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Unformatted text preview: .._.._._W_.e.¢ s.  ,.,V_~..Aﬁ W, , ﬂ/ph—Sﬂ/I‘V'ﬂr'hj 'V(r\/ /(7W KI’an/I/I/Vd FIGURE 3.4]
Flow over a cirwlnr cylinder. Re LS4. (Phalogmph by Sadatoxhi Taneda, from Van Dyke, Milton,
An Album of Fluid Motion. The Parabolic Pun, Stanford, Cal'f I982.) (0) 5/7/10!) I'Mj 0’) FIGURE 3.29
Th5: ﬂowﬁeld pictures were obtained in water, where aluminum ﬁlings were scattered on the surface In show the direction of the streamlines. (From Prandll and Tietjens, Raj? 8.) (a) Shown above is the
use for the nonspinning cylinder. (b) Spinning cylinder: peripheral velocity of the surface;= 3V.” .ﬁ...» q.— .~.'—w.... ————vW« .W..W.. A ..... ..._...... fa: FUNDAMENTALS OF AERODYNAMICS simply does not model the proper physics for drag calculations. On the other
hand, the prediction of lift via Eq. (3.140) is quite realistic. Let us return to the
windtunnel experiments mentioned at the beginning of this chapter. If a station
ary, nonspinning cylinder is placed in a lowspeed wind tunnel, the ﬂow ﬁeld
will appear as shown in Fig. 3.2%. The streamlines over the front of the cylinder
are similar to theoretical predictions, as sketched at the right of Fig. 3.21. However,
because of viscous effects, the ﬂow separates over the rear of the cylinder, creating
a recirculating ﬂow in the wake downstream of the body. This separated ﬂow
greatly contributes to the ﬁnite drag measured for the cylinder. On the other
hand, Fig. 3.29a shows a reasonably symmetric ﬂow about the horizontal axis,
and the measurement of lift is essentially zero. Now let us spin the cylinder in
a clockwise direction about its axis. The resulting ﬂow ﬁelds are shown in Fig.
3.2% and c. For a moderate amount of spin (Fig. 3.2%), the stagnation points
move to the lower part of the cylinder, similar to the theoretical ﬂow. sketched
in Fig. 3.28a. Ifthe spin is sufﬁciently increased (Fig. 3.29c), the stagnation point
lifts oﬁ the surface, similar to the theoretical ﬂow sketched in Fig. 3.28c. And
what is most important, a ﬁnite lift is measured for the spinning cylinder in the
wind tunnel. What is happening here? Why does spinning the cylinder produce
lift? In actuality, the friction between the ﬂuid and the surface of the cylinder
tends to drag the ﬂuid near the surface in the same direction as the rotational
thion. Superimposed on top of the usual nonspinning ﬂow, this “extra” velocity
contribution creates a higherthanusual velocity at the top of the cylinder and
a lowerthanusual velocity at the bottom, as sketched in Fig. 3.30. These velocities
are assumed to be just outside the viscous boundary layer on the surface. Recall (c') FICURE 3.29
(c) Spinning cylinder: peripheral velocity of the surface=6Vw .‘.; (3.114) Note that since «[1:
s that the streamlines
nes are circ es. Thus,
. Also, note from Eq.
:., straight radial lines
5 and streamlines are  our four elementary blet to synthesize the
lition, we proved that
the streamline pattern
theox ally possible
lSiStCITF/With zero lift.
a cirCular cylinder—
~ cylinder. Such lifting ed by the question as
nder. Is not the body
result in a symmetric
discussed? You might
e a stationary cylinder
sfaction, you measure
section is ridiculous—
wind tunnel, and this
itively high revolutions
his time you might be
s it to curve, and spin
here are nonsymmetric
bodies. So, maybe the
llfIndeed, as you will
ider will start us on a
nerated by airfoils, as t nonlifting ﬂow over a
7. The stream function V w 4 __..._,,. ﬁ+= l
Nonlifting flow Vortex of ' I _
over a c linder stre ‘f" Lifting ﬂow over
y d « "ngm a cylinder J FIGURE 3.27   v ‘
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): . R2
[/1] = (er sin 6)(1r—2> (3.92)
The stream function for a vortex of strength F 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 F ¢2=Eln r+c0nst (3.115) Since the value of the constant is arbitrary, let P
_ Const= ——ln R (3.116)
277
Combining Eqs. (3.115) and (3.116), we obtain
I‘ r
=——l — .
((12 2” n R (3117) Equation (3.117) is the stream function for a vortex of strength F and is just as
valid as Eq. (3.114) obtained earlier; the only diﬁerence 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 11’ = Wi'i' ‘1’2
, R2 F r 7
or ¢=(Vwrsm 6)<1——2—)+—ln— (3.118)
r 27r R From Eq. (3.118), if r= R, then 1/1 = 0 for all values of 9. Since {/1 =: constant
is the equation of a streamline, r= R is therefore a streamline of the ﬂow, but 3.118) is a valid stream
ular cylinder of radius
.115 result given by Eq. is sketched at the right
symmetrical about the
:ct (correctly) that the
owever, the streamlines
.s a result the drag will
ortex of strength I‘ has
is now ﬁnite and equal Eq. (3.118). An equally
elocity ﬁeld of a vortex
because of the linearity
d elementary ﬂows add
.ng ﬂow over a cylinder
we have, for the lifting (3.119) (3.120) = Ve=0 in Eqs. (3.119)
): (3.121) 0 (3.122) 1. (3.122) and solving for (3.123) e in the third and fourth
m the bottom half of the
3.28a. These points are
Never, this result is valid
.123) has no meaning. If
e surface of the cylinder,
3.28b. For the case of FUNDAMENTALS OF lNVlSCID. INCOMPRESSIBLE FLOW Twila/Mal? c (a) I‘ < 41erR 71< (b)I‘=41erR FIGURE 3.28 (c) 1‘ > 41IV00R
Stagnation points for the lifting ﬂow over a circular cylinder. 7" y V
> "I .0 ll F/47erR > 1, return to Eq. (3.121). We saw earlier that it is satisﬁed by r= R;
however, it is also satisﬁed by 0 = 7r/2 or —77/2. Substituting 0 = —1r/2 into Eq.
(3.122), and solving for r, we have T l" 2 2
= i __ I
r 4w... (4.1V...) R (3 124) Hence, for F/47rVaoR> 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 41:0, is just one of the allowed streamlines of the ﬂow.
There is a theoretical ﬂow inside the cylinder—flow 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 body—our circular cylinder—and 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/41rVooR > 1.] The results shown in Fig. 3.28 can be visualized as follows. Consider the
inviscid incompressible ﬂow of given freestream velocity V00 over a cylinder of
given radius R. If there is no circulation, i.e., if r=o, 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 F (477mG
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 = 477VwR. The ﬂow sketched in Fig. 3.28b will result, ...
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 Summer '07
 Ruffin

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