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 K n 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 K n 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 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 Interacting dipoles
FORC ISOTROPIC DISS.
FORC Inertial Gotoh
FORC Inertial Jimenez
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
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
LAMB dipoles I.C. Self preserving vortex Traslating with U Solution 2D Euler
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 Strain in the principal axes Simulation shows that S 2 prop ω 2
Vorticity amplification
Enstrophy prod. amplification
Taylor-Green Spectra CB
Taylor-Green Spectra Or Spectra during evolution do not have a power law
Taylor-Green max
T-G Compensated Spectra
T-G Spectra
T-G Enstrophy Prod.
Spectra of the fields
Vortical structures Lamb
Vortical structures T-G
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
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 –Kolmogorov with n=-5/3 –Why?
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 Eduction of sheets Largest eigenvalues of Red sheets, yellow tubes
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