Basic Hydraulics for the FE Civil Exam: Manning Equation, Bernoulli and Open-Channel Flow

NCEES FE Civil spec: Water Resources and Environmental Engineering, B. Basic hydraulics (e.g., Manning equation, Bernoulli theorem, open-channel flow). FE Reference Handbook: Civil Engineering chapter, Hydrology/Water Resources section (Manning, specific energy, hydraulic jump) and the Fluid Mechanics chapter.

What the FE Civil Spec Covers

Section B, basic hydraulics, lists the Manning equation, the Bernoulli theorem and open-channel flow. Typical questions ask for the discharge or velocity in a channel or partly full pipe, the depth or slope needed to carry a flow, whether flow is subcritical or supercritical, and the energy balance between two points in a pipe system. It overlaps with the Fluid Mechanics topic, but here the focus is on civil applications: channels, culverts, sewers and pipelines.

Key Concepts

Hydraulic radius $R_{H} = A / P$ , flow area divided by wetted perimeter. For a circular pipe flowing full or exactly half full, $R_{H} = D /4$ .
Uniform flow: depth and velocity constant along the channel, so the energy grade line is parallel to the bed and S equals the bed slope.
Specific energy $E = y + V^{2} /2 g$ , measured from the channel bottom. Its minimum occurs at critical depth.
Froude number $F r = V / g D$ , where hydraulic depth $D = A / T$ and T is the top width.
Energy equation: Bernoulli plus pump head and head losses between two points.

Core Formulas

Manning equation (K = 1.486 for US units, 1.0 for SI):

Q = \frac{K}{n} A R_{H}^{2/3} S^{1/2} V = \frac{K}{n} R_{H}^{2/3} S^{1/2}

Trapezoidal channel with bottom width b, depth y and side slope z horizontal to 1 vertical:

A = (b + z y) y P = b + 2 y 1 + z^{2} T = b + 2 z y

Critical depth of a rectangular channel (q = Q/b, flow per unit width):

y_{c} = (\frac{q ^{2}}{g})^{1/3}

Energy equation between points 1 and 2:

\frac{p _{1}}{γ} + z_{1} + \frac{V _{1}^{2}}{2 g} + h_{p} = \frac{p _{2}}{γ} + z_{2} + \frac{V _{2}^{2}}{2 g} + h_{L}

Where It Is in the FE Reference Handbook

The Manning equation, specific energy and hydraulic jump relations are in the Civil Engineering chapter, Hydrology/Water Resources section. Search "Manning". The Bernoulli and energy equations, Darcy-Weisbach friction loss and the Moody diagram are in the Fluid Mechanics chapter. Typical n values are tabulated near the Manning equation.

Worked Example

A trapezoidal earth channel has a 10 ft bottom width, 2H:1V side slopes, n = 0.025 and a bed slope of 0.002. Find the uniform-flow discharge at a depth of 3 ft and check whether the flow is subcritical.

Step 1, geometry:

A = (10 + 2 \times 3) (3) = 48 ft^{2}

P = 10 + 2 (3) 1 + 2^{2} = 10 + 13.42 = 23.42 ft

R_{H} = \frac{48}{23.42} = 2.050 ft

Step 2, velocity and discharge:

V = \frac{1.486}{0.025} (2.050)^{2/3} (0.002)^{1/2} = 59.44 (1.614) (0.04472) = 4.29 ft/s

Q = V A = 4.29 (48) \approx 206 cfs

Step 3, Froude number: top width T = 10 + 2(2)(3) = 22 ft, so D = 48/22 = 2.18 ft.

F r = \frac{4.29}{32.2 ( 2.18 )} = 0.51 < 1 \Rightarrow subcritical

Answer: about 206 cfs, subcritical flow.

How FE Questions Are Framed

Most Manning questions give three of the four quantities (Q, geometry, n, S) and ask for the fourth. Solving for slope or n is direct algebra. Solving for depth is not, because depth appears inside both A and $R_{H}$ ; on a multiple choice question, the fastest approach is to test each answer choice in the equation and pick the one that returns the given Q. For circular pipes that flow partly full, use the partial flow (hydraulic elements) graph in the handbook to get the ratio of actual to full flow.

Energy equation questions in this section usually involve two reservoirs or a reservoir and a nozzle. Write the equation from a point where you know everything (a free surface, where pressure is atmospheric and velocity is about zero) to the point of interest, and include the head loss term. When a question mentions a hydraulic jump, the depth before and after the jump are conjugate depths, related through the upstream Froude number.

Common Mistakes

Using 1.0 instead of 1.486 with US units, which makes Q about 33% too low.
Using the diameter instead of D/4 as the hydraulic radius of a full pipe.
Including the free surface in the wetted perimeter. Only the wetted channel boundary counts.
Using flow depth y instead of hydraulic depth A/T in the Froude number for non-rectangular channels.
Entering the slope as a percent (0.2) instead of a decimal (0.002).

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

A rectangular open channel has width b = 3 m, water depth y = 1 m, Manning roughness n = 0.015, and uniform-flow energy slope S = 0.001. What is its approximate discharge in SI units? See the worked solution or practice the Fluids & Hydraulics deck
A 300 mm concrete sewer (n = 0.013) is laid at a slope of 0.005 and flows full. Using Manning's equation, its capacity is most nearly: See the worked solution or practice the Environmental & Water Quality deck
A rectangular channel 4 m wide carries 12 m³/s. Just upstream of a hydraulic jump the flow is supercritical at a depth of 0.5 m. The depth immediately downstream of the jump is most nearly: See the worked solution or practice the Fluids & Hydraulics deck
For steady incompressible flow along a streamline, neglecting losses and pump or turbine work, Bernoulli's equation expresses conservation of: See the worked solution or practice the Fluids & Hydraulics deck
For a pipe with Darcy friction factor f = 0.02, length L = 100 m, diameter D = 0.20 m, and average velocity V = 2 m/s, what is the Darcy-Weisbach head loss? Use g = 9.81 m/s². See the worked solution or practice the Fluids & Hydraulics deck

Related Calculators and Examples

Frequently Asked Questions

What is the constant in the Manning equation for US units?

Use 1.486 (often rounded to 1.49) when lengths are in feet and discharge is in cubic feet per second. In SI units with meters and cubic meters per second the constant is 1.0.

What slope goes into the Manning equation?

The slope of the energy grade line in ft/ft or m/m. For uniform flow it equals the channel bed slope, so you can use the bed slope given in the problem.

How do I tell if open-channel flow is subcritical or supercritical?

Compute the Froude number, V divided by the square root of g times the hydraulic depth. Less than 1 is subcritical (deep and slow), greater than 1 is supercritical (shallow and fast), and exactly 1 is critical.

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