CR-07-1:Chapter3

Introduction

Accurate prediction of sediment transport rates is an essential element in morphological studies of coastal, marine, and river environments. Sediment transport mainly occurs in two modes: bed load and suspended load. The bed load is the part of the total load that is traveling directly above the bed and is supported by intergranular collisions rather than fluid turbulence (Wilson 1966). The suspended load, on the other hand, is the part of the load which is primarily supported by the fluid turbulence (Fredsøe and Deigaard 1994). Thus, bed load includes mainly sediment transport for coarse materials (saltation) or fine material on plane beds (saltation at low shear stresses and sheet flow at high shear stresses), although both types of transport can occur together and the limit is not always easy to define. The earliest formulas (models) proposed to estimate bed-load transport were mainly based on the concept that the sediment transport rate can be related to the bottom shear stress (Meyer-Peter and Müller 1948; Einstein 1950) and these formulas were valid for steady, unidirectional flows. In coastal and marine environments, the process of sediment transport becomes increasingly complex due to the presence of oscillatory flows, and the interaction between steady and oscillatory flows. For example, in longshore sediment transport, the influence of the short waves is expressed through wave-induced sediment stirring that increases the bed shear stress (Bijker 1967, 1968; Watanabe 1982; Van Rijn 1993).

However, in the case of cross-shore sediment transport, a wave-averaged approach is not adequate because of the dominant role of the time-dependent oscillatory orbital motion near the sea bed, induced by the short waves. For example, the residual cross-shore transport of sand due to short-wave asymmetry is generally described using an intrawave model concept, resolving the unsteady transport process through the wave cycle. Bagnold (1966) developed an "energetics transport model" in which the instantaneous transport through the wave cycle is related to the instantaneous energy dissipation rate due to bottom friction. Bagnold's (1966) approach has formed the basis for several bed-load transport models (Bailard and Inman 1981; Ribberink 1998) as well as total load transport models (Bailard 1981; Madsen and Grant 1976). These models use the bed-shear stress concept for river flow, and time-dependent transport through the wave cycle is treated in a quasi-steady way. That is, time-history effects from previous phases of the wave cycle or from previous wave cycles are neglected.

The development of practical sediment transport models still has a strong empirical character and relies on physical insights and quantitative data obtained in laboratory and field studies. The objective of this study is to develop a reliable, robust, and general formula for predicting bed-load transport for a wide range of coastal, marine, and river conditions. Many data sets are incorporated in the model development including steady and unsteady (oscillatory) flows.

For the bed-shear stress concept, a major problem is to estimate the total shear stress at the bottom. As a first step in this study, only conduit, flume, and river data for steady flows over a plane bed were examined to implement the bed-load formula. For these kinds of experimental studies, the bed shear stress may be estimated from energy slope measurements. Then, the formula was generalized to oscillatory flows and combined steady and oscillatory flows. However, for these more complex flows, the bed shear stress cannot be estimated directly from the measurements, but the stress must be calculated from theoretical models of the bed roughness. Thus, comparisons with experimental data were made by means of the skin friction as proposed by Soulsby (1997) and a semi-empirical function to estimate the bottom roughness in the sheet flow regime from Wilson (1989a).

Previous studies on bed-load transport under wave and current interaction

The bed-load transport corresponds to the net flux near the bottom, which then includes bed load and sheet-flow transport. Many studies (analytic and experimental) have been performed for bed-load and sheet-flow transport, which are often related to the bottom shear stress.

This section presents several sediment transport formulas that include both current and wave effects.

1. The Bijker (1968) and Van Rijn (1984a, b, 1989) formulas. The formulation of bed-load transport comes from a method used in the river environment and adapted to the coastal environment.

2. The Bailard (1981) bed-load formula, which involves an energetics description of sediment transport due to waves.

3. The Dibajnia and Watanabe (1992) sheet-flow formula, which is based on the instantaneous velocity taking into account wave and current interaction and the associated induced movement of sediment (calibrated with experiments on sheet-flow transport).

4. The Ribberink (1998) formula. Bed-load transport rate is estimated by integrating the instantaneous bed-load rate q_sb(t) over a wave period.

Bijker formula

One of the first formulas for bed-load transport that is still often applied was proposed by Bijker (1967, 1968). It was derived from the Frijlink (1952) formula for current only with a modification of the bottom shear stress using a wave-current model. The direction of sediment fluxes is always that of the current because this formula is proposed to estimate the longshore transport rate. The Bijker formula is written:

${{q}_{sb}}={{C}_{b}}\,d{\sqrt {\frac {{{\mu }_{c}}{{\tau }_{c}}}{\rho }}}\exp \left[-0.27{\frac {\left({{\rho }_{s}}-\rho \right)gd}{{{\mu }_{c}}{{\tau }_{cw}}}}\right]$

(59)

where q_sb is the sediment volume flux for bed load, C_b a breaking wave parameter, _c the ripple parameter, τ_c the shear stress due to current only, and τ_cw the shear stress including the wave-current interaction.

The ripple parameter introduced by Bijker (1968) is defined by the following equation:

${{\mu }_{c}}={{\left({\frac {{f}_{ct}}{{f}_{c}}}\right)}^{3/2}}$

(60)

where f_ct is the total friction coefficient due to the current and f_c the skin friction coefficient due to the current. The breaking wave coefficient is defined by the following relationships:

${\begin{aligned}&{{C}_{b}}=2\quad \quad \quad \quad \quad \quad \quad \quad {\text{ }}\ {\text{if}}\quad {{H}_{w}}/h<0.05\\&\quad \ =2+3\left({{H}_{w}}/h-0.05\right)\quad {\text{if}}\quad 0.05<{{H}_{w}}/h<0.4\\&\quad \ =5\quad \quad \quad \quad \quad \quad \quad \quad {\text{ }}\ {\text{if}}\quad 0.4<{{H}_{w}}/h\\\end{aligned}}$

(61)

where H_w is the wave height. The shear stress arising from the wave-current interaction is computed following the method proposed by Bijker (1968) where the effect of the waves is described through a stirring factor:

${{\tau }_{cw}}=\left[1+0.5{{\left({{\xi }_{B}}{\frac {{U}_{w}}{{U}_{c}}}\right)}^{2}}\right]{{\tau }_{c}}$

(62)

where

${{\xi }_{B}}={\sqrt {{{f}_{wt}}/{{f}_{ct}}}}$

is a parameter for wave-current interaction and f_wt the total friction coefficient due to waves.

Bailard formula

Bagnold (1966) introduced an energetics model for sediment transport, where the main idea is that the solid flux is proportional to the energy flux (local rate of energy dissipation):

${{q}_{sb}}\quad \propto \quad \Omega$

(63)

$\Omega ={\frac {1}{2}}\,\rho \,{{f}_{cw}}\ \left|{\overrightarrow {u(t)}}\left|^{3}\right.\right.$

(64)

where Ω is the energy flux from waves and currents, f_cw the friction coefficient due to the waves and current combined, u(t) the instantaneous velocity vector, in which u(t) = U_c + u_w(t) with U_c being the current velocity averaged over the depth and u_w(t) the instantaneous wave velocity.

The Bailard and Inman (1981) formula was derived from Bagnold (1966) model concepts. It can take into account the effect of the instantaneous velocity profile from waves and current combined. For a horizontal bed, the formula can be written as a vector yielding the sediment volume transport:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\vec{q}}_{sb}}= & \frac{1}{2}\frac{{{f}_{cw}}}{g\left( s-1 \right)}\frac{{{\varepsilon }_{b}}}{\tan \varphi }\left. < \right|{{\left. {\vec{u}} \right|}^{2}}\vec{u}>}

(65)

where _b is the bed-load efficiency, and < > represents the average over several wave periods. The bed-load efficiency is slightly different from the one given by Bagnold (1966). Bailard (1981) suggested based on calibration against field data that _b = 0.1. One difficulty concerning the formula is to estimate the friction coefficient for waves and current combined. Bailard did not specify any expression for the friction coefficient. Indeed, friction coefficients due to a current or to waves only display considerably different values (f_w/f_c  10 - 100). Madsen and Grant (1976) suggested the following relationship for the combined friction factor:

${{f}_{cw}}=Y\ {{f}_{c}}+\left(1-Y\right){{f}_{w}}\quad {\text{with}}\quad Y={\frac {{\theta }_{c}}{{{\theta }_{c}}+{{\theta }_{w}}}}$

(66)

where θ_c and θ_w are the Shields parameters due to current and waves, respectively.

Van Rijn formula

The Van Rijn (1989) formula is defined in the same way as the Bijker (1967, 1968) formula, and the bed-load transport rate is written as follows:

${{q}_{sb}}=0.25\ {{d}_{50}}\ d_{*}^{-0.3}{{\left({\frac {{\tau }_{ct}}{\rho }}\right)}^{0.5}}{{\left({\frac {{{\tau }_{cw}}-{{\tau }_{cr}}}{{\tau }_{cr}}}\right)}^{1.5}}$

(67)

where ${{\tau }_{ct}}={{\mu }_{f}}\ {{\alpha }_{cw}}\ {{\tau }_{c}}$ the total shear stress due to current only (taking into account the influence of bed forms and waves), τ_cr the critical shear stress for sediment transport, ${{\mu }_{f}}={{f}_{c}}/{{f}_{ct}}$ the shape factor, and α_cw a coefficient due to the presence of waves (which can affect the mean shear stress).

Dibajnia and Watanabe formula

Similar to the formulas of Bailard and Ribberink (discussed next), the Dibajnia and Watanabe (1992; see also Dibajnia 1995) formula divides the sediment transport into two half-cycles due to the presence of waves (Figure 18). During the first half-cycle, sediment moves in the direction of wave propagation, and then it moves in the opposite direction during the second half-cycle. An advantage of the formula is that it takes into account a possible quantity of sand still in suspension after each half-cycle that moves in the opposite direction. This phenomenon is referred to as a "phase lag," introduced by Dohmen-Janssen (1999). Another advantage is that the formula (similar to the Bailard (1981) and Ribberink (1998) formulas) allows for the description of nonlinear waves.

Figure 18. Definition of (a) wave and current directions, and (b) horizontal time-dependent velocity variation at bottom in direction of wave propagation
(after Dibajnia and Watanabe 1992).

