Direct Integration of Manning-Based GVF Equation in Trapezoidal Channels
Publication: Journal of Hydrologic Engineering
Volume 17, Issue 3
Abstract
The direct integration method is used to compute free surface profiles in gradually varied flow (GVF) along the length of a prismatic open channel. Analytical solutions of the GVF equation, based on the Manning equation, are available in the technical literature only for the special case of triangular and wide rectangular channels. No closed-form (direct) solution is available for this equation for the case of trapezoidal channels. Open channels with trapezoidal cross sections are widely used as drainage, irrigation, urban stormwater, water transmission, and power channels. In the current study, by applying the Manning equation, a semianalytical solution to compute the length of the GVF profile for trapezoidal channels is derived. This solution, which allows an accurate computation of the flow profiles with minimal computational effort and time, should be a useful tool for direct quantitative analysis and evaluations of trapezoidal channels and thus should be of interest to practitioners in the water engineering community.
Introduction
Sketching of water surface profiles in open channels with gradually varied flow (GVF) has been discussed in many textbooks. However, the computations of the GVF profiles are of considerable importance to hydraulic and hydrologic engineers. The most widely used methods for computing the flow profiles are classified into step methods and direct integration methods. The step methods (Chow 1959) are numerical in nature and can be used when two flow depths are given and the distance between them is required (direct step method) or when the flow depth at a specified location is required (standard step method). The direct integration methods involve the integration of the GVF governing equation and may be performed using analytical, semianalytical, or numerical procedures. Numerical integration of the GVF dynamic equation is primarily used in nonprismatic channels. In some prismatic channels, the governing equation is simplified and analytical or semianalytical integration can be applied. In such cases, the integration is straightforward and the total length of the profile can be calculated in a single computational step. However, such integration methods do not directly provide the depth of flow at a specific distance along the channel length. The concentration of this research is on direct integration methods.
Bresse (1868) derived an analytical solution for wide rectangular channels using the Chézy equation, but the roughness coefficient was considered constant. In general, however, the roughness coefficient (-coefficient) depends on hydraulic radius and channel bed slope. Bakhmeteff (1932) proposed a direct integration that is applicable to any channel shape. His approximate integration method requires dividing the channel length into short reaches. Gunder (1943) integrated the GVF equation of a wide rectangular channel considering the variation of the Chézy coefficient with depth. Chow (1955) developed an extension of Bakhmeteff’s method that eliminated its computational complexity. Gill (1976) derived an analytical solution for a wide rectangular channel using the Manning equation. The solutions presented by Gunder (1943) and Gill (1976) are preferable to Bresse’s solution for wide rectangular channels. Gill (1976) also derived some exact solutions for channels with different constant values of the hydraulic exponent. A direct integration for general rectangular and triangular channels has also been proposed by Kumar (1978) using the Chézy equation, but the Chézy coefficient was considered constant. Dubin (1999) proposed an approximate semianalytical solution for a rectangular channel using the Manning equation. Ramamurthy et al. (2000) integrated the series expansion of the dynamic equation for GVF flows. However, a large number of terms should be used for obtaining accurate results. Vatankhah (2010a) proposed an analytical solution for triangular channels using the Manning equation. Venutelli (2004) and Vatankhah (2010b) presented analytical solutions for a wide rectangular channel using the Manning equation. Vatankhah and Easa (2011) derived a semianalytical solution for general rectangular channels using the Manning equation. Vatankhah (2011) also derived a semianalytical solution for parabolic channels.
From the preceding literature review, it is clear that there is no direct (semianalytical) solution to compute the GVF profiles for trapezoidal channels. Trapezoidal open channels are used in many hydrology and hydraulic applications (Chow 1959; Chaudhry 2006; Haltas and Kavvas 2009; Wong and Zhou 2006). Open channels with trapezoidal cross section are widely used as urban stormwater, drainage, irrigation, water transmission, and power channels (Das 2007). In this study, the direct integration is used to determine the flow profiles in a trapezoidal channel based on the Manning equation. The following sections present the governing equation and the proposed semianalytical integration procedure. The practical applications are then presented, followed by the conclusions.
Governing Equation
The governing equation of steady GVF in open channels, when the Manning equation is used for the computation of the energy slope, is given by (Chow 1959; Subramanya 1986)where = depth of flow (m); = distance along the channel, measured positive in the downstream direction (m); = slope of the free surface at any location ; = longitudinal slope of the channel bottom; = velocity correction factor; = water discharge (); = Manning roughness coefficient (); = cross-sectional area of flow (); = wetted perimeter (m); = width of water surface (m); and = gravitational acceleration (). Eq. (1) is a first-order ordinary differential equation based on the Manning equation that is generally used in engineering practice.
