We treat a turbulent velocity field as a message in the same way as a book or a picture. All messages can be described by their entropy per symbol h, defined as in Shannon's theory of communication. In a turbulent flow, as the Reynolds number Re increases, more correlated degrees of freedom are excited and participate in the turbulent cascade. Experiments in a turbulent soap film suggest that the spatial entropy density h is a decreasing function of Re, namely h[proportionality]-logRe + const. In the logistic map, also analyzed here, increasing the control parameter r increases h. A modified logistic map with additional coupling to past iterations suggests the significance of correlations.

The Reynolds number dependency of intermittency for 2D turbulence is studied in a flowing soap film. The Reynolds number used here is the Taylor microscale Reynolds number R_{\lambda}, which ranges from 20 to 800. Strong intermittency is found for both the inverse energy and direct enstrophy cascades as measured by (a) the pdf of velocity differences P(\delta u(r)) at inertial scales r, (b) the kurtosis of P(\partial_x u), and (c) the scaling of the so-called intermittency exponent \mu, which is zero if intermittency is absent. Measures (b) and (c) are quantitative, while (a) is qualitative. These measurements are in disagreement with some previous results but not all. The velocity derivatives are nongaussian at all R_{\lambda} but show signs of becoming gaussian as R_{\lambda} increases beyond the largest values that could be reached. The kurtosis of P(\delta u(r)) at various r indicates that the intermittency is scale dependent. The structure function scaling exponents also deviate strongly from the Kraichnan prediction. For the enstrophy cascade, the intermittancy decreases as a power law in R_{\lambda}. This study suggests the need for a new look at the statistics of 2D turbulence.Comment: 15 pages, 11 figure

As everyone knows who has opened a kitchen faucet, pipe flow is laminar at low flow velocities and turbulent at high flow velocities. At intermediate velocities, there is a transition wherein plugs of laminar flow alternate along the pipe with "flashes" of a type of fluctuating, nonlaminar flow that remains poorly understood. In the 19th century, Osborne Reynolds sought to connect these states of flow with quantitative "laws of resistance," whereby the fluid friction is determined as a function of the Reynolds number. While he succeeded for laminar and turbulent flows, the laws for transitional flows eluded him and remain unknown to this day. By properly distinguishing between laminar plugs and flashes in the transitional regime, we show experimentally and numerically that the law of resistance for laminar plugs corresponds to the laminar law and the law of resistance for flashes is identical to that of turbulence.

We study the third-order longitudinal structure function, S 3 (r), in two-dimensional turbulence. In three dimensions, there is considerable theoretical, experimental, and numerical consensus regarding the validity of Kolmogorov's arch-famous " 4 5 th law" for S 3 (r). By contrast, in two dimensions, two disparate cascades, changed dissipation anomalies, a large-scale drag, and other factors conspire to create several versions of the S 3 (r) "law." This single quantity can vary considerably when viewed from different perspectives, reminiscent of the "Rashomon effect" in anthropology. After reviewing the history and usage of S 3 (r) in two-dimensional turbulence, we show that S 3 (r) generically embodies a mixture of energy and enstrophy fluxes. Building on this result, we derive S 3 (r) laws for freely decaying and forced two-dimensional turbulent flows, where we also account for the effects of a large-scale drag, an inextricable feature of quasi two-dimensional turbulence in experimental and atmospheric flows. We draw attention to the caution needed in interpreting S 3 (r) in two-dimensional turbulence.

Turbulent flows are not only everywhere, but every turbulent flow is the same at small scales. The extraordinary simplification engendered by this “small-scale universality” is a hallmark of turbulence theory. However, on the basis of the restrictive assumptions invoked by A. N. Kolmogorov to demonstrate this universality, it is widely thought that only idealized turbulent flows conform to this framework. Using experiments and simulations that span a wide range of Reynolds number, we show that small-scale universality governs the spectral structure of a class of flows with no apparent ties to the idealized flows: transitional pipe flows. Our results not only extend the universality of Kolmogorov’s framework beyond expectation but also establish an unexpected link between transitional pipe flows and Kolmogorovian turbulence.