《数值传热学》研究生课程教学资源（课件讲稿）Chapter 8 Numerical Simulation for Turbulent Flow and Heat Transfer

热流科学与工程西步文源大堂G教育部重点实验室8.2Time-averaged governing equation forincompressibleconvectiveheattransfer8.2.1Timeaveragegoverning equation1. Continuity equationouavauavawawa(u+u)o(v+v),a(w+w)=0OzayaxayaxaxOz0zdy=0=0Both time average velocity and time average fluctuationvelocity satisfy continuity condition2.Momentum equationTaking x-direction as an example:ΦCFD-NHT-EHT11/76CENTER

11/76 8.2 Time-averaged governing equation for incompressible convective heat transfer 1. Continuity equation Both time average velocity and time average fluctuation velocity satisfy continuity condition. Taking x-direction as an example： ' ' ' ( ) ( ) ( ) u u v v w w x y z             8.2.1 Time average governing equation 2. Momentum equation ' ' ' 0 u v w u v w x y z x y z                   ＝0 ＝0

热流科学与工程西步文源大堂G教育部重点实验室a(u+u)?1 a(p+ p)a(u+u)a(u+u)(v+v)o(u+u)(w+w)axazataxdyp(u+u), (u+u)a(u+u)OOAccording to the above characteristics, yieldingoua(uouvouwa(u)?a(uv).o(uw)ataxayOzaxOvozuauau1apMovedtorighthand sideax?Qy20z2axpandcombinedwiththe2correspondingviscousouvouwaua(utermataxayOzaau1apaauoua(u))(uv)]+(uw))1azaxaxayOzaypaxΦCFD-NHT-EHT12/76CENTER

12/76 ' ' 2 ' ' ' ' ' ( ) ( ) ( )( ) ( )( ) 1 ( ) u u u u u u v v u u w w p p t x y z x                         2 ' 2 ' 2 ' 2 2 2 ( ) ( ) ( ) [ ] u u u u u u x y z             According to the above characteristics, yielding 2 ' 2 ' ' ' ' u u uv uw u u v u w ( ) ( ) ( ) ( ) t x y z x y z                      222 2 2 2 1 ( ) p u u u x x y z                Moved to right hand side and combined with the corresponding viscous term 2 u u uv uw ( ) t x y z             1 ' 2 ' ' ' ' [ ( ) ] [ ( )] [ ( )] p u u u u u v u w x x x y y z z                         

热流科学与工程西步文源大堂G教育部重点实验室RewritteninatensorforminCartesiancoordinate:aapQui(pu) , a(puiuj)puu)(i=1,2,3)nataxaxOxOx;3.Other scalar (标量)variablesaaa(pd), a(pujd))+ SouOxatax,ax4.Discussion on the time averaged quantity(1) Linear term remains unchanged during time averagewhileproductterm（乘积项）generatesproductoftwofluctuations, representing the additional transport causedby fluctuationΦCFD-NHT-EHT13/76CENTER

13/76 Rewritten in a tensor form in Cartesian coordinate： ' ' ( ) ( ) ( ) ( 1,2,3) i i j j j j i i j u u u p u u u i t x x x x                     3. Other scalar (标量) variables ' ' ( ) ( ) ( ) j j j j j u u S t x x x                    4. Discussion on the time averaged quantity (1) Linear term remains unchanged during time average, while product term (乘积项) generates product of two fluctuations, representing the additional transport caused by fluctuation

热流科学与工程西步文源大学G教育部重点实验室(2) Equations are not closed:for 3-D problem, there arefive equations, with 14 unknown variables:Five time average variables-u,v,w,p,pNine products of fluctuationsuu,(i, j=1,2,3) (6 terms); uΦ(i=1,2,3)(3 terms)In order to close the above equations,additionalrelations must be added.Such additionalrelations arecalledturbulencemodel,orclosuremodel（封闭模型）The concept of closure model was actually 1st proposedbyProf.PYChouin1945.8.2.2Ways of determiningadditional terms中CFD-NHT-EHT14/76CENTER

14/76 In order to close the above equations, additional relations must be added. Such additional relations are called turbulence model, or closure model ( 封闭模型)。 (2) Equations are not closed：for 3-D problem, there are five equations, with 14 unknown variables： u v w p , , , , ,  Nine products of fluctuations ' ' ( , 1,2,3) 6 terms ; i j u u i j  （ ） ' '( 1,2,3) 3 terms i u i   （ ） Five time average variables－ 8.2.2 Ways of determining additional terms The concept of closure model was actually 1st proposed by Prof. P Y Chou in 1945

热流科学与工程西步文源大学E教育部重点实验室1.Reynolds stress methodFor the nine additional variables deriving their own governingequations.However, in the derivation process new additional terms ofhigher order (product of three variables, four variables, etc...) areintroduced.; If we still go along this direction then equations formuch higher order products should be derived. 2222: Thus we haveto terminate such process at certain level.Historically somecomplicatedmodelswithmorethan2oequationshavebeenderivedrelation w;w', were known. But unfortunately the equation of continuity (4.4) and thegeneral dynamical equation of double correlation (7.6) are insufficient to yield a definite solution for ww'r, because of the presence of the triple correlation ww,w's in(7.6).CFD-NHT-EHTΦ15/76CENTER

15/76 However, in the derivation process new additional terms of higher order (product of three variables, four variables, etc.) are introduced.；If we still go along this direction then equations for much higher order products should be derived. ,. Thus we have to terminate such process at certain level. Historically some complicated models with more than 20 equations have been derived. 1.Reynolds stress method For the nine additional variables deriving their own governing equations