Onset of Turbulence in Pipe Flow
More than 125 years ago Osborne Reynolds launched the quantitative study of turbulent transition as he sought to understand the conditions under which fluid flowing through a pipe would be laminar or turbulent1. Since laminar and turbulent flow have vastly different drag laws2, this question is as important now as it was in Reynolds' day.
Reynolds understood how one should define ``the real critical value'' for the fluid velocity beyond which turbulence will persist indefinitely3. He also appreciated the difficulty in obtaining this value. For years this critical Reynolds number, as we now call it, has been the subject of study, controversy, and uncertainty.
Now, more than a century after Reynolds pioneering work, we know that the onset of turbulence in shear flows is properly understood as a statistical phase transition4 5 6. How turbulence first develops in these flows is more closely related to the onset of an infectious disease than to, for example, the onset of oscillation in the flow past a cylinder or the onset of motion in a fluid layer heated from below.
Through the statistical analysis of large samples of individual decay and proliferation events, we at last have an accurate estimate of the real critical Reynolds number for the onset of turbulence in pipe flow7, and with it, an understanding of the nature of transitional turbulence.
The work is joint with: Kerstin Avila, David Moxey, Marc Avila, Alberto de Lozar, and Björn Hof
Determining the Critical Reynolds Number
low Re turbulence occurs in localized patches known as
It is well established that puffs are metastable and decay
with a characteristic time scale9.
This time scale increases rapidly with Re, but does not
diverge at finite Re10.
This is not the full story however, as puffs may also split
and generate new puffs11.
Puff splitting also has a characteristic time scale which
decreases with Re. The critical Reynolds number is determined
as the point where these time scale are equal.
The figure on left shows time scales for decay and spreading turbulence as a function of Reynolds number. Data from both experiment and direct numerical simulation (DNS) are shown. The decay data on the left branch is primarily taken from past studies. The spreading times cales have been determined as part of the current study.
The experimental time scales of over 107 advective time units is achieved from more than 106 individual events in a pipe 4000 diameters long.
Onset of Sustained Turbulence
Re larger than crossing point shown above, individual puffs are
more likely to split than decay. In the absence of interactions
between puffs, this would mark exactly the real critical
Reynolds number for the onset of sustained turbulence.
Correlations between decay and splitting events, due to the
interaction between puffs, is expected to shift the real
critical point slightly from the crossing point. At present the
onset of sustained turbulence has not been directly measured in
experiment or simulation. For this we rely on a model12
that reproduces decay and splitting of puffs in pipe flow.
The upper part of this figure shows the decay (green) and splitting (blue) time scales for model puffs. The model time scales behave qualitatively as those in real pipe flow, with a crossing at R×= 2040.
In the model the onset of sustained turbulence is directly accessible. In the lower figure the equilibrium turbulence fraction Ft is shown as a function of model Reynolds number R. Ft is the mean fraction of turbulence in the thermodynamic limit (mean over space, time, and ensemble, with length and time going to infinity).
The onset of sustained turbulence is continuous (in the directed percolation universality class) with a true critical value of Rc=2046.