Gradually varied flow is a steady non-uniform flow in which the depth variation in the direction of motion is gradual enough that the transverse pressure distribution can be considered hydrostatic. This allows the flow to be treated as one dimensional with no transverse pressure gradients other than those created by gravity. The methods developed should not be applied to regions o f highly curvilinear flow, such as can be found in the vicinity of an ogees spillway crest for example, because the centripetal acceleration in curvilinear f low alters the transverse pressure distribution so that it no longer is hydrostatic, and the pressure head n no longer can be represented by the depth of flow.(4)
The flow in an open-channel is termed as gradually varied flow (GVF) when the depth of flow varies gradually with longitudinal distance. Such flows are encountered both on upstream and downstream sides of control sections. Analysis and computation of gradually varied flow profiles in open-channels are important from the point of view of safe and optimal design and operation of any hydraulic structure.(6)
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Even with the assumption of gradually varied flow, an exact solution for the depth profile exists only in the case of a wide, rectangular channel. The solution of the equation of gradually varied flow in this case is called the Bresse function, which provides useful approximations of water surface profile lengths subject to the assumptions of a very wide channel and a constant value of Chezy's C. The solutions to all other problems in the past, were obtained graphically or from tabulations of the varied flow function based on hydraulic exponents as developed by Bakhmeteff(1932) and Chow( 1959). (4)
In analyzing the steady GVFs in open channels, flow resistance, and changes in bottom slope, channel shape, discharge and surface conditions are taken into account. It is usual to denote d the actual depth (i.e. the non-uniform flow depth), do the normal depth (i.e. uniform flow depth) and dc the critical depth.(6)
In many practical problems of hydraulic engineering concerning open channels, a correct evaluation of the water elevation in various sections is required. The objective, achieved by sketching the gradually-varied-flow profile, requires integration of the governing equations. The process can be carried out either numerically or analytically.(5)
Several integration methods exist: analytical solutions are rare but numerous numerical integration methods exist. We propose to develop the standard step method (distance computed from depth). This method is simple, extremely reliable and very stable. It is strongly recommended to practicing engineers who are not necessarily hydraulic experts.(6)
Bresse ~1868! obtained the classical direct solution for a wide rectangular channel. However, like most hydraulic engineers of his time, he assumed that the roughness coefficient of the CheÂ´zyformula, used for the calculation of the velocity, is constant. In fact, the first experimental observations showing that the roughness coefficient depends on the size and depth of the channels, started by Darcy and completed by Bazin in 1865, date back to the same period.(5)
Later, Masoni ~1900! proposed a direct integration with the same approximation, which is valid for a common rectangular section, whereas Bakhmeteff ~1932! proposed a direct integration that is applicable to all shapes of channels that in practice. However, Bakhmeteff carried out only an approximate integration, whose values are reported in tables. Moreover, the method requires a division of the channel length into short reaches. Chow ~1955! developed an extension of the method which allows one to avoid that computational difficulty. (5)
Taking the same function introduced by Bakhmeteff, Chow assumed the computational criteria and provided tables with a wider range. A direct integration for rectangular and triangular channels has also been proposed by Kumar ~1978!. CheÂ´zy's formula was assumed in that work, but the roughness coefficient was still considered constant.(5)
In order to avoid such approximations and to provide a more accurate representation of the flow profiles, Manning's formula is used for evaluation of the motion in non-uniform flow also. Then a relation obtained by direct integration of the gradually-varied-flow equation and referred to wide rectangular channels is proposed.(5)
(LAST)Many computer programs are available for computation of backwater curves. The most general and widely used programs 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 programs can be used to compute water surface profiles for both natural and artificial channels.
CLASSIFICATION OF CHANNELS FOR GRADUALLY-VARIED FLOW
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Open channels are classified as being mild, steep, critical, horizontal, and adverse in gradually-varied flow studies. If for a given discharge the normal depth of a channel is greater than the critical depth, the channel is said to be mild.
If the normal depth is less than the critical depth, the channel is called steep. For a
critical channel, the normal depth and the critical depth are equal. If the bottom
slope of a channel is zero, the channel is called horizontal. A channel is said to
have an adverse slope if the channel bottom rises in the flow direction.(5)
Mild channels yn > yc
Steep channels yn < yc
Critical channels yn = yc
Horizontal channels Sâ‚’=0
Adverse channels Sâ‚’<0
where yn= normal depth and yc= critical depth.(5)
Basic Assumptions in GVF Analysis(5)
1. The gradually varied flow to be discussed here considers only steady flows. This
implies that (i) flow characteristics do not change with time, and (ii) pressure distribution is hydrostatic over the channel section.
