/2 THE COUPLED LES - SUBGRID STOCHASTIC ACCELERATION MODEL LES-SSAM OF A HIGH REYNOLDS NUMBER FLOWS Vladmr Sabel nov DEFA/EFCA ONERA, France In collaboraton wth: Anna Chtab CORIA, Unversté de Rouen, France and Mhael Gorohovs LMFA, Ecole Centrale de Lyon, France Worshop and Mncourse Conceptual Aspects of Turbulence: Mean Felds vs. Fluctuatons, Wolfgang Paul Insttute, Venna, -5 February 28
2/2 OBJECTIVE OF STUDY Tae nto account n LES computaton of a hgh Re flow ntermttency Re number dependent at sub-grd nonresolved scales Concern: mpact of ntermttency on combuston n engnes - spontaneous extncton and gnton - evaporaton, coagulaton and burstng of lqud drops, etc
3/2 CONTENT Impetus Resdual acceleraton: order estmaton New approach. Prncpal ponts Formulaton of LES-SSAM - startng pont - assumptons - stochastc model of resdual acceleraton Valdaton tests - statonary sotropc turbulence - decayng turbulence Conclusons
IMPETUS: observatons of a hgh Reynolds number statonary homogeneous turbulence Mordant et al., 2; La Porta et al., 2; Voth et al.,998, 22: washng machne by Mordant et al. PRL, 245 2 Re = 74 u m / s =4ìm = p eau d p,2ms =,6 = 25ìm 4/2 measurements of Lagrangan acceleraton revealed a strong ntermttency at small length /tme scales non-gaussan shape wth stretched tals n dstrbuton wth ncreasng of tme lag, the Lagrangan acceleraton was rapdly decorrelated PDF of acceleraton one component however, autocorrelaton of ts norm exposed a long memory of order of few ntegral tmes for hgher turbulent Reynolds number, the ntermttency was stronger manfested
5/2 RESIDUAL ACCELERATION: ORDER ESTIMATION Δ - the flter wdth L - the ntegral turbulent scale, η - Kolmogorov scale, σ u - rms velocty u# # $ 2 u u a$ $ a / aa # t follows: / L 2/ 3 2/3 /, consequently: when ntermttency s target n SGS model, the SGS model should comprse the non-resolved acceleraton dependent of Reynolds number = L Re 3 L/ 4 L ; Re = u L L a a / a a >>, f Re >>
6/2 NEW APPROACH PRINCIPAL POINTS focusng on acceleraton of flud partcle reconstructon of non-resolved velocty by nstantaneous model-equaton, n whch the acceleraton comprses two parts: the fltered total acceleraton correspondng to classcal LES the sub-grd non-resolved resdual acceleraton of flud partcle the non-resolved acceleraton s modelled stochastcally n a way that ntermttency at sub-grd scales s taen nto account REMARK: Our LES-SSAM loos smlar to A LES-Langevn model for turbulence, Eur. Phys. J. B 49, 47-48 26, J.-P. Laval and Dubrulle. But the physcs and the way of smulatng the non-resolved scales are qute dfferent.
7/2. Startng step FORMULATION OF LES-SSAM /3 Naver-Stoes equatons: d u P a = = # + $ d t x u, u x = By flterng the Naver-Stoes equatons, we have two sets: d u ' & $ % P a = ' + d t $ = # * x u, u x = ; a - s the resolved total acceleraton * d u ' # P u a = = $ + u, d t % + = ; & # x x a - s the non-resolved acceleraton a = a + a - corresponds to orgnal Naver-Stoes equaton
8/2 FORMULATION OF LES-SSAM 2/3 2. Assumptons the orgnal equaton for resdual acceleraton s replaced by a model equaton: from presumed stochastc process guarantee the velocty vector to be solenodal the orgnal fltered equaton s descrbed by classcal LES wth the Smagornsy model unfltered hypothetcal velocty s represented by a model equaton: a x P d t d u ˆ ˆ / + = # $ $ % & ' ' ˆ = x u, t u x P d t u d a + + = # = $ % % &, x u t d t d + =, ˆ ˆ ˆ ˆ ˆ ˆ tur a x u x u x x P d t d u a + # $ % % & ' + + = = * +, u x P d t d u a + # # = $ % % & ' * = +
9/2 FORMULATION OF LES-SSAM 3/3 3. Stochastc model of resdual acceleraton Kolmogorov Obouhov, 962: 3/ 2 / 2 a a # = const # / j j Pope-Chen s 99 log-normal stochastc process for Stochastc equaton of the norm of acceleraton ' aˆ 3 $ 2 3 2 daˆ = aˆ % ln * T dt + aˆ 2 T a 6 * & # 4 T # = tur 2 ; Re = t ; 3/ 4 a = # ; / 4 2 # A = + µ d W t ; ln Re 3/ 4 ; a = ˆ a d W by Ito transformaton: t = ; 2 t d t d W = A -.863, µ.25 ; ; values of constants accordng A.G. Lamorgese et al. JFM 27 components of acceleraton are computed by ntroducng the unt vector e t wth random drecton n tme ths hypotheses can be modfed: t aˆ t e t aˆ =
/2 VALIDATION TESTS. Statonary forced homogeneous sotropc turbulence 2. Decayng homogeneous sotropc turbulence Parameters: Integraton tme step : n both cases s about of the Kolmogorov tme scale grd resoluton: statonary turbulence 32 3, decayng turbulence - 64 3 Smagornsy constant was adjusted to obtan a correct energy dsspaton rate and energy decay 2 part of energy dsspaton s due to stochastc acceleraton term n current verson of SSAM
/2 STATIONARY ISOTROPIC TURBULENCE /6 b Evoluton of the turbulent energy averagng over a computatonal doman 53 5 2 2 turb / 2 54 turb cm /s a Evoluton of the Reynolds number averagng over a computatonal doman 52 LES-SSAM 5 5 5.5 t s.5 Standard LES LES-SSAM: reproduces spotty ntermttent turbulent felds.5 t s.5
2/2 = ;.5; STATIONARY ISOTROPIC TURBULENCE 2/6 PDF of the Lagrangan velocty ncrement at dfferent tme lag.3;.6;.2; 2.5; 4.9; 9.8; 2 et 39 ms ntal condtons from Mordant et al. PRL, 245, 2 - LES-SSAM Experment, ENS/Lyon standard LES - -2 log PDF -3-4 32 3-2 log PDF -3-4 -5-5 -6-6 -7-2 - 2 U/< U 2 > /2-7 -2-2 U/< U 2 > /2 LES-SSAM: gaussan dstrbuton at tme lags of order of ntegral tme scale 45 ms strong non-gaussanty wth stretched tals at small tme lags
3/2 STATIONARY ISOTROPIC TURBULENCE 3/6 PDF of the Lagrangan acceleraton ntal condtons from Mordant et al. PRL, 245, 2-4 LES-SSAM Experment, ENS/Lyon Standard LES -4-5 -6 log PDF -7-8 -9 - -5 log PDF -6-7 -8 - -4-2 a cm/s 2 2 4 5 a cm/s 2-5 - LES-SSAM: the small ampltude acceleraton events are alternatng wth events of very large acceleraton
4/2.8.6 R L a.4.2 STATIONARY ISOTROPIC TURBULENCE 4/6 Auto-correlaton of acceleraton ntal condtons from Mordant et al. PRL, 245, 2..2.3.4.5.6 LES-SSAM: s R a R a R = rapd decorrelaton of acceleraton vector component long memory of acceleraton norm R and of ts norm qualtatve reproducton of effects observed n experment a.8.6 R L a, R L a.4.2 R a R L a, R L a R a.8.6.4.2 /T L R a -.2 2 / -.2 2 4 6 8
5/2 STATIONARY ISOTROPIC TURBULENCE 5/6 R Cross-correlaton and auto-correlaton of acceleraton ntal condtons from Mordant et al. PRL, 245, 2 a a j LES-SSAM Experment, ENS/Lyon R a.8.8.6.6 R L a.4.2 R R a a a j R L a.4.2 R a a j -.2 2 / R a R a a j -.2 2 3 /T L LES-SSAM: mmedate decorrelaton between cross-components of acceleraton vector qualtatve agreement wth experment
6/2 STATIONARY ISOTROPIC TURBULENCE 6/6 R Cross-correlaton and auto-correlaton R a of modulus of acceleraton a a j ntal condtons from Mordant et al. PRL, 245, 2 LES-SSAM Experment, ENS/Lyon.8.8.6 R L a.4.6 R L a.4 R a a j.2 2 / R a a j.2 R a R a 2 4 6 8 /T L LES-SSAM: long-tme correlaton of modulus of cross-components of acceleraton qualtatve agreement wth experment
7/2 DECAYING TURBULENCE 3 2 E, cm 3 s -2 -, cm - From the measured spectra at.2s upper lne, the ntal gaussan velocty feld s computed and then the energy spectra are compared at.284s and.655s LES-SSAM: flled symbols standard LES: empty symbols Comte-Bellot & Corrsn: contnuous lnes 5/3 law: dscontnuous lnes
8/2 CONCLUSIONS Novel coupled LES Subgrd Stochastc Acceleraton Model LES-SSAM s constructed LES-SSAM targetng the small scale turbulence structure Resdual acceleraton s as a ey varable of the SGS model. It s modelled by stochastc process The effects of ntermttency, observed recently by measurements of the Lagrangan acceleraton n a hgh Reynolds number turbulence, are reproduced by the model proposed
9/2 WORK IN PROGRESS Elaboraton of SSAM : - to control the turbulence energy balance - drect modellng spatal non-resolved acceleraton correlatons Adaptaton of SSAM for smulaton of ntermttency only n nertal range to perform calculaton wth larger tme step and low computatonal cost
2/2 Than you Questons?