Module 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur

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1 Module The Sciece of Surface ad Groud Water Versio CE IIT, Kharagpur

2 Lesso 8 Flow Dyamics i Ope Chaels ad Rivers Versio CE IIT, Kharagpur

3 Istructioal Objectives O completio of this lesso, the studet shall be able to lear: 1. The physical dyamics of water movemet i ope chaels ad rivers. The mathematical descriptio of flow processes i the above cases 3. Differet types of free surface flows: uiform, o uiform, etc. 4. Differet chael shapes ad cross sectios ad their represetatios 5. Computatio steps for gradually varied water surface profiles.8.0 Itroductio It is commo for water resources egieers to desig a water system ivolvig flow of water from oe place to aother, usually passig a variety of structures o the way some of them meat for cotrollig the flow quatity. Rivers ad artificial chaels, like caals, covey water with a free surface, that is, the surface of water beig exposed to air as opposed to flow of water i pipes. It is easy to visualize that for ay such ope chael flow, as they are called; the presece or absece of a hydraulic structure cotrols the positio of the free surface of water. Kowig the mathematical descriptio of flowig water, it is possible to compute the water surface profile, which is importat for example i desigig the height of the chael walls of the water coveyig system. Aother example, the case of river flow obstructio by the presece dam may be metioed. The water level of the river icreases o costructio of the dam ad it is essetial to kow the maximum possible rise, perhaps durig the maximum flood, i order to kow the degree of submergece of the lad behid the dam. Barrages are low height structures, ad hece, the rise of water will ot be occurrig uiformly across the river, agai due to the differece of gate operatio. I this lesso, the behavior ad correspodig mathematical descriptio of flow i ope chaels are reviewed i order to utilize them i desigig water resources systems. Versio CE IIT, Kharagpur

4 .8.1 Flow i atural rivers Figure1 shows a river carryig a low discharge. Whe the water surface of the river just touches its baks, the discharge flowig through the river at this stage is called the bak full discharge. It is also sometimes called the domiat discharge. If the discharge i the river icreases, the water will overflow the baks ad would spill over to the adjacet lad, called the flood plais (Figure ). Versio CE IIT, Kharagpur

5 Though the amout of discharge flowig through the river is of iterest to the water resources egieer it caot be measured directly by ay istrumets. Rather, a idirect method is used which requires kowledge of the velocity distributio i a river or a ope chael. If we plot the velocity profile across a river, as show i Figure 1, it would actually vary i three dimesios. Figure 3 shows the variatio of velocity at the water surface. Versio CE IIT, Kharagpur

6 It may be observed that velocity is highest at the ceter of the river but is zero at the baks. If a velocity profile were plotted o aother horizotal plae at certai depth of the river, there too the velocity profile would be foud to be similar i shape, but smaller i magitude (Figure 4). Versio CE IIT, Kharagpur

7 Similarly the velocity profile of the river flowig i flood would be as show i Figure 5, showig that the velocities over the flood plais is smaller compared to the mai stream flow. If we ow take a look at the variatio of velocity i a vertical plae withi a river, ad we plot them alog differet vertical lies across the river, the we may fid the velocity profiles similar to those show i Figure 6. Versio CE IIT, Kharagpur

8 I order to measure the discharge beig coveyed i a river, the velocity profile or the average velocity at a umber of equally spaced sectios are measured, as i Figure 6. The total discharge is the approximately take equal to the sum of the discharges passig through each segmet. Aother way of depictig the velocity variatio across a river cross-sectio is to plot Isovels, which are actually the locus of poits havig equal velocity (Figure 7). Versio CE IIT, Kharagpur

9 It has bee observed through experimets that a plot of velocity i the vertical plae would show that the maximum velocity occurs slightly below the surface (Figure 8) for a typical river flow. Versio CE IIT, Kharagpur

10 It has further bee observed that a equivalet average velocity is almost equal to the actual velocity measured at 0.6 depth..8. Variatio of discharge with river stage The water level i a river is sometimes called the stage ad as this varies, there is a proportioal chage i the total discharge coveyed. For each poit of a river, the relatio betwee stage ad discharge is uique but a geeral form is foud to be as show i Figure 9. Versio CE IIT, Kharagpur

11 The geeral mathematical descriptio for the stage-discharge relatio is give as: Q ) m = k ( h h 0 (1) Where h is the gauge correspodig to a discharge Q ad h 0 is the correspodig to zero discharge k ad m are costats. If the variables (Q ad H) are plotted o a log-log graph, the it geerally plots i a straight lie as: log Q = m log ( h h ) + log k () Flow variatio alog river legth It may be iterpreted from Figures 4 or 6 that the velocity i a river cross sectio actually varies from bak to bak ad from riverbed to free water surface ad hece, ca be called a two dimesioal variatio i a vertical plae. However, for egieerig purposes it is, sufficiet, geerally, to use a equivalet velocity i the directio of river motio (perpedicular to river cross sectio) which may be Versio CE IIT, Kharagpur

12 obtaied by dividig the total discharge by the cross sectioal area. I a atural river, therefore, these flow velocities may vary from sectio to sectio (Figure 10). If we ow cosider a axis alog the legth of the river, the total eergy (H) is give as: H V = Z + h + (3) g We may plot the total eergy as show i Figure 11, where the variables are as follows: Z: Height of riverbed above a datum h: Depth of water V: Average velocity at a sectio Versio CE IIT, Kharagpur

13 V : Kietic eergy head g Sice the cross sectio, bed slope ad flow resistace vary alog a river legth, the depth ad velocity would vary correspodigly. However, if a short stretch of a river sectio is take, the the variatios i riverbed, water surface ad the total eergy may be cosidered as liear (Figure 1). Versio CE IIT, Kharagpur

14 I Figure 1, three slopes have bee marked, which are: S 0 : Riverbed slope S: Water surface slope S f : Eergy surface slope Sice the total eergy of flowig water reduces alog the river legth due to frictio the eergy surface slope is geerally termed as the frictio slope. The eergy loss i a river or a ope chael occurs mostly due to the resistace at the chael sides, as the turbulet characteristics of the flowig water implies a smaller loss iterally withi the water body itself. It has bee early 00 years whe scietists first attempted to mathematically express (or model ) the frictio slope i terms of kow variables like average velocity, cross sectio properties ad riverbed slope. Oe of the earliest models for frictio slope S f or, i effect, the chael resistace was derived from the cosideratios of uiform flow (Figure 13) where the flow variables ad cross sectio are supposed to remai costat over a short reach. Versio CE IIT, Kharagpur

15 If we take small volume of fluid from these two sectios we may make a free body diagram of the forces actig o it (Figure 14). Versio CE IIT, Kharagpur

16 The variables represeted i the figure are as follows W: Weight of water cotaied i the cotrol volume V: Iflow velocity, which is the same as the outflow velocities θ: Agle of slope river bed, which is also equal to that water surface ad frictio slopes τ 0 : Shear stress due to frictio actig o the cotrol volume of fluid from the river bed ad all alog the periphery, though i Figure 14 oly the resistace due to the riverbed is show. Equatig the forces ad otig that the iflowig ad out flowig mometa are equal as well as the pressure forces at either ed of the cotrol volume oe obtais: τ 0 P L = W siθ = ρ g A L siθ (4) Versio CE IIT, Kharagpur

17 Where the remaiig variables are: P: wetted perimeter A: Cross sectio of flow area L: Legth of cotrol volume Assumig θ to be very small ad early equal to bed slope, we have τ 0 = ρ g R S 0 (5) Assumig a state of rough turbulet flow, as is the case for atural rivers ad chaels, oe may write τ 0 α V or τ 0 = kv (6) Substitutig ito (4), This may be writte as ρ g V = R S0 k (7) V = C R S (8) This is kow as Chezy equatio after the Frech hydraulic egieer. Atoie Chezy who first proposed the formula aroud 1768 while desigig a caal for Paris water supply. The costat C i equatio (8) actually varies depedig o Reyolds umber ad boudary roughess. I 1869, Swiss egieers, Gaguillet ad Kutter proposed a elaborate formula for Chezy s C which they derived from actual discharge data from the river Mississippi ad a wide rage of atural ad artificial chaels i Europe. The formula, i metric uits, is give as S 0 C = 0.55 (9) S0 R Where is a coefficiet kow as Kutter s, ad is depedet solely o the boudary roughess. I 1889, Robert Maig s, a Irish egieer proposed aother formula for the evaluatio of the Chezy coefficiet, which was later simplified to: Versio CE IIT, Kharagpur

