ANDERSON_Fundamentals_of_Aerodynamics_2nd_ed_1991
.pdfFUNDAMENTALS OF INVISCID. INCOMPRESSIBLE FLOW 201
see how the superposition of flows involving such vortices leads to cases with finite lift.
Consider a flow where all the streamlines are concentric circles about a given point, as sketched in Fig. 3.26. Moreover, let the velocity along any given circular streamline be constant, but let it vary from one streamline to another inversely with distance from the common center. Such a flow is called a vortex flow. Examine Fig. 3.26; the velocity components in the radial and tangential directions are V, and Va, respectively, where V, = 0 and Ve = constant/ r. It is easily shown (try it yourself) that (1) vortex flow is a physically possible incom
pressible flow, i.e., V . V = 0 at every point, 
and (2) vortex flow is irrotational, 

i.e., V x V = 0, at every point except the origin. 


From the definition of vortex flow, we have 


const 
C 
(3.104) 
Vo =  =  

r 
r 

To evaluate the constant C, take the circulation around a given circular streamline of radius r:
or 
(3.105) 
Comparing Egs. (3.104) and (3.105), we see that 

r 
(3.106) 
C=  
21T
Therefore, for vortex flow, Eg. (3.106) demonstrates that the circulation taken
FIGURE 3.26
Vortex flow.
202 FUNDAMENTALS OF AERODYNAMICS
about all streamlines is the same value, namely, f = 21TC. By convention, f is called the strength of the vortex flow, and Eq. (3.105) gives the velocity field for a vortex flow of strength f. Note from Eq. (3.105) that Vo is negative when f is positive; i.e., a vortex of positive strength rotates in the clockwise direction. (This is a consequence of our sign convention on circulation defined in Sec. 2.13, namely, positive circulation is clockwise.)
We stated earlier that vortex flow is irrotational except at the origin. What happens at r = o? What is the value of V x V at r = o? To answer these questions,
recall Eq. (2.128) relating circulation to vorticity: 

f =  f f (V x V) . dS 
(2.128) 
s 

Combining Eqs. (3.106) and (2.128), we obtain 

21TC = ff(V x V) . dS 
(3.107) 
s 

Since we are dealing with twodimensional flow, the flow sketched in Fig. 3.26 takes place in the plane of the paper. Hence, in Eq. (3.107), both V x V and dS are in the same direction, both perpendicular to the plane of the flow. Thus, Eq. (3.107) can be written as
21TC = ff(V x V) . dS = ffIv x vi dS 
(3.108) 




s 
s 

In Eq. (3.1 08), the surface integral is taken over the circular area inside the streamline along which the circulation f = 21TC is evaluated. However, f is the same for all the circular streamlines. In particular, choose a circle as close to the origin as we wish; i.e., let r~ o. The circulation will still remain f = 21TC. However, the area inside this small circle around the origin will become infinitesimally small, and

ffIv x vi dS ~Iv x vi dS 
(3.109) 




s 

Combining Eqs. (3.108) and (3.109), in the limit as r~O, we have 


or 
21TC 
(3.11 0) 
IVxvl= 


dS 

204 FUNDAMENTALS OF AERODYNAMICS
Integrating Eqs. (3.113a and b), we have
I ~~271'r In, I 
(3.114) 

Equation (3.114) is the stream function for vortex flow. Note that since '" = constant is the equation of a streamline, Eq. (3.114) states that the streamlines of vortex flow are given by r = constant; i.e., the streamlines are circles. Thus, Eq. (3.114) is consistent with our definition of vortex flow. Also, note from Eq. (3.112) that equipotential lines are given by () = constant, i.e., straight radial lines from the origin. Once again, we see that equipotential lines and streamlines are mutually perpendicular.
At this stage, we summarize the pertinent results for our four elementary flows in Table 3.1.
3.15LIFTING FLOW OVER A CYLINDER
In Sec. 3.13, we superimposed a uniform flow and a doublet to synthesize the flow over a circular cylinder, as shown in Fig. 3.21. In addition, we proved that both the lift and drag were zero for such a flow. However, the streamline pattern shown at the right of Fig. 3.21 is not the only flow that is theoretically possible around a circular cylinder. It is the only flow that is consistent with zero lift. However, there are other possible flow patterns around a circular cylinderdifferent flow patterns which result in a nonzero lift on the cylinder. Such lifting flows are discussed in this section.
Now you might be hesitant at this moment, perplexed by the question as to how a lift could possibly be exerted on a circular cylinder. Is not the body perfectly symmetric, and would not this geometry always result in a symmetric flow field with a consequent zero lift, as we have already discussed? You might be so perplexed that you run down to the laboratory, place a stationary cylinder in a lowspeed tunnel, and measure the lift. To your satisfaction, you measure no lift, and you walk away muttering that the subject of this section is ridiculousthere is no lift on the cylinder. However, go back to the wind tunnel, and this time run a test with the cylinder spinning about its axis at relatively high revolutions per minute. This time you measure a finite lift. Also, by this time you might be thinking about other situations: spin on a baseball causes it to curve, and spin on a golf ball causes it to hook or slice. Clearly, in real life there are nonsymmetric aerodynamic forces acting on these symmetric, spinning bodies. So, maybe the subject matter of this section is not so ridiculous after all. Indeed, as you will soon appreciate, the concept of lifting flow over a cylinder will start us on a journey which leads directly to the theory of the lift generated by airfoils, as discussed in Chap. 4.
Consider the flow synthesized by the addition of the nonlifting flow over a cylinder and a vortex of strength r, as shown in Fig. 3.27. The stream function
FUNDAMENTALS OF INVISCID. INCOMPRESSIBLE FLOW 205
+
Nonlifting flow 
Vortex of 

over a cylinder 
strength r 
Lifting flow over 


a cylinder 
FIGURE 3.27
The synthesis of lifting flow over a circular cylinder.
for nonlifting flow over a circular cylinder of radius R is given by Eq. (3.92):
0/1 = (Voor sin 0) ( 1 _ ~22) 
(3.92) 
The stream function for a vortex of strength r is given by Eq. (3.114). Recall that the stream function is determined within an arbitrary constant; hence, Eq. (3.114) can be written as
0/2 
r 