2. The head loss in a reach may be computed using an equation applicable to uniform flow having the same velocity and hydraulic mean radius of the section. This implies that the slope of energy grade line may be evaluated using a uniform flow formula such as Manning equation and Chezy equation, with the corresponding roughness coefficient applicable primarily for uniform flow.
3. Channel bottom slope is small. This implies that the depth of flow measured vertically is same as depth of flow measured perpendicular to channel bottom.
4. There is no air entrainment. Advanced text books may be referred to study the effects of air entrainment.
5. The velocity distribution in the channel section is invariant. This implies that the
energy correction factor, Î± , is a constant and does not vary with distance.
6. The resistance coefficient is not a function of flow characteristics or depth of flow. It does not vary with distance.
7. Channel is prismatic.
GRADUALLY-VARIED FLOW COMPUTATIONS (5)
To obtain an expression for gradually-varied flow, we refer to Total energy head Equation which define the total energy head, H, as
H = zb + y + V2/2g (1.1)
where zb=elevation of the channel bottom, y=flow depth, V= average cross-sectional velocity, and g=gravitational acceleration. Now, referring 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 differentiate both sides of Equation 4.3 with respect to x to obtain
dH /dx = dzb / dx + dE / dx (1.4)
where x is the displacement in the flow direction. By definition, Sf=-dH/dx,
and Sâ‚’=-dzb/dx. By substituting these into Equation 1.4 and rearranging,
we obtain one form of the gradually-varied flow equation as
dE /dx = S0 - Sf (1.5)
We can obtain another form of the gradually-varied flow equation by expanding
the left-hand side of Equation 1.5 to
dE /dx= dy/dx +d(VÂ²/2g)/dx=dy/dx + V/g dv/dx=dy/dx + V/g dv/dy dy/dx
= dy/dx+V/g dy/dx d(Q/a)/dy = dy/dx+ V/g dy/dx Qd(1/A)/dy
where Q=constant discharge and A=area. Further mathematical manipulation
by using the definitions T=top width=dA/dy, D=hydraulic depth=A/T, and
Fr=Froude number=V/(gD)0.5 will lead 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)/AÂ² = dy/dx (1 - VÂ²/gD) =dy/dx(1 - FÂ²r)
dE/dx = dy/dx (1 - FÂ²r)( (1.6)
By substituting Equation 1.6 into 1.5 and rearranging, we obtain
dy/dx = (Sâ‚’ -Sf) / (1-FÂ²r ). (1.7)
We can solve either Equation 1.5 or Equation 1.7 in order to determine the gradually-varied flow depths at different sections along a channel. However, we
find Equation 1.5 more convenient for this purpose. As we pointed out before, this is a differential equation; a boundary condition is required for solution. It is very important to remember that subcritical flow is subject to downstream control.
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Therefore, if flow in the channel is subcritical, then a downstream boundary condition must be used to solve Equation 1.5 given Q. Conversely, supercritical
flow is subject to upstream control, and an upstream boundary condition is needed to solve Equation 1.5 for supercritical flow. By boundary condition, we generally mean a known flow depth associated with a known discharge.(4)
Analytical solutions to Equation 1.5 are not available for most open-channel flow
situations typically encountered. In practice, we apply a finite difference approach to calculate the gradually-varied flow profiles. In this approach, the channel is divided into short reaches and computations are carried out from one end of the reach to the other.(4)
Consider the channel reach shown in Figure 1.11 having a length of âˆ†X. Sections U and D denote the flow sections at the upstream and downstream ends of the reach, respectively. Using the subscripts U and D to denote the upstream and downstream sections, we can write Equation 1.5 for this reach in finite difference form as
ED - EU/ âˆ†X = Sâ‚’ - Sfm (1.9)
FIGURE 4.11Definition sketch for gradually flow froumulation.jpg
Figure 4.5 Definition sketch for gradually-varied flow formulation
where Sfm=average friction slope in the reach, approximated as
Sfm = Â½ (SfU + SfD) (1.10)
By rearranging the Manning formula, the friction slopes at sections U and D are
SfU = nÂ²/kÂ²n VÂ²U/Ru^4/3 (1.11)
SfD = nÂ²/kÂ²n VÂ²d/Rd^4/3 (1.12)
In the past direct and graphical solution methods have been used to solve these, however these method have been superseded by numerical methods which are now be the only method used.
