1. POWER LAWS and VORTICAL STRUCTURES P. Orlandi G.F. Carnevale, S. Pirozzoli Universita' di Roma “La Sapienza” Italy

2. POWER LAWS Non-linear terms create wide E(k)‏ Triadic interaction in K space Vortical structures in physical space At high Re and at low K K n Turbulence ≠ 0 high K exp(-  K)‏ Kolmogorov n=-5/3 Analysed structures with strong  Worms or tubular structures

3. INVISCID FLOWS Lack of dissipation Possibility of a FTS E(k) varies in time Before singularity n=-3 Initial conditions important Interacting Lamb dipoles n=-6 Taylor-Green t=0 E(1)‏

4. COMPARISON Comparison viscous inviscid Difference in n related to structures Filtering the fields Possibility to isolate structures Selfsimilarity in the range K n Shape of structures related to n

5. NUMERICAL TOOLS 2° order accuracy more than sufficient Stable Physical principle reproduced in discrete Mass conservation Energy conservation inviscid Finite difference simple Reproduce all the requirements IMPORTANT to resolve the flow NOT the accuracy

6. time reversibility Duponcheel et al. 2008 Taylor-Green Forward up to t=10 V(t,X)=-V(t,X)‏ From t=10 to t=20 equivalent To backward At t=20 V(20,X)=V(0,x)‏ Comparison R-K-low storage FD2 with FD4 and Pseudospectral

7. RESULTS time reversibility

8. RESULTS Grafke et al. 2007 Interacting dipoles

9. FORC ISOTROPIC DISS.

10. FORC Inertial Gotoh

11. FORC Inertial Jimenez

12. INVISCID SOLUTION Question? Has Euler a Finite Time Singularity Does it depend on the Init. Condit. Several simulations in the past Pumir, Peltz, Kerr, Brachet Interest on the Euler equations Mostly by Pseudospectral Init. condition Ortogonal Dipoles Taylor-Green Kida-Peltz

13. SOLID PROOF Infinite space-time resolution near singularity From well resolved simulations indications E.G. derive one model equation having FTS dx/dt= x 2

14. LAMB dipoles I.C. Self preserving vortex Traslating with U Solution 2D Euler

15. LAMB DIPOLES

16. LAMB spectra LD1

17. Compensated SPECTRA LD1

18. Lamb Evolution t=1

19. Vorticity amplification

20. INITIAL CONDITIONS

21. SPECTRA near FTS

22. VORTICITY near FTS

23. Component along S_2 Strain in the principal axes Simulation shows that S 2 prop ω 2

24. Vorticity amplification

25. Enstrophy prod. amplification

26. Taylor-Green Spectra CB

27. Taylor-Green Spectra Or Spectra during evolution do not have a power law

28. Taylor-Green  max

29. T-G Compensated Spectra

31. T-G Spectra

32. T-G Enstrophy Prod.

33. Spectra of the fields

34. Vortical structures Lamb

35. Vortical structures T-G

36. Filtering To investigate the strutures in a range of K From physical to wave number Set u(K)=0 for Kl < K < Ku Back from wave to physical

37. Filtering Lamb  (max)=50

38. Filtering Lamb  (max)=410

39. Filtering Lamb  (max)=240

40. Filtering Lamb  (max)=225

41. Lamb self-similarity

42. Filtering T-G  (max)=4.2

43. Filtering T-G  (max)=13.8

44. Filtering T-G  (max)=20.7

45. Filtering T-G  (max)=17.6

46. Filtering T-G  (max)=12.5

47. T-G selfsimilarity

48. Pdf 

49. Lamb Pdf 

50. T-G Pdf 

51. FORCED ISOTROPIC –Kolmogorov with n=-5/3 –Why?

52. DNS with SMOOTH I.C Comprehension non linear terms - Inviscid leads to FTS (personal view)‏ - I would like to know which is a convincing proof Well resolved leads to n=-3 - Viscous lead to n=-5/3 No FTS for N-S (personal view) Different equations Small ν leads to exp range in E(k)‏ R‏esolution important

53. ONE LAMB viscous and inviscid

54. ENSTROPHY

55. Spectra before FTS

56. Spectra after FTS

57. LAMB COUPLES Re=3000

58. Three LAMB viscous and inviscid

59. SPECTRA Enstr. amplification

60. SPECTRA Enstr. max

61. SPECTRA Enstr. decay

62. ENSTROPHY Eq.

63. ENSTROPHY balance

64. ENSTROPHY production

65. Enstrophy prod. Princ. axes

66. Rate enstrophy prod.

67. Jpdf Enstr. Prod. ; Rs amplification

68. Jpdf Enstr. Prod. ; Rs maximum

69. Jpdf Enstr. Prod. ; Rs decay

70. STRUCTURES Eduction of tubes Swirling strength criterium Eduction of sheets Largest eigenvalues of Red sheets, yellow tubes

71. Lamb weak interaction

72. Lamb strong interaction

73. Lamb max enstrophy

74. Kolmogorov range formation Before t* vortex sheets and tubes Amplification stage sheets formations At t* intense curved sheets After t* tubes form from sheet roll-up Tubes interact with sheets Sheets more compact K- 5/3 Bottleneck forms At large times K -3/2 –

75. Lamb vs Isotropic Energy and enstrophy

76. Lamb vs Isotropic Spectra

77. Lamb vs Isotropic Velocity derivatives skewness

78. Lamb vs Isotropic Velocity derivatives flatness

79. Conclusions EULER have a FTS Navier-Stokes do not have FTS View of engineers from DNS Of different smooth I.C. Lamb dipole a good I.C. Shape preserving Spectra evolve maintaining power law Interaction with matematician necessary To find the relevant proofs Necessity of large CPU (common effort)‏

80. Vortical structures Forc Turb

81. Filtering Isot. Turb.  (max) =64

82. Filtering Isot. Turb.  (max) =106

83. Filtering Isot. Turb.  (max) =114

84. Filtering Isot. Turb.  (max) =144

85. Iso. Turb. Pdf 