18 1 6 R C = (10) From Equatio (8), the Maig equatio may be writte as: V = R S 0 (11) Where the Maig is umerically equivalet to Kutter s. May research workers have experimetally foud the value of, ad for atural rivers, the followig books may be cosulted: 1. Chow, V T (1959) Ope Chael Hydraulics, McGraw Hill.. Chaudhry, M H (1994) Ope Chael Flow, Pretice Hall of Idia..8.4 Uiform flow i chaels of simple cross sectio For problems cocerig the steady uiform flow i rivers ad ope chaels, the Maig s equatio is commoly used i Idia. The depth of water correspodig to a discharge i a chael or river uder uiform flow coditios is called ormal depth. By combiig the cotiuity equatio with that of Maigs, oe obtais Q = A R S (1) Where the variables have bee defied i the earlier sectios. Oe may also write equatio (1) as follows Q = K S (13) A R Where K = 3, also called Coveyace, is ofte ecessary to fid out the ormal depth of flow correspodig to a discharge Q, flowig i a chael for which equatio (11) may be rearraged as 3 A R = Q 1 S (14) Versio CE IIT, Kharagpur

19 I equatio (14), the right had side terms are kow where as those i left had are ukow ad are fuctios of water depth. For a few commoly ecoutered sectios the parameters A ad R are give i the table below. Flow Area, A Wetted Perimeter, P Hydraulic Radius, R Free surface width, B Rectagle Trapezoid Circle b.h b +h bh b h (b+my).y 1 ( φ siφ) 8 b +h. 1+ m 1 φ D (b + my).y 1 siφ (1 )D b + h 1+ m 4 φ + b b +mh (si φ )D I the table, m stads for the side slope of a trapezoidal chael ad stads for the agle subteded at the cetre by the water surface chord lie. As see from the above table except for the very simple rectagular sectio it is ot possible directly to evaluate h, correspodig to Q as the left had side of equatio 13 is oliear i terms of h. Oe way of solvig is by Newto s method, where equatio (14) is writte as Q 3 f ( h) = AR = 0 (15) 1 S For usig Newto s method the derivative of the fuctio is required 3 ' d ( ) A Q f h = A = 0 (16) 1 dh 3 P S ' f ( h) = 5 da 5 5 dp P A P A (17) 3 dh 3 dh ' f ( h) = 5 3 dp 3 3 BR R (18) 3 dh da dp Where we have used = B.Similarly the expressio may be evaluated for dh dh ay sectio. Startig with a realistic value h i the iteratio may be carried out as give below: Versio CE IIT, Kharagpur

20 h i+1 = h i i f ( h ) - ' i f ( h ) (19) Where h i+1 is the value of h at ext iteratio, which is a improvemet of iitial guess h i. The iteratio may be cotiued till a desired accuracy is achieved..8.5 Uiform flow i chaels of compoud cross sectio A compoud sectio may be defied as a sectio i which various portios of the cross-sectio have differet flow properties, like surface roughess or chael depth. (Figure 15) I order to use the uiform flow formula i compoud chaels oe way may be to divide the flow sectio ito sub areas (Figure 16) ad treat the flow i each area separately. Versio CE IIT, Kharagpur

21 However, it has bee foud that this method may lead to errors by as much as ± 0% or eve more (Chadwick et al 004). The error is largely due to the eglectig of mass ad mometum iterchage betwee adjacet sub-areas. The curret solutio would however be more complex by usig a two or eve three-dimesioal model. I aother method, the eergy coefficiet (α) ad frictio slope S f are evaluated i terms of coveyace K of the sub areas. With these expressios, the flow i compoud sectio may be computed without kowig the idividual flows i each sub area. For a compoud chael divided ito N sectios. (For example N = 3 i Figure 15). The eergy coefficiet,α, is foud out as: N 3 V i i = 1 = N 3 V m i = 1 A i α (0) A Where V m is the mea flow velocity i the etire sectio ad is give as follows i V m V i i = (1) A A i Where V i = Q i /A i ad A i is the area of its i th sub-area. Equatio (18) ow ca be writte as Versio CE IIT, Kharagpur

22 3 Q i / A i ( Ai ) α = () ( Q ) 3 Now, the flow i sub-areas i may be writte as i i i 1 f i Q = K S (3) 1 f i S = Qi (1) K i Here, a assumptio has bee made that S f has the same value for all subareas, which is ot quite correct sice the velocities of each of these areas beig differet, would ot give equal velocity heads. Where as, the water surface is almost level over the etire cross sectio. Q 1 = K1 K Q = = Q 1 = Costat = f K S (4) It follows from equatio (3) that Q 1= K Q K Q = K Q = Q K K Q K (5) (6) Addig all the above equatio yields Q Q = Q i = i = K i = 1 1 K i (7) By substitutig this expressio for Qi = the equatio, oe obtais Q K i K ito equatio (7) ad simplifyig Versio CE IIT, Kharagpur

