Turbulent Flow MCQ with answer and solution

Concept

Equivalent Roughness Height in Turbulent Pipe Flow

Formula

Yes. For smooth pipes: u / u\* = 5.75 log₁₀(y · u\* / ν) + 5.5 For rough pipes: u / u\* = 5.75 log₁₀(y / kₛ) + 8.5 Where: - u = local velocity at distance y from the wall - u\* = friction velocity (√(τ₀ / ρ)) - y = distance from the pipe wall - ν = kinematic viscosity - kₛ = equivalent roughness height

Step-by-step solution

- Compare the velocity profile equation for smooth pipes: u / u\* = 5.75 log₁₀(y · u\* / ν) + 5.5 with that for rough pipes: u / u\* = 5.75 log₁₀(y / kₛ) + 8.5. - Observe that the kinematic viscosity (ν) term inside the logarithm for smooth pipes is replaced by the equivalent roughness height (k⛛) for rough pipes. - Conclude that equivalent roughness height (k⛛) is used in place of kinematic viscosity (ν).

Explanation

In smooth pipes, the near-wall velocity distribution depends on viscosity, using a viscous scale governed by kinematic viscosity (ν). In rough pipes, the physical roughness elements break up the viscous sublayer, so the equivalent roughness height (ks) replaces kinematic viscosity to account for the wall resistance.

Shortcut

For smooth pipes, the near-wall velocity depends on fluid viscosity (ν). For rough pipes, the physical roughness height (ks) breaks the viscous layer and takes its place. Shortcut rule: Smooth = Viscosity (ν), Rough = Roughness (ks).

Common mistake

Many students mistakenly select Pipe diameter D or Roughness Reynolds number Re_k because they associate relative roughness (ks/D) directly with pipe dimensions or flow parameters, rather than recognizing that ks replaces the viscous length scale governed by kinematic viscosity (ν) in the velocity distribution equation.