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, . The choice of the
direction is
immaterial by virtue of statistical isotropy. In the
Newtonian case the distribution is remarkably close
to the sub-Gaussian density
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.
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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 .
In the case of a slow forcing, with a decorrelation time
longer that the large-eddy turnover time
,
those contributions are correlated, producing a ballistic
growth of velocity
up to the time
when the non-linear term starts to transfer energy down in the cascade.
Thus we obtain the dimensional estimate
, or equivalently
,
which can be used to link the pdf of the forcing to the pdf of velocities.
If the statistics of forcing is Gaussian
,
the resulting pdf for velocities is
Since we are using a Gaussian, -correlated in time forcing,
we are in the opposite limit of fast forcing, with
.
In this case the different contributions are independent,
producing a diffusive growth
for
. Substituting
the resulting dimensional estimate
in the Gaussian pdf of forcing we end with the prediction
for the pdf of single point velocities: