solu0003

solu0003 - 1mx11ct3T(11yq g1c3cec&1n i y y d f } k...

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Unformatted text preview: 1mx11ct3T(11yq g1c3cec&1n i y y d f } k y k y x w 11y 31y11|uigi d k y f y eTxcg11ggcVdgqm31fgw3l1r3ggc11 y y y x y y n y r y f n "11r1gVTec (1fc i1 d d y y xd y d y 3pc$c$y31c {3T(cVzgc3T(11gT3$fVc31Ti3cz Qg1f yi1c d y r y y xd i d y d j x y k y xd m yd VTi1Vs 1fv&c3T(d11gT3 i y y xd d f xd x m d cVs1cVg$3u$yV$eqV1Tg3t333c$g1$qlc3TV1$gT3qc1c1&3cs" pc qcV3o yd f y x f y i d y y k f x d y xd y r r 1g3VcVd$c e111Vg1V1cVh1g3(g11cecVd$ g 3T(11c c c&w y y x y i y f yd f y xd d f y f y x y x y y y x x d (Tei1Vd $f&11y 31cg11qpn3y 3 3k 31cg$1g3T(d3l$1 Vdg1ccVd""$g1c1klT$Vdc31Ti3cx m y d k y f y o k y y f m f y y x f y y i f j vW3ETUVRS1Hts1Hr3E"RP 1H Uq$ph3EQIiD3EVh3G$fg11dec&a s f u D f D H W H H b @"@1' A 4 ` B B C - Solution A @ "96 147$4 231&)0(&# 8 6 5 ' ' % # $!" 0=0 f u 2 u = ( 2)+ x t z 0=0 1 p f z 0=- + z u v w + + =0 x y z f u u u u Du 1 p 2 u 2 u 2 u +u +v +w = =- + ( 2 + 2 + 2)+ x x x t x y z Dt y z 2 2 2 fy v v v v Dv 1 p v v v +u +v +w = =- + ( 2 + 2 + 2)+ t x y z Dt y x y z f w w w w Dw 1 p 2 w 2 w 2 w +u +v +w = =- + ( 2 + 2 + 2 )+ z z x t x y z Dt y z equation: 3 x f x = 0 f x / = -g $g1tc3T(11qc g1c3ecV 11g3V33cecVl3x~ k f c3T$3Tvc R1Q$H 3(C H P I G F E D u 2u = 2 t z 1 p = -g z t =0 u=0 x, y - v z=0 cVs111Vg0V1cV 1s xT(T3c31gi1c3cvc 3 xT(T3c31l$ (T gc& = u =0 z u = 0. x, y z = - cVx 1x11qi1c0TsT1 exgTxci11gl3y3""3ccVl1c1x 1 "1g1tTQ3Tg 01gV&11 w Y X W EUR SG E 3TVT3P olution: t - z 2 / 4t u ( z, t ) = z[1 + erf ( z / 2 t )] + 2 e 0 / 2 z -t 2 erf {z ) = e dt , 0 2 -. 15 1 -. 05 - .5 00 Fig.1 u ( z, t ) t - = z[1 + erf (-)] + 2 e = z[1 - 1] + 2 0 0 = 0 . 0 / Fig.2 t =0 z<0 z - t - z 2 / 4t - 2 z u 0 2 - z 2 / 4t 1 = [1 + erf ( z / 2 t )] + z +2 e e z 4t 2 t 2 2 z z = 0 [1 + erf ( z / 2 t )] + e - z / 4t e - z / 4t - t t = 0 [1 + erf ( z / 2 t )] u z =0. z =0 eG!Vd99T#G9GY#!93#'STH#3PH#EC `S c b a F U ` X F W V U F D R Q F D Q F I G F D 0B #A@ z=0 d 2 -z2 erf {z ) = e dz 2 gT(q3T311 z = - , t -t u ( z, t ) = -[1 + erf (-)] + 2 e = - 0 0 / lim f ( z ) = 0 f ( z ) = z[1 + erf ( z )] A @ "96 147$4 231&)0(&# 8 6 5 ' ' % 875 2 00 96431 ) Te1 ~ # $!" C - Solution -. 