Turbulent Pipe Flow – Roughness vs laminar sublayer thickness Water (ρ = 1000 kg/m^3, ν = 1.0×10^-6 m^2/s) flows in a commercial pipe with equivalent sand roughness k_s = 0.12 mm. If the average wall shear stress is τ_w = 600 N/m^2, compute k_s / δ′, where δ′ is the laminar sublayer thickness.

Introduction / Context:In turbulent flow near walls, a viscous (laminar) sublayer exists adjacent to the boundary. If the roughness height protrudes beyond this sublayer, the flow behaves as transitionally rough or fully rough. Comparing the equivalent sand roughness k_s with the sublayer thickness δ′ helps decide the roughness regime and informs friction factor estimation.

Given Data / Assumptions:

• ρ = 1000 kg/m^3, ν = 1.0×10^-6 m^2/s.
• k_s = 0.12 mm = 1.2×10^-4 m.
• Average wall shear stress τ_w = 600 N/m^2.
• Laminar sublayer thickness δ′ ≈ 11.6 * ν / u_* (hydraulic smooth-wall estimate), where u_* is friction velocity.

Concept / Approach:

Compute the friction velocity u_* = sqrt(τ_w / ρ). Then compute δ′ = 11.6 * ν / u_. Finally, form the ratio k_s / δ′ to compare roughness to the sublayer thickness.

Step-by-Step Solution:

Verification / Alternative check:

Since k_s/δ′ ≫ 1, the boundary is in the fully rough regime, consistent with high shear stress and commercial roughness magnitude.

Why Other Options Are Wrong:

0.25 and 0.50 imply hydraulically smooth behavior (k_s << δ′); 6.0 underestimates the ratio from the computed values.

Common Pitfalls:

Confusing kinematic viscosity with dynamic viscosity, or forgetting to convert k_s to metres; using τ in Pa but ρ in inconsistent units leads to erroneous u_*.

Final Answer:

8.0