Ensemble modeling of stochastic unsteady open-channel flow in terms of its time–space evolutionary probability distribution – Part 2: numerical application

Abstract

Abstract. The characteristic form of the Saint-Venant equations is solved in a stochastic setting by using a newly proposed Fokker–Planck Equation (FPE) methodology. This methodology computes the ensemble behavior and variability of the unsteady flow in open channels by directly solving for the flow variables' time–space evolutionary probability distribution. The new methodology is tested on a stochastic unsteady open-channel flow problem, with an uncertainty arising from the channel's roughness coefficient. The computed statistical descriptions of the flow variables are compared to the results obtained through Monte Carlo (MC) simulations in order to evaluate the performance of the FPE methodology. The comparisons show that the proposed methodology can adequately predict the results of the considered stochastic flow problem, including the ensemble averages, variances, and probability density functions in time and space. Unlike the large number of simulations performed by the MC approach, only one simulation is required by the FPE methodology. Moreover, the total computational time of the FPE methodology is smaller than that of the MC approach, which could prove to be a particularly crucial advantage in systems with a large number of uncertain parameters. As such, the results obtained in this study indicate that the proposed FPE methodology is a powerful and time-efficient approach for predicting the ensemble average and variance behavior, in both space and time, for an open-channel flow process under an uncertain roughness coefficient.

Ensemble modeling of stochastic unsteady open-channel flow in terms of its time–space evolutionary probability distribution – Part 1: theoretical development

“…This provided the equations that may be solved numerically, through only one simulation, in order to determine the ensemble behavior and variability of a system described by the stochastic open-channel flow equations. The application and validation of this methodology, applied to an open-channel flow problem with an uncertain roughness coefficient, is provided in a companion paper by Dib and Kavvas (2018), in which the statistical results of the proposed FPE methodology are compared against the results obtained by the MC approach.…”

“…This proposed FPE methodology can also be expanded to problems with uncertainties in other channel or flow parameters. The performance of the proposed FPE methodology is evaluated in a companion paper by Dib and Kavvas (2018) which compares its results to those obtained by the MC approach.…”