8.2Time-averagedgoverningequationforincompressibleconvectiveheattransfer8.2.1Timeaveragegoverningequation8.2.2Waysfordeterminingadditionalterms8.2.3Governingequationswithturbulent viscosity11/82
11/82 8.2.1 Time average governing equation 8.2.2 Ways for determining additional terms 8.2 Time-averaged governing equation for incompressible convective heat transfer 8.2.3 Governing equations with turbulent viscosity
8.2Time-averaged governing equation forincompressibleconvectiveheattransfer8.2.1Timeaveragegoverningequation1. Continuity equationouavavawQuaw(u+u).o(v+v),a(w+w=0ayOzO2axayOzaxaxdy=0=0Both time average velocity and time average fluctuationvelocity satisfy continuity condition.2.Momentum equationTaking x-direction as an example:12/82
12/82 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
a(u+u)?a(u+u)1 a(p+p)a(u+u)(v+v)a(u+u)(w+w)axOzatoyaxpa(u+u), a(u+u)a(u+u)TOOzAccording to the above characteristics, yieldinga(uQuuvouwa(u)?a(uv).a(uw)ataxayOzaxavOza2uouau1apMoved to right hand sideax?Oz?ay2ax0andcombinedwiththe2correspondingviscoustermouvouwoua(uataxoyOzou1 apaou0aua(u)(uv)]+(uw))1azaxaxayOzoypax13/82
13/82 ' ' 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 2 corresponding viscous term 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 − + − + − + −
（张量）form inCartesiancoordinate:Rewritten inatensor.aapouia(pu), a(puiu,)puu,)(i= 1,2,3)nataxax,axaxDouble subscript means summation over the subscript3. Other scalar (标量) variablesadaa(pp), a(pu;g)pu,Φ')+sataxax,ax,4.Discussion on the time averaged quantity(1) Linear term remains unchanged during time average, whileproduct term（乘积项）generates product of two fluctuationsrepresenting the additional transport caused by fluctuation.14/82
14/82 Rewritten in a tensor （张量）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. Double subscript means summation over the subscript
2) Equations are not closed:for 3-D problem, there areonlyfive equations, with 14 unknown variables:Five time average variables-u,v,w,p,p,Nine 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.PYChou（周培源）in1945.8.2.2Waysofdeterminingadditionalterms15/82
15/82 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 only 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 (周培源）in 1945