(1)
Proposed Direct Solution
In a regular trapezoidal channel with equal side slopes (symmetric cross section), Eq. (1) takes the formin which = bottom width of the channel; = dimensionless depth of flow (); and = side slopes of the channel ( horizontal to 1 vertical). Eq. (2) can be written in a dimensionless form aswhere = dimensionless distance along the channel; and , , and = functions of channel geometric and flow parameters. These variables are given by
(2)
(3)
(4)
(5)
(6)
(7)
Note that the proposed method is also applicable to a nonsymmetrical trapezoidal channel with different side slopes and . In this case, and .
Different approximations of Eq. (9) can be considered for integration of Eq. (8). However, the most suitable form that results in an analytical solution of Eq. (8) should be sought. In the current study, an appropriate approximation is proposed for a given side slope, , using a polynomial of degree 6, aswhere is an approximation to . The coefficients (, 2 … 6) are determined through curve fitting and presented in Table 1. As noted, the maximum percentage difference of (i.e., ), for practical ranges of and , is less than 0.1%. Hence, is in very good agreement with .Table 1. Proposed Coefficients for in Trapezoidal Channels for Application Ranges of and
(10)
Maximum relative error (%) | |||||||
---|---|---|---|---|---|---|---|
0.25 | 2.6703 | 0.6820 | 0.1149 | 0.0023 | 0.013 | ||
0.50 | 2.8200 | 0.6110 | 0.1548 | 0.0035 | 0.014 | ||
0.75 | 3.0916 | 0.5865 | 0.1814 | 0.0042 | 0.020 | ||
1.00 | 3.4480 | 0.6143 | 0.2359 | 0.0060 | 0.022 | ||
1.25 | 3.8652 | 0.6682 | 0.3060 | 0.0085 | 0.026 | ||
1.50 | 4.3289 | 0.7038 | 0.3191 | 0.0082 | 0.042 | ||
1.75 | 4.8205 | 0.7702 | 0.3929 | 0.0112 | 0.055 | ||
2.00 | 5.3292 | 0.8718 | 0.4885 | 0.0150 | 0.057 | ||
2.25 | 5.8515 | 0.9921 | 0.5751 | 0.0164 | 0.068 | ||
2.50 | 6.3906 | 1.1029 | 0.7014 | 0.0212 | 0.068 | ||
2.75 | 6.9429 | 1.1736 | 0.7125 | 0.0197 | 0.083 | ||
3.00 | 7.4930 | 1.3233 | 0.8739 | 0.0255 | 0.094 |
The proposed direct solution depends on and its application ranges ( and ). Thus, it cannot be converted to the solution for a rectangular channel () or a triangular channel (). Other strategies should be used for these cases. For a rectangular channel, Eq. (8) reduces to
(11)
Vatankhah and Easa (2011) showed that Eq. (11) can be analytically solved for a given by applying . Also, for a triangular channel, making nondimensional the depth using the uniform flow depth, Eq. (1) can be analytically integrated as shown by Vatankhah (2010a).
Replacing into Eq. (8) leads to
(12)
Let , then Eq. (12) becomes
(13)
As noted, analytical integration of Eq. (13) is now possible by using the partial fraction expansions. Integrating both sides of Eq. (13) yields
(14)
To integrate the integrand, the denominator of the integrand needs to be factorized. The denominator is a polynomial of degree 19 with real coefficients. According to the complex conjugate root theorem, if a polynomial with real coefficients has a complex root ( is the principal square root of ; ), then its complex conjugate is also a root of this polynomial. To factorize the denominator of the integrand of Eq. (14), each complex conjugate pair of roots, , should be combined to produce one real factor as []. A real root, , is also factorized as (). Thuswhere , are real roots of the denominator and , are conjugate pairs of complex roots of the denominator. The denominator of the integrand of Eq. (14) can be factorized with the aid of mathematical software. For example, this can be done by using the Maple factor command.
(15)
Now Eq. (16) can be analytically integrated with the aid of popular software such as Maple, Mathematica, Matlab, and Mathcad.
Considering the condition in the control section , the integration constant can be eliminated as follows:in which
(17)
(18)
Using Eqs. (17) - (18), the GVF profiles of a trapezoidal channel can be determined directly for a given boundary condition.
