--边界层理论剖析课件

Mechanics of Fluid12 Chapter 8 Boundary Layer Theory 81 Introduction86 Laminar Boundary Layer of smooth BoardChapter 8 Exercises85 The Momentum Equation of Boundary Layer and Friction shear Stress84 Variable Thickness of Boundary Layer83 The Motion Differential Equation of Boundary Layer82 Basic Concept of Boundary Layer 3第八章第八章 边界层理论边界层理论81 引言引言86 光滑平板上的层流边界层光滑平板上的层流边界层 第八章第八章 习题习题85 边界层的动量方程式和摩擦切应力边界层的动量方程式和摩擦切应力84 边界层中的各种厚度边界层中的各种厚度 83 边界层的运动微分方程式边界层的运动微分方程式82 边界层的基本概念边界层的基本概念4Chapter 8 Boundary Layer Theory8-1 8-1 Introduction The last chapter introduces Navier-Stokes equation and Reynolds equation,the differential continuity and these two equations form basic differential equation which find the solution of viscosity fluid dynamics.As for they are nonlinear second-order partial differential equations,Reynolds equation is not still be closed,usually we can not obtain the accuracy solution,people turn to seek approximate solution.5第八章第八章 边界层理论边界层理论8-1 8-1 引言引言 上章介绍了纳维斯托克斯方程与雷诺方程，它们与连续性微分方程一起构成求解粘性流体动力学的基本微分方程。由于它们是非线性的二阶偏微分方程，雷诺方程还无法封闭，所以在一般情况下不易得到它们的精确解，所以人们转向寻求近似的解答。6This chapter introduces boundary layer of plane board for the approximate of linear surrounding flow and plane board surrounding flow.We can obtain the analytic approximate solution when viscosity fluid moves by two methods as following,one is called“creeping flow theory”which we neglect inertia force and make basic concept linearity when .The other is to find boundary layer theory of resistance force of surrounding flow when ,it only considers flow viscosity inside the boundary layer,and the outside can be considered the potential flow of the ideal fluid.7 本章主要研究平板上的边界层，因为流线体绕流与平板绕流相接近。粘性流体运动时的解析近似解至今在两种情况下才能获得，一种是 时，可忽略惯性力，使基本方程线性化，这就是所谓蠕流理论；另一种是 时，求解物体绕流阻力的边界层理论，它对流体的粘性仅局限于边界内考虑，而边界层之外的广大主流区，可当作理想流体的势流。88-28-2 The Basic Concept of Boundary Layer The Basic Difference Between Viscid Fluid and Ideal Fluid:viscid fluid has viscosity.When viscid fluid flows on stationary fixed boundary,its velocity is 0,with the increasing of the distance to the fixed boundary,the effect of fixed boundary or viscosity on flow will be decreased,flow velocity increases,finally approaches the arrival flow velocity .The flow thickness which the velocity increase from 0 to 0.99 is called the thickness of boundary layer .When Reynolds number of arrival flow is greater,the extent which has variable velocity is limited to the thinnest layer near the fixed boundary,which is called boundary layer.Definition:98-28-2 边界层的基本概念边界层的基本概念 粘性流体与理想流体的根本区别粘性流体与理想流体的根本区别：粘性流体具有粘滞性。当粘性流体在静止固定边界上流动时，流体在固定边界上的速度为零，随与固体边界距离的增大，固体边界或粘性对流动的影响逐渐减小，流速逐渐增大，最后接近来流流速 。当来流的雷诺数较高时，具有速度变化 的范围只限于靠近固体边界的极薄的一层内，此薄层称为边界层。流速由 0 增加到0.99 处流体的厚度称为边界层的厚度 。