Power laws and vortical structures



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POWER LAWS and VORTICAL STRUCTURES

    • P. Orlandi
    • G.F. Carnevale, S. Pirozzoli
    • Universita' di Roma “La Sapienza” Italy

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 Kn

  • Turbulence  ≠ 0

  • high K exp(-K)‏

  • Kolmogorov n=-5/3

  • Analysed structures with strong

  • Worms or tubular structures



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)‏



COMPARISON

  • Comparison viscous inviscid

  • Difference in n related to structures

  • Filtering the fields

  • Possibility to isolate structures

  • Selfsimilarity in the range Kn

  • Shape of structures related to n



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



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



RESULTS time reversibility



RESULTS

  • Grafke et al. 2007 Interacting dipoles



FORC ISOTROPIC DISS.



FORC Inertial Gotoh



FORC Inertial Jimenez



INVISCID SOLUTION



SOLID PROOF



LAMB dipoles I.C.



LAMB DIPOLES



LAMB spectra LD1



Compensated SPECTRA LD1



Lamb Evolution t=1



Vorticity amplification



INITIAL CONDITIONS



SPECTRA near FTS



VORTICITY near FTS



Component along S_2



Vorticity amplification



Enstrophy prod. amplification



Taylor-Green Spectra CB



Taylor-Green Spectra Or



Taylor-Green max



T-G Compensated Spectra



T-G Compensated Spectra



T-G Spectra



T-G Enstrophy Prod.



Spectra of the fields



Vortical structures Lamb



Vortical structures T-G



Filtering



Filtering Lamb (max)=50



Filtering Lamb (max)=410



Filtering Lamb (max)=240



Filtering Lamb (max)=225



Lamb self-similarity



Filtering T-G (max)=4.2



Filtering T-G (max)=13.8



Filtering T-G (max)=20.7



Filtering T-G (max)=17.6



Filtering T-G (max)=12.5



T-G selfsimilarity



Pdf 



Lamb Pdf 



T-G Pdf 



FORCED ISOTROPIC



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



ONE LAMB viscous and inviscid



ENSTROPHY



Spectra before FTS



Spectra after FTS



LAMB COUPLES Re=3000



Three LAMB viscous and inviscid



SPECTRA Enstr. amplification



SPECTRA Enstr. max



SPECTRA Enstr. decay



ENSTROPHY Eq.



ENSTROPHY balance



ENSTROPHY production



Enstrophy prod. Princ. axes



Rate enstrophy prod.



Jpdf Enstr. Prod. ; Rs amplification



Jpdf Enstr. Prod. ; Rs maximum



Jpdf Enstr. Prod. ; Rs decay



STRUCTURES

  • Eduction of tubes

  • Swirling strength criterium



Lamb weak interaction



Lamb strong interaction



Lamb max enstrophy



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



Lamb vs Isotropic

  • Energy and enstrophy



Lamb vs Isotropic

  • Spectra



Lamb vs Isotropic

  • Velocity derivatives skewness



Lamb vs Isotropic

  • Velocity derivatives flatness



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)‏



Vortical structures Forc Turb



Filtering Isot. Turb. (max)=64



Filtering Isot. Turb. (max)=106



Filtering Isot. Turb. (max)=114



Filtering Isot. Turb. (max)=144



Iso. Turb. Pdf 




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