Fluid Dynamics: Energy & Momentum

Fluid dynamics considers the forces causing fluid motion in hydraulic systems, such as flow through pipes, channels, and pumps. The analysis relies on three fundamental laws of physics applied to fluid mechanics.

Euler's equation is the fundamental differential equation describing the motion of an inviscid (frictionless) fluid. It forms the basis for deriving Bernoulli's equation.

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.

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.

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 ($V^2/2g$).

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.

The terms in the Navier-Stokes equations represent distinct physical forces acting on the fluid volume:

• Inertial Forces (left-hand side): Includes local acceleration $\partial \vec{V} / \partial t$ (changes over time) and convective acceleration $\vec{V} \cdot \nabla \vec{V}$ (changes over space).
• Pressure Forces ($-\nabla P$): Forces arising due to the pressure gradient across the fluid.
• Viscous Forces ($\mu \nabla^2 \vec{V}$): Forces arising from fluid shear stress and internal friction (viscosity).
• Body Forces ($\rho \vec{g}$): External forces acting on the entire bulk of the fluid, primarily gravity in civil engineering applications.

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

Suppose a Pitot tube is placed in a river and the difference between stagnation and static pressure heads is measured to be 0.15 m. Using the Pitot Tube Velocity formula:

$V = \sqrt{2(9.81\text{ m/s}^2)(0.15\text{ m})} \approx 1.71\text{ m/s}$

This real-world application shows how static and dynamic heads combine to offer a direct measurement of fluid velocity.

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.

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

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.

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

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.