4.2 Hydraulics & Hydrologic Systems
Key Takeaways
- Manning's equation (SI) is V = (1/n)R^(2/3)S^(1/2); use the 1.486 constant for US customary units.
- Hydraulic radius R = A/P (flow area over wetted perimeter); for a full circular pipe R = D/4.
- The Rational Method peak runoff is Q = CiA (US: cfs with i in in/hr, A in acres); C is the runoff coefficient.
- Darcy's law for groundwater is Q = kiA, where k is hydraulic conductivity and i = dh/dL is the gradient.
- Critical flow occurs at Froude number Fr = 1; specific energy E = y + V²/2g is minimized at critical depth.
Why This Area Carries Weight
Hydraulics & Hydrologic Systems is one of the most heavily weighted FE Civil areas at 8–12 questions. It applies the fluid mechanics fundamentals to open channels, watersheds, storm infrastructure, and groundwater. All formulas appear in the NCEES FE Reference Handbook, but many have separate SI and US customary constants — picking the wrong constant is the dominant exam mistake.
Open-Channel Flow — Manning's Equation
Uniform open-channel flow uses Manning's equation:
| Unit system | Equation | Constant |
|---|---|---|
| SI | V = (1/n) R^(2/3) S^(1/2) | 1.0 |
| US customary | V = (1.486/n) R^(2/3) S^(1/2) | 1.486 |
Here n is the Manning roughness coefficient (e.g., n ≈ 0.013 for finished concrete, 0.025 for natural earth channels), S is the channel slope, and R is the hydraulic radius:
R = A / P (flow area / wetted perimeter)
For a pipe flowing full, R = D/4. Discharge follows Q = VA.
Specific Energy and Critical Depth
Specific energy is the energy head relative to the channel bottom:
E = y + V²/2g = y + Q²/(2gA²)
The Froude number Fr = V/√(gy) classifies flow: subcritical (Fr < 1, deep/slow), critical (Fr = 1), and supercritical (Fr > 1, shallow/fast). Critical depth occurs where specific energy is minimum for a given discharge. A hydraulic jump dissipates energy when supercritical flow transitions to subcritical.
The Hydrologic Cycle and the Rational Method
The hydrologic cycle moves water through precipitation, infiltration, evapotranspiration, surface runoff, and groundwater recharge. For small urban watersheds, peak runoff uses the Rational Method:
Q = CiA
- C = dimensionless runoff coefficient (0.1 pervious turf → 0.95 pavement)
- i = rainfall intensity at the time of concentration t_c
- A = drainage area
In US customary units, with i in in/hr and A in acres, Q comes out in cfs (1 acre-in/hr ≈ 1 cfs). The rainfall intensity is read at the time of concentration from an Intensity-Duration-Frequency (IDF) curve.
Hydrographs and Detention
A hydrograph plots discharge versus time for a runoff event. A detention basin stores peak inflow and releases it slowly, so outflow peak < inflow peak; routing balances inflow, outflow, and change in storage (ΔS = I − O over Δt).
Storm Sewer and Culvert Design
Storm sewers are sized with Manning's equation for the design storm. Culverts operate under inlet control (entrance geometry governs, common for steep culverts) or outlet control (barrel friction and tailwater govern). The exam asks you to identify which control governs and apply the corresponding headwater relation from the handbook.
Groundwater — Darcy's Law
Flow through a saturated porous medium follows Darcy's law:
Q = kiA or v = ki
where k = hydraulic conductivity (permeability), i = dh/dL = hydraulic gradient, A = gross cross-sectional area, and v = Darcy (apparent) velocity. The actual seepage velocity equals v/n_e (effective porosity). Steady radial flow to a well uses the Theim/Dupuit equations in the handbook.
A rectangular concrete channel (n = 0.013) is 2 m wide, flows 1 m deep, on a slope S = 0.001. Using Manning's equation (SI), the hydraulic radius is closest to:
A 5-acre site has a runoff coefficient C = 0.6. During the design storm the rainfall intensity at the time of concentration is 3 in/hr. Using the Rational Method, the peak runoff is approximately:
Groundwater flows through a sand aquifer with hydraulic conductivity k = 0.001 m/s. The head drops 2 m over a 100 m flow path, and the cross-sectional area is 50 m². Using Darcy's law, the discharge Q is: