Computational models for dune time-lag: calculations using Stein's rule for dune height

Abstract

A linear dependence of dune height upon flow depth under equilibrium conditions (Yalin's rule) is abandoned as the basis for calculating the characteristics of dune populations under unsteady conditions, since it is apparently but a first approximation to a more complex dependence. In this new relationship, derived from the results of Stein (1965), dune height relative to flow depth becomes a non-linear function of the nondimensional bed shear stress, declining towards the upper and lower bounds of the dune existence field from an intermediate maximum. Population characteristics for three contrasted regimes of behaviour are accordingly obtainable from the model, depending on the hydraulic and dune parameters chosen. In regime I (relatively low shear stresses), dune wavelength and height phase differences are of the same comparatively small order and increase up from zero with increasing time ratio. Regime II (relatively high shear stresses) shows dune height to be initially π rad out of phase with discharge, whereas dune wavelength has an initial phase difference of zero. Regime III is defined by bed shear stresses which range about the value for which the relative dune height is a maximum. The wavelength phase difference is initially zero in this regime but the initial height phase difference is 3 π/4 rad, rising to a maximum at moderate time ratios, and doubly periodic behaviour is possible. Calculations from the model using Stein's rule afford results in semi-quantitative agreement with the character of the best-known dune population. This suggests that the model is significantly improved compared to earlier versions. Aside from its important effect on the dependence of the phase difference upon the time ratio, the use of this rule instead of Yalin's has generally no substantial influence on the characteristics or behaviour of the experimental dune populations, which are found to be similar to those previously calculated. The main differences seem to be that in Regime III the creation of dispersion is unrelated to changes in population size, while the dispersion itself can vary on as little as one-quarter of the period of the discharge.