Author: Krishnendu G S

Reynolds Number (Re)

The Reynolds number is a dimensionless quantity that represents the ratio of inertial forces to viscous forces in a fluid flow. It is a key parameter in fluid mechanics used to predict flow behavior, determining whether it will be laminar, transitional, or turbulent.

Internal Flow:

Internal flow refers to fluid motion that is confined within solid boundaries, such as flow through pipes, ducts, or channels. The defining characteristic of internal flow is that the fluid is fully enclosed, leading to the development of a boundary layer along the walls. As the flow progresses, these boundary layers grow and eventually merge, resulting in a fully developed flow profile.

$\text{Reynolds Number for internal flow,} \quad Re = \frac{\rho \cdot U \cdot D_h}{\mu}$

where $\rho$ is the fluid density (kg/m³), $U$ is the free-stream velocity (m/s), $D_h$ is the hydraulic diameter, and $\mu$ is the dynamic viscosity (Pa.s).

Hydraulic Diameter ( $D_h$ ) Calculation

Circular Duct,

$D_h=D$

where D is the diameter of the circular duct.

Rectangular Duct,

$D_h=\frac{2WH}{W+H}$

where W and H are the Width and Height of the rectangular duct

Square Duct,

$D_h=L$

where L is the side length of the duct

External Flow:

External flow describes fluid motion around a solid object or over surfaces, such as airflow over an airplane wing or water flowing past a submarine. Unlike internal flow, the fluid in external flow is unconfined, allowing the boundary layer to develop along the object’s surface.

$\text{Reynolds Number for external flow,} \quad Re = \frac{\rho \cdot U \cdot L}{\mu}$

where L is the characteristic length.

Flow Regimes Based on Reynolds Number:

Flow Type	Reynolds Number Range
Flow over a flat plate	Laminar: Re<5 $\times 10^5$ , Turbulent: Re>5 $\times 10^5$
For flow in a pipe	Laminar: Re<2300, Turbulent: Re>2900

When inertial forces increase relative to viscous forces, due to higher fluid velocity, larger characteristic length (such as pipe diameter or object size), increased fluid density, or decreased viscosity, the Reynolds number becomes high. This leads to inertial forces dominating, resulting in turbulent flow with chaotic patterns and vortices, as seen in high-speed airflow over an airplane wing. In contrast, when viscous forces are more significant, the Reynolds number remains low, leading to smooth and orderly laminar flow, such as the slow movement of oil through a narrow tube.

References:

[1] White, F. M. (2006). Viscous Fluid Flow (3rd ed.). McGraw-Hill Education.

[2] Fox, R. W., McDonald, A. T., & Pritchard, P. J. (2015). Introduction to Fluid Mechanics (8th ed.). Wiley.

[3] Kundu, P. K., Cohen, I. M., & Dowling, D. R. (2015). Fluid Mechanics (6th ed.). Academic Press.

[4] Cengel, Y., & Cimbala, J. (2013). Ebook: Fluid mechanics fundamentals and applications (si units). McGraw Hill

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Reynolds Number (Re)

Internal Flow:

Hydraulic Diameter () Calculation

External Flow:

Flow Regimes Based on Reynolds Number:

References:

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Hydraulic Diameter ( $D_h$ ) Calculation