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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