Stochastic modeling and uncertainty investigation of unsteady open channel flows

Abstract

The flow conditions in practical open channel systems such as mountainous rivers can be very complex with uncertainties due to natural and artificial factors. Moreover, the increasing occurrence of extreme weather due to climate change and advanced human activities, leading to more uncertain events of rainfalls and droughts, which makes more difficulties in the predictions and analysis of flow process in such open channel systems. Numerous theoretical and experimental research works have been done in this field to study the physics of open channel flows (including theories, models and measurements), which were however focused usually on specific channel conditions rather than complex uncertainty situations. Therefore, it is necessary to further develop theory and model to capture such uncertainty characteristics and their influences in complex open channel flows. This research is conducted to better understand the stochastic features and uncertainty propagation in unsteady open channel flows, and to examine how the system parameters and flow conditions influence flow uncertainties in the open channel systems. To this end, a one-dimensional (1D) stochastic model is firstly developed in this research, consisting of zeroth-order base flow equations and first-order covariance equations. This stochastic model is derived by applying the perturbation method to the 1D Saint-Venant equations with lateral flows, so as to express the uncertainty propagation of open channel flow responses induced by different random factors (including channel width, bed slope, roughness, boundary inflow and lateral inflow). Several assumptions are taken to conduct the model developing, including: (a) rectangular and wide channel, (b) mild and uniform bed slope, (c) hydrostatic water pressure, (d) same friction resistance for initial steady flow, and (e) incompressible and homogeneous water with constant density and viscosity. Solution methods are illustrated in which the software EPA SWMM is employed for base flow computation and a combination of finite difference scheme and Gauss elimination for covariance computation. Based on this developed stochastic model, extensive numerical applications are then performed for systematic analysis of different factors affecting the uncertainty evolution in the open channel flow process. To demonstration, all the random factors are assumed to be exponentially correlated in both temporal and spatial domains. The results show that: (1) upstream inflow uncertainty has the most significant on flow variability and a linear positive relation is found; (2) the channel width uncertainty reduces the flow uncertainty growth but has little effect on the final flow uncertainty; (3) bed slope uncertainty slows down initially the flow uncertainty growth but increases greatly finally the uncertainty magnitude; (4) roughness uncertainty which is represented by the Manning's n weakens the wave variability during initial stage but increases flow uncertainty finally; and (5) lateral inflow decreases the system uncertainty response since it increases the base flow discharge, which is also found to affect upstream flow properties in subcritical flow. Finally, the effect of the combination of all these uncertainty factors is investigated for their significance rankings of influence on the unsteady open channel flows.