POWER LAWS and VORTICAL STRUCTURES

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POWER LAWSandVORTICAL STRUCTURES

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

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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

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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 t0 E(1)?

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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

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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

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time reversibility

• Duponcheel et al. 2008
• Taylor-Green
• Forward up to t10
• V(t,X)-V(t,X)?
• From t10 to t20 equivalent
• To backward
• At t20 V(20,X)V(0,x)?
• Comparison R-K-low storage
• FD2 with FD4 and Pseudospectral

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RESULTS time reversibility
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RESULTS

• Grafke et al. 2007 Interacting dipoles

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FORC ISOTROPIC DISS.
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FORC Inertial Gotoh
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FORC Inertial Jimenez
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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
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SOLID PROOF
Infinite space-time resolution near
singularity From well resolved simulations
indications E.G. derive one model equation
having FTS dx/dt x2
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LAMB dipoles I.C.
Self preserving vortex Traslating with U
Solution 2D Euler
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LAMB DIPOLES
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LAMB spectra LD1
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Compensated SPECTRA LD1
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Lamb Evolution t1
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Vorticity amplification
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INITIAL CONDITIONS
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SPECTRA near FTS
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VORTICITY near FTS
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Component along S_2
Strain in the principal axes Simulation shows
that S2 prop ?2
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Vorticity amplification
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Enstrophy prod. amplification
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Taylor-Green Spectra CB
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Taylor-Green Spectra Or
Spectra during evolution do not have a power law
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Taylor-Green ??max
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T-G Compensated Spectra
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T-G Compensated Spectra
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T-G Spectra
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T-G Enstrophy Prod.
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Spectra of the fields
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Vortical structures Lamb
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Vortical structures T-G
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Filtering
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Filtering Lamb ?(max)50
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Filtering Lamb ?(max)410
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Filtering Lamb ?(max)240
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Filtering Lamb ?(max)225
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Lamb self-similarity
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Filtering T-G ?(max)4.2
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Filtering T-G ?(max)13.8
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Filtering T-G ?(max)20.7
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Filtering T-G ?(max)17.6
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Filtering T-G ?(max)12.5
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T-G selfsimilarity
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Pdf ???

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Lamb Pdf ???
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T-G Pdf ???
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FORCED ISOTROPIC

• Kolmogorov with n-5/3

• Why?

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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 ? leads to exp range in E(k)?
• R?esolution important

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ONE LAMB viscous and inviscid
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ENSTROPHY
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Spectra before FTS
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Spectra after FTS
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LAMB COUPLES Re3000
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Three LAMB viscous and inviscid
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SPECTRA Enstr. amplification
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SPECTRA Enstr. max
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SPECTRA Enstr. decay
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ENSTROPHY Eq.
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ENSTROPHY balance
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ENSTROPHY production
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Enstrophy prod. Princ. axes
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Rate enstrophy prod.
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Jpdf Enstr. Prod. Rs amplification
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Jpdf Enstr. Prod. Rs maximum
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Jpdf Enstr. Prod. Rs decay
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STRUCTURES

• Eduction of tubes
• Swirling strength criterium

• Eduction of sheets
• Largest eigenvalues of

• Red sheets , yellow tubes

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Lamb weak interaction
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Lamb strong interaction
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Lamb max enstrophy
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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

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Lamb vs Isotropic

• Energy and enstrophy

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Lamb vs Isotropic

• Spectra

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Lamb vs Isotropic

• Velocity derivatives skewness

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Lamb vs Isotropic

• Velocity derivatives flatness

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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)?

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Vortical structures Forc Turb
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Filtering Isot. Turb. ?(max)64
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Filtering Isot. Turb. ?(max)106
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Filtering Isot. Turb. ?(max)114
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Filtering Isot. Turb. ?(max)144
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Iso. Turb. Pdf ???