Reynolds Number

The Reynolds number is a key concept for anyone interested in biofluid dynamics. Russ Hobbie and I discuss the Reynolds number in Section 1.18 (Turbulant Flow and the Reynolds Number) of the 4th edition of Intermediate Physics for Medicine and Biology.

The importance of turbulence (nonlarminar) flow is determined by a dimensionless number characteristic of the system called the Reynolds number N_R. It is defined by

N_R = L V ρ/η

where L is a length characteristic of the problem, V a velocity characteristic of the problem, ρ the density, and η the viscosity of the fluid. When N_R is greater than a few thousand, turbulence usually occurs…

When N_R is large, inertial effects are important. External forces accelerate the fluid. This happens when the mass is large and the viscosity is small. As the viscosity increases (for fixed L, V, and ρ) the Reynolds number decreases. When the Reynolds number is small, viscous effects are important. The fluid is not accelerated, and external forces that cause the flow are balanced by viscous forces… The low-Reynolds-number regime is so different from our everyday experience that the effects often seem counterintuitive.”

Steven Vogel, in his fascinating book Life in Moving Fluids, describes the importance of the Reynolds number more elegantly.

The peculiarly powerful Reynolds number [is] the center piece of biological (and even nonbiological) fluid mechanics. The utility of the Reynolds number extends far beyond mere problems of drag; it’s the nearest thing we have to a completely general guide to what’s likely to happen when solid and fluid move with respect to each other. For a biologist, dealing with systems that span an enormous size range, the Reynolds number is the central scaling parameter that makes order of a diverse set of physical phenomena. It plays a role comparable to that of the surface-to-volume ratio in physiology.

The Reynolds number is named after the British engineer Osborne Reynolds (1842–1912). He developed the Reynolds number as a simple way to understand the transition from laminar to turbulent flow of fluids in a pipe. Perhaps it is fitting to let Reynolds have the last word. Below he describes experiments in which he added a filament of dye to the fluid (as quoted by Vogel in Life in Moving Fluids).

When the velocities were sufficiently low, the streak of colour extended in a beautiful straight line across the tube. If the water in the tank had not quite settled to rest, as sufficiently low velocities, the streak would shift about the tube, but there was no appearance of sinuosity. As the velocity was increased by small stages, at some point in the tube, always at a considerable distance from the trumpet or intake, the colour band would all at once mix up with the surrounding water. Any increase in the velocity caused the point of break-down to approach the trumpet, but with no velocities that were tried did it reach this. On viewing the tube by the light of an electric spark, the mass of colour resolved itself into a mass of more or less distinct curls showing eddies.