van Rijn (1984)

van Rijn (1984) published separate formulas for bed-load¹ and suspended load².

The van Rijn model is not a multi-fraction model but is extended in the standard manner to multi-fraction by multiplying the suspended load with the local fraction weight in the active layer. The formula itself is used with the sediment parameters for the individual fractions.

First the so-called transport stage parameter (\(T\)) is defined as:

\[ T = \frac{\theta'}{\theta_{c}} - 1 \]

Equation: van Rijn transport stage parameter.

The skin friction Shields parameter should be calculated from "Law of the wall, full depth" (set in the GUI) when using van Rijn. When using the van Rijn model the critical Shields parameter given in the GUI is not used. Instead we use the fit to the Shields curve:

\[ \theta_{c} = \begin{cases} 0.24\ D_{*}^{-1}, & D_{*} < 4 \\ 0.14\ D_{*}^{-0.64}, & 4 < D_{*} < 10 \\ 0.04\ D_{*}^{-0.1}, & 10 < D_{*} < 20 \\ 0.013\ D_{*}^{0.29}, & 20 < D_{*} < 150 \\ 0.055, & 150 < D_{*} \end{cases} \]

Equation: Fit to the Shields curve for the critical Shields parameter.

The non-dimensional grain diameter \(D_{*}\) is defined by:

\[ D_{*} = d\left( \frac{(s - 1)g}{\upsilon^{2}} \right)^{1/3} \]

Equation: Non-dimensional grain diameter.

Where \(\nu\) is the kinematic viscosity of the water, which can be set in the GUI. Since the grain size and density are specific for each fraction, the non-dimensional grain size is also specific for each fraction.

Bed-load

The van Rijn (1984) bed-load model yield the simple formula for the non-dimensional bed-load:

\[ \phi_{b} = 0.053\ \frac{T^{2.1}}{D_{*}^{0.3}} \]

Equation: van Rijn non-dimensional bed-load.

Suspended load

For suspended load van Rijn (1984) is a bit more complicated. First, we check whether the sediment can go into suspension by checking the criterion:

\[ u_{f} > w_{s}\ \max\left( 0.4;\ \frac{4}{D_{*}} \right) \]

Equation: Criterion for whether sediment goes into suspension.

Here we note that \(u_{f}\) is the total friction velocity, which is more appropriate to use because we are comparing the fall velocity to the vertical turbulent fluctuations.

The bed concentration applies at the distance \(a\) from the bed defined by:

\[ a = \max(0.01h; 2d) \]

Equation: Reference level at which the bed concentration applies.

And the volumetric bed concentration at the distance \(a\) is given by:

\[ c_{a} = 0.015\ \frac{d}{a}\ \frac{T^{1.5}}{\left( D_{*} \right)^{0.3}} \]

Equation: Volumetric bed concentration at the reference level.

A special Rouse number \(Z'\) is used in the formula (in this case \(Z'\) does not mean skin friction), and again we use the total friction velocity to represent the concentration profile:

\[ Z' = \frac{w_{s}}{\beta\kappa u_{f}} + \varphi \]

Equation: Modified Rouse number used in the van Rijn suspended load formula.

This is seen to be a special version of the standard Rouse number:

\[ Z = \frac{w_{s}}{\kappa u_{f}} \]

Equation: Standard Rouse number.

In the modification \(\beta\) is given by:

\[ \beta = 1 + 2\left( \frac{w_{s}}{u_{f}} \right)^{2} \]

Equation: Beta factor in the modified Rouse number.

And the following is added to the Rouse number:

\[ \varphi = \frac{5}{2}\ \left( \frac{w_{s}}{u_{f}} \right)^{0.8}\left( \frac{c_{a}}{c_{0}} \right)^{0.4} \]

Equation: Additive term in the modified Rouse number.

Where \(c_{0} = 0.65\) is maximum possible volumetric concentration.

The volumetric suspended load is calculated from:

\[ q_{s} = F(Z')\ c_{a}\ Vh \]

Equation: van Rijn volumetric suspended load.

Where \(h\) is the water depth and \(V\) the depth-integrated flow velocity. The correction factor \(F(Z')\) is calculated from:

\[ F(Z') = \frac{\left( \frac{a}{h} \right)^{Z'} - \left( \frac{a}{h} \right)^{1.2}}{\left( 1 - \frac{a}{h} \right)^{Z'}(1.2 - Z')} \]

Equation: Correction factor in the van Rijn suspended load formula.

For graded sediment the fractional loads are modified using the surface weight \(F_{i}\).

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1. van Rijn, Leo C. (1984a) "Sediment transport, part I: bed load transport." Journal of Hydraulic Engineering 110.10 (1984): 1431-1456. ↩

2. van Rijn, L.C. (1984b) "Sediment Transport Part II: Suspended load transport", Journal of Hydraulic Engineering, Vol. 110, No. 11, November, 1984. ↩