Our analysis of both ideal and viscous fluids assumed that the flow was laminar, or smooth, with fluid particles moving along parallel streamlines. With laminar flow, pressure and velocity change gradually in predictable ways. The layers in a fluid—“laminar” means “arranged in layers”—remain distinct and do not mix.

Not all fluid flow is laminar, however. Many flowing fluids exhibit turbulence, which is pretty much the opposite of laminar flow. Turbulent flow is characterized by chaotic and erratic changes in pressure and velocity. Whirlpool-like eddies and vortices are common, and the fluid is well mixed. Figure 14.24 shows the dramatic difference when a flow transitions from laminar to turbulent.

What determines whether a fluid has nicely behaved laminar flow or irregular and unpredictable turbulent flow? The Reynolds number! In Chapter 6 we introduced the Reynolds number $Re=\rho vL/\eta$ as the ratio of inertial forces to viscous forces. Here $L$ is a characteristic size or dimension. $L$ is the diameter $D=2r$ both for a liquid flowing through a circular tube, the subject of this section, and for a sphere moving through a fluid, the situation we considered in Chapter 6.

Viscous fluids, like honey, tend to flow very smoothly, so we expect laminar flow for low values of the Reynolds number. Flow at high values of the Reynolds number, where inertia dominates, is like a stampede of thousands of closely packed cattle: fine as long as every cow maintains absolutely the same velocity, but the slightest stumble or irregularity causes the flow to break up into a chaotic mess. So, as the Reynolds number increases, there comes a point where the slightest irregularity—even just molecular diffusion—causes a flow to become turbulent.

There is no satisfactory theory of turbulence—it remains an unsolved problem of classical physics—and thus no way to predict the range of Reynolds numbers that characterize laminar or turbulent flow. Nonetheless, experimental measurements have established the relevant Reynolds numbers for many situations of practical importance. FIGURE 14.36 illustrates laminar and turbulent flows for a fluid flowing through a circular tube and for a sphere moving through a fluid.

FIGURE 14.36 Laminar and turbulent fluid flows.

We see that Poiseuille’s equation for flow through a circular tube, which assumes laminar flow, is valid only when $Re<2000$. The flow is fully turbulent for $Re>4000$, and in between is a complicated transition region where flows exhibit both laminar and turbulent characteristics.

Recall from Chapter 6 that an object moving through a fluid experiences linear drag when $Re<1$ and quadratic drag when $Re>1000$. Now we can see why. For low Reynolds number, the fluid flow around a sphere is laminar; the streamlines part and then come back together, leaving no trace of the sphere’s passage. In contrast, the sphere leaves a fully turbulent wake behind it at high Reynolds number. You’re familiar with this from boats; water moves smoothly around a boat that is gliding slowly through the water, but a fast boat leaves a turbulent wake behind. Quadratic drag, which is common for objects moving through the air, occurs only when the object creates a turbulent wake.