Applications in Open Channel Flow and Hydraulic Structures

TL;DR

You'll see how the full Navier-Stokes equations collapse, step by step, into the Saint-Venant equations that run every flood model, and further into the simple formulas engineers use for weirs, sluice gates, and hydraulic jumps. You'll work a real hydraulic jump calculation with actual numbers. You'll walk away knowing exactly which assumption gets dropped at each simplification stage — and why that matters for design accuracy.

1. The Mental Model

Open channel flow is Navier-Stokes with gravity fighting a free surface instead of walls on every side. Because channels are shallow compared to their length, engineers integrate the vertical momentum equation away and assume the pressure just stacks up like a column of still water (hydrostatic). Everything from the Saint-Venant equations to the weir formula is really Navier-Stokes with weaker and weaker assumptions stacked on top. Every "simple" hydraulics formula is a shortcut through the same fluid physics — just with more corners cut.

2. The Core Material

2.1 From Navier-Stokes to the Saint-Venant Equations

Start with the incompressible Navier-Stokes equations for a fluid parcel:

$$\rho\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u}\right) = -\nabla p + \mu\nabla^2\mathbf{u} + \rho\mathbf{g}$$

For a river or canal, the depth $h$ is tiny compared to the length $L$ over which flow changes — often $h/L < 0.01$. That ratio lets you throw out the vertical acceleration term in the $z$-momentum equation entirely. What's left is:

$$\frac{\partial p}{\partial z} = -\rho g$$

Integrate that from the bed to the free surface, and pressure becomes purely hydrostatic: $p(z) = \rho g (h - z)$. This one assumption is the single biggest simplification in all of open channel hydraulics — you've turned a 3D pressure field into something you compute with a ruler.

Now take the streamwise ($x$) momentum equation and integrate it over the channel's cross-section $A(x,t)$, using Leibniz's rule to move the derivative outside the integral, and applying two boundary conditions: zero shear at the free surface, and a bed shear stress $\tau_0$ at the channel bottom (this absorbs the visc