4.2 Fluid Dynamics: Continuity, Bernoulli & Momentum

Continuity — Conservation of Mass

Conservation of mass for steady flow says mass in = mass out: ρ₁A₁V₁ = ρ₂A₂V₂. For steady, incompressible flow (constant ρ, as for water) this reduces to constant volumetric flow rate Q = A·V along a streamtube, so A₁V₁ = A₂V₂. Here A is the cross-sectional area and V is the average (mean) velocity over that area. The immediate consequence: where a pipe narrows, velocity must rise, V₂ = V₁·(A₁/A₂). For a circular pipe A = πD²/4, so halving the diameter quarters the area and quadruples the velocity — and because losses scale with V², small pipes are costly in head.

Units must be consistent — Q in ft³/s (cubic feet per second, cfs) or m³/s, with A in ft² or m² and V in ft/s or m/s. A frequent trap is mixing gallons per minute (gpm) with cfs; remember 1 cfs ≈ 448.8 gpm. When two streams merge or split, total volumetric flow is conserved: Q_in = ΣQ_out. The continuity and energy equations together solve the vast majority of FE pipe and channel problems, so set them up first whenever areas change.

The Bernoulli / Energy Equation

Along a streamline for ideal flow, the Bernoulli equation states that total head is constant:

p/γ + V²/2g + z = constant

Each term is a head (length): p/γ = pressure head, V²/2g = velocity head, z = elevation head. For real flow between points 1 and 2, add machine and loss terms (the extended energy equation):

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

where h_pump is head added by a pump, h_turbine is head extracted, and h_L is the total head loss (friction + minor losses). Each term has units of length (energy per unit weight), which is why we speak of 'feet of head' or 'meters of head.' Multiply any head by γ to get a pressure, or by Q·γ to get power. The Bernoulli equation assumes steady, incompressible, frictionless flow along a streamline; the extended energy equation relaxes the frictionless and machine assumptions and is the form you actually use for design.

Common traps: forgetting that velocity head depends on V² (a 2× velocity is a 4× head), using gauge versus absolute pressure inconsistently between the two points, mixing up the sign of the pump and turbine terms, and trying to apply plain Bernoulli straight across a pump (you cannot — energy is added there, so you must include h_pump). At a large reservoir surface the velocity head and gauge pressure are both ~0, which simplifies many problems to elevation and loss terms only.

Reynolds Number & Flow Regime

The Reynolds number is the dimensionless ratio of inertial to viscous forces, and it governs whether flow is smooth or chaotic:

Re = ρVD/μ = VD/ν

where D is the pipe diameter (use the hydraulic diameter D_h = 4R = 4A/P for non-circular sections) and ν = μ/ρ. For flow in a circular pipe the regimes are: laminar below Re ≈ 2100, transitional from roughly 2100 to 4000, and turbulent above ≈ 4000. In laminar flow, fluid moves in orderly layers and the velocity profile is parabolic with V_max = 2·V_avg; in turbulent flow, eddies mix the fluid and the profile is flatter. The regime controls how you obtain the friction factor in Darcy-Weisbach: for laminar flow the friction factor is exactly f = 64/Re, but for turbulent flow you must read the Moody diagram.

Because most water-distribution and channel flows in practice are turbulent, expect to use the Moody chart on the exam. Always compute Re first to decide which path to take.

Momentum Equation & Flow Measurement

The linear momentum equation gives the resultant force a flowing fluid exerts (e.g., on a pipe bend, nozzle, or vane):

ΣF = ρQ(V₂ − V₁) (applied component-by-component, x and y).

This is how you size thrust blocks on bends and reaction forces on nozzles. Work each component separately and include the pressure forces (p·A) on the inlet and outlet faces plus the body's reaction; the net of these equals ρQ·ΔV. Sign convention matters — pick a positive direction and stay consistent.

Flow meters all derive from Bernoulli + continuity:

• Pitot tube measures stagnation vs static pressure: V = √(2g·Δh) (or √(2Δp/ρ)).
• Venturi / orifice meters use Q = C_d·A_throat·√[2g·Δh/(1 − (A₂/A₁)²)], where C_d is a discharge coefficient (orifices ≈ 0.6, venturis ≈ 0.98).

Worked Bernoulli Example

Water flows in a horizontal pipe that contracts from 0.10 m to 0.05 m diameter. The upstream velocity is 2 m/s and upstream gauge pressure is 150 kPa. Find the downstream pressure (neglect losses).

• A₁ = π(0.10)²/4 = 7.85×10⁻³ m²; A₂ = π(0.05)²/4 = 1.96×10⁻³ m².
• Continuity: V₂ = V₁(A₁/A₂) = 2(4) = 8 m/s (area drops 4×).
• Horizontal so z₁ = z₂. Bernoulli: p₂ = p₁ + ρ(V₁² − V₂²)/2 = 150,000 + 1000(2² − 8²)/2 = 150,000 − 30,000 = 120 kPa.