23 3 1 1 = = = i i i i K A α, = i i i A K 1 3 (8) Elimiatio of K Q from equatios (4) ad (6) ad squarig both sides give = i i f K Q S (9) f S = K i Q (30) Thus, expressios for α ad S f have bee evaluated for ay give stage without explicitly determiig the flow i each sub areas, Q i. I additio, equatio (30) may be used i the procedure for determiig varied flow profiles as discussed i Sectio.8.6. Versio CE IIT, Kharagpur

24 .8.6 No uiform i chaels There are quite a few examples of o-uiform flow i rivers or ope chaels that may be ecoutered by a water resources egieer. Some of these have bee illustrated i Figure 17. I this lesso we shall discuss the procedure to evaluate water surface profiles for steady, gradually varyig flow situatios. For steady, rapidly varyig ad usteady flow situatios, referece may be made to followig or similar texts o hydraulics of ope chael flow, like Raga Raju (003) or Subramaya (00). Versio CE IIT, Kharagpur

25 .8.7 No-uiform gradually varied flow calculatio A represetative o-uiform gradually varied flow is show i Figure 18. Over the icremetal distace Δx, the depth ad velocity are kow to chage slowly. The slope of the eergy grade lie is desigated as α i cotrast to uiform flow, the slopes of the eergy grade lie, water surface, ad chael bottom are o loger parallel. Sice the chages i the water depth h ad velocity V are gradual, the eergy lost over the icremetal Δx ca be represeted by maig equatio. This meas that equatio 11, which is valid for uiform flow ca also be used to evaluate S for a gradual varied flow situatio, ad that the roughess coefficiets discussed i Sectio.8.3 are applicable. Versio CE IIT, Kharagpur

26 Additioal assumptio icludes a regular cross sectio, small chael slope, hydrostatic pressure distributio ad oe-dimesioal flow. Applyig the equivalece of eergy betwee locatios 1 ad, ad assumig the loss term as h L give by S f Δx oe obtais Z V1 V + h1 + α = Z + h + + S f Δx (31) g g 1 α I the above equatio, Δx is the distace betwee two cosecutive sectios x 1 ad x such that Δx=x - x 1. V The eergy coefficiet α has bee used alog with the term, as it may be g much differet from 1.0 for atural sectios. The term i equatio (31) may be evaluated by the expressio for uiform flow, equatio (11), where S0 may be replaced by S. Sice equatio (31) relates the eergy betwee the sectios, S f may be take either of the followig: 1 Arithmetic mea: S f = ( S f 1 + S f ) (3) Geometric mea: S f = S f 1 S f (33) Harmoic mea: S f f 1 f S f S f 1 S f = (34) S + S f Where ad are the frictio slopes evaluated at sectio 1 ad by usig S f 1 S f the Maigs formula equatio (1). Equatio (9) may be used by startig from oe ed of the chael where the flow depth ad velocity are kow ad workig backward or forward i steps. Here, two, methods are used of which we shall discuss oe, called the stadard step method. Avery popular computer program called HEC- developed by hydrologic egieerig ceter of the US Army Corps of Egieers is based o this method. It may be freely dowloaded from the website: legacysoftware/hec/hec-dowload.htm. I the stadard step method, for ay give discharge the depth of flow would be kow at the cotrol sectio. It is the required to calculate the depth of flow at the sectio immediately ext to the cotrol sectio. Two examples are illustrated i Figure 19. Versio CE IIT, Kharagpur

27 The distace betwee the two successive sectios (i ad i+1) is take as costat, say Δx. It may be observed from the Figure 19a sice the water is flowig above the dam the water depth above the dam crest ca be foud out for the give discharge. Hece the water level at the cotrol sectio just upstream of Versio CE IIT, Kharagpur

