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    Gradually

    Varied FlowComputations FLUID MECHANICS ||

    CIVIL ENGINEEING!

    " # H  SEMES#E

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    INTRODUCTION Computations o$ %radually &aried 'ow (GVF) are part o$ t*e routine pra+ti+e o$ *ydrauli+ en%ineerin%, #*e GVF e-uation des+ri.es steady %radually &aried 'ow in open +*annels (C*ow /0"01 Henderson /022), #*e

    +on&entional GVF e-uation is e3pressed in terms o$ .edslopeSo! $ri+tion slope Sf ! and Froude num.er F , Herein! t*e GVF e-uation is alternati&ely e3pressed in terms o$ .ed slope So! +riti+al slope Sc! and Froude num.er F , Analysis o$ t*is e-uation re&eals t*at t*e 'ow5dept* %radient dy 6dx  is stri+tly limited to &alues outside t*e

    ran%e en+ompassed .y So and Sc, #*is impro&es and +ompletes t*e de7nition o$ 'ow5dept*5%radient ran%es in t*e analysis o$ water sur$a+e pro7les,

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    GRADUALLY VARIED

    FLOW EQUATIONS  #*e GVF e-uation is8

     dx   So - Sf  

     ___ =  ___________________   (1)

     dy 1 - [(Q 2 T  ) / (g A 3)]

     in w*i+* y  9 'ow dept*! x  9 distan+e alon% t*e +*annel! dy 6dx  9 'ow5dept* %radient! Q 9 dis+*ar%e! T  9 top widt*! A 9 'ow area! and g 9 %ra&itational a++eleration, #*is e-uation is &alid $or small .ed slopes (So : ;,/)! w*i+* is usually t*e +ase,

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    GRADUALLY VARIED

    FLOW EQUATIONS  #*e $ri+tion slope in terms o$ t*e C*e)  C 2  A 2 R

     in w*i+* R = A/P 9 *ydrauli+ radius! and P 9 wetted perimeter,

      #*e Froude num.er in terms o$ dis+*ar%e is   Q 2 T   F 2  =  _______

    (?)

      g  A  3

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    GRADUALLY VARIED

    FLOW EQUATIONS Com.inin% E-uations > and ? leads to8  Sf   = (P / T  ) (g / C 

    2) F 2

    (@)

     At normal +riti+al 'ow! F  9 /! and t*e $ri+tion slope at +riti+al state! i,e,! t*e +riti+al slope! is8

     Sc  = (Pc / T c ) (g / C  2) (")

    Com.inin% E-uations /! @! and "8

    dy   So - Sc F  2 

    ___ =  ___________

    (2)

     dx 1 - F 2

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    GRADUALLY VARIED

    FLOW EQUATIONS w*i+* is stri+tly &alid $or t*e $ollowin% +ondition8 P 6T  9 Pc 6T c , #*is latter +ondition is %enerally satis7ed in a *ydrauli+ally wide +*annel! $or w*i+* T  is asymptoti+ally e-ual to P,

    For ease o$ e3pression! t*e 'ow5dept* %radient is renamed S y  9 dy 6dx , Sol&in% $or Froude num.er $rom E-uation 28

      S o

     - S y 

      F 2  = _________

    ()

      Sc - Sy 

     Sin+e F  > B ;! t*e 'ow5dept* %radient must satis$y t*e $ollowin% ine-ualities8

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    GRADUALLY VARIED

    FLOW EQUATIONS So  S y   Sc (8)  So  S y   Sc (9)

    w*i+* ee+ti&ely limits t*e 'ow5dept* %radient to &alues outside t*e

    ran%e en+ompassed .y So and Sc, Furt*ermore! E-uation 2 +an .ealternately e3pressed as $ollows8

      (/;)

    E-uation /; is t*e GVF e-uation in terms o$ .ed slope So ! +riti+al

    slope Sc ! and Froude num.er F , #*e .ed slope +ould .e positi&e (steep!

    +riti+al! or mild)!

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      WATER-SURFACE

    ROFILES E-uation /; is used to de&elop a +lassi7+ation o$ water5 sur$a+e pro7les .ased solely on t*e t*ree dimensionless parameters8 Sy 6S+ ! So 6S+ ! and F, For t*e sae o$ +ompleteness! su.+riti+al 'ow is de7ned as t*at $or w*i+*

    t*e 'ow dept* is %reater t*an t*e +riti+al dept* (F > : /)arallelin% t*is widely a++epted de7nition! su.normal 'ow is de7ned as t*at $or w*i+* t*e 'ow dept* is %reater t*an t*e normal dept* F > : So 6S+ , Supernormal 'ow is de7ned as t*at $or w*i+* t*e 'ow dept* is smaller t*an t*e normal dept* F > B So 6S+  (USDA SCS /0/), #a.le

    / s*ows t*e possi.le types o$ water5sur$a+e pro7les,

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      WATER-SURFACE

    ROFILES

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      WATER-SURFACE

    ROFILES  #*e +lassi7+ation $ollows dire+tly $rom t*e %o&ernin% e-uation (E-uation /;), It is seen t*at t*e %eneral type o$ pro7le (#ype /! >! or ?) determines t*e si%n o$ Sy 6S+ (Column >) and t*us! t*e +lassi7+ation o$ eit*er .a+water or drawdown (Column ?), Also! t*e %eneral type o$ pro7le determines t*e $easi.le ran%e o$ So 6S+

    (Column @) and t*us! t*e e3isten+e o$ spe+i7+ pro7les types (Steep! Criti+al! Mild! Hori,

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    SUMMAK  #*e %radually &aried 'ow e-uation is e3pressed in terms o$ t*e +riti+al slope S+ , In t*is way! t*e 'ow dept* %radient dy6d3 is s*own to .e stri+tly limited to &alues outside o$ t*e ran%e en+ompassed .y So and S+, #*is

    +ompletes t*e de7nition o$ dept*5%radient ran%es $or allwater5sur$a+e pro7les, For instan+e! t*e 'ow5dept* %radient $or t*e S? pro7le de+reases $rom S+ (a 7nite positi&e &alue) to ; (asymptoti+ to normal dept*), Liewise! t*e 'ow dept* %radient $or t*e C/ and C? pro7les is +onstant and e-ual to So 9 S+, #a.le ? s*ows a

    summary o$ water5sur$a+e pro7les, Jnline +al+ulators are pro&ided to round up t*e e3perien+e,

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     #HAN KJU

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