热流科学与工程西步文源大堂E教育部重点实验室4.General form of the governing equationsa(pp)div(ppU) = div(Tegrad(Φ)) + SatTransientConvectionDiffusionSourceThe differences between different problems:(1) Different boundary and initial conditions;(2）Different nominal source（名义源项）terms;(3) Different physical properties (nominal diffusioncoefficients，a/Pr，名义扩散系数中CFD-NHT-EHT11/57CENTER
11/57 4. General form of the governing equations * * ( ) div U div grad S ( ) ( ( )) t The differences between different problems: （1）Different boundary and initial conditions； （2）Different nominal source（名义源项） terms； Transient Convection Diffusion Source （3）Different physical properties (nominal diffusion coefficients, Pr, 名义扩散系数)
热流科学与工程西步文源大堂G教育部重点实验室（说明）5. Some remarks1.The derived transient 3DNavier-Stokes equations canbeappliedforbothlaminarandturbulentflows2. When a HT & FF problem is in conjunction with (与..有关）mass transfer process, the component（组份）conservation equation should be included in thegoverning equations3.Although c,is assumed constant, the above governingequation can also be applied to cases with weaklychangedc，（比热略有变化）4.Radiativeheattransfer（辐射换热）is governed by adifferential-integral1(微分-积分）equation, and itsnumerical solution will not be dealt with here中CFD-NHT-EHT12/57CENTER
12/57 2. When a HT & FF problem is in conjunction with (与. 有关）mass transfer process, the component（组份） conservation equation should be included in the governing equations. 5. Some remarks（说明） 1. The derived transient 3D Navier-Stokes equations can be applied for both laminar and turbulent flows. 3. Although cp is assumed constant, the above governing equation can also be applied to cases with weakly changed cp (比热略有变化）. 4. Radiative heat transfer （辐射换热） is governed by a differential-integral （微分-积分）equation, and its numerical solution will not be dealt with here
热流科学与工程西步文源大堂C教育部重点实验室1.1.2 Conditions for unique solution(taking energy eq. as example)1. Initial condition(初始条件)t =O,T= f(x,y,z)2.Boundarycondition(边界条件)First kind (Dirichlet): Tβ =T(1)given(2)Second kind (Neumann):qβ =-)aiveOn(3)Third kind (Rubin)： Specifying （规定）the relationshipbetween boundary value and its first-order normal derivative:q = h(T,- T.) or q = h(T- T,)aTB= h(Tβ - T,)Forthe 3rd kind boundary conditiononheat flux at the boundary is not known!3. Fluid thermo-physical properties and source term of the process.中CFD-NHT-EHT13/57CENTER
13/57 1.1.2 Conditions for unique solution(taking energy eq. as example) 1. Initial condition (初始条件) t T f x y z 0, ( , , ) 2. Boundary condition (边界条件) （1） First kind (Dirichlet)： （2） Second kind (Neumann)： （3） Third kind (Rubin)：Specifying（规定） the relationship between boundary value and its first-order normal derivative: 3. Fluid thermo-physical properties and source term of the process. T T B given ( ) B B given T q q n ( ) ( ) B B f h T T n T ( ) or ( ) w w q T q h T h T T For the 3rd kind boundary condition heat flux at the boundary is not known!
热流科学与工程西步文源大堂E教育部重点实验室1.1.3Exampleof mathematical formulation1.Problem and assumptionsConvective heat transfer in a sudden expansion region:2D,steady-state,incompressiblefluid,constantproperties, neglecting gravity and viscous dissipation(粘性耗散)进口边界出口边界Tin中心线dSolutiondomainTTTTCFD-NHT-EHTΦ14/57固体边界CENTER
14/57 1.1.3 Example of mathematical formulation 1. Problem and assumptions Convective heat transfer in a sudden expansion region： 2D, steady-state, incompressible fluid, constant properties, neglecting gravity and viscous dissipation （粘性耗散）. Solution domain
热流科学与工程西步文源大堂E教育部重点实验室2. Governing equationsOuv=0axOy1 opCaa(uu)a(vu)u1CompleteXV2Oxayp axaaxsetofgovernin1 opa(uv)a(vv)Og2Oxayp ayayaxequations2a?Ta?T元1a(uT)a(vT)Qax?axayOyPrpcpc,nCFD-NHT-EHTΦ15/57CENTER
15/57 2. Governing equations 0 u v x y 2 2 2 2 ( ) ( ) 1 ( ) uu vu p u u x y x x y 2 2 2 2 ( ) ( ) 1 ( ) uv vv p v v x y y x y 2 2 2 2 ( ) ( ) ( ) uT vT T T a x y x y p a c Complete set of governin g equations 1 Pr p c