Turbulent flow, velocity profiles and friction factor

Why it matters

Almost all water, steam, gas and low-viscosity solvent lines in a plant run in turbulent flow. Their pressure drop, pump power, heat-transfer coefficients and mixing all depend on turbulence, and the friction factor correlations in this topic are the ones you will use to size every such line. Turbulence also explains why heat and mass transfer are so much faster in turbulent than in laminar flow.

Key ideas

Nature of turbulence. Turbulent flow is unsteady, three-dimensional and rotational, with eddies over a wide range of sizes. At any point the velocity fluctuates about a time-mean: u = ū + u′. Eddies carry momentum, heat and mass across the flow far faster than molecular diffusion. In pipes, flow becomes turbulent above about Re = 4000 (transitional between about 2100 and 4000).

Reynolds stresses and eddy viscosity. Time-averaging the Navier–Stokes equations produces extra apparent stresses, −ρ·u′v′ (overbar meaning time-average), called Reynolds stresses. Boussinesq modelled them with an eddy viscosity μ_t, which is a property of the flow, not the fluid. Prandtl's mixing-length model writes μ_t = ρ·l²·|dū/dy|, with l ≈ κy near the wall (κ ≈ 0.41 von Kármán constant).

Structure of turbulent pipe flow near the wall. Using the friction velocity u* = √(τ_w/ρ) and the wall coordinates u⁺ = ū/u*, y⁺ = y·u*/ν:

• Viscous sublayer (y⁺ < about 5): viscous stress dominates, u⁺ = y⁺.
• Buffer layer (about 5 < y⁺ < 30): both stresses matter.
• Turbulent core / log region (y⁺ > about 30): u⁺ = 2.5·ln y⁺ + 5.5 for smooth pipes (universal velocity profile). The thin sublayer is why wall roughness matters: when roughness elements protrude through it, they shed eddies and raise friction.

Mean velocity profile. The turbulent profile is much flatter than the laminar parabola, with a steep gradient at the wall. A convenient approximation is the 1/7th power law, ū/u_max = (y/R)^(1/7), giving V/u_max ≈ 0.82 (49/60). The ratio rises towards 1 as Re increases. Kinetic-energy and momentum correction factors are close to 1 (α ≈ 1.05, β ≈ 1.02).

Friction factor. Dimensional analysis gives f = φ(Re, ε/D). Regimes:

• Hydraulically smooth: roughness buried in the sublayer; f depends only on Re (Blasius, Prandtl–Kármán).
• Transitional: f depends on both Re and ε/D (Colebrook–White).
• Fully rough: f depends only on ε/D; head loss ∝ V². Typical absolute roughness ε (take from your data book): drawn tubing about 0.0015 mm, commercial steel about 0.045 mm, cast iron about 0.26 mm.

Darcy and Fanning. Two friction factors are in use: Darcy f_D (Moody chart, civil and mechanical texts) and Fanning f_F = f_D/4 (many chemical engineering texts and Perry). Always check which one a correlation or chart gives.

Formulas

• Re = ρ·V·D/μ — turbulent above about 4000.
• h_f = f_D·(L/D)·V²/(2g), Δp = f_D·(L/D)·ρV²/2 = 4·f_F·(L/D)·ρV²/2 — Darcy–Weisbach; h_f (m), Δp (Pa), L, D (m).
• τ_w = f_D·ρ·V²/8 = f_F·ρ·V²/2 — wall shear stress (Pa).
• u* = √(τ_w/ρ) — friction velocity (m/s); y⁺ = y·u*/ν, u⁺ = ū/u*.
• u⁺ = y⁺ (y⁺ < 5); u⁺ = 2.5·ln y⁺ + 5.5 (y⁺ > 30) — smooth-pipe universal profile.
• ū/u_max = (y/R)^(1/7) — power-law profile; y distance from wall (m), R radius (m).
• f_D = 0.316·Re^(−0.25) (Fanning f_F = 0.079·Re^(−0.25)) — Blasius, smooth pipes, 4000 < Re < 10⁵.
• 1/√f_D = −2.0·log₁₀[(ε/D)/3.7 + 2.51/(Re·√f_D)] — Colebrook–White, implicit; ε roughness (m).
• f_D = 0.25/[log₁₀((ε/D)/3.7 + 5.74/Re^0.9)]² — Swamee–Jain explicit approximation (within about 1 %), 5000 < Re < 10⁸.

