2 The governing equations. 3 Statistical description of turbulence. 4 Turbulence modeling. 5 Turbulent wall bounded flows
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1 1 The turbulence fact : Definition, observations an universal features of turbulence 2 The governing equations PART VII Homogeneous Shear Flows 3 Statistical escription of turbulence 4 Turbulence moeling 5 Turbulent wall boune flows 6 Homogeneous Isotropic Turbulence 7 Homogeneous Shear Flows 8 Results base on the equations of the ynamics in fully evelope turbulence HST// thomas.gomez@univ-lille1.fr 335/374 HST// thomas.gomez@univ-lille1.fr 336/374 7 Homogeneous Shear Flows Definition an observations Slow an fast terms Simplification of the buget equations Rapi Distortion Theory Definition an observations Isotropy broken by the existence of non uniform mean velocity fiel ū. Homogeneity is preserve, Craya in 1958 : the mean velocity graient matrix A A(t) rū(t) (43) is uniform in space, but can eventually be a time-varying quantity. Incompressibility constrain applie to ū =) A ii =. The mean velocity fiel is then given by ū(x,t)= A(t)x + u(,t) (44) HST// thomas.gomez@univ-lille1.fr 337/374 HST/Definition an observations/ thomas.gomez@univ-lille1.fr 338/374
2 Definition an observations (cont ) Definition an observations (cont ) The mean velocity fiel is then given by ū(x,t)= A(t)x + u(,t) (45) Rmq : iverge when x! +1!Not physical! Inee no characteristic length-scales for ū, only a time-scale usually estimate as 1/kAk. The homogeneity constrain implies that all the terms of the buget equation, appearing as turbulent spatial fluxes, are zero. The main ifference with the isotropic case is that the prouction terms (of Reynols stresses, kinetic energy, turbulent scalar flux, scalar variance,...) are non zero. The fiel ū being anisotropic, the turbulence prouction will be also (at all the scales affecte by the prouction phenomena), an this anisotropy will propagate via the non-linear interactions (turbulent cascae). Ientify the effects irectly riven by the prouction an those riven by the non-linear mechanisms of the cascae. HST/Definition an observations/ thomas.gomez@univ-lille1.fr 339/374 HST/Definition an observations/ thomas.gomez@univ-lille1.fr 34/374 Definition an observations (cont ) All the matrices are not amissible to efine an mean velocity graient. Inserting the velocity expression from (45) in the mean momentum equation, we obtain the following compatibility conition for A t A + A2 is symmetric (46) Hereafter, we consier the constant pure shear S 1 A A (47) where S = ū 1 /x 2 is the shear rate. Slow an fast terms Non uniform mean fiel =) Fast terms, which explicitly inclue ū. Slow terms, which contain only the fluctuating fiel u. The fast terms are instantaneously moifie if ū (so here A) is perturbe. The slow terms are moifie with a elay, associatetothe characteristic timescale of the non linear mechanisms which propagate this perturbation. Momentum equation (in absence of external force) + A jkx k +A ij u j iu j)= j {z } D Dt u + (48) HST/Definition an observations/ thomas.gomez@univ-lille1.fr 341/374 HST/Slow an fast terms/ thomas.gomez@univ-lille1.fr 342/374
3 Slow an fast terms Momentum equation (in absence of external + A jkx k +A ij u j iu j)= j {z } D Dt u + (49) The only slow term base on the velocity is the non linear (u i u j ). All the fast terms are linear in u. Slow an fast Pressure terms Pressure terms given by Poisson equation p =2A m m (5) By linearity, the pressure p is ecompose into : a fast p f an a fast p s fiel p f =2A m, Then, eq. (49) can be rewritten as p s m +A jkx k +A ij u j (u iu s + (52) HST/Slow an fast terms/ thomas.gomez@univ-lille1.fr 343/374 HST/Slow an fast terms/ thomas.gomez@univ-lille1.fr 344/374 Slow an fast Pressure terms Slow an fast Pressure terms Time scales Characteristic Fast time scale base on ū : f S 1 Characteristic Slow time scale base on the kinetic energy K an the issipation " : s K/" Characteristic time scale ratio : shear rapiity Pure shear flows Momentum + Sx 2 + Su 2 i1 iu j)= j Fast pressure component : Poisson s + (54) p f 1 (55) s f SK " (53) HST/Slow an fast terms/ thomas.gomez@univ-lille1.fr 345/374 HST/Slow an fast terms/ thomas.gomez@univ-lille1.fr 346/374
