Publcaton 2006/01 Transport Equatons n Incompressble URANS and LES Lars Davdson Dvson of Flud Dynamcs Department of Appled Mechancs Chalmers Unversty of Technology Göteborg, Sweden, May 2006
Transport Equatons n Incompressble URANS and LES L. Davdson Dvson of Flud Dynamcs Dept. of Appled Mechancs Chalmers Unversty of Technology SE-412 96 Göteborg October 16, 2014 1 The Transport Equaton for the Reynolds Stresses The fltered Naver-Stoes equaton for ū reads ū t + (ū ū ) = 1 p +ν 2 ū τa g β t ρx τ = u u ū ū, τ a = τ 1 (1) 3 δ τ Whereτ denotesmodelledsgsstressoruransstress. TheSGS/URANS turbulent netc energy s defned as T = 0.5τ. Decompose ū and p nto a tme-averaged (or ensemble-averaged) value and a resolved fluctuaton as and nsert ths n Eq. 1 so that ū = U +ū, p = P + p, t = T + t U = ū, P = p, T = t ū t + ((U +ū x )(U +ū )) = 1 (P + p ) +ν 2 (U +ū ) ρ x Tme (ensemble) averagng of Eq. 3 yelds τa g β(t + t ) (U U ) = 1 P +ν 2 U ( ū ρx x ū +τa ) g βt (4) Now subtract Eq. 4 from Eq. 3 ū t + (U ū x +U ū +ū ū ) = 1 p +ν 2 ū + (5) ū ρx x ū +τa τ a g }{{} β t τ a 2 (2) (3)
Multply Eq. 5 wth ū and a correspondngequaton for ū by ū, add them together, and tme (ensemble) average ū (U ū x +U ū +ū ū ) + ū (U ū x +U ū +ū ū ) = ū ρ p x p ū ρ x ū τ a +ν ū ū τ a 2 ū +ν ū 2 ū g β ū t g β ū t The two frst lnes correspond to the usual u u equaton n conventonal Reynolds decomposton. The two last terms on lne 2 can be re-wrtten as ν ū ū +ν ū ū ū ū 2ν 2 ū = ν x ū ū 2ν ū (6) (7) The two frst terms on the last lne n Eq. 6 can be rewrtten as (ū x τa + ) τ a ū (ū x τa + ) τ a ū (8) Fnally, we can now wrte the transport equaton for ū ū as (U ū x ū ) = ū ū U ū x ū U 1 ū p 1 ū p + ρ x ρ x ū x ū ū +ν 2 ū ū x ū ū 2ν g β ū x t g β ū t (ū x τa ) (ū x τa ) + τ a ū + τ a ū (9) where the two last lnes nclude all terms related to the SGS/URANS stresses. The thrd lne represents dffuson transport by SGS/URANS stresses and the fourth lne represents dsspaton by SGS/URANS stresses. For an eddyvscosty SGS/URANS model τ a = 2ν T s, s = 1 ( ū + ū ) (10) 2 x x 3
1.1 Resolved Turbulent Knetc Energy Now we wll derve the transport equaton for the resolved turbulent netc energy = ū ū /2. Tae the trace of Eq. 9 and dvde by two (U ) = ū x ū U x x ν ū x ū x ( 1 ū ρ p + 1 ) 2 ū ū ū g β ū t (ū x τ ) a + +ν 2 x x τ a ū x (11) The pressure-velocty term was re-wrtten as ū p x = ū x p p ū x (12) where the last term s zero due to contnuty. Thelast termneq.11canbebothpostveandnegatve. However, fwe ntroduce an eddy-vscosty model t can be shown that t s predomnantly negatve. If the approxmaton (usng Eq. 10) τ a = τa τa = 2(ν T s ν T s ) 2ν T s (13) s made we fnd that the term s always negatve. Ths s easly seen when nsertng Eq. 13 nto the last term of Eq. 11 τ a ū 2 ν T s x (s +ω ) = 2 ν T s s < 0 (14) where ω = 0.5(ū /dx ū /dx ). In Eq. 14 we have used the fact that the product of a symmetrc and ant-symmetrc tensor s zero. The terms n Eq. 11 have the followng physcal meanng. The term on the left-hand sde s the advecton. The terms on the rght-hand sde are producton of, transport of by resolved fluctuatons, vscous transport of, vscous dsspaton of, producton/destructon of by buoyancy, transport of by SGS/URANS turbulence and producton/destructon of by SGS/URANS turbulence. 1.2 Modelled Turbulent Knetc Energy T The equaton for the modelled turbulent SGS/RANS netc energy reads T t + (ū T ) = [ (ν +ν T ) ] T +2ν T s s ε (15) x x x The terms on the rght-hand sde represent vscous and turbulent dffuson, producton and vscous dsspaton. 4
