Gradually varied flow is a steady non-uniform flow in which the deepness fluctuation in the way of gesture is gradual plenty that the cross force per unit area distribution can be considered hydrostatic. This allows the flow to be treated as one dimensional with no cross force per unit area gradients other than those created by gravitation. The methods developed should non be applied to parts o f extremely curvilineal flow, such as can be found in the locality of an ogees wasteweir crest for illustration, because the centripetal acceleration in curvilineal degree Fahrenheit low alters the cross force per unit area distribution so that it no longer is hydrostatic, and the force per unit area caput n no longer can be represented by the deepness of flow. ( 4 )

The flow in an open-channel is termed as bit by bit varied flow ( GVF ) when the deepness of flow varies bit by bit with longitudinal distance. Such flows are encountered both on upstream and downstream sides of control subdivisions. Analysis and calculation of bit by bit varied flow profiles in open-channels are of import from the point of position of safe and optimum design and operation of any hydraulic construction. ( 6 )

Even with the premise of bit by bit varied flow, an exact solution for the deepness profile exists merely in the instance of a broad, rectangular channel. The solution of the equation of bit by bit varied flow in this instance is called the Bresse map, which provides utile estimates of H2O surface profile lengths subject to the premises of a really broad channel and a changeless value of Chezy ‘s C. The solutions to all other jobs in the yesteryear, were obtained diagrammatically or from tabular matters of the varied flow map based on hydraulic advocates as developed by Bakhmeteff ( 1932 ) and Chow ( 1959 ) . ( 4 )

In analysing the steady GVFs in unfastened channels, flow opposition, and alterations in underside incline, channel form, discharge and surface conditions are taken into history. It is usual to denote d the existent deepness ( i.e. the non-uniform flow deepness ) , do the normal deepness ( i.e. unvarying flow deepness ) and dc the critical deepness. ( 6 )

In many practical jobs of hydraulic technology refering unfastened channels, a right rating of the H2O lift in assorted subdivisions is required. The aim, achieved by chalk outing the gradually-varied-flow profile, requires integrating of the regulating equations. The procedure can be carried out either numerically or analytically. ( 5 )

## Background

Several integrating methods exist: analytical solutions are rare but legion numerical integrating methods exist. We propose to develop the standard measure method ( distance computed from deepness ) . This method is simple, highly dependable and really stable. It is strongly recommended to practising applied scientists who are non needfully hydraulic experts. ( 6 )

Bresse ~1868! obtained the classical direct solution for a broad rectangular channel. However, like most hydraulic applied scientists of his clip, he assumed that the roughness coefficient of the CheA?zyformula, used for the computation of the speed, is changeless. In fact, the first experimental observations demoing that the raggedness coefficient depends on the size and deepness of the channels, started by Darcy and completed by Bazin in 1865, day of the month back to the same period. ( 5 )

Subsequently, Masoni ~1900! proposed a direct integrating with the same estimate, which is valid for a common rectangular subdivision, whereas Bakhmeteff ~1932! proposed a direct integrating that is applicable to all forms of channels that in pattern. However, Bakhmeteff carried out merely an approximative integrating, whose values are reported in tabular arraies. Furthermore, the method requires a division of the channel length into short ranges. Chow ~1955! developed an extension of the method which allows one to avoid that computational trouble. ( 5 )

Taking the same map introduced by Bakhmeteff, Chow assumed the computational standards and provided tabular arraies with a wider scope. A direct integrating for rectangular and triangular channels has besides been proposed by Kumar ~1978! . CheA?zy ‘s expression was assumed in that work, but the roughness coefficient was still considered changeless. ( 5 )

In order to avoid such estimates and to supply a more accurate representation of the flow profiles, Manning ‘s expression is used for rating of the gesture in non-uniform flow besides. Then a relation obtained by direct integrating of the gradually-varied-flow equation and referred to broad rectangular channels is proposed. ( 5 )

( LAST ) Many computing machine plans are available for calculation of backwater curves. The most general and widely used plans are, HEC-2, developed by the U.S. Army Corps of Engineers ( 1982 ) and Bridge Waterways Analysis Model ( WSPRO ) developed for the Federal Highway Administration. These plans can be used to calculate H2O surface profiles for both natural and unreal channels.

