Shear stress and sediment transport calculations for sheet flow under waves

A simple method is provided for calculating transport rates of not too fine (d50≥0.20 mm) sand under sheet flow conditions. The method consists of a Meyer-Peter-type transport formula operating on a time-varying Shields parameter, which accounts for both acceleration-asymmetry and boundary layer streaming. While velocity moment formulae, e.g.., <f><ovl type="bar" STYLE="S">Qs</ovl>=Constant×<ovl type="bar" STYLE="S">u∞3</ovl></f>, calibrated against U-tube measurements, fail spectacularly under some real waves (Ribberink, J.S., Dohmen-Janssen, C.M., Hanes, D.M., McLean, S.R., Vincent, C., 2000. Near-bed sand transport mechanisms under waves. Proc. 27th Int. Conf. Coastal Engineering, Sydney, ASCE, New York, pp. 3263–3276, Fig. 12), the new method predicts the real wave observations equally well. The reason that the velocity moment formulae fail under these waves is partly the presence of boundary layer streaming and partly the saw-tooth asymmetry, i.e., the front of the waves being steeper than the back. Waves with saw-tooth asymmetry may generate a net landward sediment transport even if <f><ovl type="bar" STYLE="S">u∞3</ovl>=0</f>, because of the more abrupt acceleration under the steep front. More abrupt accelerations are associated with thinner boundary layers and greater pressure gradients for a given velocity magnitude. The two “real wave effects” are incorporated in a model of the form Qs(t)=Qs[θ(t)] rather than Qs(t)=Qs[u∞(t)], i.e., by expressing the transport rate in terms of an instantaneous Shields parameter rather than in terms of the free stream velocity, and accounting for both streaming and accelerations in the θ(t) calculations. The instantaneous friction velocities u*(t) and subsequently θ(t) are calculated as follows. Firstly, a linear filter incorporating the grain roughness friction factor f2.5 and a phase angle ϕτ is applied to u∞(t). This delivers u*(t) which is used to calculate an instantaneous grain roughness Shields parameter θ2.5(t). Secondly, a constant bed shear stress is added which corresponds to the “streaming related bed shear stress” <f>−ρ(<ovl type="bar" STYLE="S">u˜w˜</ovl>)∞</f>. The method can be applied to any u∞(t) time series, but further experimental validation is recommended before application to conditions that differ strongly from the ones considered below. The method is not recommended for rippled beds or for sheet flow with typical prototype wave periods and d50<0.20 mm. In such scenarios, time lags related to vertical sediment movement become important, and these are not considered by the present model. [Copyright &y& Elsevier]