Fluid Dynamics: Energy & Momentum - Theory & Concepts

Fluid Dynamics: Energy & Momentum

Dynamics of fluid flow including Bernoulli's equation, energy lines, the impulse-momentum principle, and practical applications.

Overview

Fluid dynamics considers the forces causing fluid motion. The analysis is based on three fundamental laws:

Procedure

STEP-BY-STEP

Conservation of Mass (Continuity Equation)
Conservation of Energy (Bernoulli's Equation)
Conservation of Momentum (Impulse-Momentum Equation)

Euler's Equation of Motion

Euler's equation is the differential equation describing the motion of an inviscid fluid.

Euler's Equation

Applying Newton's Second Law to an infinitesimal fluid particle along a streamline, assuming zero viscosity (inviscid flow), yields Euler's equation of motion.

Euler's Equation

The differential equation describing the motion of an inviscid fluid along a streamline.

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

Variables

Symbol	Description	Unit
$P$	Pressure	Pa
	Fluid density	$k g / m^{3}$
$V$	Flow velocity	m/s
$g$	Acceleration due to gravity	$m / s^{2}$
$z$	Elevation	m

Note

Integrating Euler's equation for steady, incompressible flow yields the well-known Bernoulli's Equation.

Concept Overview

Bernoulli's equation relates pressure, velocity, and elevation for an inviscid, incompressible fluid in steady flow along a streamline. It states that the total mechanical energy per unit weight (head) remains constant.

Bernoulli's Equation

The mathematical statement of the conservation of mechanical energy for steady, incompressible, frictionless flow along a streamline.

\frac{P_1}{\gamma} + \frac{V_1^2}{2g} + z_1 = \frac{P_2}{\gamma} + \frac{V_2^2}{2g} + z_2 = H

Variables

Symbol	Description	Unit
$P$	Pressure	Pa
	Specific weight of the fluid	$N / m^{3}$
$V$	Flow velocity	m/s
$g$	Acceleration due to gravity	$m / s^{2}$
$z$	Elevation above datum	m
$H$	Total head (constant along streamline)	m

Bernoulli Head Components

The components of Bernoulli's equation represent distinct forms of energy per unit weight:

Pressure Head: $P/\gamma$ (Work done by pressure forces)
Velocity Head: $V^2/2g$ (Kinetic Energy)
Elevation Head: $z$ (Potential Energy)
Total Head ( $H$ ): Sum of the three components.

Note

Restrictions:

Steady flow
Incompressible flow ( $\rho = \text{constant}$ )
Frictionless flow (Inviscid)
Along a streamline

Venturi Meter Simulation

Use the interactive Venturi Meter simulation below to visualize how pressure changes as velocity changes due to varying cross-sectional area. Observe the trade-off between kinetic energy and pressure energy.

Bernoulli's Principle (Venturi Meter)

Inlet Diameter (

D_1

): 200 mm

Throat Diameter (

D_2

): 100 mm

Inlet Velocity (

V_1

): 2 m/s

Inlet (1)

150.0 kPa

2.00 m/s

Throat (2)

0.0 kPa

0.00 m/s

As the area decreases at the throat, velocity must increase (Continuity). This increase in kinetic energy causes a drop in pressure potential energy (Bernoulli). If pressure drops below vapor pressure, cavitation occurs.

Graphical Representation of Head

The Energy Line and Hydraulic Grade Line provide a graphical representation of the energy state along a fluid flow path.

Energy Line (EL): Represents the total head ( $H$ ). In ideal flow, EL is horizontal. In real flow, EL slopes downward due to friction loss ( $h_f$ ).
Hydraulic Grade Line (HGL): Represents the sum of pressure and elevation heads (piezometric head). The HGL is always below the EL by the velocity head ( $V^2/2g$ ).

Energy Line (EL)

Calculates the total head at any point along the streamline.

EL = \frac{P}{\gamma} + \frac{V^2}{2g} + z

Variables

Symbol	Description	Unit
$E L$	Energy Line (total head)	m
$P$	Pressure	Pa
	Specific weight of fluid	$N / m^{3}$
$V$	Flow velocity	m/s
$g$	Acceleration due to gravity	$m / s^{2}$
$z$	Elevation above datum	m

Hydraulic Grade Line (HGL)

Calculates the piezometric head, representing the height to which liquid would rise in a piezometer tube.

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

Variables

Symbol	Description	Unit
$H G L$	Hydraulic Grade Line	m
$P$	Pressure	Pa
	Specific weight of fluid	$N / m^{3}$
$z$	Elevation above datum	m

Viscous Flow Overview

While Bernoulli's Equation assumes inviscid (frictionless) flow, the Navier-Stokes Equations describe the motion of real, viscous fluid substances. They are derived from the conservation of momentum (Newton's second law) applied to a fluid particle, adding terms for internal viscous stresses.

Navier-Stokes Equations

These non-linear partial differential equations represent the balance between inertial, pressure, viscous, and body forces in a fluid. They are the fundamental equations of fluid dynamics.

Navier-Stokes Equations

The vector representation of the conservation of momentum for incompressible Newtonian viscous flow.

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

Variables

Symbol	Description	Unit
	Fluid density	$k g / m^{3}$
	Velocity vector	m/s
$t$	Time	s
$P$	Pressure	Pa
	Dynamic viscosity	Pa·s
	Gravitational acceleration vector	$m / s^{2}$

Navier-Stokes Force Terms

The terms in the equation represent distinct physical forces:

Inertial Forces (left-hand side): Includes local acceleration $\partial \vec{V} / \partial t$ and convective acceleration $\vec{V} \cdot \nabla \vec{V}$ .
Pressure Forces ( $-\nabla P$ ): Forces arising due to the pressure gradient.
Viscous Forces ( $\mu \nabla^2 \vec{V}$ ): Forces arising from fluid shear stress and viscosity.
Body Forces ( $\rho \vec{g}$ ): External forces acting on the bulk fluid (primarily gravity).

Note

Solving the Navier-Stokes equations analytically is one of the biggest challenges in mathematics and physics (a Millennium Prize Problem). They are typically solved numerically using Computational Fluid Dynamics (CFD).

Applications of Bernoulli's Equation

Bernoulli's equation supports common velocity and flow measurement tools such as orifices, Venturi meters, and Pitot tubes.

Torricelli's Theorem

The principle stating that the velocity of efflux of a fluid from an orifice under gravity is equivalent to the velocity acquired by a body falling freely from rest through the same height.

Torricelli's Theorem

Calculates the theoretical velocity of a liquid exiting an orifice.

V = \sqrt{2gh}

Variables

Symbol	Description	Unit
$V$	Velocity of efflux	m/s
$g$	Acceleration due to gravity	$m / s^{2}$
$h$	Depth of fluid above the orifice	m

Venturi Meter

A flow measurement device consisting of a converging section, a throat, and a diverging section. It uses the pressure difference between the inlet and the throat to determine volumetric flow rate.

Venturi Meter Discharge

Calculates the flow rate (discharge) through a Venturi meter based on pressure differential.

Q = C_d A_1 A_2 \sqrt{\frac{2g(h_1-h_2)}{A_1^2 - A_2^2}}

Variables

Symbol	Description	Unit
$Q$	Volumetric flow rate (discharge)	$m^{3} / s$
$C_{d}$	Coefficient of discharge (typically 0.96 - 0.99)	dimensionless
$A_{1}$	Cross-sectional area of inlet pipe	$m^{2}$
$A_{2}$	Cross-sectional area of throat	$m^{2}$
$g$	Acceleration due to gravity	$m / s^{2}$
$h_{1}$	Piezometric head at inlet	m
$h_{2}$	Piezometric head at throat	m

Pitot Tube

An instrument used to measure the local flow velocity at a specific point in a fluid stream by converting kinetic energy into pressure energy at a stagnation point.

Pitot Tube Velocity

Calculates point velocity using the difference between stagnation and static pressures.

V = \sqrt{2g \Delta h}

Variables

Symbol	Description	Unit
$V$	Local velocity	m/s
$g$	Acceleration due to gravity	$m / s^{2}$
	Difference between total (stagnation) head and static head	m

Impulse-Momentum Principle

Derived from Newton's Second Law ( $F=ma$ ). It states that the sum of external forces acting on a fluid control volume equals the rate of change of momentum.

Impulse-Momentum Equation

Calculates the forces exerted by or on a fluid control volume due to momentum changes.

\sum \vec{F} = \rho Q (\vec{V}_{out} - \vec{V}_{in})

Variables

Symbol	Description	Unit
	Sum of external forces acting on the fluid	N
	Fluid density	$k g / m^{3}$
$Q$	Volumetric flow rate	$m^{3} / s$
	Outlet velocity vector	m/s
	Inlet velocity vector	m/s

Note

Analysis guidelines for the impulse-momentum principle:

$\sum \vec{F}$ includes pressure forces, weight, and reaction forces from boundaries (e.g., pipe bends).
Vector equation: Solve for X and Y components separately.