定义定义：10Define friction resistance force of aircraft and naval vessel;Define coefficient value of theoretical flow velocity on overflow dam;Define the incorporation point of high velocity flow in steep groove;Define the flow resistance force and water head loss.Because with the length increase of plane board,friction loss is also increased,fluid internal energy is decreased,so flow velocity is,in order to meet the continuity requirement,the thickness of boundary layer is increased.Engineering Application of Boundary Layer Theory:1、The thickness of boundary layer is smaller than characteristic length of object,that is extreme thickness of boundary layer.Characteristics of Boundary Layer:2、The thickness of boundary layer is increased in the flow direction on plane board.11 飞机和舰船的摩擦阻力确定；溢流坝面理论流速系数值的确定；陡槽中高速水流掺气点的确定；水流阻力与水头损失的确定。1、边界层的厚度 与物体的特征长度 相比是非常小的，即边界层极薄。因为随着平板长度的增加，摩擦损失亦增加，流体内部的能量减少，流速亦减少，为了满足连续条件，边界层的厚度增大。边界层理论在实际工程中的应用边界层理论在实际工程中的应用：边界层的特点边界层的特点：2、边界层的厚度 在平板上沿流动方向增加。12 3、laminar flow section,transition section and turbulent flow section also exists in the boundary layer,under the transition section and turbulent flow section,there also exists a bottom layer of laminar flow.As shown in Fig.8-1.Laminar FlowBoundary LayerTransition SectionTurbulent Flow Boundary LayerBottom Layer of Laminar Flow Fig.8-1 Boundary Layer Structure13 3、边界层中也存在着层流区、过渡区和紊流区，过渡区和紊流区下面也存在一个层流底层 。如图81所示。层流边界层过渡区紊流边界层层流底层图 81 边 界 层 结 构14With the thickness increase of boundary layer,restriction effect of viscosity on flow in the boundary layer is decreased,and inertia function is increased.Laminar flow will turn into turbulent flow when viscosity can not control water particle motion,just like the flow in the cylindrical pipe,this phenomenon is called transition of boundary layer,and there exists a bottom layer of flow under the transition section and turbulent flow section .Assuming flow velocity in main flow is ,the distance to the front of plane is x,the Renault number is （81）Usually we take Reynolds number at transition point is（82）15 随着边界层厚度的增加，粘性对边界层内流体的约束作用减小，而惯性作用增大。当粘性作用控制不住水质点的运动时，就和流体在圆管中流动一样，由层流转变成紊流，此现象称为边界层转捩，并且在过渡区和紊流区下面存在一层流底层 。假设主流中流速为 ，到平板前端的距离为 x，这时的雷诺数为（81）一般取转捩点的雷诺数为（82）16 4、The flow extent of viscid flow is classified into tow sections with different properties by boundary layer.The section which is out of boundary layer is looked as ideal flow section,otherwise is viscid flow section.17 4、边界层将粘性流体的流动范围分成性质完全不同的两个区。边界层以外的区可视为理想流动区，边界层内视为粘性流动区。188-38-3 The Motion Differential Equation of Boundary LayerAssume：（a）（b）（c）（83）Hence,NS motion equation and continuity equation are changed into Flow is steady;Plane flow;Gravity function exists which can be neglected;Incompressible flow.198-38-3 边界层的运动微分方程式边界层的运动微分方程式 假设假设：（a）（b）（c）（83）于是，NS运动方程式和连续方程式变为 流动是恒定的；平面流动；质量力只有重力作用，且可以忽略；不可压缩流体。20（84）According to formula（83c），then（85）According to formula（83c）（86）According to formula（8-6）we know，is equivalent to the quantity class of ,according to formula（8-5）we know，is equivalent to the quantity class of y,so y is equivalent to the quantity class of 。thenIn order to simplify the differential group above,analyze the quantity class of equation terms,neglect the low quantity class terms,shown as Fig.8-2.Assuming the characteristic length of object is ,the main flow velocity is ,the thickness of boundary layer is define the equivalent quantity class with as 1,define the equivalent quantity class with as ,and ,then we have 21（84）由式（83c），得（85）又由式（83c）（86）由式（86）可知，相当于 的量级，由式（85）可知，相当于 y 的量级，因此 y 也相当于 量级。