Special Case:
Factorizing a polynomial and symbolic integration can be easily implemented using commercial mathematical software (if available) or free Internet websites (e.g., http://www.quickmath.com). In this study, the previously noted website was used. The analytical integration of Eq. (20) can be performed by first obtaining the denominators of the integrand as follows:
(21)
Then, integrating Eq. (20) yields
(22)
Eq. (22) provides the dimensionless distance along the channel length (which is a function of ) for any given (which is a function of ). A similar procedure can be followed for other side slopes.
Practical Applications
Three applications of the proposed method are presented. The first application illustrates the direct solution for possible profiles in a horizontal trapezoidal channel using a numerical example to familiarize the reader with the proposed method. The second application shows how the proposed method is used to estimate the water discharge. The third application shows dimensionless profiles for subcritical and supercritical flow in both a mild-slope and a steep-slope trapezoidal channel.
Direct Solution for Possible Profiles in a Horizontal Trapezoidal Channel
For a horizontal trapezoidal channel () with () and (, = critical depth), possible profiles ( and ) are calculated and shown in Fig. 1. The and profiles are obtained by applying as a boundary condition (at ) into Eq. (22) as
(23)
The dimensionless distance along the channel length, , is considered positive in the downstream direction and represents a dimensionless critical depth. As seen, at or near the critical depth, the flow becomes so curvilinear or rapidly varied that the equation (definition) of GVF will introduce large errors. Thus, the GVF equation cannot be used to compute accurately the flow profile near the critical depth of flow (Vatankhah 2010a). However, this equation can be used to compute the free water surface curve, which is far from the critical depth.
Computation of Channel Discharge in a Horizontal Trapezoidal Channel
Besides a direct solution for free surface profile in a trapezoidal channel, the proposed semianalytical method can also be used to compute the channel discharge. Consider two depths and that are far from the critical depth. By measuring these depths and the distance between them, , the variable and hence the discharge of the channel can be explicitly determined. The following example illustrates the application of the GVF equation to compute the channel discharge in a horizontal trapezoidal channel.
A horizontal trapezoidal channel is lined with rough concrete () and has a bottom width of 4 m and side slopes . In a reach 60 m long, the water depths at the upstream and downstream ends are 2 and 1.95 m, respectively. Compute the discharge in the channel.
Applying Eqs. (22) - (25) yields
(26)
Thus, and .
Water Surface Profiles in a Trapezoidal Channel
For the given channel geometry and flow parameters, the normal-depth and critical-depth lines divide the space in a channel into three zones. The flow profile may be classified into different types according to these zones. In a channel with mild slope there are three profiles: , , and . Similarly, in a channel with steep slope, there are three profiles: , , and . The slope of channel, which carries a given discharge as a uniform flow at the critical flow depth, is called the critical slope (Abdulrahman 2010). The critical value of (i.e., ) for a trapezoidal channel can be obtained by setting the denominator of the integrand of Eq. (3) to zero for the critical depth, , as follows:where . The proposed semianalytical procedure can be used to determine the dimensionless profiles for a given boundary condition. For this, the expressions for the and are written as follows:where = normal flow depth; and represents dimensionless normal depth. The profiles for subcritical () and supercritical flow () in mild-slope ( or ) and steep-slope ( or ) channels are presented in the following examples.
(27)
(28)
(29)
Dimensionless Profiles in a Mild-Slope Channel
Consider a trapezoidal channel with side slopes , the dimensionless critical flow depth , and the dimensionless normal flow depth . Using Eqs. (28) - (29) yields and , respectively. Also, using Eq. (27), . As noted, the slope of the channel is mild ( or ). Using Table 1, Eq. (14) reduces to
(30)
The denominator of the integrand can be factorized as follows:
(31)
For this mild-slope channel, possible profiles are calculated and shown in Fig. 2. The profile is obtained using as a boundary condition (at ). The and profiles are obtained using as the boundary condition (at ). The GVF equation is not valid for estimating flow depth at or near the critical depth because the water surface profile is very curvilinear in this region.
Dimensionless Profiles in a Steep-Slope Channel ()
In this case, consider a trapezoidal channel with side slopes , the dimensionless critical flow depth , and the dimensionless normal flow depth . Using these values results in , , and . As seen, the slope of the channel is steep ( or ). Using Table 1, Eq. (14) reduces to
(33)
The denominator of the integrand of Eq. (33) can be factorized as follows:
(34)
For this steep-slope channel, possible profiles are calculated and shown in Fig. 3. The profile is obtained using as a boundary condition (at ). The and profiles are obtained using as the boundary condition (at ).