## Categorization OF CHANNELS FOR GRADUALLY-VARIED FLOW

Open channels are classified as being mild, steep, critical, horizontal, and inauspicious in gradually-varied flow surveies. If for a given discharge the normal deepness of a channel is greater than the critical deepness, the channel is said to be mild.

If the normal deepness is less than the critical deepness, the channel is called steep. For a

critical channel, the normal deepness and the critical deepness are equal. If the underside

incline of a channel is zero, the channel is called horizontal. A channel is said to

hold an inauspicious incline if the channel underside rises in the flow way. ( 5 )

## In drumhead:

Mild channels yn & gt ; yc

Steep channels yn & lt ; yc

Critical channels yn = yc

Horizontal channels Sa‚’=0

Adverse channels Sa‚’ & lt ; 0

where yn= normal deepness and yc= critical deepness. ( 5 )

## Basic Assumptions in GVF Analysis ( 5 )

1. The bit by bit varied flow to be discussed here considers merely steady flows. This

implies that ( I ) flow features do non alter with clip, and ( two ) force per unit area distribution is hydrostatic over the channel subdivision.

2. The head loss in a range may be computed utilizing an equation applicable to uniform flow holding the same speed and hydraulic average radius of the subdivision. This implies that the incline of energy classs line may be evaluated utilizing a unvarying flow expression such as Manning equation and Chezy equation, with the corresponding raggedness coefficient applicable chiefly for unvarying flow.

3. Channel bottom incline is little. This implies that the deepness of flow measured vertically is same as deepness of flow measured perpendicular to impart underside.

4. There is no air entrainment. Advanced text books may be referred to analyze the effects of air entrainment.

5. The speed distribution in the channel subdivision is invariant. This implies that the

energy rectification factor, I± , is a changeless and does non vary with distance.

6. The opposition coefficient is non a map of flow features or deepness of flow. It does non vary with distance.

7. Channel is prismatic.

## GRADUALLY-VARIED FLOW COMPUTATIONS ( 5 )

To obtain an look for gradually-varied flow, we refer to Total energy caput Equation which define the entire energy caput, H, as

H = zb + y + V2/2g ( 1.1 )

where zb=elevation of the channel underside, y=flow deepness, V= mean cross-sectional speed, and g=gravitational acceleration. Now, mentioning to the definition of specific energy as

E =y + V2 / 2g ( 1.2 )

Equation 1.1 can be expressed as

H = zb + E ( 1.3 )

Let us distinguish both sides of Equation 4.3 with regard to x to obtain

dH /dx = dzb / dx + dE / dx ( 1.4 )

where ten is the supplanting in the flow way. By definition, Sf=-dH/dx,

and Sa‚’=-dzb/dx. By replacing these into Equation 1.4 and rearranging,

we obtain one signifier of the gradually-varied flow equation as

Delaware /dx = S0 – Sf ( 1.5 )

We can obtain another signifier of the gradually-varied flow equation by spread outing

the left-hand side of Equation 1.5 to

Delaware /dx= dy/dx +d ( VA?/2g ) /dx=dy/dx + V/g dv/dx=dy/dx + V/g dv/dy dy/dx

= dy/dx+V/g dy/dx vitamin D ( Q/a ) /dy = dy/dx+ V/g dy/dx Qd ( 1/A ) /dy

where Q=constant discharge and A=area. Further mathematical use

by utilizing the definitions T=top width=dA/dy, D=hydraulic depth=A/T, and

Fr=Froude number=V/ ( soman ) 0.5 will take to

dE/dx=dy/dx+V/g dy/dx Qd ( 1/A ) /dy = dy/dx – V/g dy/dx Q ( dA/dy ) /A2

= dy/dx – V/g dy/dx Q ( T ) /AA? = dy/dx ( 1 – VA?/gD ) =dy/dx ( 1 – FA?r )

Therefore,

dE/dx = dy/dx ( 1 – FA?r ) ( ( 1.6 )

By replacing Equation 1.6 into 1.5 and rearranging, we obtain

dy/dx = ( Sa‚’ -Sf ) / ( 1-FA?r ) . ( 1.7 )

We can work out either Equation 1.5 or Equation 1.7 in order to find the gradually-varied flow deepnesss at different subdivisions along a channel. However, we

happen Equation 1.5 more convenient for this intent. As we pointed out before, this is a differential equation ; a boundary status is required for solution. It is really of import to retrieve that subcritical flow is capable to downstream control.

