Continuity, Momentum & Energy Equations

Forces acting on a flowing fluid

• Gravity — weight of the fluid.
• Pressure — from the surrounding fluid.
• Viscous — internal friction.
• Turbulent — Reynolds stresses in turbulent flow.
• Compressibility and surface tension — usually negligible in hydraulic problems.

Newton's second law with gravity and pressure only gives Euler's equation; adding viscous forces gives the Navier–Stokes equations; adding turbulence terms gives the Reynolds equations.

Euler's equation of motion

For steady flow of an ideal fluid along a streamline:

Bernoulli's equation

Integrating Euler's equation for an incompressible fluid:

Assumptions

1. Ideal fluid — non-viscous (no friction losses).
2. Steady flow.
3. Incompressible fluid.
4. Flow along a streamline (between two points on the same streamline; for irrotational flow, between any two points).
5. No energy added or removed (no pumps or turbines), no heat transfer.
6. Velocity uniform across the section (one-dimensional).

Real fluids

For real flow between sections 1 and 2 with head loss $h_L$, pump head $H_p$ and turbine head $H_t$:

Kinetic energy correction factor $\alpha$ accounts for non-uniform velocity across the section:

$\alpha$ = 2.0 for laminar flow in pipes; about 1.01–1.1 for turbulent flow (often taken as 1). The momentum correction factor $\beta =\frac{1}{A{V}^2}\int u^{2}dA$ is 4/3 for laminar pipe flow and about 1.01–1.04 for turbulent flow.

Energy line and hydraulic gradient line

• Total energy line (TEL) — plots total head $\frac{p}{\rho g}+\frac{V^2}{2g}+z$ along the flow. It always slopes downward in the direction of flow (losses), except where a pump adds energy.
• Hydraulic gradient line (HGL) — plots piezometric head $\frac{p}{\rho g}+z$; lies below the TEL by the velocity head.
• Where the HGL falls below the pipe, pressure is negative (sub-atmospheric) — risk of air entry and cavitation.
• At a sudden enlargement, the HGL may rise (velocity head converts to pressure) while the TEL drops by the loss.

Impulse-momentum principle

Newton's second law applied to a flowing fluid: the net external force on the fluid in a control volume equals the rate of change of momentum of the fluid passing through it.

• The momentum equation is a vector equation — resolve into components.
• Unlike Bernoulli's equation, it needs no information about internal losses; it is therefore used where losses are unknown (hydraulic jump, sudden enlargement, jets).
• For non-uniform velocity, multiply momentum flux by the momentum correction factor $\beta$ (4/3 laminar pipe flow; ≈1.0–1.04 turbulent).

Force on pipe bends and reducers

For a bend turning the flow through angle $\theta$ (horizontal plane), with inlet pressure $p_1$, area $A_1$, velocity $V_1$ and outlet $p_2$, $A_2$, $V_2$, the force exerted by the fluid on the bend:

• For a reducer (straight, $\theta$ = 0): $F_x=\rho Q(V_1-V_2)+p_1{A}_1-p_2{A}_2$.
• These forces are resisted by anchor (thrust) blocks at bends, tees, dead ends and reducers in water mains.

Force exerted by a jet

Jet of area $a$, velocity $V$, density $\rho$.

Stationary plates

Moving plates

If the plate moves in the jet direction with velocity $u$, the relative velocity is $(V-u)$:

Jets on curved vanes with inlet and outlet angles

For turbines, velocity triangles at inlet and outlet give the work done per second per unit weight:

($V_w$ = whirl component; + when the outlet whirl is opposite to vane motion). Detailed in Hydraulic Turbines.

Describing fluid motion

Kinematics describes velocity and acceleration of fluid without considering forces.

• Lagrangian method — follows an individual fluid particle as it moves.
• Eulerian method — observes velocity at fixed points in space as a function of position and time: $\vec{V}=\vec{V}(x,y,z,t)$. Used almost universally in fluid mechanics.

Types of flow

Combinations: steady uniform (flow at constant rate in a long straight pipe of constant diameter), steady non-uniform (constant flow in a tapering pipe), unsteady uniform (accelerating flow in a constant-diameter pipe), unsteady non-uniform (accelerating flow in a tapering pipe).

Flow patterns

• Streamline — an imaginary line whose tangent at every point gives the direction of velocity at that instant. No flow crosses a streamline; streamlines cannot intersect. Equation: $\frac{dx}{u}=\frac{dy}{v}=\frac{dz}{w}$.
• Pathline — the actual path traced by one particle over time.
• Streakline — the locus of all particles that have passed through a fixed point (e.g. dye injected continuously).
• Streamtube — a tube formed by streamlines through a closed curve; fluid cannot cross its walls.

In steady flow, streamlines, pathlines and streaklines coincide.

Continuity equation

Conservation of mass:

Acceleration

The acceleration of a fluid particle has two parts:

(similarly $a_y$, $a_z$).

• Local acceleration — due to unsteadiness; zero in steady flow.
• Convective acceleration — due to change of velocity with position; zero in uniform flow.
• Along a streamline: tangential $a_s=\frac{\partial V}{\partial t}+V\frac{\partial V}{\partial s}$; normal $a_n=\frac{V^2}{r}$.