Mixing length, Hagen-Poiseuille, Nikuradse, Colebrook

(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.