Fluid Dynamics: Energy & Momentum - Theory & Concepts

Fluid Dynamics: Energy & Momentum

$Learning Objectives$

Connect fluid motion to pressure, gravity, viscous stress, body force, and shaft work.
State the assumptions and limitations of Euler's equation, Bernoulli's equation, and the Navier-Stokes equations.
Apply the extended mechanical-energy equation with pumps, turbines, losses, and kinetic-energy correction factors.
Construct and interpret the hydraulic grade line and energy grade line for real pipe systems.
Apply linear momentum to bends, reducers, nozzles, jets, plates, and vanes using consistent control-volume signs.
Distinguish a stationary plate, a single moving plate, a continuous series of moving vanes, and an ideal Pelton runner.
Apply angular momentum to hydraulic machinery and distinguish absolute, blade, and relative velocity concepts.
Evaluate stagnation pressure, siphon pressure, vapor-pressure limits, and cavitation using absolute pressure.

$Fluid dynamics$

The study of fluid motion together with the forces, stresses, and energy transfers that produce or accompany that motion.

$Mass, Energy, and Momentum Answer Different Questions$

Continuity determines compatible mass flow rates and section velocities.
Mechanical energy relates pressure, elevation, velocity, shaft work, and irreversible loss.
Linear momentum determines resultant forces and reactions caused by changes in velocity and pressure forces.
Angular momentum relates torque to changes in moment of momentum.

No single conservation equation replaces the others. For example, an energy equation can determine pressure or head, but a support reaction generally requires momentum.

$Euler equation$

The inviscid differential momentum relation obtained from Newton's second law when viscous stresses are neglected.

Euler Equation Along a Streamline

Differential momentum balance along a streamline for steady inviscid flow.

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

Variables

Symbol	Description	Unit
$p$	Static pressure	Pa
$\rho$	Fluid density	$kg/m^3$
$V$	Speed along the streamline	m/s
$z$	Elevation above the chosen datum	m

$Bernoulli equation$

The integrated mechanical-energy relation for steady, constant-density, inviscid flow along a streamline when no shaft work is exchanged between the selected points.

Classical Bernoulli Equation

Relates pressure, velocity, and elevation head under the classical Bernoulli assumptions.

\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 speed or appropriate section velocity	m/s
$z$	Elevation above datum	m
$H$	Total mechanical head	m

Bernoulli energy components

Elevation, pressure, and velocity heads along a streamline.

$Bernoulli Is an Ideal Special Case$

Classical Bernoulli requires steady flow, constant density, negligible viscous dissipation, no pump or turbine work between the selected points, and use along a streamline. It can be extended across streamlines only when the flow is irrotational. Real pipe systems normally require the extended mechanical-energy equation.

$Interactive Bernoulli Demonstration$

The Venturi simulation illustrates continuity and ideal horizontal Bernoulli behavior. Interpret low-pressure results using absolute pressure when checking vapor-pressure limits.

Bernoulli's Principle: Venturi Meter

Learning objective: Observe how continuity couples velocity and pressure through a contraction while total-head accounting separates ideal conversion from losses.

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 dimensionless factor that corrects the mean-velocity kinetic-energy term so it represents the actual kinetic-energy flux through a nonuniform velocity profile.

Kinetic-Energy Correction Coefficient

Corrects kinetic-energy flux computed from section-average velocity.

\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 cross section	m/s
$V$	Section-average velocity	m/s
$A$	Cross-sectional area	$m^2$

$Typical Energy-Correction Behavior$

For fully developed laminar flow in a circular pipe, $\alpha=2$ . In fully developed turbulent pipe flow, the flatter profile usually makes $\alpha$ much closer to unity. A value of exactly one is an approximation that should be stated when profile nonuniformity may matter.

Extended Mechanical-Energy Equation

Relates two sections through pressure, elevation, velocity, shaft work, and irreversible loss.

