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Engelund-Fredsøe

Engelund-Fredsøe refers to two separate formulas for bed-load and suspended load1.

Bed-load

Bed-load is calculated from:

\[ \phi_{b} = 5p\ \left( \sqrt{\theta'} - 0.7\sqrt{\theta_{c}} \right) \]

Equation: Engelund-Fredsøe non-dimensional bed-load.

Where \(p\) is interpreted as the probability of sediment movement, and calculated as:

\[ p\left( \theta' \right) = \left( 1 + \left( \frac{\frac{\pi}{6}\ \mu_{d}}{\theta' - \theta_{c}} \right)^{4} \right)^{-1/4}, \quad \theta' > \theta_{c} \]

Equation: Probability of sediment movement in the Engelund-Fredsøe bed-load formula.

The \(p\)-value is zero at the critical Shields parameter and increases towards unity for higher Shields parameters.

The extension to graded sediment is standard for Engelund-Fredsøe, and the grain size and porosity are specific for each sediment fraction.

Engelund-Fredsøe uses the GUI specified critical Shields parameter and reacts to Egiazaroff if activated.

Note

When using Engelund-Fredsøe, the skin friction model should be Law of the wall (skin friction depth). The Engelund-Fredsøe model was developed for that particular skin friction model.

Suspended load

In the Engelund-Fredsøe model the suspended load is calculated by using a Rouse profile and a logarithmic velocity profile. The bed concentration applies \(2d_{50}\) above the bed, which makes the calculations challenging.

The velocity profile is given by:

\[ u(y) = \frac{u_{f}}{\kappa}\log\frac{30y}{k_{N}} \]

Equation: Logarithmic velocity profile used in the Engelund-Fredsøe suspended load formula.

The biggest problem with Engelund-Fredsøe is that the lower limit for the concentration profile may be below the lower limit for the velocity profile. In order to avoid that we use the velocity profile defined from the median grain size, i.e. the standard roughness definition:

\[ k_{N} = 2.5d_{50} \]

Equation: Nikuradse roughness scaled with the median grain size.

This is not necessarily identical to the roughness used in the skin friction, usually derived from \(d_{90}\). However, the concentration profile starts at \(2d_{50}\), so we need to ensure that the velocity profile starts below \(2d_{50}\), guaranteed by scaling the Nikuradse roughness with the \(d_{50}\) as well.

Note

For single fraction models the use of \(d_{50}\) for roughness and lower limit for the concentration profile means that the model will be consistent with the normal use for single fraction. The version developed for MIKE 21C and MIKE 1D is mathematically for multi-fraction models, but it is better to use a model like Garcia & Parker (1991) for multi-fraction, as it is essentially formed on the same theory, but based on multi-fraction data.

Now we can write the velocity profile as (\(k_{0} = d_{50}/12\)):

\[ u(y) = \frac{u_{f}}{\kappa}\log\frac{y}{k_{0}} \]

Equation: Velocity profile expressed through the roughness length k₀.

In order to get rid of the friction velocity, so the profile is solely described by \(k_{0}\), we calculate the depth-integrated velocity (\(V\)):

\[ V = \frac{1}{h - k_{0}}\ \frac{u_{f}}{\kappa}\ \int_{k_{0}}^{h}{\log\frac{y}{k_{0}}\ dy} \]

Equation: Depth-integrated velocity from the logarithmic profile.

Rearranging:

\[ V = \frac{k_{0}}{h - k_{0}}\ \frac{u_{f}}{\kappa}\ \left\lbrack \frac{h}{k_{0}}\log\frac{h}{k_{0}} - \frac{h}{k_{0}} + 1 \right\rbrack \]

Equation: Depth-integrated velocity after evaluating the integral.

Rearranging:

\[ \frac{u_{f}}{\kappa} = V\ \frac{\frac{h}{k_{0}} - 1}{\frac{h}{k_{0}}\log\frac{h}{k_{0}} - \frac{h}{k_{0}} + 1} \]

Equation: Friction velocity expressed through the depth-integrated velocity.

Inserting into the velocity profile:

\[ \frac{u(\eta)}{V} = \frac{\frac{h}{k_{0}} - 1}{\frac{h}{k_{0}}\log\frac{h}{k_{0}} - \frac{h}{k_{0}} + 1}\ \left( \log\eta + \log\frac{h}{k_{0}} \right) \]

Equation: Velocity profile normalised by the depth-integrated velocity.

Engelund-Fredsøe is based on the Rouse profile:

\[ c(\eta) = c_{b}\ \left( \frac{\eta}{b}\ \frac{1 - b}{1 - \eta} \right)^{-Z} \]

Equation: Rouse concentration profile.

Where:

\[ b = 2\frac{d_{50}}{h} \]

Equation: Relative reference level of the concentration profile.

Rouse number:

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

Equation: Rouse number.

The Zyserman & Fredsøe (1994) bed concentration is applied2:

\[ c_{b}(\theta') = \frac{0.331\ (\theta' - \theta_{c})^{1.75}}{1 + \frac{0.331\ (\theta' - \theta_{c})^{1.75}}{0.46}} \]

Equation: Zyserman & Fredsøe bed concentration.

The suspended load calculated as the integral of the product of the velocity and concentration profiles:

\[ q_{s}\left( \frac{h}{k_{0}},\ Z,\ \theta' \right) = q\ c_{b}\left( \theta' \right)\int_{b}^{1}{\frac{\frac{h}{k_{0}} - 1}{\frac{h}{k_{0}}\log\frac{h}{k_{0}} - \frac{h}{k_{0}} + 1}\ \left( \log\eta + \log\frac{h}{k_{0}} \right)}\left( \frac{\eta}{b}\ \frac{1 - b}{1 - \eta} \right)^{-Z}d\eta \]

Equation: Suspended load as the integral of the velocity and concentration profiles.

The integral is now called \(f(h/k_{0}, Z)\) and therefore:

\[ q_{s}\left( \frac{h}{k_{0}},\ Z,\ \theta' \right) = q\ c_{b}\left( \theta' \right)\ f\left( \frac{h}{k_{0}},\ Z \right) \]

Equation: Suspended load transport capacity expressed through the tabulated integral.

This is the suspended load transport capacity, extension to graded sediment is done the usual manner by multiplying with the surface weight \(F_{i}\).

Note

Engelund-Fredsøe is not a true graded sediment model, so it is recommended to use e.g. Garcia & Parker (1991) for graded sediment.

  1. Engelund, F. and Fredsøe, J. (1976) "A sediment transport model for straight alluvial channels". Nordic Hydrology, Vol. 7, No. 5. ↩

  2. Zyserman, J. A., & Fredsøe, J. (1994). Data analysis of bed concentration of suspended sediment. Journal of Hydraulic Engineering, 120(9), 1021-1042. ↩