NUMERICAL METHODS (7)
Numerical integration is primarily used in non-prismatic channels, such as natural streams. In prismatic channels, such as artificial ones, the structure of the basic equation is simplified, and then direct integrations can also be applicable. Direct methods have the advantage of providing independent solutions of the previous computational steps. The total lengths of the profile can be evaluated with a single computation. On the other hand, these
methods have the disadvantage of not evaluating the depth of the flow at a specific longitudinal distance.(5)
There are two basic numerical methods that can be used:
Direct step - distance from depth
Standard step method - depth from distance
DIRECT STEP METHOD (distance from depth) (5,7))
This method will calculate ( by integrating the gradually varied flow equation) a distance for a given change in surface height.
In the direct step method, we write Equation 1.9 as
âˆ†X= Ed-Eu/Sâ‚’-Sfm = ((yd+VÂ²d/2g)-(yu+ VÂ²u/2g))/Sâ‚’-Sfm (1.13)
In a typical subcritical flow problem, the condition at the downstream section 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 calculate the corresponding VU, SfU, and Sfm. Then we calculate _X from Equation 1.13
Conversely, where supercritical flow is involved, conditions at section U are
known. In this case, we pick a value for yD to calculate the reach length.
This method is called the direct step method, since the reach length is obtained directly from Equation 1.13 without any trial and error. These calculations are repeated for the subsequent reaches to determine the water surface profile.
For subcritical flow calculations, we start from the downstream end of a channel and proceed in the upstream direction. In other words, the first reach considered is at the downstream end of the channel, and the downstream section of this reach coincides with the downstream extremity of the channel.
At the downstream extremity, yD is known from the boundary condition.
Using the known discharge and the cross-sectional properties, we first calculate
VD and SfD. Next we pick a value for yU and calculate the corresponding VU and SfU. Then, from Equation 1.13, we determine the channel reach _X.
This process is repeated for further upstream reaches until the entire length of the channel is covered. Note that yU of any reach becomes yD for the reach considered next. Also, we must be careful in picking the values for yU.
These values depend on the type of the profile that will occur in the channel.
For example, if an M2 profile is being calculated, yU must satisfy the inequalities yU>yD and yn>yU>yc. Likewise, for an S1 profile, yU<yD and yU>yc>yn.
For supercritical profiles, we start at the upstream end and proceed in the downstream direction. For the first reach, yU is known from the upstream boundary condition.We choose a value for yD and calculate the reach length, âˆ†X,
using Equation 1.13. This process is repeated for further downstream reaches until the length of the channel is covered. The yD of any reach becomes yU of the subsequent reach. The values of yD must be chosen carefully in the process.
For instance, for M3 profiles, yD>yU and yD<yc<yn. Likewise, for S2 profiles,
yD<yU and yn<yD<yc.
In certain situations, the flow depths at both ends of a surface profile will be known and we can perform the calculations to determine the total length of the profile. In such a case we can start from either the upstream end or the downstream end, regardless of whether the flow is subcritical or supercritical.
However, a downstream boundary condition is always known for subcritical flow, and an upstream boundary condition is always known for supercritical flow.
Therefore, it is reasonable to adopt the general rule that subcritical flow calculations start at the downstream end, and supercritical flow calculations start
at the upstream end.
STANDARD STEP METHOD (depth from distance) (5,7)
This method will calculate ( by integrating the gradually varied flow equation ) a depth at a given distance up or down stream.
In the standard step method, the flow depths are calculated at specified locations. As in the direct step method, we know the flow depth and velocity at one end of a channel reach. We then choose the reach length, âˆ†X, and calculate the depth at the other end of the reach.