Therefore, if flow in the channel is subcritical, so a downstream boundary status must be used to work out Equation 1.5 given Q. Conversely, supercritical

flow is capable to upstream control, and an upstream boundary status is needed to work out Equation 1.5 for supercritical flow. By boundary status, we by and large mean a known flow deepness associated with a known discharge. ( 4 )

Analytic solutions to Equation 1.5 are non available for most open-channel flow

state of affairss typically encountered. In pattern, we apply a finite difference attack to cipher the gradually-varied flow profiles. In this attack, the channel is divided into short ranges and calculations are carried out from one terminal of the range to the other. ( 4 )

See the channel range shown in Figure 1.11 holding a length of a?†X. Sections U and D denote the flow subdivisions at the upstream and downstream terminals of the range, severally. Using the inferiors U and D to denote the upstream and downstream subdivisions, we can compose Equation 1.5 for this range in finite difference signifier as

ED – EU/ a?†X = Sa‚’ – Sfm ( 1.9 )

FIGURE 4.11Definition study for bit by bit flow froumulation.jpg

Figure 4.5 Definition study for gradually-varied flow preparation

where Sfm=average clash incline in the range, approximated as

Sfm = A? ( SfU + SfD ) ( 1.10 )

By rearranging the Manning expression, the clash slopes at subdivisions U and D are

obtained as

SfU = nA?/kA?n VA?U/Ru^4/3 ( 1.11 )

and

SfD = nA?/kA?n VA?d/Rd^4/3 ( 1.12 )

In the past direct and graphical solution methods have been used to work out these, nevertheless these method have been superseded by numerical methods which are now be the lone method used.

## NUMERICAL METHODS ( 7 )

Numeric integrating is chiefly used in non-prismatic channels, such as natural watercourses. In prismatic channels, such as unreal 1s, the construction of the basic equation is simplified, and so direct integratings can besides be applicable. Direct methods have the advantage of supplying independent solutions of the old computational stairss. The entire lengths of the profile can be evaluated with a individual calculation. On the other manus, these

methods have the disadvantage of non measuring the deepness of the flow at a specific longitudinal distance. ( 5 )

There are two basic numerical methods that can be used:

Direct measure – distance from deepness

Standard measure method – deepness from distance

## Direct STEP METHOD ( distance from deepness ) ( 5,7 ) )

This method will cipher ( by incorporating the bit by bit varied flow equation ) a distance for a given alteration in surface tallness.

In the direct measure method, we write Equation 1.9 as

a?†X= Ed-Eu/Sa‚’-Sfm = ( ( yd+VA?d/2g ) – ( yu+ VA?u/2g ) ) /Sa‚’-Sfm ( 1.13 )

In a typical subcritical flow job, the status at the downstream subdivision D

is known. In other words, yD, VD, and SfD are given. We pick an appropriate value ( depending on the type of flow profile we have predicted ) for yU, and cipher the corresponding VU, SfU, and Sfm. Then we calculate _X from Equation 1.13

Conversely, where supercritical flow is involved, conditions at subdivision U are

known. In this instance, we pick a value for yD to cipher the range length.

This method is called the direct measure method, since the range length is obtained straight from Equation 1.13 without any test and mistake. These computations are repeated for the subsequent ranges to find the H2O surface profile.

For subcritical flow computations, we start from the downstream terminal of a channel and continue in the upstream way. In other words, the first range considered is at the downstream terminal of the channel, and the downstream subdivision of this range coincides with the downstream appendage of the channel.

At the downstream appendage, yD is known from the boundary status.

Using the known discharge and the cross-sectional belongingss, we foremost cipher

VD and SfD. Next we pick a value for yU and cipher the corresponding VU and SfU. Then, from Equation 1.13, we determine the channel range _X.

This procedure is repeated for farther upstream ranges until the full length of the channel is covered. Note that yU of any range becomes yD for the range considered following. Besides, we must be careful in picking the values for yU.

