Direct Integration of Manning-Based GVF Equation in Trapezoidal Channels

Vatankhah, Ali R.

doi:10.1061/(ASCE)HE.1943-5584.0000460

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TECHNICAL NOTES

Jun 14, 2011

Direct Integration of Manning-Based GVF Equation in Trapezoidal Channels

Author: Ali R. Vatankhah [email protected]Author Affiliations

Publication: Journal of Hydrologic Engineering

Volume 17, Issue 3

https://doi.org/10.1061/(ASCE)HE.1943-5584.0000460

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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 (

C

-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)

\frac{d y}{d x} = \frac{S_{0} - \frac{n^{2} Q^{2} P^{4 / 3}}{A^{10 / 3}}}{1 - α \frac{Q^{2} T}{g A^{3}}}

(1)

where

y

= depth of flow (m);

x

= distance along the channel, measured positive in the downstream direction (m);

d y / d x

= slope of the free surface at any location

x

;

S_{0}

= longitudinal slope of the channel bottom;

α

= velocity correction factor;

Q

= water discharge (

m^{3} / s

);

n

= Manning roughness coefficient (

m^{- 1 / 3} \cdot s

);

A

= cross-sectional area of flow (

m^{2}

);

P

= wetted perimeter (m);

T

= width of water surface (m); and

g

= gravitational acceleration (

m / s^{2}

). Eq. (1) is a first-order ordinary differential equation based on the Manning equation that is generally used in engineering practice.

Proposed Direct Solution

In a regular trapezoidal channel with equal side slopes (symmetric cross section), Eq. (1) takes the form

\frac{n^{2} Q^{2}}{B^{19 / 3}} d x = {\frac{1 - \frac{Q^{2}}{g B^{5}} (\frac{1 + 2 z η}{η^{3} (1 + z η)^{3}})}{\frac{S_{0} B^{16 / 3}}{n^{2} Q^{2}} - \frac{(1 + 2 \sqrt{1 + z^{2}} η)^{4 / 3}}{η^{10 / 3} (1 + z η)^{10 / 3}}}} d η

(2)

in which

B

= bottom width of the channel;

η

= dimensionless depth of flow (

= y / B

); and

z

= side slopes of the channel (

z

horizontal to 1 vertical). Eq. (2) can be written in a dimensionless form as

d χ = {\frac{1 - δ_{1} (\frac{1 + 2 z η}{η^{3} (1 + z η)^{3}})}{δ_{2} - \frac{(1 + z_{*} η)^{4 / 3}}{η^{10 / 3} (1 + z η)^{10 / 3}}}} d η

(3)

where

χ

= dimensionless distance along the channel; and

z_{*}

,

δ_{1}

, and

δ_{2}

= functions of channel geometric and flow parameters. These variables are given by

χ = x n^{2} Q^{2} / B^{19 / 3}

(4)

z_{*} = 2 \sqrt (1 + z^{2})

(5)

δ_{1} = Q^{2} / g B^{5}

(6)

δ_{2} = S_{0} B^{16 / 3} / (n^{2} Q^{2})

(7)

Note that the proposed method is also applicable to a nonsymmetrical trapezoidal channel with different side slopes

z_{1}

and

z_{2}

. In this case,

z = (z_{1} + z_{2}) / 2

and

z_{*} = \sqrt (1 + z_{1}^{2}) + \sqrt (1 + z_{2}^{2})

.

Eq. (3) can be written as

d χ = {\frac{η^{10 / 3} (1 + z η)^{3} - δ_{1} η^{1 / 3} (1 + 2 z η)}{δ_{2} η^{10 / 3} (1 + z η)^{3} - F (η)}} d η

(8)

in which

F (η) = \frac{(1 + z_{*} η)^{4 / 3}}{(1 + z η)^{1 / 3}}

(9)

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,

z

, using a polynomial of degree 6, as

F (η) ≅ F_{*} (η) = 1 + a_{1} η + a_{2} η^{2} + a_{3} η^{3} + a_{4} η^{4} + a_{5} η^{5} + a_{6} η^{6}

(10)

where

F_{*} (η)

is an approximation to

F (η)

. The coefficients

a_{i}

(

i = 1

, 2 … 6) are determined through curve fitting and presented in Table 1. As noted, the maximum percentage difference of

F_{*} (η)

(i.e.,

| F_{*} (η) / F (η) - 1 | \times 100

), for practical ranges of

0 \leq η \leq 3

and

0.25 \leq z \leq 3

, is less than 0.1%. Hence,

F_{*} (η)

is in very good agreement with

F (η)

.

