Abstract
We review some advances in the theory of homogeneous, isotropic turbulence. Our emphasis is on the new insights that have been gained from recent numerical studies of the three-dimensional Navier Stokes equation and simpler shell models for turbulence. In particular, we examine the status of multiscaling corrections to Kolmogorov scaling, extended self similarity, generalized extended self similarity, and non-Gaussian probability distributions for velocity differences and related quantities. We recount our recent proposal of a wave-vector-space version of generalized extended self similarity and show how it allows us to explore an intriguing and apparently universal crossover from inertial- to dissipation-range asymptotics.
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References
M Van Dyke,An album of fluid motion (The Parabolic Press, Stanford, California, 1982)
K R Sreenivasan and R A Antonia,Ann. Rev. Fluid Mech. 29, 435 (1997)
E D Siggia,Ann. Rev. Fluid Mech. 26, 137 (1994)
M Nelkin,Adv. Phys. 43, 143 (1994)
V L’vov and I Procaccia,Phys. World 35, (1996)
W D McComb,The physics of fluid turbulence, (Oxford University Press, Oxford, 1991)
U Frisch,Turbulence: the legacy of A N Kolmogorov (Cambridge University Press, Cambridge, 1995)
T Bohr, M H Jensen, G Paladin, A Vulpiani,Dynamical systems approach to turbulence, to be published (Cambridge University Press, Cambridge, 1997)
G K Batchelor,The theory of homogeneous turbulence (Cambridge University Press, Cambridge, 1953)
H Tennekes and J L Lumley,A first course in turbulence (MIT Press, Cambridge, Massachusetts, 1972)
A S Monin and A M Yaglom,Statistical fluid mechanics (MIT Press, Cambridge, Massachusetts, 1975)
G I Taylor,Proc. R. Soc. (London) A151, 421 (1935)
G S Saddoughi and S V Veeravalli,J. Fluid Mech. 268, 333 (1994)
N Goldenfeld,Lectures on phase transitions and the renormalization group (Addison-Wesley, New York, 1992)
To the best of our knowledge, a direct mapping of a deterministic partial differential equation (PDE) with spatiotemporal chaos onto a stochastic PDE has been carried out only for the Kuramoto-Sivashinsky (KS) equation. This mapping uses a numerical coarse-graining procedure and shows, both in one and two spatial dimensions, that the KS equation is in the universality class of the Kardar-Parisi-Zhang (KPZ) equation, i.e., the long-distance and long-time behaviours of their correlations functions are the same. (See: S Zaleski,Physica D34, 427 (1989); F Hayot, C Jayaprakash and Ch Josserand,Phys. Rev. E47, 911 (1993); C Jayaprakash, F Hayot and R Pandit,Phys. Rev. Lett. 71, 15 (1993).) The case of the deterministically forced Navier-Stokes equation is considerably more subtle; however, in the absence of a direct mapping, it has been conjectured that the appropriate PDE is the NS equation with an additive, Gaussian white noise whose variance has a power-law dependence on the wave-vector [16,17,18]
C DeDominicis and P C Martin,Phys. Rev. A19, 419 (1979)
V Yakhot and S A Orszag,J. Sci. Comput. 1, 3 (1986);Phys. Rev. Lett. 57, 1722 (1986)
J K Bhattacharjee,J. Phys. A21, L 551 (1988);Phys. Rev. A40, 6374 (1989);Phys. Fluids A3, 879 (1991)
