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Dynamics of velocity gradient tensor in turbulent flows

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Unraveling the origin of non-Gaussian 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 strain-rate with highly non-Gaussian statistics. Quantitatively, intermittency is manifested in highly elongated tails in the probability density functions of the velocity increments between pairs of points. A long-standing open issue has been to predict the origins of intermittency and non-Gaussian statistics from the Navier-Stokes equations. We 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 (for details, see Li & Meneveau (2005). From this system we are 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 (click here for a "news-release" 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 non-dimensionalization:

We call this system the "advected delta-vee" system. The first term on the right-hand-side (rhs) of the equation for du also occurs in 1D Burgers equation (the self-amplification effect of negative velocity gradients). The second term indicates that the transverse velocity (rotation) tends to counteract the self-amplification process. For dv, the term on the rhs suggests exponential growth of dv at a rate 2|du| when du<0. This "cross-amplification'' mechanism can lead to very large values of dv. Figure 1 shows the phase-space portrait of this simple system.

Figure 1: Phase-space portrait of the "advected delta-vee 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 flare-up 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; dash-dotted: t=0.09; dash-double-dotted: t=0.12; long-dashed: t=0.15; long-dash-dotted: 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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News release on our new wind-energy research project

Previously show-cased research features:

Dynamics of velocity gradient tensor in turbulent flows

Large Eddy Simulation of Dispersion in Urban Areas

 
Charles Meneveau | Department of Mechanical Engineering | Johns Hopkins University
3400 N. Charles Street | Baltimore, MD 21218 | USA
Phone: 1 (410) 516-7802 | Fax: 1 (410) 516-7254 | Email: meneveau@jhu.edu
 
Last update: December 13, 2013