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When a viscous fluid flows steadily through a straight pipe far from the inlet, the velocity varies across the cross-section in a pattern determined by the Reynolds number. At low Reynolds numbers (Re < ∼2300) the flow is laminar: fluid moves in orderly concentric layers described by the Poiseuille equation, giving a parabolic velocity profile with maximum speed at the centreline and zero at the wall (no-slip condition).

Above Re ≈ 2300 the flow enters a transitional regime and above ∼4000 becomes fully turbulent. Turbulent eddies mix momentum vigorously across the cross-section, producing a much flatter profile well approximated by the 1/7 power law: u(r)/umax ≈ (1−r/R)1/7. Use the Compare view to see both profiles side-by-side for the same mean velocity.

The difference matters enormously in engineering: laminar flow gives exactly twice the pressure drop for the same mean velocity (Hagen-Poiseuille), turbulent flow transfers heat and mass far more efficiently, and the transition determines mixing quality in chemical reactors and heat exchangers.

Часті запитання

Poiseuille flow is steady, fully-developed laminar flow through a pipe driven by a pressure difference. The velocity profile is parabolic: u(r) = (ΔP/4ηL)(R²−r²), where r is radial distance from the axis, R is pipe radius, ΔP is pressure drop, η is dynamic viscosity, and L is pipe length. The peak speed at the centreline is exactly twice the mean flow velocity. It was first described by Jean Léonard Marie Poiseuille in 1840.
Re = ρvD/μ is the ratio of inertial to viscous forces, where ρ is fluid density, v is mean velocity, D is pipe diameter and μ is dynamic viscosity. In a pipe, flow is laminar for Re < ∼2300, transitional between ∼2300 and ∼4000, and fully turbulent above ∼4000. The exact transition depends on inlet disturbances and wall roughness.
In fully turbulent pipe flow the time-averaged velocity profile follows u(r)/umax ≈ (1−r/R)1/7. Turbulent eddies redistribute momentum radially, giving a flat core and steep wall gradient. The exponent n varies from about 1/6 to 1/10 with Reynolds number; 1/7 is the standard engineering approximation. Adjust the n slider to explore different exponents.
For laminar Poiseuille flow v̄ = umax/2 exactly. For turbulent flow with the 1/n power law, v̄/umax = 2n²/((n+1)(2n+1)). For n=7 this gives v̄ ≈ 0.817 umax. The flatter turbulent profile carries proportionally more flux near the walls.
Q = πR⁴ΔP/(8ηL). Flow rate scales as the fourth power of radius: doubling the pipe radius increases flow 16-fold for the same pressure gradient. This applies only for laminar flow (Re < ∼2300). Viscosity enters linearly: a 10-fold more viscous fluid flows 10 times slower at the same pressure.
Turbulent eddies transfer momentum radially far more effectively than molecular viscosity alone, homogenising velocity across most of the cross-section. Only in the thin viscous sublayer near the wall does speed drop steeply to zero. The result is the characteristic blunt turbulent profile compared with the smooth laminar parabola.
Immediately adjacent to the wall (y⁺ < ∼5 in wall units) turbulent fluctuations are suppressed by viscosity and the flow behaves almost as laminar. Outside is the buffer layer, then the log-law region. The 1/7 power law implicitly blends these regions but does not resolve sublayer detail; direct numerical simulation is required for that.
In smooth pipes the Blasius friction factor f = 0.316 Re−1/4 applies for Re < 100 000. Rougher pipes increase friction per the Colebrook–White equation (Moody chart). Above a critical Re, friction depends only on relative roughness ε/D and becomes Re-independent — the fully rough turbulent regime.
ΔP = f (L/D) (ρv²/2), where f is the Darcy friction factor, L is pipe length, D is diameter, ρ is density and v is mean velocity. For laminar flow f = 64/Re; for turbulent flow f comes from the Moody diagram. This is the standard equation for pipeline pressure-drop calculations.
Water distribution networks, oil and gas pipelines, HVAC ductwork, blood flow in arteries (Womersley flow), microfluidic chips, chemical reactors and hydraulic actuators all depend on pipe flow analysis. The regime (laminar or turbulent) controls heat transfer rates, mixing efficiency, pressure losses and the onset of flow-induced vibration or erosion.