Reynolds Number in Internal vs. External Flows: Transition from Laminar to Turbulent Flow

The Reynolds number (Re) is a dimensionless quantity used to predict flow patterns in different fluid flow situations. It is defined as:

Re (frac{rho v L}{mu}) (frac{v L}{ u})

where:

(rho) - Fluid density (v) - Flow velocity (L) - Characteristic length, like diameter for pipes (mu) - Dynamic viscosity ( u) - Kinematic viscosity, ( u frac{mu}{rho})

Transition from Laminar to Turbulent Flow

Internal Flows (e.g. Pipes)

In internal flows such as those in pipes, the transition from laminar to turbulent flow typically occurs at a Reynolds number around Re ≈ 2000. The flow is influenced by the boundary conditions imposed by the walls of the conduit. In laminar flow, fluid particles move in smooth layers and disturbances can easily dampen out due to the viscous forces acting on them. The flow is stable, and it takes a relatively low level of inertia represented by the Re to disrupt this stability and induce turbulence.

External Flows (e.g. Around Objects)

In external flows such as air flowing over a flat plate, the transition to turbulence occurs at much higher Reynolds numbers typically around Re ≈ 5 × 10^5 to 1 × 10^6. In external flows, the fluid interacts with the surrounding environment, creating more complex flow patterns and instabilities. The inertia of the flow must overcome not just viscous effects but also the influence of external forces and pressures. The flow can remain laminar over a longer distance due to the presence of pressure gradients, leading to higher Reynolds numbers before turbulence sets in.

Summary

In summary, the transition from laminar to turbulent flow occurs at much lower Reynolds numbers for internal flows compared to external flows because:

Internal flows are constrained by walls, which help maintain laminar flow with lower inertia. External flows experience more complex interactions with the environment, requiring higher inertia to induce turbulence.

This difference is fundamentally due to the influence of boundary conditions, flow geometry, and the nature of disturbances in the respective flow regimes.