A pipe carrying water operates at Re = 5×10⁶ and relative roughness ε/D = 0.05 (fully rough regime). An engineer doubles the flow velocity, which doubles the Reynolds number to 10⁷. What is the best prediction for the Darcy friction factor?
AIt decreases significantly, because higher Re means a thinner viscous sublayer and less friction
BIt stays approximately the same, because in the fully rough regime f depends only on ε/D, not Re
CIt doubles, because friction scales with velocity in turbulent flow
DIt falls to zero, because viscous effects become negligible at very high Re
In the fully rough regime, roughness elements protrude through the viscous sublayer so thoroughly that viscous effects no longer influence friction. The Moody diagram shows horizontal asymptotes on the right — f depends only on relative roughness ε/D, not Re. Doubling Re changes nothing. The common misconception is that more turbulence always means more friction variation, but the fully rough regime is defined precisely by this independence from Re.
Question 2 Multiple Choice
A reference gives the Fanning friction factor f_F = 0.005 for a pipe flow. What Darcy friction factor should be used in the Darcy-Weisbach equation h_f = f(L/D)(V²/2g)?
A0.005 — the two friction factors are identical
B0.00125 — the Darcy factor is one-quarter of the Fanning factor
C0.010 — the Darcy factor is twice the Fanning factor
D0.020 — the Darcy factor is four times the Fanning factor
f_Darcy = 4 × f_Fanning. This is one of the most consequential unit-convention errors in fluid mechanics — confusing the two introduces a factor-of-4 error in calculated head loss. Always check which convention a reference uses. The Darcy-Weisbach equation as written above uses the Darcy (Moody) friction factor.
Question 3 True / False
In laminar pipe flow (Re < 2000), a smoother pipe wall will produce a lower friction factor than a rougher one.
TTrue
FFalse
Answer: False
In laminar flow, the viscous sublayer is thick enough to completely cover all surface roughness. The flowing fluid 'sees' a smooth wall regardless of actual surface condition. The friction factor f = 64/Re exactly in laminar flow — independent of roughness. Roughness only matters once the flow is turbulent and the viscous sublayer thins enough for roughness elements to protrude through it.
Question 4 True / False
In the fully turbulent (fully rough) regime, the Darcy friction factor depends only on relative roughness ε/D and becomes independent of Reynolds number.
TTrue
FFalse
Answer: True
Correct. This is the defining characteristic of the fully rough regime: roughness elements fully protrude beyond the viscous sublayer, generating turbulent eddies and pressure drag that dominate friction. Viscous effects — which depend on Re — become negligible. On the Moody diagram, each ε/D curve converges to a horizontal asymptote at high Re, reading a constant f determined solely by ε/D.
Question 5 Short Answer
Why must the Colebrook equation be solved iteratively rather than directly for the friction factor, and what does this imply about engineering practice?
Think about your answer, then reveal below.
Model answer: The Colebrook equation is implicit in f — f appears inside the square root on the right-hand side as well as on the left. It cannot be algebraically rearranged to give f explicitly. In practice, you start with an initial guess (e.g., f = 0.02), substitute into the right side, solve for a new f, and repeat until convergence (typically 2–3 iterations). The explicit Swamee-Jain approximation avoids iteration at the cost of a small error (~3%). The Moody diagram is a graphical solution to the same equation.
The iterative nature reflects that turbulent friction involves a genuine self-consistent relationship: the friction factor depends on flow conditions that in turn depend on the friction factor (through velocity and head loss). Engineers in practice either use the Moody diagram graphically, the Swamee-Jain approximation explicitly, or run 2–3 Colebrook iterations — all converge quickly because f varies weakly with Re in the turbulent regime.