gradually varied flow computations

8/16/2019 Gradually Varied Flow Computations

1/17

Gradually

Varied FlowComputationsFLUID MECHANICS ||

CIVIL ENGINEEING!

" # H SEMES#E


2/17

INTRODUCTION Computations o$ %radually &aried 'ow (GVF) are part o$t*e routine pra+ti+e o$ *ydrauli+ en%ineerin%, #*e GVFe-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 int*e analysis o$ water sur$a+e pro7les,


3/17

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 9top widt*! A 9 'ow area! and g 9 %ra&itationala++eleration, #*is e-uation is &alid $or small .ed slopes(So : ;,/)! w*i+* is usually t*e +ase,


4/17

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 wettedperimeter,

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

(?)

g A 3


5/17

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


6/17

GRADUALLY VARIED

FLOW EQUATIONS w*i+* is stri+tly &alid $or t*e $ollowin% +ondition8P 6T 9 Pc 6T c , #*is latter +ondition is %enerally satis7ed ina *ydrauli+ally wide +*annel! $or w*i+* T is asymptoti+allye-ual to P,

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

So

- Sy

F 2 = _________

()

Sc - Sy

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


7/17

GRADUALLY VARIED

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

w*i+* ee+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)!


8/17

WATER-SURFACE

ROFILES E-uation /; is used to de&elop a +lassi7+ation o$ water5sur$a+e pro7les .ased solely on t*e t*ree dimensionlessparameters8 Sy 6S+ ! So 6S+ ! and F, For t*e sae o$+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 'owis de7ned as t*at $or w*i+* t*e 'ow dept* is %reater t*ant*e normal dept* F > : So 6S+ , Supernormal 'ow isde7ned as t*at $or w*i+* t*e 'ow dept* is smaller t*ant*e normal dept* F > B So 6S+ (USDA SCS /0/), #a.le

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


9/17

WATER-SURFACE

ROFILES


10/17

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,


11/17


12/17


13/17


14/17

SUMMAK #*e %radually &aried 'ow e-uation is e3pressed in termso$ 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 &aluesoutside 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 7nitepositi&e &alue) to ; (asymptoti+ to normal dept*),Liewise! 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 arepro&ided to round up t*e e3perien+e,


15/17


16/17


17/17

#HAN KJU

gradually varied flow computations

Documents