于是 为简化上面的微分方程组，对式中各项进行量级分析，忽略其低量级量，如图82所示。假设物体的特征长度为 ，主流流速为 ，边界层厚度为 ，与 相当的量级定为 1，与 相当的量级定为 ，且 ，于是有22（87）As forHence（88）23（87）因为所以（88）24and（89）25又（89）26 Substitute the quantity class of formula（8-4）（8-9）into（8-3a）and（8-3b），then（810）（811）27 将式（84）（89）的量级代入式（83a）和（83b）中，则得（810）（811）28 Eq.（8-10）the first term which is relative with and the second term can be neglected in the left parenthesis.Assuming that the effect of pressure term,viscosity term and inertia term are same,their quantity classes must be same.So we obtain the quantity class of is .If substitute the quantity class of into Eq.(8-11),then terms are and exclude,which compare with the quantity class of pressure term ,they all can be neglected.Finally we get the motion equation of boundary layer and boundary condition,these are（a）（b）（c）（d）（812）29 式（810）左端圆括号中的第一项同第二项相比可以忽略。假设在边界层中的压力项、粘性项和惯性项的影响相同，则它们的量级也应该相同。于是，得 的量级为 ，若将 的量级代入式（811），则压力项以外各项的量级分别为 和 ，同压力项的量级 相比，均可以忽略掉。最后，得边界层的运动方程式及边界条件为（a）（b）（c）（d）（812）30 （1）From Eq.（8-12b）,we know，pressure P in the boundary layer is unchangeable in the direction of y,and it can be determined by the motion equation of x-direction which flows outside boundary layer.When mass force is neglected,and is invariable flow,we would have U（x）is the ideal flow or potential flow velocity in the,simplifies U，as for boundary layer on plane board is ,the internal of the boundary layerWe would have（the internal of the boundary layer）（813）Conclusion：so(on the boundary of boundary layer)integrate31 （1）由式（812b）可知，在边界层内压强P沿 y 方向不变化，可由边界层外部流动的 x 方向的运动方程式确定。当忽略质量力，且为恒定流时得 U（x）为边界层边界上理想流动或势流的流速，以后简记为U，对于平板上的边界层就是 ，由于在边界层内部 所以有（在边界层内部）（813）结论：结论：所以（在边界层边界上）积分得32 （2）Consulting motion equation of boundary layer(8-12a)obtaining variable regulation of boundary layer,the quantity class of two ends of(8-12)（814）Due to Renault number is great,we may know the boundary layer thickness is small.33 （2）由边界层运动方程式（812a）求得边界层厚度变化规律。（812）两端的量级为（814）由于雷诺数 很大，可见边界层厚度 是很小的。348-48-4 Variable Thickness of Boundary Layer Expressing boundary layer thickness with ,it is the distance from this point to solid surface when a certain point velocity on cross-section of boundary layer is equal to the 99%of arrival flow velocity 1.Boundary Layer Thickness Definition：Ideal Velocity the thickness which is equivalent to the decreased flow rate in the boundary layer,is called pressed thickness ,also called thickness of flow rate losses.2.Displacement ThicknessDefinition：Classification：Boundary Layer Thickness Displacement Thickness Momentum Thickness358-48-4 边界层中的各种厚度边界层中的各种厚度种类：种类：边界层厚度一般用 表示，它是边界层横断面上某点的流速 等于来流流速 的99%时，此点到固体表面的距离。理想流速 通过边界层中减小的流量所相当的厚度称为排挤厚度 ，也称为流量损失厚度。一、边界层厚度一、边界层厚度 定义：定义：二、排挤厚度二、排挤厚度 定义：定义：36or（815a）（815b）Shown in Fig.8-3a,decreased flow rate in the boundary;the vertical hatch area represents the flow rate through the cross-section which Displacement thickness is at ideal velocity .The first right term of Eq.