\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 to the fluid by a pump	m
$h_t$	Head extracted from the fluid by a turbine	m
$h_L$	Irreversible head loss from section 1 to section 2	m
$\alpha_1,\alpha_2$	Kinetic-energy correction coefficients	-

$Head-Loss and Shaft-Work Signs$

When sections are ordered in the actual flow direction, $h_L$ is nonnegative. In the equation above, $h_p$ is positive when a pump adds energy to the fluid and $h_t$ is positive when a turbine removes energy from the fluid. Reversing section labels without revisiting flow direction is a common sign error.

$Hydraulic grade line$

The locus of piezometric head $z+p/\gamma$ along a flow system.

Hydraulic Grade Line

Defines the piezometric head represented by a piezometer for the connected fluid.

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

Variables

Symbol	Description	Unit
$HGL$	Hydraulic grade-line elevation	m
$z$	Elevation head	m
$p/\gamma$	Pressure head	m

$Energy grade line$

The locus of total mechanical head including the kinetic-energy correction term.

Energy Grade Line

Defines total mechanical head at a section.

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

Variables

Symbol	Description	Unit
$EGL$	Energy grade-line elevation	m
$\alpha V^2/(2g)$	Corrected velocity head	m

$Interpreting HGL and EGL$

In a passive pipe segment, the EGL falls in the flow direction by the irreversible head loss.
Across a pump, the EGL rises by the head delivered to the fluid.
Across a turbine, the EGL falls by the head extracted in addition to losses.
The HGL can rise through a diffuser even while the EGL falls because velocity head is partly converted into pressure head.
If the HGL falls below the pipe centerline, gauge pressure is negative; physical acceptability still depends on absolute pressure, vapor pressure, air entry, and structural limits.

Energy and hydraulic grade lines

EGL and HGL track velocity head and loss along a system.

Hydraulic Power Associated with Head

Converts fluid head and discharge into a rate of mechanical energy transfer.

P_h=\rho gQH=\gamma QH

Variables

Symbol	Description	Unit
$P_h$	Hydraulic power	W
$Q$	Volumetric discharge	$m^3/s$
$H$	Relevant pump, turbine, or loss head	m

$Navier-Stokes equation$

The differential linear-momentum balance for a Newtonian fluid, combining inertia, pressure, viscous stress, and body forces.

Navier-Stokes Equation for Constant Viscosity

Vector momentum equation for a Newtonian fluid with constant dynamic viscosity.

\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}

Variables

Symbol	Description	Unit
$\rho$	Fluid density	$kg/m^3$
$\mu$	Dynamic viscosity	Pa·s
$\mathbf{g}$	Body-force acceleration, commonly gravity	$m/s^2$

$Turbulent Momentum Requires Closure$

Time-averaging turbulent Navier-Stokes equations introduces Reynolds stresses that are not determined by the mean velocity alone. Turbulence models, friction factors, roughness correlations, and laboratory coefficients provide practical closure.

$Linear momentum equation$

Newton's second law applied to a fluid control volume, equating external force to momentum accumulation plus net momentum flux.

General Linear Momentum for a Fixed Control Volume

Integral momentum balance for a control volume fixed in space.

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

Variables

Symbol	Description	Unit
$\sum\mathbf{F}_{\text{ext}}$	Vector sum of external forces acting on the fluid	N
$\mathbf{n}$	Outward unit normal on the control surface	-

$Momentum correction coefficient$

A dimensionless factor that corrects momentum flux computed from section-average velocity when the actual velocity profile is nonuniform.

Momentum Correction Coefficient

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

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

Variables

Symbol	Description	Unit
$\beta$	Momentum correction coefficient	-
$u$	Local normal velocity	m/s
$V$	Section-average velocity	m/s

Steady One-Dimensional Momentum

Section-average momentum balance for discrete inlets and outlets.

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

Variables

Symbol	Description	Unit
$\beta$	Momentum correction coefficient	-
$Q$	Volumetric discharge at an inlet or outlet	$m^3/s$

$Energy and Momentum Correction Factors Are Different$

For fully developed laminar flow in a circular pipe, $\alpha=2$ while $\beta=4/3$ . Turbulent profiles usually bring both factors closer to one, but they remain conceptually distinct because one corrects energy flux and the other momentum flux.

