Non-Newtonian turbulence: viscoelastic fluids and binary mixtures.

18 1. Newtonian turbulence
The dimensionless constants Cp are not known, except for the universal value
C3 = −4/5.
An interesting case corresponds to p = 2; the K41 prediction for the second
order structure function then is
S2(ℓ) ∼ ɛ 2/3 ℓ 2/3
The above scaling implies the power law energy spectrum:
E(k) = C2ɛ 2/3 k −5/3
where C2 is called Kolmogorov constant.
(1.54)
(1.55)
Figure 1.2: Turbulent spectra in the time domain for data from the S1 wind tunnel
ONERA [9] (Rλ = 2720, left) and a low temperature helium gas flow between
counter-rotating cylinders [10] (Rλ = 1200, right).
Longitudinal structure functions are convenient also for an experimental approach.
Indeed, longitudinal velocity increments can be obtained by means of
standard techniques such as, e. g., hot wire anemometry. Let us suppose to have
a velocity field u which can be decomposed in a mean flow U = (U, 0, 0) and
a turbulent fluctuating part u ′ = u − U whose intensity is assumed to be small
compared to the mean flow 〈|u ′ | 2 〉 1/2 ≪ U. Due to the cooling produced by the
flow, the resistance of a hot wire perpendicular to the mean flow, say parallel to
the z direction, is reduced. From the measure of this reduction, it is possible to
obtain the time series of the velocity integrated in the direction of the wire:
uN = [(u ′ x + U) 2 + u ′2
y ] 1/2 = U[1 + u′ x
U + O(u′2 x
U 2)]
(1.56)
where it has been supposed that amplitudes of fluctuations in the two directions
perpendicular to the wire are of the same order u ′ x ∼ u′ y . Within Taylor hypothesis,
18