Fluid Mechanics and Hydraulics

G. Fluid dynamics

Continuity, Bernoulli, and the Energy Equation

Conservation of mass and energy along a streamline, the head form with pump, turbine, and friction terms, EGL/HGL, and the impulse-momentum principle.

Almost every hydraulics problem on the FE Environmental exam rests on two conservation laws applied to a flowing fluid: mass cannot accumulate in a steady pipe, and mechanical energy is conserved except for what friction degrades and what pumps add or turbines remove. Master the head form of the energy equation and you can size a pump, find a pressure at a junction, or predict a velocity at a nozzle without ever leaving the NCEES FE Reference Handbook ("Fluid Mechanics — Principles of One-Dimensional Fluid Flow"). Points are lost here by mixing energy units, dropping a velocity-head term, or putting the pump on the wrong side of the equation.

Continuity: conservation of mass

For steady one-dimensional flow the mass flow rate

\dot m = \rho A v

is the same at every cross section. When the fluid is incompressible — true for water and essentially true for the low-pressure air streams in environmental work — density cancels and the volumetric flow

Q = Av

is constant. The immediate consequence you use constantly: where a pipe narrows, the velocity must rise in inverse proportion to area, so halving the diameter quadruples the velocity.

\dot m_1 = \dot m_2 \;\Rightarrow\; \rho_1 A_1 v_1 = \rho_2 A_2 v_2, \qquad Q = A_1 v_1 = A_2 v_2

The energy equation in head form

Dividing the steady-flow energy balance by weight expresses every term as a length — a "head" in metres or feet. Pressure head

p/\gamma

, velocity head

v^2/2g

, and elevation head

z

at section 1 equal the same three at section 2, plus any friction loss

h_f

Bernoulli as the frictionless special case

When no pump or turbine sits between the two sections and friction is negligible, the energy equation collapses to the Bernoulli equation: the sum of pressure, velocity, and elevation head is constant along the streamline. Use Bernoulli for short accelerating reaches — a nozzle, a venturi throat, flow from a tank — where the friction loss is genuinely small. The moment a long pipe, a fitting, or a pump enters the picture, return to the full energy equation.

\frac{p_1}{\gamma} + \frac{v_1^2}{2g} + z_1 = \frac{p_2}{\gamma} + \frac{v_2^2}{2g} + z_2 = \text{constant}

EGL and HGL

The energy grade line (EGL) plots the total head

p/\gamma + v^2/2g + z

above a datum; the hydraulic grade line (HGL) plots only

p/\gamma + z

. The two are separated everywhere by exactly the velocity head

v^2/2g

The impulse-momentum principle

Energy methods give heads and velocities but never the force a moving fluid exerts on a bend, nozzle, or plate — for that you need the momentum equation. The resultant external force in any direction equals the rate of change of momentum flux,

\Sigma F = \rho Q (v_{\text{out}} - v_{\text{in}})

taken component by component. A free jet striking a fixed plate reverses or stops, so the plate feels

F = \rho Q v

Putting it together

A reliable routine: (1) sketch the system and number two sections where you know the most; (2) write the energy equation and strike the zero terms — an open reservoir surface has

p=0

gauge and

v\approx 0

; (3) solve for the unknown head, velocity, or pressure; (4) if a force is asked, switch to momentum with the velocities you just found. Keep

\gamma_{\text{water}} = 9{,}810\ \text{N/m}^3

Exam strategy

Decide first whether friction or a machine is present: if yes, use the full energy equation; if not, Bernoulli is faster. Pump head is added on the upstream side (it raises the EGL), turbine head is subtracted. When two sections are both open to atmosphere, the pressure-head terms drop and the problem reduces to elevation and velocity heads plus losses. For any force question, abandon energy and use

\Sigma F = \rho Q\,\Delta v

— and remember the pressure-times-area terms on a bend. Carry units explicitly; a velocity head left in

\text{m}^2/\text{s}^2

instead of metres is the classic silent error.

Key equations

Continuity (volumetric)

Q = A_1 v_1 = A_2 v_2

Worked examples

Velocity and pressure drop in a contraction

Problem. Water at $1000\ \text{kg/m}^3$ flows at $Q = 0.10\ \text{m}^3/\text{s}$ through a horizontal contraction from $300\ \text{mm}$

Common pitfalls

•Mixing energy units: every term in the head form must be a length (m or ft). Leaving $v^2/2g$ as $\text{m}^2/\text{s}^2$ (forgetting to divide by $g$

References

NCEES FE Reference Handbook — Fluid Mechanics
NCEES FE Reference Handbook — Civil Engineering
Munson, Young & Okiishi, Fundamentals of Fluid Mechanics — energy line / control-volume momentum background

A. Fluid statics

Fluid Statics and Manometry

B. Closed conduits

Pipe Friction: Darcy–Weisbach, Hazen–Williams, and Minor Losses

C. Open channel

Open-Channel Flow and Pump Application

E. Flow measurement

Flow Measurement: Weirs, Orifices, and Venturi Meters

F. Blowers

Blowers, Aeration, and Unsteady Flow