Non-Newtonian turbulence: viscoelastic fluids and binary mixtures.

24 1. Newtonian turbulence
all h and u0, weighted with their respective probabilities, [τη(h, u0)/τℓ0(u0)] [3−D(h)]
(1−h)
and p(u0). The latter, as before, is reasonably approximated by a gaussian of standard
deviation σ2 u = 〈u20 〉. Integration over u0 gives
p(a) ∼
h∈I
�
h∈I
�
× exp
dha [h−5+D(h)]/3 ν [7−2h−2D(h)]/3 ℓ D(h)+h−3
0 σ −1
u
− a2(1+h)/3ν2(1−2h)/3ℓ2h 0
2σ2 u
�
(1.70)
From (1.70) the dependence of the acceleration moments on Rλ can be derived.
For example, in the limit of large Rλ the second order moment is given by 〈a2 〉 ∼
R χ
λ , where χ = suph{2[D(h) −4h −1]/(1+h)}. The precise value of χ depends
on the choice of D(h), but it is typically close to the K41 value χK41 = 1. In terms
of the dimensionless acceleration ã = a/σa, (1.70) becomes
�
p(ã) ∼ dhã [h−5+D(h)]/3 R y(h)
�
λ exp − 1
2 ã2(1+h)/3R z(h)
�
λ (1.71)
where y(h) = χ[h−5+D(h)]/6+2[2D(h)+2h−7]/3 and z(h) = χ(1+h)/3+
4(2h − 1)/3. Let us observe that the K41 prediction (1.67) can be recovered from
(1.71) with h = 1/3, D(h) = 3 and χ K41 = 1. Comparison with data from high
resolution DNS, as reported in [24], shows that the multifractal prediction (1.71)
captures the shape of the acceleration pdf much better than its K41 counterpart
(see fig. 1.6).
Figure 1.6: Log-linear plot of the acceleration pdf. The crosses are DNS data, the
solid line is the multifractal prediction, and the dashed line is the K41 prediction.
The DNS statistics was calculated along the trajectories of 2 × 10 6 Lagrangian
particles amounting to 1.06 × 10 10 events in total. Inset: ã 4 p(ã) for DNS data
(crosses) and the multifractal prediction [24].
24