C-Y Mou and P B Weichman,Phys. Rev. Lett. 70, 1101 (1993)
J K Bhattacharjee,Pramana — J. Phys. 48, 365 (1997)
P Bak, C Tang and K Wiesenfeld,Phys. Rev. Lett. 59, 381 (1987);Phys. Rev. A38, 364 (1988)
G Stolovitzky and K R Sreenivasan,Phys. Rev. E48, R33 (1993)
S Douady, Y Couder and M E Brachet,Phys. Rev. Lett. 67, 983 (1991)
O Cadot, S Douady and Y Couder,Phys. Fluids 7, 630 (1995)
E Villermaux, E Sixou and Y Gagne,Phys. Fluids 7, 2008 (1995)
L F Richardson,Weather prediction by numerical process (Cambridge University Press, Cambridge, 1922)
A N Kolmogorov,C.R. Acad. Sci. USSR 30, 301 (1941)
N Cao, S Chen and K R Sreenivasan,Phys. Rev. Lett. 77, 3799 (1996)
H L Grant, R W Stewart and A Moilliet,J. Fluid Mech. 12, 241 (1962)
K R Sreenivasan,Phys. Fluids 7, 2778 (1995)
F Anselmet, Y Gagne, E J Hopfinger and R A Antonia,J. Fluid Mech. 140, 63 (1984)
R Benzi, S Ciliberto, R Trippiccione, C Baudet, F Massaioli and S Succi,Phys. Rev. E48, R29 (1993)
J Herweijer and W van de Water,Phys. Rev. Lett. 74, 4651 (1995)
T Katsuyama, Y Horiuchi and K Nagata,Phys. Rev. E49, 4052 (1994)
S Grossman, D Lohse, V L’vov and I Procaccia,Phys. Rev. Lett. 73, 432 (1994)
V S L’vov and I Procaccia,Phys. Rev. Lett. 74, 2690 (1994)
G I Barenblatt and N Goldenfeld,Phys. Fluids 7, 3078 (1995)
G Zocchi, J Maurer, P Tabeling and H Williame,Phys. Rev. E50, 3693 (1994)
B Chabaud, A Naert, J Peinke, F Chilla, B Castaing and B Hebral,Phys. Rev. Lett. 73, 3227 (1994)
V Emsellem, L P Kadanoff, D Lohse, P Tabeling and J Wang, chao-dyn/9604009,Phys. Rev. E to appear
A N Kolmogorov,J. Fluid Mech. 13, 82 (1962)
A A Praskovsky,Phys. Fluids A4, 2589 (1992)
G Stolovitzky, P Kailasnath and K R Sreenivasan,J. Fluid Mech. 297, 275 (1995)
G Stolovitzky and K R Sreenivasan,Rev. Mod. Phys. 66, 229 (1994)
S T Thoroddsen and C W Van Atta,Phys. Fluids A4, 2592 (1992)
Y Gagne, M Marchand and B Castaing,J. Phys. 4, 1 (1994)
S Chen, G D Doolen, R H Kraichnan and L P Wang,Phys. Rev. Lett. 74, 1755 (1995)
K R Sreenivasan and P Kailasnath,Phys. Fluids A5, 512 (1993)
U Frisch, P L Sulem and M Nelkin,J. Fluid Mech. 87, 719 (1978)
G Parisi and U Frisch, inTurbulence and predictability in geophysical fluid dynamics edited by M Ghil, R Benzi and G Parisi (North-Holland, Amsterdam, 1985) pp. 84–87
C Meneveau and K R Sreenivasan,J. Fluid Mech. 224, 429 (1991)
Z S She and E Leveque,Phys. Rev. Lett. 72, 336 (1994)
E D Siggia,J. Fluid Mech. 107, 375 (1981)
Z S She, E Jackson and S A Orszag,Nature (London) 344, 226 (1990)
G R Chavarria, C Baudet and S Ciliberto,Phys. Rev. Lett. 74, 1986 (1995)
B Dubrulle,Phys. Rev. Lett. 73, 959 (1994)
D Segel, V L’vov and I Procaccia,Phys. Rev. Lett. 76, 1828 (1996)
C Meneveau,Phys. Rev. E54, 3657 (1996)
R Benzi, L Biferale, S Ciliberto, M Struglia and R Tripiccione,Europhys. Lett. 32, 709 (1995)
S Chen, G Doolen, J R Herring, R H Kraichnan, S A Orszag and Z S She,Phys. Rev. Lett. 70, 3051 (1993)
S Chen, G D Doolen, R H Kraichnan and Z S She,Phys. Fluids A5, 458 (1993)
R H Kraichnan,J. Fluid Mech. 5, 497 (1959)
D Lohse and A Müller-Groeling,Phys. Rev. Lett. 74, 1747 (1995);Phys. Rev. E54, (1996)
