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 certainlevel.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).Copied From the original paper of Prof. P. Y. Chou.16/82
16/82 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. Copied From the original paper of Prof. P. Y. Chou
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 variablesProf. LX Zhou （周力行)in Tsinghua university contributeda lot in this regard.2.Turbulentviscosity methodThe product of fluctuations of two velocities is expressed viaturbulentviscosity17/82
17/82 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
(1)DefinitionofturbulentviscosityIn 1877Boussinesg introduced following equation, bymimicking(比拟）theconstitutionequation（本构方程）oflaminar fluid flow:au0au2(tu),=-puui,=(-p.0. ) + n(x,3n,diy-dxP,=p[(u)+(+(w]-pk k=[(u+(+(w))(2) Definition of turbulent diffusivity of other scalar variablesadPr, ---turbulent Prandtl number.n-pu.dusually treated as a constant.Pr,Ox18/82
18/82 (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
Format ImprovementoftheGeneral GoverningEquationThe G.E. we just learned in the previous chapters is:a(pp)+ div(ppU) = div(T*grad) + S(1)atIp*S$*dEquationp100Continuityequationpappfr-Momentumeqn.(xdirection)puμdxap0Momentumeqn.(ydirection)pupfy-%TA/CpST/CpEnergyequationpThistypeofgeneralgoverning eguationleadsto someinconvenienceforpractical applications.Newequationwillbeusedinourteachingcode:a(p Φ)+ div(p pU) = div(T gradp) + Sat19/75
19/75 * * ( ) div U div grad S ( ) ( ) t + = + This type of general governing equation leads to some in convenience for practical applications. New equation will be used in our teaching code: （1） Format Improvement of the General Governing Equation The G.E. we just learned in the previous chapters is: * * ( ) div U div grad S ( ) ( ) t + = +
So*dEquationTp0100Continuity equation0apMomentumeqn.(xdirection)upfx1RoxoMomentumeqn.(ydirection)0upfy-oy0TST入EnergyequationOCThat is, here we regard pCp in the energy equation as a general density:a(pc,T)a(pcpuT)aa(Oc,UT)aaTaT+ST.axataxayadxduSuch a treatment is much better than taking / cp as a nominaldiffusion coefficient and Sr / cp as a nominal source term in energy eqLi W, Yu B, Wang Y, et al. Communications in Computational Physics, 2012, 12(5): 1482-149420/75
20/75 That is, here we regard in the energy equation cp as a general density： Such a treatment is much better than taking as a nominal diffusion coefficient and as a nominal source term in energy eq. / p c / T p S c Li W, Yu B, Wang Y，et al. Communications in Computational Physics, 2012, 12(5): 1482-1494