mid-term_2_solution

mid-term_2_solution - 1 A uniform ﬂow is described by the...

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Unformatted text preview: 1. A uniform ﬂow is described by the complex potential U(1 - i); (a) What is the magnitude of the velocity and what angle does the [low make to the .r-axis? (b) \N’hat is the complex potential for this flow around a Circle of unit radius centered at the origin? (e) Where are the stagnation points located? . = 7(l-LD'3 4) 71¢ Vvoéaé WA“; M Maw-«,4 ‘7 5.7!: u-L'l/ = U/I-L) 0/2 m w“: U— M Vt U 1005: 30:4 M17.) in}. .4» ~42) c f/e)... %¥(I/e) NW ﬂ) =. V(/+L)-2x a fo/g) z U(/+i)/é 72% “(g-.2): 7//~c)-Z 4 77(21? i C) m vogug M éﬁ/ 5 “'0V I: (7-(/_ ‘9, U'K/J-i) 4312' -E Qfa=1lc 0 r. /+(: . («2 e 2-: if. = (/46); 9‘; .2 _ ([+‘~) /-£ 2 4/2. g 0’65; Mv/gﬁw ion/D M 42%,? 42’ z:4é(l.,,;) M ,, / (Au) 2. For steady developed flow in an annulus between two circular cylinders of radii a. and b, b > a, under the action of at constant pressure gradient dp/dar. the ar—iiioiiieiitum equation reduces to _ dp pd _du O——da:+rdr<7dr> where u (7") is the velocity in the x—direction. Show that the velocity profile is given by _ b2 (1])1 7'2 1 i ln(b/r) u‘ Linda; I)2 b2 ln(b/a) . 1’ Mg g rag 00‘ all- #47:, r664 = f—C, Jr 2/4 x g 4—“: lid/Ma d” 92/“ x. f Mu; 134,344.4m .Lc:z #412: NM (4—- 0 6.144,“ r=a. war=é j; = ﬁzz/[+91né +62— . \$44512: WC 0=£0¢+cl\$~a +cz guik m»: 0 = (é‘m‘) of +¢,~&(b/a.) £4641“: WWW“;— = ~43? 6" dx 9/10.. -52 R 4x5 97‘}; a f‘ L L7 - gﬁ/r/A 5}“ “(‘33 b3. dz 4 (6/4 ) «(MA/r) “(ﬂ(b/4)' 3. The drag force D on a. ship is assumed to depend on the density p and coefficient of viscosity p. of sea water, the ship’s velocity U, its length L , and the gravitational acceleration g. D _f l1, 5% pU2L2 ‘ pUL" U2 (b) If a 1/10th scale model is tested in sea water and it is possible to vary the gravitational acceleration by testing in a centrifuge, what is the ratio of the required acceleration to the standard ac- celeration y if dynamic similarity is to be maintained? (a) Show that Note: The dimensions of n are [ML‘It‘I] 3) a) M 4 M 3 A-MW.M Lﬁm M 3 WW5 N I955. aw”; 8217/ “~01 L M voti— mg 7/ :- F6477"ch w [0] = [MLé-"ﬂML‘ﬂ‘ﬁe'J‘fL’J" O:/+¢ vol a=-/ .ICwL: 0:1—3a+b+c «.1 b+c=~7¢ AIY‘E': 0:: -2-—b “'2 =__2 anJC=_2_ £7 77,: F 4"“: 0:: l+a~ «14:4 Al O=~/’5 Qb='/ )CL/ L' 0=~/»3a+b+c .uc=—/ 773 = ﬁ- U2— gr IE = 7! ’ if: } C'Uz‘Lz evL UP 5) Ar 0‘9 ch; ﬁ'm'éwv’gj .«é’tt. [Zane/ab M Muécc MAJ) . fume/5c Wa(%v% Mﬁlw. "" 5.: >UL; Cay-LN grlavdc ﬂay! paw M lb W ﬂ = ’2! U; LM . 5 2 3 gm / 12—1) 5 b; 17:: = Z1: . 1:; g E— : I000 '/ D"; V‘ "‘ f ‘Djf‘ LM L,..3 4. A laminar boundary layer on a flat plate (zero pressure gradient) is approximated by a linear velocity proﬁle i- y/5 21:6 U00” 1 y>6 Show that, (a) 6* = 6/2 (b) 6 = 6/6 (c) 6/1‘ : 3.464Rc_;1/2 4v vﬁ. yam Vale 5:6 fmff’zl.) 1 I 2’ or = J/(z—w7 ‘ org—.3} z 5 a a 2 ‘4: J"! = 0.58 W- ” (SW/we, , = 5. ~H ﬂ 5/le ~ J 12 73 / 5/4 C 2'3}, = ‘° _6’= 5/4 6) M944 cf: .z/ru/ M T a R u: a 60-062 éy/o 4“] (7%=219 c a/x. 6&7; 5‘ “1913 g _/ in; 3'sz 3 x ar _ égj . 60¢ - 7,0 “ 5:2. 5. The exact skin friction coefﬁcient for the Blasius boundary layer is given by ~ 2 lTwl _ 9U; Consider a flat plate of length L : 10m. The fi'eestl‘eani velocity is 30 ni/s, 1/ = 1.5 x 10—51112/s, and p = 1.2kg/in3. c, = 0.6641103” (a) Derive an expression for the total drag based on the skin friction coefﬁcient. (b) What is the total drag force per unit width on the plate? (Include both sides of the plate.) ll 0 6‘7» g- M , 9c ng to m viva/W ﬁM/W mama/“ca x241? "0W5 L D : ﬂTv/dx o E d ’44” . ﬂip“: Myaaﬁv 36¢ la’w/ 3mm 4 1 a. 72. ‘3 ‘= ’2 633:6 C346: = CUM/0.5651 Jr. .2 0:02; 0 9 . i L =67?) 6156?! ’ ,C'Vux 77:0 o z j) /z [1 L :62” (aéé¢)(__ 21x /0 03¢ ’/a_ u. g = emLL (0.664) ELL A/o-cJ 7-7—11. _ &L (35300) = 2 x/o? . q) '0-5X/o’{ . I/ . f (D: 0.2)(309‘Klo)/0.5543( 2, m7) ‘ :pé u/m ...
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mid-term_2_solution - 1 A uniform ﬂow is described by the...

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