The solid volume flux is given by the following equation:

${{\vec {q}}_{sb}}={{A}_{dw}}\ {{W}_{s}}\ d{\frac {\vec {\Gamma }}{\Gamma }}{{\Gamma }^{{B}_{dw}}}$

(68)

where the coefficients of calibration are A_dw = 0.001 and B_dw = 0.55, and:

(69)

in which T_w, T_wc, T_wt are the period and half-periods, respectively, of the wave taking into account the influence of the current (see Figure 18), _c and _t the amount of sand entrained and settled during the half-period T_wc of the wave crest and T_wt of the wave trough, respectively, ${{{\Omega }'}_{c}}$ and ${{{\Omega }'}_{t}}$ the amount of suspended sand remaining from the positive (crest) and the negative (trough) half-cycle, respectively, and u_wc and u_wt the quadratic velocity (wave and current combined) over each half-period calculated from:

$u_{wj}^{2}={\frac {2}{{T}_{wj}}}\int _{t}^{t+{{T}_{wj}}}{{{u}^{2}}\left(t\right)}dt+2\ U_{c}^{2}{{\sin }^{2}}\varphi$

(70)

where j can be c or t, $u(t)={{U}_{c}}\cos \varphi +{{u}_{w}}(t)$ , with u_w(t) being the instantaneous wave orbital velocity, and  the angle between the wave and current directions (Figure 18). The nondimensional quantities in Equation 69 are defined as:

${\begin{aligned}&{\text{if }}{{\omega }_{j}}\leq {{\omega }_{cr}}\quad {\text{then}}\quad {{\Omega }_{j}}={{\omega }_{j}}{\frac {2\ {{W}_{s}}\ {{T}_{wj}}}{d}}\\&\ \ \ \ \quad \quad \quad \quad \quad {\text{and}}\quad \ \ {{{\Omega }'}_{j}}=0,\\&{\text{if }}{{\omega }_{j}}\geq {{\omega }_{cr}}\quad {\text{then }}\quad {{\Omega }_{j}}={\frac {2\ {{W}_{s}}\ {{T}_{wj}}}{d}}{{\omega }_{cr}}\\&\ \quad \quad \quad \quad \quad {\text{ and}}\quad \ \ {{{\Omega }'}_{j}}=\left({{\omega }_{j}}-{{\omega }_{cr}}\right){\frac {2\ {{W}_{s}}\ {{T}_{wj}}}{d}}\\\end{aligned}}$

(71)

with:

${{\omega }_{j}}={\frac {u_{wj}^{2}}{2\left(s-1\right)g\ {{W}_{s}}\ {{T}_{wj}}}}$

(72)

where j can be c or t as before. The parameter ω_cr characterizes the ripples (a function of the total shear stress including waves and current):

${\begin{aligned}&{{\omega }_{cr}}=0.03\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ \ \ \,{\text{if}}\quad {{\theta }_{cw,\max }}\leq 0.2\\&\quad \ \,=1-0.97{{\left[1-{{\left({\frac {{{\theta }_{cw,\max }}-0.2}{0.4}}\right)}^{2}}\right]}^{0.5}}{\text{if}}\quad 0.2<{{\theta }_{cw,\max }}<0.6\\&\quad \ \,=1\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\quad \ \,\,{\text{if}}\quad 0.6<{{\theta }_{cw,\max }}\\\end{aligned}}$

(73)

where θ_cw,max is the maximum Shields parameter due to waves and current combined (following Soulsby 1997, pp. 87-95).

Ribberink formula

Ribberink (1998) proposed a model for the quasi-steady bed-load transport for which the instantaneous solid flux is assumed to be a function of the difference between the actual time-dependent bed shear stress and the critical bed shear stress (Figure 19). This formula has been calibrated against flume data for waves and current combined in the plane-bed mode (suspended load negligible), as well as for field data for unidirectional flow in rivers.

Figure 19. Time variation of bottom velocity (a) in wave direction, and (b) induced shear stress for waves and current combined (after Ribberink 1998).

Thus, Ribberink (1998) proposed the following transport formula:

${{\vec {q}}_{sb}}={{m}_{Rib}}{\sqrt {\left(s-1\right)g\ {{d}^{3}}}}<{{\left({\overrightarrow {\left|\theta \right.\left(t\right)}}\left|-{{\theta }_{cr}}\right.\right)}^{{n}_{Rib}}}{\frac {\left[{\overrightarrow {\theta \left(t\right)}}\right]}{\left|\theta \left.\left(t\right)\right|\right.}}$

(74)

where ${\vec {\theta }}\left(t\right)=0.5\ {{f}_{cw}}\left|u\left(t\right)\right|{\vec {u}}\left(t\right)/\left[\left(s-1\right)\ g\ d\right]$ is the time-dependent Shields parameter (Figure 19) with the instantaneous velocity ${\vec {u}}\left(t\right)={{\vec {U}}_{c}}+{{\vec {u}}_{w}}\left(t\right)$ , the wave-current friction factor f_cw computed according to Madsen and Grant (1976), θ_cr the critical Shields parameter, < > the time-average over several wave periods, and m_Rib = 11, n_Rib = 1.65 calibration coefficients. Ribberink (1998) proposed to compute the total roughness from ${{k}_{st}}=\max \left\{{{k}_{s}}{\text{;}}d\left[1+6\left(<\left|\theta \left(t\right)\right|>/{{\theta }_{cr}}-1\right)\right]\right\}$ , where k_s is the Nikuradse skin roughness.

Bed-load transport by currents

Here, bed load refers mainly to the rolling, sliding, and jumping grains in almost continuous contact with the bed. Also, it follows the ideas of Wilson (1966) about upper-regime transport where a layer with a thickness of several grain diameters is transported and where intergranular collision is important (sheet flow).

Existing formulas

The bed-load transport is often represented by the following non-dimensional parameter:

$\Phi ={\frac {{q}_{sb}}{\sqrt {\left(s-1\right)g\ {{d}_{50}}^{3}}}}$

(75)

where q_sb is the volumetric bed-load sediment transport rate per unit time and width. Some authors have also proposed to use the parameter ${{\Phi }_{b}}={{q}_{sb}}/\left({{W}_{s}}{{d}_{50}}\right)$ . If d₅₀ > 10⁻³ mm, _b   because the settling velocity is proportional to the square-root of the median diameter (according to Van Rijn 1984a), whereas for d₅₀ < 10⁻³ mm, Φ_b becomes smaller than Φ, implying that Φ_b introduces a characterization of the sediment transport rate that is more sensitive for fine sediments. Several relationships for estimating the bed-load sediment transport under a steady current have been proposed where the rate is related to the dimensionless bottom shear stress or Shields parameter:

${{\theta }_{c}}={\frac {{\tau }_{c}}{\left({{\rho }_{s}}-\rho \right)g{{d}_{50}}}}={\frac {{\frac {1}{2}}{{f}_{c}}U_{c}^{2}}{\left(s-1\right)g{{d}_{50}}}}$

(76)

Here, τ_c is the shear stress at the bottom due to the current. Three commonly applied formulas were investigated in this study:

${\text{Meyer-Peter }}\!\!\And \!\!{\text{ }}{\text{M }}\!\!{\ddot {\mathrm {u} }}\!\!{\text{ ller (1948)}}\quad \Phi =8{{\left({{\theta }_{c}}-{{\theta }_{cr}}\right)}^{3/2}}$

(77)

${\text{Nielsen (1992)}}\quad \Phi =12\ \theta _{c}^{0.5}\left({{\theta }_{c}}-{{\theta }_{cr}}\right)$

(78)

${\text{Ribberink (1998)}}\quad \Phi =11{{\left({{\theta }_{c}}-{{\theta }_{cr}}\right)}^{1.65}}$

(79)

where θ_cr is the critical Shields parameter. The formula for θ_cr proposed by Soulsby and Whitehouse (1997; see also Soulsby (1997), p. 104-110) was used in the present study (Equation 49).

Comparison with data

To investigate bed-load transport formulas, available data sets on sediment transport covering a wide range of unidirectional steady currents were compiled and analyzed, including both laboratory and field data. Table 5 summarizes these data sets with the type of flow motion and sediment properties listed. It may be noted that many data sets come from the compilation made by Brownlie (1981), where limited information was given about the type of sediment transport (bed load and/or suspended load), although comments on the presence of bed forms were provided. From these data, only plane-bed cases were selected, where bed load should prevail. Typically, for fine sediment, suspended load is not negligible if bed forms appear. Thus, bed-load transport in the presence of bed forms was not investigated. The data base compiled covers a wide range of median grain sizes, many of them considerably coarser than what would be found at a coastal inlet, but the range provides an informative test on the limits of the formulas. Also, the currents included in the data base covered a broad range. Figures 20a and 20b display histograms of the median grain size and mean current speed for the data base employed. The existing formulas selected for comparison with the data relate the sediment transport rate to the bed shear stress, and they were developed for bed-load transport. In the employed database, it is uncertain what the ratio was between bed load and suspended load in each individual case. For the cases with coarser sediment, bed load must have dominated, with the grains moving through rolling, sliding, and jumping for lower shear stresses or in a sheet-flow layer for higher shear stresses. Even where suspended load might have been significant, experience with the Watanabe (1989) formula shows that the total load can often be related to the bed shear stress. This implies that the tested formulas may have some predictive potential in such a situtation, although it is in principle outside their range of applicability.

---

Table 5. Data base compiled to study bed-load sediment transport in steady current (plane bed cases only).

Author(s)	Flow Type	Number	Material	S	D'50 '(mm)
Gilbert (1914)	Steady uniform flow, flume, plane bed	250	Sand	2,65	0.3-4.9
U.S. Army Engineer Waterways Experiment Station (1935-1936) (in Brownlie 1981)	Steady uniform flow, flume, plane bed	162	Sand	2.65	0.18-4.1
Willis et al. (1972)	Flume, plane bed	43	Sand	2.65	0.1
Brownlie (1981)	Various experimental data Plane bed	297	Plastic Sand	1.30-1.41 2.49-2.67	2.2-20.2 0.088-20
Brownlie (1981)	Various field data, plane bed	40	Sand	2.65	0.84-7.0
Smart (1984, 1999 with Nikora and Smart 1997)	Exp. for steep channels and field data, plane bed	140	Sand Gravel	2.65 2.65	2.0-10.5 53-200
Nnadi & Wilson (1992)	Pressurized closed conduit	105	Bakelite Sand Nylon	1.55 2.67 1.14	0.67-1.05 0.70 3.94

(a)	(b)

Figure 20. Distribution of median grain size and mean current speed for database compiled on sediment transport under steady current.

Figure 21 illustrates a comparison between the sediment transport rate calculated with the Meyer-Peter and Müller (1948) formula and the compiled database. The other studied formulas yield similar agreement and exhibit the same general behavior, although the Nielsen (1992) formula showed slightly less agreement with the data. From Figure 21, two observations can be made: (1) for the Willis et al. (1972) data, the formulas underpredict the transport rate, which is probably due to the fine sediment used in the experiment and probable significant transport in suspension that the formulas fail to accurately quantify; and (2) for small transport rates, close to θ_cr, the formulas overpredict the transport rate. With regard to the latter observation, it should be noted that all individual cases where the predictions implied a zero transport because θ_c < θ_cr were not plotted in the figure because of the log-log scale.

Figure 21. Comparison between Meyer-Peter and Müller formula and compiled database on sediment transport rates.

Because marked uncertainties exist in the measurements of the bed-load sediment transport (especially for field measurements), a prediction within a factor 2 of the measurements is considered to be a satisfactory result. In Table 6, the percentage of predicted values included within a factor of 2 or a factor of 5 deviation is presented (Px2 and Px5, respectively), as well as the root-mean-square error defined as:

${{E}_{rms}}={\frac {1}{n}}\sum {{\left[{\text{Log}}\left({\frac {{q}_{sb,pred}}{{q}_{sb,meas}}}\right)\right]}^{2}}$

(80)

To avoid any distortion in the calculation of E_rms for low transport rates (where q_sb,pred or q_sb,meas could be found equal to zero), a minimum value for q_sb,pred and q_sb,meas was specified (q_sb,min = 10^-8m²/sec). It appears that the Meyer-Peter and Müller (1948) formula, which was originally calibrated with coarse sediment data, gives fairly good results in comparison to the measurements, even for large shear stresses and fine sediment. An underestimation is, however, observed for the Willis et al. (1972) data, but as sediment used in this experiment was very fine (0.1 mm), some suspended transport probably occurred. The three formulas studied show similar behavior for large shear stresses with a slightly better prediction skill for the Ribberink (1998) formula. For smaller Shields parameter (θ_c close to θ_cr, where θ_cr is estimated using the Soulsby and Whitehouse (1997) formula), all formulas tend to overestimate the bed-load rate with errors up to one order of magnitude for the Nielsen (1992) formula, which significantly increases the E_rms value. Thus, it seems that the use of the critical Shields parameter as a limit for specifying no transport is not sufficiently accurate for the existing formulas.