$Control-Volume Force Analysis$

STEP-BY-STEP

Draw a control volume around the fluid and define coordinate axes.
Determine signed inlet and outlet velocity components from continuity.
Draw pressure forces acting inward on each control-surface face.
Include weight, wall reaction, shear, atmospheric pressure, and other forces when they are not negligible or canceled.
Apply the momentum equation independently in each coordinate direction.
Solve for the force exerted by the pipe, support, vane, or nozzle on the fluid.
Reverse that vector to obtain the force exerted by the fluid on the structure.

Control-volume momentum balance

Velocity and external-force vectors define a momentum balance.

$Pressure Force Direction$

Pressure is compressive. On a control-surface face with outward normal $\mathbf{n}$ , the pressure force on the fluid is $-pA\mathbf{n}$ when pressure is uniform over the section. Drawing the normal vector is safer than memorizing inlet and outlet signs.

$Stationary Jet on a Flat Plate$

For a free jet at uniform atmospheric pressure striking a stationary flat plate normally, if the outgoing water has negligible velocity component normal to the plate, the normal force magnitude is

F=\rho QV=\rho A V^2

where $A$ and $V$ describe the incoming jet. The relation follows from normal momentum change, not from Bernoulli alone.

$Single Moving Flat Plate$

Consider a single flat plate moving in the jet direction at speed $u$ , with an ideal uniform jet of area $A$ and speed $V>u$ . The plate intercepts mass at the relative rate

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

and the normal force is

F=\rho A(V-u)^2

if the plate removes the jet's velocity component normal to itself. The mechanical power transferred to the plate is therefore

P=Fu=\rho A\,u(V-u)^2

This is a single-plate interception model: not all nozzle discharge necessarily strikes the same moving plate continuously.

Maximum Power for a Single Moving Flat Plate

Maximizes the single-plate power relation P proportional to u(V-u)^2.

\frac{d}{du}\left[u(V-u)^2\right] = (V-u)(V-3u)

u_{\text{opt}}=\frac{V}{3}

Variables

Symbol	Description	Unit
$u_{\text{opt}}$	Plate speed for the nontrivial maximum of the single-plate model	m/s
$V$	Jet speed	m/s

$Series of Moving Vanes and an Ideal Pelton Runner$

A wheel with a series of buckets can continuously intercept essentially the full nozzle discharge. That changes the power model from the single moving-plate case.

For an idealized Pelton runner that reverses the relative tangential jet velocity by $180^\circ$ with no relative-speed loss,

V_{2x}=2u-V

so the tangential force on the runner is

F_x=\rho Q(V-V_{2x})=2\rho Q(V-u)

and the runner power is

P=2\rho Q\,u(V-u)

which is maximized at $u=V/2$ . Real Pelton buckets have finite outlet angle, bucket friction, splitter geometry, losses, and additional efficiency effects, so real optimum speed ratio is a design result rather than a universal exact constant.

$Do Not Mix Moving-Vane Models$

The results $u=V/3$ and $u=V/2$ come from different physical systems. The first belongs to the ideal single moving flat plate model with intercepted mass rate proportional to $V-u$ . The second belongs to an idealized continuous Pelton-type bucket series that receives the full nozzle discharge. Changing only the keyed answer without changing the physical model is incorrect.

$Curved Vanes and Velocity Triangles$

For a moving curved vane, distinguish:

Absolute velocity $\mathbf{V}$ , measured in the stationary frame.
Blade velocity $\mathbf{u}$ , the vane or bucket speed.
Relative velocity $\mathbf{V}_r=\mathbf{V}-\mathbf{u}$ , measured with respect to the moving vane.

Vane geometry redirects the relative velocity, while momentum equations use the absolute inlet and outlet velocity vectors. Bucket friction can be represented by reducing the relative outlet speed from its inlet value when that model is explicitly stated.

