Environmental Hydraulics: Fluid Statics, Flow in Pipes, Flow Measurement and Channel Hydraulics

A fluid deforms continuously under any shear stress, however small. Its properties are density ρ (water about 1000 kg/m³), specific weight γ = ρg (9.81 kN/m³ for water), dynamic viscosity μ, which relates shear stress to the velocity gradient in Newton’s law τ = μ du/dy (water about 1.0 × 10⁻³ Pa·s at 20 °C, falling as it warms), kinematic viscosity ν = μ/ρ, compressibility (bulk modulus), vapour pressure, and surface tension σ (about 0.073 N/m for water in air), which gives the capillary rise h = 4σ cos θ/(ρgd) in a tube of diameter d. The concepts of mechanics carried in from statics and dynamics are the free-body diagram, Newton’s second law, and the conservation of mass, momentum and energy — the three that become the continuity, momentum and energy equations of fluid flow.

In a fluid at rest pressure increases linearly with depth, p = p₀ + ρgh, and acts equally in all directions (Pascal’s law). Pressure is measured as absolute or as gauge (relative to the atmosphere) by a piezometer, a U-tube manometer (for a manometric liquid of specific gravity S_m reading a deflection x against water, the head difference is x(S_m − 1)), a differential or inclined manometer, or a Bourdon gauge. The hydrostatic force on a plane surface is the pressure at its centroid times its area, F = ρg h̄ A, and it acts at the centre of pressure, below the centroid by I_G/(A h̄): h_cp = h̄ + I_G/(A h̄). For a vertical rectangle of depth d with its top at the free surface, h_cp = 2d/3. On a curved surface the horizontal component is the force on its vertical projection and the vertical component is the weight of fluid above it.

Buoyancy is Archimedes’ principle: a body immersed in a fluid is pushed up by a force equal to the weight of the fluid it displaces, acting through the centre of buoyancy B, the centroid of the displaced volume. A floating body is in equilibrium when that force equals its weight. Its stability against rolling depends on the metacentre M: for a small heel the buoyancy acts through M, and the body is stable if M lies above the centre of gravity G. The metacentric height is GM = BM − BG, with BM = I/V, where I is the second moment of the waterline area about the tilting axis and V the displaced volume. For a rectangular pontoon of width b and draft d, BM = b²/(12d).

Kinematics describes the motion without its causes. Flow is steady if conditions at a point do not change with time and uniform if they do not change along a streamline. A streamline is everywhere tangent to the velocity; a pathline is the track of one particle; a streakline joins all particles that passed one point; in steady flow the three coincide. The acceleration of a particle has a local part ∂V/∂t (zero in steady flow) and a convective part V ∂V/∂s (zero in uniform flow). A flow is irrotational if its vorticity, the curl of velocity, is zero; then a velocity potential φ exists with V = ∇φ. In two-dimensional incompressible flow a stream function ψ exists with u = ∂ψ/∂y, v = −∂ψ/∂x, and the difference of ψ between two streamlines is the discharge between them.

The continuity equation conserves mass: for an incompressible fluid in a pipe Q = A₁V₁ = A₂V₂, and in differential form ∂u/∂x + ∂v/∂y + ∂w/∂z = 0. The energy equation along a streamline for steady, incompressible, frictionless flow is Bernoulli’s equation, p/(ρg) + V²/(2g) + z = constant — pressure head, velocity head and datum head; with friction and machines it becomes p₁/ρg + V₁²/2g + z₁ + h_pump = p₂/ρg + V₂²/2g + z₂ + h_L. The momentum equation is Newton’s second law for a control volume: the net force equals the rate of momentum outflow minus inflow, ΣF = ρQ(V₂ − V₁), resolved in each direction. It gives the force on a pipe bend or a nozzle, the thrust of a jet on a plate, and — because it needs no knowledge of the energy lost — the hydraulic jump.

The Reynolds number Re = ρVD/μ = VD/ν compares inertia with viscosity. In a pipe, flow is laminar below about Re = 2000, turbulent above about 4000, and transitional between. Laminar pipe flow has a parabolic velocity profile with the maximum velocity twice the mean, and the Hagen–Poiseuille result for its head loss, h_f = 32μVL/(ρgD²). The general head-loss law for any regime is Darcy–Weisbach, h_f = fLV²/(2gD), where the friction factor is f = 64/Re in laminar flow and depends on Re and the relative roughness ε/D in turbulent flow (the Moody diagram; the Colebrook equation). Minor losses at entrances, bends, valves and expansions are written kV²/(2g); a sudden expansion loses (V₁ − V₂)²/(2g), and a pipe discharging into a tank loses its whole velocity head. The Hazen–Williams formula is the empirical alternative used for water mains.

