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

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), gives Euler's equation:

Euler's Equation

The differential equation describing the motion of an inviscid fluid.

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

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

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.

Concept Overview

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

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

Bernoulli Simulation (Venturi Meter): 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.

Energy Line (EL) and Hydraulic Grade Line (HGL)

Graphical Representation of Head

Energy Line (EL): Represents the total head ( $H$ ). In ideal flow, EL is horizontal. In real flow, EL slopes downward due to friction loss ( $h_f$ ).
Graphical Representation of Head
<strong>Bernoulli Simulation (Venturi Meter):</strong> Visualize how pressure changes as velocity changes due to varying cross-sectional area. Ob...
$EL = \frac{P}{\gamma} + \frac{V^2}{2g} + z$
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 ( $V^2/2g$ ).
$HGL = \frac{P}{\gamma} + z$

Navier-Stokes Equations

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. For incompressible Newtonian fluids, they can be written in vector form as:

Navier-Stokes Equations

\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}

Left Side: Inertial forces (local and convective acceleration).
$-\nabla P$ : Pressure gradient force.
$\mu \nabla^2 \vec{V}$ : Viscous forces.
$\rho \vec{g}$ : Body forces (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

Torricelli's Theorem

The velocity of efflux from an orifice at a depth

h

below the free surface is:

Torricelli's Theorem

V = \sqrt{2gh}

Venturi Meter

A device used to measure flow rate in a pipe. By measuring the pressure difference between the inlet and the throat, the discharge can be calculated.

Venturi Meter

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

Variables

Symbol	Description	Unit
$C_d$	Coefficient of discharge (typically 0.96 - 0.99).	-

Pitot Tube

Used to measure velocity at a point by converting kinetic energy into pressure energy (stagnation pressure).

Pitot Tube

V = \sqrt{2g \Delta h}

Variables

Symbol	Description	Unit
$\Delta h$	Difference between total head and static head.	-

Impulse-Momentum Equation

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 Principle

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

$\sum \vec{F}$ 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

Applying the impulse-momentum principle to calculate the force of a fluid jet striking a surface.

Jet Deflection

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.

Stationary Flat Plate Normal to Jet: All forward momentum is destroyed.
Jet Deflection
Applying the impulse-momentum principle to calculate the force of a fluid jet striking a surface.
$F = \rho A V^2 = \rho Q V$
Stationary Curved Vane: The jet is deflected by an angle $\theta$ from its original direction. Assuming negligible friction (velocity magnitude $V$ remains constant):
$F_x = \rho Q (V - V \cos\theta) = \rho Q V (1 - \cos\theta)$
$F_y = \rho Q (0 - V \sin\theta) = -\rho Q V \sin\theta$

Moving Vanes (Pelton Wheel Principle)

If the vane is moving in the same direction as the jet with velocity

u

(

u < V

), the relative velocity (

V_r = V - u

) dictates the force.

The mass flow rate striking the vane is based on the relative velocity:

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

Moving Vanes (Pelton Wheel Principle)

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

The mechanical power generated is

P = F_x u

. This is the fundamental operating principle of impulse turbines like 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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