Stage-discharge relationship developing for multi-stage compound channels based on 1D and 2D models

Background and objectives
Flood is a phenomenon during which the water fills the main channel of the river and spreads on to the floodplains. In some natural rivers as well as artificial canals in cities may be more than one floodplain flank the main channel, which is called multi-stage compound channels. In these channels, when the flood occurs and the main channel is filled, the first floodplain is activated, and then when this floodplain overflows, the second floodplain is activated immediately. One of the hydraulic aspects of these channels is the stage-discharge relationship, which is used to estimate the flow discharge for any given flow depth and hence is an important tool in the river design and management during floods. In this study, the one-dimensional model of interacting divided channel and two-dimensional model of Shiono and Knight, which were previously proposed to calculate the flow discharge in classic compound channels, are developed for multi-stage compound channels.

Materials and methods
For one-dimensional model, using Newton's second law and by considering into account the apparent shear stresses at the interface of the main channel and the first flood plain, as well as the interface between first and second floodplains, a linear equation system was derived to simultaneously estimate the average flow velocities in adjacent flow compartments of these sections. Huthoff et al. equation was used for the involvement of apparent shear stresses in the interfaces. In this method, the momentum exchange coefficients were calibrated based on the laboratory data from Singh (2021) in a three-stage rectangular compound channel and using the nonlinear generalized reduced gradient optimization algorithm. To derive the semi two-dimensional model of Shiono and Knight, by depth integrating of the Navier-Stokes equations, a differential equation was obtained in terms of shear stress. Then by applying several appropriate assumptions for Reynolds stresses and secondary flows, a simple equation was obtained in terms of depth-averaged velocity. This equation was solved numerically using finite difference method.

Results
The results of the one-dimensional flow interacting model showed that this method has a good efficiency in estimating the total flow discharges with an average and maximum error of about 2.6 and 5.7 percent, respectively. Meanwhile, the vertical divided channel method which is widely used in water engineering packages, does not provide reliable results with an average error of 9.1% and a maximum error of 17.4%. The results of the numerical solution of Shiono and Knight model showed that there is a good agreement between the observed and calculated lateral velocity distributions. It was also found that the effect of eddy viscosity and secondary currents in this channel is significant and should be considered. The mean and maximum error of this model in estimating the total flow discharges was 2.4% and 4.1%, respectively.

Conclusion
The results of both one- and two-dimensional models proposed in this research showed that these models have a suitable ability to estimate the total flow discharge as well as the subdivision flow discharges in multi-stage compound channels. Investigations showed that in terms of the intensity of turbulence and the strength of the secondary currents, the interface plane between the main channel and the first floodplain is much more affected than the interface between first and second floodplains.