Pipes in series carry the same flow and their head losses add; pipes in parallel share the same head loss and their flows add. For parallel pipes of equal length and f, h_f ∝ Q²/D⁵, so Q ∝ D^2.5: doubling the diameter multiplies the flow by 2^2.5 = 5.66. A pipe network — the looped grid of a distribution system — must satisfy continuity at every node and zero net head loss round every loop. The Hardy Cross method solves it iteratively: assume flows that satisfy continuity, compute the loop correction ΔQ = −Σh_f/(nΣ|h_f/Q|) (n = 2 for Darcy–Weisbach, 1.85 for Hazen–Williams), apply it to every pipe of the loop, and repeat until the corrections are small. A body in a flowing fluid experiences drag parallel to the flow and lift normal to it, F_D = C_D A ρV²/2 and F_L = C_L A ρV²/2. Drag is part skin friction and part pressure (form) drag; for a small sphere at low Re Stokes’ law gives F_D = 3πμVd, i.e. C_D = 24/Re — the law behind the settling of particles in Section 5.

In pipes, the venturimeter, orifice meter and flow nozzle all apply continuity and Bernoulli between a full section and a constriction and read the pressure drop: Q = C_d A₁A₂√(2gh)/√(A₁² − A₂²), where h is the difference in piezometric head. The venturimeter’s gradual diverging cone recovers most of the pressure, so C_d is about 0.95–0.99; the orifice plate is cheap but loses much more, with C_d about 0.6. The Pitot tube reads the stagnation head at a point, and V = √(2gΔh) gives the local velocity; electromagnetic and ultrasonic meters are used where no loss can be tolerated. In open channels discharge is measured by weirs and notches: a rectangular weir of length L gives Q = (2/3)C_d L √(2g) H^(3/2), and a triangular (V-) notch of angle θ gives Q = (8/15)C_d √(2g) tan(θ/2) H^(5/2) — more sensitive at low flows, which is why V-notches meter small effluent streams. The Parshall flume forces critical flow in a throat and gives Q from a single upstream depth, with little head loss and no pond for solids to settle in, which suits raw sewage; the current meter and area–velocity method gauge rivers.

Uniform flow in an open channel, where the depth is constant and gravity balances friction, is described by Manning’s equation, V = (1/n)R^(2/3)S^(1/2), with R = A/P the hydraulic radius, S the bed slope and n the roughness (about 0.013–0.015 for concrete). The specific energy is the energy above the bed, E = y + V²/(2g) = y + q²/(2gy²) for a rectangular channel carrying q per unit width. For a given q, E has a minimum at the critical depth y_c = (q²/g)^(1/3), where E_min = 1.5y_c and the Froude number Fr = V/√(gy) = 1. Deeper, slower flow (Fr < 1) is subcritical; shallower, faster flow (Fr > 1) is supercritical or rapid. Two depths — alternate depths — share each E above the minimum, which is why a small hump in the bed can lower a subcritical surface and raise a supercritical one.

A hydraulic jump is the abrupt rise from supercritical to subcritical flow, as below a spillway or sluice gate, and it dissipates energy in intense turbulence — which is also why it is used as a rapid mixer for coagulants. The momentum equation gives the sequent depth, y₂ = (y₁/2)(√(1 + 8Fr₁²) − 1), and the energy lost is ΔE = (y₂ − y₁)³/(4y₁y₂). Rapidly varied flow — jumps, falls, flow over weirs and under gates — changes depth over a short distance; gradually varied flow changes it slowly under the balance of friction and slope, dy/dx = (S₀ − S_f)/(1 − Fr²), and its water-surface profiles (backwater behind a weir is the M1 curve on a mild slope) are classified by the slope (mild, steep, critical, horizontal, adverse) and by where the depth lies relative to the normal and critical depths. Lined channels are designed for a non-erodible boundary: the section is chosen for the most economical, or best hydraulic, section — the one of least wetted perimeter for a given area, which is half a hexagon for a trapezium and, for a rectangle, b = 2y with R = y/2 — and then checked for a velocity high enough not to deposit silt and a freeboard above the design depth.