Reynolds Number

For flow in a circular pipe, the accepted transition range is 2,300 < Re < 4,000, with fully turbulent flow above 4,000. For flow over a flat plate, transition from laminar to turbulent boundary layer occurs at Re_x ≈ 5×10⁵ (based on distance from the leading edge). For flow past a smooth circular cylinder, the boundary layer separates symmetrically up to Re ≈ 2×10⁵, then asymmetrically in the "drag crisis" region. For flow past a sphere, the drag crisis occurs near Re ≈ 3×10⁵. For flow in an open channel, transition is at Re ≈ 500 (based on hydraulic radius). For flow through packed beds of spheres, transition is at Re ≈ 10 based on particle diameter.

Dynamic viscosity μ (also called absolute viscosity) is the proportionality constant in Newton's law of viscosity, τ = μ · du/dy — it has units of Pa·s and represents the fluid's intrinsic resistance to shear. Kinematic viscosity ν = μ / ρ has units of m²/s and represents how quickly momentum diffuses through the fluid. Both are important in fluid mechanics: μ appears in the force balance (stresses), ν appears in the momentum equation (acceleration). The Reynolds number can be written either as ρvD/μ or as vD/ν — the two forms are identical. Water at 20 °C has μ = 1.0×10⁻³ Pa·s and ν = 1.0×10⁻⁶ m²/s; air at the same conditions has μ = 1.8×10⁻⁵ Pa·s but ν = 1.5×10⁻⁵ m²/s, an order of magnitude higher because air is much less dense.

For non-circular ducts — rectangular air conditioning ducts, annular gaps in heat exchangers, triangular cooling channels in turbine blades — there is no single diameter, so engineers use the hydraulic diameter D_h = 4A/P, where A is the flow cross-sectional area and P is the wetted perimeter. For a circular pipe of diameter D, D_h reduces to D (since A = πD²/4 and P = πD). For a square duct of side a, D_h = a. For a rectangular duct of width w and height h, D_h = 2wh/(w+h). Using D_h in the Reynolds number and friction factor correlations gives good engineering accuracy for turbulent flow, though the approach is less reliable for laminar flow where corner effects matter.

Temperature affects both density and viscosity, usually in opposite directions. For liquids, warming decreases both ρ and μ, but μ typically changes much more — water's viscosity drops by a factor of 6 between 0 °C and 100 °C, while its density falls by only 4%. So heating a liquid dramatically increases Re. For gases at constant pressure, warming decreases ρ (via the ideal-gas law) but slightly increases μ. The net effect is that Re drops roughly as T⁻¹·⁸ for gases at constant pressure. Both effects matter in combustion, gas turbines, and atmospheric flows, where temperature varies widely.

Use the Darcy–Weisbach equation: Δp = f · (L/D) · (ρ·v²/2), where f is the Darcy friction factor and L is the pipe length. For laminar pipe flow (Re < 2300), f = 64/Re exactly. For turbulent flow in smooth pipes, the Blasius correlation f ≈ 0.316·Re⁻⁰·²⁵ is accurate up to Re ≈ 10⁵, or the more accurate Colebrook equation can be solved iteratively. For rough pipes, f depends on both Re and the relative roughness ε/D. The friction factor is what connects the Reynolds number you've computed here to the pumping power needed for a real system.

When fluid enters a pipe, the velocity profile develops from uniform at the inlet to fully-developed (parabolic for laminar, or the 1/7 power law for turbulent) over some distance called the entrance length. For laminar flow, L_e ≈ 0.06 · Re · D. For turbulent flow, L_e ≈ 4.4 · Re^(1/6) · D. In the entrance region the wall shear stress and heat-transfer coefficients are higher than in fully-developed flow, so entrance effects matter for short pipes and heat exchangers. The correlations used to predict friction factor and Nusselt number assume fully-developed flow, so if your pipe is shorter than the entrance length you must apply a correction.