2.Momentum conservationApplying the 2nd law of Newton (F=ma) to an elementalcontrolvolume（控制容积）iinthethree-dimensional coordinates[Increasingrateof momentumoftheCV(控制容积中动量的增加率）]=[SummationofexternalforcesapplyingontheCV（作用在控制容积中的外力之和）1AdoptingStokes assumption(采用斯托克斯假设）：stress islinearlyproportionaltostrain（应力与应变成线性关系），wehavefollowing governing equation for component u in x-direction :6/62
2. Momentum conservation Applying the 2 nd law of Newton （F=ma) to an elemental control volume（控制容积） in the three-dimensional coordinates: [Increasing rate of momentum of the CV(控 制容积中动量的增加率）]= [Summation of external forces applying on the CV（作用在 控制容积中的外力之和）] Adopting Stokes assumption(采用斯托克斯假设）：stress is linearly proportional to strain (应力与应变成线性关系)，we have following governing equation for component u in x-direction： 6/62
a(pu) +a(puu) +(puv) +a(puw) - - p0(MdivU + 2natOxaxaxzSourceConvectionTransientDiffusiontermterm（瞬态项）term（对流项）term（扩散项）(源项)Eguation of the variation offluid momentumin x-direction ( pu).Diffusionterm（扩散项）ndynamic viscosity(动力粘度）Q（第2分子粘度）.For gas，π=-2 fluid 2nd molecular viscosity237/62
dynamic viscosity(动力粘度） ， (u) + (uu) + (uv) + (uw) = − p + (divU + 2 u t x y z x x 3 fluid 2 nd molecular viscosity（第2分子粘度）.For gas, =- 2 Transient Convection term（瞬态项） term（对流项） Diffusion Diffusion term（扩散项） Source 7/62 term term（扩散项） （源项） Equation of the variation of fluid momentum in xdirection （ u）
It can be shown (see the notes) that the above equationcan be reformulated as（改写为）following general form ofNavier-Stokes equation for u component of fluid momentum:a(pu)div(puu)-十atTransientConvectionDiffusionSourceterm非稳态项term对流项term源项term扩散项u, y, w ----velocity components in x,y,z three directions, respectively.they are the dependent variable (因变量)to be solved;U---fluid velocity vector;, U=ui+vj+wkSu ----source term.For y , w components, similar equations can be detived.8/62
(u) + div(uU )= t It can be shown (see the notes) that the above equation can be reformulated as （改写为）following general form of Navier-Stokes equation for u component of fluid momentum: Transient term 非稳态项 Convection term对流项 Diffusion term扩散项 Source term源项 u, v, w -velocity components in x,y,z three directions, respectively, they are the dependent variable（因变量） to be solved； U -fluid velocity vector; Su -source term. U =ui +v j + wk For v , w components, similar equations can be detived. 8/62
Sourceterm in x-direction:ForincompressiblefluidaaOMoxoxSimilarly:aaddvO(idivU)+pF,oyO0d1CaaaOowOuap(AdivU)+pF.O2zOyOzOzzOOxForincompressiblefluid（不可压流体）with constantproperties the source term does not contain velocity-related part9/62
Source term in x-direction： u S = ( u ) + ( v ) + ( w x x y Sv = y ( u ) + ( v ) + ( w ) + (divU) + F − p x y y y z y y y w z S = ( u ) + ( v ) + ( w ) + (divU) + F − p x z y z z z z z Similarly： For incompressible fluid ( 不 可 压 流 体 ） with constant properties the source term does not contain velocity-related part. For incompressible fluid 9/62
3.EnergyconservationIncreasingrateofinternalenergyintheCV（控制容积内能的增加率) =[Net heattransferrategoingintotheCVl+|Powerconductedbybodyforcesand surfaceforces（进入控制容积的净传热速率+体积力与表面力作用在控制容积上的功率）Introducing Fourier's law of heat conduction and neglecting thework conducted by forces; Introducing enthalpy (焰)h = C,T , assuming C, = constant,We have:a(pT) + div(pTU)= div(2grad(T)) + SatnT;:naT2k温度梯度igrad(T)XOzaxyPrcCpnCpn10/62
3. Energy conservation [Increasing rate of internal energy in the CV（控制容积内能的增 加率）]= [Net heat transfer rate going into the CV]+[Power conducted by body forces and surface forces（进入控制容积的净传 热速率+体积力与表面力作用在控制容积上的功率）] Introducing Fourier’s law of heat conduction and neglecting the work conducted by forces；Introducing enthalpy（焓） p (T ) t + div(TU )= div( grad(T)) + S c T Pr ( ) c p c p c p h = c pT , assuming c p = constant, x y z grad(T ) = T i + T j + T k（温度梯度） We have: 10/62