Questions: Friction Factor Determination: Laminar, Transitional, and Turbulent

Question 1

An engineer increases flow velocity through a smooth pipe from Re = 500 to Re = 1,500, keeping both values in the laminar regime. What happens to the Darcy friction factor?

• A. It increases — higher velocity creates stronger shear forces at the pipe wall
• B. It remains essentially unchanged — friction factor is nearly constant across the laminar regime
• C. It decreases — in laminar flow f = 64/Re, so higher Reynolds number gives lower friction factor
• D. It depends on the pipe roughness — even in laminar flow, wall roughness controls friction

Answer: (C)

This is one of the most counterintuitive results in pipe flow. In laminar flow, the analytical derivation of f = 64/Re shows that friction factor inversely tracks Reynolds number. At Re = 500, f = 0.128; at Re = 1,500, f = 0.043. The proportional friction loss (head loss per unit velocity) actually decreases as you speed up the flow — because inertial forces grow faster than viscous shear forces in this regime. Also critical: laminar friction factor has no dependence on pipe roughness whatsoever. The roughness elements are submerged within the orderly streamlines and have no effect on the parabolic velocity profile.

Question 2

A smooth stainless steel pipe (very low roughness ε/D ≈ 0.00001) and a rough cast iron pipe (ε/D ≈ 0.002) carry turbulent flow at Re = 100,000. Which pipe has the lower friction factor?

• A. Both have the same friction factor — at Re = 100,000, turbulent friction is controlled only by Reynolds number, not roughness
• B. The smooth pipe has a lower friction factor — less wall roughness means turbulent eddies generate less additional wall shear
• C. The rough pipe has a lower friction factor — roughness disrupts the viscous sublayer and reduces the effective boundary layer thickness
• D. The smooth pipe has f = 64/Re since roughness is negligible and laminar theory applies

Answer: (B)

In turbulent flow, relative roughness ε/D is a primary determinant of friction factor alongside Reynolds number. Higher roughness means more protrusions into the turbulent boundary layer, generating more form drag and increasing wall shear stress. On the Moody chart, the smooth pipe's curve (low ε/D) sits well below the rough cast iron curve at Re = 100,000. The f = 64/Re formula applies only to laminar flow (Re < 2,300); at Re = 100,000 the flow is fully turbulent regardless of roughness, and you must use the Colebrook equation or Moody chart.

Question 3

In fully turbulent flow at very high Reynolds numbers, the friction factor eventually becomes independent of Reynolds number and depends only on the pipe's relative roughness ε/D.

• T. True
• F. False

Answer: True

This is the 'fully rough' or 'complete turbulence' limit visible on the Moody chart as the horizontal asymptote of each roughness curve at high Re. Physically, once Re is large enough, the viscous sublayer near the pipe wall becomes so thin that roughness elements protrude through it entirely. The friction is then dominated by pressure drag on those protruding elements — a purely geometric effect that depends on ε/D but not on Re. The Colebrook equation reduces to f = [−2 log(ε/3.7D)]⁻² in this limit, with no Re dependence.

Question 4

The Colebrook equation for turbulent friction factor can be solved directly by algebra to express f as an explicit function of Re and ε/D.

• T. True
• F. False

Answer: False

The Colebrook equation — 1/√f = −2 log(ε/3.7D + 2.51/Re√f) — is implicit in f because f appears on both sides (inside the square root on the right-hand side). There is no closed-form algebraic solution. In practice, engineers use the Moody diagram for graphical lookup, iterative numerical solution (converging in 2–3 iterations), or the explicit Swamee-Jain approximation f ≈ 0.25/[log(ε/3.7D + 5.74/Re⁰·⁹)]², which is accurate to within about 3% for most engineering purposes.

Question 5

Explain why pipe wall roughness has no effect on friction factor in laminar flow, but becomes a major factor in turbulent flow.

Model answer: In laminar flow, the velocity profile is a smooth parabola with organized concentric streamlines and no cross-stream momentum transport. Roughness elements on the pipe wall are submerged within this ordered flow and do not disrupt the streamlines — the flow simply curves around them. The analytical solution to the Navier-Stokes equations for this geometry yields f = 64/Re, which contains no roughness parameter. In turbulent flow, random three-dimensional eddies transport momentum across the entire pipe cross-section. Wall roughness elements protrude into the turbulent boundary layer and the viscous sublayer, generating pressure drag ('form drag') on each protrusion and disrupting the thin viscous region near the wall. The larger the ε/D ratio, the more this disruption increases wall shear stress and friction factor.

The physical distinction is between viscous-dominated flow (laminar, where roughness is irrelevant) and inertia-dominated flow (turbulent, where the boundary layer structure is thin and roughness elements have room to exert influence). This is why smooth pipes are important in turbulent-flow applications but completely unnecessary when Re < 2,300.