Fluid Dynamics: Energy & Momentum - Theory & Concepts

Fluid Dynamics: Energy & Momentum

Learning Objectives

Relate fluid motion to pressure, gravity, viscous stress, and external work.
State the assumptions behind Euler's equation, Bernoulli's equation, and the Navier-Stokes equations.
Apply the extended mechanical-energy equation with pumps, turbines, friction, minor losses, and kinetic-energy correction factors.
Construct and interpret the hydraulic grade line and energy grade line.
Apply linear momentum to pipe bends, nozzles, jets, reducers, gates, and moving vanes using a consistent force sign convention.
Use momentum correction factors and angular momentum where velocity profiles or rotating machinery require them.
Recognize cavitation, vapor-pressure, absolute-pressure, and model-validity limits.

Fluid dynamics connects the kinematic description of motion to the forces and energy transfers that produce it. Civil-engineering applications combine conservation of mass, mechanical energy, and momentum to analyze pipes, meters, pumps, turbines, gates, bends, jets, spillways, and hydraulic structures.

Three Complementary Conservation Laws

Mass: Determines how discharge and velocity are related between sections.
Energy: Relates pressure, elevation, velocity, shaft work, and irreversible losses.
Momentum: Determines resultant forces and reactions from changes in velocity and pressure.

These equations answer different questions. An energy equation does not directly provide a support reaction, and a momentum equation does not by itself determine head loss.

Euler's Equation Along a Streamline

For steady, inviscid flow, Newton's second law applied tangentially to a differential fluid element gives

\frac{dp}{\rho}+V\,dV+g\,dz=0

Integrating for constant density produces Bernoulli's equation. The differential form makes clear that pressure gradients, acceleration, and gravity balance one another along the streamline.

Bernoulli Equation

Mechanical-energy relation for steady, incompressible, inviscid flow along a streamline with no shaft work.

\frac{p}{\gamma}+\frac{V^2}{2g}+z=H=\text{constant}

Variables

Symbol	Description	Unit
$p$	Static pressure	Pa
$\gamma$	Specific weight	$N/m^3$
$V$	Local or appropriately averaged flow velocity	m/s
$z$	Elevation above a selected datum	m
$H$	Total mechanical head	m

Bernoulli Assumptions

The basic Bernoulli equation requires steady flow, constant density, negligible viscous loss, no pump or turbine work between sections, and application along a streamline. It may be applied across streamlines only when the flow is irrotational. Real engineering systems normally require the extended energy equation.

Interactive Venturi Demonstration

The simulation applies continuity and ideal horizontal Bernoulli flow while using absolute pressure to identify vapor-pressure limits.

Bernoulli's Principle: Venturi Meter

Inlet Diameter (

D_1

): 200 mm

Throat Diameter (

D_2

): 100 mm

Inlet Velocity (

V_1

): 2.0 m/s

Inlet Absolute Pressure (

P_1

): 150 kPa

Inlet (1)

150.0 kPa abs

2.00 m/s

Throat (2)

120.0 kPa abs

8.00 m/s

This ideal horizontal-flow model neglects losses. Cavitation is assessed using absolute pressure and the approximate vapor pressure of water at 20°C. Once vapor pressure is reached, the single-phase Bernoulli model is no longer physically valid.

Kinetic-Energy Correction Coefficient

A cross section rarely has perfectly uniform velocity. Replacing the true velocity distribution with mean velocity requires the coefficient $\alpha$ in the kinetic-energy term.

Kinetic-Energy Correction Coefficient

Corrects mean-velocity kinetic energy to equal the actual energy flux.

\alpha=\frac{1}{A V^3}\int_A u^3\,dA

Variables

Symbol	Description	Unit
$\alpha$	Kinetic-energy correction coefficient	-
$u$	Local velocity normal to the section	m/s
$V$	Section-average velocity	m/s

Typical Values of $\alpha$

For fully developed laminar flow in a circular pipe, $\alpha=2$ . For ordinary turbulent pipe flow, the flatter profile commonly gives $\alpha$ near 1.03 to 1.10. Using $\alpha=1$ is often adequate but should be stated, especially near entrances, outlets, contractions, or strongly nonuniform channels.

Extended Mechanical-Energy Equation

Relates two sections through shaft work and irreversible losses.

