Back to basics: On the proper determination of free-surface slope (FSS) in gradually varied open channel flows

Abstract

This study is a fundamental evaluation of the fluvial wave propagation in river reaches affected by hysteresis, one of the most complex open-channel topics, materialized in loops and lags among hydraulic variables. Hysteresis processes are still understudied as measurements in natural streams for the whole wave propagation duration are hardly available, while the data from existing gaging sites (almost exclusively relying on stage-discharge relationships) can deviate up to 65 % from the actual flows. A better understanding of hysteresis in general and its impact on streamflow monitoring in unsteady flows can be obtained if the free-surface slope (FSS) is determined and analyzed for its variation during wave propagation. Reliable FSS replication in such flows requires a robust understanding of the spatial-temporal sampling constraints. The study addresses the basic, but still weakly resolved, issue of tracing the FSS for waves of different magnitudes and durations. We do so by translating theoretical concepts on oscillatory waves to fluvial counterparts and observing rules for sampling continuous-time signals with discrete-time measurements. The conceptual understanding is verified with numerical simulations and experimental data represented in Eulerian and Lagrangian observation frameworks. We demonstrate that sampling stream stages with spatial and temporal resolutions (expressed in terms of fractions of the wavelength, dx_i/λ_R, and duration, ${\Delta T}_i/T_R$ for the flood wave to reach its peak) between approximately${{0.0075\le dx}_i/\lambda }_R\le 0.01$ and, 0.004 $\le {\Delta T}_i/T_R\le 0.06$, respectively, are required to properly trace FSS for subsequent usage in experimental or numerical simulation contexts.