Two pipes of the same diameter carry the same fluid: Pipe A at Re = 1,000 (laminar) and Pipe B at Re = 100,000 (turbulent). Which requires more diameters of length before the flow is fully developed?
APipe B — turbulent flow carries more momentum, requiring a longer distance to redistribute it
BThey are approximately equal — the entrance length in diameters is insensitive to Reynolds number for both flow regimes
CPipe A — laminar entrance length scales as 0.05·Re, giving 50 diameters, while turbulent entrance length is only 10–60 diameters regardless of Re
DPipe B — at Re = 100,000, the laminar entrance formula gives 5,000 diameters
For laminar flow, L_e/D ≈ 0.05·Re, so Pipe A needs about 50 diameters. For turbulent flow, L_e/D ≈ 10–60 diameters regardless of Re — turbulent mixing is so efficient at redistributing momentum that the profile develops quickly. This seems counterintuitive: higher Re in laminar flow means *longer* entrance length, while turbulent flow (which occurs at high Re) needs far *fewer* diameters. Option D applies the laminar formula to turbulent flow, which is the classic misconception.
Question 2 Multiple Choice
In the hydrodynamic entrance region of a pipe, the wall shear stress is higher than in the fully developed region. Why?
AThe fluid velocity is higher in the entrance region because the pipe has not yet expanded to its full diameter
BThe boundary layer is thin near the inlet, creating a steeper velocity gradient at the wall, which produces higher shear stress
CThe flow is turbulent in the entrance region even when the fully developed flow is laminar
DPressure is highest at the inlet and drives extra shear at the wall through the Navier-Stokes equation
Wall shear stress τ_w = μ (∂u/∂r) at the wall. Near the pipe inlet, the boundary layer is thin — the high-velocity core extends almost to the wall, producing a very steep velocity gradient at the wall surface. As the boundary layer grows downstream and the parabolic profile develops, the velocity gradient at the wall decreases and shear stress drops, asymptoting to the fully developed value. This is why the friction factor is highest near the inlet and decreases with axial distance. The flow regime doesn't change (A is wrong), and pipe diameter doesn't change (C is irrelevant here).
Question 3 True / False
For laminar pipe flow, the hydrodynamic entrance length is proportional to the Reynolds number — a flow at Re = 2,000 requires roughly 100 pipe diameters to fully develop.
TTrue
FFalse
Answer: True
The laminar hydrodynamic entrance length formula is L_e/D ≈ 0.05·Re. At Re = 2,000 (near the laminar-turbulent transition), this gives L_e ≈ 100 diameters. This proportionality arises because higher Re means the viscous boundary layer grows more slowly relative to the convective transport of momentum — the fluid 'outruns' the diffusion of viscous effects inward. In a 2 cm pipe, 100 diameters is 2 meters of pipe before fully developed conditions can be assumed — a non-trivial engineering consideration.
Question 4 True / False
The hydrodynamic entrance length and thermal entrance length of a pipe are generally equal, since both depend on the same boundary layer growth process.
TTrue
FFalse
Answer: False
The hydrodynamic entrance length governs velocity profile development (driven by viscous diffusion), while the thermal entrance length governs temperature profile development (driven by thermal diffusion). These are equal only when the Prandtl number Pr = ν/α = 1, meaning viscous and thermal diffusivities are identical. For common engineering fluids, Pr departs significantly from 1: liquid metals have Pr ≪ 1 (thermal diffusivity dominates, so temperature develops faster), while oils have Pr ≫ 1 (viscous diffusivity dominates). Treating the two entrance lengths as interchangeable leads to errors in heat transfer calculations.
Question 5 Short Answer
Explain why increasing Re in laminar pipe flow increases the entrance length, while turbulent flow — which occurs at higher Re — actually has a much shorter entrance length in diameters.
Think about your answer, then reveal below.
Model answer: In laminar flow, velocity profile development is driven purely by viscous diffusion: the boundary layer grows inward as viscosity slows down near-wall fluid. At higher Re, convective momentum is stronger relative to viscous diffusion, so the boundary layer grows more slowly inward per diameter traveled — the fluid moves through more pipe lengths before diffusion can reach the centerline. Turbulent flow is governed by a completely different mechanism: turbulent mixing transports momentum radially far more efficiently than molecular viscosity. Even though turbulence occurs at high Re, the turbulent eddies redistribute momentum so rapidly that the profile develops in only 10–60 diameters regardless of Re.
The counterintuitive result stems from confusing two different transport mechanisms. Laminar entrance length scales with Re because viscous diffusion competes with convection. Turbulent entrance length is controlled by eddy mixing, which overwhelms the laminar scaling entirely. Students who apply the laminar formula (0.05·Re) to turbulent flows predict absurdly long entrance lengths — understanding why the formula doesn't apply requires recognizing the mechanism change.