Energy Cascade in Turbulent Flows: Quantifying Effects of Reynolds Number and Local and Nonlocal Interactions

Transcription

1 Energy Cascade in Turbulent Flows: Quantifying Effects of Reynolds Number and Local and Nonlocal Interactions J.A. Domaradzi University of Southern California D. Carati and B. Teaca Universite Libre Bruxelles Supported by ULB Council of International Relations

2 Classical result -5/3 Kolmogoroff 1941 theory: assumption of local energy cascade & dimensional analysis Local energy flux implies no influence of the energy containing range and no influence of the dissipation range on the inertial range eddies E() depends only on and the energy transfer rate Π = ε E ( ) = CKε 2/3 5/3

3 Incompressible turbulence inetic energy vorticity

4 Energy and transfer spectra forcing 10 1 Energy Spectrum E Transfer Spectrum T

5 Basic Quantities 2 1 ui ν u 2 i = p u j ui t xj ρ xi xj Spectral description 1 u() = d x u(x) e 3 (2 π ) Energy spectrum 2 S ( ) 3 ix 1 E ( ) = u() 2 Nonlinear term 3 u i(p) u j(-p) d p

6 Triad structure of nonlinear interactions Et (, ) t ν Et (, ) = Tt (, ) T ( ) u ( ) upu ( ) ( pdp ) Triad decomposition of T() T ( ) = T ( pq, ) = p q= p p P ( p) allows to assess role of all interacting scales on energy transfer p -p

7 Function T( p,q) 0.5 x 10 3 Symmetrized Transfer Spectra for interacting bands (16) 0 T( p,q) Local energy transfer through nonlocal interactions

8 Questions and Controversies T( p,q) consistent with local transfer and nonlocal interactions because of strong dependence on the energy containing range (JAD & Rogallo, Yeung & Brasseur, Ohitani & Kida, Alexais, Mininni & Pouquet) Is this an artefact of using bands defined by sharp spectral filters (Waleffe, Eyin)? Will smooth bands change the picture? No! (JAD & Carati) Does T( p,q) have a physical interpretation or should only its integrals be used to assess scale dependence of nonlinear interactions? (Zhou & Rogallo, Zhou, Eyin)? Energy flux: 0 0 p+ q= ' Π ( ) = T ( ') d ' T ( ' p, q) dpdqd ' SGS transfer: T ( ) = T( ) T( ), < SGS c c c

9 Interpretation of observed behavior of transfer function T( p,q) Banded energy spectra (tanh filter) E

10 Interpretation: energy production and redistribution T ( pq, ) ( ui ) u + ( ui ) u p q q p T α,α u Interactions between bands 3 and u x 1 3 u U u u u 3 1 u x, ' redistribution production 0.1 band 3 advective band 1 advective

11 Analyzes of integrals of T( p,q) 512^3 DNS data; 13 bands (sharp and smooth). Analysis for c=32 and 90. Banded energy spectra (tanh filter) 7 E

12 c 0 Energy flux: p+ q= T ( p, q) ddpdq T( p,q) summed over p and q such that ½< q/p <2 gives local component. T( p,q) summed over p and q such that ½>q/p >2 gives nonlocal component local/nonlocal decomposition of flux across 7 band Energy range 1 local nonlocal interacting band ir1 + total c Total mostly local Energy containing range always provides a large nonlocal contribution that is canceled by other interactions leaving the total flux dominated by local interactions local/nonlocal decomposition of flux across 10 band Observed cancellations consistent with theoretical predictions of Waleffe (1991) and Eyin (2005) and numerical results of Zhou (1993) local nonlocal interacting band ir1 + total c

13 Subgrid-scale (SGS) energy flux Et (, ) t + ν = 2 2 Et (, ) Tt (, ) 10 1 < E ( c ) Energy Spectrum E 10 5 E < ( c) + t c = + ( ) ( ) 2 < < < 2 ν E( c) T( c) T T c At least one interacting scale has >c T SGS ( ) c

