Applications in Open Channel Flow and Hydraulic Structures

TL;DR

You'll see how the full Navier-Stokes equations collapse into the Saint-Venant equations once you average over channel depth. You'll learn to classify flow as subcritical or supercritical using the Froude number, and apply momentum conservation to size hydraulic jumps, weirs, and spillway stilling basins. By the end you can compute conjugate depths and energy loss across a jump by hand.

1. The Mental Model

Open channel flow is Navier-Stokes with the vertical dimension squashed flat. Because channels are shallow compared to how far the water travels, pressure is nearly hydrostatic and vertical acceleration barely matters. That lets you swap a nasty 3D PDE for two much friendlier 1D equations you can actually solve by hand. Depth-averaging turns a fluid mechanics problem into a bookkeeping problem: track mass and momentum along the channel, and the structure's behavior falls out.

2. The Core Material

2.1 From Navier-Stokes to the Saint-Venant Equations

Start with the incompressible Navier-Stokes momentum equation in the flow direction $x$:

$$\rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)=-\frac{\partial p}{\partial x}+\rho g_x+\mu\nabla^2 u$$

Three physical facts about open channels let you simplify this drastically:

1. Shallow-water assumption: the channel length scale is much larger than the depth, so vertical velocity and vertical acceleration are negligible. This kills the vertical momentum equation down to hydrostatic pressure: $p = \rho g(h-z)$, where $h$ is the free-surface elevation.
2. Hydrostatic pressure gradient: because pressure is hydrostatic, $\partial p/\partial x = \rho g\,\partial h/\partial x$ — a term you can compute directly from the water-surface slope.
3. Turbulent friction replaces viscous stress: real channel flow is turbulent, so the viscous term $\mu\nabla^2u$ is replaced by an empirical friction slope $S_f$, usually from Manning's or Chézy's formula.

Integrating the momentum and continuity equations over the cross-sectional area $A$ and using the kinematic condition at the free surface (no flow crosses it) plus no-slip at the bed, you land on the Saint-Venant equations:

Continuity: $\dfrac{\partial A}{\partial t}+\dfrac{\partial Q}{\partial x}=0$

Momentum: $\dfrac{\partial Q}{\partial t}+\dfrac{\partial}{\partial x}\left(\dfrac{Q^2}{A}\right)+gA\dfrac{\partial h}{\partial x}=gA(S_0-S_f)$

Here $Q$ is discharge, $S_0$ is bed slope, $S_f$ is friction slope. Notice what happened: the pressure gradient term became $gA\,\partial h/\partial x$ (hydrostatic assumption), and viscosity became a friction slope term. Everything else is exact bookkeeping of mass and momentum flux. This is the equation set behind every 1D hydraulic model (HEC-RAS, MIKE 11) used to design channels, bridges, and levees.

flowchart TD
    A["3D Navier-Stokes (incompressible)"] --> B["Assume hydrostatic pressure + shallow depth"]
    B --> C["Depth-average over cross-section"]
    C --> D["Saint-Venant equations (1D unsteady)"]
    D --> E["Steady, gradually varied flow"]
    D --> F["Rapidly varied flow / hydraulic jump"]
    E --> G["Weir & spillway discharge design"]
    F --> H["Stilling basin & energy dissipator design"]

2.2 Specific Energy, Critical Flow, and the Froude Number

For steady flow, define specific energy relative to the channel bed:

$$E = y + \frac{V^2}{2g} = y + \frac{Q^2}{2gA^2}$$

For a fixed discharge $Q$, plotting $E$ against depth $y$ gives a curve with a minimum. Setting $dE/dy = 0$ for a rectangular channel of width $b$ (so $A = by$, $Q = qb$ per unit width) gives the critical depth:

$$y_c = \left(\frac{q^2}{g}\right)^{1/3}$$

At this depth the flow is exactly at the boundary between two regimes, characterized by the Froude number:

$$Fr = \frac{V}{\sqrt{gy}}$$

• $Fr < 1$: subcritical — deep, slow, gravity-dominated, disturbances travel upstream.
• $Fr > 1$: supercritical — shallow, fast, inertia-dominated, disturbances only travel downstream.
• $Fr = 1$: critical — minimum energy for that discharge.

This single dimensionless number, derived directly from the momentum balance, tells you everything about how a structure will behave. A sluice gate forces supercritical flow downstream; a broad-crested weir forces critical flow over its crest (which is exactly why weirs make reliable flow meters — at critical depth, $Q$ depends only on $y_c$, not on downstream conditions).

2.3 Momentum Conservation at Hydraulic Structures

Energy isn't conserved across rapid transitions like hydraulic jumps — turbulence eats it. But momentum is always conserved in a control volume with no external horizontal force other than hydrostatic pressure and bed friction (negligible over a short jump length). This is just Newton's second law applied to the Navier-Stokes momentum flux, integrated over the jump:

$$\frac{Q^2}{gA_1} + A_1\bar{y}_1 = \frac{Q^2}{gA_2} + A_2\bar{y}_2$$

where $\bar{y}$ is the depth to the centroid of the cross-section. For a rectangular channel this reduces to the Bélanger equation:

$$\frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1+8Fr_1^2}-1\right)$$

This tells you the downstream ("conjugate") depth $y_2$ needed to force a supercritical flow of depth $y_1$ back to subcritical. It's the single most important design equation for stilling basins below spillways and sluice gates — get it wrong and the jump either doesn