Models and Approximate Solutions for Open-Channel Flows: Application to a Free Sharp-Crested Weir

https://github.com/QuentinMale/sharp-crested-weir

Slide deck available at https://quentinmale.github.io/sharp-crested-weir

Rectangular sharp-crested weir

Application: Irrigation canals & water distribution

• No moving parts
• Converts an easy measurement (water level) into flow rate

Question to answer today

Given a certain water level , what is the water flow rate ?

Bernoulli principle (energy budget)

• Inviscid flow (no viscous dissipation)
• Steady flow
• Incompressible fluid

The Bernoulli equation holds along a streamline (everywhere if irrotational):

Weir-discharge equation

• Contraction is neglected
• ()

Corrected weir equation

Interpretation: (real/ideal) is a correction factor.
Incorporate: viscous friction losses, upstream velocity head, and contraction.

How is determined?

1. Measure head

2. Measure discharge

3. Compute from the definition

4. Repeat for many heads and/or geometries

You now have a set of points and can fit an empirical correlation.

Dimensional analysis (Buckingham theorem)

At sufficiently high Reynolds number and Weber number:

• viscous effects are confined to thin boundary layers
• surface tension effects are negligible

Discharge coefficient

Linear fit (Kandaswamy & Rouse, 1957):

Summary (what to remember)

• Simplified derivation from first principles: .
• Real flows deviate due to losses + contraction + approach effects.
• Correction determined by calibration:
  measure , measure , compute .
• Over practical ranges (high enough , ), the dominant dependence collapses to
  .

Exercise

1. Run the code in the basilisk folder to get .

2. Fit a linear discharge coefficient law given the data points

3. Given , find the water level to achieve a certain flow rate.