1. EULERIAN AND LAGRANGIAN STATISTICS FROM HIGH RESOLUTION DNS
LUCA BIFERALE
Dept. of Physics, INFN and CNISM. University of Tor Vergata, Rome.
ICTR
Collaboration
Thanks to:
M. Cencini, A.S. Lanotte, F. Toschi, J. Bec, R. Benzi, E. Bodenschatz, G. Boffetta, E. Calzavarini, A. Celani,
B. Devenish, R. Fisher, T. Gotoh, L. Kadanoff, D. Lamb, S. Musacchio, N. Ouellette, H. Xu.

2. CORRELATION BETWEEN LAGRANGIAN AND EULERIAN MEASUREMENTS.
INTERMITTENCY OF VELOCITY INCREMENTS --STRUCTURE FUNCTIONS.
HIGH RESOLUTION DNS AND EXPERIMENTAL DATA.
INERTIAL RANGE AND VISCOUS RANGE.

3. LINK EULERIAN- LAGRANGIAN
K41 NOT ENOUGH !

4. INTERMITTENCY INTERMITTENCY 2048^3 16 Mega particles 3==Gaussian
[ DNS Rel = 600; R. Benzi, L.B. R. Fischer, D. Lamb,L. Kadanoff, F. Toschi] PRL 100,
2048^3
16 Mega particles
3==Gaussian

5. IN LOG-LOG ALL COWS ARE BLACK!
DEF
MAGNIFYING GLASS
IN LOG-LOG ALL COWS ARE BLACK!

6. EULERIAN STATISTICS: LONGITUDINAL VS TRANSVERSE
DEF
DEF
LOCAL SLOPES: LONGITUDINAL AND TRANSVERSE:
[ DNS Rel = 600
2048^3 R. Benzi, L.B. R. Fischer,
L. Kadanoff, F. Toschi]
PRL 100,
good agreement with data from Gotoh (PoF 2002)

7. INERTIAL RANGE : LONGITUDINAL VS TRANSVERSE
LOCAL SLOPES: LONGITUDINAL AND TRANSVERSE:
[ DNS Rel = 600
2048^3 R. Benzi, L.B. R. Fischer,
L. Kadanoff, F. Toschi]
PRL 100,
good agreement with data from Gotoh (PoF 2002)

8. 1st MESSAGE: LONGITUDINAL AND TRANSVERSE SCALE DIFFERENTLY
P=10
K41=10/3

9. EULERIAN
USE TWO DIFFERENT MULTIFRACTAL D(H) TO FIT SEPARATELY LONGITUDINAL AND TRANSVERSE EULERIAN SCALING
Parisi & Frisch (1983)

10. going Lagrangian

11. LINK EUL-LAG
SAME!!!!!

12. (i) INTERMITTENCY; (ii) UNIVERSALITY; (iii) ANISOTROPY WE LEARN ABOUT:
K41
[PRL 100, (2008), ICTR:
WE LEARN ABOUT:
(i) INTERMITTENCY; (ii) UNIVERSALITY; (iii) ANISOTROPY
WE LEARN ABOUT:
(i) INTERMITTENCY; (ii) UNIVERSALITY; (iii) ANISOTROPY
see also: L.B. E. Bodenschatz, M. Cencini, A. Lanotte, N. Ouellette, F. Toschi and H. Xu, PoF 2008

13. (i) INTERMITTENCY; (ii) UNIVERSALITY; (iii) ANISOTROPY
K41
[PRL 100, (2008), ICTR:
WE LEARN ABOUT:
(iv) MULTIFRACTALITY
WE LEARN ABOUT:
(i) INTERMITTENCY; (ii) UNIVERSALITY; (iii) ANISOTROPY
LINK EUL-LAG
see also: L.B. E. Bodenschatz, M. Cencini, A. Lanotte, N. Ouellette, F. Toschi and H. Xu, PoF 2008

14. (v) ROLE OF VORTEX FILAMENTS WE LEARN ABOUT:
K41
[PRL 100, (2008), ICTR:
WE LEARN ABOUT:
(v) ROLE OF VORTEX FILAMENTS
WE LEARN ABOUT:
(i) INTERMITTENCY; (ii) UNIVERSALITY; (iii) ANISOTROPY
see also: J. Bec, L.B. M. Cencini, A.S. Lanotte and F. Toschi, PoF 2006

15. CHECK THE LINK EUL-LAG TO HIGHER LAGRANGIAN STATISTICS.
INERTIAL RANGE: .
p=8
p=10
[ DNS Rel = 600
2048^3; 16MParticles; R. Benzi, L.B. R. Fischer, L. Kadanoff, F. Toschi]
PRL 100,
p=6
p=4

16. LAGRANGIAN SCALING AS PREDICTED FROM THE
EULERIAN D(H)’s
LINK EUL-LAG

17. take-home messages:
Eulerian homogeneous and isotropic statistics: longitudinal and transverse scaling differ at high moments. Is this true also at higher Reynolds? (waiting from Kaneda’s data); Is this a problem for theory? (see also L.B. and I. Procaccia, Phys. Rep 2005)
dimensional Eulerian-Lagrangian link, extended to Multifractal phenomenology, gives a prediction for Lagrangian statistics once given the Eulerian one (and --to some extent-- viceversa).
Prediction is satisfactory, WITHIN ERROR BARS, up to p=10, hints that at even higher orders things can change (wait and see).
Universality of Lagrangian statistics: good overlap between DNS and EXP, WITHIN ERROR BARS.

18. http://cfd.cineca.it Thanks to: M. Cencini, A.S. Lanotte, F. Toschi
J. Bec, R. Benzi, E. Bodenschatz, G. Boffetta, E. Calzavarini, A. Celani, B. Devenish, R. Fisher, T. Gotoh,
L. Kadanoff, D. Lamb, S. Musacchio, N. Ouellette, H. Xu.
see Toschi & Bodenschatz, ARFM 2009 for a review.