Worked examples

Example 1 (standard). Water (ρ = 998 kg/m³, μ = 1.002 × 10⁻³ Pa·s) flows at 1.5 m/s through a smooth 50 mm tube, 50 m long. Find Re, f_D, Δp, τ_w, u* and the viscous sublayer thickness.

1. Re = 998 × 1.5 × 0.05 / 1.002 × 10⁻³ = 74 700 → turbulent, within the Blasius range.
2. f_D = 0.316 × 74 700^(−0.25) = 0.316/16.53 = 0.0191 (Colebrook for ε = 0 gives the same, 0.0191).
3. Δp = f_D(L/D)(ρV²/2) = 0.0191 × 1000 × 1122.75 ≈ 21 450 Pa ≈ 21.5 kPa.
4. τ_w = f_D·ρV²/8 = 0.0191 × 998 × 2.25 / 8 = 5.37 Pa.
5. u* = √(5.37/998) = 0.0733 m/s.
6. Sublayer edge y⁺ = 5: y = 5ν/u* = 5 × 1.004 × 10⁻⁶ / 0.0733 = 6.8 × 10⁻⁵ m. Re ≈ 74 700, f_D ≈ 0.0191, Δp ≈ 21.5 kPa, τ_w ≈ 5.4 Pa, sublayer ≈ 0.07 mm. Roughness smaller than this keeps the tube hydraulically smooth.

Example 2 (GATE level). Water (ν = 1 × 10⁻⁶ m²/s) flows at 3 m/s in a 150 mm commercial steel pipe (ε = 0.045 mm). Find f_D by Colebrook and Swamee–Jain, and the head loss per 100 m.

1. Re = VD/ν = 3 × 0.15 / 10⁻⁶ = 4.5 × 10⁵; ε/D = 0.045/150 = 3.0 × 10⁻⁴.
2. Swamee–Jain: 5.74/Re^0.9 = 4.69 × 10⁻⁵; (ε/D)/3.7 = 8.11 × 10⁻⁵; log₁₀(1.280 × 10⁻⁴) = −3.893; f_D = 0.25/3.893² = 0.0165.
3. Colebrook, iterating from f = 0.0165: converges to f_D = 0.0164.
4. h_f = f(L/D)V²/2g = 0.0164 × (100/0.15) × 9/19.62 = 5.02 m per 100 m. f_D ≈ 0.0164, h_f ≈ 5.0 m per 100 m of pipe.

Common mistakes

• Mixing Darcy and Fanning factors — an error of exactly 4.
• Using Blasius beyond Re = 10⁵ or for rough pipes.
• Taking ε (absolute roughness, mm) instead of ε/D.
• Assuming turbulent friction is independent of Re at moderate Re; only the fully rough regime is.
• Applying the 1/7 power law at the wall or centre to get τ_w (its slope is infinite at the wall and non-zero at the centre).

For GATE CH

Expect friction-factor and pressure-drop numericals (Blasius, Colebrook or a given chart value), the effect of changing velocity or diameter on Δp in turbulent flow (Δp ∝ V^1.75 in the Blasius range and ∝ V² when fully rough), universal velocity profile and friction-velocity questions, and conceptual MCQs on Reynolds stresses and eddy viscosity. Be fluent with both Darcy and Fanning forms.

Quick check

1. What is the Fanning friction factor if f_D = 0.024?
2. In the Blasius range, how does Δp scale with velocity at fixed D?
3. Is eddy viscosity a fluid property?
4. Approximately what is V/u_max for the 1/7th power-law profile?

Answers: 1. 0.006. 2. Δp ∝ V^1.75. 3. No, it depends on the flow and position. 4. About 0.82.