01 - .5 01 -. 02 B B 3T(T3c31i1c3t xT(T3c31$ (Tg #'% ! "#! (& $ # e4 8g!14 1c #b" % 0 56% ' % UU0h244 &$ % SR % 9" @ &$ 6WTSR % V 0@ % 0 % `ba0!(6W0& % " G ! % 8 4 % G 4 G " % ' ' 4 SR % V %0@ $ G0Q1SR % V 50@ % 0 % h#"T$ 0 % R % 0@ R0%G &$ 6)A'"B8RS % V 0@ 5 0A'0G % " R G i D " % ` G 8 " % G &4 g # " % ' " " 9G"W 8 $ 9"S494 b('f&f Y'"aeeG " $ "9S4dd" 5 &94 "(0A(Y1c 4 # ' ' # G # # 8 ') 4 ' '" D $ SR % "5 0@ % 0 % &0& $ QD $ 9"S4#30 % 643'"5 b 6@% 0U" %% 0% 0a0"2 % 7 ' %0@ YT% G0&@ $ % ` P ' # G ' G C ` X ' ) # G 94 ("03'"W0 $ " % 0)VQD64 ( 0!('9U 0C(TRS 9" QP P " ' 8 4 ' 4 8 solution. - 8 " 8 C G 5 % " % 00 @ " 5 " % 0 % 5 & $ G t -0 e 2 DE" % 000 % ( @ 0B'"" % 094A % @6% &" % 0900" 7 C ! ' " ! # " ! 4" 8 64 65% ! % 403 % & % 2"10(& &$ ! ) ' % 2 t 3 2 B B C - Solution A @ "96 147$4 231&)0(&# 8 6 5 ' ' % 0 = = 0.002 air (Vwind - u 0 ) 2 u 0 = 0.004 air (Vwind - u 0 ) 2 t / . air = 1.21kg / m 3 , = 1000kg / m 3 Vwind = 6m / s 8 '!G IH ' ') (000A % (0(0&& $ 8 u 0 = u (0, t ) = 0 / 0[1 + erf (0 / 2 t )] + 2 1 1 ) 2 - 144 (12 + 0.0029 t 0.0029 t u0 = 2 1 1 ) 2 - 144 - (12 + 12 + 0.0029 t 0.0029 t u0 = 2 t = 60 s u 0 = 0.64m / s t = 3600s u 0 = 2.3m / s . 8 ! 4 ` g8"1' 1e4 # " ! = u 2u = 2 t z 2 2u 0 2 - z 2 / 4t 1 1 u 0 = = 0 = [1 + erf ( z / 2 t )] e e - z / 4t . 2 z t z 2 t 1 - z 2 / 4t t - z 2 / 4t - z 2 - 1 2 - z 2 / 4t z - 1 u 0 = [z +2 +2 ] e e e 2 t 4 t 2 t 2 t 2t # $!" 2.90 10 (6 - u 0 ) 8 4 5 4 8 e@"p11g14 F Q#G` = 2u u - 2 = 0 t z 0 -z e t 0 - z 2 -z [ e 2 t 3 12 + 0 % 8 -3 2 / 4t 2 / 4t -z e t 2 t = u0 + : 2 / 4t -z e t 0 - 2 / 4t 0 1 -z e t + z2 / 4t e-z 2 =0 3 / 4t / 4t = u0 < 6 , 0 t 2 0 2 = t / . = 1.01e - 6 . 8 F #!" @ 8 8 '"g2"g#"014 # @ ! ` 8 pqg"81' $4s' s) B B C - Solution A @ "96 147$4 231&)0(&# 8 6 5 ' ' % # $!" -0.25 -0.15 -0.05 -0.25 -0.15 -0.05 -0.2 -0.1 -0.2 -0.1 -0.01 0 0 0 t=0 0 t=0 t=60 0.01 0.2 0.02 t=60 0.4 0.03 0.04 u(z,t)/(0/) t=3600 /0 0.6 : t=3600 0.05 0.8 4 0.06 1 0.07 0.08 z z ...
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This note was uploaded on 03/19/2012 for the course CIVL 000 taught by Professor Kk during the Spring '10 term at HKUST.

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