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

• Lectures 13 Intermittency and Anomalous Scaling
• Velocity structure functions and experimental evidence
for intermittency & multiscaling
• Dissipative scaling exponents µn and “Dissipative” bridge
• Phenomenological models of multi-scaling: from K62 Log-normal
model to β− and multifractal model
• Dynamical “shell models” of multiscaling: GOY, Sabra and others
• Toward analytical theory of multiscaling
1

Log E(R)
Re =
ν
~ u L 1 1
1
Energy
Containing
Interval
K41: Richardson-Kolmogorov cascade picture of turbulence:
u 2
10 7
L 2
u 1
Re =
Re > Re > Re > .... >Re > Re ~ 100
1 2 3 n cr
Inertial interval
Energy Flux
R 2/3
Log(1/R)
u n L n
ν
n 3
L
1
u 3
L
Dissipative
interval
R 2
Energy
Dissipation
I. Universality of small scale statis-
tics, isotropy, homogeneity;
II. Scale-by-scale “locality” of the
energy transfer;
III. In the inertial interval of scales
the only relevant parameter is the
mean energy flux ε .
Dimensional reasoning ⇒ V ℓ (εℓ) 1/3
1. Turbulent energy of scale ℓ in
inertial interval E ℓ ρ ε 2/3 ℓ 2/3 ,
2. Turnover and life time of
ℓ-eddies: τ ℓ ε −1/3 ℓ 2/3
3. Viscous crossover scale
η ε −1/4 ν 3/4 , N ∼ Re 3/4 . . .
2

Structure functions and experimental evidence for intermittency
– Velocity difference across
separation r gives velocity
of ”r-eddies:”
Longitudinal velocity structure functions S ℓ
n (r) = W ℓ n r
Wr = v(r, t) − v(0, t) ,
– Longitudinal velocity:
W ℓ r = Wr · r
r
∝ r ζn.
In particular: S2(r) – Energy of r-eddies,
S3(r) = − 4
ε r (Kolmogorov-41) – Energy flux on scale r,
5
. S4(r) − 3 S 2 2 (r) – Deviation from the Gaussian statistics,
...
S2n(r)/S n 2 (r) – Statistics of very rare events
S ℓ
n (r) = Cn εr n/3 r ζn−n/3 , L − renormalization length .
L
3

Scaling exponents µn for the energy dissipation field ε(r) = ν|∇v(r)| 2
Knε(Rij) = 〈ε 11 ′ε 22 ′ . . . ε nn ′〉 ∝ R −µn , εij ≡ ε(ri) − ε(rj) , Rij ≡ ri − rj .
Straightforward K41 phenomenology predicts µ2 = 8
3 .
Experiment: µ2 0.3 ? ⇒ Viscous anomaly:
S ℓ 2 (R) =
〈ε(r)〉
⇒
v α kvβ k ′
= (2π) 3 δ(k + k ′ )F αβ
2 (k) , ⇒ K41 ⇒ F2(k) ε2/3
k
dk
(2π) 3|1 − exp(ik · R)|2F ℓℓ
2 (k) ε2/3
(ε R) 2/3
ν dk
(2π) 3k2 F2(k) ν ε 2/3
η ε −1/4 ν 4/3
∞
0
k 2 d k
1
k11/3 −1
11/3 .
d x
π2 sin2 k R x
2
due to UV & IR convergence of the integral .
1/η
0
k 4 d k
k
11/3 νε2/3
1/η
0
k 1/3 dk νε2/3
⇒
η4/3 ⇒ ε , ν − independent ⇒ the viscous anomaly!
ε(r) is the viscous scale object and Knε(Rij) cannot be evaluated in the
K41 manner via inertial-interval parameters!
4

• Exact “Dissipative-bridge” relationships, (L’vov-Procaccia-96)
NSE:
∂v
+ (v · ∇)v + ∇p = ν ∆v
∂t
⇒
∂v
∂t + P
(v · ∇)v
= ν ∆v
P: transversal projector, Wr: r-separated velocity increment.
NSE for Wr schematically :
Wr ∂Wr
∂t + Wr P
(Wr · ∇)Wr
∂Wr
∂t + P
(Wr · ∇)Wr
= ν ∆Wr ⇒
= Wrν ∆Wr ⇒ −ν(∇Wr) ·(∇Wr) −ε(r) .
Thus, inside of the average operator, 〈. . .〉, the viscous range object ε(r)
can be evaluated as the inertial range object Wr P
(Wr · ∇)Wr
ε(r) ⇒ W3 r
r
W 3 r /r:
⇒ µn = n − ζ3n ⇒ µ2 = 2 − ζ6 0.3 ÷ 0.4
Exact “dissipative-bridge” relations µn = n − ζ3n
are known in liter-
ature as a consequence of Kolmogorov “Refined Similarity” hypothesis.
5

