Computation of gradually varied flow in compound open channel networks

Abstract

Although, natural channels are rarely rectangular or trapezoidal in cross section, these cross sections are assumed for the computation of steady, gradually varied flow in open channel networks. The accuracy of the computed results, therefore, becomes questionable due to differences in the hydraulic and geometric characteristics of the main channel and floodplains. To overcome these limitations, an algorithm is presented in this paper to compute steady, gradually varied flow in an open-channel network with compound cross sections. As compared to the presently available methods, the methodology is more general and suitable for application to compound and trapezoidal channel cross sections in series channels, tree-type or looped networks. In this method, the energy and continuity equations are solved for steady, gradually varied flow by the Newton–Raphson method and the proposed methodology is applied to tree-type and looped-channel networks. An algorithm is presented to determine multiple critical depths in a compound channel. Modifications in channel geometry are presented to avoid the occurrence of multiple critical depths. The occurrence of only one critical depth in a compound cross section with modified geometry is demonstrated for a tree-type channel network.

Appendix I. Partial derivatives of energy equation

1.1 Partial derivatives of energy equation

For each energy equation, there are four non-zero partial derivatives, namely the partial derivatives with respect to flow depth and with respect to the discharge at the section under consideration as well as partial derivatives with respect to the corresponding variables for the adjacent section. Thus for an energy equation, F _{i, k} , between section j and j+1 of channel i, the following non-zero partial derivatives are obtained.

$$ \frac{\partial F_{i,k} }{\partial y_{i,j} }=-1-\frac{\partial a_{1}}{\partial y_{i,j} }\frac{Q_{t_{i,j}^{2}} }{2g}+\frac{\Delta x_{i}}{2}\frac{\partial S_{F_{i,j}}}{\partial y_{i,j}} $$

(A1)

$$ \frac{\partial F_{i,k} }{\partial y_{i,j+1} }=1+\frac{\partial a_{2}}{\partial y_{i,j+1} }\frac{Q_{t_{i,j+1}^{2}}}{2g}+\frac{\Delta_{i}}{2}\frac{\partial S_{F_{i,j+1}}}{\partial y_{i,j+1} } $$

(A2)

$$ \frac{\partial F_{i,k} }{\partial Q_{t_{i,j} } }=-\frac{2a_{1} }{2g}Q_{t_{i,j}} +\frac{\Delta x_{i}}{2}\frac{\partial S_{F_{i,j}}}{\partial Q_{i,j} } $$

(A3)

$$ \frac{\partial F_{i,k} }{\partial Q_{t_{i,j+1} } }=-\frac{2a_{2} }{2g}Q_{t_{i,j+1}} +\frac{\Delta x_{i}}{2}\frac{\partial S_{F_{i,j+1}}}{\partial Q_{i,j+1} } $$

(A4)

$$\frac{\partial S_{F_{i,j}}}{\partial Q_{i,j} }=\frac{2Q_{t_{i,j}}}{K_{t_{i,j}}^{2}} $$

(A5)

$$ \frac{\partial S_{F_{i,j+1}}}{\partial Q_{i,j+1} }=\frac{2Q_{t_{i,j+1}}}{K_{t_{i,j+1}}^{2}}. $$

(A6)

The following partial derivates of the terms a and S _F with respect to y in Eqs. (A1) to (A4) are obtained by using the chain rule.

$$ \frac{da}{dy}=\frac{da}{dA_{f} }\frac{dA_{f} }{dy}+\frac{da}{dA_{m} }\frac{dA_{m} }{dy}+\frac{da}{dR_{f} }\frac{dR_{f} }{dy}+\frac{da}{dR_{m} }\frac{dR_{m} }{dy} $$

(A7)

$$ \frac{dS_{F} }{dy}=\frac{dS_{F} }{dA_{f} }\frac{dA_{f} }{dy}+\frac{dS_{F} }{dA_{m} }\frac{dA_{m} }{dy}+\frac{dS_{F} }{dR_{f} }\frac{dR_{f} }{dy}+\frac{dS_{F} }{dR_{m} }\frac{dR_{m} }{dy}. $$

(A8)

Expressions for individual terms in Eqs. (A7) and (A8) are presented below.

$$ \frac{dA_{f} }{dy}=B_{f} +S_{f} \left( {y-Z} \right) $$

(A9)

$$ \frac{dA_{m} }{dy}=B_{m} +2Zs_{m} $$

(A10)

$$ \frac{dR_{f} }{dy}=\frac{\sqrt {s_{f}^{2} +1} \left( {\frac{s_{f} \left({Z^{2}+y^{2}} \right)}{2}-s_{f} Zy} \right)+B_{f}^{2} +B_{f} s_{f} \left({y-Z} \right)}{P_{f^{2}} } $$

(A11)

$$ \frac{dR_{m} }{dy}=\frac{B_{m} +2Zs_{m} }{P_{m} } $$

(A12)

$$ \frac{dS_{F} }{dA_{f} }=-\frac{4{Q_{t}^{2}}R_{f}^{2 / 3}}{n_{m} \left( {2K_{f} +K_{m} } \right)^{3}} $$

