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Unraveling the origin of nonGaussian statistics in hydrodynamic turbulence Yi Li & Charles Meneveau 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 strainrate with highly nonGaussian statistics. Quantitatively, intermittency is manifested in highly elongated tails in the probability density functions of the velocity increments between pairs of points. A longstanding open issue has been to predict the origins of intermittency and nonGaussian statistics from the NavierStokes equations. We have derived, from simplified forms of the NavierStokes equations, a remarkably simple nonlinear dynamical system for the Lagrangian evolution of longitudinal and transverse velocity increments (for details, see Li & Meneveau (2005). From this system we are able to show that the ubiquitous nonGaussian tails in turbulence have their origin in the inherent self amplification of longitudinal velocity increments, and cross amplification of the transverse velocity increments (click here for a "newsrelease" about this work). Let du and dv be the longitudinal and transerse velocity increments over some specified (fixed) distance between two points. Without loss of generality dimensions can be scaled so that the fixed distance is unity. Now we evaluate the rate of change of du and dv if we follow the fluid as well as the orientation of the initial displacement as it is advected and tilted by the flow. We neglect effects of friction and pressure. The advective terms can then be simplified to read, after proper nondimensionalization: We call this system the "advected deltavee" system. The first term on the righthandside (rhs) of the equation for du also occurs in 1D Burgers equation (the selfamplification effect of negative velocity gradients). The second term indicates that the transverse velocity (rotation) tends to counteract the selfamplification process. For dv, the term on the rhs suggests exponential growth of dv at a rate 2du when du<0. This "crossamplification'' mechanism can lead to very large values of dv. Figure 1 shows the phasespace portrait of this simple system. Figure 1: Phasespace portrait of the "advected deltavee system" Figure 2 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 flareup of long tails in the pdfs of transverse velocity increment. The next challenge is to understand the efffects of the neglected pressure and viscous terms on these dynamics, and to use the insights gained to develop improved turbulence models. Figure 2: Evolution of the pdf of velocity increments in time: (a) longitudinal velocity increment, (c) transverse velocity increment vector component. Dotted line: Gaussian; solid: t=0.03; dashed: t=0.06; dashdotted: t=0.09; dashdoubledotted: t=0.12; longdashed: t=0.15; longdashdotted: t=0.18 The approach has recently been generalized to passive scalars and including some effects of pressure, in a paper recently published in the J. Fluid Mechanics (Yi & Meneveau, 2006). 

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Charles
Meneveau  Department of Mechanical Engineering  Johns Hopkins
University 3400 N. Charles Street  Baltimore, MD 21218  USA Phone: 1 (410) 5167802  Fax: 1 (410) 5167254  Email: meneveau@jhu.edu 

Last
update:
December 13, 2013