Table 6. Prediction of bed-load transport rate within factor of 2 and 5 of measured values and root-mean-square errors using current only data.

Author(s)	All Data					All Data Except Willis et al.
	Px2 (%)	Px5 (%)	Erms	Px2 (%)	Px5 (%)	Erms
Meyer-Peter & Müller	62	83	0.34	66	87	0.30
Nielsen	54	72	0.48	57	75	0.46
Ribberink	66	85	0.28	69	89	0.25
Equation 81	73	89	0.19	78	93	0.15

New formula for bed-load transport

To better understand the influence of the critical value of the Shields parameter on the sediment transport, Φ-values from the measurements are plotted versus the ratio θ_c/θ_cr in Figure 22. The studied formulas are also plotted with a constant value for θ_cr = 0.04 (mean value for the data set).

Figure 22. Effect of critical Shields parameter on bed-load transport rate: comparison between data and studied formulas.

All the predictive formulas are in fairly good agreement with the data if θ_c > 5 θ_cr, but they tend to slightly overestimate Φ when θ_cr < θ_c < 5 θ_cr. For this range of values, the Meyer-Peter and Müller (1948) and Ribberink (1998) formulas exhibit a better behavior than the Nielsen (1992) formula. However, if Failed to parse (unknown function "\raise"): {\displaystyle {{\theta }_{c}}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{<}{{\theta }_{cr}}} , significant deviations in the predictions occur. All these formulas predict no sediment transport, although small sediment transport is often observed. The prediction of the critical Shields parameter is obviously associated with marked uncertainty (so is the sediment transport estimation for these low values). Thus, to avoid such errors, a new approach is introduced by including an exponential relationship for the effect of the critical Shields parameter, which allows low sediment transport rates if   _cr. Calibration against the measurements led to the following bed-load formula:

$\Phi =12\ {{\theta }_{c}}^{1.5}\exp \left(-4.5{\frac {{\theta }_{cr}}{{\theta }_{c}}}\right)$

(81)

Cheng (2002) proposed a formula similar to Equation 81, but without taking into account the effect of a varying critical Shields parameter. The Cheng (2002) formula seems to include a constant value of _cr  0.05. The effect of a varying critical value of the Shields parameter is, however, not negligible as observed by Meyer-Peter and Müller (1948). Figures 22, 23, and 24 indicate that this new approach of describing the effect of the critical Shields parameter significantly improves the prediction skill for weak shear stresses. Predictions within a factor-2 deviation reach 70 percent (even 80 percent without the Willis et al. (1972) data), which increase the agreement by 10 percent compared to previous formulas (Table 6). The new relationship tends to overestimate the transport rate if ${{\theta }_{c}}$ is much smaller than ${{\theta }_{cr}}$ (see Figure 22), but reproduces trends well for small measured rates.

Figure 23. Influence of critical Shields parameter on bed-load transport rate illustrated through data and Equation 81 (for different values on θ_cr).

Figure 24 plots measured and calculated transport rate values for the database using the new formula (Equation 81). The plot confirms the improvement in the prediction of the bed-load transport.

Figure 24. Comparison between bed-load transport for current only predicted
by new formula (Equation 81) and measurements.

Bed-load transport by waves

Existing formulas

Several relationships for bed-load transport by waves have been proposed that relate the transport rate to the wave orbital velocity at the bottom U_w or the wave Shields parameter θ_w defined as:

${{\theta }_{w}}={\frac {{\frac {1}{2}}{{f}_{w}}U_{w}^{2}}{\left(s-1\right)g{{d}_{50}}}}$

(82)

where f_w is the dimensionless wave friction factor. Assuming that the rough turbulent regime is fully developed, the friction factor can be estimated, for example, with the formula suggested by Swart (1974) (see Equation 25). Transport formulas have been employed to estimate the wave half-cycle sediment transport Φ_1/2 (Madsen and Grant 1976; Soulsby et al. 1993; Soulsby 1997). The net transport can then be calculated as the difference between the half-cycle transport beneath the crest and beneath the trough (the shear stress is calculated using the maximum and minimum values of the wave velocity at the bottom, u_w,max and u_w,min, respectively, instead of the wave orbital velocity U_w). In some other studies (Bailard and Inman 1981; Ribberink 1998), a relationship based on the instantaneous wave velocity was introduced for the bed load Φ(t) that can be integrated over a wave period. The following formulas are among the most common ones for calculating sediment transport under waves:

${\text{Madsen }}\!\!\And \!\!{\text{ Grant (1976)}}\quad {{\Phi }_{1/2}}=12.5{\frac {{W}_{s}}{\sqrt {\left(s-1\right)g\ {{d}_{50}}}}}\theta _{w}^{3}$

(83)

${\text{Bailard }}\!\!\And \!\!{\text{ Inman (1981)}}\quad \Phi \left(t\right)={\frac {{{\varepsilon }_{b}}{{f}_{w}}}{{{\left(s-1\right)}^{2}}{{g}^{2}}\tan \phi {{d}_{50}}}}\left|{{u}_{w}}\left(t\right)\left|^{2}\ {{u}_{w}}\left(t\right)\right.\right.$

(84)

${\text{Dibajnia }}\!\!\And \!\!{\text{ Watanabe (1992)}}\quad \Phi =0.001{\frac {{W}_{s}}{\sqrt {\left(s-1\right)g\ {{d}_{50}}}}}{{\Gamma }^{0.55}}{\frac {\Gamma }{\left|\Gamma \right|}}$

(85)

${\text{Soulsby (1997)}}\quad {{\Phi }_{1/2}}=5.1{{\left({{\theta }_{w}}-{{\theta }_{cr}}\right)}^{3/2}}$

(86)

${\text{Ribberink (1998)}}\quad \Phi \left(t\right)=11{{\left(\left|{{\theta }_{w}}\left(t\right)\right|-{{\theta }_{cr}}\right)}^{1.65}}{\frac {{{\theta }_{w}}\left(t\right)}{\left|{{\theta }_{w}}\left(t\right)\right|}}$

(87)

It should be noted that Madsen and Grant (1976) as well as Dibajnia and Watanabe (1992) used the parameter Φ_b instead of the parameter Φ (see previous discussion), which explains the coefficient ${{W}_{s}}/{\sqrt {\left(s-1\right)g\ {{d}_{50}}}}$ in their formulas. Bailard and Inman (1981) proposed a coefficient value of (_b = 0.13, and ( is the internal friction angle of the sediment (( ( 30 deg). Dibajnia and Watanabe (1992) defined Γ as a function of the half periods T_wc and T_wt and the amount of sediment entrained that settle in each half-period, because some of the sediment might still be in suspension from the previous half-period (see also Camenen and Larroudé 2003).

Development of new formula

As Madsen (1991) and Ribberink (1998) proposed, the instantaneous sediment transport rate may be related to the instantaneous shear stress in the same manner as for the steady case. Following the idea of Dibajnia and Watanabe (1992), a simplified velocity variation at the bottom may be considered to estimate the effect of the wave asymmetry on the sediment transport. Thus, the net sediment transport over a wave period is estimated for each half-period using a characteristic value for the quadratic velocity, or equivalently on the shear stress (if the friction coefficient is assumed to be constant). Thus, the mean value of the instantaneous shear stress over half a period may be used (Figure 25):

${\begin{aligned}&{{\theta }_{w,onshore}}={\frac {1}{{T}_{wc}}}\int _{0}^{{T}_{wc}}{{{\theta }_{w}}\left(t\right)dt}\\&{{\theta }_{w,offshore}}={\frac {1}{{T}_{wt}}}\int _{{T}_{wc}}^{{T}_{w}}{{{\theta }_{w}}\left(t\right)dt}\\\end{aligned}}$

(88)

where T_wc and T_wt (= T_w - T_wc) are the half periods where the instantaneous velocity u_w(t) (or instantaneous Shields parameter) is onshore (u_w(t) > 0) or offshore (u_w(t) < 0), respectively, and the instantaneous shear stress is defined as follows:

${{\theta }_{w}}\left(t\right)={\frac {{\frac {1}{2}}{{f}_{w}}\left|{{u}_{w}}\left(t\right)\right|{{u}_{w}}\left(t\right)}{\left(s-1\right)g{{d}_{50}}}}$

(89)

Figure 25. (a) Typical wave velocity variation, and (b) instantaneous Shields parameter variation over wave period in direction of wave propagation.

Figure 25 presents a typical velocity variation and the associated instantaneous Shields parameter variation over a wave period. In the case of an asymmetric wave, a maximum in the shear stress occurs during onshore flow (in the direction of the wave propagation) that is larger than the minimum during offshore flow, which induces a net sediment transport in the direction of the waves. A constant value on the friction coefficient over the wave period was assumed in calculating the Shields parameter (Equation 89). Drake and Calantoni (2001) and Antunes Do Carmo et al. (2003) showed that the wave friction coefficient depends also on the acceleration of the fluid near the bottom. This dependence was previously noted by Trowbridge and Madsen (1984a, b), who made an analytical study of the turbulent wave boundary layer, highlighting the contribution of a time-varying eddy viscosity. However, it is assumed here that the predictions of the shear stresses using a constant friction factor are sufficiently accurate for application in practical sediment transport calculations. Using the same approach as for the steady current, an equation for the net sediment transport under waves may be derived similar to Equation 81. The net sediment transport under waves is expressed as:

$\Phi ={{a}_{w}}{\sqrt {{{\theta }_{w{\text{,}}onshore}}+{{\theta }_{w{\text{,}}offshore}}}}{{\theta }_{w{\text{,}}m}}\ \exp \left(-{{b}_{w}}{\frac {{\theta }_{cr}}{{\theta }_{w}}}\right)$

(90)

where a_w and b_w are coefficients $\left(\left|{{\theta }_{w{\text{,}}onshore}}\right|\right.$ is always supposed to be larger than $\left.\left|{{\theta }_{w{\text{,}}offshore}}\right|\right)$ and ${{\theta }_{w{\text{,}}m}}=<\left|{{\theta }_{w}}\left(t\right)\right|>$ is the time-averaged absolute value of the instantaneous Shields parameter. A conceptual model that supports this type of formulation would be based on the following assumptions:

• The transport in the bottom layer is the product of the typical speed of the layer and the layer thickness, where the former is denoted U_s and the latter _m.
• The speed U_s is assumed proportional to the net shear velocity over a wave period, which gives a dependence on
• ${{\left|{{\theta }_{w{\text{,}}onshore}}+{{\theta }_{offshore}}\right|}^{1/2}}$
• $\left({{u}_{*}}\propto {{\theta }^{1/2}}\right)$
• .
• _m is assumed proportional to the mean wave shear stress, which gives the dependence on θ_w,m.

