Publication 2006/01. Transport Equations in Incompressible. Lars Davidson

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1 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

2 Transport Equatons n Incompressble URANS and LES L. Davdson Dvson of Flud Dynamcs Dept. of Appled Mechancs Chalmers Unversty of Technology SE Göteborg October 16, 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)

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

4 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

5 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

6 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

7 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

8 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