Open Channel Flow for the FE Exam
Open channel flow occurs when a fluid flows with a free surface exposed to the atmosphere, as in rivers, canals, storm sewers, and irrigation channels. Unlike pipe flow where pressure drives the flow, open channel flow is driven by gravity. Manning's equation is the primary tool for analyzing open channel flow on the FE exam.
Key Definitions
- Flow area (A): The cross-sectional area of the flowing water.
- Wetted perimeter (P): The length of the channel boundary in contact with water.
- Hydraulic radius (R): R = A / P. A measure of channel efficiency.
- Top width (T): The width of the channel at the water surface.
- Hydraulic depth (D): D = A / T. The average depth of flow.
Manning's Equation
Manning's equation relates flow velocity to channel properties and slope:
V = (1/n) * R^(2/3) * S^(1/2) (SI units, V in m/s)
V = (1.486/n) * R^(2/3) * S^(1/2) (US customary units, V in ft/s)
Where n is Manning's roughness coefficient, R is the hydraulic radius, and S is the channel slope (for uniform flow, this equals the bed slope).
Flow rate: Q = V * A = (1/n) * A * R^(2/3) * S^(1/2)
Manning's Roughness Coefficient
The roughness coefficient n depends on the channel material:
- Smooth concrete: n = 0.012 to 0.013
- Unfinished concrete: n = 0.014 to 0.017
- Corrugated metal: n = 0.022 to 0.025
- Earth channel (clean): n = 0.022 to 0.025
- Natural stream (clean): n = 0.030 to 0.035
- Natural stream with vegetation: n = 0.035 to 0.050
These values are available in the reference handbook.
Common Channel Shapes
For a rectangular channel of width b and depth y:
- A = b * y
- P = b + 2*y
- R = b*y / (b + 2*y)
For a trapezoidal channel with bottom width b, depth y, and side slopes z:1 (horizontal:vertical):
- A = (b + z*y) * y
- P = b + 2*y*sqrt(1 + z^2)
For a circular pipe flowing partially full, the geometry depends on the flow depth relative to the pipe diameter. Tables relating depth ratio to area and wetted perimeter ratios are in the reference handbook.
Normal Depth
Normal depth is the depth at which flow is uniform (velocity, depth, and area are constant along the channel). Finding normal depth requires solving Manning's equation for depth, which is usually iterative because the equation cannot be solved explicitly for depth in most channel shapes.
Critical Flow and the Froude Number
Critical flow occurs when the Froude number equals 1:
Fr = V / sqrt(g * D) = 1
At critical flow, the specific energy is minimized for a given discharge, and the discharge is maximized for a given specific energy.
- Subcritical flow (Fr less than 1): Slow, deep flow. Controlled from downstream. Disturbances can travel upstream.
- Supercritical flow (Fr greater than 1): Fast, shallow flow. Controlled from upstream. Disturbances cannot travel upstream.
Critical Depth
For a rectangular channel, critical depth is:
y_c = (Q^2 / (g * b^2))^(1/3)
Or equivalently: y_c = (q^2 / g)^(1/3), where q = Q/b is the flow rate per unit width.
Specific Energy
Specific energy is the energy per unit weight measured relative to the channel bottom:
E = y + V^2 / (2g) = y + Q^2 / (2g * A^2)
The specific energy diagram (E vs. y) shows that for any specific energy greater than the minimum, there are two possible depths: a subcritical depth (deep, slow) and a supercritical depth (shallow, fast).
Hydraulic Jump
A hydraulic jump occurs when flow transitions abruptly from supercritical to subcritical. It is characterized by turbulence, energy loss, and an increase in depth. For a rectangular channel:
y_2 / y_1 = (1/2) * [-1 + sqrt(1 + 8 * Fr_1^2)]
Where y_1 and Fr_1 are the depth and Froude number upstream of the jump, and y_2 is the depth downstream. These are called sequent or conjugate depths.
Energy loss in the jump: delta_E = (y_2 - y_1)^3 / (4 * y_1 * y_2)
Gradually Varied Flow
When flow conditions change gradually along the channel (changes in slope, cross section, or obstructions), the depth varies gradually from one section to another. The water surface profiles are classified by channel slope and depth relative to normal and critical depths. The most common classifications are M1, M2, M3 (mild slope), S1, S2, S3 (steep slope), and others.
Practice Strategy
- Practice calculating hydraulic radius for rectangular and trapezoidal channels
- Master Manning's equation for both SI and US customary units
- Practice finding critical depth for rectangular channels
- Know the hydraulic jump equations
- Understand the difference between subcritical and supercritical flow
Open channel flow problems are formula-driven and very scoreable on the FE exam. Know the equations, practice the calculations, and use the reference handbook tables for Manning's n values and channel geometry.