\frac{p_1}{\gamma} +\alpha_1\frac{V_1^2}{2g} +z_1 +h_p -h_t = \frac{p_2}{\gamma} +\alpha_2\frac{V_2^2}{2g} +z_2 +h_L

Variables

Symbol	Description	Unit
$h_p$	Head added by a pump	m
$h_t$	Head extracted by a turbine	m
$h_L$	Total irreversible major and minor head loss from section 1 to 2	m
$\alpha_1,\alpha_2$	Kinetic-energy correction coefficients	-

Loss Head Is Directional and Nonnegative

With sections ordered in the actual direction of flow, $h_L\ge0$ . Reversing section labels without changing the assumed flow direction is a common sign error. Pump head is positive when energy is transferred to the fluid; turbine head is positive in the equation above when energy is extracted from the fluid.

Hydraulic Grade Line, HGL

The locus of piezometric head,

HGL=z+\frac{p}{\gamma}

In a piezometer open to atmosphere, the liquid level indicates the local HGL for the connected fluid.

Energy Grade Line, EGL

The locus of total mechanical head,

EGL=z+\frac{p}{\gamma}+\alpha\frac{V^2}{2g}

The vertical separation between EGL and HGL is the corrected velocity head.

EGL and HGL Behavior

Both lines fall in the direction of flow through passive components because of loss.
The EGL rises across a pump and falls across a turbine.
The HGL can rise through a diffuser even while the EGL falls, because velocity head is converted to pressure head with some loss.
Where the HGL drops below the pipe centerline, gage pressure is negative.
Absolute pressure must still remain above vapor pressure and above any structural or air-entry limit.

Power Associated with Head

Converts pump, turbine, or loss head to a rate of energy transfer.

P=\rho gQH=\gamma QH

Variables

Symbol	Description	Unit
$P$	Fluid power associated with head H	W
$Q$	Volumetric flow rate	$m^3/s$

Navier-Stokes Equation

For a Newtonian fluid with constant viscosity, the differential linear-momentum equation is

\rho\left( \frac{\partial\mathbf{V}}{\partial t} +(\mathbf{V}\cdot\nabla)\mathbf{V} \right) = -\nabla p +\mu\nabla^2\mathbf{V} +\rho\mathbf{g}

The left side is mass density times material acceleration. The right side contains pressure force, viscous diffusion of momentum, and body force. Most practical hydraulic formulas are exact solutions, approximations, or empirical closures derived from this balance.

Turbulent Flow Requires Closure

Time-averaging turbulent Navier-Stokes equations introduces Reynolds stresses that are not known from the mean velocity alone. Friction factors, roughness formulas, turbulence models, and laboratory coefficients supply the required closure information.

Control-Volume Linear Momentum

For a fixed control volume, the sum of external forces equals the rate of momentum accumulation inside plus the net momentum flux through the control surface.

General Linear-Momentum Equation

Integral momentum balance for a fixed control volume.

\sum\mathbf{F}_{ext} = \frac{d}{dt}\int_{CV}\rho\mathbf{V}\,dV + \int_{CS}\rho\mathbf{V}(\mathbf{V}\cdot\mathbf{n})\,dA

Steady One-Dimensional Momentum

Section-average form for discrete inlets and outlets.

\sum\mathbf{F}_{ext} = \sum_{out}\beta\rho Q\mathbf{V} - \sum_{in}\beta\rho Q\mathbf{V}

Variables

Symbol	Description	Unit
$\beta$	Momentum correction coefficient	-
$\mathbf{F}_{ext}$	All external forces acting on the fluid control volume	N

Momentum Correction Coefficient

Corrects mean-velocity momentum flux to equal the actual flux.

\beta=\frac{1}{A V^2}\int_A u^2\,dA

Typical Values of $\beta$

For fully developed laminar flow in a circular pipe, $\beta=4/3$ . For ordinary turbulent pipe flow, $\beta$ is commonly close to 1.01 to 1.05. Do not interchange $\alpha$ and $\beta$ : one corrects energy flux and the other momentum flux.

Control-Volume Force Analysis

STEP-BY-STEP

Draw a control volume around the fluid, not around the pipe material.
Choose coordinate axes and assign signed velocity components.
List all forces on the fluid: pressure, weight, wall/support reaction, shear when important, and atmospheric pressure if not canceled by consistent gage pressure.
Apply continuity to obtain missing flow rates or mean velocities.
Apply momentum separately in each coordinate direction.
Solve for the force exerted by the pipe or support on the fluid.
Reverse the sign to obtain the force exerted by the fluid on the pipe, bend, nozzle, gate, or support.

