西安交通大学：《数值传热学》课程教学资源（PPT课件）第一章 绪论 Numerical Heat Transfer

2. Governing equationsOu. Ov+axa?ua21 opa(uu)a(uv)1Complete+VOx2axayOyp axset ofgoverning1 opa(vu)a(vv)equationsar?Qy2axdyp aythermal元a2Ta?Ta(uT)a(vT)diffusivity+a=ax2Qy2Oxdypc（热扩散率）16/62

2. Governing equations u + v  2 u  2 u + = − + ( + ) x (uu) (uv) 1 p  x x 2 y 2  2 v  2 v + = − + ( + ) x y (vu) (vv) 1 p  y x 2 y 2 x y (uT ) (vT)  2T  2T + x y x 2 y 2 p  c = a( + ) a = Complete set of governing equations 16/62 thermal diffusivity (热扩散率)

(3) Center line:3.Boundary conditionsu_=0, =0(1) Inlet: specifyingaydy(说明）variations ofu,yT(4) Outlet: Mathematically thewithy ;distributions of u.yT or their生口边界first-order derivatives(一阶导数)出口边界Tin中心线care required. Actually,numericalapproximations mustbe made.77Can we regard this质体聚街boundary consasHT &FF overa圖体边界backward step?(2) Solid B.C.: No slip(滑移)(跳跃)in velocity, no jumpintemp17/62

17/62 3. Boundary conditions x y （1）Inlet：specifying (说明）variations of u,v,T with y ； （2）Solid B.C.：No slip（滑移） in velocity, no jump（跳跃） in temp. (4）Outlet：Mathematically the distributions of u,v,T or their first-order derivatives(一阶导数) are required. Actually， numerical approximations must be made. Can we regard this boundary cons as HT & FF overa backward step? （3）Center line： u = T = 0; v =0 y y

Can we regard this boundary formulation as heat transferand fluid flow over a backward step?上壁面绝热Adiabatic[进口InletU...T.Outlet出口下壁面等温Constanttemperature18/62

Can we regard this boundary formulation as heat transfer and fluid flow over a backward step? Tw U0,T0 上壁面绝热 出口 18/62 进口 Inlet Outlet Adiabatic 下壁面等温 Constant temperature

Notes to Section 1.1In the left hand sidea(pu + a(puv) + (pu) - div(puu)axayOzThe right hand side :Ovaaauauw]+pFop.(adivU+ 2nOu07n++oxOzaxOzaxayayaxaxaaaauava%()+%(n%raxayaxaxdiv(grad(u))u i+ u i+ u kgrad (u)=dp= div(ngradu) + S,pFOxOza2 (2u)+ 2 (2u)+ 2(2u)div(grad(u) = Thus we have:OyOzOzOx axoya(ou+ div(puu)= div(ngradu)+SNavier-Stokesat19/62

The right hand side : Thus we have: x − p = x  (divU + 2 u ) +  [( v + u )] +  [( u + w )]+  F x x y x y z z x u (u) + div(uU ) = div(gradu) + S t grad (u) = u i + u j + u k x y z div(grad(u)) =  ( u ) +  ( u ) +  ( u ) x x y y z z = u div(gradu) + S   ( u ) +  ( u ) +  ( u ) +  ( u ) +  ( v ) +  ( w x x y x p  Fx −  Navier-Stokes x x y y z z div(grad(u)) Su (uu) + (uv) + (uw) = div(uU ) x y z Notes to Section 1.1 In the left hand side 19/62

Gradientofa scalar（标量的梯度)is avector:grad(u)=++ukaxyzDivergenceofavector（矢量的散度）isa scalar：ou:Ououkdiv(grad(u)) = div(Ozayax?Qudiv(grad(u) = (u)(nu)+(nu)+三(nou)div( ngrad(u)) =axOxoyayozozEnd of Notes to Section 1.120/62

Gradient of a scalar (标量的梯度) is a vector： grad(u) = u i + u j + u k x y z Divergence of a vector (矢量的散度) is a scalar： div(grad(u)) = div( u i + u j + u k) x y z div(grad(u)) =  ( u ) +  u div(grad(u)) =  ( u ) +  ( u ) +  ( u ) x x y y z z End of Notes to Section 1.1 20/62