# CR-07-1:Chapter6

## Conclusions

This report presents an extensive study of the bed load and suspended load for the case of current and wave interaction. The unified formulation was developed for clastic material in sand range or coarser, and the formulas transition smoothly among differing degrees of waves and currents and between breaking and nonbreaking waves.

A formula for bed-load sediment transport was developed and presented that includes interaction between waves and current. This formula is based on the assumption that the sediment transport is proportional to the total Shields parameter to the power 3/2. For purely oscillatory ﬂows, the mean Shields parameter for each half period (for *u* > 0 and for *u* 0) is computed to take into account the effect of wave asymmetry. The new formula provides satisfactory agreement with the extensive data set that was compiled, and the best agreement compared to other formulas previously proposed.

The effect of the critical Shields parameter (θ_{cr}) was examined, and an exponential function of the ratio θ_{cr}/θ_{cw} was proposed (θ_{cw} is the maximum Shields parameter for the specific flow situation). This relationship signiﬁcantly improves agreement with data for both steady current and oscillatory flow (wave) cases.

The net sediment transport by waves produced a transport coefficient that was smaller than expected. A coefficient value of *a* = 6 was found (Equation 90), although it reached *a* = 12 for a steady current (Equation 81). This value for the waves may be due to difficulties in estimating the bed roughness and to the influence of phase lag between fluid and sediment. Thus, it seems that phase-lag effects might be present even for small wave orbital velocities and coarser sediment, which introduce a weaker net sediment transport over a wave period.

Some discrepancy remains because the total shear stress is unknown for many of the experiments. For the experiments with oscillatory ﬂows, the total shear stress must be estimated based on theoretical values. Two calculation approaches were presented, either using the Wilson (1966) formula to compute the Nikuradse roughness or using the Nikuradse skin roughness (even if it is known that the roughness increases strongly if sheet ﬂow occurs). Depending on the data set, one or the other of these formulas gives the best agreement. This result emphasizes how important it is to accurately estimate the bottom shear stress if sheet ﬂow occurs to predict the sediment transport rate accurately.

Because the bed-load formula does not take into account the effect of phase lag, adding a coefficient quantifying this effect should increase its accuracy. The phase-lag phenomenon is the main nonsteady effect due to oscillatory ﬂows: a quantity of sand can still remain mobilized in the bed layer after each half-cycle of the wave velocity profile, and hence, move in the other direction. Dibajnia and Watanabe (1992) introduced a semi-empirical formula that allows estimation of phase-lag effects. Dohmen-Janssen (1999) and Camenen and Larroudé (2003) also proposed some semi-empirical coefficients to estimate the phase lag.

Regarding suspended load, a study of sediment concentration profiles using a large data set showed that an exponential profile overall gives a correct prediction of the profile shape. Assuming that the time-averaged current velocity is constant over the depth, the resulting sediment transport may be estimated from a simple equation. The two main parameters in the concentration profile are the mean sediment diffusivity over the depth and the bottom reference concentration. Comparison with measured velocity and concentration profiles over the depth showed that the assumption of a constant velocity over the depth does not significantly affect the results for most situations, especially for a steady current. If complex ﬂows occur, such as undertow in the surf zone, the results are, in general, more scattered, but still good.

A prediction equation for the sediment diffusivity was proposed assuming a linear combination of the mixing generated by breaking waves and the mixing by energy dissipation in the bottom boundary layer due to the mean current and waves. For the dissipation in the boundary layer, the dissipation by the current/waves was expressed as the product between a force (bottom shear stress) and a velocity (shear velocity) in order to be coherent with the classical mixing length approach (where = /6*u*_{*}*h*; Rouse 1938; Dally and Dean 1984). An estimation of the Schmidt number was proposed for the current data and wave data separately. For the mixing due to breaking waves, an efficiency coefficient was introduced characterizing the energy dissipation due to breaking waves, and its value was determined through calibration against experimental data.

Following the approach by Madsen (1993), the reference concentration was found to be proportional to the mean Shields parameter including the effect of the critical Shields parameter. The results showed scatter mainly because of uncertainty in the prediction of the total Shields parameter including the effects of the bed forms. The formulas proposed by Van Rijn (1989) were adopted to estimate the ripple geometry and the Nikuradse roughness if no measurements were available.

Furthermore, as Van Rijn (1993) noted, the dimensionless grain size *d*_{*} should be taken into account in calculation of the reference concentration. It was introduced in the formula as a reduction factor for larger grain size. A study of sediment transport by breaking waves on a sloping beach showed that plunging breaking waves may increase the reference concentration. This effect was taken into account using an empirically derived formula based on the Irribaren parameter.

If ripples are present, a strong phase lag in the suspended load may be observed (Van der Werf and Ribberink 2004). One of the main parameters controlling this phase lag appears to be the ratio between the ripple height and the median grain size. A similar approach as for describing the phase lag in bed-load transport for the sheet-flow regime was proposed, which overall yielded good results.

The resulting formula for the suspended load appears to be robust and effective. It gives the best results among the studied formulas for most data sets. Also, because this formula is physically based, any improvement in knowledge concerning sediment transport processes (for example, estimation of the total shear stress) could be taken into account and thus is expected to improve its predictive capability.

Chapter 5 - Unified Sediment Transport Formula for Coastal Inlet Applications | References |