Table 1. Proposed Coefficients for

F_{*} (η)

in Trapezoidal Channels for Application Ranges of

0 \leq η \leq 3

and

0.25 \leq z \leq 3.00

$z$	$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$	$a_{5}$	$a_{6}$	Maximum relative error (%)
0.25	2.6703	0.6820	$- 0.3177$	0.1149	$- 0.0249$	0.0023	0.013
0.50	2.8200	0.6110	$- 0.3716$	0.1548	$- 0.0362$	0.0035	0.014
0.75	3.0916	0.5865	$- 0.4102$	0.1814	$- 0.0433$	0.0042	0.020
1.00	3.4480	0.6143	$- 0.4906$	0.2359	$- 0.0595$	0.0060	0.022
1.25	3.8652	0.6682	$- 0.5925$	0.3060	$- 0.0812$	0.0085	0.026
1.50	4.3289	0.7038	$- 0.6321$	0.3191	$- 0.0816$	0.0082	0.042
1.75	4.8205	0.7702	$- 0.7352$	0.3929	$- 0.1061$	0.0112	0.055
2.00	5.3292	0.8718	$- 0.8785$	0.4885	$- 0.1369$	0.0150	0.057
2.25	5.8515	0.9921	$- 1.0403$	0.5751	$- 0.1561$	0.0164	0.068
2.50	6.3906	1.1029	$- 1.2163$	0.7014	$- 0.1969$	0.0212	0.068
2.75	6.9429	1.1736	$- 1.2779$	0.7125	$- 0.1916$	0.0197	0.083
3.00	7.4930	1.3233	$- 1.5119$	0.8739	$- 0.2419$	0.0255	0.094

The proposed direct solution depends on

F_{*} (η)

and its application ranges (

0 \leq η \leq 3

and

0.25 \leq z \leq 3

). Thus, it cannot be converted to the solution for a rectangular channel (

z = 0

) or a triangular channel (

η \to \infty

). Other strategies should be used for these cases. For a rectangular channel, Eq. (8) reduces to

d χ = {\frac{η^{10 / 3} - δ_{1} η^{1 / 3}}{δ_{2} η^{10 / 3} - (1 + 2 η)^{4 / 3}}} d η

(11)

Vatankhah and Easa (2011) showed that Eq. (11) can be analytically solved for a given

δ_{2}

by applying

t = (2 + 1 / η)^{1 / 3}

. 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

F_{*} (η)

into Eq. (8) leads to

\begin{matrix} (12) & d χ = {\frac{η^{10 / 3} (1 + z η)^{3} - δ_{1} η^{1 / 3} (1 + 2 z η)}{δ_{2} η^{10 / 3} (1 + z η)^{3} - (1 + a_{1} η + a_{2} η^{2} + a_{3} η^{3} + a_{4} η^{4} + a_{5} η^{5} + a_{6} η^{6})}} d η \end{matrix}

(12)

Let

t = η^{1 / 3}

, then Eq. (12) becomes

d χ = {\frac{3 t^{12} (1 + z t^{3})^{3} - 3 δ_{1} t^{3} (1 + 2 z t^{3})}{δ_{2} t^{10} (1 + z t^{3})^{3} - (1 + a_{1} t^{3} + a_{2} t^{6} + a_{3} t^{9} + a_{4} t^{12} + a_{5} t^{15} + a_{6} t^{18})}} d t

(13)

As noted, analytical integration of Eq. (13) is now possible by using the partial fraction expansions. Integrating both sides of Eq. (13) yields

χ = \int {\frac{3 t^{12} (1 + z t^{3})^{3} - 3 δ_{1} t^{3} (1 + 2 z t^{3})}{δ_{2} t^{10} (1 + z t^{3})^{3} - (1 + a_{1} t^{3} + a_{2} t^{6} + a_{3} t^{9} + a_{4} t^{12} + a_{5} t^{15} + a_{6} t^{18})}} d t + const

(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

a + b i

(

i

is the principal square root of

- 1

;

i = \sqrt - 1

), then its complex conjugate

a - b i

is also a root of this polynomial. To factorize the denominator of the integrand of Eq. (14), each complex conjugate pair of roots,

a \pm b i

, should be combined to produce one real factor as [

(t - a)^{2} + b^{2}

]. A real root,

r

, is also factorized as (

t - r

). Thus

δ_{2} t^{10} (1 + z t^{3})^{3} - (1 + a_{1} t^{3} + a_{2} t^{6} + a_{3} t^{9} + a_{4} t^{12} + a_{5} t^{15} + a_{6} t^{18}) = (t - r_{1}) (t - r_{2}) \dots [(t - a_{1})^{2} + b_{1}^{2}] [(t - a_{2})^{2} + b_{2}^{2}] \dots

(15)

where

r_{1}

,

r_{2} \dots

are real roots of the denominator and

a_{1} \pm b_{1} i

,

a_{2} \pm b_{2} i \dots

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.

