# How to model bed load transport in channels of non-rectangular cross sections

All the formulas presented in the relevant technical literature refer to channels of rectangular cross section — the formulas assume a constant depth of flow right across the channel section.

I need to calculate the bed load in a channel of roughly triangular cross section. I do not think that calculating the bed load discharge with the average depth of flow is appropriate or theoretically justified, so I am looking for a special hydraulic analysis of the situation which may be applied to channels of triangular or trapezoidal cross section.

## 1 Answer

There are several approaches to answer this question. One is simple, but not that accurate. This approach would be to to discretize points along the perimeter and calculate a per point boundary shear stress. The equation would be the same as the depth averaged, except you would solve for each individual depth h1, ..., hn. One thing to note is that the bed load will mostly be concentrated at the lowest point in the channel.

The easiest accurate method to implement is the WTA method, which uses the law of the wall to calculate boundary shear stress. The papers outlining the method are Wobus et al. (2006, 2008) titled "Self-formed bedrock channels", and "Modeling the evolution of channel shape: Balancing computational efficiency with hydraulic fidelity" . I have a python code that implements this, but in closed conduit conditions. It can be modified to work in the open channel case.

References:

• Wobus et al., 2006: Self-formed bedrock channels. doi: 10.1029/2006GL027182
• Wobus et al., 2008: Modeling the evolution of channel shape: Balancing computational efficiency with hydraulic fidelity. doi: 10.1029/2007JF000914
• I cannot find Wobus et al. (2006) with the title Modeling the geometry of bedrock river channels. There are Wobus et al. (2006), Self-formed bedrock channels, doi: 10.1029/2006GL027182 and Nelson & Seminara (2011), Modeling the evolution of bedrock channel shape with erosion from saltating bed load, doi: 10.1029/2011GL048628. – daniel.heydebreck Dec 15 '17 at 8:57
• Yes, sorry, there is Wobus et al. (2006) Self-formed bedrock channels, and Wobus et al. (2008) Modeling the evolution of channel shape: Balancing computational efficiency with hydraulic fidelity. These both discuss the same model and both are needed to really understand the algorithm. – MaxC Mar 13 '18 at 19:55