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Unraveling the origin of non-Gaussian statistics in hydrodynamic turbulence
Figure 1(a) Highly intermittent signal of velocity gradient in turbulent flow, as measured in the Corrsin wind-tunnel at the Johns Hopkins University (b)
Figure 2(a): Time evolution of probability density functions of velocity increments, as predicted by the advected delta-vee system. (b) shows the phase-space trajectories of the system showing that initially small velocity increments du tend to become negative, and then dv may grow to very high values.
Description
Turbulent flows are notoriously difficult to describe and understand based on first principles. One reason is that turbulence contains highly intermittent bursts of vorticity and strain-rate with highly non-Gaussian statistics. Figure 1(a) shows a signal of velocity gradient measured in the Corrsin windtunnel (Fig. 1b). Quantitatively, intermittency is manifested in highly elongated tails in the probability density functions of the velocity increments between pairs of points. This has also challenged computational simulations of turbulence. A long-standing open issue has been to predict the origins of intermittency and non-Gaussian statistics from the Navier-Stokes equations. Professor Meneveau and graduate student Yi Li have derived, from simplified forms of the Navier-Stokes equations, a remarkably simple nonlinear dynamical system for the Lagrangian evolution of longitudinal and transverse velocity increments (Li & Meneveau 2005, 2006). From this system, called the “advected delta-vee system”, they were able to show that the ubiquitous non-Gaussian tails in turbulence have their origin in the inherent self amplification of longitudinal velocity increments, and cross amplification of the transverse velocity increments. Figure 2(a) shows the evolution of the system when it is initialized with random (Gaussian) initial conditions. It is immediately clear that the two main qualitative trends observed in turbulence naturally evolve from the solution of the system: the skewness towards negative values of longitudinal velocity increment, and the noticeable flare-up of long tails in the pdfs of transverse velocity increment. These trends can be understood from the phase-space portrait of the system, shown in Fig. 2(b). Prof. Meneveau and his group continue their research to understand the effects of the neglected pressure and viscous terms on these dynamics, and to use the insights gained to develop improved computational turbulence models.
References
Li, Y. & C. Meneveau, "On the origin of non-Gaussian statistics in hydrodynamic turbulence" (2005), Phys. Rev. Letts. 95, 164502.
Li, Y. & C. Meneveau, "Intermittency trends and Lagrangian evolution of non-Gaussian statistics in turbulent flow and scalar transport" (2006), J. Fluid Mech. 558, 133-142.