Substituting Eq. (15) into Eq. (14) yields

χ = \int {\frac{3 t^{12} (1 + z t^{3})^{3} - 3 δ_{1} t^{3} (1 + 2 z t^{3})}{(t - r_{1}) (t - r_{2}) \dots [(t - a_{1})^{2} + b_{1}^{2}] [(t - a_{2})^{2} + b_{2}^{2}] \dots}} d t + const

(16)

Now Eq. (16) can be analytically integrated with the aid of popular software such as Maple, Mathematica, Matlab, and Mathcad.

Considering the condition

η = η (0)

in the control section

χ = χ (0)

, the integration constant can be eliminated as follows:

χ - χ (0) = I (η) - I [η (0)]

(17)

in which

I (η) = \int {\frac{3 t^{12} (1 + z t^{3})^{3} - 3 δ_{1} t^{3} (1 + 2 z t^{3})}{(t - r_{1}) (t - r_{2}) \dots [(t - a_{1})^{2} + b_{1}^{2}] [(t - a_{2})^{2} + b_{2}^{2}] \dots}} d t

(18)

Using Eqs. (17) - (18), the GVF profiles of a trapezoidal channel can be determined directly for a given boundary condition.

Special Case: $δ_{2} = 0$

For a horizontal trapezoidal channel (

S_{0} = 0

), Eq. (7) yields

δ_{2} = 0

and Eq. (14) reduces to

χ = \int {\frac{3 δ_{1} t^{3} (1 + 2 z t^{3}) - 3 t^{12} (1 + z t^{3})^{3}}{1 + a_{1} t^{3} + a_{2} t^{6} + a_{3} t^{9} + a_{4} t^{12} + a_{5} t^{15} + a_{6} t^{18}}} d t + const

(19)

For a channel with side slope

z = 1.5

and using Table 1, Eq. (19) takes the form

χ = \int {\frac{3 δ_{1} t^{3} (1 + 3 t^{3}) - 3 t^{12} (1 + 1.5 t^{3})^{3}}{1 + 4.3289 t^{3} + 0.7038 t^{6} - 0.6321 t^{9} + 0.3191 t^{12} - 0.0816 t^{15} + 0.0082 t^{18}}} d t + const

(20)

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:

1 + 4.3289 t^{3} + 0.7038 t^{6} - 0.6321 t^{9} + 0.3191 t^{12} - 0.0816 t^{15} + 0.0082 t^{18} = 0.0082 (t + 0.6240) (t + 1.1270) (t^{2} - 3.4710 t + 3.0650) (t^{2} - 2.7442 t + 2.3009) \times (t^{2} - 1.1270 t + 1.2701) (t^{2} - 0.6240 t + 0.3894) (t^{2} + 0.2520 t + 2.3008) \times (t^{2} + 1.3368 t + 3.0650) (t^{2} + 2.1342 t + 3.0650) (t^{2} + 2.4922 t + 2.3008)

(21)

Then, integrating Eq. (20) yields

χ = I (η) + const = - 308.6890 t^{4} - 0.0165 t^{3} + 0.0206 t^{2} - 14757.1585 t - 0.9977 δ_{1} + (- 0.0439 δ_{1} - 0.0006) \ln (t + 0.6240) + (- 0.5069 δ_{1} - 0.6813) \ln (t + 1.1270) + (- 0.5761 δ_{1} - 4609.8667) \ln (t^{2} - 3.4710 t + 3.0650) + (0.4345 δ_{1} - 253.7097) \ln (t^{2} - 2.7442 t + 2.3009) + (0.2535 δ_{1} + 0.3407) \ln (t^{2} - 1.1270 t + 1.2701) + (0.0219 δ_{1} + 0.0003) \ln (t^{2} - 0.6240 t + 0.3894) + (- 0.6832 δ_{1} + 470.8214) \ln (t^{2} + 0.2520 t + 2.3008) + (- 0.1932 δ_{1} + 4498.4746) \ln (t^{2} + 1.3368 t + 3.0650) + (0.7693 δ_{1} + 111.4439) \ln (t^{2} + 2.1342 t + 3.0650) + (0.2487 δ_{1} - 217.1490) \ln (t^{2} + 2.4922 t + 2.3008) + (1.5535 δ_{1} + 5452.7006) {tan}^{- 1} (0.6180 t + 0.4131) + (- 0.2146 δ_{1} + 42.0373) {tan}^{- 1} (0.6615 t + 0.0833) + (0.4420 δ_{1} + 10517.7373) {tan}^{- 1} (0.7205 t + 0.7688) + (- 0.8779 δ_{1} - 1.1796) {tan}^{- 1} (1.0246 t - 0.5774) + (- 1.2906 δ_{1} + 836.5001) {tan}^{- 1} (1.1562 t + 1.4408) + (1.0761 δ_{1} - 794.6938) {tan}^{- 1} (1.5463 t - 2.1216) + (- 0.0760 δ_{1} - 0.0010) {tan}^{- 1} (1.8505 t - 0.5774) + (1.1106 δ_{1} - 5058.7196) {tan}^{- 1} (4.3421 t - 7.5357) + const