（7-15a）represents the flow rate of ideal fluid through cross-section which thickness is .The second term represents the flow rate through same cross-section in the boundary layer.The difference between two terms represents the decreased flow rate for the existence of boundary layer.37或（815a）（815b）如图83a 所示，图中斜影线面积表示边界层中减小的流量；图中铅直影线面积表示以理想流速 通过排挤 厚度 的流量。式（715a）右边第一项表示理想流体通过厚度为 断面的流量。第二项表示边界层中同一断面通过的流量。两者之差表示由于边界层的存在所减小的流量。38（a）Streamline（b）Fig.8-3 Where is supposed as the flow rate of external of boundary layer,h is the front distance between streamline and the plane board,due to the flow velocity u inside the boundary layer is less than ,in order to keep the flow rate constant,streamline should offset a distance.Now we confirm this offset distance is Displacement thickness ,supposing the distance between streamline and plane board is ,from the continuity equation,we have In addition,press thickness determines the offset distance of streamline,shown in Fig.8-3(b),the flow rate between plane board and streamline keeps invariable.39（a）流线（b）图 8 3 设边界层边界外的流速处处为 ，流线和平板间的距离在平板前端为 h，由于边界层内的流速 u 小于 ，为了保持通过的流量不变，流线应该向外偏移一个距离。现证明这个偏移距离就是排挤厚度 ，设此时流线与平板间的距离为 ，由连续方程 另外，排挤厚度 决定了流线的偏移距离。如图83（b）所示，通过平板与流线间的流量应保持不变。40Demonstration is over.The thickness which is equivalent to the decreased momentum when ideal velocity pass through boundary layer,called momentum thickness ,also called momentum loss thickness.（816a）（816b）3.Momentum Thickness Definition：41证毕。理想流速 通过边界层中减小的动量所相当的厚度，称为动量厚度 ,又称动量损失厚度。（816a）（816b）三、动量厚度三、动量厚度 定义：定义：42 Example8-1Assuming the flow velocity in the boundary distributes according the following exponential regulation.Find：displacement thickness and momentum thickness of boundary layer when n=7.Solution displacement thicknessmomentum thickness43 例题例题81假设边界层中的流速按下面指数规律分布试求：n=7时边界层的排挤厚度和动量厚度。解解 排挤厚度动量厚度448-58-5 The Momentum Equation of Boundary Layer and Friction shear Stress When fluid flows on the solid wall,like what said before,in the direction of vertical wall,velocity is changed from 0 on the wall to the main velocity ,the reason which formed this distribute is the result that resistance force on the wall take effect on the flow motion direction conversely.This section introduces how to find the shear stress of friction on the wall in terms of momentum laws .Shown in Fig.8-4,dashed line OAD is the separation line of boundary layer and main flow,the boundary region between dashed line and solid wall has obvious function,viscid function outside the dashed region can be neglected,it can be looked as flow region of ideal flow.As for the velocity in the boundary layer of plane board is constant ,and for curve wall,it is an variable number .Fig.8-4458-58-5 边界层的动量方程式和摩擦切应力边界层的动量方程式和摩擦切应力 当流体沿固体壁面流动时，如前所述在垂直壁面方向，速度由壁面上的零变为主流流速 ，形成这种速度分布的原因是由于壁面上的阻力逆流体流动方向作用的结果。本节我们应用动量定律来求壁面的摩擦切应力 。如图84所示，虚线OAD为边界层与主流的分界线、虚线与固体壁面间为边界层区域，粘性作用显著，虚线外区域忽略流体中的粘性作用，视为理想流体流动区域，对于平板边界层上的流速为常数 ，对于曲线壁面，边界层边界上的流速为变数 图 8 446 If we take the control body ABCD with length ,we express the boundary thickness of section AB with ,action pressure with p,shown in Fig.Plane ABPlane CDPlane ADAction force in the x-direction are as following:a、two dimension steady flow；b、mass force only has gravity,can be neglected;c、curvature of section is small,can be looked as straight line.Assume：47 如图所示，取长为 的控制体ABCD、AB断面处边界层厚度为 ，作用压强为 p。在 x 方向的作用力如下：AB面CD面AD面 a、二元恒定流动；b、质量力只有重力，且忽略不计；c、段壁面曲率小，可视为直线。假设假设：48Plane BCResultant force（1）Momentum change in x direction is displayed as following:Momentum flow-in on plane ABMomentum flow-out on plane AB49BC面合力（1）x 方向的动量变化如下：AB面流入的动量CD面流出的动量50Momentum calculation on plane ADTotal momentum change is：（2）mass flow-in rate on plane AB mass flow-out rate on plane CD mass flow-in rate on plane AD momentum flow-in on plane AD51AD面流入的动量计算总动量变化为：（2）52or（3）Pressure change in the boundary in flow direction is given by Eq.(8-13)（4）The first term on the right of Eq.（3）can be rearranged according to step differential：BecauseHence（5）Substitute Eq.(1)and Eq.(2)into momentum equation ,notice in the boundary layer p=p（x），hence ,yields 53或者（3）边界层内的压强在流动方向上的变化由式（813）为（4）式（3）右端的第一项根据分步积分变形如下：因为所以（5）将式（1）和（2）代入动量方程 ，并注意到在边界层内p=p（x），所以 ，于是得54 Substitute Eq.（4）and Eq.（5）into Eq.（3）,thatDivide both sides by ,the result is（817）Eq.(8-17)is called Karlmn Momentum Equation.55 将式（4）和式（5）代入式（3）得两端除以 ，得（817）式（817）称为卡门动量方程式卡门动量方程式。56 As for the boundary layer of plane board,we consider its existence can not take effect on external flow for thin plane board.So we have ,Eq.(8-17)is rearranged as（818）Assuming width of plane board is b,length is ,then total friction force which action force acts on single side of plane board is（819）Application Extent：Laminar Boundary Layer Turbulent Boundary Layer Plane Board or Curve Wall57适用范围适用范围：对于平板上的边界层，一般由于平板很薄，可以认为由于它的存在而不影响外部流动。因此 ，这样式（817）就变为（818）假设平板的宽度为 b，长度为 ，则作用力在单侧平板上的总摩擦阻力为（819）588-6 8-6 Laminar Boundary Layer of smooth Board Definition：Assuming flow velocity of invariable uniform flow is ,we put a plane board in the side of parallel flow which length is ,width is ,then its resistance force is（820）When viscid fluid flows on the plane board,resistance force acts the plane board,it is called friction resistance force.whereresistance force coefficient of friction598-6 8-6 光滑平板上的层流边界层光滑平板上的层流边界层 定义定义：假设恒定均匀流的流速为 ，平行流动方向放置一长为 ，宽为 b 的平板，则平板一侧所受的阻力（820）式中 当粘性流体在平板上流动时，平板将受到阻力作用，此阻力称为摩擦阻力。601.Distribute Equation of Flow Velocity of Laminar Boundary LayerAssuming（821）As for invariable flow,motion equation of boundary layer（822）On the wall of plane board（y=0）,then when y=0,（823）61一、层流边界层的流速分布公式一、层流边界层的流速分布公式设（821）对于恒定流，边界层的运动方程式为（822）因为在平板的壁面上（y=0），所以 y=0时，（823）62 From(8-1),the pressure change inside the boundary is（824）As for the boundary layer of plane board,then ，substitute into Eq.(8-23),we would have the result when y=0,（825）Continuous partial differential with Eq.(8-21)twice,let it equal to zero,we haveWe also have B=0.and at （826）63 由式（813），边界层内部的压强变化为（824）对于平板上的边界层，于是上式中 ，代入式（823），得 y=0时，（825）对式（821）连续偏导两次，并令其等于零，得由此得B=0。又在 处有（826）64We still use Eq.(8-21),we haveWe would find Substitute A、B、C above into Eq.(8-2),we would obtain flow velocity distribute of boundary layer of laminar flow,that is（827）65仍应用式（821）得解得 将上面的A、B、C各值代入式（821），得层流边界层内的流速分布为（827）662.Thickness Equation of Boundary Layer of Laminar Flow （828）On the other hand,from Eq.