Typically, the laminar-to-turbulence transition is studied mathematically by linearizing the Navier-Stokes equations, the governing equations of fluid dynamics, then perturbing the system. These perturbations will gradually disappear in laminar flow, but if the flow is turbulent, they’ll grow and produce chaotic motion. The transition, then, is the critical point between these two.
However, for pipe flows, this linearized approach shows that the perturbations decay for all Reynolds numbers, even though this doesn’t happen in actual experiments. In the real world, as the Reynolds number increases, small, turbulent puffs begin to split and interact, and their lifetimes increase. Eventually, these puffs carry enough turbulence to transition the flow entirely. # (submitted by David T)