(22)

Eq. (22) provides the dimensionless distance along the channel length

χ

(which is a function of

x

) for any given

t

(which is a function of

y

). 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 (

δ_{2} = 0

) with

z = 1.5

(

z_{*} = 2 \sqrt 3.25

) and

δ_{1} = 0.0110

(

η_{c} = y_{c} / B = 0.2

,

y_{c}

= critical depth), possible profiles (

H_{2}

and

H_{3}

) are calculated and shown in Fig. 1. The

H_{2}

and

H_{3}

profiles are obtained by applying

η = η_{c} = 0.2

as a boundary condition (at

χ = 0

) into Eq. (22) as

χ = I (η) - I (0.2)

(23)

Fig. 1. Possible profiles in a horizontal trapezoidal channel ( $η_{c} = 0.2$ )

The dimensionless distance along the channel length,

χ

, is considered positive in the downstream direction and

η_{c} = 0.2

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

y_{1}

and

y_{2}

that are far from the critical depth. By measuring these depths and the distance between them,

L

, the variable

δ_{1} = Q^{2} / (g B^{5})

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 (

n = 0.015

) and has a bottom width of 4 m and side slopes

z = 1.5

. 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 Eq. (22) to the profile, then

χ_{L} = I (η_{2}) - I (η_{1})

(24)

where

η_{1} = y_{1} / B = 0.5

;

η_{2} = y_{2} / B = 0.4875

; and

χ_{L} = {Ln}^{2} Q^{2} / B^{19 / 3} = δ_{1} (g {Ln}^{2} / B^{4 / 3})

. Thus

δ_{1} \times \frac{g {Ln}^{2}}{B^{4 / 3}} = I (0.4875) - I (0.5)

(25)

Applying Eqs. (22) - (25) yields

0.0209 δ_{1} = - 0.0075 δ_{1} + 0.0019

(26)

Thus,

δ_{1} = 0.0676

and

Q = \sqrt (g B^{5} δ_{1}) = 26.06 m^{3} / s

.

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:

M_{1}

,

M_{2}

, and

M_{3}

. Similarly, in a channel with steep slope, there are three profiles:

S_{1}

,

S_{2}

, and

S_{3}

. The slope of channel, which carries a given discharge as a uniform flow at the critical flow depth, is called the critical slope

S_{c}

(Abdulrahman 2010). The critical value of

δ_{2}

(i.e.,

δ_{2 c}

) for a trapezoidal channel can be obtained by setting the denominator of the integrand of Eq. (3) to zero for the critical depth,

y_{c}

, as follows:

δ_{2 c} = \frac{(1 + z_{*} η_{c})^{4 / 3}}{η_{c}^{10 / 3} (1 + z η_{c})^{10 / 3}}

(27)

where

η_{c} = y_{c} / B

. The proposed semianalytical procedure can be used to determine the dimensionless profiles for a given boundary condition. For this, the expressions for the

δ_{1}

and

δ_{2}

are written as follows:

δ_{1} = \frac{η_{c}^{3} (1 + z η_{c})^{3}}{1 + 2 z η_{c}}

(28)

δ_{2} = \frac{(1 + z_{*} η_{0})^{4 / 3}}{η_{0}^{10 / 3} (1 + z η_{0})^{10 / 3}}

(29)

where

y_{0}

= normal flow depth; and

η_{0} = y_{0} / B

represents dimensionless normal depth. The profiles for subcritical (

η > η_{c}

) and supercritical flow (

η < η_{c}

) in mild-slope (

η_{0} > η_{c}

or

δ_{2} < δ_{2 c}

) and steep-slope (

η_{0} < η_{c}

or

δ_{2} > δ_{2 c}

) channels are presented in the following examples.

