西安交大高等传热学热对流第三章

高等传热学AdvancedHeatTransferChap.3Laminarexternalboundarylayers高等传热学AdvancedHeatTransfer1.TheGoverningEqs&BCs§3-1laminarforcedconvectionoveraflatplate研究对象：常物性，不可压缩流体，2D，忽略黏性耗散，无内热源，无体积力，u∞，T∞＝constxy0lxdu∞主流区边界层区高等传热学AdvancedHeatTransfer22tttuvaxyy22)duuuuuvuxydxy（0yvxu7个BC：0:,0:0,0,():,wxuuTTyuvTTyuuTTd高等传热学AdvancedHeatTransfer2.Theflowsolutionx1与x2处，层流速度并不相似，但都从0-u∞引入：uuyxd1RexxLxuxddxxud1uyxyyxuxd不唯一高等传热学AdvancedHeatTransfer引入流函数1uuuy,uvyxuuux相似解若存在，则fuu11uyuxfthequationof?~f高等传热学AdvancedHeatTransfer'ffufuyfyfy112uuyxyxxxxvfuxxx,ufyxux1'2uffx12uffuxxx高等传热学AdvancedHeatTransfer'ufuxx'ufuyy222'''''uuuuffyyxx22uuuuvxyy代入1'''2ufufxx'''ufuufyx高等传热学AdvancedHeatTransfer1'''''02fff:0(0):'0,uBCyfu1'002uvfffx():'1yf黏性力惯性力无量纲切向速度无穷级数（1908，Blasius）；数值积分解(Runge-Kutta)BlasiusEq.高等传热学AdvancedHeatTransfer平板层流边界层的布拉修斯解uuxyfffvxuu0000.3320604.42.692380.975870.038975.03.283290.991550.015915.43.680940.996160.007938.06.279231.000000.000010.860388.46.679231.000000.000000.86038.……………………高等传热学AdvancedHeatTransfer5.0uxdxxRe0.5d,0,wxyxuy1,220.664Re2wxfxxcu12,0121.328ReLffxfLLccdxcL上述值与实验测定值符合，证明了Prandtl边界层理论3322,0,0''0.332wxyxuuufyxx',uufyux高等传热学AdvancedHeatTransfer3.Theheattransfersolution：引入：wwTTTTuyx:wTTorTTxx'uyyx22''uyyyx111''22uxyxx'ufu1'2uvffx高等传热学AdvancedHeatTransfer1''Pr'02fPohlhausenEq.:0:0:1BC1''Pr'02f1'''''02fff二阶线性常微分方程三阶非线性常微分方程wwTTTTuuPr122uvxyy高等传热学AdvancedHeatTransfer直接积分求解：1200Prexp()2CfddC20(fromBC:Tw)C1001Prexp()2(fromBC:T)Cfdd0000Prexp()2Prexp()2fddfdd(,Pr)F高等传热学AdvancedHeatTransfer壁面热流：,0,xwxyxThTTy00,xyxhudydx10dCd1200ReRe,PrPrexp()2xxxxhxNuffdd高等传热学AdvancedHeatTransfer分段拟合：xNu12120.564RePr,Pr0.05x12130.332RePr,Pr0.610x12130.339RePr,Pr10x012LmxLhhdxhL2mLNuNu12,,xxhxxh?0,xxh高等传热学AdvancedHeatTransfer§3-2laminarforcedconvectionwithpressuregradients1.TheGoverningEqs&BCs研究对象：常物性，2D，低速层流22tttuvaxyy22)duxuuuuvuxxydxy（0yvxu0:0,0,():,0:,wyuvTTyuuxTTxuuxTTd高等传热学AdvancedHeatTransfer存在相似解muxcx2m12mmduxdpmuxcxmcxudxdxx0,00,,,dpmpvdx顺压梯度流动0,00,,,dpmpvdx逆压梯度流动0,00,dumdx＝＝外掠平板流动xy,uT高等传热学AdvancedHeatTransfer2.Flowsolutions()12uxmyxfuufuyfy''udfufuyydy222121'''2mumcxfy11''''2mmumcxfcxmffxx11221121'212mmvcxfxxmmmcxffm1uy112muyux()21'12muxdfmuxcxfmdx331()11''''22muxmmcxuffx高等传热学AdvancedHeatTransfer:0:0(0),'0(0):'1BCfvfuf由由2'''''1'0ffffFalkner-SkanEq.三阶非线性常微分方程引入相似变量不同所致0代入动量方程'''''0fff()12uxmyxuyx1'''''0?2fff高等传热学AdvancedHeatTransfer解的三种特例0,0,,muconst＝＝外掠平板流动1,1,,dumuxcxcdx二维滞止流动，00.1988,0,uyy＝边界层分离0.1988,流动边界层从壁面脱离并在紧靠壁面处产生回流15.03.62Rexuxxddd高等传热学AdvancedHeatTransferuu0()12uxyx高等传热学AdvancedHeatTransfer33,0,1''02mwxyxumcxfyx33,,2212''022mwxfxmcxfxcuu获得速度分布后muxcx0,0,''(0)0.4696mf＝＝1212,21Re''00.664Refxxxcmf1221Re''0xmf高等传热学AdvancedHeatTransfer3.Heattransfersolutions22uvxyy()12uxmyx1'2mxxx11'2mmcxywwTTTT2121''2mmcxy22tttuvaxyy高等传热学AdvancedHeatTransfer代入边界层能量方程''Pr'0f:0:0:1BC二阶齐次线性常微分方程''Pr'0Pr'ddffdd0Pr1'fdce0Pr120fdcedc20c01Pr01fdced00Pr0Pr0Pr,fdfdedFed高等传热学AdvancedHeatTransfer''Pr'0fPr=1?2'''''1'0ffffPr=1,β=0高等传热学AdvancedHeatTransfer壁面热流：,0,xwxyxThTTy00,()12xyxhuxdmydx0dd1200Re1,Re,Pr2exp(Pr)xxxxhxmNufmfdd01Pr01fdcedmuxcx12mxhx120,0,xmhx1,1,xmhconst高等传热学AdvancedHeatTransfer0pNuNu11.60.2Pr0,00,,,dpmpvdx边界层外主流加速导致0,00,,,dpmpvdx边界层外主流减速导致高等传热学AdvancedHeatTransfer应用边界层概念应注意的问题:(1)上述边界层概念及分析是以沿平板的无界外部流动为例进行介绍的，内部流动的边界层情况不同(2)在平板前缘很短的一段距离内，边界层理论不适用(3)若出现边界层脱体，或发生回流情况，边界层的特性也将改变高等传热学AdvancedHeatTransfer通过引入适当的相似变量，变换边界层动量方程、能量方程与边界条件，消除其对x的依赖关系，将偏微分方程转化为常微分方程。但相似解存在条件苛刻。求解不相似层流边界层问题数值求解，将偏微分方程离散成代数方程局部相似解和局部不相似解边界层积分方程对于工程实际情况，复杂壁面，复杂BC，任意变化的位流速度，依赖于近似解；但积分方程所包含的动量、热量以及质量传递信息比边界层微分方程要少，基本上已被高精度的数值计算所代替。高等传热学AdvancedHeatTransfer§3-3IntegralEquation一、边界层积分方程组1.基本思想边界层微分方程:要求对边界层内每一个微元体都满足守恒定律边界层积分方程:对包括固体边界及边界层外边界在内的有限大小的控制容积满足动量及能量守恒定律即可。xxxWF高等传热学AdvancedHeatTransfer能量平衡0Hx