(3.115) 
=In r+const 


21T 



Since the value of the constant is arbitrary, let 





r 
R 
(3.116) 
Const =  In 



21T 


Combining Eqs. (3.115) and (3.116), we obtain 




r 
r 

(3.117) 

0/2 = 21T In R 

Equation (3.117) is the stream function for a vortex of strength r and is just as valid as Eq. (3.114) obtained earlier; the only difference between these two equations is a constant of the value given by Eq. (3.116).
The resulting stream function for the flow shown at the right of Fig. 3.27 is
or 




(3.118) 






From Eq. (3.118), if r = R, then 0/ = 0 for all values of O. Since 0/ = constant 

is the equation of a streamline, 
r = R is therefore a streamline of the flow, but 
FUNDAMENTALS OF INVISCID, INCOMPRESSIBLE FLOW 207
(a) r<47TVooR 

(b) r =47TVooR 

FIGURE 3.28 
(e) r> 47TVooR 


Stagnation points for the lifting flow over a circular cylinder. 

f / 417'Vo;)R > 1, return to Eq. (3.121). We saw earlier that it is satisfied by r = R; however, it is also satisfied by e= 17'/2 or 17'/2. Substituting e= 17'/2 into Eq. (3.122), and solving for r, we have
(3.124)
Hence, for f /417' VooR > 1, there are two stagnation points, one inside and the other outside the cylinder, and both on the vertical axis, as shown by points 4 and 5 in Fig. 3.28c. [How does one stagnation point fall inside the cylinder? Recall that r = R, or '" = 0, is just one of the allowed streamlines of the flow. There is a theoretical flow inside the cylinderflow that is issuing from the doublet at the origin superimposed with the vortex flow for r < R. The circular streamline r = R is the dividing streamline between this flow and the flow from the freestream. Therefore, as before, we can replace the dividing streamline by a solid bodyour circular cylinderand the external flow will not know the difference. Hence, although one stagnation point falls inside the body (point 5), we are not realistically concerned about it. Instead, from the point of view of flow over a solid cylinder of radius R, point 4 is the only meaningful stagnation point for the case f/417'VocR> 1.]
The results shown in Fig. 3.28 can be visualized as follows. Consider the inviscid incompressible flow of given freestream velocity Vo;) over a cylinder of given radius R. If there is no circulation, i.e., if f = 0, the flow is given by the sketch at the right of Fig. 3.21, with horizontally opposed stagnation points A and B. Now assume that a circulation is imposed on the flow, such that r = 417' Vo;)R. The flow sketched in Fig. 3.28a will result; the two stagnation points will move to the lower surface of the cylinder as shown by points 1 and 2. Assume that f is further increased until f = 417'V",R. The flow sketched in Fig. 3.28b will result,
FUNDAMENTALS OF INVISCID, INCOMPRESSIBLE FLOW 209
The limits of integration in Eq. (3.129) are explained as follows. In the first integral, we are integrating from the leading edge (the front point of the cylinder), moving over the top surface of the cylinder. Hence, e is equal to 17' at the leading edge and, moving over the top surface, decreases to 0 at the trailing edge. In the second integral, we are integrating from the leading edge to the trailing edge while moving over the bottom surface of the cylinder. Hence, e is equal to 17' at the leading edge and, moving over the bottom surface, increases to 217' at the trailing edge. In Eq. (3.129), both Cp," and Cp,1 are given by the same analytic expression for Cp , namely, Eq. (3.126). Hence, Eq. (3.129) can be written as

If1T 
Cp 
cos 
IJ21T 
Cp 
cos 
e de 


Cd =  

e de  
1T 


2 
0 


2 




or 
1 f21T 

e de 



(3.130) 

Cd =  

Cp cos 





2 
0 







Substituting Eq. (3.126) into (3.130), and noting that 







21T 









f o 
cos e de = 0 

(3.131a) 




21T 
sin2 







f o 
e cos e de = 0 

(3.131b) 




21T 









fo 
sin e cos e de = 0 

(3.131c) 






we immediately obtain
(3.132)
Equation (3.132) confirms our intuitive statements made earlier. The drag on a cylinder in an inviscid, incompressible flow is zero, regardless of whether or not the flow has circulation about the cylinder.
The lift on the cylinder can be evaluated in a similar manner as follows. From Eq. (1.15) with cf = 0,


1 fe 
1 fe 
Cp," dx 

(3.133) 

CI = Cn = 
Cp,1 dx  





C 0 
C 
0 



Converting to polar coordinates, we obtain 





x = R cos e 
dx =  
R sin e de 

(3.134) 

Substituting Eq. (3.134) into (3.133), we have 





lJ2rr 

lJO 
Cp," sin 
e de 
(3.135) 

CI = "2 
1T 
Cp,l sin e de +"2 
1T 