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Last modified
1/25/2010 7:10:12 PM
Creation date
10/5/2006 2:34:31 AM
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Floodplain Documents
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Statewide
Title
Hydraulic Engineering volume 1
Date
1/1/1994
Prepared By
American Society of Civil Engineers
Floodplain - Doc Type
Educational/Technical/Reference Information
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<br />. <br /> <br />,... <br /> <br />BYI)RAUI.I(: EN(HNEERING '94 <br /> <br />the hydl'aullc radius we get for the wide rectangular channel <br />a R' <br /> <br />/8 R S <br /> <br />(6a) <br /> <br />and for the <br /> <br />v -- <br />r (p+l J yP <br />o <br />se.i-circular seclion <br />2(P.1Ja RP ----- <br />/8 R S = F.V <br />, <br /> <br />(7.) <br /> <br />v <br />.. <br /> <br />where r is the ~hape <br /> <br />Ip'~) y' <br />o <br />factor <br />f : 2(P+U <br /> <br />(BI <br /> <br />(p.1! <br /> <br />(p'21 <br />The difference of 21X between the two model shapes for a l.miform <br />flow In an open channel shows that an influence of a channel shape <br />on the Hanning n u.y be quite significant. Proceedlna: in a similar <br />way with the Prandtl-von Karman universal logarithmic velocity <br />distribution law the shape factor can be shown to be <br /> <br />F = (-In 7 + 1'72. 2.10 2)/(-10 7 + 7 - 1) <br />B <br />where 'FY /R. For 1/1 = 103. 10.. 105 and 106 the f equals 1.404. <br />. <br />1.291. 1.2~1 and t.186 respectively_ <br />(d) For other shapes the dislrlbuti.::n of shear along the wet led <br />pt!f'lmeter Is not unlfor. and the lsovels will not be sirnllar as in <br />tl.lO special cases above. <br />THE FlPST AXIOM VIOLATION. Let us assUlne the flQW area- flow <br />dIscharge curve above the initial ievei ho in the form <br /> <br />Q = Q (A fA )- <br />. . <br />Then the exponent m is the raHo of kinematic wave spet:l1 to the <br />average velocity of flow <br />. . (dQ/dAI/(Q/AI . 1111 <br />Using the geometric relationships for a symmetric channel and <br />assuming the constant SF one can express the variables in equation <br />(10) by means of the width of channel water surface (T). the water <br />depth (0) and tha initial level parameters <br />h 211\-.11'(..-1)1 <br />dT/dh = :!:( (A +ST dh) T~/ 02 - 4}0.5 <br />. 0 <br />. <br /> <br />(91 <br /> <br />( 10) <br /> <br />(12) <br /> <br />with T=T <br />. <br /> <br />for h=h and <br />. <br /> <br />B ={(r_1}/(r_adJ.(A(..-all'lr-t}p I. <br />. .. <br />The negative root in equation (12) corresponds to the narrowing <br />surface w-idth with lncr-easlng depth. whIle the positive one to the <br />widening surface width with deplh. Equation (121 has been soived by <br />the Runge-Kulla fourlh order Method. The cross sectioO$ profiles for <br />the semi"'circular initial shape of lwo meters wiele. the Hanning <br />friction and the III values in the vicinity of zero are displayed at <br />Fig.i. Obviously a negative value of. contradicts the first axlo.. <br />THE SECOND AXIOM VIOLATION. Let us assuae that above the lnll tal <br />level there exists the alternate shape (2) to the original one (il <br /> <br />((31 <br /> <br />- <br /> <br />- <br /> <br />CHEZY FORMULA V. THEORY <br /> <br />.605 <br /> <br />C.lh)=C,lh). (14) <br />Then th~ solution can be obtain~d numerically froll the equivalence <br />of Lerma <br /> <br />dCrdA <br /> <br />Ir-I! dP <br /> <br />dh <br />for the original channel <br /> <br />((51 <br /> <br />A dh P <br />(1).nd the <br />dC dC <br />_2 _I <br /> <br />dh dh <br /> <br />dh <br />al ternate <br /> <br />channel (2) <br /> <br />(16) <br /> <br />h-h,(m) <br />.30 <br /> <br />0.2 <br /> <br />.30 <br /> <br /> <br />0.0 <br />-Q.2' <br /> <br />m <br /> <br />.00 <br />. <br /> <br />2 <br /> <br />. <br />(T-T,)/2 <br /> <br />. <br />(m) <br /> <br />. <br /> <br />10 <br /> <br />Figure 1. Channel shapes for small values of m. <br />For simplicity let the -,rlginal channel be with constant side <br />slopes Z OVer an initial level ho~' so that the \talue of the right <br /> <br />hand side of equation (16) for the original channel can be <br />determined for any value of h <br /> <br />dG <br />_I <br /> <br />2(r-1 Ill..' <br /> <br />rlT +2.hl <br />_0 <br />A +T h+zha <br />. . <br /> <br />(17) <br /> <br />P ... 2h /1+ z'2 <br />. <br />Using the geOMetric relationships for a symmetric channel one can <br />express the variables in equation (16) by means of T. h and the <br />initial level paramelers getting the differential-integral equation: <br />dT 2 (rT)' T <br />dh . (hI'. IAI l-A -2rA..3(h)-.:(h)..:(A)'.~lhl (IB) <br />where <br /> <br />dh <br /> <br />.,(h) '= 2 (A +l hi-zh2) "/C".I}(p "'2h~+z~) <br />o 0 0 <br />IfIz(A) = (r ... 1 ) A-O(..-IJ <br /> <br />"3(h}-r(T +2zhl/(A +T h+zha)-2(r-l)~+z2/(P +2hvf...z21 <br />o 0 0 0 <br /> <br />(l9a) <br /> <br />(19b) <br /> <br />liBel <br />
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