Pressure Forces Act Inward on the Control Surface

Pressure is compressive. At an inlet or outlet, the pressure-force vector acting on the fluid is opposite the outward control-surface normal: $\mathbf{F}_p=-pA\mathbf{n}$ . Memorizing “inlet positive, outlet negative” without drawing normals often fails for inclined or multiport systems.

Forces on Bends and Reducers

A bend changes the momentum-vector direction and may also change its magnitude. The support reaction must balance momentum-flux change, pressure forces at every section, fluid weight when significant, and wall shear when it cannot be neglected. Three-dimensional bends require vector components in all relevant directions.

Jets Striking Stationary Surfaces

For a free jet at atmospheric pressure, pressure forces at exposed jet sections cancel in gage form. The force on a stationary plate or vane follows from the change in momentum. A normal flat plate that stops the jet's normal velocity component experiences approximately

F=\rho QV

when splash leaves with negligible normal velocity.

Moving Vanes and Relative Velocity

For a vane moving at speed $u$ , the mass flow that actually enters the vane is based on relative speed. For a jet of area $A$ moving in the same direction as the vane,

\dot m=\rho A(V-u)

Momentum analysis must use the absolute inlet and outlet velocities, while the vane geometry commonly controls relative velocity. Neglecting this distinction produces major force and power errors.

Angular-Momentum Equation

Relates torque to the change in moment of momentum, central to pumps and turbines.

\sum M_O = \frac{d}{dt}\int_{CV}\rho(\mathbf{r}\times\mathbf{V})\,dV + \int_{CS}\rho(\mathbf{r}\times\mathbf{V})(\mathbf{V}\cdot\mathbf{n})\,dA

For steady one-inlet/one-outlet turbomachinery,

T=\dot m(r_2V_{\theta2}-r_1V_{\theta1})

Variables

Symbol	Description	Unit
$T$	Torque exerted on the fluid	$N\cdot m$
$V_\theta$	Tangential or whirl component of absolute velocity	m/s

Torricelli, Pitot, and Venturi Applications

Torricelli: For a large reservoir discharging ideally to atmosphere, $V\approx\sqrt{2gH}$ .
Pitot tube: Stagnating a streamline converts velocity head to pressure head, so $V=C\sqrt{2g\Delta h}$ after calibration.
Venturi meter: Continuity and pressure-head difference determine discharge, modified by a discharge coefficient.

Each device inherits Bernoulli assumptions and requires appropriate coefficients, pressure references, tap elevations, and fluid-column conversions.

Cavitation

Formation and collapse of vapor cavities when local absolute pressure falls to or below the liquid vapor pressure at the operating temperature.

Use Absolute Pressure for Cavitation

Gage pressure cannot be compared directly with vapor pressure unless atmospheric pressure is first added. Once vapor forms, a single-phase incompressible Bernoulli calculation no longer represents the actual flow. Cavitation can cause noise, vibration, erosion, loss of capacity, and unstable operation.

Energy–Momentum Problem Strategy

STEP-BY-STEP

Establish the actual flow direction and pressure reference: absolute or gage.
Apply continuity and calculate section-average velocities.
Use the extended energy equation to find head, pressure, discharge, losses, or shaft work.
Construct HGL/EGL checks and confirm that absolute pressure remains physically acceptable.
Use momentum, not energy alone, to obtain reactions or forces.
Include $\alpha$ and $\beta$ when profiles are strongly nonuniform or laminar.
Check units, signs, and whether empirical coefficients match the device and flow regime.

Key Takeaways

Bernoulli's equation is an inviscid special case; real systems use the extended energy equation with pumps, turbines, losses, and often $\alpha$ .
HGL represents piezometric head, while EGL includes corrected velocity head.
Momentum analysis requires a clearly drawn fluid control volume, inward pressure forces, vector velocity components, and a final action–reaction sign reversal for structural loads.
$\alpha$ corrects kinetic-energy flux and $\beta$ corrects momentum flux; they are not interchangeable.
Navier-Stokes provides the local momentum balance, while engineering friction factors and coefficients supply practical closures.
Cavitation checks always use absolute pressure and vapor pressure at the actual temperature.

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