14 0.1 local/nonlocal decomposition of SGS flux across band 7 0 SGS energy transfer excludes self-interactions of scales below c, accounting for cancellation effects in the energy flux nonlocal contributions are of the same sign local nonlocal interacting band ir1 + total c total local/ nonlocal 0.4 local/nonlocal decomposition of SGS flux across band Asymptotic locality despite total dominated by nonlocal interactions total nonlocal 0.8 local c nonlocal interacting band ir1 + total

15 Kraichnan infrared and ultraviolet locality functions Π ( c ) Π ( c ) Π ( ) c =ir Infrared part of total flux due to scales <ir c Ultraviolet part of total flux due to scales >uv =uv Classical results for the inertial range spectrum (Kraichnan, Eyin); 4/3 slope observed in LES (Zhou)). Π( c) c c 4/3 c Π( c ) c 4/3

16 10 1 Kraichnan locality function (ultraviolet) Π( cut ) cut / Banded energy spectra (tanh filter) E

17 10 1 Kraichnan locality function (infrared) Π( cut ) / cut Asymptotic infrared nonlocality stronger than the theoretical prediction: scales less than ½ c contribute 50-70% of the total flux, i.e., such nonlocal interactions dominate reconciles numerical conclusion using the locality factor /c=2 with the theoretical prediction. 1/2

18 10 1 Kraichnan locality function (ultraviolet) 0.4 Π( cut ) 0.01 cut / 1/2 Asymptotic ultraviolet nonlocality weaer than the theoretical prediction: scales greater than 2 c contribute 1-40% of the total flux, i.e., the local interactions with <2c dominate.

19 Reynolds number dependence forcing Energy Spectra E ^(-5/3) constraint Re = 10 5 Re=

20 10 1 Kraichnan locality function (infrared) DNS Π( cut ) Kraichnan locality 10function 1 (infrared) 10 1 / cut Constrained inertial range simulation Π( cut ) / cut

21 10 1 Kraichnan locality function (ultraviolet) DNS Π( cut ) Π( cut ) Kraichnan locality 10function 1 (ultraviolet) 10 1 cut / Constrained inertial range simulation / cut

22 10 1 Kraichnan locality function (ultraviolet) DNS 0.4 Π( cut ) 0.02 / cut 1/2 Energy Spectra E

23 Π( cut ) 10 1 Kraichnan locality function (ultraviolet) Constrained inertial range simulation cut / Energy Spectra E

24 10 1 Kraichnan locality function (infrared) Energy Spectra Π( cut ) / cut E 10 1 Kraichnan locality function (ultraviolet) Π( cut ) cut /

25 Truncated N-S dynamics (spectral space) E() Large physical scales (on coarse mesh): computed by N-S eqns. Estimated scales (on fine mesh): Artificial energy accumulation due to absence of (natural or eddy) viscosity. Unresolved scales c 2 c Filter small-scales at fixed interval and replenish using estimation model TNS=Sequence of DNS runs with periodic processing of high modes

26 Decaying High Reynolds Number Turbulence Energy spectrum of TNS with VEP compared with C-L and original TNS model Energy decay: TNS with VEP compared with C-L and original TNS model

27 High Reynolds number LES

28 Conclusions Integrated quantities obtained from T( p,q) (energy transfer, energy flux, SGS energy transfer) exhibit asymptotic infrared and ultraviolet locality In DNS data infrared nonlocality for /c<1/2 is stronger than the theoretical prediction, i.e., the numerical results can be interpreted as being infrared nonlocal for the locality parameter s=2 Ultraviolet nonlocality for /c>2 is weaer than the theoretical prediction, i.e., the numerical results can be interpreted as strongly ultraviolet local for the locality parameter s=2 Numerical data constrained by the inertial range spectrum show clear trend toward the theoretical ^(4/3) result

29 Conclusions The locality results imply that a range of subgrid scales adjacent to the resolved range dominates dynamics of the resolved eddies These subgrid scales can be estimated in terms of the resolved scales (estimation model) Dynamics of the resolved eddies is approximated by Truncated Navier-Stoes equations for resolved and estimated scales The method consists of a sequence of underresolved DNS and a periodic processing of the solution Approach successful in LES of high Reynolds number turbulence of different flows