4.General form of the governing equationsa(pd)+ div(pU )= div(Tgrad() + SatDiffusionSourceConvectionTransient@ is a general dependent variable: u, v w and T ;Thedifferences between different problems1Different boundary and initial conditions;2Different nominal source（名义源项）terms;3Different physical properties (nominal diffusion2/Pr)coefficients，名义扩散系数11/62
4. General form of the governing equations * * () + div(U )= div( grad()) + S t Transient Convection Diffusion Source is a general dependent variable: u, v, w and T ； The differences between different problems: 1 Different boundary and initial conditions； 2 Different nominal source（名义源项） terms； 3 Different physical properties (nominal diffusion Pr coefficients, 名义扩散系数, ) 11/62
（说明）5.Some remarks1.The derived transient 3D Navier-Stokes equations can be(湍流）applied for both laminar and turbulent flows2.WhenaHT&FFproblemisinconjunctionwith（与...有关）mass transfer process, the component （组 份）conservationequation should be included in the governing equations.3.Although c, is assumed constant, the above governingequation can also be applied to cases with weakly changed cp（比热略有变化）4.Radiativeheat transfer（辐射换热）isgoverned byadifferential-integral（微分-积分）equation，anditsnumericalsolution will not be dealt with (处理)here12/62
5. Some remarks（说明） 1.The derived transient 3D Navier-Stokes equations can be applied for both laminar and turbulent flows （湍流）. 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. 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 (处理) here. 12/62
1.1.2 Conditions for unique solution(taking energy eq. as example)1. Initial condition (初始条件）t =O,T = f2.Boundarycondition（边界条件）(1)First kind(Dirichlet): Tg = Tgiven(2)Second kind (Neumann): qε =-Loinedn(3)Third kind (Rubin): Specifying （规 定）the relationshipbetween boundary value and its first-order normal derivative （法向导数):q = h(T- T.) or q= h(T- Tw))β= h(Tβ- T,)Forthe3rdkindboundaryconditionheat flux atthe boundary is not known!3. Fluid thermo-physical properties and source term of the process13/62
1.1.2 Conditions for unique solution(taking energy eq. as example) 1. Initial condition (初始条件) t = 0, T = f 2. Boundary condition (边界条件) （3） Third kind (Rubin)：Specifying（ 规 定 ） the relationship between boundary value and its first-order normal derivative（法向导 数）: 3. Fluid thermo-physical properties and source term of the process. （1） First kind(Dirichlet)： TB =Tgiven （2） Second kind (Neumann B B given )： q = −( T ) = q n − B ( T ) = h(TB − Tf ) n q = h(Tw −T ) or q = h(T −Tw ) For the 3 rd kind boundary condition heat flux at the boundary is not known! 13/62
1.1.3 Example of mathematical formulation1.Problemandassumptions1（突扩区Convective heat transfer in a sudden expansion region域 ） : 2D, steady-state, incompressible fluid, constant properties,(粘性耗散neglecting gravity and viscous dissipation进口边界出口边界Tin中心线Solutiondomain（求解区域）7固体边界14/62
1.1.3 Example of mathematical formulation 1. Problem and assumptions Convective heat transfer in a sudden expansion region（突扩区 域 ）：2D, steady-state, incompressible fluid, constant properties, neglecting gravity and viscous dissipation（粘性耗散）. Solution domain （求解区域） 14/62
2D Sudden expansion region開件车固体边界15/62
2D Sudden expansion region 15/62