28 the dam is kow. Similarly, i Figure 19b, sice the water is flowig dow from the reservoir ito the steep chael critical depth correspodig to the give discharge would exist at the cotrol sectio. Here two, the water level at the cotrol sectio is the kow. Startig at the cotrol sectio (i =1), the total eergy of water is foud out to be H 1 = Z V1 + h1 + (35) g 1 α Next, cosider the first reach, that is, betwee sectios i =1 ad i =. A depth of flow is assumed at sectio ad the eergy there, that is, H = Z V + h + (36) g α is evaluated. Now, oe of the equatios for fidig slope) i the reach is foud out by, say equatio 31. S f (the average frictio As may be observed from Figure 18 the umerical value of H foud from equatio (33) should be equal to that of h 1 foud from equatio (33) +. If the depth at the sectio has bee correctly assumed if the two do t match, a ew depth h is assumed ad the calculatios are repeated till the two values match. Oce a correct depth is foud at sectio, a similar procedure is used to fid the depth at sectio 3, ad so o. These are the two other methods to fid out water surface profiles of gradually varied flow situatios, amely; method of direct itegratio ad method of graphical itegratio. Iterested reader may refer to stadard textbooks o Hydraulics of ope chael flow, like the followig for details about these methods. 1. Raga Raju (003). Subramaya (00) S f.8.8 Gradually varied flow profiles I may flow problems it is eough to make a qualitative sketch of water surface profile for a give flow that is takig place betwee two locatios. It is ot ecessary therefore to fid out the exact level of water at differet poits but the geeral shape of the free surface has to be draw as accurately as possible. A Versio CE IIT, Kharagpur

29 aalysis of water surface profile may be doe by studyig the goverig equatio, which ca be derived from the sketch i Figure 0. The total eergy H at a chael sectio is give as Where H = Z V + h +α (37) g H: Elevatio of eergy lie above the datum Z: Elevatio of chael bottom above datum h: Flow depth V: Mea flow velocity α: Velocity head coefficiet Cosiderig x as the space coordiate, take positive i the directio of flow oe obtais by differetiatig both sides of the equatio (36) with respect to x ad expressig V i terms of discharge Q. Versio CE IIT, Kharagpur

30 dh dx Agai, we kow by defiitio: dz dh Q d α (38) dx dx g dx A = dh = - Sf dx (39) dz Ad = - So dx (40) I which S f : Slope of the eergy grade lie S o : Slope of the chael bottom. The egative sig of S f ad S o idicates that both H ad Z decrease as x icreases. I equatio (37) a expressio for the derivative of A - may be foud out as follows: Sice da = B dh d 1 d 1 da = dx A da A dx (41) d 1 da dh = da A dh dx (4) B dh = 3 A dx (43) By substitutig equatios (39), (40) ad (43) ito equatio (38), ad rearragig the resultig equatio oe obtais dh dx S S = (44) 0 f 3 1 ( α BQ ) / ga If the chael is ot prismatic, the the cross sectioal area A chages with distace, ad may be expressed as: da = dx A x + A y dh dx (45) The above chage would modify equatios (40) ad (43) accordigly. Versio CE IIT, Kharagpur

31 We may express equatio (43), which describes the variatio of h with x, i terms of the Froude Number (Fr) if we ote the followig: α B Q 3 g A ( Q / A) = = Fr ( g A) / ( α B) (46) Hece, equatio (39) may be writte as dh S0 S f = (47) dx 1 Fr Equatio (47) ca give a geeral idea about the ature of the curve if oe kows the relative icliatios of the chael bed slope ad frictio slope (S f ) ad the Froude Number (Fr). This may be doe by observig the water flow depth (h) with respect to ormal depth (h ) ad critical depth (h c ) for a give discharge, the followig figures show the relative chages of h ad h c as chael bed slope is icreased gradually from horizotal. It may be observed that the for a give discharge h c does ot chage but h goes o decreasig startig from a ifiite value for a flat slope. Versio CE IIT, Kharagpur

32 Versio CE IIT, Kharagpur

33 Versio CE IIT, Kharagpur

34 I water resources projects, oe geerally ecouters slopes of chaels that are either of the followig: Mild, where h > h c (Figure ) Steep, where h < h c (Figure 4) Critical, where h = h c (Figure 3) Flat, where h = (Figure 1) Adverse, where the slope is reversed (Figure 5) For each of these slopes, the actual water surface would vary depedig upo a cotrol that exist either at the upstream or dowstream ed of the chael. Some examples of cotrols are give below Weir or spillway (Figures 6 ad 7) Gate (Figure 8) Free overfall (Figure 9) Versio CE IIT, Kharagpur

