Fluid Dynamics and Bernoulli's Equation

Conservation of Mass (Continuity)

For steady, incompressible flow, the volumetric flow rate Q is constant:

A₁V₁ = A₂V₂ = Q (m³/s)

The mass flow rate is ṁ = ρAV = ρQ (kg/s). Because A = πD²/4 for a round pipe, velocity scales with the inverse square of diameter: halve D and V quadruples.

Worked example — continuity in a contraction

Water flows in a pipe narrowing from D₁ = 10 cm to D₂ = 5 cm; V₁ = 2 m/s. Then:

V₂ = V₁ (D₁/D₂)² = 2 × (10/5)² = 2 × 4 = 8 m/s.

Bernoulli's Equation

For steady, inviscid, incompressible flow along a streamline:

p₁/γ + V₁²/2g + z₁ = p₂/γ + V₂²/2g + z₂

Every term has units of length (head):

• p/γ = pressure head
• V²/2g = velocity head
• z = elevation head

Their sum, the total head, is constant along a streamline. Physically, where velocity rises (a constriction), pressure falls — the basis of venturi meters and airfoil lift.

Applications

Pitot tube (velocity from stagnation pressure): V = √(2(p_stag − p_static)/ρ).

Venturi meter (flow from a constriction's pressure drop): Q = C_d A₂ √[2gΔh / (1 − (A₂/A₁)²)].

Free jet / Torricelli's theorem (tank draining through a hole at depth h): V = √(2gh) — identical to the speed of an object falling from height h.

Worked example — Torricelli

Water exits a tank through a hole 5 m below the surface: V = √(2gh) = √(2 × 9.81 × 5) = √98.1 = 9.9 m/s.

Reynolds Number and Flow Regime

The Reynolds number is the ratio of inertial to viscous forces:

Re = ρVD/μ = VD/ν

Worked example — Reynolds number

Water (ν = 10⁻⁶ m²/s) flows at 1.5 m/s in a 0.1 m pipe:

Re = VD/ν = (1.5)(0.1)/10⁻⁶ = 0.15/10⁻⁶ = 150,000 → fully turbulent.

Other dimensionless numbers

The Energy Equation (Real Pipe Flow)

Real flow loses energy to friction, so Bernoulli gains a head-loss term:

p₁/γ + V₁²/2g + z₁ = p₂/γ + V₂²/2g + z₂ + h_L

where h_L = major (friction) losses + minor (fitting) losses.

Major Losses — Darcy-Weisbach

h_f = f (L/D)(V²/2g)

f is the Darcy friction factor. For laminar flow it is exact and roughness-independent:

f = 64/Re

For turbulent flow, read f from the Moody diagram using Re and the relative roughness ε/D (or solve the Colebrook equation). Beware the Fanning friction factor, which is ¼ of the Darcy factor — the FE Handbook uses Darcy.

Worked example — Bernoulli with a pressure drop

Water accelerates from V₁ = 2 m/s to V₂ = 8 m/s through a horizontal contraction (z₁ = z₂). With p₁ = 200 kPa, find p₂ (inviscid):

p₂ = p₁ + (γ/2g)(V₁² − V₂²) ... or more directly p₂ = p₁ + ½ρ(V₁² − V₂²) = 200,000 + ½(1,000)(2² − 8²) = 200,000 + 500(4 − 64) = 200,000 − 30,000 = 170 kPa.

The pressure drops 30 kPa as the fluid speeds up — velocity head is bought with pressure head, the core Bernoulli trade-off.

Minor Losses (Fittings, Valves, Entrances)

h_m = K (V²/2g)

Each fitting has a loss coefficient K:

Globe valves throttle well but waste head; gate valves are for isolation. Total h_L = Σh_f + Σh_m for the whole line.

Worked example — friction head loss

Water flows at V = 2 m/s through a 100 m pipe of D = 0.05 m with f = 0.02:

h_f = f(L/D)(V²/2g) = 0.02 × (100/0.05) × (2²/(2×9.81)) = 0.02 × 2,000 × 0.204 = 8.2 m of head loss.

Open-Channel Flow

Manning's equation gives velocity in an open channel (SI):

V = (1/n) R_h^(2/3) S^(1/2)

where n is Manning's roughness, R_h = A/P (area/wetted perimeter) is the hydraulic radius, and S is the energy-grade-line slope. The Froude number Fr = V/√(gD_h) classifies the flow: Fr < 1 subcritical (deep, slow), Fr = 1 critical, Fr > 1 supercritical (shallow, fast).

Common Traps

• Bernoulli's assumptions. It is inviscid — once friction matters (long pipes), switch to the energy equation with h_L.
• f = 64/Re only for laminar flow. For turbulent flow, f needs the Moody diagram.
• Velocity head uses V², not V. Doubling velocity quadruples velocity head and friction loss.

Quick reference

• Continuity: A₁V₁ = A₂V₂; ṁ = ρQ.
• Bernoulli: p/γ + V²/2g + z = const.
• Re = VD/ν; laminar <2,300, turbulent >4,000.
• h_f = f(L/D)(V²/2g); laminar f = 64/Re. h_m = K(V²/2g).