Turbulence is one of the great mysteries of modern science. It is also one of the most important, as most of the flows we're interested in are turbulent. In some applications, such as industrial reactors, turbulence is desirable due to its mixing properties; in many others, we want to avoid turbulence due to the additional friction it causes.

The transition from laminar (smooth, predictable flow) to turbulent (chaotic, randomly fluctuating flow) is currently not well understood, although empirical information allows us to estimate the point at which many flows will make this transition. Flow through a pipe, however, is a rather important example that is not easily predicted. A paper recently published in Science (by a group we reported on in the past) has tried to identify just where the transition to turbulence occurs in pipes.

Pipe flows are rather important for industrial applications such as in oil pipelines, where additional energy is required to overcome the increased drag due to turbulence. They are also scientifically interesting, as fluid dynamicists have been studying pipe flows (without resolution) since Osborne Reynolds, one of the fathers of turbulent flow research, first looked at the transition problem in the 1880s.

Whether a flow is laminar or turbulent is typically characterized using the dimensionless Reynolds number, defined as a characteristic velocity V times a characteristic length L divided by the kinematic viscosity ν, or Re=VL/ν. This represents the ratio of inertial to viscous forces in the flow. In the case of pipe flow, V is the mean velocity and L is the pipe diameter.

In general, the transition from laminar to turbulent flow occurs between Reynolds numbers of 1,700-3,000, but the exact number varies not only between experimental facilities but also between different runs on the same equipment.

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.

The authors of the current paper introduced turbulent puffs into fully developed (that is, not varying in time) flow using a small water jet, which enabled them to create one puff at a time. Previous experimental work used obstacles located near the entrance of the pipe, but this continuous source of turbulence made it difficult to control the transition.

Here, by finely controlling the Reynolds number (±5), the lifetime of the turbulent puffs could be compared with the time it takes for puffs to split. As the Reynolds number goes up, the lifetimes should extend while the time between splits drops—the transition to turbulence occurs when these two quantities meet. After performing a large number of measurements (at least 2,000, and as many as 60,000 near the transition) for each data point—the splitting of puffs is highly stochastic, so multiple runs are necessary—the authors determined the critical Reynolds number to be 2040±10.

The current study is noteworthy not only because it pinpoints the Reynolds number where laminar-to-turbulent transition occurs in pipe flow (and with some statistical confidence due to the large amount of measurements), but also because of the novel approach used. Many other flow types have transitions that are tricky to analyze, such as boundary layers, Couette flow, and other shear-driven flows. The method used here may lead to better understanding of the transition to turbulence in those cases as well.

Science, 2011. DOI: 10.1126/science.1203223 (About DOIs).

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