Lagrangian tetrad dynamics and the phenomenology of turbulence

Physics of Fluids

Volumn 11, Issue 8, 1999, Pages 2394-2410

  Chertkov, Michael ^a,b   Pumir, Alain ^c   Shraiman, Boris I ^b  

^a PRINCETON UNIVERSITY   (United States)

^b LUCENT TECHNOLOGIES   (United States)

^c INSTITUT NON LINÉAIRE DE NICE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000657080     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.870101     Document Type: Article

References (79)

1. U. Frisch, Turbulence: The Legacy of AN Kolmogorov (Cambridge University Press, Cambridge, 1995).
2. K. Sreenivasan and R. A. Antonia, "The phenomenology of small scale turbulence," Annu. Rev. Fluid Mech. (1997).
3. R. A. Antonia and C. W. van Atta, J. Fluid Mech. 84, 561 (1978).
4. P. Tabeling, G. Zocchi, F. Belin, J. Maurer, and H. Willaime, "Probability density function, skewness and flatness in large Reynolds number turbulence," Phys. Rev. E 55, 1613 (1994).
5. M. Nelkin, "In what sense is turbulence an unsolved problem?," Science 255, 566 (1992).
6. R. H. Kraichnan, "Models of intermittency in turbulence," Phys. Rev. Lett. 65, 575 (1990).
7. E. Siggia, "Numerical study of intermittency in 3-D turbulence," J. Fluid Mech. 107, 375 (1981).
8. W. Ashurst, A. Kerstein, R. Kerr, and C. Gibson, "Alignment of vorticity and scalar gradient with strain rate in simulated Navier-Stokes turbulence," Phys. Fluids 30, 2343 (1987).
9. S. Douady, Y. Couder, and M. E. Brachet, "Direct observation of intense vortex filaments in turbulence," Phys. Rev. Lett. 67, 983 (1991).
10. Z.-S. She, E. Jackson, and S. Orszag, "Structure and dynamics of homogeneous turbulence: Models and simulations," Proc. R. Soc. London, Ser. A 434, 101 (1991).
11. A. Vincent and M. Meneguzzi, "The spatial structure and statistical properties of homogeneous turbulence," J. Fluid Mech. 225, 1 (1991).
12. J. Jimenez, A. Wray, P. Saffman, and R. Rogallo, "The structure of intense vorticity in homogeneous isotropic turbulence," J. Fluid Mech. 255, 65 (1993).
13. J. Jimenez, "Kinematic alignment effects in turbulent flows," Phys. Fluids A 4, 652-654 (1992).
14. A. Tsinober, E. Kit, and T. Dracos, "Experimental investigation of the field of velocity gradients in turbulent flows," J. Fluid Mech. 242, 169 (1992).
15. P. Constantin, "Geometrical statistics in turbulence," SIAM (Soc. Ind. Appl. Math.) Rev. 36, 73 (1994).
16. R. H. Kraichnan, "Anomalous scaling of a randomly advected passive scalar," Phys. Rev. Lett. 72, 1016 (1994).
17. R. H. Kraichnan, V. Yakhot, and S. Chen, "Scaling relations for a randomly advected passive scalar field," Phys. Rev. Lett. 75, 240 (1995).
18. K. Gawedzki and A. Kupiainen, "Anomalous scaling of the passive scalar," Phys. Rev. Lett. 75, 3834 (1995).
19. M. Chertkov, G. Falkovich, I. Kolokolov, and V. Lebedev, "Normal and anomalous scaling of the fourth-order correlation function of randomly advected passive scalar," Phys. Rev. E 52, 4924 (1995).
20. B. I. Shraiman and E. D. Siggia, "Anomalous scaling of a passive scalar in a turbulent flow," C. R. Acad. Sci. 321, 279 (1995).
21. A. Polyakov, "Turbulence without pressure," Phys. Rev. E 52, 6183 (1995).
22. W. E. K. Khanin, A. Mazel, and Y. Sinai, "Probability distribution for the random forced Burgers equation," Phys. Rev. Lett. 78, 1904 (1996).
23. J. P. Boucheau, M. Mezard, and G. Parisi, "Scaling and intermittency in Burgers turbulence," Phys. Rev. E 52, 3656 (1995).
24. R. H. Kraichnan, "Small-scale structure of a scalar field convected by turbulence," Phys. Fluids 11, 945 (1968); "Convection of a passive scalar by a quasi-uniform random straining field," J. Fluid Mech. 64, 737 (1974).
25. R. H. Kraichnan, "Small-scale structure of a scalar field convected by turbulence," Phys. Fluids 11, 945 (1968); "Convection of a passive scalar by a quasi-uniform random straining field," J. Fluid Mech. 64, 737 (1974).