Dimensionless Profiles in a Mild-Slope Channel

Consider a trapezoidal channel with side slopes

z = 1

, the dimensionless critical flow depth

η_{c} = 0.2

, and the dimensionless normal flow depth

η_{0} = 0.3

. Using Eqs. (28) - (29) yields

δ_{1} = 0.0099

and

δ_{2} = 52.3463

, respectively. Also, using Eq. (27),

δ_{2 c} = 211.6258

. As noted, the slope of the channel is mild (

η_{0} > η_{c}

or

δ_{2} < δ_{2 c}

). Using Table 1, Eq. (14) reduces to

χ = \int {\frac{3 t^{12} (1 + t^{3})^{3} - 0.0297 t^{3} (1 + 2 t^{3})}{(52.3463 t^{10} (1 + t^{3})^{3} - 1 - 3.4480 t^{3} - 0.6143 t^{6} + 0.4906 t^{9} - 0.2359 t^{12} + 0.0595 t^{15} - 0.0060 t^{18})}} d t + const

(30)

The denominator of the integrand can be factorized as follows:

52.3463 t^{10} (1 + t^{3})^{3} - 1 - 3.4480 t^{3} - 0.6143 t^{6} + 0.4906 t^{9} - 0.2359 t^{12} + 0.0595 t^{15} - 0.0060 t^{18} = 52.3463 (t - 0.6694) (t + 0.6245) (t + 1.0623) (t^{2} + 1.9816 t + 0.9919) (t^{2} + 1.0940 t + 0.4648) (t^{2} + 0.4266 t + 0.4522) (t^{2} - 0.5101 t + 0.4406) (t^{2} - 0.7223 t + 0.7879) (t^{2} - 1.0057 t + 1.1222) (t^{2} - 1.1237 t + 0.4867) (t^{2} - 1.1579 t + 1.0883)

(31)

Substituting Eq. (31) into Eq. (30) and integrating Eq. (30) yields

1, 000 χ = 19.1035 t^{3} + 3.2845 \times 10^{- 3} t^{2} + 0.7459 \times 10^{- 6} t + 1.2033 \ln | t - 0.6694 | - 1.4059 \ln (t + 0.6245) - 1.8263 \ln (t + 1.0623) - 0.2379 \ln (t^{2} - 1.1579 t + 1.0883) - 0.2250 \ln (t^{2} - 1.1237 t + 0.4867) - 1.2140 \ln (t^{2} - 1.0057 t + 1.1222) + 1.7013 \ln (t^{2} - 0.7223 t + 0.7879) - 1.8594 \ln (t^{2} - 0.5101 t + 0.4406) + 1.3694 \ln (t^{2} + 0.4266 t + 0.4522) + 1.2786 \ln (t^{2} + 1.0940 t + 0.4648) + 0.2015 \ln (t^{2} + 1.9816 t + 0.9919) - 1.1679 {tan}^{- 1} (1.0725 t - 0.5393) - 1.3991 {tan}^{- 1} (1.1523 t - 0.6671) + 7.3041 {tan}^{- 1} (1.2333 t - 0.4454) + 2.6880 {tan}^{- 1} (1.5681 t + 0.3344) - 2.9314 {tan}^{- 1} (1.6318 t - 0.4162) - 3.2761 {tan}^{- 1} (2.4183 t - 1.3587) - 3.6417 {tan}^{- 1} (2.4575 t + 1.3443) - 5.5126 {tan}^{- 1} (9.8827 t + 9.7918) + const

(32)

For this mild-slope channel, possible profiles are calculated and shown in Fig. 2. The

M_{1}

profile is obtained using

η = 1.2 η_{0}

as a boundary condition (at

χ = 0

). The

M_{2}

and

M_{3}

profiles are obtained using

η = (1 \pm 0.01) η_{c}

as the boundary condition (at

χ = 0

). 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.

Fig. 2. Possible profiles in a mild-slope trapezoidal channel ( $η_{c} = 0.2$ and $η_{0} = 0.3$ )

Dimensionless Profiles in a Steep-Slope Channel ( $δ_{2} > δ_{2 c}$ )

In this case, consider a trapezoidal channel with side slopes

z = 1

, the dimensionless critical flow depth

η_{c} = 0.2

, and the dimensionless normal flow depth

η_{0} = 0.15

. Using these values results in

δ_{1} = 0.0099

,

δ_{2} = 560.8203

, and

δ_{2 c} = 211.6258

. As seen, the slope of the channel is steep (

η_{0} < η_{c}

or

δ_{2} > δ_{2 c}

). Using Table 1, Eq. (14) reduces to

χ = \int {\frac{3 t^{12} (1 + t^{3})^{3} - 0.0297 t^{3} (1 + 2 t^{3})}{(560.8203 t^{10} (1 + t^{3})^{3} - 1 - 3.4480 t^{3} - 0.6143 t^{6} + 0.4906 t^{9} - 0.2359 t^{12} + 0.0595 t^{15} - 0.0060 t^{18})}} d t + const

(33)

The denominator of the integrand of Eq. (33) can be factorized as follows:

560.8203 t^{10} (1 + t^{3})^{3} - 1 - 3.4480 t^{3} - 0.6143 t^{6} + 0.4906 t^{9} - 0.2359 t^{12} + 0.0595 t^{15} - 0.0060 t^{18} = 560.8203 (t - 0.5313) (t + 0.5213) (t + 1.0342) (t^{2} + 1.9707 t + 0.9727) (t^{2} + 0.8650 t + 0.2857) (t^{2} + 0.3307 t + 0.2833) (t^{2} - 0.3411 t + 0.2782) (t^{2} - 0.8594 t + 0.2859) (t^{2} - 0.9176 t + 0.9114) (t^{2} - 0.9955 t + 1.0602) (t^{2} - 1.0768 t + 1.0289)

(34)

Substituting Eq. (34) into Eq. (33) and integrating Eq. (33) leads to

1, 000 χ = 1.7831 t^{3} + 2.8614 \times 10^{- 5} t^{2} + 2.3487 \times 10^{- 9} t - 0.1196 \ln | t - 0.5313 | - 0.3366 \ln (t + 0.5213) - 0.4573 \ln (t + 1.0342) + 0.3323 \ln (t^{2} - 1.0768 t + 1.0289) - 0.1287 \ln (t^{2} - 0.9955 t + 1.0602) - 0.1953 \ln (t^{2} - 0.9176 t + 0.9114) + 0.1467 \ln (t^{2} - 0.8594 t + 0.2859) - 0.1690 \ln (t^{2} - 0.3411 t + 0.2782) + 0.0227 \ln (t^{2} + 0.3307 t + 0.2833) + 0.2278 \ln (t^{2} + 0.8650 t + 0.2857) + 0.2203 \ln (t^{2} + 1.9707 t + 0.9727) - 0.7276 {tan}^{- 1} (1.1095 t - 0.5522) + 0.1709 {tan}^{- 1} (1.1632 t - 0.6263) + 0.5877 {tan}^{- 1} (1.1945 t - 0.5480) + 0.4918 {tan}^{- 1} (1.9765 t + 0.3268) - 0.4351 {tan}^{- 1} (2.0035 t - 0.3417) + 0.2042 {tan}^{- 1} (3.1429 t - 1.3506) - 0.3543 {tan}^{- 1} (3.1846 t + 1.3773) - 0.8159 {tan}^{- 1} (23.7729 t + 23.4246) + const

(35)

For this steep-slope channel, possible profiles are calculated and shown in Fig. 3. The

S_{1}

profile is obtained using

η = 2 η_{c}

as a boundary condition (at

χ = 0

). The

S_{2}

and

S_{3}

profiles are obtained using

η = (1 \pm 0.01) η_{0}

as the boundary condition (at

χ = 0

).

Fig. 3. Possible profiles in a steep-slope trapezoidal channel ( $η_{c} = 0.2$ and $η_{0} = 0.15$ )

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

30 m^{3} / s

and has a bottom width of 10 m, equal side slopes

z = 2

, and

α = 1

. 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 (

y_{1} = 3 m

) 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,

y_{b}

, at the control structure and proceed in the upstream direction. By considering the location at the control structure as

x_{b} = 0

, the computed values of

x

will be negative.

Using the boundary condition

y (x_{b}) = y_{b}

, prescribed water depth, the direct step method will be (Chow 1959; Subramanya 1986)

x_{2} = x_{b} + \frac{E_{2} - E_{b}}{S_{0} - \frac{1}{2} (S_{f 2} + S_{f b})}

(36)

in which

E = y + \frac{Q^{2}}{2 g A^{2}}

(37)

S_{f} = \frac{Q^{2} n^{2} P^{4 / 3}}{A^{10 / 3}}

(38)

where

E

is specific energy; and

(S_{f 2} + S_{f b}) / 2

is mean friction slope. Thus, by knowing a specified flow depth,

y_{2}

, the location of section 2,

x_{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.

Used in this example is decrement of flow depth, that is,

(3 - 1.2) / J m

, where

J

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

J = 500

(number of segments) as 2,137.91 m.

