Skip to content

Wilcock & Crowe (2003)

Wilcock & Crowe (2003) give the following non-dimensional bed-load1:

\[ W_{i}^{*} = \frac{\left( s_{i} - 1 \right)gq_{b,i}}{F_{i}\ \left( u_{f}' \right)^{3}} \]

Equation: Definition of the non-dimensional bed-load for a sediment fraction.

Where:

  • \(W_{i}^{*}\) — non-dimensional bed-load of fraction \(i\)
  • \(s_{i}\) — relative density of fraction \(i\)
  • \(g\) — acceleration of gravity
  • \(q_{b,i}\) — volumetric bed-load of fraction \(i\)
  • \(F_{i}\) — relative weight of the fraction (\(i\)) in the bed surface
  • \(u_{f}'\) — skin friction velocity

The non-dimensional bed-load is calculated from:

\[ W_{i}^{*}(\phi_{i}) = \begin{cases} 0.002\ \phi_{i}^{7.5}, & \phi < 1.35 \\ 14\ \left( 1 - \dfrac{0.894}{\phi_{i}^{0.5}} \right)^{4.5}, & \phi \geq 1.35 \end{cases} \]

Equation: Wilcock & Crowe non-dimensional bed-load.

Where \(\phi_{i}\) is given by:

\[ \phi_{i} = \frac{\tau'}{\tau_{ri}} \]

Equation: Ratio of skin friction shear stress to the reference shear stress.

Where \(\tau'\) is the bed shear stress (skin friction) and \(\tau_{ri}\) a reference shear stress for fraction \(i\). The reference shear stress for fraction \(i\) is given by:

\[ \tau_{ri} = \tau_{rg}\left( \frac{d_{i}}{d_{g}} \right)^{b} \]

Equation: Reference shear stress for a sediment fraction.

Where:

  • \(d_{g}\) — geometric mean grain diameter
  • \(d_{i}\) — grain size for fraction \(i\)
  • \(\tau_{rg}\) — reference shear stress for the geometric grain size
  • \(b\) — exponent

The exponent \(b\) is calculated from:

\[ b = \frac{0.67}{1 + \exp\left( 1.5 - \frac{d_{i}}{d_{g}} \right)} \]

Equation: Exponent in the reference shear stress relation.

The reference shear stress for the geometric mean grain size is defined by:

\[ \tau_{rg} = \theta_{rg}\ \left( s_{g} - 1 \right)\rho gd_{g} \]

Equation: Reference shear stress for the geometric mean grain size.

Where:

  • \(s_{g}\) — representative relative density for the geometric mean diameter
  • \(\theta_{rg}\) — critical Shields parameter for the geometric mean grain size

It is noted that the original model did not assume any variation in the \(s\)-values for the sediment fractions. In the present implementation we use the average \(s\)-value for the non-cohesive sediment fractions. The critical Shields parameter for the geometric mean grain size is calculated from:

\[ \theta_{rg} = 0.021 + 0.015\ \exp( -20F_{s}) \]

Equation: Critical Shields parameter for the geometric mean grain size.

Where:

  • \(F_{s}\) — relative sand content in the bed surface

\(F_{s}\) is calculated by summing the mass in the active layer of all fractions satisfying the two conditions:

  • Fraction is non-cohesive
  • Fraction has \(d_{i} < 1\) mm

This is divided by the total mass in the active layer.

Notes regarding Wilcock & Crowe

Wilcock & Crowe is only selectable for bed-load. The Wilcock & Crowe formula was developed for multi-fraction bed-load calculations. The \(F_{i}\) weights are already in the formula, and no other modifications apply.

Wilcock & Crowe can be used also for single fraction bed-load.

Egiazaroff does not apply to Wilcock & Crowe; it uses its own hiding function.

The GUI critical Shields parameter does not apply.

  1. Wilcock, P. R., & Crowe, J. C. (2003). Surface-based transport model for mixed-size sediment. Journal of Hydraulic Engineering, 129(2), 120-128. ↩