1.3 Mean Knetc Energy K The equaton for the netc energy K = 1 2 U U s derved by multplyng the tme-averaged (ensemble-averaged) momentum equaton, Eq. 4, by U so that U (U U ) = 1 x ρ U P 2 U ( +νu U τ a x x x dx +ū ū ) U g βt The left-hand sde of Eq. 16 can be rewrtten as (16) U (U U U )U U = U (U U ) 1 x x x 2 U (U U ) x = 1 2 U (U U ) = (U K) x x (17) The frst term on the rght-hand sde of Eq. 16 can be wrtten as U P x = x (U P). (18) The vscous term n Eq. 16 s rewrtten n the same way as the vscous term n Eq. 7,.e. νu 2 U x x = ν 2 K x x ν U x U x (19) The turbulent term s rewrtten as U dx ( τ a +ū ū ) = dx [ U ( τ a +ū ū )] ( τ a +ū ū ) U dx (20) Now we can assemble the transport equaton for K by nsertng Eqs. 17, 18, 19 and Eq. 20 nto Eq. 16 (U K) = ν 2 K (U P) [ ( U τ a x x x x dx +ū ū )] + ( τ a +ū ū ) U dx ν U x U x g βu T (21) We recognze the usual transport term on the left-hand sde due to advecton. On the rght-hand sde we have vscous dffuson, transport of K by tme-averaged (ensemble-averaged) pressure-velocty nteracton. The term n square bracets represents transport by nteracton between the tme-averaged (ensemble-averaged) velocty feld and turbulence. The term ū ū U /dx s the usual producton term of the resolved netc energy 0.5ū ū whch usually s negatve. Ths term appears n Eq. 11 but wth opposte sgn. The term τ au /dx s the producton term n the turbulent netc energy equaton T = 0.5τ. Ths term s usually referred to as 5
the SGS/URANS dsspaton term, and for an eddy-vscosty model we fnd (cf. Eqs. 13 and 14) τ a U dx = 2ν T s (S +Ω ) 2ν T s S = 2ν T S S < 0 (22) It s nterestng to compare ths SGS dsspaton term wth the vscous dsspaton term n Eq. 20. If ν sgs ν, the SGS dsspaton s much larger than the vscous one. If ths s not the case, then we re dong a DNS! T K 2ν T S S u u U /dx εt 2 ν T s s ν(u /x )(U /x ) ν(u /x )(u /x ) T Fgure 1: Transfer of netc turbulent energy. Often the energy s transferred n both drectons. A double arrow ndcated n whch drecton the net energy s transferred. 2 The Transport Equaton for the Heat Fluxes The fltered temperature equaton for t reads Use Eq. 2 n Eq. 23 so that t t + (ū t) = ν 2 t h h = u tū t (23) t (T + t )+ ((U +ū x )(T + t )) = ν 2 (T + t ) h (24) Tme (ensemble) averagng of Eq. 24 yelds (U T) = ν 2 T ( ū t +h ) (25) 6
Now subtract Eq. 25 from Eq. 24 t t + (ū x T +U t +ū t ) = ν 2 t + ū x t +h h }{{} h (26) Multply Eq. 26 wth ū and multply Eq. 5 wth t, add them together and tme (ensemble) average ū (ū x T +U t +ū t )+ t (U ū x +U ū +ū ū ) t p = + ν ū 2 t +ν t 2 ū g β t t (27) ρ x ū h t τa The two frst lnes correspond to the conventonal heat flux equaton. The two terms n the mddle on lne 2 can be re-wrtten as ν ū t ν ū t +ν t ū ū ν t = ν ū t +ν ( t ū ν + ν ) ū t (28) Usng Eq. 28 n Eq. 27 and at the same tme re-wrtng the SGS/URANS terms we get + ν ū t = ū ū T +ν ū h ū x t U t ū t ρ ( ν + ν + h ū p x ) ū t t τ a U ū t ū ū t g β t 2 + τ a t (29) The SGS/URANS heat fluxes are commonly obtan from an eddy-vscosty model h = ν T t (30) T x 7
3 The Transport Equaton for the Temperature Varance Multply Eq. 26 wth t and tme (ensemble) average t (ū x T +U t +ū t ) = ν t 2 t t h (31) The frst term on the rght-hand sde can be re-wrtten as ν ( t t ) ν t t = 1 ν 2 2 t 2 ν t t (32) Usng Eq. 32 and re-wrtng the SGS/URANS term, Eq. 31 can now be wrtten as + 1 ν 2 2 t 2 ν Multply Eq. 31 by 2 and we get + ν 2 t 2 2 ν 1 U t 2 = ū 2 t T 1 ū 2 t 2 t t t h + h t (33) U t 2 = 2 ū x t T ū t 2 t t 2 t h +2 h t (34) 8