Fig. 4. Total length of backwater curve versus number of segments ( $J$ )

Solution: Proposed Method

Using Eqs. (6) - (7) yields

δ_{1} = 0.0009

and

δ_{2} = 1, 221.3349

, respectively. The denominator of the integrand of Eq. (18) can be factorized as follows:

1, 221.3349 t^{10} (1 + 2 t^{3})^{3} - 1 - 5.3292 t^{3} - 0.8718 t^{6} + 0.8785 t^{9} - 0.4885 t^{12} + 0.1369 t^{15} - 0.0150 t^{18} = 9, 770.679 (t + 0.842) (t + 0.485) (t - 0.485) (t^{2} + 1.550 t + 0.604) (t^{2} + 0.786 t + 0.241) (t^{2} + 0.309 t + 0.237) (t^{2} - 0.319 t + 0.246) (t^{2} - 0.670 t + 0.529) (t^{2} - 0.787 t + 0.698) (t^{2} - 0.811 t + 0.248) (t^{2} - 0.901 t + 0.664)

(39)

The proposed solution is

1, 000 χ = f (t) = 0.8188 t^{3} + 1.8855 \times 10^{- 6} t^{2} + 25.7239 \times 10^{- 12} t + 0.0136 \ln (t - 0.4847) - 0.0416 \ln (t + 0.4850) - 0.0428 \ln (t + 0.8421) + 0.0080 \ln (t^{2} - 0.9013 t + 0.6641) + 0.0037 \ln (t^{2} - 0.8111 t + 0.2482) - 0.0233 \ln (t^{2} - 0.7871 t + 0.6982) + 0.0207 \ln (t^{2} - 0.6696 t + 0.5295) - 0.0358 \ln (t^{2} - 0.3193 t + 0.2458) - 0.0203 \ln (t^{2} + 0.3094 t + 0.2366) + 0.0254 \ln (t^{2} + 0.7864 t + 0.2414) + 0.0164 \ln (t^{2} + 1.5501 t + 0.6043) - 0.0125 {tan}^{- 1} (1.3566 t - 0.5339) - 0.0388 {tan}^{- 1} (1.4727 t - 0.6636) + 0.0692 {tan}^{- 1} (1.5478 t - 0.5182) - 0.0093 {tan}^{- 1} (2.1306 t - 0.3402) + 0.0554 {tan}^{- 1} (2.1686 t + 0.3354) - 0.0648 {tan}^{- 1} (3.3940 t + 1.3346) - 0.0365 {tan}^{- 1} (3.4558 t - 1.4014) - 0.1040 {tan}^{- 1} (16.5871 t + 12.8559)

(40)

Applying boundary values yields

0 - 1, 000 χ = f ({0.3}^{1 / 3}) - f ({0.12}^{1 / 3}) = 0.022972725 + 0.152065959 - 1000 \frac{x n^{2} Q^{2}}{B^{19 / 3}} = 0.175038685 x = - 2, 137.81 m

(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 (

B \neq 0

and

z \neq 0

) 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:

$A$: cross-sectional area of flow ( $m^{2}$ );
$B$: bottom width of the channel (m);
$d y / d x$: slope of the free surface at any location;
$E$: specific energy (m);
$g$: gravitational acceleration ( $m / s^{2}$ );
$L$: length of a channel reach (m);
$n$: Manning roughness coefficient ( $m^{- 1 / 3} \cdot s$ );
$P$: wetted perimeter (m);
$Q$: water discharge ( $m^{3} / s$ );
$S_{0}$: longitudinal slope of the channel bottom;
$S_{f}$: friction slope;
$T$: width of water surface (m);
$t$: dimensionless parameter [ $= η^{1 / 3}$ ];
$x$: distance along the channel, considered positive in the downstream direction (m);
$y$: depth of flow (m);
$y_{0}$: normal depth of flow (m);
$y_{c}$: critical depth of flow (m);
$z$ , $z_{1}$ and $z_{2}$: side slope of the channel;
$z_{*}$: dimensionless parameter [ $= 2 \sqrt (1 + z^{2})$ or $\sqrt (1 + z_{1}^{2}) + \sqrt (1 + z_{2}^{2})$ ];
$α$: velocity correction factor;
$δ_{1}$: dimensionless discharge ( $Q^{2} / g B^{5}$ );
$δ_{2}$: dimensionless function of channel geometric and flow parameters ( $S_{0} B^{16 / 3} / (n^{2} Q^{2})$ );
$δ_{2 c}$: critical value of $δ_{2}$ ;
$η$: dimensionless depth of flow ( $y / B$ );
$η_{0}$: dimensionless critical depth of flow ( $y_{0} / B$ );
$η_{1}$: dimensionless depth of flow $y_{1} (y_{1} / B)$ ;
$η_{2}$: dimensionless depth of flow $y_{2} (y_{2} / B)$ ;
$η_{c}$: dimensionless critical depth of flow ( $y_{c} / B$ ); and
$χ$: dimensionless distance along the channel ( $= x n^{2} Q^{2} / B^{19 / 3}$ ).

Acknowledgments

The author would like to thank four anonymous reviewers for providing valuable comments on this work.

References

Abdulrahman, A. (2010). “Limit slope in uniform flow computations: Direct solution for rectangular channels.” J. Hydrol. Eng., 15(10), 808–812.

