Instabilities and Turbulent Flows

Abstract

When a viscous fluid flows through long straight tubes at reasonably high speeds, the Hagen–Poiseuille law (4.45), according to which the pressure drop is linearly proportional to the volume of fluid flowing through the pipe, is replaced by another law, in which the pressure drop is significantly greater, and almost proportional to the square of the volume flow rate of fluid. At the same time it is found that the flow field, which is smooth and straight (or laminar) in the Hagen–Poiseuille regime, becomes at higher velocities full of irregular eddying motions (or turbulent). This may be seen clearly in the case of a fluid flowing through glass tubes if a dye is introduced through a small injector at the inlet (Figures 6.1, 4.52). The colored filament is straight and smooth for low speeds but breaks off and disperses almost uniformly when turbulence develops. As a second example, introduced in Chapter 1, consider a jet of water that emerges from a circular orifice into a tank of still water. At very low speeds of the fluid the jet is smooth and steady. For higher speeds, it develops swirls of various sizes amidst avalanches of complexity.