These values depend on the type of the profile that will happen in the channel.

For illustration, if an M2 profile is being calculated, yU must fulfill the inequalities yU & gt ; yD and yn & gt ; yU & gt ; yc. Likewise, for an S1 profile, yU & lt ; yD and yU & gt ; yc & gt ; yn.

For supercritical profiles, we start at the upstream terminal and proceed in the downstream way. For the first range, yU is known from the upstream boundary condition.We choose a value for yD and cipher the range length, a?†X,

utilizing Equation 1.13. This procedure is repeated for farther downstream reaches until the length of the channel is covered. The yD of any range becomes yU of the subsequent range. The values of yD must be chosen carefully in the procedure.

For case, for M3 profiles, yD & gt ; yU and yD & lt ; yc & lt ; yn. Likewise, for S2 profiles,

yD & lt ; yU and yn & lt ; yD & lt ; yc.

In certain state of affairss, the flow deepnesss at both terminals of a surface profile will be known and we can execute the computations to find the entire length of the profile. In such a instance we can get down from either the upstream terminal or the downstream terminal, irrespective of whether the flow is subcritical or supercritical.

However, a downstream boundary status is ever known for subcritical flow, and an upstream boundary status is ever known for supercritical flow.

Therefore, it is sensible to follow the general regulation that subcritical flow computations start at the downstream terminal, and supercritical flow computations start

at the upstream terminal.

## Standard STEP METHOD ( deepness from distance ) ( 5,7 )

This method will cipher ( by incorporating the bit by bit varied flow equation ) a deepness at a given distance up or down watercourse.

In the standard measure method, the flow deepnesss are calculated at specified locations. As in the direct measure method, we know the flow deepness and speed at one terminal of a channel range. We so choose the range length, a?†X, and cipher the deepness at the other terminal of the range.

For subcritical flow, the conditions at the downstream subdivision will be known. For this instance, to ease the computations, we will rearrange Equation 1.13 as

yU + VA?U/2g – A? ( a?†X ) Sfu = yD + VA?D/2g + A? ( a?†X ) SfD – ( a?†X ) Sa‚’ ( 1:14 )

For a changeless discharge, we can show VU and ( Sf ) U in footings of yU. Therefore,

the lone unknown in Equation 1.14 is yU. However, the look is inexplicit in yU, and we can work out it by usage of an iterative technique. We try different values

for yU until Equation 1.14 is satisfied. Because of the iterative nature of the process the standard measure method is non suited for computation by manus, and we usually employ a computing machine plan. However, in the absence of such a plan we can better the guessed values of yU in each loop utilizing

( yU ) k+1 = ( yU ) K -a?†yk ( 1:15 )

The standard measure method can be used for non-prismatic channels as good, with

some alterations.

## Categorization OF GRADUALLY-VARIED FLOW PROFILES

A gradually-varied flow profile or gradually-varied H2O surface profile is a line

bespeaking the place of the H2O surface. It is a secret plan of the flow deepness as a map of distance along the flow way. A sound apprehension of possible profiles under different flow state of affairss is indispensable before we can obtain numerical solutions to gradually-varied flow jobs. ( 5 )

See a mild channel as shown in Figure 1.1. By definition, yn & gt ; yc. The

channel underside, the critical deepness line, and the normal deepness line divide the

channel into three zones in the perpendicular dimension, viz. M1, M2, and M3

( M stands for mild ) . The solid lines in the figure represent the forms of the possible flow profiles in these three zones. Obviously, the normal deepness line itself

would stand for the H2O surface if the flow in the channel were normal. In zone

M1, the H2O surface is above the normal deepness line. Therefore, in this zone Y & gt ; yn and accordingly Sf & lt ; Sa‚’ . Besides, y & gt ; yc and therefore Fr & lt ; 1.0 in zone M1. ( 5 )

Therefore, both the numerator and the denominator of Equation 1.7 are positive measures, and ( dy/dx ) & gt ; 0. In other words, the flow deepness must increase in the flow way in zone M1. We can analyze the zones M2 and M3 in a similar mode, and conclude that ( dy/dx ) & lt ; 0 in zone M2 and ( dy/dx ) & gt ; 0 in zone M3.