Crossref

Google Scholar

Bakhmeteff, B. A. (1932). Hydraulics of open channels, McGraw-Hill, New York.

Google Scholar

Bresse, J. A. Ch. (1868). Cours de mécanique appliquée, 2e partie, hydraulique, Mallet-Bachelier, Paris.

Google Scholar

Chaudhry, M. (2006). Open-channel flow, Springer, New York.

Google Scholar

Chow, V. T. (1955). “Integrating the equation of gradually varied flow.” Proceedings Paper 838, ASCE, New York, 1–32.

Google Scholar

Chow, V. T. (1959). Open-channel hydraulics, McGraw-Hill, New York.

Google Scholar

Das, A. (2007). “Flooding probability constrained optimal design of trapezoidal channels.” J. Irrig. Drain Eng., 133(1), 53–60.

Crossref

Google Scholar

Dubin, J. R. (1999). “On gradually varied flow profiles in rectangular open channels.” J Hydraul. Res., 37(1), 99–106.

Crossref

Google Scholar

Gill, M. A. (1976). “Exact solution of gradually varied flow.” J. Hydr. Div., 102(9), 1353–1364.

Google Scholar

Gunder, D. F. (1943). “Profile curves for open-channel flow.” Trans. Am. Soc. Civ. Eng., 108, 481–488.

Google Scholar

Haltas, I., and Kavvas, M. L. (2009). “Modeling the kinematic wave parameters with regression methods.” J. Hydrol. Eng., 14(10), 1049–1058.

Crossref

Google Scholar

Kumar, A. (1978). “Integral solutions of the gradually varied equation for rectangular and triangular channels.” Proc., Inst. Civ. Eng., 65(3), 509–515.

Crossref

Google Scholar

Ramamurthy, A. S., Saghravani, S. F., and Balachandar, R. (2000). “A direct integration method for computation of gradually varied flow profiles.” Can. J. Civ. Eng., 27(6), 1300–1305.

Crossref

Google Scholar

Subramanya, K. (1986). Flow in open channels, Tata McGraw-Hill, New Delhi, India.

Google Scholar

Vatankhah, A. R. (2010a). “Analytical integration of the equation of gradually varied flow for triangular channels.” Flow Meas. Instrum., 21(4), 546–549.

Crossref

Google Scholar

Vatankhah, A. R. (2010b). “Exact sensitivity equation for one-dimensional steady-state shallow water flow (Application to model calibration).” J. Hydrol. Eng., 15(11), 939–945.

Crossref

Google Scholar

Vatankhah, A. R. (2011). “Direct integration of gradually varied flow equation in parabolic channels.” Flow Meas. Instrum., 22(3), 235–241.

Crossref

Google Scholar

Vatankhah, A. R., and Easa, S. M. (2011). “Direct integration of Manning-based gradually varied flow equation.” Proc., ICE, J. Water Manage., 164(5), 257–264.

Crossref

Google Scholar

Venutelli, M. (2004). “Direct integration of the equation of gradually varied flow.” J. Irrig. Drain Eng., 130(1), 88–91.

Crossref

Google Scholar

Wong, T. S., and Zhou, M. C. (2006). “Kinematic wave parameters for trapezoidal and rectangular channels.” J. Hydrol. Eng., 11(2), 173–183.

Crossref

Google Scholar

Information & Authors

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Published In

Journal of Hydrologic Engineering

Volume 17 • Issue 3 • March 2012

Pages: 455 - 462

Copyright

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Received: Oct 19, 2010

Accepted: Jun 10, 2011

Published online: Jun 14, 2011

Published in print: Mar 1, 2012

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Ali R. Vatankhah [email protected]

Assistant Professor, Dept. of Irrigation and Reclamation Engineering, Univ. College of Agriculture and Natural Resources, Univ. of Tehran, P.O. Box 4111, Karaj, 31587-77871, Iran. E-mail: [email protected]

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Abstract

Introduction

Governing Equation

Proposed Direct Solution

Special Case: δ2=0

Practical Applications

Direct Solution for Possible Profiles in a Horizontal Trapezoidal Channel

Computation of Channel Discharge in a Horizontal Trapezoidal Channel

Water Surface Profiles in a Trapezoidal Channel

Dimensionless Profiles in a Mild-Slope Channel

Dimensionless Profiles in a Steep-Slope Channel (δ2>δ2c)

Comparison between the Proposed Method and the Direct Step Method

Solution: Direct Step Method

Solution: Proposed Method

Conclusions

Notation

Acknowledgments

References

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Special Case: $δ_{2} = 0$

Dimensionless Profiles in a Steep-Slope Channel ( $δ_{2} > δ_{2 c}$ )