The nearbed sediment concentration and vertical sediment flux are important in sediment transport mechanics, but they are known much less than the horizontal sediment flux, especially for fine sediment in unsteady flows. A developed twophase model is applied to study the nearbed sediment concentration, vertical sediment flux, and the relevant total sediment amount for the velocityskewed oscillatory sheet flow transport. With the sediment concentration hindered fall velocity, the classical reference concentration formulas conducted by Engelund and Fredsoe (1976) and Zyserman and Fredsoe (1994) are utilized for the comparison with the twophase model and illustration of the phaselag and sediment size effects in nearbed sediment concentration and vertical sediment flux. The concentration and vertical flux predicted by the twophase model agree well with experimental data and are better than empirical formulas. Furthermore, the sediment size effect for pickup flux function over starved bed is shown to be quite different from that containing sufficient sediment in oscillatory flows.
The phenomenon of sediment entrainment and settlement is universal in river and coastal environment and extraordinarily important in the subaquatic geomorphology. The vertical sediment fluxes consist of pickup flux due to turbulence, settling flux due to gravity, and their summation denoting the variation rate of total sediment amount. Associated with the sediment concentration, they represent the vertical sediment exchange strength and are important for the study of sediment settling down to deposition and picked up into suspension. Most sediment models need a type of nearbed condition associated with vertical sediment flux for bottom boundary conditions. So the predictions of nearbed concentration, vertical sediment flux, pickup flux, and settling flux are very fundamental for such sediment models.
In fully developed steady flows, the pickup flux and settling flux are in balance and the total vertical flux is zero. Numerous steady flow formulas have been carried out to predict the vertical sediment flux over loose sediment bed in developed equilibrium steady flows. Generally, some are straight driven by the pickup function based on clear water flow over starved bed without considering the influence of suspended sediment [
The nearbed sediment concentration and vertical sediment fluxes in unsteady flows are much more complicated than that in steady flows due to the phaselag effect, and formulas in unsteady flows are lacking. So the nearbed concentrations denoting vertical flux strength in steady flows [
In unsteady flows, the vertical sediment flux can be derived from the variation rate of sediment amount in movement [
Present study utilizes a twophase model to study the nearbed sediment concentration and vertical fluxes in velocityskewed oscillatory sheet flows containing sufficient amount of sediment. The results mainly exhibit and explain the phaselag characteristics in concentration and vertical flux at the reference height, nearimmobile bed, and very low concentration area for difference sediment sizes.
Chen and Yu [
The twophase model includes Reynolds averaged continuity and momentum equations:
Equations (
At the tunnel top,
Free stream velocity and acceleration.
Nonslip condition is assumed at the bottom that
In the lateral boundaries, the horizontal gradients of
The sediment amount in movement is defined by Nielsen et al. [
The total vertical flux is
For
Reference
For
Computations are based on the O’Donoghue and Wright [
Computation conditions.
Test case 





Fine: FA5010  1.5  0.13  5.0  0.0126 
Medium: MA5010  1.5  0.27  5.0  0.0387 
Coarse: CA5010  1.5  0.46  5.0  0.0672 
The references
Reference
The EF76 and ZF94 are strictly in phase with
Rootmeansquare (RMS) errors at
Cases 

 

FA5010  MA5010  CA5010  FA5010  MA5010  CA5010  
Twophase model  0.051  0.039  0.042  0.037  0.038  0.053 
EF76  0.138  0.094  0.086  0.091  0.100  0.118 
ZF94  0.148  0.094  0.068  0.095  0.102  0.106 
Corresponding to the concentration due to phaselag, the experimental
Near the flow crest, there are hollows in EF76 and ZF94 that predicted
Against the settling down induced by gravity, the pickup effect induced by turbulence maintains the suspension of sediment, and the total vertical sediment flux (
Vertical sediment flux and immobile bed surface.
Corresponding to
For comparison without the effect of suspended hindered sediment, van Rijn [
Pickup function of van Rijn.
Sediment amount above
In developed oscillatory sheet flow, the averaged
The nearbed flux at the locations above or below the reference height is essential for better understanding of the nearbed sediment exchange. Figure
Computed vertical flux at
FA5010,
FA5010,
MA5010,
MA5010,
CA5010,
CA5010,
Also,
Sediment amount above
Above
Above
The present work studies the nearbed conditions in velocityskewed oscillatory sheet flows with a twophase model and classical empirical formulas. The phaselag and sediment size effects in concentration, vertical sediment flux, and sediment amount are focused on the area near the initial bed, the immobile bed, and the low concentration area.
Due to the phaselag, the reference concentration and the settling flux never drop to zero in a period even at the flow reversal, and the relevant magnitudes between onshore and offshore are similar. The twophase model computed reference concentration and settling flux by twophase model agree with experiment very well and are better than empirical formulas. Under the same flows, reference concentration magnitudes decrease with the increment of sediment size due to the increment of fall velocity. While, for settling flux, as the product of concentration and fall velocity, the magnitude increases first and then decreases with the increment of sediment size.
The pickup flux and settling flux are in periodic averaged balance in oscillatory sheet flow, and their magnitudes are around each other with opposite direction. The pickup flux increases and reaches a local maximum value before flow peak and decreases to a local minimum value before flow reversal both in the onshore and in the offshore durations. Corresponding to the settling flux, the pickup flux also increases first and then decreases with the increment of sediment size.
The sediment amount magnitude increases with the decrement of sediment size. Phaselag effect increases the sediment amount magnitude’s ratio between onshore and offshore. As the sediment amount variation rate, the total vertical flux is much smaller than the pickup flux and varies almost following the pickup flux. The vertical sediment flux magnitude decreases from the initial bed toward immobile bed and top of tunnel.
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Science Foundation of China (Grants no. 51609244 and no. 11472156) and National ScienceTechnology Support Plan of China (Grant no. 2015BAD20B01).