26. A. Tsinober, L. Shtilman, and H. Vaisburd, "A study of properties of vortex stretching and enstrophy generation in numerical and laboratory turbulence," Fluid Dyn. Res. 21, 477 (1997); A. Tsinober, "Is concentrated vorticity that important?," Eur. J. Mech. B/Fluids 17, 421 (1998).
27. A. Tsinober, L. Shtilman, and H. Vaisburd, "A study of properties of vortex stretching and enstrophy generation in numerical and laboratory turbulence," Fluid Dyn. Res. 21, 477 (1997); A. Tsinober, "Is concentrated vorticity that important?," Eur. J. Mech. B/Fluids 17, 421 (1998).
28. A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, MA, 1975).
29. 2 and observing that upon coarse-graining over region of size R it reduces approximately to a sum over many uncorrelated volumes (of size R). For large "d" the number of uncorrelated R-volumes in a shell surrounding the central - local - region is large.
30. J-D. Fournier, U. Frisch, and H. Rose, "Infinite-dimensional turbulence," J. Phys. A 11, 187 (1978).
31. V. Yakhot, "Strong turbulence in D-dimensions," chao-dyn/9805027.
32. S. Pope, "Lagrangian PDF methods for turbulent flow," Annu. Rev. Fluid Mech. 26, 23 (1994).
33. W. C. Reynolds and S. C. Kassinos, "A one-point model for the evolution of Reynolds stress and structure tensors in rapidly deformed homogeneous turbulence," Proc. R. Soc. London, Ser. A 451, 87 (1995).
34. P. R. van Slooten and S. B. Pope, "PDF modeling for inhomogeneous turbulence with exact representation of rapid distortions," Phys. Fluids 9, 1085 (1997).
35. P. Vieillefosse, "Internal motion of a small element of a fluid in inviscid flow," Physica A 125, 150 (1984).
36. J. Léorat, Thèse de Doctorat, Université Paris-VII, 1975.
37. B. J. Cantwell, "Exact solution of a restricted Euler equation for the velocity gradient tensor," Phys. Fluids A 4, 782 (1992).
38. B. Cantwell, "On the behavior of the velocity gradient tensor invariants in DNS of turbulence," Phys. Fluids A 5, 2008 (1993).
39. H. Blackburn, N. Mansour, and B. Cantwell, "Topology of fine-scale motions in turbulent channel flow," J. Fluid Mech. 310, 269 (1996).
40. a,b to the shape of the tetrad given by (3c) is presented here as a model assumption. Alternatively, one could start with the pressure anisotropy tensor (defined over spherical volume of size R) and study its Lagrangian dynamics which, if the present assumption is correct, should be consistent with (3b) and (3c).
41. V. Borue and S. A. Orszag, "Local energy flux and subgrid scale statistics in three-dimensional turbulence," J. Fluid Mech. 336, 1 (1998).
42. i,jδ(t) term in (5b). It, however, plays no role as long as one works with pure inertial range, coarse-grained quantities.
43. B. I. Shraiman and E. D. Siggia, "Lagrangian path integrals and fluctuations in random flow," Phys. Rev. E 49, 2912 (1994).
44. †)→0.
45. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, New York, 1960).
46. W. Reynolds, "The potential and the limitations of direct and large eddy simulation," in Whither Turbulence?, edited by J. L. Lumley (Springer, New York, 1990), p. 313.
47. C. M. Meneveau, "Statistics of turbulence subgrid stresses," Phys. Fluids 6, 815 (1994).
48. Liu et al., "On the properties of similarity subgrid models as deduced from measurements in a turbulent jet," J. Fluid Mech. 275, 83 (1994).
49. These sorts of singular solutions have been earlier noted by others (e.g., Childress and Siggia - private communications).
50. 2 which underlies the singularity (Ref. 33).
51. By ribbon, we mean configuration with two unequal stretching directions.
52. A. Betchov, "An inequality concerning the production of vorticity in isotropic turbulence," J. Fluid Mech. 1, 497 (1956).
