# 6.5.3: Instantaneous bed load transport

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)As an example of a bed-load transport formulation of the form of Eq. 6.5.2.2 we mention the quasi-steady bed-load transport formulation according to Ribberink (1998). The author assumed that the sand transport rate is a function of the difference between the actual time-dependent non-dimensional bed shear stress and the critical bed shear stress:

\[\Phi_b (t) = \dfrac{S_b (t)}{\sqrt{(s - 1)g D_{50}^3}} = 9.1 \dfrac{\beta_s}{(1 - p)} \{|\theta' (t)| - \theta_{cr} \}^{1.8} \dfrac{\theta' (t)}{|\theta' (t)|}\label{eq6.5.3.1}\]

The values of the coefficients in this equation were derived using various datasets of sediment transport in oscillatory flow over horizontal beds. Therefore the formulation is supplemented with a correction \(\beta_s\) to account for slope effects on the transport (see Intermezzo 6.2). The modulus is used to obtain the correct direction of transport, i.e. in the direction of the instantaneous bed shear stress. The transport magnitude is dependent on the shear stress magnitude to the power 1.8. The transport rate is given in volumes per unit time and width of deposited material (see the factor (\(1 - p\)) in Eq. \(\ref{eq6.5.3.1}\)).

An often neglected transport component is the one due to gravitation along a sloping bed. In the case of a sloping bed, not only the effects of slope on the initiation of motion (see Eq. 6.3.2.3) have to be taken into account, but also the transport directly induced by gravity when the grains have been set in motion. The bed load formulations are mostly derived (from data sets) for a horizontal bed and do not automatically include the effect of slope. Nevertheless, we expect moving grains to rather go downhill than uphill. This gives an additional transport component which is directed downhill. Therefore, sometimes a slope correction parameter is introduced (after Bagnold, see also Sect. 6.7.2) which increases the transport rates for downslope transport and decreases the transport rates for upslope transport.

The downhill gravitational transport component has a smoothing effect on the bed topography. This effect is of great importance for the morphological stability of the bed and for the equilibrium state to which the bed topography tends (but which it probably never reaches, since this state is a function of the ever changing input conditions).

To calculate \(S_b (t)\) we need to compute the instantaneous dimensionless bed shear stress (Eq. 6.5.2.3). As already indicated in Ch. 5, the computation of the time-averaged – let alone the instantaneous – bed shear stress under a combined wave-current motion is not straightforward at all. Without detailed modelling of the vertical velocity structure and turbulence, the computation of \(\theta' (t)\) is most easily done using a quadratic friction law (see Sects. 5.4.3 and 5.5.5).

Grant and Madsen (1979) suggested expressing the bed shear stress as a quadratic function of the combined wave/current velocity at some height \(z\) above the bed. With the velocity \(u_0 (t)\) at the top of the wave boundary layer (\(z = \delta\)), we have:

\[\tau_b (t) = 1/2 \rho f_{cw}' |u_0 (t)| u_0 (t)\label{eq6.5.3.2}\]

in which \(f_{cw}'\) is a (skin) friction factor for the combined wave-current motion and \(u_0\) is the time-dependent (intra-wave, that is: within the wave period) near-bottom horizontal velocity vector of the combined wave-current motion. Compare Eq. \(\ref{eq6.5.3.2}\) with Eqs. 5.4.3.4 and 5.5.5.5. In principle the problem is 2DH since waves and currents may interact under an arbitrary angle. The bed-shear stress \(\tau_b\) and near-bed velocity \(u_0\) are vectors in the same direction with varying magnitudes and varying directions during the wave cycle.

The velocity \(u_0\) should be representative for irregular wave groups and therefore con- tain contributions due to wave skewness and asymmetry, wave group related amplitude modulation and bound long waves. These contributions to \(u_0(t)\) were described in detail in Ch. 5 and follow from an appropriate wave theory. In addition, the mean flow at the top of the wave boundary layer must be taken into account, for instance by solving the mean flow in the vertical. Wave-induced contributions to the near-bed mean velocity are Longuet-Higgins streaming, undertow and longshore current, as already discussed in Ch. 5. The near-bed velocity vector is then computed as the vector addition of the near-bed oscillatory velocity signal and the time-averaged velocity at the same height.

We then have for the time-dependent \(\theta' (t)\):

\[\theta' (t) = \dfrac{1/2 \rho f_{cw}' |u_0 (t)| u_0 (t)}{(\rho_s -\rho) gD_{50}}\]

We have now reduced the problem to the determination of the skin friction factor \(f_{cw}'\). This is the major unknown and therefore the bottleneck in many transport computations. It is dependent on amongst others the bed roughness, which is highly variable in nature and not easily measured in practical applications. This is why results from laboratory experiments are still widely used to find a relationship between the friction factor and the bed roughness.

Note that since the friction factor \(f_{cw}'\) is a skin-friction factor (related to grains only and not to the bed forms) we need to use roughness heights related to the grain size and not the height of bed forms.

Grant and Madsen determine the friction coefficient \(f_{cw}'\) for currents and waves in combination by first computing the (skin) friction factors for ‘waves alone’ and ‘currents in the presence of waves’. This can be done using relationships as presented in Sect. 5.4.3 (but with \(f_c'\) evaluated using the mean velocity at the top of wave boundary layer). Note further that currents in the presence of waves experience an increased roughness due to the wave boundary layer. Next, \(f_{cw}'\) is calculated by weighting \(f_w'\) and \(f_c'\) linearly with the relative strength of the near-bed net current and oscillatory velocity amplitude. Others propose different models, resulting in different relative contributions of waves and currents to the bed shear stress.

The net wave-averaged bed-load transport rate can be obtained by averaging of the time-dependent transport vector \(S_b (t)\) over the duration of the imposed near bottom velocity time series. It includes the transport by the mean current as well as the net transport as a result of the oscillatory wave motion. Due to the non-linear relation between velocity and shear stress, the latter contribution will be zero only for a completely symmetric velocity signal.

At small shear stresses this bed load transport formulation represents the transport occurring as individual particles moving over a rippled bed, while at higher shear stresses the formulation represents the sheet-flow phenomenon where particles move as bed load in several layers (sheets) over a plane bed.