For subcritical flow, the conditions at the downstream section will be known. For this case, to facilitate the calculations, we will rearrange Equation 1.13 as
yU + VÂ²U/2g - Â½ (âˆ†X)Sfu = yD + VÂ²D/2g + Â½ (âˆ†X)SfD -(âˆ†X)Sâ‚’ (1:14)
For a constant discharge, we can express VU and (Sf)U in terms of yU. Therefore,
the only unknown in Equation 1.14 is yU. However, the expression is implicit in yU, and we can solve it by use of an iterative technique. We try different values
for yU until Equation 1.14 is satisfied. Because of the iterative nature of the procedure the standard step method is not suitable for calculation by hand, and we normally employ a computer program. However, in the absence of such a program we can improve the guessed values of yU in each iteration using
( yU)k+1 = (yU)k -âˆ†yk (1:15)
The standard step method can be used for non-prismatic channels as well, with
CLASSIFICATION OF GRADUALLY-VARIED FLOW PROFILES
A gradually-varied flow profile or gradually-varied water surface profile is a line
indicating the position of the water surface. It is a plot of the flow depth as a function of distance along the flow direction. A sound understanding of possible profiles under different flow situations is essential before we can obtain numerical solutions to gradually-varied flow problems.(5)
Consider a mild channel as shown in Figure 1.1. By definition, yn > yc. The
channel bottom, the critical depth line, and the normal depth line divide the
channel into three zones in the vertical dimension, namely M1, M2, and M3
(M stands for mild). The solid lines in the figure represent the shapes of the possible flow profiles in these three zones. Obviously, the normal depth line itself
would represent the water surface if the flow in the channel were normal. In zone
M1, the water surface is above the normal depth line. Therefore, in this zone y> yn and consequently Sf<Sâ‚’. Also, y> yc and thus Fr<1.0 in zone M1.(5)
Therefore, both the numerator and the denominator of Equation 1.7 are positive quantities, and (dy/dx)>0. In other words, the flow depth must increase in the flow direction in zone M1. We can examine the zones M2 and M3 in a similar manner, and conclude that (dy/dx)<0 in zone M2 and (dy/dx)>0 in zone M3.
The behavior of the water surface profile near the zone boundaries can also be
examined. From Equation 1.7, as yâ†’ âˆž we can see that Frâ†’0 and Sfâ†’0. Thus
(dy/dx)â†’Sâ‚’, meaning the water surface will approach a horizontal line asymptotically as yâ†’âˆž. Likewise, as yâ†’ yn, by definition Sfâ†’Sâ‚’ and thus (dy/dx)â†’0. Therefore, the surface profile approaches the normal depth line asymptotically. Near the critical depth line, yâ†’ yc and Frâ†’1.0. Thus (dy/dx)â†’1, and the water surface will approach the critical depth line at an angle close to a right-angle. Near the bottom of the channel, as yâ†’0, both Sfâ†’âˆž, and Frâ†’âˆž. Therefore, the water surface will approach the channel (5)
Flow profile in mild channel.jpg
Figure 1.1 Flow profiles in mild channels(5)
bottom at a finite positive angle. The magnitude of this angle depends on the
friction formula used and the specific channel section.
Based on this qualitative examination of Equation 1.7 near the zone boundaries,
we conclude that in zone M1 the water surface profile is asymptotical to the normal depth line as yâ†’yn and is asymptotical to a horizontal line as yâ†’âˆž.
The M2 profile is asymptotical to the normal depth line, and it makes an angle
close to a right-angle with the critical depth line. The M3 profile makes a positive
angle with the channel bottom and an angle close to a right-angle with the critical
depth line. The water surface profiles sketched in Figure 1.1 reflect these
We should note that a flow profile does not have to extend from one zone boundary to another. For example, an M2 profile does not have to begin at the
normal depth line and end at the critical depth line. It is possible that an M2 profile begins at a point below the normal depth line and ends at a point above the critical depth line.