The behaviour of the H2O surface profile near the zone boundaries can besides be

examined. From Equation 1.7, as ya†’ a?z we can see that Fra†’0 and Sfa†’0. Thus

( dy/dx ) a†’Sa‚’ , intending the H2O surface will near a horizontal line asymptotically as ya†’a?z . Likewise, as ya†’ yn, by definition Sfa†’Sa‚’ and therefore ( dy/dx ) a†’0. Therefore, the surface profile approaches the normal deepness line asymptotically. Near the critical deepness line, ya†’ yc and Fra†’1.0. Thus ( dy/dx ) a†’1, and the H2O surface will near the critical deepness line at an angle stopping point to a right-angle. Near the underside of the channel, as ya†’0, both Sfa†’a?z , and Fra†’a?z . Therefore, the H2O surface will near the channel ( 5 )

Flow profile in mild channel.jpg

Figure 1.1 Flow profiles in mild channels ( 5 )

underside at a finite positive angle. The magnitude of this angle depends on the

clash expression used and the specific channel subdivision.

Based on this qualitative scrutiny of Equation 1.7 near the zone boundaries,

we conclude that in zone M1 the H2O surface profile is asymptotical to the normal depth line as ya†’yn and is asymptotical to a horizontal line as ya†’a?z .

The M2 profile is asymptotical to the normal deepness line, and it makes an angle

near to a right-angle with the critical deepness line. The M3 profile makes a positive

angle with the channel underside and an angle stopping point to a right-angle with the critical

depth line. The H2O surface profiles sketched in Figure 1.1 reflect these

considerations. ( 5 )

We should observe that a flow profile does non hold to widen from one zone boundary to another. For illustration, an M2 profile does non hold to get down at the

normal deepness line and terminal at the critical deepness line. It is possible that an M2 profile begins at a point below the normal deepness line and terminals at a point above the critical deepness line.

For a steep channel, yn & gt ; yc by definition. The channel underside, the normal deepness line, and the critical deepness line divide the channel into three zones in the perpendicular dimension, viz. S1, S2, and S3 ( S stands for steep ) as shown in Figure 1.2. As earlier, the solid lines in the figure represent the forms of the possible flow profiles in these three zones. If the flow were normal in this channel, the normal deepness line itself would stand for the H2O surface. In zone S1 the H2O surface is above the critical deepness line, hence in this zone Y & gt ; yc and therefore Fr & lt ; 1.0. Besides, y & gt ; yc & gt ; yn, and accordingly Sf & lt ; Sa‚’ . Therefore, both the numerator and the denominator of Equation 1.7 are positive measures, and in zone S1 ( dy/dx ) & gt ; 0. In other words, the flow deepness must increase in the flow way. We can analyze the zones S2 and S3 in a similar mode, and conclude that ( dy/dx ) 50 in zone S2 and ( dy/dx ) & gt ; 0 in zone S3. ( 5 )

The behaviour of the surface profile near the zone boundaries examined for mild

channels is valid for steep channels as good, since Equation 1.7 is applicable to

both steep and mild channels. Consequently, the S1 profile makes an angle stopping point to

the right-angle with the critical deepness line, and it approaches to a horizontal line

flow profiles in steep channels.jpg

Figure 1.2 Flow profiles in steep channels ( 5 )

asymptotically as ya†’a?z . The S2 profile makes an angle stopping point to the right-angle with the critical deepness line, and it approaches the normal deepness line asymptotically. The S3 profile will do a positive angle with the channel underside, and it will near the normal deepness line asymptotically. ( 5 )

The possible profile types that can happen in horizontal, inauspicious, and critical channels are shown in Figure 1.3 These profiles are sketched by analyzing the mark of ( dy/dx ) with the aid of Equation 1.7, and sing the behaviour of the profile near the zone boundaries. Note that for horizontal and inauspicious channels normal flow is non possible, therefore yn is non defined, and zones H1 and A1 do non be. Likewise, for critical channels yn=yc, and hence zone C2 does non be. It is besides deserving observing that the flow is subcritical in zones M1, M2, S1, H2, A2, and C1, and it is supercritical in zones M3, S2, S3, H3, A3, and C3. ( 5 )

## Equations for GVF: backwater computation ( 6 )

The term ‘backwater computations ‘ refers more by and large to the computations of the longitudinal free-surface profile for both sub- and supercritical flows.

The backwater computations are developed assuming:

[ H1 ] a non-uniform flow,

[ H2 ] a steady flow,

[ H3 ] that the flow is bit by bit varied,

[ H4 ] that, at a given subdivision, the flow opposition is the same as for an unvarying flow for the

same deepness and discharge, irrespective of tendencies of the deepness.

The GVF computations do non use to uniform equilibrium flows, nor to unsteady flows, nor to RVFs. The last premise [ H4 ] implies that the Darcy, Chezy or Gauckler-Manning equation may be used to gauge the flow opposition, although these equations were originally developed for unvarying equilibrium flows merely.

Significance OF FROUDE NUMBER IN GRADUALLY-VARIED FLOW CALCULATIONSflow profiles in horizontal adverse, and critical channels.jpg

The gradually-varied flow equation ( Equation 1.6 or 1.7 ) is a differential equation, and we need a boundary status to work out it. Mathematically, the flow deepness at any given flow subdivision can be used as a boundary status.

However, for right representation of open-channel flow the boundary status will be prescribed at either the upstream or the downstream terminal of the channel, depending on whether the flow in the channel is supercritical or subcritical. The undermentioned observation is presented to explicate the ground for this.

A pebble thrown into a big still organic structure of H2O will make a perturbation, which will propagate outward in the signifier of homocentric circles as shown in Figure 1.4a.

Figure 1.3 flow profiles in horizontal inauspicious and critical channels ( 5 )

The velocity with which the perturbation propagates is called quickness ( or quickness of

gravitation moving ridges in shallow H2O ) , and it is evaluated as

degree Celsiuss = a?sgD

where degree Celsius =c elerity, g = gravitative acceleration, and D = hydraulic deepness.

If the pebble is thrown into a organic structure of H2O traveling with a speed V the moving ridge

extension will no longer be in the signifier of homocentric circles. Mentioning to the definition of Froude figure, we can compose that Fr = ( V ) / a?sgD = V /c

## Significance of Froude figure in gradually-varied flow computations

E.O.F.N.jpg

Figure 1.4 Consequence of Froude figure on extension of a perturbation in unfastened channel ( 5 )

For subcritical flow, Fr & lt ; 1 and V & lt ; c. On the other manus, for supercritical flow, Fr & gt ; 1 and V & gt ; c. Obviously, for critical flow V=c. Therefore, if the flow is subcritical, the perturbations will propagate upstream at a velocity ( c-V ) and downstream at a velocity ( c+V ) , as shown in Figure 1.4b. If the flow is critical, so the upstream border of the moving ridge will be stationary while the downstream extension will be at a velocity 2c, as shown in Figure 1.4c. If the flow is supercritical, so the extension will be in the downstream way merely every bit shown in Figure 1.4d, with the dorsum and forepart borders traveling with velocities ( V-c ) and ( V+c ) , severally.

It is of import for us to retrieve that a perturbation in subcritical flow will propagate upstream every bit good as downstream to impact the flow in both farther upstream and downstream subdivisions. However, in supercritical flow the extension will be merely in the downstream way and the flow at upstream subdivisions will non be affected. Besides, as shown in Figure 1.4d, in the instance of supercritical flow the lines tangent to the moving ridge foreparts lie at an angle I?=arc wickedness ( c/V ) =arc wickedness ( 1/Fr ) with the flow way. ( 5 )

Because the perturbations can propagate upstream in subcritical flow, the conditions at the downstream terminal of a channel affect flow in the channel.

In other words, subcritical flow is capable to downstream control. Therefore, a downstream boundary status is needed to work out the gradually-varied flow

equations for subcritical flow profiles. On the other manus, because perturbations

in supercritical flow can non propagate upstream, supercritical flow in a channel

is non affected by the conditions at the downstream terminal every bit long as the flow remains supercritical. Therefore, supercritical flow is capable to upstream control, and we need an upstream boundary status to work out the gradually-varied flow equations. ( 5 )