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Statistics of velocity fluctuations

The effect of polymer additives cannot be merely represented by a rescaling of velocity fluctuations by a given factor. In Fig. 4.8 we show the probability distribution of a velocity component, $u_x$. The choice of the $x$ direction is immaterial by virtue of statistical isotropy. In the Newtonian case the distribution is remarkably close to the sub-Gaussian density $N\exp(-c\vert u_x\vert^3)$ stemming from the balance between forcing and nonlinear terms in the Navier-Stokes equation, in agreement with the prediction by Falkovich and Lebedev [79]. On the contrary, the distribution in the viscoelastic case is markedly super-Gaussian, with approximately exponential tails. This strong intermittency in the velocity dynamics is due to the alternation of quiescent low-velocity phases ruled by polymer feedback and bursting events where inertial nonlinearities take over.

Figure 4.8: Intermittency of velocity fluctuations induced by polymer additives. The probability density function $P(u_x)$ of the velocity component $u_x$ for the Newtonian (solid line) and for the viscoelastic case with strong feedback (dashed line). Same parameters as in Fig. 4.6. Also shown the distribution $\Gamma(2/3) 3^{3/2} \exp(-c\vert u_x\vert^3)/(4\pi c)$ with $c=2.1\cdot 10^{-3}$ (dotted line).
\includegraphics[draft=false,scale=0.7]{P_pdfvelocita.eps}

The sub-Gaussian shape of the pdf of velocities in the Newtonian case can be predicted from the statistics of the forcing with the following dimensional argument. The values of velocity can be considered as the result of summation of forcing contributions during one large-eddy turnover time $t_L \sim L/V$.

In the case of a slow forcing, with a decorrelation time $\tau_f$ longer that the large-eddy turnover time $\tau_f > t_L$, those contributions are correlated, producing a ballistic growth of velocity $V(t) \sim f t$ up to the time $t_L \sim L/V$ when the non-linear term starts to transfer energy down in the cascade. Thus we obtain the dimensional estimate $V \sim f t_L \sim f L/V$, or equivalently $V^2 \sim L f$, which can be used to link the pdf of the forcing to the pdf of velocities. If the statistics of forcing is Gaussian $P(f) = N e^{-f^2/ 2 F_0}$, the resulting pdf for velocities is $P(V) = N e^{-\beta V^4 }$

Since we are using a Gaussian, $\delta$-correlated in time forcing, we are in the opposite limit of fast forcing, with $\tau_f < t_L$. In this case the different contributions are independent, producing a diffusive growth $V^2(t) \sim f^2 \tau_f t$ for $t \le t_L \sim L/V$. Substituting the resulting dimensional estimate $V^3 \sim f^2 \tau_f L$ in the Gaussian pdf of forcing we end with the prediction for the pdf of single point velocities:

\begin{displaymath}
P(V) = N e^{-\beta V^3}
\end{displaymath} (4.19)

which is consistent with our measurement.


next up previous contents
Next: Lagrangian chaos reduction Up: Active polymers Previous: Energy balance   Contents
Stefano Musacchio 2004-01-09