4 Simplification of the buget equations Simplification of the buget equations Reynols stress R ij Reynols stress tensor R ij t R ij = with ij = p S ij, 2R 12 R 22 R 23 R 22 R 23 1 A + ij " ij (56) " ij k (57) Reynols stress R ij For an initially isotropic conition, one has 8 t R 11 = 2SR " 11 >< t R 22 = 22 " 22 t R 33 = 33 " 33 t R 12 = SR " 12 >: t K = SR 12 " (58) HST/Simplification of the buget equations/ thomas.gomez@univ-lille1.fr 347/374 HST/Simplification of the buget equations/ thomas.gomez@univ-lille1.fr 348/374 Energetic coupling Energetic coupling Prouction of kinetic energy Prouce by the coupling between the mean fiel an the cross correlation R 12. The prouction is anisotropic. Because governe by R 12 which is ientically zero in the isotropic case. It appears only in the R 11 buget equation. There is no pressure contribution in the buget of K(t) (via ij ). Inee, ii is ientically zero, because of the incompressibility conition on u. The pressure role only consists in reistributing the kinetic energy between the components of the Reynols tensor, without lost or creation. homogeneous shear case HST/Simplification of the buget equations/ thomas.gomez@univ-lille1.fr 349/374 HST/Simplification of the buget equations/ thomas.gomez@univ-lille1.fr 35/374
5 Measure of the anisotropy Experimental an numerical observations Anisotropy tensor b b ij =in the isotropic case Traceless tensor : b ii =. b ij = R ij 2K ij 3. (59) From isotropic state First stage : R 12,initiallyzero,isstillsmall.K ecreases. Secon stage : The anisotropy an the kinetic energy prouction strongly increase. At this moment, one observes that K start to increase. Final stage characterize by a universal asymptotic regime of exponential growth of the fluctuating kinetic energy. Invariant in time : SK ", SR 12 SK =2b 12 (6) " " Equation (58) =) K(t) =K()e St SK, = 2b b 12 = cste (61) " HST/Simplification of the buget equations/ thomas.gomez@univ-lille1.fr 351/374 HST/Simplification of the buget equations/ thomas.gomez@univ-lille1.fr 352/374 Sheare homogeneous turbulence Experimental results : Exponential growth Sheare homogeneous turbulence Experimental results Figure: Time evolution of K (at left) an p R 11 (at right) (extracte from Rohr et al. (1988)) Figure: Evolution of the spatial scales (L k :Kolmogorov sscale, :Taylor s scale, l :integralscale),thesolilineenotestheevolutionsintheisotropiccase. (extracte from Rohr et al. (1988)) HST/Simplification of the buget equations/ thomas.gomez@univ-lille1.fr 353/374 HST/Simplification of the buget equations/ thomas.gomez@univ-lille1.fr 354/374
6 Sheare homogeneous turbulence EDQNM results Sheare homogeneous turbulence EDQNM results.4 Transient regime Asymptotic regime.4 Transient regime Asymptotic regime bij(t) ansr(t).2.2 b 11 b 22 b 33 b 13 S R St bij(t) ansr(t).2.2 b 11 b 22 b 33 b 13 S R St Figure: Anisotropy tensor b ij(t) an shear rapiity S R(t) with S =1 1.(a) =2.(b) =4. HST/Simplification of the buget equations/ thomas.gomez@univ-lille1.fr 355/374 HST/Simplification of the buget equations/ thomas.gomez@univ-lille1.fr 356/374 Sheare homogeneous turbulence Experimental/Numerical results Table: Summary of global quantities obtaine in DNS an experiments for shear flows, classifie by ate. Authors Kin Date Re () S () S en b 13 (St) max Tavoularis et.al. Exp / Shirani et.al. DNS / 7 Tavoularis et.al. Exp / Tavoularis et.al. Exp / Lee et.al. DNS / 12 De Souza et.al. Exp / De Souza et.al. Exp / Ferchichi et.al. Exp / / / Schumacher DNS / / 1 Brethouwer DNS / Isaza et.al. DNS Isaza et.al. DNS Sukheswalla et.al. DNS Sukheswalla et.al. DNS Average Stanar eviation Rapi Distortion Theory Analysis of the fast physical mechanisms Non zero mean velocity fiel. Is the behavior observe are ue to rapi (linear in u )orslow (non-linear) effects? =) Strong importance for the evelopment of new turbulence moels. RDT : short time response analysis to the instantaneous variation of A. Justifie approach if the shear rapiity parameter SK/" f s Equations after elimination of slow + A jkx k + A ij u j = 1, f + (62) p f =2A m (63) HST/Simplification of the buget equations/ thomas.gomez@univ-lille1.fr 357/374 HST/RDT/ thomas.gomez@univ-lille1.fr 358/374
7 Rapi Distortion Theory Comparison between Experiment an RDT Table: Asymptotic behaviors for the case of shear homogeneous flows exp. measures RDT RDT without pressure K(St 1) / e St / St / (St) 2 b 11 (St 1), 23 2/3 2/3 b 22 (St 1) -,143-1/3-1/3 b 33 (St 1) -,6-1/3-1/3 b 12 (St 1) -,15 RDT : algebraic growth an not exponential... =) Exponential growth ue to non linear effects! Stabilizing effect of pressure : t 2! t. Ba preiction of anisotropy behavior at large time, as expecte... HST/RDT/ thomas.gomez@univ-lille1.fr 359/374
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