---
title: "Mixing length, Hagen-Poiseuille, Nikuradse, Colebrook"
subject: "Hydraulics"
chapter: 1
key: "semester_4/hydraulics:pipe-flow-expressions-describe"
source: https://www.bctengineer.com/q/semester_4/hydraulics/pipe-flow-expressions-describe
site: BCT Engineering
---

# Mixing length, Hagen-Poiseuille, Nikuradse, Colebrook

Hydraulics · Chapter 1: Pipe flow

- Asked in: 2069 Bhadra, 8 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/pipe-flow-expressions-describe
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-1

## Question

Describe with appropriate expressions (a) Prandtl's mixing length theory (b) Hagen-Poiseuille equation (c) Nikuradse's experiments and (d) Colebrook-White equation.

## Short answer

Prandtl's mixing length theory gives τ_t = ρ(κy)²(du/dy)² and log laws. Hagen-Poiseuille: Q = πD⁴Δp/(128μL), f = 64/Re. Nikuradse's sand-roughened pipe experiments identified smooth, transition, and fully rough regimes; the Colebrook-White equation 1/√f = -2 log₁₀(k/(3.7D) + 2.51/(Re√f)) unifies smooth and rough friction.

## Answer

#### (a) Prandtl's mixing length theory
Turbulent eddies move lumps of fluid across the layers; a lump keeps its identity over the **mixing length** $l$ before mixing. The velocity fluctuations are $u' \approx v' \approx l\,\dfrac{du}{dy}$, so the turbulent shear stress is
$$
\tau_t = -\rho\overline{u'v'} = \rho\,l^2\left(\frac{du}{dy}\right)^2, \qquad l = \kappa y,\ \kappa \approx 0.4
$$
The total stress is $\tau = \mu\dfrac{du}{dy} + \rho l^2\left(\dfrac{du}{dy}\right)^2$. For constant $\tau = \tau_0$ near the wall, it leads to the logarithmic velocity law $\dfrac{u}{u_*} = 5.75\log_{10}\dfrac{u_*y}{\nu} + 5.5$ (smooth) and $5.75\log_{10}\dfrac{y}{k} + 8.5$ (rough), where $u_* = \sqrt{\tau_0/\rho}$.

#### (b) Hagen-Poiseuille equation
For steady laminar flow in a circular pipe of diameter $D$ and length $L$ the discharge and head loss are
$$
Q = \frac{\pi D^4\,\Delta p}{128\,\mu L}, \qquad h_f = \frac{32\mu L V}{\gamma D^2}
$$
The velocity is parabolic, $u = \dfrac{\Delta p}{4\mu L}(R^2 - r^2)$, with $u_{max} = 2V$. Comparing with Darcy-Weisbach gives $f = 64/Re$. It is valid for $Re < 2000$.

#### (c) Nikuradse's experiments
Nikuradse (1933) measured head loss in pipes artificially roughened with uniform sand grains of size $k$, for relative roughness $k/D$ from 1/30 to 1/1014 and $Re$ from $10^3$ to $10^6$. Plotting $\log f$ against $\log Re$ he found:
- Laminar flow: all pipes lie on $f = 64/Re$, independent of roughness.
- Smooth turbulent: $f$ follows Blasius/Prandtl smooth curve, depends on $Re$ only.
- Transition: $f$ depends on both $Re$ and $k/D$.
- Fully rough: $f$ becomes constant for given $k/D$, depends on $k/D$ only; $\dfrac{1}{\sqrt f} = 2\log_{10}\dfrac{D}{k} + 1.14$.

#### (d) Colebrook-White equation
For commercial pipes, Colebrook and White combined the smooth and rough laws into one transition formula:
$$
\frac{1}{\sqrt f} = -2\log_{10}\left(\frac{k}{3.7D} + \frac{2.51}{Re\sqrt f}\right)
$$
- When $k \to 0$ (smooth): $\dfrac{1}{\sqrt f} = 2\log_{10}(Re\sqrt f) - 0.8$.
- When $Re \to \infty$ (rough): $\dfrac{1}{\sqrt f} = 2\log_{10}\dfrac{D}{k} + 1.14$.
The equation is implicit in $f$ and is solved by iteration; it is the basis of the Moody chart.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the turbulent shear stress in Prandtl's mixing length theory?
   A. tau_t = mu du/dy
   B. tau_t = rho l^2 (du/dy)^2 with l = kappa y
   C. tau_t = rho l du/dy
   D. tau_t = rho kappa y^2 du/dy
   Answer: B. With fluctuations u' = v' = l du/dy, tau_t = rho l^2 (du/dy)^2, and l grows with distance from the wall as l = 0.4 y. (Key point: Mixing length shear stress)

2. Which statement about Hagen-Poiseuille flow is correct?
   A. Velocity is logarithmic and f = 0.316/Re^0.25
   B. Velocity is uniform and f depends on k/D
   C. Velocity is parabolic, u_max = 2V, f = 64/Re for Re < 2000
   D. Velocity is parabolic, u_max = V, f = 16/Re
   Answer: C. Laminar pipe flow has a parabolic profile with centreline velocity twice the mean, and compares with Darcy-Weisbach to give f = 64/Re. (Key point: Hagen-Poiseuille)

3. In Nikuradse's fully rough region, f depends on:
   A. Re only
   B. both Re and k/D
   C. neither Re nor k/D
   D. k/D only
   Answer: D. At high Re the curves become horizontal, so f is constant for a given k/D, with 1/sqrt f = 2 log(D/k) + 1.14. (Key point: Nikuradse regions)

4. What does the Colebrook-White equation reduce to when Re tends to infinity?
   A. 1/sqrt f = 2 log(D/k) + 1.14
   B. 1/sqrt f = 2 log(Re sqrt f) - 0.8
   C. f = 64/Re
   D. f = 0.316/Re^0.25
   Answer: A. The term 2.51/(Re sqrt f) vanishes, leaving the fully rough law that depends only on k/D. (Key point: Colebrook-White limits)

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