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Unformatted text preview: Probiem Set #4 PART A: Short Answers 1. Complete the foliowing statements: a) If you took a long time—exposure photograph of a luminous particle moving with
the ﬂow in a river, the image would be a ( pathline , streakline , streamline ). b) A is a curve in a ﬂow field which has the property that
the velocity vector of each particle on the curve is tangent to the curve. 0) There are a ( ﬁnite , variable , inﬁnite ) number of streamlines in a ﬂow ﬁeid. d) If all"? is the inﬁnitesimal displacement vector along a streamline and l7 is the
velocity of a particle on the streamline, then dR x I7 a e) If you continuously inject dye at a fixed point into a ﬂowing fluid in such a way
that the dye particles essentially match the speed of the ﬂuid as they enter the
ﬂow, the pattern of dye you see in a snapshot of the ﬂow is a ( pathline ,
streakiine , streamline ). f) A streamline is an ( Eulerian , Lagrangian) concept. g) A pathline is an ( Eulerian , Lagrangian) concept. 2. If you saw a plot of a flow in which two initially separate streamlines eventualiy
intersected with each other at a point, what would that necessarily tell you about the fluid
velocity at that point? Why? DI? BI? 3. We know that E = 5+ I?  V 17. Using vector identities, show that the following
M H 2
expressing is equivalent: €35; = %::+ —— (V x I7)X V . Note that V2 m V . I? _ 4. In a certain river with a velocity ﬁeld 1301:) with respect to the ground, the
concentration of dissolved oxygen is given by the function f(5c‘,t). A small ﬁsh is in the river and has a velocity (.70) with respect to the ground. The time rate of change of f experienced by the ﬁsh is: The ﬁsh lays eggs that are very small and neutrally buoyant. The time rate of change off
experienced by the eggs is: Some of the eggs eventually become trapped among rocks and remain there. The time
rate of change of f experienced by the eggs is then: 5. In an incompressible ﬂow where 38 m a)?“ + I7 ~ V730 2 O , the density p in generai Dr 3: a) may vary in time at a particular point in the ﬂow ﬁeld
b) must be a constant at all points in the ﬂow field
0) may vary in space but not time throughout the ﬂow ﬁeid 6. For a steady onedimensionai ﬂow with a velocity magnitude u, a) the velocity gradient Bat/3): must be zero at any point
b) the density ,0 must be the same at all points in space 0) the rate of change of the density of a particle as it moves through the ﬂow
depends oniy on the spatial variation of the density PART B : BASIC CONCEPTS Problem 1: Buoyancy (3 pts.) An AUV is designed as shown below. it has three, hollow, cylindrical canisters arranged
in an upside down triangle. The cyiindricai canisters are constructed out of aluminum,
speciﬁc gravity of 2.7, and have a wail thickness of 1 cm, length 2 meters, and an inner
diameter 14 em. What is the maximum load (kg) this vehicie can carry without sinking, ignoring the additional weight and buoyancy of the struts? (Think neutrain buoyant!)
Assume the vehicle will be operating in SALT water ( p w 1025 kg 1' m3 ). AUV Design. Probiem 2: Basic Fiuids & Math (3 pts.) 3.) Aithough ﬂuids, such as water, are really made up of discreet molecules, we can
describe their behavior by differential equations by virtue of the hypothesis.
b) Discuss the difference between a pathiine and a streamiine. Under what conditions are they the same? 0) Show by expanding and collecting terms that (‘7 0 WV = A»??? ' i7) for irrotationai
2 — é‘w (with; aw)~. 8v ﬂu
dz é'x flow. (Hint: §xV= ———— a + ]+ ww— Eachohoi for
3y dz 6x 8y irrotationai ﬂow.) Problem 3: Velocity Field (8 pts.) 3) b)
c) d)
e) s) h) A velocity field is given by l7 = (x2  y2 )5 —2xy}. Is this a valid velocity ﬁeld for an incompressible ﬂuid? (Hint: It must satisfy the Continuity Equation.)
Is the flow steady? Sketch arrows depicting the velocity vectors at each point on the graph below. (Hint: There is no ﬂow across the dashed lines.)
Sketch the streamlines on your graph.
Determine the angle 9‘ (Recall, streamlines are everywhere tangent to the ﬂow, so ﬂ = 3— along a streamline.)
u
. . . m m 3v 8a A .
Is the flow irrotattonal? (Hint: V x V a (3m 3;]16 for this 2D flow.)
x Sketch “Bob the Fluid Blob” at four different points along his path through this flow.
Sketch his acceleration vector for each point. If we assume that pressure is the only force acting on Bob, then where is the pressure
the highest? Bob starts here Problem 4: Hydrostatics The lock system between the Charles River and Boston harbor is designed to control
the ﬂow of water from the river due to an extreme change in the level of the water in
Boston harbor between low and high tide. The tides in Boston Harbor change 15
feet each day from low to high tide. The level of the river is a constant 18 feet deep
ail daylong and at low tide the harbor is 14 feet deep. Chamber The lock consists of two swinging gates that are 22 feet wide and extend to the
river/harbor seabed. No water can leak through the locks when they are closed. The
two walls are 50 feet apart as shown in the drawing. When a ship wants to go from the harbor to the river one side of the locks is opened
to allow the water in the middie chamber to equalize to the same height as the
harbor. When the water in the chamber is at the same level then the ship enters the
locks. The gate is closed behind him. A pump then begins to ﬁll or drain the chamber
in order to equalize the depth in the chamber to the depth in the river. Once the
water level in the chamber is equai to the height of the water in the river, the gate on
the river side is opened and the vessel can exit safeiy. When the vessel has exited the
gate closes and the water in the chamber remains at that height until the next vessel
goes through. a) Determine the maximum possible force on the ate adjacent to the harbor
when it is closed. Does this occur at high or low tide and what ievel is the
water in the middle chamber? (Hint: Ignore the dynamics forces due to ﬁlling
the chamber ad/or currents in the harbor) b] Where is this force acting (center of pressure)? c} You are asked to redesign the lock gates and are given three options sketched
below. Use your intuition and knowiedge of hydrostatics to determine which wouid be the most feasible design. It is possible that more than one design
would work. Harbor (A) Swinging Grater; \i 1%
f? 1?“ River I I Harbor (B) Awinging Double Gates; Harbor ((22) Sliding Gatee ...
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This note was uploaded on 04/02/2010 for the course ME 152a taught by Professor Meinhart,c during the Spring '08 term at UCSB.
 Spring '08
 Meinhart,C

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