Comparison of Average Energy Slope Estimation Formulas for One-dimensional Steady Gradually Varied Flow

Abstract

To find the steady flow water surface profile, it is possible to use Bernoulli’s equation, which is a discrete form of the differential energy equation. Such an approach requires the average energy slope between cross-sections to be estimated. In the literature, many methods are proposed for estimating the average energy slope in this case, such as the arithmetic mean, resulting in the standard step method, the harmonic mean and the geometric mean. Also hydraulic averaging by means of conveyance is commonly used. In this study, water surface profiles numerically computed using different formulas for expressing the average slope were compared with exact analytical solutions of the differential energy equation. Maximum relative and mean square errors between numerical and analytical solutions were used as measures of the quality of numerical models. Experiments showed that all methods gave solutions useful for practical engineering purposes. For every method, the numerical solution was very close to the analytical one. However, from the numerical viewpoint, the differences between the methods were significant, as the errors differed up to two orders of magnitude.

Patterns of Difference between Physical and 1-D Calibrated Effective Roughness Parameters in Mountain Rivers

“…First, the selection of the RFSM methodology does not influence most of the GLUE results except for cascades and step-pools at low flows (see Table 1). The cascade reach in this research could be considered similar to the rapid flow case in Artichowicz and Mikos-Studnicka [30]. However, in our study, the best RFSM for cascade at low flow was AFSE (Equation ( 6)), unlike the HMFSE (Equation ( 8)) obtained by Artichowicz and Mikos-Studnicka [30].…”

“…The cascade reach in this research could be considered similar to the rapid flow case in Artichowicz and Mikos-Studnicka [30]. However, in our study, the best RFSM for cascade at low flow was AFSE (Equation ( 6)), unlike the HMFSE (Equation ( 8)) obtained by Artichowicz and Mikos-Studnicka [30]. A possible explanation for this difference might be that the Artichowicz and Mikos-Studnicka [30] test is performed in a prismatic flume without bed material, while cascade has boulders and cobbles interacting with the flow.…”

“…According to this analysis, AFSE is the best and safest methodology to predict water levels. Artichowicz and Mikos-Studnicka [30] tested four theoretical cases in a prismatic channel (three tranquil flows and one rapid flow) using the solution of the differential energy equation as validation data. In this study, the analyzed RFSM included all the methodologies available in HEC-RAS and some additional methods available in the literature.…”

Impact of Energy Slope Averaging Methods on Numerical Solution of 1D Steady Gradually Varied Flow

“…When the harmonic mean is used, the resulting method is not 0-stable, and therefore not convergent. Artichowicz and Mikos-Studnicka (2014) showed that all the methods of averaging the energy slope give almost identical solutions when applied to the SGVF equation. The analyses presented in this paper show that the rule with arithmetic averaging seems to be the best choice for solving the SGVF equation.…”

“…Laurenson's (1986) analysis was based on an arbitrarily assumed slope friction function, which was not an analytical solution of the flow equation. Artichowicz and Mikos-Studnicka (2014) compared the numerical solutions of the one-dimensional energy flow equation with analytical solutions, and concluded that differences between the outcomes of formulas 2, 5, 6, 7 and 8 are insignificant from the practical viewpoint. None of the researchers presents detailed discussion of the origin of those methods or the consequences and formalism of their usage.…”