Jet deflection by a curved vane

Changing jet momentum produces an equal-and-opposite reaction.

$Angular momentum equation$

A control-volume relation equating external torque to the rate of change of moment of momentum.

Steady Turbomachinery Angular Momentum

Relates shaft torque to tangential components of absolute velocity at one inlet and one outlet.

T=\dot m(r_2V_{\theta 2}-r_1V_{\theta 1})

Variables

Symbol	Description	Unit
$T$	Torque exerted on the fluid	N·m
$\dot m$	Mass flow rate	kg/s
$r$	Radius from the rotation axis	m
$V_\theta$	Tangential component of absolute velocity	m/s

$Stagnation pressure$

The pressure obtained when a moving fluid is brought to rest along a streamline under the assumptions used by the stagnation relation.

Incompressible Stagnation Relation

Relates static and stagnation pressure when elevation change and losses through the probe are negligible.

p_0-p=\frac{1}{2}\rho V^2

Variables

Symbol	Description	Unit
$p_0$	Stagnation pressure	Pa
$p$	Static pressure	Pa
$V$	Local flow speed	m/s

$Pitot, Venturi, and Torricelli Applications$

Pitot-static tube: Converts local velocity head to a measured stagnation-minus-static pressure; a calibration coefficient may be required.
Venturi meter: Uses continuity and a measured piezometric-head difference to infer discharge; a discharge coefficient represents real losses and installation effects.
Torricelli relation: For a large reservoir discharging ideally to the same pressure environment, $V=\sqrt{2gH}$ follows from Bernoulli.

All three applications inherit the assumptions of their governing energy model and must use consistent pressure references and elevations.

$Cavitation$

The formation and subsequent collapse of vapor cavities when local absolute liquid pressure reaches or falls below the vapor pressure corresponding to the operating temperature.

$Cavitation Uses Absolute Pressure$

Vapor pressure is an absolute thermodynamic pressure. A negative gauge pressure is not itself a cavitation criterion; convert to absolute pressure first. Once significant vapor forms, a single-phase incompressible Bernoulli model no longer describes the actual two-phase flow.

$Siphon Pressure Limits$

A siphon can convey liquid over an intermediate crest only while a continuous liquid column is maintained. Increasing crest elevation, velocity head, or upstream loss lowers the crest pressure. The practical limit depends on atmospheric pressure, liquid vapor pressure, dissolved gas release, leakage, transient effects, and available priming.

$Energy and Momentum Workflow$

STEP-BY-STEP

Establish the actual flow direction, datum, and pressure reference.
Apply continuity to calculate section-average velocities.
Use the extended energy equation for pressure, head, losses, pump work, or turbine work.
Check the HGL and EGL for consistency and inspect absolute-pressure limits.
Use linear momentum for reactions and forces, keeping pressure-force directions explicit.
For moving vanes, separate absolute, blade, and relative velocities before applying momentum.
Choose the correct physical model before optimizing jet power; do not transfer a speed ratio from one vane system to another.
Include $\alpha$ , $\beta$ , empirical coefficients, and real losses when their neglect would materially affect the result.

Key Takeaways

Classical Bernoulli is an inviscid special case; real systems use the extended energy equation with shaft work, losses, and sometimes $\alpha$ .
HGL is piezometric head; EGL adds corrected velocity head and falls through passive losses.
Momentum analysis requires a fluid control volume, signed velocity components, inward pressure forces, and action-reaction reversal for structural loads.
A single moving flat plate with $P\propto u(V-u)^2$ has the nontrivial optimum $u=V/3$ .
An idealized continuous Pelton bucket series uses a different power model and gives $u=V/2$ only under its stated ideal assumptions.
Moving-vane analysis must distinguish absolute velocity, blade velocity, and relative velocity.
$\alpha$ corrects energy flux and $\beta$ corrects momentum flux; they are not interchangeable.
Cavitation checks compare local absolute pressure with vapor pressure at the actual temperature.

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