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aer205_q3_2006_solutions

# aer205_q3_2006_solutions - AER205F FLUID MECHANICS and...

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Unformatted text preview: AER205F FLUID MECHANICS and TRANSPORT PHENOMENA Quiz 3 17 November 2006 9:10 pm - 10:00 am Closed Book, no aid sheets Non-programmable calculators allowed Instructor: J. W. Davis Family Name: .8 Kb bouts. Given Name: (ml wl‘k (JV-4. Student #: FOR MARKER USE ONLY )—A O ,_. O O —_0 —- —_ Useful data: pair— — 1.23 kg/m3 pm: 1. 8 X 10'5 Ns/m2 pwater— — 1000 kg/m3 l’l’water— -— L] X 10 3 Ns/m2 Page 1 of 7 1) Indicate whether the statement is True or False. (10 marks) b) d) g) h) j) The Reynolds number is a measure of the relative importance of inertial and viscous forces. The speed of sound is higher in summer than in winter. In a converging duct, an incompressible ﬂuid will be accelerated more than a compressible ﬂuid. For an irrotational ﬂow that is steady and incompressible, the Bernoulli equation applies between any two points in the ﬂow. The Mach number is an example of a dimensionless number. Sound waves are very small amplitude pressure pulses that travel at the”speed of sound”. The 2ml law of thermodynamics does not apply in a shock wave. The average velocity of water ﬂowing in a pipe will increase by a factor of 4 if the pipe diameter decreases by a factor of 2. the Froude number is a measure of the relative importance of pressure and gravitational forces. The speed at which a tsunami travels depends on the depth of the water. Page 2 of 7 2) Starting with the momentum of a ﬂuid particle entering a control volume along a streamline and the momentum of a ﬂuid particle leaving the control volume, derive the following expression which relates changes in momentum to the sum of all external forces acting on a ﬂuid element: mf(‘72 _ ‘71): FE (10 marks) St a -" -‘ . - u‘ —U" Page30f7 __ F; .. WEEK 2.. ‘) 3) Water at a pressure of pgage = 10 kPa, ﬂows in a horizontal pipe, which has a 180° bend as shown in the ﬁgure. Given q1 = 10 m/s, and a constant cross-sectional area A = 0.01 m2, what force must be applied by the clamps to hold the bend in place. (7 marks) A = constonT ﬁlm‘tys - ﬁPMM‘ — 17E = Q-?-l\ﬁ+ ZS?- PM. f = Z (@000) (0.0%: 1— 2. (torso) (ltdL (OJ-9‘»: A = 2.100li E. Dal-c: “(cm-Alums {0 +1“ 34%me ﬁuakm‘ H1 'PTCMLLft mu he ml and: +quﬁwu4 H“ Clam. Page 4 of 7 4) The energy equation for adiabatic, frictionless, steady ﬂow is given by: & Lie- ds— p2 ds _ Show that for an ideal gas, where p = pRT and e = C VT, this equation leads to the relation: p:C1p7 (6marks) d o «it? - 9.? is. 6': Q'T '= Q”: #7 f = S d5 :2. 5 5\ E: if _£C:? -£)é_§ :0 SI d5 g R- 52 5 ii? ._ Q .L + _?_~_ {5 ? d5. S Q03 \$ ,_ l n+2) his Qv § CLS ,. H5 5 M \X l y In Jr COM—1+ let-(grep V3 W? " S Page 5 of 7 5) Given the unsteady, two-dimensional, incompressible ﬂow ﬁeld: mar) = 2xy(l + at)? + (ax2 + by2)(1+ et + ftz )j' Choose the constants a, b, e and f so that (i) the equation of continuity is satisﬁed, and (ii) the ﬂow is irrotational, that is, has vorticity V x \7 = 0. Find the scalar potential ¢, such that 9: V¢. (7marks) UL: Zx‘SlH-okk) u- =(ax‘+le~11)ll+e.¥ + 5H?) at v g; = lwlwc) 173' 7.51 hut +t;+f) Com‘x'tkmh'g. :.~%§ :0 =7 ‘3: '1 u- : (Ont-1ws1‘) (Hurt) Q:- 0 3—“ = ZAL‘+H~5\ AU‘ _ “Zn-x Ll+\$£) ‘8 "5.1 ‘ FM 9’ 5+. am: 9:; . u- “ix—KUWQ _) 9f: lleHMc) + Okla-lit) um 25’ 3 tr : ;EUUM) ._.> 9’= (M— a )ll-H-Q + Cilia) +1 94-.- (,szs -3“ mt) + 030,) Page 6 of 7 6) A viscous liquid of constant pand ,u falls due to gravity between two inﬁnite plates a distance 2h apart. The ﬂow is fully developed, with a single velocity component in the z-direction, v = -w(x)k. Find the velocity proﬁle between the plates. Assume atmospheric pressure everywhere. (10 marks) 2h 4x. lv[X] "C‘ X £1: Kora Z Lx In stead: MW, Hm Dual ”fa-Akita w‘\\\ m be welaalrté. 4’le €51»?in «FL £91m: balance: at?- 3151 - ”L \$23.8 — \$3 51%;“; =0 (Dc-m aux z 493 smite: Qt _._._£‘ AT. 3 3 ,7) ___...— :33 L “QM-3W0». Flu-ital.- °C=jx§§ 1..) #933; 7.3% lli'ﬂtiroA-KNS: Q.“ a! EL+Q .3 W‘- ﬂxli-Q‘x‘l-CZ 41:. ,u. ‘ Z,» _ k1 BowmAa-Wi C S! L“ [k\ =0 : 5-3 lb; * CJ‘J‘ CL=O C?" lg “3—13".- 19" Page 7 of 7 ...
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