Forces Exerted by Jets

The impulse-momentum principle is applied to calculate the force of a fluid jet striking a surface.

Jet Deflection Overview

When a fluid jet of cross-sectional area $A$ and velocity $V$ strikes a stationary flat plate or curved vane, it exerts a force due to the change in momentum. We analyze the forces for two main cases:

Stationary Flat Plate Normal to Jet: Where all forward momentum is destroyed in the direction of the jet.
Stationary Curved Vane: Where the jet is deflected by an angle $\theta$ from its original direction.

Stationary Flat Plate Normal to Jet

Calculates the force exerted by a jet striking a flat stationary surface at a right angle.

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

Variables

Symbol	Description	Unit
$F$	Normal force on the plate	N
	Fluid density	$k g / m^{3}$
$A$	Cross-sectional area of the jet	$m^{2}$
$V$	Velocity of the jet	m/s
$Q$	Volumetric flow rate	$m^{3} / s$

Stationary Curved Vane - Horizontal Force

Calculates the horizontal component of the force exerted by a deflected jet on a stationary curved vane.

F_x = \rho Q V (1 - \cos\theta)

Variables

Symbol	Description	Unit
$F_{x}$	Force component in the direction of the jet	N
	Fluid density	$k g / m^{3}$
$Q$	Volumetric flow rate	$m^{3} / s$
$V$	Velocity of the jet	m/s
	Deflection angle of the vane	degrees

Stationary Curved Vane - Vertical Force

Calculates the vertical component of the force exerted by a deflected jet on a stationary curved vane.

F_y = -\rho Q V \sin\theta

Variables

Symbol	Description	Unit
$F_{y}$	Force component perpendicular to the jet direction	N
	Fluid density	$k g / m^{3}$
$Q$	Volumetric flow rate	$m^{3} / s$
$V$	Velocity of the jet	m/s
	Deflection angle of the vane	degrees

Moving Vanes (Pelton Wheel Principle)

If a curved vane moves in the same direction as the jet with velocity $u$ (where $u < V$ ), the force is dictated by the relative velocity of the fluid relative to the vane ( $V_r = V - u$ ). The mass flow rate actually striking the vane is also reduced to $\dot{m} = \rho A (V - u)$ .

Force on a Moving Curved Vane

Calculates the force exerted in the direction of vane movement.

F_x = \rho A (V - u)^2 (1 - \cos\theta)

Variables

Symbol	Description	Unit
$F_{x}$	Force component in the direction of motion	N
	Fluid density	$k g / m^{3}$
$A$	Cross-sectional area of the jet	$m^{2}$
$V$	Absolute velocity of the jet	m/s
$u$	Velocity of the moving vane	m/s
	Deflection angle of the vane	degrees

Jet Power on Moving Vanes

The mechanical power generated by the jet striking the moving vane is $P = F_x u$ . This is the fundamental operating principle of impulse turbines such as the Pelton wheel.

Key Takeaways

Euler's Equation is the differential form of motion for an inviscid fluid. Integrating it yields Bernoulli's equation.
Bernoulli's Equation applies to steady, incompressible, frictionless flow along a streamline, acting as a statement of conservation of mechanical energy.
It states that the sum of pressure head, velocity head, and elevation head is constant.
The Energy Line (EL) represents total head and is horizontal for ideal flow, while the Hydraulic Grade Line (HGL) represents piezometric head and is lower than the EL by the velocity head.
The Navier-Stokes Equations extend fluid dynamic analysis to real, viscous fluids by including internal friction (viscous forces).
They are based on Newton's Second Law and represent a complex balance of inertial, pressure, viscous, and body forces.
Torricelli's Theorem shows that fluid exiting an orifice has a velocity proportional to the square root of the depth ( $V = \sqrt{2gh}$ ), similar to a freely falling body.
Venturi Meters measure flow rate accurately by creating a constriction that increases velocity and decreases pressure, with minimal permanent head loss.
Pitot Tubes measure local point velocity by bringing the fluid to a dead stop and measuring the resulting stagnation pressure.
The Impulse-Momentum Equation is used to calculate forces exerted by moving fluids on solid boundaries, such as pipe bends, nozzles, or turbine blades.
It is a vector equation, meaning forces and velocities must be analyzed in their respective X, Y, and Z components.

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