Reynolds number and flow regime in pipes

Definition and meaning

The Reynolds number is the ratio of inertia force to viscous force:

$$Re=\frac{\rho V L}{\mu}=\frac{VL}{\nu}$$

For pipe flow $L=D$. A small $Re$ means viscous forces dominate and disturbances are damped (laminar flow); a large $Re$ means inertia dominates, disturbances grow and the flow becomes turbulent.

Typical limits for pipes: laminar for $Re<2300$, transitional for about 2300 to 4000, turbulent for $Re>4000$.

(a) Reynolds number

$$Re=\frac{998\times0.9\times0.04}{1.002\times10^{-3}}=3.59\times10^{4}$$

Since $Re\gg4000$, the flow is turbulent.

(b) Velocity for $Re=2300$

$$V=\frac{Re\,\mu}{\rho D}=\frac{2300\times1.002\times10^{-3}}{998\times0.04}=0.0577\ \text{m/s}$$

So only a very slow flow (5.8 cm/s) is laminar in this pipe.

(c) Importance in CFD

• It decides whether a laminar solver or a turbulence model is required.
• It sets the mesh needs: higher $Re$ gives thinner boundary layers, so finer wall meshes (small $y^+$) are needed.
• It is the similarity parameter: a CFD model at small scale with the same $Re$ represents the full-size case.
• Drag coefficient, friction factor and transition behaviour are all functions of $Re$.

Answer: $Re=3.59\times10^4$ (turbulent); $V_{Re=2300}=0.0577$ m/s.