, a circle of radius c centered at the origin of the z plane produced a £at-plate airfoil while a slightly larger circle centered a small distance along the real axis from the origin produced a thin symmetrical airfoil. It will now be shown that a circle whose radius is slightly larger than c and whose center is located on the imaginary axis of the z plane produces an airfoil that has no thickness but has curvature of camber.
Referring to Fig. 4.17a, consider a circle of radius a> c in the z plane such that the center of the circle is located a distance m along the positive imaginary axis. Since the trailing edge of the airfoil should be sharp, the circle should pass through the critical point z¼ c as before. Then, in this case, the circle will also pass through the other critical point, z¼ c.
The equation of the airfoil in the z plane may be obtained by substitut- ing z¼ Reininto the Joukowski transformation,where, on the circumference
of the circle in the z plane, R is a function of n. This substitution gives
z¼ Reinþc 2 Re in ¼ R þc2 R cos nþ i R c 2 R sin n
Thus the parametric equations of the airfoil pro¢le are x¼ R þc 2 R cos n y¼ R c 2 R sin n
The variable R may be eliminated from these equations as follows:
x2sin2n y2cos2n¼ R þc 2 R 2 sin2n cos2n R c 2 R 2 sin2n cos2n ¼ 4c2sin2n cos2n
This is the equation of the airfoil surface in the z plane, but it still contains the variable n. 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 radius a, the coordinate R, and the imaginary z axis. From this it follows that
a2 ¼ R2þ m2 2Rm cos p 2 n
c2þ m2 ¼ R2þ m2 2Rm sin n
where the fact that a2¼ c2þ m2 has been used. Solving this equation for
sin n, it follows that
sin n¼R
2 c2
2Rm
But it was shown above that y¼ ½ðR2 c2Þ sin n=R; hence it follows that
sin n¼ y 2m sin n or sin2n¼ y 2m and so cos2n¼ 1 y 2m
Using these results to eliminate n, the equation of the airfoil surface becomes x2 y 2m y 2 1 y 2m ¼ 4c2 y 2m 1 y 2m
Collecting like terms, this equation may be put in the form x2þ y2þ 2 c
2
m m
y¼ 4c2
Completing the square in y shows that the equation of the airfoil surface is
x2þ y þ c c m m c h i2 ¼ c2 4þ c m m 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- meter e¼ m=c will again be assumed to be small compared with unity.Then, linearizing in e, the equation of the airfoil surface becomes
x2þ y þc 2 m 2 ¼ c2 4þ c2 m2
That is, correct to the ¢rst order in e, the center of the circle in the z plane is located at y¼ c2=m and the radius of the circle is cpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ c2=m2.
The characteristic parameters of the airfoil are the chord l and the camber height h, 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 z plane, namely, c and m. Since the ends of the circular- arc airfoil lie on the real axis y¼ 0, the foregoing equation of the airfoil shows that the corresponding values of x are 2c. That is, the chord of the airfoil is
l¼ 4c
This is the same chord length as for the two previous airfoils.
The simplest way of establishing the camber h of the airfoil is to use the fact that, in view of the result that the center of the circular arc is at x¼ 0, the maximum value of y will occur when x¼ 0. But the parametric equation x¼ ðR þ c2=RÞ cos n shows that this corresponds to n ¼ p=2. Then the other
parametric equation, namely, y¼ 2m sin2n, shows that the maximum value of y is 2m. That is,
h¼ 2m
Using the foregoing results, the z-plane parameters c and m may be replaced by the z-plane parameters l=4 and h=2, respectively. Then the equation of the airfoil surface in the z plane may be written in the form
x2þ y þl 2 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 than a, the angle of the free stream. By rotat- ing through the angle a, the rear stagnation point will be located on the sur- face of the circle in the z plane 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 distance m
above the real z axis. Thus, in order to be located at the point z¼ c, the rear stagnation point must rotate through a further angle given by
tan1m c ¼ tan
1e
¼ e þ Oðe2Þ
That is, in order to comply with the Kutta condition, the rear stagnation point must rotate through the angle aþ m=c,to the ¢rst order in e.Then, from Eq. (4.16), the required circulation is
G ¼ 4pUa sin a þm c
but a¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2þ m2so that, to ¢rst order in e; a ¼ c. Hence
G ¼ 4pUc sin a þm c
Then, from the Kutta-Joukowski law, the lift force is