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Unformatted text preview: 530.327  Intro” to Fluid Mechanics u 811
Final Exam Name gﬁumws 180 points Short answers. (5 problems, total 40 pts) As usual, justify any assumptions, explain your
answers, and be as brief as possible. Problem 1. (8 pts) One idea. for reducing ﬂow separation from, say, a. cylinder is to punch small
holes in the surface and generate suction through the holes“ If the zero degree point is the most
upstream point on the cylinder and 180° is the most downstream point, what angular position
(approximately) would be the best place to put these suction holes? What’s the physical reasoning
about why they would w01k to reduce separation"? an» / $5.?AZAI’WN ‘PT (Foil ex AM? be) —>
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back bumper“ The wheels protrude a. little bit into the airstream on either side of the car. If you
want to minimize the drag caused by the bike, should you lock the wheels in piece, or allow them to rotate freely? W“
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d, and a streamlined shape consisting of a half—cylinder (with diameter 03) followed by a long tail
on the downstream side, as shown in the ﬁgure. When the airstream ﬂows from left to right,
the streamlined shape has signiﬁcantly less drag than the cylinder. Now suppose that the ﬂow is
turned around so that it goes from righttoIeft and strikes the sharp end, not the blunt end, of the
streamlined shape. How does the drag of the streamlined shape compare to that of the cylinder
now? (The air is, of course, treated as viscous.) .3 CD site” 093912th occugzs towpgrm or “EB more“ tom . 1F ‘foxx "were
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relying on the wind to blow the dust away? Tam; A tamer wav— vm m Caz}, Seem was at»: umcmes
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Suppose you have a. spherical weather baiioon, of diameter 2 111, whose skin is rigid (ie the balloon’s
shape and volume are ﬁxed), totally leakproof‘, and weightlessw in your sea—level garage you ﬁll the
balloon with helium at the same temperature and pressure as the atmosphere“ Helium has density
0. 138 times that of air at the same thermodynamic conditions” You attach an instrument package
(with mass m = 605 kg) to the baﬁoon that transmits meteorological data back to you, then you
go outside and let go of the balloons. If there is no wind, describe What happens to the balioon as
time passes, and state as precisely as you can where the balloon will end up in inﬁnite time. N61 d9vJA1LD were : .. 3 k a
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pipes (the ones without the valves), you may have found that you got values for the friction
factor, f, that led to nonsensical values for the relative roughness, e/ D (that is, the intersection
of the f value and the Reynolds number, Re, was below the ‘smooth pipe’ curve on the Moody
diagram). Does it make sense to blame this result on buildup in the pipes, which would reduce
the effective diameter of the pipes? Be as mathematically precise as you can. b) (25 pts) [11 Lab 3, suppose that instead of a. cylinder, we put a triangular wedge into the wind
tunnel, where the wedge engie is 60° and the wedge height is h. Suppose the freestream tunnel
speed is U , the tunnel height is H m 10h, and its width is w in the out~oprlane direction.
Now suppose that the boundary layer on the wedge is inﬁnitely thin, so that upstream of the
back of the wedge (region 1 in the ﬁgure), the velocity is uniform on tunnel cross—sections; also
suppose that the ﬂow separates right at the back corner of the wedge, and that the pressure is
uniform everywhere downstream of the wedge (region 2), With these assumptions, what’s the
drag coefﬁcient of the wedge? , Tin
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of the tank opens into a, horizontal Spﬁlwey; the spiliway has a. square cross—section that measures
2h tall by 2k in the out—oprlane direction. A massless gate, which tums on a horizontal hinge at
its midpoint, sepmetes the water in the tank from the air in the spillwey. In the closed position,
the gate makes an angle of 6 from the horizontal. Let d = 5m and h = 1 In, and use a. value for the gravitational acceleration of g "2 9313/52" pmmh'll ’V a) (15 pts) What’s the tOIque (magnitude and direction) you need to apply at the hinge to open
the gate when 6 = 90°? 1)) (20 pts) What’s the torque (magnitude and diz‘ection) you need to apply at the hinge to open
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side“ The tank sits on a very accurate scale“ Water (at 20°C) ﬂows into the tank, at volumetric
ﬂow rate Q = 02 ﬁteIs per second, from a faucet with diameter d m 1.5cm, whose outlet is located
H = 1m above the bottom of the tank" You went to put 50 liters of water in the tank, which
(using 9 = 9.8 1321/52 and p = 998 kg/ 1113) weighs 489 N; you decide to do this by running the faucet
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u, 2 2x3; + y in meters per second when m and y are in meters, and you’re given that v = 4m/s
when :1: = 1 m and y m 2111“ Specify a form of v 2 v(m, y) (doesn’t have to be the simplest) that
satisﬁes the given conditions“ b) (10 pts) Consider a turbulent pipe ﬂow, in a round pipe with radius R, where the average
velocity proﬁle happens to be given by V0“) 2 Vgcos (1 —~ What would the kinetic energy coefﬁcient, 05 (which shows up in the head loss equation), be for
this case? c) (15 pts) In HW 9, we analyzed the ﬂow of viscous fluid between palatial plates, Where the top
plate moved with speed U1 and the bottom plate moved with speed U2“ For the case where
dp/da: = D, the velocity ﬁeid came out to be My) = U2 + (U1 W U2)%« (1) Write the Navier—Stokes equation for the wcomponest of velocity Which term describes the
viscous forces? Show why, even though we know the ﬂuid is viscous, the viscous terms go to
zero for the solution in Eq. 1, and explain what’s going on" Q9 .3
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