Phenomenology of turbulence according to K41

let us consider a scale l associated with a velocity scale

𝑣𝑙 = 𝛿 𝑣∥ 𝑙2

or 𝑣𝑙= 𝑣 𝑟 + 𝑙 ∗ 𝑣(𝑙)

small scales large scales (>integral scale)

the eddy turnover time of the scale l is given by:

𝑡𝑙 =𝑙

𝑣𝑙

note that the scale l is subject to a distortion induced by a velocity difference vl. So tl is not an advection time scale, rather a time scale of energy transfer

for continuity as 1,2 diverge 3,4 get closer, thus within tl some turbulent motion is transferred to smaller scales

The energy flux from scales ~ l to smaller scale can be formulated as:

Π 𝑙 ~𝑣𝑙

2

𝑡𝑙=

𝑣𝑙3

𝑙

In the inertial range, there is no energy input or direct energy dissipation into heat so the energy flux should be independent of l and equal to finite mean dissipation rate ε

Π 𝑙 ~ 𝑣𝑙

3

𝑙 ~ 𝜖 𝑣𝑙 = 𝑙1/3𝜖1/3

On the upper limit of the inertial range, for l ~ integral scale

𝜖 ~ 𝑣0

3

𝑙0

𝑡𝑙 = 𝑙2/3𝜖−1/3

At the lower limit of the inertial range, where viscous effects are dominant, the diffusive time scale reads:

𝑡𝑙 =𝑙2

𝜐

These two term s are equal for

𝑙 = 𝜐3/4𝜖−1/4 = 𝜂

K41 : Space filling eddies vs coherent but anisotropic motions

l=l0rn

Violation of K41

any energy flux bypassing the inertial range will result in a unconventional (non K41) statistical behavior of structure functions and into a change in the exponents Sp (intermittency) e.g. in flows in complex terrain, or past a wake, where the turbulence does not have time-space to adjust itself to dissipate at the required rate following the standard cascade.

remember the assumptions: high Re, away from boundaries(!!!) a small scale homogeneity and self similarity

original K41 first universality assumption: at very high but not infinite Reynolds number, all the small scale statistical properties are uniquely, universally determined by the scale l, the mean energy dissipation ε and by the kinematic viscosity ν original K41 second universality assumption: in the limit of infinite Reynolds number, all the small scale statistical properties are uniquely, universally determined by the scale l and the mean energy dissipation ε

Critic to K41 universality: Landau objection

Landau 1944 I might be thought that the possibility exists in principle of obtaining a universal formula, applicable to any turbulent flow, which should give S2(l) for all distance l smaller than the integral scale l0 . In fact however there can be no such formula: the instantaneous value of (δv(l))2 might in principle be expressed as a universal function of the dissipation ε at the instant considered. When we average that expression however an important part will be played by the manner of variation of ε over times of the order of the period of the larger eddies, of

order of l0 , and this variation is different in different flows. the result of such averaging

therefore cannot be universal.

Kraichnan 1974 The slope of the structure function power law Cp for p ≠ 3 is not universal as it depends on the detailed geometry of production of turbulence

The H1,2,3 hypothesis are based on self similarity and symmetry rather than on universality and still are sufficient to derive the 2/3 law. Universality is not necessary and in fact, K41 predictions hold reasonably well (though non perfectly). The manifestation of non – universality is in the intermittent property of the flow

Intermittency is quantified by the flatness of the distribution (PDF) of the quantity of interest: high flatness implies strong tails and large kurtosis

𝐹=< 𝑣4 >

(< 𝑣2 >)2

self similarity intermittency

Intermittency in turbulence is assessed by checking the slopes of the structure functions at different order

𝑆𝑝(𝑙) = 𝛿 𝑣∥ 𝑙𝑝

Because of self-similarity, 𝑆𝑝 𝑙 𝛼 𝑙 𝑝/3

In the premultiplied form we have:

Estimates of the structure functions exponents using Extended Self Similarity

From SLTEST data

Guala , Mezger McKeon, 2009

Benzi 1993 Extended Self Similarity: flow past a cylinder

jet flow

Using ESS the linear fit in the log log phase space is much easier. It does not depend on the correct identification of the inertial range scaling region

flow past a cylinder

jet flow

Atmospheric Surface Layer

any energy flux bypassing the inertial range as in the case of strong scale interaction will result in a unconventional (non K41) statistical behavior of structure functions and into a change in the exponents Sp (intermittency)

Phenomenologically, it occurs when the turbulence does not have time-space to adjust itself to dissipate at the required rate following the standard cascade.