For a steep channel, yn>yc by definition. The channel bottom, the normal depth line, and the critical depth line divide the channel into three zones in the vertical dimension, namely S1, S2, and S3 (S stands for steep) as shown in Figure 1.2. As before, the solid lines in the figure represent the shapes of the possible flow profiles in these three zones. If the flow were normal in this channel, the normal depth line itself would represent the water surface. In zone S1 the water surface is above the critical depth line, therefore in this zone y>yc and thus Fr<1.0. Also, y>yc>yn, and consequently Sf<Sâ‚’. Therefore, both the numerator and the denominator of Equation 1.7 are positive quantities, and in zone S1 (dy/dx)>0. In other words, the flow depth must increase in the flow direction. We can examine the zones S2 and S3 in a similar manner, and conclude that (dy/dx)50 in zone S2 and (dy/dx)>0 in zone S3.(5)
The behavior of the surface profile near the zone boundaries examined for mild
channels is valid for steep channels as well, since Equation 1.7 is applicable to
both steep and mild channels. Accordingly, the S1 profile makes an angle close to
the right-angle with the critical depth 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 yâ†’âˆž. The S2 profile makes an angle close to the right-angle with the critical depth line, and it approaches the normal depth line asymptotically. The S3 profile will make a positive angle with the channel bottom, and it will approach the normal depth line asymptotically. (5)
The possible profile types that can occur in horizontal, adverse, and critical channels are shown in Figure 1.3 These profiles are sketched by examining the sign of (dy/dx) with the help of Equation 1.7, and considering the behavior of the profile near the zone boundaries. Note that for horizontal and adverse channels normal flow is not possible, thus yn is not defined, and zones H1 and A1 do not exist. Likewise, for critical channels yn=yc, and therefore zone C2 does not exist. It is also worth noting 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 calculation(6)
The term 'backwater calculations' refers more generally to the calculations of the longitudinal free-surface profile for both sub- and supercritical flows.
The backwater calculations are developed assuming:
[H1] a non-uniform flow,
[H2] a steady flow,
[H3] that the flow is gradually varied,
[H4] that, at a given section, the flow resistance is the same as for an uniform flow for the
same depth and discharge, regardless of trends of the depth.
The GVF calculations do not apply to uniform equilibrium flows, nor to unsteady flows, nor to RVFs. The last assumption [H4] implies that the Darcy, Chézy or Gauckler-Manning equation may be used to estimate the flow resistance, although these equations were originally developed for uniform equilibrium flows only.
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 condition to solve it. Mathematically, the flow depth at any given flow section can be used as a boundary condition.
However, for correct representation of open-channel flow the boundary condition will be prescribed at either the upstream or the downstream end of the channel, depending on whether the flow in the channel is supercritical or subcritical. The following observation is presented to explain the reason for this.
A pebble thrown into a large still body of water will create a disturbance, which will propagate outward in the form of concentric circles as shown in Figure 1.4a.
Figure 1.3 flow profiles in horizontal adverse and critical channels(5)
The speed with which the disturbance propagates is called celerity (or celerity of
gravity waves in shallow water), and it is evaluated as
c = âˆšgD
where c =c elerity, g = gravitational acceleration, and D = hydraulic depth.
If the pebble is thrown into a body of water moving with a velocity V the wave
propagation will no longer be in the form of concentric circles. Referring to the definition of Froude number, we can write that Fr = (V)/ âˆšgD = V /c
Significance of Froude number in gradually-varied flow calculations
Figure 1.4 Effect of Froude number on propagation of a disturbance in open channel(5)
For subcritical flow, Fr<1 and V<c. On the other hand, for supercritical flow, Fr>1 and V>c. Obviously, for critical flow V=c. Therefore, if the flow is subcritical, the disturbances will propagate upstream at a speed (c-V) and downstream at a speed (c+V), as shown in Figure 1.4b. If the flow is critical, then the upstream edge of the wave will be stationary while the downstream propagation will be at a speed 2c, as shown in Figure 1.4c. If the flow is supercritical, then the propagation will be in the downstream direction only as shown in Figure 1.4d, with the back and front edges moving with speeds (V-c) and (V+c), respectively.
It is important for us to remember that a disturbance in subcritical flow will propagate upstream as well as downstream to affect the flow in both further upstream and downstream sections. However, in supercritical flow the propagation will be only in the downstream direction and the flow at upstream sections will not be affected. Also, as shown in Figure 1.4d, in the case of supercritical flow the lines tangent to the wave fronts lie at an angle Î²=arc sin (c/V)=arc sin (1/Fr) with the flow direction.(5)
Because the disturbances can propagate upstream in subcritical flow, the conditions at the downstream end of a channel affect flow in the channel.
In other words, subcritical flow is subject to downstream control. Therefore, a downstream boundary condition is needed to solve the gradually-varied flow
equations for subcritical flow profiles. On the other hand, because disturbances
in supercritical flow cannot propagate upstream, supercritical flow in a channel
is not affected by the conditions at the downstream end as long as the flow remains supercritical. Therefore, supercritical flow is subject to upstream control, and we need an upstream boundary condition to solve the gradually-varied flow equations.(5)