Non-Newtonian turbulence: viscoelastic fluids and binary mixtures.

22 1. Newtonian turbulence
pushed to the left by a burst of acceleration and comes nearly to a stop before
being rapidly accelerated upward by a fluctuation roughly equal to 30 times the
root mean square value.
The acceleration a of a fluid particle in a turbulent flow is given by the Navier-
Stokes equation:
a ≡ du
= −∇p + ν∆u + f (1.63)
dt
Provided f is a large scale forcing and Re is large enough, the statistics of a
is essentially determined by that of pressure gradients. Experimental data [18]
indicate that the acceleration is an extremely intermittent variable and the shape
of its pdf is a stretched exponential (fig. 1.5).
Figure 1.5: Probability distribution functions of a normalized acceleration component
at three Reynolds numbers. The solid line is a stretched exponential parameterization
of the highest Rλ data, the inner dotted line is a gaussian for reference.
Inset: flatness F = 〈a 4 〉/〈a 2 〉 2 as a function of Rλ [18].
Let us now derive the shape of the acceleration pdf by means of simple phenomenological
arguments. By definition
δuτ
a ≡ lim
τ→0 τ
� δuτη
τη
(1.64)
where τη is the eddy turnover time associated with the Kolmogorov dissipative
scale η. The velocity fluctuations along a particle trajectory may be considered as
the superposition of different contributions from eddies of all sizes. In a time lag
τ the contributions from eddies smaller than a given scale ℓ are uncorrelated and
one may then write δuτ ∼ δuℓ. We assume that ℓ and τ are linked by the typical
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