The overall effect of the critical shear stress on the sediment transport over a wave period is estimated using the same approximation as for a steady current. However, because Soulsby (1997, pp. 104-106) proposed to compute the critical shear stress based on the maximum shear stress, the maximum Shields parameter is used in Equation 90. A difficulty encountered in applying the formula (and any other formulas based on the shear stress) is to estimate the total Shields parameter (or total friction coefficient) for the sheet flow mode. The work by Wilson (1989a) on wave-induced sheet-flow roughness was employed in this study, where he proposed to use the same equation as for steady current, but with the maximum wave-induced Shields parameter (k_s = k_s,sfw = 5 θ_w d₅₀). Because most of the results presented by Ribberink (1998) were based on the skin friction, to be consistent, two computations were made, first using the skin friction (k_s = 2d₅₀) and then the Wilson formula (Equation 41). It should be noted that the Wilson formula requires an iterative approach in solving for k_s.

Comparison with experimental data

To investigate bed-load transport under waves only, a wide range of existing data sets was compiled and analyzed. Table 7 summarizes these data sets, in which the type of experiment, sediment properties, and wave properties are listed. It can be observed that most of the data are from oscillating water tunnels (OWT). The OWT experiments have two advantages for this type study: large orbital velocities can be achieved, and bed-load transport prevails. Previously, experimental studies were often carried out using an oscillating tray (OT: oscillating bed in a tank of still water; Manohar 1955; Kalkanis 1964; Abou-Seida 1965; and Sleath 1978). More recently, Dohmen-Janssen and Hanes (2002) carried out an experiment in a large wave flume (LWF).

Table 7. Data summary for bed-load sediment transport experiments carried out in oscillatory flow with and without current.

Author(s)	Exp. Facil.	Cycle	Number	s	d50 (mm)	Uc '(m/sec)'	Uw,max (m/sec)	Tw '(sec)'
Kalkanis (1964)	OT	Half	25	2.63	1.68-2.82	0	0.28-0.71	3.2-6.2
Abou Seida (1965)	OT	Half	34 9	2.65 2.23	0.14-2.61 0.70	0 0	0.35-1.28 0.41-0.80	1.7-5.1 2.0-4.8
Sleath (1978)	OT	Half	22 12	2.60 1.138	1.89, 4.24 3.04	0 0	0.20-0.68 0.07-0.17	0.5-2.7 1.3-9.0
Horikawa et al. (1982)	OWT	Half	6	2.66	0.2	0	0.76-1.27	2.6-6.0
Sawamoto & Yamashita (1986)	OWT	Half	22	1.58 2.65	1.5 0.2-1.8	0 0	0.74-1.25 0.46-1.25	3.8 3.8
Ahilan & Sleath (1987)	OWT	Half	5 4	1.137 1.44	4.0 4.3	0 0	0.3-0.5 1.1-1.2	3.6-3.7 4.7-4.9
Watanabe & Isobe (1990)	OWT	Full	11 65	2.65 2.65	0.18, 0.87 0.18, 0.87	0 -0.3-0.25	0.27-0.43 0.27-0.52	3.0, 6.0 3, 6
King (1991)	OWT	Half	178	2.65	0.1-1.1	0	0.3-1.2	2.0-12.0
Dibajnia & Watanabe (1992)	OWT	Full	25 76	2.65 2.65	0.20 0.20	0 -0.26-0.22	0.6-1.0 0.61-1.24	1.0-4.0 1-4
Ribberink & Chen (1993)	OWT	Full	4	2.65	0.128	0	0.6-1.2	6.5
Ribberink & Al Salem (1994)	OWT	Full	10	2.65	0.21	0	0.7-1.4	5.0-12.0
Delft Hydraulics (1993-1999; see Van Rijn et al. 2001)	OWT	Full	52	2.65	0.13-0.24	-0.45-0.56	0.37-1.49	5, 12
Dohmen-Janssen (1999)	OWT	Full	27	2.65	0.13-0.32	0.23-0,45	0.46-1.85	4-12
Dohmen-Janssen & Hanes (2002)	LWF	Full	4	2.65	0.21	-0.05 - -0.03	0.88-1.5	6.5, 9.1
Ahmed & Sato (2003)	OWT	Full	15	2.65	0.21-0.74	0	1.16-1.85	3.0

Figure 27 shows the calculated and measured bed-load transport for the studied experiments using values on the empirical coefficients of a_w = 6 and b_w = b = 4.5 (see Figure 26 for the calibration of a_w). This new fit shows that if the effect of the critical Shields parameter on the mean sediment transport rate over a wave period is similar to steady flow, the total net rate is a function of the Shields parameter to the power 1.5 with a smaller value on the coefficient a than for steady current. Soulsby (1997) found similar results using the Meyer-Peter and Müller (1948) equation (Equations 77 and 86 include the coefficient values 8 and 5.1, respectively). This lower value is partly due to the time averaging. It may also contain contributions owing to a phase lag between instantaneous sediment concentration and the velocity at the bottom. In Figure 27(a), the predicted bed load using Equation 90 is plotted versus the measurements encompassing waves only. Even if some scatter occurs, the obtained results are seen to be good. Around 60 percent of the cases are predicted within a factor of 2 of the measured values. An additional difficulty in comparing the transport formula with the wave measurements is that no shear stress may be derived from the measurements, but the shear stress has to be calculated based on an estimate of the bed roughness. This introduces an extra element of uncertainty in the calculations of the sediment transport rates. The formulas proposed by Wilson (1966, 1989a, b) were employed to calculate the Nikuradse roughness:

${\frac {{k}_{s,sfc}}{{d}_{50}}}=5\ {{\theta }_{c}}$

(91)

${\frac {{k}_{s,sfw}}{{d}_{50}}}=5\ {{\theta }_{w}}$

(92)

Both equations require an iterative method in solving for the Nikuradse roughness. An underestimation for the Ahilan and Sleath (1987) data set and an overestimation for the Sawamoto and Yamashita (1986) data set is observed. One explanation could be that large uncertainties are induced by this kind of iterative formula (Bayram et al. 2003; Camenen et al. 2006). Using the Nikuradse skin roughness shows better agreement for the Sawamoto and Yamashita (1986) data set, but worse agreement for the Ahilan and Sleath (1987) data set (Figure 27(b)). Finally, it may be noted that Equation 90 slightly underestimates the sediment transport rate for the Kalkanis (1964) data set and overestimates the rate for the Abou-Seida (1965) and the Sleath (1978) data sets. Thus, it appears that the OT set-up does not yield the same behavior concerning the sediment transport as the OWT set-up.

Figure 26. Calibration of coefficient '''aw 'using all experimental data with waves only (Nikuradse roughness calculated using Wilson (1989a) formula).

---

(a)
(b)

Figure 27. Comparison between bed-load transport predicted by Equation 90 and experimental data with waves (a) where '''ks 'is calculated using Wilson (1989a) formula, and (b) where '''ks '= 2'''d50.

Table 8 lists the percentages of predicted values within a factor of 2 deviation or a factor of 5 deviation and the root-mean-square error for the six studied formulas and all the data, as well as for the data encompassing complete wave cycles, using the Wilson (1989a) formula to compute the Nikuradse roughness (k_s = k_s,sfw). The same calculations were also performed for all the data with the skin friction (k_s = 2d₅₀) to compute the Shields parameter.

Table 8. Predictive capability of bed-load transport rate within factor of 2 and 5 of measured values and root-mean-square errors, data from waves only.

Author(s)	All Data ('''ks '''''= ks,sfw)							Full Cycle Data ('''ks '''''= ks,sfw)	All Data ('''ks '''''= '''2'''d50)
	Px2 (%)	Px5 (%)	Erms	Px2 (%)	Px5 (%)	Erms	Px2 (%)	Px5 (%)	Erms
Madsen & Grant	37	64	1.25	06	14	5.22	45	75	0.99
Bailard & Inman	47	75	0.92	38	57	3.81	45	73	0.93
Dibajnia & Watanabe	36	75	0.49	49	85	0.79	36	75	0.49
Soulsby (coef. 2.5)	36	75	0.98	20	52	3.92	39	76	0.97
Ribberink	29	73	1.08	20	46	4.14	36	76	1.03
Equation 90	59	86	0.77	34	58	3.84	57	86	0.78

Equation 90 yields the best overall results. For the full cycle data, because the phase lag is not taken into account, the agreement is not as good, and the scatter is larger (E_rms = 3.8). The Madsen and Grant (1976) formula tends to overestimate the sediment transport for large values of the shear stress. This formula is proportional to the Shields parameter to the power 3 and calibrated with lower shear stress data. Soulsby (1997) suggested to use the maximum shear stress in his formula, which causes a large overestimation of the transport rates. Replacement of a coefficient value of 2.5 instead 5.1 improves the results (results in Table 8), but it still indicates an overestimation of the sediment transport for low shear stresses. If the mean shear stress is used instead of the maximum shear stress (with the coefficient 5.1), results are better and similar to those given by Equation 90, except for data with low sediment transport rates, which are typically overestimated. The Bailard and Inman (1981), Dibajnia and Watanabe (1992), and Ribberink (1998) formulas also overestimate the weak sediment transport rate data. This means that the effect of the critical Shields parameter is not taken into account properly (not included in the Bailard and Inman (1981) and Dibajnia and Watanabe (1992) formulas). The Ribberink (1998) formula typically overestimates the sediment transport rate. Using the coefficient values (7.9, 1.97) instead of (11, 1.65) or using k_s = d50, as Ribberink (1998) proposed, improves the results. However, this is not in accordance with the formula originally proposed for a steady current (Equation 79) or with physical values of the Nikuradse roughness. The Dibajnia and Watanabe (1992) formula, because it takes the phase lag into account, allows the sediment transport to be in the opposite direction to the waves for fine sediment. Thus, it presents the overall best results for the full cycle data and a much lower scatter (E_rms < 1). On the other hand, it yields poor results for the half-cycle data. Finally, it should be noted that results from the OT data are typically more scattered. This may be a result of the shear stresses being close to the critical shear stress, inducing additional randomness, as well as uncertainty in the measurements. The main theoretical difference between the Ribberink (1998) relationship (Equation 87) and Equation 90 is that the former equation is a mean value of the instantaneous sediment transport over the wave period, whereas the latter calculates the sediment transport from the mean shear stress. Using the mean value over the wave period of the instantaneous sediment transport induces a sediment transport rate 20 percent larger than using the same equation with the mean shear stress over the period. Moreover, as was observed previously, phase-lag effects seem to be non-negligible, even for relatively small wave orbital velocities. This may explain the overestimation observed for the Ribberink formula. Another remark to be made (Figure 28) is that the direction of the net sediment transport may be opposite to the direction of the waves. Ribberink and Chen (1993) and Dibajnia and Watanabe (1992) observed this phenomenon for highly asymmetric waves and fine sediment (d₅₀  0.2 mm). It is caused by the phase lag in the response of the sand to the fluid motion. The quantity of sediment in suspension depends not only on the instantaneous velocity, but also on the settling velocity. In the case of oscillatory flow, not all the sand grains put into suspension during the first half-period settle during the same half-period. The proportion still in suspension is then carried away in the opposite direction during the second half-period. Dohmen-Janssen (1999) and Dohmen-Janssen and Hanes (2002) introduced a parameter to describe the phase lag.

${{p}_{pl}}={\frac {2\pi \ {{\delta }_{s}}}{{{W}_{s}}{{T}_{w}}}}$

(93)

where δ_s is the thickness of the sheet-flow layer. Following Asano (1992) and Dohmen-Janssen and Hanes (2002), δ_s = 10 d₅₀ _w. They observed that phase-lag effects occur if p_pl > 0.3-0.4. Figure 28 plots the cases where strong phase lag occurs (circles). These data points correspond to an overestimation (and/or wrong direction) of the predicted values using Equation 90. Thus, it appears that the proposed formula is mainly restricted to cases without significant phase lag. This phenomenon is not included in Equation 90, but could be approximated by adding a correction factor (Dohmen-Janssen 1999; Dohmen-Janssen and Hanes 2002; Camenen and Larroudé 2003).

Figure 28. Comparison between bed-load transport predicted by Equation 90 and experimental data over full wave cycle ('''ks 'calculated using Wilson (1989a) formula,
'''ks '= '''ks,sfw). Bed-load transport by waves and currents Stirring by waves in the presence of a steady current tends to increase the total sediment transport significantly. Also, as shown in the previous section, asymmetric waves can transport sediment. Thus, to generalize the proposed formula to encompass both waves and current, a modification of the formula is proposed including the total shear stress obtained from the interaction between waves and current.

Development of new formula

The conceptual model proposed for the case of waves only (Equation 90) can be extended to the interaction between waves and current. Assuming that U_s is proportional to the shear velocity at the bottom, a dependence on ${{\left|{{\theta }_{cw,onshore}}+{{\theta }_{cw,offshore}}\right|}^{1/2}}$ may be assumed for U_s, where the interaction between waves and current is taken into account. The representative shear stresses θ_cw,onshore and θ_cw,offshore are defined based on the instantaneous Shields parameter in the direction of the wave for positive and negative values of θ_cw(t), respectively (Figure 29). For an arbitrary angle ( between the waves and the current, this yields the same equations as Equation 88, where θ_w is replaced by θ_cw, and T_wc and T_wc are the half-periods where the instantaneous velocity $u\left(t\right)={{U}_{c}}\cos \varphi \ +{{u}_{w}}\left(t\right)$ (or instantaneous Shields parameter) is onshore (u(t) > 0) or offshore (u(t) < 0), respectively (Figure 29). The representative shear stresses θ_cw,onshore and θ_cw,offshore are defined as quadratic values of the instantaneous Shields parameter in the direction of the wave for positive and negative values of θ_cw(t), respectively (Figure 29). For an arbitrary angle ( between the waves and the current, this yields:

${{\theta }_{cw,onshore}}={\frac {1}{{T}_{wc}}}\int _{0}^{{T}_{wc}}{{{\theta }_{cw}}\left(t\right)dt}$

(94)

${{\theta }_{cw,offshore}}={\frac {1}{{T}_{wt}}}\int _{{T}_{wc}}^{{T}_{w}}{{{\theta }_{cw}}\left(t\right)dt}$

(95)

where T_wc and T_wt are the half-periods where the instantaneous velocity u = U_c cos + u_w(t) (or instantaneous Shields parameter) is onshore (> 0) or offshore (< 0), respectively. The instantaneous Shields parameter is defined as follows:

${{\theta }_{cw}}\left(t\right)={\frac {{\frac {1}{2}}{{f}_{cw}}\left|{{U}_{c}}\cos \varphi +{{u}_{w}}\left(t\right)\right|\left({{U}_{c}}\cos \varphi +{{u}_{w}}\left(t\right)\right)}{\left(s-1\right)g{{d}_{50}}}}$

(96)

where f_cw is the friction coefficient for an interaction between wave and current. This coefficient is assumed to be constant over the wave period as a first approximation but, as already noted, a temporal variation exists. Madsen and Grant (1976) suggested that f_cw could be obtained as a linear combination of the current-related (f_c) and wave-related (f_w) friction coefficients according to Equation 66.

(a) (b)

Figure 29. Definition sketches for (a) wave and current interaction, and (b) typical velocity variation over wave period in direction of wave propagation including effect of steady current.

Appendix B compiles analytical expressions for θ_cw,onshore and θ_cw,offshore for different wave theories or approximations. The assumption that the moving sediment layer thickness Δ_s is proportional to the total mean bottom shear stress gives a dependence on θ_cw,m for the transport rate (the stirring at the bottom by the waves and current together is mobilizing the layer). For an arbitrary angle  between the waves and the current, the mean and maximum combined Shields parameters at the bottom θ_cw,m and θ_cw, respectively, are written:

${{\theta }_{cw,m}}={{\left(\theta _{c}^{2}+\theta _{w,m}^{2}+2\ {{\theta }_{w,m}}\ {{\theta }_{c}}\ \cos \varphi \right)}^{1/2}}$

(97)

${{\theta }_{cw}}={{\left(\theta _{c}^{2}+\theta _{w}^{2}+2\ {{\theta }_{w}}\ {{\theta }_{c}}\ \cos \varphi \right)}^{1/2}}$

(98)

where the magnitude of the mean (maximum) shear stresses from the waves and current were added (vector addition). A more sophisticated approach would include the time variation of the waves in this addition. In the derivation of a transport formula where waves contribute to the transporting velocity, it is convenient to determine the transport rate parallel and perpendicular to the direction of wave propagation. Then, for an arbitrary angle  between the waves and the current, Equation 81 can be written in the following manner.

$\left\{{\begin{aligned}&{{\Phi }_{w}}={{a}_{w}}{\frac {{{\theta }_{cw,onshore}}+{{\theta }_{offshore}}}{\sqrt {\left|{{\theta }_{cw,onshore}}+{{\theta }_{offshore}}\right|}}}{{\theta }_{cw,m}}\ \exp \left(-b{\frac {{\theta }_{cr}}{{\theta }_{cw}}}\right)\\&{{\Phi }_{n}}={{a}_{n}}{\frac {{\theta }_{cn}}{\sqrt {\left|{{\theta }_{cn}}\right|}}}{{\theta }_{cw,m}}\ \exp \left(-b{\frac {{\theta }_{cr}}{{\theta }_{cw}}}\right)\\\end{aligned}}\right.$

(99)

where w and n correspond, respectively, to the wave direction and the direction normal to the wave direction, ${{\theta }_{cn}}={\frac {1}{2}}{{f}_{c}}U_{c}^{2}\sin \varphi \left|\sin \varphi \right|/\left(\left(s-1\right)g{{d}_{50}}\right)$ , and a_w, a_n, and b are empirical coefficients as before. The same value b = 4.5 is retained. For consistency with previous results obtained for current only and waves only, the following relationship is proposed for the coefficient a_w:

${{a}_{w}}=6+6Y{\text{ with }}Y={\frac {{\theta }_{c}}{{{\theta }_{c}}+{{\theta }_{w}}}}$

(100)

Also, we take a_n = 12. The overall effect of the critical shear stress on the sediment transport over a wave period may be estimated using the same approximation as for the steady current. However, because Soulsby (1997, pp. 104-106) proposed to compute the critical shear stress based on the maximum shear stress, the maximum Shields parameter θ_cw is used. For  = 90 deg and ${{\theta }_{w}}\gg {{\theta }_{c}}$ , ${{\theta }_{cw}}\approx {{\theta }_{w}}$ , that is, the waves are perpendicular to the current and control the stirring of the sediment. If θ_c = 0 (no current) with sinusoidal waves, θ_cw = θ_w and θ_cw,onshore + θ_cw,offshore = 0, which gives no bed load unless wave asymmetry is included.

Comparison with experimental data

To investigate bed-load transport where waves and current are interacting, several existing data sets were compiled and analyzed (Table 7). Most of the data are from OWT experiments implying large orbital velocities with prevailing bed-load transport. More recently, Dohmen-Janssen and Hanes (2002) carried out experiments in a large-scale wave flume (LWF). For nonbreaking waves, they found that the bed-load sediment transport represents around 90 percent of the total load. These experiments showed that results are consistent with those observed in oscillating water tunnels (OWTs), although differences in the suspended sediment concentration and the total sediment transport rate are apparent. Net transport rates under waves were found to be about a factor 2.5 larger than in uniform horizontal oscillatory flows. They explained this by referring to the differences between boundary layer flows in OWT sand under free surface gravity waves. This difference may partly be attributed to the onshore-directed boundary layer streaming that is present under waves and is absent in horizontal oscillatory flow. Figure 30(a) plots the bed-load transport predicted by Equation 99 against experimental data for current and waves interacting (with U_c > 0.02 m/sec). Good agreement is observed, and more than 50 percent of the data points are predicted within a factor of 2 of the measured values. A major part of the discrepancy (E_rms  2, see Table 9) can be explained by the difficulty in predicting roughness values for oscillatory sheet flow (no measured values available) and because phase-lag effects are not included in the formula (some large overestimations or even incorrect transport directions are observed if strong phase lag occurs, as indicated in Figure 28). The sediment transport rates for the Delft Hydraulic data set are slightly overestimated if the Wilson formula is used and better predicted using the Nikuradse skin roughness (Figure 30(b)). On the other hand, the sediment transport rates for the Dibajnia and Watanabe (1992) data set are underestimated if the skin friction is used and better predicted if the Wilson formula is employed. The four data points from the large wave flume experiment are well predicted (within a factor of 2 of the measured values). It seems that the differences observed by Dohmen-Janssen and Hanes (2002) between the LWF and OWT experiments for the linear relationship between the net transport rates and the third-power velocity moment are reduced by inclusion of the friction coefficient (within the Shields parameter). Finally, contrary to comparisons with other formulas, good agreement is observed for the Watanabe and Isobe (1990) data set, corresponding to small wave orbital velocities and periods. This means that the effect of the critical Shields parameter is well described. It should be noted that for this data set, ripples were sometimes observed, which induced stronger suspension and phase-lag effects. Appearance of ripples may explain the opposite direction of the sediment transport observed for some of the data points as pointed out by Dibajnia and Watanabe (1992).

Comparison with existing formulas for waves and current

Only the Bailard and Inman (1981), Dibajnia and Watanabe (1992), and Ribberink (1998) formulas include the effects of a current. Table 9 lists the percentage of predicted values, within a factor of 2 or a factor of 5 deviation from the measurements, for the four studied formulas presented for all the data where a mean current is present using the Wilson relationship (k_s = k_s,sfw) or the skin friction (k_s = 2d₅₀) to compute the Nikuradse roughness. As seen in Table 9, a larger discrepancy exists compared to the steady current data, because the total shear stress has to be estimated instead of using measured values. Equation 99 still yields the best overall results. The Bailard and Inman (1981) formula overestimates the sediment transport rate for low shear stresses (e.g., the Watanabe and Isobe (1990) data set), and underestimates the rate for the Dibajnia and Watanabe (1992) data set and for all the cases where strong phase lag occurs. Nevertheless, it gives good overall agreement with the data. As already observed, the Ribberink (1998) formula overestimates the transport rates when waves are present. Results are improved using the Nikuradse skin roughness, but they remain poor compared to the other predictive formulas. Finally, although the Dibajnia and Watanabe (1992) formula shows good overall results, it overestimates the data from Delft Hydraulics and seems not to be adequately sensitive to the bottom shear stress.

