Laminar and Turbulent Flow

We can distinguish the difference between these two classifications of flow by using a simple apparatus as shown by the figure (Reynold’s dye experiment) – injecting dye into a pipe in which water flowed due to gravity.

Laminar flow: Low $V$. Characterized by smooth, orderly fluid motion. Dye streaks remain well-defined and straight, showing layers of fluid sliding past each other with minimal mixing.

Transitional flow: Medium $V$. Flow starts to become unstable; some oscillations and disturbances in dye streak.

Turbulent flow: High $V$. Fluid particles move chaotically with mixing and swirling. They dye streak disperses rapidly,.

The flow scenario depends on velocity $V$, pipe diameter $D$, fluid viscosity $\mu$, and fluid density $\rho$.

The transition between these flow types is governed by the dimensionless number called the Reynolds Number, defined as

$$\text{Re}=\frac{\rho VD}{\mu }$$

Essentially, the Reynolds number serves as a ratio that compares inertial forces to viscous forces:

$$\text{Re}=\frac{\rho VD}{\mu }\cdot \frac{du/dy}{du/dy}=\frac{\rho Vdu}{\tau }=\frac{\rho Vdt\frac{du}{dt}}{\tau }=\frac{\rho La}{\tau }=\frac{\rho La}{\tau }\cdot \frac{L^2}{L^2}=\frac{F_{\text{inertial}}}{F_{\text{viscous}}}$$

We have:

• Laminar Flow: $Re<2100$.
• Transitional Flow: $2100<Re<4000$.
• Turbulent Flow: $Re>4000$.