Estimation of time-varying discharge and cumulative volume in individual overtopping waves

• Published in: Coastal engineering (Amsterdam) Vol. 117; pp. 191 - 204
• Main Authors: Hughes, Steven A., Thornton, Christopher I.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-11-2016
• Subjects: Cumulative overtopping volume; Dikes; Discharge; Empirical equations; Estimates; Individual overtopping wave parameters; Mathematical analysis; Mathematical models; Shape factor; Time series; Time-varying discharge; Wave overtopping; Weibull equation; Weibull equation; Cumulative overtopping volume; Individual overtopping wave parameters; Wave overtopping; Time-varying discharge
• ISSN: 0378-3839; 1872-7379
• DOI: 10.1016/j.coastaleng.2016.08.006

The time variation of discharge per unit dike length in an overtopping wave is characterized by a rapid increase to a maximum discharge that can be several times greater than the mean discharge, followed by a slower decrease in discharge until overtopping for that wave ceases. Measurements of wave overtopping acquired during the European small-scale FlowDike experiments were analyzed to identify individual overtopping waves using a two-step “supervised” procedure that combines the best features of automated wave determination augmented with manual error correction and validation. The result was a well-vetted data set of 5799 individual overtopping waves represented by time-series of flow depth and velocity near the seaward edge of the dike crest. The model dikes had planar seaward dike slopes of either 1V-on-3H or 1V-on-6H. Instantaneous discharge time series were calculated as the product of the flow thickness and velocity time series. In this paper, the two-parameter Weibull probability density function is adopted to represent the time variation of instantaneous discharge in an overtopping wave. Values of the Weibull scale factors, a, and shape factors, b, are obtained through nonlinear best-fitting of the Weibull equation to all 5799 waves. Best fits were also performed for the simpler Rayleigh version of the Weibull equation when b=2. An empirical equation was developed for scale factor, a, in terms of predicable parameters of the overtopping waves. The shape factor, b, could not be successfully parameterized, but it was found that the shape factors are narrowly distributed about the Rayleigh value of b=2. Predictions of time-varying discharge made using the Weibull equation with b=2 (i.e., Rayleigh equation) are assessed in terms of the root-mean-square errors between predictions and measurements. The estimates are reasonable for most of the waves. The capability to estimate the time-varying discharge in individual overtopping waves will improve the art of full-scale wave overtopping simulation, and the resulting empirical equations will contribute to methodologies aimed at quantifying the resiliency of dike erosion protection.
•A total of 5,799 high-quality time series of individual overtopping wave discharge are analyzed.•Best-fits of the two-parameter Weibull distribution were performed using time series of discharge and cumulative overtopping volume as targets.•An empirical relationship was established for the Weibull scale factor, a, in terms of predictable overtopping parameters.•The Weibull shape factor, b, was found to be narrowly distributed about the Rayleigh value of b = 2.•Comparisons of individual wave discharge time series to predictions made using the relationship in this paper were reasonable.•Application of the new relationships to a methodology aimed at parameterizing earthen levee grass resiliency is described.