(a)
(b)

'Figure 30. Comparison between bed-load transport predicted by Equation 99 and experimental data with current ('''Uc '> 0.02 m/sec) (a) where '''k_s 'is calculated using Wilson (1989a) formula, and (b) where '''ks '= 2'''d₅₀.

Table 9. Prediction of bed-load transport rate within factor of 2 and 5 of
measured values and root-mean-square errors using data on waves
and current combined ('''Uc'''_' > 0.02 m/sec).

Author(s)	All Data ('''ks '''''= ks''','''sfw)					All Data ('''ks '''''= '''2'''d50)
	Px2 (%)	Px5 (%)	Erms	Px2 (%)	Px5 (%)	Erms
Bailard & Inman	47	70	2.07	41	68	2.10
Dibajnia & Watanabe	41	72	1.74	41	72	1.74
Ribberink	18	44	4.13	32	52	4.01
Equation 99	54	77	2.03	46	74	2.05

Phase-lag effects on sediment transport in sheet flow

Introduction

Manohar (1955) performed one of the first studies on the inception of sheet flow (disappearance of the wave ripples) by using an oscillatory tray facility. In more recent times, sheet flow processes have typically been studied in oscillating water tunnels because field measurements close to the bed are not possible during storms, when sheet flow should prevail (e.g., Horikawa et al. 1982; Sawamoto and Yamashita 1986; King 1991; Asano 1992; Dibajnia and Watanabe 1992; Ribberink and Chen 1993; Ribberink and Al-Salem 1994; Li and Sawamoto 1995; Zala Flores and Sleath 1998; Dohmen-Janssen 1999; and Dohmen-Janssen et al. 2002). These studies considerably improved knowledge on the thickness of the sheet-flow layer, the time-dependent concentration profile inside the sheet-flow layer, and the resulting sediment transport rates. Ribberink (1998) showed that a quasi-steady formula based on the Meyer-Peter and Müller (1948) concept successfully describes the net sediment transport rates with the exception of relatively fine sands (d₅₀ < 0.2 mm) and short wave periods (T_w < 3 sec). Dibajnia and Watanabe (1992) carried out experiments with short period waves and found that in many cases a quasi-steady transport model failed to describe the magnitude and the direction of the net transport rate. They hypothesized that the phase lag between velocity and concentration was responsible for this discrepancy. Similar observations were made by Ribberink and Chen (1993) with fine sediment (d₅₀ = 0.13 mm) and large asymmetric waves. Dohmen-Janssen et al. (2002) extended the data set on net transport rates and time-dependent velocities and concentrations under sheet-flow conditions during sinusoidal oscillatory flow combined with a net current for sand with different grain sizes (i.e., d₅₀ = 0.13, 0.21, and 0.32 mm). They suggested that this limitation of quasi-steady models may be due to a phase lag, such as delayed entrainment and settling of sand grains, and proposed a simple model to take these effects into account. The lag between the sediment concentration and the flow was characterized by the ratio between the fall time of the sediment particle (which may be represented by the ratio between the sheet-flow layer thickness and the settling velocity) and the wave period. Although knowledge on phase lag in sheet flow has improved recently, only a few models allow for a quantitative description of this phenomenon. The formula proposed by Dibajnia and Watanabe (1992) appears to be the only one that shows a correct behavior in the sheet-flow regime (Camenen and Larroudé 2003). More recently, Dohmen-Janssen et al. (2002) proposed a correction of the sediment transport (based on the Ribberink (1998) formula if a phase lag occurs that allows for improved estimates of the net bed-load transport rate. Because sheet flow sediment transport occurs near the bed, it is often assumed that the response time of the sediment is much shorter than the wave period. However, in practice, a certain delay exists for the sand to respond to the fluid. Thus, the quantity of sediment in suspension is primarily related to the instantaneous velocity, but it also depends on the settling velocity. In oscillating flow, not all sand grains put into suspension during the first half-period are transported and settle during this period. The proportion still in suspension is then carried away in the opposite direction during the second half-period.

A simple conceptual model

A simple conceptual model was introduced by Dohmen-Janssen (1999) to include the phase lag in sediment transport: the instantaneous bed-load transport is assumed to be proportional to the instantaneous sediment concentration at the bottom (taken constant over the sheet-flow layer) multiplied by the instantaneous horizontal velocity at the bottom. Assuming that the instantaneous sediment concentration is a function of the instantaneous velocity to the power 2, but with a possible phase lag _s, the effect of this phase lag on the sediment transport may be estimated. For a sinusoidal wave, the sediment transport reduction due to a phase lag may be expressed as follows:

${\begin{aligned}&{{r}_{pl1}}={\frac {{q}_{s,net}}{{q}_{s,net,{{\phi }_{s}}=0}}}={\frac {\int _{0}^{{T}_{w}}{\left({{r}_{u}}+\cos \omega t\right){{\left[{{r}_{u}}+\cos \left(\omega t+{{\phi }_{s}}\right)\right]}^{2}}dt}}{\int _{0}^{{T}_{w}}{{{\left({{r}_{u}}+\cos \omega t\right)}^{3}}dt}}}\\&\quad \,\ \,={\frac {r_{u}^{2}+1/2+X}{r_{u}^{2}+3/2}}\\\end{aligned}}$

(101)

where r_u = U_c/U_w, in which U_c is the mean current averaged over the depth, and X = cos _s. An analytical solution also exists for a second-order wave assuming u_w(cos t + r_w 2t) (see also Figure 33):

${\begin{aligned}&{{r}_{pl2}}={\frac {\int _{0}^{{T}_{w}}{\left({{r}_{u}}+\cos \omega t+{{r}_{w}}\cos 2\omega t\right)}{{\left[{{r}_{u}}+\cos \left(\omega t+{{\phi }_{s}}\right)+{{r}_{w}}\cos 2\left(\omega t+{{\phi }_{s}}\right)\right]}^{2}}dt}{\int _{0}^{{T}_{w}}{{{\left({{r}_{u}}+\cos \omega t+{{r}_{w}}\cos 2\omega t\right)}^{3}}dt}}}\\&\quad \ \,\,={\frac {r_{u}^{3}+{{r}_{u}}\left[1/2+X+r_{w}^{2}\left(2{{X}^{2}}-1/2\right)\right]+{{r}_{w}}\left({{X}^{2}}+X-1/2\right)/2}{r_{u}^{3}+3/2{{r}_{u}}\left(1+r_{w}^{2}\right)+3/4\ {{r}_{w}}}}\\\end{aligned}}$

(102)

Equations 101 and 102 are functions only of the wave profile (i.e., of the coefficients r_u and r_w) and the phase lag of the sediment suspension _s. Figure 31 shows the effect of the coefficients r_u and _s in Equation 101 (r_w = 0) displayed. As observed by Ribberink and Chen (1993) and Ahmed and Sato (2003), sediment transport in the direction opposite to the waves is possible for large values on _s and if ${{r}_{u}}<{\sqrt {2}}/2$ (sinusoidal waves) or $r_{u}^{3}+{{r}_{u}}\left(3r_{w}^{2}-1\right)/2-1/4{{r}_{w}}<0$ (second-order wave). It should be noted that Equation 102 does not have a solution if the denominator $Den=r_{u}^{3}+3/2\ {{r}_{}}\left(1+r_{w}^{2}\right)+3/4\ {{r}_{w}}=0$ (the net sediment transport rate equals zero if no phase lag is assumed). The asymmetry factor r_w modifies the results significantly, particularly if Den is close to zero where the function diverges (Figure 32). If Equation 101 yields a minimum value for r_pl equal to -0.35, Equation 102 produces much smaller values on r_pl if the mean current is relatively weak (min(r_pl) ( 0.5 when r_u = 0) and opposite to the waves (r_pl ( -1.1, if r_u = 0.05, r_w = 0.20 and X = cos (_s ( (/2). Furthermore, Equation 102 is not a symmetric function of r_u and r_pl, and it may be greater than 1 if Den < 0 and close to zero (Figure 32).

(a)	(b)

Figure 31. Phase-lag effect on sediment transport for sinusoidal wave with superimposed current when phase lag _s 'is introduced for concentration at bottom: (a) instantaneous variation in velocity, concentration, and bed-load rate for '''Uc/'''Uw '= 0.2 and _s '''''''= -0.2,
and (b) effect on sediment transport for varying s 'and ratio '''Uc/'''Uw.

(a)	(b)

'Figure 32. Phase-lag effects on sediment transport for second-order Stokes wave with (a) positive, or (b) negative, and adding current introducing phase lag s '''''''for concentration at bottom and with '''rw '= 0.2.

Dohmen-Janssen (1999) and Dohmen-Janssen et al. (2002) proposed a more complex model where the sediment concentration varies over the sheet-flow layer following an exponential law. The solution obtained is similar to Equation 101 (or Equation 102, if a 2^nd order Stokes wave is used) in which X = cos _s is replaced by a function of the phase-lag parameter p_pl defined as in Equation 93. However, the model does not allow for small values on r_pl (nor negative values) for large phase lag (the minimum value of r_pl is 0.35 for a sinusoidal wave, and it is 0.0 for a second-order Stokes wave), but may indicate a possible relationship between _s and p_pl.

Dibajnia and Watanabe formula

Dibajnia and Watanabe (1992) proposed a sediment transport model that accounts for the phase lag. They considered the amount of sand that is entrained during a positive half-wave cycle. They determined the part that will be transported directly by the positive velocity during the first half cycle and the part that will still be in suspension as the flow reverses and therefore transported by the negative velocity during the following half cycle (see Figure 33). The same concept was proposed for the negative half cycle.

Figure 33. Notation for colinear wave and current interaction.

Dibajnia and Watanabe (1992) assumed that phase-lag effects occur as soon as sheet flow appears. Thus, following the criterion for the inception of sheet flow developed by Dibajnia (1991, Equation 55), they proposed to use the parameter ω_pl (Equation 54) for each half period:

${{\omega }_{pl,j}}={\frac {{\frac {1}{2}}U_{wj}^{2}}{\left(s-1\right)g{{W}_{s}}{{T}_{wj}}}}$

(103)

where U_wj is the representative velocity for the positive or negative half period T_wj, and the subscript j should be replaced either by onshore (direction of the wave) or offshore (opposite direction to the wave). Dibajnia and Watanabe (1992) assigned the indices c for crest and t for trough, instead of onshore and offshore, respectively. According to their data, sheet flow occurs as soon as ω_pl,j > 1. If no phase lag occurs, their formula is proportional to the mobility parameter for both half periods Ψ_j (where U_w is replaced by U_wj) as Ω_j = W_sT_j/d₅₀ and ${{{\Omega }'}_{j}}=0$ . For the case where sheet flow is reached during both half periods, the phase-lag effects may thus be quantified as follows:

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Double exponent: use braces to clarify"): {\displaystyle {{r}_{pl,DW}}={{\left[{\frac {{{T}_{wc}}{{U}_{wc}}\left(\Omega _{c}^{3}+{\Omega }'_{t}^{\,3}\right)-{{T}_{wt}}{{U}_{wt}}\left(\Omega _{t}^{3}+{\Omega }'_{c}^{\,3}\right)}{{{T}_{wc}}{{U}_{wc}}\Psi _{c}^{3}-{{Y}_{wt}}{{U}_{wt}}\Psi _{t}^{3}}}\right]}^{0.55}}}

(104)

where subscript c indicates crest and t trough, and:

${{\Omega }_{J}}=\max \left({{\omega }_{pl,j}},1\right){\frac {{{W}_{s}}{{T}_{j}}}{{d}_{50}}}{\text{ and }}{{{\Omega }'}_{j}}=\left({{\omega }_{pl,j}}-1\right){\frac {{{W}_{s}}{{T}_{j}}}{{d}_{50}}}$

(105)

Dibajnia and Watanabe (1992) slightly modified their formula to take into account the phase lag occurring in the ripple regime by introducing the parameter ω_cr. Thus, phase lag occurs if ω_pl,j > ω_cr, where ω_cr = 1 for the sheet-flow regime, and ω_cr = 0.03 for the ripple regime.

