Study on Conveyance Coefficient Influenced by Momentum Exchange Under Steady and Unsteady Flows in Compound Open Channels

Abstract

Many natural compound channels with differential stages play a vital role during high flow events in real-time. When a flood occurs, and water flows into floodplains, the flow structure becomes more complex because of the momentum transfer between sub-segments of the compound channel, caused by the large difference of velocities in different sub-segments. The conventional methods of discharge calculation based on conveyance coefficients of a single channel do not consider momentum transfer, resulting in inaccurate prediction for compound channels. This paper uses a new method of determining conveyance coefficient in compound channels to be incorporated in the two-dimensional analytical solution of the Reynolds averaged Navier–stokes equations for stage-discharge and hydrographs prediction. The proposed conveyance model for flood routing is obtained by solving 1D unsteady flow equations. The flow calculation considers the interaction between sub-segments of compound channels using the momentum equation for shallow water. The proposed model was evaluated to show that incorporating the momentum flux improves the predicted maximum discharge and flow depth in the output hydrographs of the unsteady flow. This result suggests that the proposed method can effectively determine the conveyance coefficient of the compound channel in steady and unsteady flow prediction.

Appendix

Knight and Tang (2008) used the equation in Shiono and Knight method (1988, 1991) to calculate shear stress on the boundaries of sub-segments. Shiono and Knight (1988, 1991) gave the analytical solution of the RANS equation to calculate the lateral distribution of the depth-averaged velocity (${U}_{d}$) in a given depth in each part of the channel segment, shown in Fig. 11. 

Fig. 11

Trapezoidal section of a compound channel with two floodplains and seven plates

For the channel segment with fixed depth $H$:

$${U}_{d}={({A}_{1}{e}^{\gamma y}+{A}_{2}{e}^{-\gamma y}+\kappa )}^\frac{1}{2}$$

(21)

where $y$ is the lateral distance from the central line of the channel, and A₁ to A₂ are unknown coefficients related to the panels created over each channel zone with constant friction factor, eddy viscosity coefficient, and secondary parameter. The other constants in Eq. (21) are established as follows (Shiono and Knight 1988, 1991):

$$\kappa =\frac{8g{S}_{0}H}{f}\left(1-{\beta }_{0}\right)$$

(22)

$$\gamma =\sqrt{\frac{2}{\lambda } }{\left(\frac{f}{8}\right)}^\frac{1}{4} \frac{1}{H}$$

(23)

$${\beta }_{0}=\frac{\Gamma }{\rho gH{S}_{0}}$$

(24)

where $\lambda$ is dimensionless eddy viscosity coefficient, $H$ is the flow depth in the main channel. $\Gamma$ is the depth-averaged lateral gradient of the secondary flow force per unit length, and $f$ is the friction factor.

For a segment with a side slope (1:s) where the depth changes linearly (Shiono and Knight 1991):

$${U}_{d}={({A}_{3}{\varsigma }^{\alpha }+{A}_{4}{\varsigma }^{-\left(\alpha +1\right)}+\omega \varepsilon +\eta )}^\frac{1}{2}$$

(25)

where A₃ to A₄ are respectively integral constants related to the divided panels (e.g. (2), (3), (6,) and (7) in Fig. 11), $\varsigma$ is the local depth of the flow which equals to $(H-\frac{y-b}{s})$ for $y>0$ and $(H+\frac{y+b}{s})$ for $y<0$. Constants ${A}_{3}$ and ${A}_{4}$ can be obtained from specific boundary conditions related to the velocity gradient with respect to $y$, which is zero at the center of the main channel and increases toward the surface. The other constants in Eq. (25) are established as follows:

$$\alpha =-\frac{1}{2}+\frac{1}{2}\sqrt{1+\frac{s\sqrt{1+{s}^{2}}}{\lambda }\sqrt{8f}}$$

(26)

$$\omega =\frac{g{S}_{0}}{\frac{\sqrt{1+{s}^{2}}}{s}\left(\frac{f}{8}\right)-\frac{\lambda }{{s}^{2}}\sqrt{\frac{f}{8}} }$$

(27)

$$\eta =\frac{-\Gamma }{\rho \sqrt{1+\frac{1}{{s}^{2}}} \left(\frac{f}{8}\right)}$$

(28)

Since $f, \lambda ,$ and $\Gamma$ in each channel zone are calculated, Knight and Tang (2008) used $f$ values based on research laboratory data and $\lambda$ values for all segments equal to 0.07, which can be constant for overbank flow. Since $\Gamma$ plays a more significant role in overbank flows, they calculated $\Gamma$ for estimating the effects of secondary flow term in the main channel and floodplain, respectively, from the equations $\frac{\Gamma }{\rho gH{S}_{0}}=0.15$ and $\frac{\Gamma }{\rho g(H-h){S}_{0}}=-0.25$, where h is the channel depth of bank full (Abril and Knight 2004). Shiono and Knight (1988, 1991) proposed these equations based on the data from SERC-FCF (Flood channel facility of the University of Birmingham).