35 Versio CE IIT, Kharagpur

36 Apart from the above a ormal depth may be assumed to exist withi a very log chael, for which the coditios at the far ed may be eglected (Figure 30). The situatio show i Figure 30 is used ofte while aalyzig flow i, say, at the tail ed of log irrigatio chaels or i a log river. Examples illustratig the use Versio CE IIT, Kharagpur

37 of equatio (4) ad a kow cotrol sectio i determiig flow profiles where for a mildly slope chael. Similar profiles may be qualitatively sketched for other chaels too..8.9 Dowstream cotrol raisig the water level above ormal depth This situatio is commo for spillways of large dams. The flow profile i a mildly sloped chaels where h > h > h c as show i Figure 31 is kow as the M 1 curve. Now, for uiform flow, S f =S= S o whe h = h. Hece it is clear from Maigs formula (equatio 11), that for a give discharge, Q, S f < S o if h > h Thus, i equatio (47) i.e., dh S0 S f = (47) dx 1 Fr Versio CE IIT, Kharagpur

38 The umerator is positive Fr<1 sice h>h c. Therefore, the deomiator of equatio (47) is positive as well. Hece, it follows from this eq that dh S0 S f + = = = + dx 1 Fr + This meas that h icreases with distace x. Comparig with Figure 9 it may be iferred that quite some distace upstream of the spillway the flow depth early equals ormal depth. Ad, sice dh/dx for this profile is positive which meas that the water depth goes o icreasig towards the spillway, the flow depth becomes early horizotal. However very close to spillway the flow profile agai chages which is due to the fact that the flow here is ot really oe-dimesioal (Figure 3). Versio CE IIT, Kharagpur

39 .8.10 Dowstream cotrolled raisig water level above critical depth but below ormal depth The flow profile i a mildly slopig chael where h >h>h c, has bee show i Figure 33 is kow as the M curve. I this case S f >S o sice h<h (from Maigs formula). Thus the umerator i equatio (46) is egative. However, the deomiator is positive, sice Fr<1 because h>h c hece it follows from equatio 46 that dh S0 S f = = = dx 1 Fr Thus h decreases as x icreases for upstream of this spillway cotrol sectio the flow depth would be asymptotic to ormal depth h. Versio CE IIT, Kharagpur

40 .8.11 Upstream cotrol causig water depth to be less tha both ormal ad critical depths This situatio is show i Figure 34 for flow takig place below a sluice gate. The reader is advised to check the tred of water surface profile usig equatio (47) i this case..8.1 Importat terms, defiitios ad procedures This lesso has used certai terms, which are discussed to some detail here. Newto s Method This method is useful i fidig a simple root of the fuctio f(x) = 0, whe the derivative of f(x) is easily obtaiable. The iteratio formula used i the method ca be derived by the Taylor s series expasio of f(x) about x=x 0, the approximate value of the desired root. We have Versio CE IIT, Kharagpur

41 f ' h '' x0 + h) = f ( x0 ) + h f ( x0 ) + f ( x ) +...! ( 0 Where h is the small correctio to the root. Now if h is relatively small, we may eglect terms cotaiig ad higher powers of h. The, we get ' f ( x0) + h f ( x0) = 0 f ( x0 ) This gives h = ' f ( x ) Thus, we ca take the improved value of the root as f ( x0) x 1 = x 0 ' f ( x ) The Newto-Raphso iteratio ca thus be writte as f ( x ) x +1 = x, = 0,1,,... ' f ( x ) The sequece { Froude Number x }, if it coverges, gives the root. This measures the ratio of iertia to gravity forces. I problems where there is a iterface betwee two immiscible fluids the gravity forces are of importace. Froude umber is defied by the relatio V Fr = g D Normal depth For give values of chael roughess, discharge Q, ad the chael slope S, there is oly oe depth possible at which uiform flow occurs. It is kow as ormal depth. 0 0 Critical depth The depth of flow at which the specific eergy value is called critical depth. V E = y + attais a miimum g Versio CE IIT, Kharagpur

42 .8.13 Refereces Sturm, T W (001) Ope Chael Hydraulics, McGraw Hill Julie, P (00) River Mechaics, Cambridge Uiversity Press Chadwick, A, Morfett, J, ad Borthwick, M (004) Hydraulics i Civil ad Evirometal Egieerig, Spo Press Subramaya, K (00) Flow i Ope Chaels, Secod editio, Tata McGraw Hill Raga Raju, K G (003) Flow through Ope Chaels, Secod editio, Tata McGraw Hill Versio CE IIT, Kharagpur