热流科学与工程西步文源大学G教育部重点实验室In the Reynolds stress models, the second moment model(modelfortheproductsoftwofluctuationquantities)isquitefamous and has been applied in some engineering problems. Inthe second moment model, for the product terms with twofluctuations their equations are derived, while for the terms withthree or more fluctuations models are used to relate such termswith time average variables.Prof.LX Zhou(周力行in Tsinghua university contributeda lot in this regard.2. Turbulent viscosity methodThe product of fluctuations of two velocities is expressed viaturbulent viscosityCFD-NHT-EHTΦ16/76CENTER
16/76 2. Turbulent viscosity method The product of fluctuations of two velocities is expressed via turbulent viscosity. In the Reynolds stress models, the second moment model (model for the products of two fluctuation quantities) is quite famous and has been applied in some engineering problems. In the second moment model, for the product terms with two fluctuations their equations are derived, while for the terms with three or more fluctuations models are used to relate such terms with time average variables. Prof. L X Zhou (周力行) in Tsinghua university contributed a lot in this regard
热流科学与工程亚步文源大堂E教育部重点实验室(l)DefinitionofturbulentviscosityIn 1877Boussinesq introduced following equation, bymimicking(比拟）theconstitutionequation（本构方程）oflaminar fluid flow:auCau2(t),=-puu,=(-p,)+,3nd,diydxP,=p[(u+(+(w)]=pkk==[(u)? +(v)? +(w)](2) Definition of turbulent diffusivity of other scalar variablesPr,---turbulent Prandtl number.adn.-pu.dusually treated as a constant.Pr,axΦCFD-NHT-EHT17/76CENTER
17/76 (1) Definition of turbulent viscosity In 1877 Boussinesq introduced following equation, by mimicking(比拟) the constitution equation (本构方程) of laminar fluid flow: ' ' , , , 2 ( ) ( ) ( ) 3 i j i j i j i j j j i t t t i t u u u u p divU x x 1 2 ' 2 ' 2 ' 2 [( ) ( ) ( ) ] 3 3 t p u v w k 1 ' 2 ' 2 ' 2 [( ) ( ) ( ) ] 2 k u v w ' ' i t i u x (2) Definition of turbulent diffusivity of other scalar variables Pr t t t Prt -turbulent Prandtl number, usually treated as a constant. 0
热流科学与工程亚步文源大堂E教育部重点实验室ForlaminarheattransferwehaveLannicpn,cp=2Pr;cpniCp niCpniSimilarly:I, = 2, = n,c, / Pr,Therefore for turbulent viscosity model its major taskis to find n,Pr.The name of engineering turbulence models comesfrom the number of PDEqs. included in the model todetermine turbulence viscosity8.2.3 Governing equationsofviscocity models1.Governing equationsΦCFD-NHT-EHT18/76CENTER
18/76 For laminar heat transfer we have ( ) Pr ( ) l l p l p l p l p p l p l p l l c c c c c c c Similarly： / Pr t t t p t c 8.2.3 Governing equations of viscocity models 1. Governing equations Therefore for turbulent viscosity model its major task is to find . ,Pr t t The name of engineering turbulence models comes from the number of PDEqs. included in the model to determine turbulence viscosity
热流科学与工程西步文源大堂E教育部重点实验室For simplicity of presentation, the symbol of timeaverage“bar"is omittedhereafter:ou=0OxkNeffOpefaQu,aua(pu,u,Ll+,Pefr=p+p[(n, + n.ataxkax,OxkOxkaa(pp) , a(pud)adI+s[(T +IataxkOxOx2. Differences from laminar governing equations:(1) uj, p,Φ -Time average; (2)Replacing T by Fefr =+F(3) Replacing p by Pefr(4)In the source term S, of u,the additional terms caused by time averaging are included.---pcI, =, = nc, / Pr,ΦCFD-NHT-EHT19/76CENTER
19/76 0 k k u x ( ) [( ) ] i i k eff l t i k i i k k u p S t x x x u u u x * * ( ) ( ) [( ) ] k l t k k k u S t x x x 2. Differences from laminar governing equations: ; eff t p p p (1) , , i u p -Time average; (2)Replacing by eff t (3) Replacing eff p by p (4) of i S i In the source term u the additional terms caused by time averaging are included. eff eff For simplicity of presentation, the symbol of time average “bar” is omitted hereafter： / Pr t t t p t c * - p c
热流科学与工程西步文源大堂G教育部重点实验空In the Cartesian coordinates, the source terms of the threecomponents are:aaawaauauu:S:neffarNeffarneffarazaaayaawaaauyauU:S=MeffaNeffaMeffayazarayaaawauaauw:SNeffazNeffazNeffaazadIn laminar flow of constant properties, all source terms arezero, but for turbulent flow they are not zeroThus the above governing egs. for flow and heat transferare not complete. Eq. for determining nefr should be added3.TurbulentPrandtlnumberIts value varies within a certain range, usually is taken asa constant, and I, =c,n, / Pr,ΦCFD-NHT-EHT20/76CENTER
20/76 In the Cartesian coordinates, the source terms of the three components are： 3. Turbulent Prandtl number Its value varies within a certain range，usually is taken as a constant, and / Pr t p t t c In laminar flow of constant properties, all source terms are zero, but for turbulent flow they are not zero . Thus the above governing eqs. for flow and heat transfer are not complete. Eq. for determining should eff be added