53. R. M. Kerr, "Higher order derivative correlations and alignment of small scale structures in isotropic turbulence," J. Fluid Mech. 153, 31 (1985).
54. 2>=0. It can be easily added.
55. M. Chertkov, "Instanton for random advection," Phys. Rev. E 55, 2722 (1997).
56. G. Falkovich, I. Kolokolov, V. Lebedev, and A. Migdal, "Instantons and intermittency," Phys. Rev. E 54, 4896 (1996).
57. Z.-S. She, E. Jackson, and S. Orszag, "Intermittent vortex structures in homogeneous and isotropic turbulence," Nature (London) 344, 226 (1990).
58. This is not an essential approximation, and proper integration may be performed numerically.
59. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipies in Fortran (Cambridge University Press, Cambridge, 1992).
60. 3> + (3/4)<ω·S·ω>=0, which allows us to relate average energy flux to the average enstrophy production.
61. The degree of alignment, of course, would vary if conditioned on σ or e as an independent variable.
62. As the trajectory enters the σ<0 upper right quadrangle, matrix presented for isotropic tetrad.
63. While the question of existence of finite times' singularities is still being debated, it is clear that it has to be generated by some nonlocal effect, rather than by local stretching effect (Refs. 62-66).
64. A. Pumir and E. D. Siggia, "Collapsing solutions to the 3-D Euler equations," Phys. Fluids A 2, 220 (1990).
65. A. Pumir and E. D. Siggia, "Development of singular solutions to the axisymmetric Euler equations," Phys. Fluids A 4, 1472 (1992).
66. R. M. Kerr, "Evidence for a singularity of the three-dimensional incompressible Euler equations," Phys. Fluids A 5, 1725 (1993).
67. R. B. Pelz and Y. Gulak, "Evidence for a real time singularity in hydrodynamics from time series analysis," Phys. Rev. Lett. 79, 4998 (1997).
68. R. Grauer, C. Marliani, and K. Germaschewski, "Adaptive mesh refinement for singular solutions of the incompressible Euler equations," Phys. Rev. Lett. 80, 4177 (1998).
69. An earlier effort to construct a stochastic generalization of RE dealt with only local gradients and resulted in a very different type of model (Ref. 68).
70. S. Ghirimaghi and S. Pope, "A diffusion model for velocity gradients in turbulence," Phys. Fluids A 2, 242 (1990).
71. 2v(r)> "anomaly" contribution.
72. Note that the "tails" here refer to the intermediate asymptotics where λ is much larger than its root-mean-square (r.m.s.) value but smaller than the characteristic large scale velocity over r. The latter condition is imposed by the bound on velocity difference PDF imposed by the single point PDF (Ref. 71).
73. A. Noullez, G. Wallace, W. Lempert, R. B. Miles, and U. Frisch, "Transverse velocity increments in turbulent flow using the RELIEF technique," J. Fluid Mech. 339, 287 (1997).
74. E. D. Siggia, "Model of intermittency in three dimensional turbulence," Phys. Rev. A 17, 1166 (1978).
75. B. I. Shraiman and E. D. Siggia, "Symmetry and scaling of turbulent mixing," Phys. Rev. Lett. 77, 2463 (1996); "Anomalous scaling of passive scalar near Batchelor limit," Phys. Rev. E 57, 2965 (1998).
76. B. I. Shraiman and E. D. Siggia, "Symmetry and scaling of turbulent mixing," Phys. Rev. Lett. 77, 2463 (1996); "Anomalous scaling of passive scalar near Batchelor limit," Phys. Rev. E 57, 2965 (1998).
77. L. Mydlarsky, A. Pumir, B. I. Shraiman, E. D. Siggia, and Z. Warhaft, "Structure and multi-point correlators for turbulent advection," Phys. Rev. Lett. 81, 4373 (1998).
78. L. Mydlarsky and Z. Warhaft, "3-point statistics and anisotropy of a turbulent passive scallar,"Phys. Fluids 10, 2885 (1998).
79. J. Martin, A. Ooi, M. S. Chong, and J. Soria, "Dynamics of the velocity gradient tensor invariants in isotropic turbulence," Phys. Fluids 10, 2336 (1998).