• Phenomenological models of multi-scaling
– Kolmogorov-62 log-normal model: ⇒ K62 conjecture by analogy of
random breaking of stones with that of eddies: ln ε(r) is normally
distributed (i.e. Gaussian statistics). This gives:
µn = n − ζ3n
µn = µ2
2 n(n − 1) , ζn = n µ2
− n(n − 3) ,
3 18
Experimentally reasonable for n ≤ 6 ÷ 8 ⇒ “small n” expansion.
For large n: wrong, in particular, contradicts to exact statement dζn
dn
. (K62)
≥ 0 .
– β-model of anomalous scaling: Frisch-Sulem-Nelkin-78 conjecture:
Volume fraction Vr, occupied by r-eddies scales: Vr r β .
Energy flux (at r-scale) εr v2 r
τr
Vr v3 r
r
r
L
β = ε ⇒ vr (ε r) 1/3 L
L
r
Structure functions: Sn(r) v n r Vr = (ε r) n/3 r β(1−n/3) ⇒
L
ζn = n
3
β
+ (n − 3) β − co-dimension (β − model)
3
β/3 .

– 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”.
6

Dynamical “shell models” of multiscaling:
GOY model (Gledzer-73, Ohkinani-Yamada-89), and (born in Israel)
Sabra model (L’vov-Podivilov-Pomyalov-Procaccia-Vandembroucq-98)
. and many others
The model equation of motion mimics
NSE nonlinearity & “interaction locality”:
dun
dt = i(ak n+1u ∗ n+2 u∗ n+1 + bknu ∗ n+1 u∗ n−1
+ − ckn−1u ∗ n−1 u∗ n−2) − νk 2 n un + fn ,
Conservation of energy E =
|un| 2
requires a + b + c = 0 ⇒
conservation of“helicity”H = a c
n
n
7
n|un| 2 .

Kolmogorov-41 fixed point
¯vn =
1
[2(a − c)] 1/3 λ n/3
is unstable, giving
a random evolution
of the “shell velocities” ⇒
ζ p
Anomalous exponents in
2.5
⇓ Sabra shell model ⇓
2
1.5
1
0.5
0
Shell model
p/3
0 1 2 3 4 5 6 7
p
|u 11 |/
8.0
6.0
4.0
2.0
0.0
0.0e+00 2.0e+06 4.0e+06 6.0e+06 8.0e+06 1.0e+07
timesteps
Let S3(kn) ≡ Im un−1unun+1∗ . Exact:
S3(kn) =
1
2kn(a − c)
S2p(kn) ≡
|un| 2p
− ɛ + δ( c
a )n , ζ3 = 1 .
∝ k −ζ2p
n
.
8

“Euler” rescaling symmetry and “self-similar” propagation of “solitons”:
| u n (t)|
0.015
un(t) ≈ vλ −hn f
(t − tn)vk0λ (1−h)n
0.01
0.005
0
2.6 2.62 2.64 2.66 2.68
time
shell 15
shell 16
shell 17
shell 18
shell 19
shell 20
, h is free. Multiscaling!?
| u n (t)|/K am (n)
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
shell 15
shell 16
shell 17
shell 18
shell 19
shell 20
-0.03 -0.02 -0.01 0 0.01
(t-t max n )/K w (n)
L’vov-02: Stability of solitons @ [h mix, hmax] & asymptotical multiscaling
9

• Toward analytical theory of multiscaling:
Kraichnan-59 DIA: Direct Interaction Approximation ζ2 = 2 3 + 1 6
Kraichnan-62 “Lagrangian-History” DIA: ζ2 = 2 3 < ζ2,exp 0.701
0.83
Belinicher-L’vov-87 sweeping-free approach: K41 is an “order-by-order”
perturbation solution and intermittency is not perturbation phenomenon.
Lebedev-L’vov-94 Telescopic Multi-Step Eddy-Interaction: non-perturbation
mechanism of multiscaling ⇒ infinite re-summation of ladder diagrams.
Yakhot-Orszag-86-90 Straightforward Renormalization Group (RG)
approach (for car design, etc.) reproduces K41 scaling
Anjemyan-Antonov-Vasil’ev-89-now: modern RG ⇒ principal possibility
of anomalous scaling
10

Belinicher-L’vov-Pomyalov-Procaccia-98: Standard Gaussian decomposi-
tion, like F4 ⇒ F 2 2 , destroys Euler rescaling symmetry and fixes h = 1 3 ,
(K41). Suggested h-invariant decompositions, like F4 ⇒ F 2 3 /F2, preserve
the rescaling symmetry, leave h free, and demonstrate multiscaling in an
analytical, NS based theory (in the BL-87 sweeping-free representation)
L’vov-Procaccia-2000 Analytic calculation of anomalous exps. ζn in NS
turbulence: Using the (LP-96) fusing rules to flush out a small parameter
δ = ζ2− 2 3 0.03 in “4–eddy interaction amplitude” in the ladder diagrams
for exps. ⇒ ζn = n − 3)
− δn(n 1 + 2 δ b(n − 2)
3 2
, δ n < 1 , n ≤ 12 .
Benzi-Bifferale-Sbragaglia-Toschi-03: Anomalous scaling in shell models:
Using Fusion Rules and “time-dependent random multiplicative process”
for closure of correlation function ⇒ calculation (without free parameters)
of the anomalous scaling exps. in shell models.
T O BE CON T IN UED