A Praskovsky and S Oncley,Phys. Rev. Lett. 73, 3399 (1994)
R Benzi, L Biferale, G Paladin, A Vulpiani and M Vergassola,Phys. Rev. Lett. 67, 2299 (1991)
Z S She,Phys. Rev. Lett. 66, 600 (1991)
P Kailasnath, K R Sreenivasan and G Stolovitzky,Phys. Rev. Lett. 68, 2766 (1992)
G Stolovitzky,The statistical order of small scale turbulence, Ph.D. Thesis, Yale University, USA (1994), cited in ref. [2]
Z S She and E C Waymire,Phys. Rev. Lett. 74, 262 (1995)
S A Orszag and G S Patterson,Phys. Rev. Lett. 28, 76 (1972)
R M Kerr,J. Fluid Mech. 153, 31 (1985)
M M Rogers and P Moin,J. Fluid Mech. 176, 33 (1987)
K Yamamoto and I Hosokawa,J. Phys. Soc. Jpn. 57, 1532 (1988)
R H Kraichnan and R Panda,Phys. Fluids 31, (1988)
M Meneguzzi and A Vincent, inAdvances in turbulence 3 edited by A V Johansson and P H Alfredsson (Springer, Berlin, 1991) pp. 211–220; A Vincent and M Meneguzzi,J. Fluid Mech. 258, (1994)
S Kida and K Ohkitani,Phys. Fluids A4, 1018 (1992)
J Jimenez, A A Wray, P G Saffman and R S Rogallo,J. Fluid Mech. 255, 65 (1993)
N Cao, S Chen, and Z-S She,Phys. Rev. Lett. 77, 3711 (1996)
E B Gledzer,Sov. Phys. Dokl. 18, 216 (1973)
K Ohkitani and M Yamada,Prog. Theor. Phys. 81, 329 (1989)
A M Obukhov,Atmos. Oceanic Phys. 7, 41 (1971)
V N Desnyansky and E A Novikov,Atmos. Oceanic Phys. 10, 127 (1974)
M Yamada and K Ohkitani,J. Phys. Soc. Jpn. 56, 4210 (1987)
M H Jensen, G Paladin and A Vulpiani,Phys. Rev. A43, 798 (1991)
D Pisarenko, L Bieferale, D Courvoisier, U Frisch and M Vergassola,Phys. Fluids A5, 2533 (1993)
L Kadanoff, D Lohse, J Wang and R Benzi,Phys. Fluids 7, 617 (1995)
Sujan K Dhar, Ph.D. thesis, Indian Institute of Science, Bangalore (1996) unpublished
L Kadanoff, D Lohse and N Schörghofer,Physica D160, 165 (1997)
L Biferale, A Lambert, R Lima and G Paladin,Physica D80, 105 (1995)
S K Dhar, A Sain and R Pandit,Phys. Rev. Lett. (to appear) (1997)
L Biferale and R Kerr,Phys. Rev. E52, 61133 (1995)
R Benzi, L Biferale and G Parisi,Physica D65, 163 (1993)
N Schörghofer, L Kadanoff and D Lohse,Physica D88, 40 (1995)
Z S She, E Jackson and S A Orszag,Proc. R. Soc. London A434, 101 (1991)
M E Brachet, D I Meiron, S A Orszag, B G Nickel, R H Morf and U Frsich,J. Fluid Mech. 130, 411 (1983)
J Eggers and S Grossmann,Phys Fluids A3, 1958 (1991)
S Grossmann and D Lohse,Phys. Rev. E50, 2784 (1994)
E Leveque and Z S She,Phys. Rev. Lett. 75, 2690 (1995)
V Borue and S A Orszag,Europhys. Lett. 29, 6875 (1995)
C Foias, O Manley and L Sirovich,Phys. Fluids A2, 464 (1990)
L Biferale,Phys. Fluids A5, 428 (1993)
J Eggers and S Grossmann,Phys. Lett. A156, 444 (1991)
D Lohse,Phys. Rev. Lett. 73, 3223 (1994)
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Dhar, S.K., Sain, A., Pande, A. et al. Some recent advances in the theory of homogeneous isotropic turbulence. Pramana - J Phys 48, 325–364 (1997). https://doi.org/10.1007/BF02845638
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DOI: https://doi.org/10.1007/BF02845638
Keywords
- Isotropic turbulence
- homogeneous turbulence
- turbulence simulation and modelling
- theory and models of chaotic systems
- critical point phenomena