Notes

Section 4: Fluid Flow

Reynolds numbers

If a system is to act like a continuum fluid, then almost by definition its behavior can involve only a limited number of macroscopic quantities, such as density and velocity. And from this it follows that patterns of flow should not depend separately on absolute speeds and sizes. Instead, the character of a flow should typically be determined by a single Reynolds number, Re = U L/ν, where U is the characteristic speed of the flow (measured say in cm/sec), L is a characteristic size (measured say in cm), and ν is the kinematic viscosity of the fluid. For water, ν = 0.01, for air ν = 0.15, and for glycerine ν = 10000, all in units of cm^2/sec. In flow past a cylinder it is conventional to take L to be the diameter of the cylinder. But the fact that the form of flow should depend only on Reynolds number means that in the pictures in the main text for example it is not necessary to specify absolute sizes or speeds: one need only know the product U L that appears in the Reynolds number. In practice, moving one's finger slowly through water gives a Reynolds number of about 100 (so that a regular array of dimples corresponding to eddies are visible behind one's finger), walking in air about 10,000, a boat in the millions, and a large airplane in the billions.

The Reynolds number roughly measures the ratio of inertial to viscous effects. When the Reynolds number is small the viscous damping dominates, and the flow is laminar and smooth. When the Reynolds number is large, inertia associated with fluid motions dominates, and the flow is turbulent and complicated.

In different systems, the characteristic length used typically in the definition of Reynolds number is different. In most cases, however, the transition from laminar to turbulent flow occurs at Reynolds numbers around a hundred.

In some situations, however, Reynolds number alone does not appear to be sufficient to determine when a flow will become turbulent. Indeed, modern experiments on streams of dye in water (or rising columns of smoke) typically show a transition to turbulence at a significantly lower Reynolds number than the original experiments on these systems done by Osborne Reynolds in the 1880s. Presumably the reason for this is that the transition point can be lowered by perturbations from the environment, and such perturbations are more common in the modern mechanized world. If perturbations are indeed important, it implies that a traditional fluid description is not adequate. I suspect, however, that even though perturbations may determine the precise point at which turbulence begins, intrinsic randomness generation will dominate once turbulence has been initiated.

From Stephen Wolfram: A New Kind of Science [citation]