(A13)

$$ \frac{dS_{F} }{dA_{m} }=-\frac{2{Q_{t}^{2}}R_{m}^{2 / 3}}{n_{m} \left({2K_{f} +K_{m}} \right)^{3}} $$

(A14)

$$ \frac{dS_{F} }{dR_{f} }=-\frac{8A_{f} {Q_{t}^{2}}}{3R_{f}^{1 / 3}n_{f} \left( {2K_{f} +K_{m} } \right)^{3}} $$

(A15)

$$ \frac{dS_{F} }{dR_{m} }=-\frac{4A_{m} {Q_{t}^{2}}}{3R_{m}^{1 / 3}n_{m} \left( {2K_{f} +K_{m} } \right)^{3}} $$

(A16)

$$ \frac{da}{dA_{f} }=-\frac{2R_{f}^{2/3}n_{m} \left({4A_{f} {R_{f}^{2}}n_{m} ^{3}+3A_{m} {R_{m}^{2}}{n_{f}^{3}}} \right)-2A_{m} {R_{f}^{2}}R_{m}^{2 / 3}n_{f} {n_{m}^{3}}}{\left( {2A_{f}{R}_{f}^{2/3}n_{m} +A_{m} R_{m}^{2 / 3}n_{f} } \right)^{4}} $$

(A17)

$$ \frac{da}{dA_{m} }=-\frac{2R_{m}^{2/3} n_{f} \left( {3A_{f} {R_{f}}^{2} n_m^{3} +A_{m} R_{m}^{2} n_{f}^{3}} \right)-2A_{f} R_{f}^{2/ 3} R_{m}^{2} n_{m} n_{f}^{3} }{\left( {2A_{f} R_{f}^{2 / 3} n_{m} +A_{m} R_{m}^{2/3} n_{f} } \right)^{4}} $$

(A18)

$$ \frac{da}{dR_{f} }=-\frac{4A_{f} A_{m} n_{f} n_{m} \left( {R_{f} R_{m}^{2 / 3} n_{m}^{2} -\frac{R_{m}^{2} n_{f}^{2} }{R_{f}^{1 / 3}}} \right)}{\left( {2A_{f} R_{f}^{2 / 3} n_{m} +A_{m} R_{m}^{2 / 3} n_{f} } \right)^{4}} $$

(A19)

$$ \frac{da}{dR_{m} }=-\frac{4A_{f} A_{m} n_{f} n_{m} \left( {R_{m} R_{f}^{2 / 3} n_{f}^{2} -\frac{R_{f}^{2} n_{m}^{2} }{R_{m}^{1 / 3}}} \right)}{\left( {2A_{f} R_{f}^{2 /3} n_{m} +A_{m} R_{m}^{2 / 3} n_{f} } \right)^{4}}. $$

(A20)

Numerical experimentation revealed that $2A_{m} R_{f}^{2} R_{m}^{2 / 3} n_{f} {n_{m}^{3}}$ can be neglected without compromising the accuracy of the computations.

1.2 Partial derivatives of continuity equation

Here, the subscript k refers to the equation number and its value is not identical to that of j. Likewise for the continuity equation F _i,k+1, the only non-zero partial derivatives are those with respect to the discharges of the adjacent sections, i.e.,

$$ \frac{\partial F_{i,k+1} }{\partial Q_{t_{i,j} } }=1; \frac{\partial F_{i,k+1} }{\partial Q_{t_{i,j+1} } }=-1. $$

(A21)

1.3 Partial derivatives of junction equations

The Partial derivatives at junction of two upstream channels and one downstream channel are given as following.

$$ \frac{\partial F_{J1,1} }{\partial Q_{t_{i+1,1} } }{}={}-1; \frac{\partial F_{J1,1} }{\partial Q_{t_{i+2,1} } }{}={}-1; \frac{\partial F_{J1,2} }{\partial Q_{t_{i+1,1} } }{}={}-1; \frac{\partial F_{J1,2} }{\partial y_{i,N_{i} +1} }{}={}-1; \frac{\partial F_{J1,3} }{\partial y_{i+2,1} }{}={}-1; \frac{\partial F_{J1,3} }{\partial y_{i,N_{i} +1} }{}={}-1. $$

(A22)

Similarly, partial derivatives at junction of one upstream channel and two downstream channel and junction of two series channels can be derived.

1.4 Partial derivatives of boundary conditions

The partial derivatives of boundary conditions for one upstream channel and two downstream channels may be written as

$$ {55} \frac{\partial F_{BC,1} }{\partial Q_{t_{i,1} } }=-1; \frac{\partial F_{BC,2} }{\partial y_{i+1,N_{i+1} +1} }=-1; \frac{\partial F_{BC,3} }{\partial y_{i+2,N_{i+2} +1} }=-1. $$

(A23)

Similarly, partial derivatives may be written for single inlet and single outlet channel or any other combination of boundary conditions.