Comparison between the Proposed Method and the Direct Step Method
To show the accuracy and efficiency (minimal computational effort) obtainable with the proposed method, a comparison between the length of the water surface profile computed using the direct step method and the near exact value computed with the proposed method is performed and shown in the following example.
A trapezoidal channel is carrying a flow of and has a bottom width of 10 m, equal side slopes , and . The channel has a slope of 0.001 and a Manning roughness coefficient of 0.014. A control structure is built at the downstream end, which raises the water flow depth at the downstream end to 3.0 m. Compute the length of the backwater curve between the control structure () and where the flow depth is 1.2 m at the upstream of the control structure.
Solution: Direct Step Method
For the direct step method, the computations are started with a known depth, , at the control structure and proceed in the upstream direction. By considering the location at the control structure as , the computed values of will be negative.
Using the boundary condition , prescribed water depth, the direct step method will be (Chow 1959; Subramanya 1986)in whichwhere is specific energy; and is mean friction slope. Thus, by knowing a specified flow depth, , the location of section 2, , can be determined using Eqs. (36) - (38). This is the starting value for the next step, and thus the total length of the water surface profile will be calculated.
(36)
(37)
(38)
Used in this example is decrement of flow depth, that is, , where is the number of segments per profile that the channel is divided into based on its starting and ending depths. Increasing this number will increase the accuracy of the profile calculation but will increase the calculation time. Fig. 4 shows the total length of the backwater curve in terms of number of segments. The near exact solution is obtained using (number of segments) as 2,137.91 m.
Solution: Proposed Method
Using Eqs. (6) - (7) yields and , respectively. The denominator of the integrand of Eq. (18) can be factorized as follows:
(39)
The proposed solution is
(40)
Applying boundary values yields
(41)
Thus, the total length of the water surface profile is 2,137.81 m according to the proposed method. This solution can be obtained using the direct step method with 125 segments. As noted, the proposed solution achieves suitable accuracy with minimal computational effort and time.
Conclusions
This paper presents a semianalytical solution of the GVF equation for ordinary trapezoidal channels ( and ) based on the Manning equation. Several practical applications in which the proposed solution can be implemented are described. The proposed solution can accurately estimate the flow profiles of trapezoidal channels with minimal computational effort and as such should be a useful tool for direct quantitative analysis and evaluations. The proposed solution uses a single step for the computation of flow profiles and, as such, provides a more efficient calculation procedure compared with normal numerical methods (such as the direct step method) that depend on the number of segments. Indeed, this solution reduces the computational burden. The computational effectiveness is important when the intensive calculation of the flow profiles is required in some especial simulations and optimizations. Although the Manning-based solution presented in this study is more complex than that based on the Chézy equation, the proposed solution does provide more accurate results. The mathematical approach used in this study may be useful in addressing similar challenges in hydraulic and hydrologic engineering.
Notation
The following symbols are used in this paper:
- =
- cross-sectional area of flow ();
- =
- bottom width of the channel (m);
- =
- slope of the free surface at any location;
- =
- specific energy (m);
- =
- gravitational acceleration ();
- =
- length of a channel reach (m);
- =
- Manning roughness coefficient ();
- =
- wetted perimeter (m);
- =
- water discharge ();
- =
- longitudinal slope of the channel bottom;
- =
- friction slope;
- =
- width of water surface (m);
- =
- dimensionless parameter [];
- =
- distance along the channel, considered positive in the downstream direction (m);
- =
- depth of flow (m);
- =
- normal depth of flow (m);
- =
- critical depth of flow (m);
- , and =
- side slope of the channel;
- =
- dimensionless parameter [ or ];
- =
- velocity correction factor;
- =
- dimensionless discharge ();
- =
- dimensionless function of channel geometric and flow parameters ();
- =
- critical value of ;
- =
- dimensionless depth of flow ();
- =
- dimensionless critical depth of flow ();
- =
- dimensionless depth of flow ;
- =
- dimensionless depth of flow ;
- =
- dimensionless critical depth of flow (); and
- =
- dimensionless distance along the channel ().
Acknowledgments
The author would like to thank four anonymous reviewers for providing valuable comments on this work.
References
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© 2012 American Society of Civil Engineers.
History
Received: Oct 19, 2010
Accepted: Jun 10, 2011
Published online: Jun 14, 2011
Published in print: Mar 1, 2012
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