Intermittency and Anomalous scaling in turbulence - Victor S. L'vov ...

– Multifractal model (Parisi-Frisch-85)
The Euler equation:
∂v
∂t + P
(v · ∇)v
= 0 has the rescaling symmetry
R(λ, h)r = λr , R(λ, h)t = λ 1−h t , R(λ, h)v = λ h v , h – scaling of velocity:
Let “ℓ-eddy” v ℓ(r, t) be a solution of EE with characteristic scale ℓ. Then
v λℓ(r, t) ≡ R(λ, h)v ℓ(r, t) = λ h v ℓ(λr, λ 1−h t)
is “ λ ℓ- eddy”, an EE solution with scale λℓ. Denote as P(ℓ) the proba-
bility to meet ℓ-eddy in the turbulent ensemble. One expects, that
P(ℓ) is scale invariant: R(λ, h)P(ℓ) ≡ P(λℓ) = P(ℓ)λ β(h)
with β(h) being the “probability scaling index”, that depends on h. Now
Sn(r) V n
L
hmax
h min
dh r
L
nh+β(h) ⇒
steepest decent
V n
r ζn L L
ζn = min
h [n h + β(h)]. Geometrically: β(h) ⇒ 3−D(h), D(h) - co-dimension
of the “fractal support” of “h-turbulent cascade”.
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