Modification of Camenen and Larson formula for phase lag

Following Dibajnia and Watanabe (1992), a modification of the Camenen and Larson (2005a, b) formula is proposed to take the phase lag into account. This formula is based on the "bed-shear stress concept" (function of the Shields parameter) and allows sediment transport in the direction of the wave using characteristic values on the Shields parameter for both half periods: θ_cw,onshore (> 0) and θ_cw,offshore (< 0). These parameters correspond to the mean value of the instantaneous Shields parameter ${{\theta }_{cw}}\left(t\right)=1/2{{f}_{cw}}\left({{U}_{c}}\cos \varphi +{{u}_{w}}\left(t\right)\right)\left|{{U}_{c}}\cos \right.\varphi +{{u}_{w}}\left(t\right)\left|/\left(\left(s-1\right)g{{d}_{50}}\right)\right.$ over each half period where ( is the angle between the wave and current directions, and f_cw is obtained using the Madsen and Grant (1976) relationship (Equation 96). Thus, the net sediment transport depends on the factor θ_cw,net = θ_cw,onshore + θ_cw,offshore. In the wave direction, the bed-load sediment transport may be expressed as follows (compare with Equation 99):

${{\Phi }_{w}}={{a}_{w}}{\frac {{\theta }_{cw,net}}{\sqrt {\left|{{\theta }_{cw,net}}\right|}}}{{\theta }_{cw,m}}\exp \left(-b{\frac {{\theta }_{cr}}{{\theta }_{cw}}}\right)$

(106)

where a_w and b are coefficients, and θ_cw,m, θ_cw, and θ_cr are the mean values of the absolute instantaneous Shields parameter, the maximum Shields parameter, and the critical Shields parameter for the inception of motion, respectively. The sediment phase lag is introduced in the formula by assuming that values of the Shields parameter for both half periods are modified due to this effect. A decrease in θ_cw,onshore and an increase in θ_cw,offshore appear if the critical velocity for inception of sheet flow is reached:

${{\theta }_{cw,net}}=\left(1-{{\alpha }_{pl,b}}\right){{\theta }_{cw,onshore}}+\left(1+{{\alpha }_{pl,b}}\right){{\theta }_{cw,offshore}}$

(107)

in which _pl,b = _onshore - _offshore and:

${{\alpha }_{j}}={\frac {{{\nu }^{0.25}}U_{wj}^{0.5}}{{{W}_{s}}{{T}_{j}}^{0.75}}}\exp \left[-{{\left({\frac {{U}_{w,cr,sf}}{{U}_{cwj}}}\right)}^{2}}\right]$

(108)

where U_cwj is the root-mean-square value of the velocity (waves and current combined) over the half period T_wj, and the subscript j should be replaced either by onshore or offshore. The effect of the main parameters (i.e., U_cwj, T_j, and W_s) was examined, and they were calibrated based on the compiled data set. The exponential function, describing the effect of the critical velocity on the inception of sheet flow, is proposed to allow a possible error in the estimation of U_w,cr,sf (U_w,cr,sf is calculated following Equation 57), i.e., ${{\alpha }_{j}}\neq 0$ even if U_cwj is slightly lower than U_w,cr,sf. Strictly, U_w,cr,sf should be compared to a maximum value of the velocity and not to a quadratic mean value. However, the results would only be modified by a constant factor in the exponential function, changing the fit of the empirical function α_j. Thus, the coefficient quantifying the phase lag may be written as follows:

${{r}_{pl,CL}}={{\left[{\frac {\left(1-{{\alpha }_{pl,b}}\right){{\theta }_{w,onshore}}+\left(1+{{\alpha }_{pl,b}}\right){{\theta }_{w,offshore}}}{{{\theta }_{w,onshore}}+{{\theta }_{w,offshore}}}}\right]}^{0.5}}$

(109)

Experimental data

To investigate phase-lag effects on bed-load transport under waves and current, data sets from OWT experiments covering a wide range in values were compiled and analyzed. This kind of experiment has several advantages for the present study: large orbital velocities can be reached, bed-load transport prevails, and a strong phase lag is often observed. Table 10 summarizes the data sets employed, where the type of experiment, sediment characteristics, and wave properties are listed.

Table 10. Summary of data on bed-load sediment transport in full-cycle oscillatory flow.

Author(s)	Number	s	d50 (mm)	Uc (m/sec)	Uw,max '(m/sec)'	Tw '(sec)'
Watanabe & Isobe (1990)	65	2.65	0.18, 0.87	-0.3 - 0.25	0.27-0.52	3.0, 6.0
Dibajnia & Watanabe (1992)	76	2.65	0.20	-0.26 - 0.22	0.61-1.24	1.0-4.0
Ribberink & Chen (1993)	4	2.65	0.128	< 0.05	0.6-1.2	6.5
Ribberink & Al Salem (1994)	30	2.65	0.21	-0.11 - 0.56	0.37-1.37	5.0-12.0
Delft Hydraulics (1994-1999; see Van Rijn et al. 2001)	22	2.65	0.13-0.24	-0.45 - 0.47	0.46-1.49	4.0-12.0
Dohmen-Janssen (1999)	27	2.65	0.13-0.32	0.23-0.45	0.46-1.85	4.0-12.0
Ahmed & Sato (2003)	15	2.65	0.21-0.74	-	1.16-1.85	3.0

Calibration of conceptual model

Calibration of the conceptual model is not as easy as the modification of the Camenen and Larson (2005a, b) formula. Indeed, it is difficult to estimate what the sediment transport rate would have been without phase-lag effects. Three methods are proposed to estimate r_pl following Camenen and Larroudé (2003) by using a sensitivity analysis of the different parameters. Thus, as soon as no phase lag occurs, bed load might be:

• Independent of median grain size.
• Independent of wave period.
• Proportional to velocity moment to the power 3.

Using experimental data where all the parameters are fixed except the median grain size (or the wave period), it is possible to estimate r_pl assuming r_pl = 1 for larger values of d₅₀ (or T_w), i.e., ${{q}_{s,net}}={{q}_{s,net,{{\phi }_{s}}}}=0$ . For the third method, the relationship between the bed-load rate and the velocity moment to the power 3 is assumed exact, if U_w is small. A curve proportional to $<{{u}^{3}}>$ is fitted to the experimental value of q_s,net where the measured value of U_w is the smallest, and compared to the other experimental values of q_s,net where U_w is higher (and where phase-lag effects are expected). In total, 85 data values were derived using these three methods. It should be noted that several values do not have a solution by Equation 102; the estimated value of r_pl was found to be smaller than the minimum value obtained by the model (min (r_pl)  -0.5 when r = 0). In these cases, the _s value corresponding to the minimum of r_pl was used. This illustrates the limitations of the conceptual model. However, these errors may also be induced by some limitations in the experimental set-up. Indeed, for some data from the Dibajnia (1991) experiment, r_pl reaches -5. As discussed previously, the piston generated additional vortices for the shortest wave periods that induced more suspension, and thus artificially increased the phase-lag effect. It seems that _s is well described by the parameter ω_pl (Equation 54) introduced by Dibajnia (1991), using U_w and T_w instead of U_wj and T_wj. The following empirical relationship is proposed (curve in Figure 34(a)):

${{\phi }_{s}}=\pi \tanh \left(1.5\omega _{pl}^{1.5}\right)$

(110)

(a)	(b)

Figure 34. Calibration of conceptual model against data.

Figure 34(a) plots the points obtained for _s from the three different methods against ω_pl. Although significant scatter is observed, most of the data are correctly predicted by Equation 110 (solid line; the dashed lines correspond to Equation 110 with ${{{\omega }'}_{pl}}$ = 0.5 ω_pl and 2ω_pl). Figure 34(b) shows a comparison between the observed and predicted values on r_pl from the experiment (using Equations 102 and 110). The factor r_pl is generally well predicted; it should be remembered that large uncertainties are induced through the calculation of r_pl. Even if large underestimation occurs for extreme cases (from the wave period method) where r_pl < -0.5, the general tendency obtained by Equations 102 and 110 is encouraging.

Influence of median grain size

Strong phase-lag effects were first observed for fine sand (d₅₀ = 0.2 mm, Dibajnia 1991; d₅₀ = 0.13 mm, Ribberink and Chen 1993), where a net sediment transport opposite to the direction of the waves (and mean current) was observed. The specific study on grain-size influence on sediment transport in oscillatory sheet flow by Dohmen-Janssen (1999) clearly showed that the finer the sand is the larger the phase lag might be. The phase-lag parameter (p_pl, Equation 93) is inversely related to the settling velocity. An increasing sediment transport rate with increasing grain size is clearly observed in Figure 35(a). Bed-load formulas are in general not sensitive to the grain size, with a slight proportionality for coarser sand (see also Camenen and Larroudé 2003). The effect of the phase lag is significant for both the Dibajnia and Watanabe (1992) and the modified Camenen and Larson (2005b) formulas when d50 < 0.3 mm. The previous study by Watanabe and Isobe (1990) showed similar results for smaller wave periods (T_w = 3 sec). For coarser grain size, the Dibajnia and Watanabe (1992) formula predicts a decrease in bed-load transport (induced by the varying ω_cr) that is not observed experimentally. The effect of grain size on the phase lag was introduced in the Camenen and Larson (2005b) formula through the settling velocity of the coefficients α_j were found to be proportional to $W_{s}^{-1.0}$ .

Figure 35. Influence of grain size on bed load sediment transport (details of input parameters for cases a and b are given in Table 11).

Table 11. Experimental conditions for studied cases on median grain size effect.

Case	Data Set	h '(m)'	Uc '(m/sec)'	Uw '''''''(m/sec)	Tw '(sec)'	rw '(-)'
(a)	Dohmen-Janssen	0.80	0.24	1.08	7.2	0.00
(b)	Watanabe & Isobe	0.31	0.00	0.69	3.0	0.20

Influence of wave orbital velocity

The phase-lag effect is proportional to the wave orbital velocity, and the higher U_w is, the larger the amount of sediment put in suspension (the available energy is greater) and the larger the sheet flow layer thickness (δ_s). This implies a larger delay between the instantaneous concentration and the shear stress and fluid velocity. However, the experiments by Dohmen-Janssen (1999) do not show the effect of the wave orbital velocity clearly. Nevertheless, by comparing plots (Figure 36(a), (b)) it may be observed that the increase in the sediment transport rate with U_w is much slower for d50 = 0.13 mm (Figure 36(a)) compared to d50 = 0.21 mm. Al Salem (1993) observed that the sediment transport is approximately proportional to the velocity moment to the power 3 (for a sinusoidal wave together with a current, $<{{u}^{3}}>=U_{c}^{3}+{{U}_{c}}U_{w}^{2}$ ). If this relationship is applied to the data (Figure 36(b)) (i.e., ${{q}_{s}}\ \propto \ U_{w}^{2}$ ), it is found not to hold for case (a) where ${{q}_{s}}\ \propto \ U_{w}^{1.4}$ . For the experiments of Ribberink and Chen (1993) and Ahmed and Sato (2003), where the current was negligible (Figure 36(c) and (d)), the effect of the wave orbital velocity is obvious because the direction of the sediment transport changes with an increasing U_w. The Dibajnia and Watanabe (1992) formula, as well as the Camenen and Larson (2005b) formula with the coefficient r_pl2 (Equations 102 and 110), tends to be overly sensitive to the wave orbital velocity. Occasionally a decrease in the net sediment transport rate with an increasing wave orbital velocity is estimated, whereas the opposite behavior is observed in the experiment (Figure 36(a) and (b)). However, it seems like the sheet-flow layer and the associated phase-lag effects appear quite abruptly. Thus, predictive formulas are in general difficult to calibrate.

Figure 36. Influence of wave orbital velocity on sediment transport (values of input parameters for cases a, b, c, and d are given in Table 12).

Table 12. Experimental conditions for studied cases on wave orbital velocity effects.

Case	Data Set	d'50 '(mm)	h (m)	Uc '''''''(m/sec)	Tw '(sec)'	rw '(-)'
(a)	Dohmen-Janssen	0.13	0.80	0.24	7.2	0.00
(b)	Dohmen-Janssen	0.21	0.80	0.40	7.2	0.00
(c)	Ahmed & Sato	0.21	0.31	0.00	3.0	0.20
(d)	Ribberink & Chen	0.13	0.80	0.03	6.5	0.25

The effect of the wave orbital velocity on the phase lag was introduced in the Camenen and Larson (2005b) formula through the root-mean-square values of the velocity U_wj (wave and current) for each half-period T_wj. The coefficients α_j were found to be proportional to $U_{wj}^{0.5}$ .

Influence of wave period

The wave period is also a controlling factor for the phase lag and its effects on sediment transport: the shorter T_w is, the larger the amount of sediment still in suspension after half a period. Indeed, the delay in sediment settling before the change in the velocity direction strongly depends on the wave period. The two experiments by Dohmen-Janssen (1999) (with d₅₀ = 0.13 and 0.21 mm; Figure 37(a) and (b), respectively) clearly show an increase in sediment transport rate with an increase in wave period as soon as sheet flow occurs. A bed-load formula based on the shear stress displays an inverse behavior, because the friction coefficient f_w is inversely proportional to the wave period (because the boundary layer is getting thinner). The modified Camenen and Larson (2005b) formula exhibits the correct behavior, although the effect of f_w sometimes remains larger than the introduced phase-lag parameter (Figure 37(b)). The experiments by Dibajnia (1991) (Figure 37(c) and (d)) illustrate how strong the effect of short wave periods can be. A change in the sediment transport direction for T_w  2 sec (sediment transport was observed to be in the direction of the waves for T_w = 4 sec and U_c > -0.2 m/sec). However, this effect may be overestimated due to the limitations of the experimental set-up. Dibajnia (1991) pointed out that the piston used in his experiment could not smoothly follow the input signal for periods T_w  1.5 sec. The generated vortex caused larger suspension and increased the phase-lag effects more than what normally should occur. The Dibajnia and Watanabe (1992) and Camenen and Larson (2005b) formulas predict a negative sediment transport rate even for higher values on the wave period (Figure 37(c)). If a slight phase lag tends to decrease the absolute value of the sediment transport rate for the Dibajnia and Watanabe (1992) formula, it increases the rate for the modified Camenen and Larson (2005b) formula. This latter behavior agrees better with the measurements. However, even if it is difficult to verify the behavior with the present data, it seems like the Camenen and Larson (2005b) formula (Equations 99 and 107) is too sensitive to the wave period. In spite of this, except for case 3, the Camenen and Larson (2005b) formula with the calibrated conceptual model (Equations 101 and 110) presents better behavior than the other formulas with respect to variations in T_w. The effect of the wave period on the phase lag was introduced in the Camenen and Larson (2005b) formula through the half-periods T_wj, and the coefficients α_j were found to be proportional to $T_{wj}^{-0.75}$ .

Figure 37. Influence of wave period on sediment transport (details of input parameters for cases a, b, c, and d are given in Table 13).

Table 13. Experimental conditions for studied cases on wave period effects (for Dibajnia experiments, maximum onshore wave orbital velocity is
approximately constant).

Case	Data Set	d'50 '(mm)	h '(m)'	Uc '''''''(m/sec)	Uw '(m/sec)'	rw '(-)'
(a)	Dohmen-Janssen	0.13	0.80	0.25	1.06	0.00
(b)	Dohmen-Janssen	0.21	0.80	0.25	1.06	0.00
(c)	Dibajnia	0.20	0.22	-0.125	0.80	0.20
(d)	Dibajnia	0.20	0.22	0.125	0.80	0.20

Comparison with all data

A comparison with all the data is presented in this section. Figure 38 and Table 14 display the overall results shown for the original Camenen and Larson (2005b) formula (Figure 38(a)), the Dibajnia and Watanabe (1992) formula (Figure 38(b)), the Camenen and Larson (2005b) formula (Equations 99 and 107 and Figure 38(c)), and the Camenen and Larson (2005b) formula with the coefficient r_pl (Equations 102 and 110 and Figure 38(d)). It clearly shows how significanat the introduction of phase lag in the formulas is. Even if the overall results (in terms of predictive skill within a factor 2, Px2, or 5, Px5) are not substantially improved (except for the modified Camenen and Larson (2005b) formula where results are improved by 10 percent), the general behavior of the formulas are better, and a significant decrease in the standard deviation of Δq_s is obtained. It appears that the Dibajnia and Watanabe (1992) formula and the Camenen and Larson (2005b) formula with the coefficient r_pl tend to underestimate the absolute sediment transport rate if a phase lag occurs. However, the coefficient r_pl (Equation 102) is sensitive to the wave asymmetry if the current is negligible. Figures 35 to 37 reveal that the Dibajnia and Watanabe (1992) formula and Camenen and Larson (2005b) formula with the coefficient r_pl do not induce any change below a critical value (of ω_pl), and then tend to abruptly decrease the sediment transport rate. On the other hand, the modified Camenen and Larson (2005b) formula produces a more gradual modification of the sediment transport rate compared to the original formula (except with regard to the wave period), which seems closer to the trend in measurements. Finally, a comparison was made using the Camenen and Larson (2005b) formula with the parameter r_pl proposed by Dohmen-Janssen et al. (2002) with δ_s = α_s θ_cw d₅₀ (where α_s is a function of d₅₀: α_s = 13 when d₅₀ > 2.1 mm, and α_s = 35 when d₅₀ = 1.3 mm). Similar to Equation 102, it improves the results if a phase lag occurs. However, the effects are often not strong enough because this analytical formula does not allow for sediment transport in the opposite direction, as was observed in the Ribberink and Chen (1993) and the Ahmed and Sato (2003) experiments.

(a)	(b)
(c)	(d)

'Figure 38. Comparison between predicted and measured sediment transport rate using (a) original Camenen and Larson formula, (b) Dibajnia and Watanabe formula,
(c) modified Camenen and Larson formula (Equations 99 and 107), and
(d) Camenen and Larson formula with coefficient '''rpl'''2 '(Equations 102 and 110).

Table 14. Prediction of bed-load transport rate within factor of 2 or 5 of measured values, together with mean value and standard deviation of Δ'''qs'''_'.

0.11

0.22

Author(s)	Px'''2 (%)	Px'''5 (%)	${\overline {\Delta {{q}_{s}}}}$	Std ('''qs)
Dibajnia and Watanabe (1992)	42	75
7.0
Camenen and Larson (2005b)	47	72
9.8
Equations 99 and 107	53	81	-0.37	4.5
Camenen and Larson (2005b) with Equations 102 & 110	50	74	-0.003	7.0
Camenen and Larson (2005b) with Dohmen-Janssen et al. equation (2002) for rpl	48	71	-0.003	6.8

Concluding remarks on phase-lag effects

In the present study, a large data set on sheet-flow transport was compiled and analyzed to improve the prediction of the bed-load transport rate if a phase-lag effect occurs. A simple conceptual model was proposed as well as a modification of the Camenen and Larson (2006) formula. This modification was suggested by the Dibajnia and Watanabe (1992) formula, which is the first and only formula found that takes into account the phase lag between the sediment transport rate and fluid velocity. The conceptual model for the correction of the sediment transport rate is based on the work by Dohmen-Janssen (1999), who assumed that the sediment transport is proportional to the product of the instantaneous velocity and concentration. In turn, the instantaneous concentration is assumed to depend on the square of the instantaneous velocity with a phase lag _s > 0. This simple model allows a ratio ${{r}_{pl}}={{q}_{s,net}}={{q}_{s,net,{{\phi }_{s}}}}=0$ to be derived that can reach -0.35 for a sinusoidal wave and -0.5 for a second-order Stokes wave. However, for a second-order Stokes wave, the model may diverge if the prediction is equal to zero without the phase-lag effects. The phase lag (_s was calibrated against data, and (_s was found to be a function of the parameter ω_pl proposed by Dibajnia (1991). Even if significant scatter is observed, this function improves the behavior of the Camenen and Larson (2005b) formula markedly, and it is applicable to any other sediment transport formula. It also shows somewhat improved results compared to the analytical model by Dohmen-Janssen et al. (2002). A modification of the Camenen and Larson (2005b) formula was also proposed (Equations 107 and 108). It assumes that the characteristic Shields parameters for each half period θ_cw,onshore (> 0) and θ_cw,offshore (< 0) are modified if the sheet flow is reached. A decrease of ${{\alpha }_{pl}}\%=\left({{\alpha }_{c}}-{{\alpha }_{t}}\right)\%$ on $\left|{{\theta }_{w,onshore}}\right|$ and an increase of α_pl% on $\left|{{\theta }_{w,offshore}}\right|$ introduce a general decrease in the net sediment transport rate, and it may also change the direction of the sediment transport. The coefficients α_j (j = c or t) were calibrated with the compiled data and were found to be proportional to the root-mean-square velocity over the half period and inversely proportional to the settling velocity and the half period T_wj. These coefficient values are similar to those proposed by Dibajnia (1991, ω_pl) and Dohmen-Janssen (1999, p_pl). The new formula presents the best overall agreement with the data (Camenen and Larson 2006).

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Chapter 2 - General Sediment Transport Properties

Chapter 4 - Suspended Load