Pipe flow regimes and velocity profiles

Numerical classification (Reynolds number)

Flow in pipes is classified by the Reynolds number, the ratio of inertia force to viscous force:

$$Re = \frac{\rho V D}{\mu} = \frac{VD}{\nu}$$

Example: water ($\nu = 10^{-6}$ m²/s) at 1 m/s in a 0.1 m pipe: $Re = 10^5$, so turbulent.

Velocity profiles

 Laminar (parabolic)       Turbulent (flatter)
 |=====================    |=====================
 |-->                      |------->
 |------>                  |--------->
 |--------->  u_max = 2V   |---------->  u_max ~ 1.2V
 |------>                  |--------->
 |-->                      |------->
 |=====================    |=====================
   wall   u = 0               thin laminar sublayer

• Laminar: velocity varies parabolically with radius (Hagen–Poiseuille):

$$u = u_{max}\left(1 - \frac{r^2}{R^2}\right), \quad u_{max} = 2\bar V$$

• Turbulent: mixing evens out the velocity, so the profile is flat in the core and changes steeply near the wall; approximately the power law $u = u_{max}(y/R)^{1/7}$, with $u_{max} \approx 1.2\bar V$ and a thin laminar sub-layer at the wall.