(8-18)and Eq.(8-16),we haveFrom Newtons internal friction law67二、层流边界层厚度公式二、层流边界层厚度公式 （828）另一方面，由式（818）和式（816），有由牛顿内摩擦定律68so（829）We can find which find from Eq.(8-28)or Eq.(8-29)would be equal,hence Differentiate equation above,and notice：in front end of plane board,that is at the location of x=0,we would have（830）69所以（829）因为由式（828）或式（829）求得的 应该相等，所以得 将上式积分，并且注意：在平板前端，即 x=0 处，于是得（830）703.Friction Resistance Equation of Laminar Boundary Layer（831）（832）Friction resistance force which acts on the single side of plane board which width is b length is ,from Eq.(8-19),we have Combining Eq.(8-32)and Eq.(8-20),yielding friction resistance coefficient（833）Renault number is Substitute Eq.(8-30)into Eq.(8-28),then71三、层流边界层的摩擦阻力公式三、层流边界层的摩擦阻力公式（831）（832）作用在宽为 b，长为 单侧平板上的摩擦阻力，由式（819）为 联立式（832）和式（820），得摩擦阻力系数（833）式中雷诺数 。将式（830）代入式（828）得72 This is only a approximate solution,precise solution is got from Blasius Equation according to boundary layer equation(8-12),limited by mathematical reason,we give the result:（834）Notice：This result only applies to laminar boundary layer,not turbulent boundary layer.73 这只是一种近似解，严密解是由勃拉休斯根据边界层基本公式（812）求得的，但是限于数学原因，这里只给出其结果：（834）注意：注意：此结果只适用于层流边界层，对于紊流边界层不适用。74 Example8-2 In the water which motion viscosity coefficient is ,the plane board which length is 2m and width is 2m moves at the velocity of 0.2m/s,find precise solution according to Blasius equation （1）friction stress with 0.5m to the front end of plane board;（2）force which can tow the plane board.Solution （1）Calculation of friction stress Because ,then laminar boundary layer generates on plane board,it is right to use Blasius equation.when x=0.5 m75 例题例题82在运动粘滞系数 的水中，长2m，宽2m的平板以0.2m/s的速度运动，试用勃拉休斯严密解求：（1）距平板前端0.5m处的摩擦应力；（2）拖动此平板所需的力。解解 （1）摩擦应力计算 因为 ，所以整个平板上产生层流边界层，故采用勃拉休斯公式是正确的。当 x=0.5 m 时76（2）Drag force calculationFriction resistance on sides of plane boardHence Drag force which can drag the plane board is36N.Friction resistance coefficient 77（2）拖力计算平板两侧的摩擦阻力为故拖动平板所需的力为0.336N。摩擦阻力系数784.Calculation Equation of Turbulent Boundary Layer（confirmation is neglected.）5.Calculation Equation of mixed Boundary Layer of plane board（Confirmation is neglected.）79四、紊流边界层的计算公式四、紊流边界层的计算公式（推导略）（推导略）五、平板混合边界层计算公式五、平板混合边界层计算公式（推导略）（推导略）80(4)Calculation Equation of Transition Pointadditional layer of laminar flow would be changed into additional layer of turbulent flow at a certain point,this point is called transition point,the Reynolds number which take of this point as characteristic length is called transition(critical)Reynolds number,using to express,and Then transition location8182 Chapter 8 Exercises8-1 There is a rectangular thin board which is 1.5 m4.5 m,dray at the velocity of 3 m/s in the direction of board surface,we know viscosity coefficient of air motion is v=1.5105 m2/s,density is r=1.2 kg/m3。Find resistance force along short edge and long edge respectively.Solution：（1）find motion resistance force in short edge direction.83 第八章第八章 习题习题 81 有一块 1.5 m4.5 m 的矩形薄板在空气中以 3 m/s速度沿板面方向拖动，已知空气运动粘性系数为 v=1.5105 m2/s,密度为 r=1.2 kg/m3。试求薄板沿短边方向和长边方向运动时，各自的摩擦阻力。解：（1）求沿短边方向运动的阻力。8485868788