(英汉双语)流体力学第三章流体动力学基础.ppt

1,Chapter 3Basis of Fluid Dynamics,Fluid Mechanics,2,第三章 流体动力学基础,3,Chapter 3 Basis of Fluid Dynamics,34 Continuity Equation,31 Preface,32 Methods to Describe Fluid Motion,33 Basic Concepts of Fluid Motion,35 Motion Differential Equation of Ideal Fluid,36 Bernoulli Equation and Its Application,37 System and Control Volume,38 Momentum Equation,39 Moment of Momentum Equation Exercises of Chapter 3,4,第三章 流体动力学基础,34 连续方程式,31 引言,32 描述流体运动的方法,33 流体运动的基本概念,35 理想流体的运动微分方程,36 伯努利方程及其应用,37 系统与控制体,38 动量方程,39 动量矩方程 第三章 习 题,5,Chapter 3 Basis of Fluid Dynamics,3-1 Preface,Basis of Fluid Dynamics,The backgrounds,fundamentals and fundamental equations of fluid dynamics all have certain relations with each part of engineering fluid mechanics,so this chapter is the emphases in the whole lessons.,6,第三章 流体动力学基础,3-1 引言,流体动力学基础,流体动力学的基础知识，基本原理和基本方程与工程流体力学的各部分均有一定的关联，因而本章是整个课程的重点。,7,3-2 Methods to Describe the Fluid Motion,Methods to describe the fluid motion：,1.Lagranges method,Definition：,Lagranges method is to consider the fluid particles as research objects and to research the motion course of each particle,and then gain the kinetic regulation of the whole fluid through synthesizing motion instances of all being researched objects.The essential of lagrangian method is a method of particle coordinates.,Basis of Fluid Dynamics,8,3-2 描述流体运动的方法,描述流体运动的方法：,一、拉格朗日法,定义：,把流体质点作为研究对象，研究各质点的运动历程，然后通过综合所有被研究流体质点的运动情况来获得整个流体运动的规律，这种方法叫做拉格朗日法。实质是一种质点系法。,流体动力学基础,9,when we use lagranges method to describe the fluid motion the position coordinates of motion particles are not independent variables but functions of original coordinate a,b,c and time variable t,that is,（31）,In this formula,a,b,c and t are all called lagrangian variables.Different particles have different original coordinates.,Difficulties will be met when using lagranges method to analyze fluid motion on math except for fewer instances(such as researching wave motion).Eulers method is used mostly in fluid motion.,Basis of Fluid Dynamics,10,用拉格朗日法描述流体的运动时，运动质点的位置坐标不是独立变量，而是起始坐标a、b、c和时间变量 t 的函数，即,（31）,式中a，b，c，t 统称为拉格朗日变量，不同的运动质点，起始坐标不同。,用拉格朗日法分析流体运动，在数学上将会遇到困难。除少数情况外（如研究波浪运动），在流体运动中多采用欧拉法。,流体动力学基础,11,2.Eulers method,Definition：,When we use Eulers method to describe fluid motion the motion factors are continuous differential functions of space coordinates x,y,z and time variable t.x,y,z and t are called Eulers variables.So the,velocity field can be expressed by the following formulas:,（32）,With a view to the space points in the fluid field(the space full of motion fluid)without researching the moving course of each particle.It is to synthesize enough space points to gain the regulation of the whole fluid by observing the regulations of motion factors of particle flowing via each space point changing with time which is called Eulers method(fluid field method).,Basis of Fluid Dynamics,12,二、欧拉法,定义：,用欧拉法描述流体的运动时，运动要素是空间坐标x，y，z和时间变量t的连续可微函数。x，y，z，t 称为欧拉变量，因此,速度场可表示为：,（32）,不研究各个质点的运动过程，而着眼于流场（充满运动流体的空间）中的空间点，即通过观察质点流经每个空间点上的运动要素随时间变化的规律，把足够多的空间点综合起来而得出整个流体运动的规律，这种方法叫做欧拉法（流场法）。,流体动力学基础,13,Pressure field and density field can be expressed as:,In the formula（32）x，y and z are motion coordinates of fluid particles at time t and namely are functions of time variable t.So according to the principle of compound function differentiate and also think over the following formulas:,The acceleration components in direction of space coordinates of x,y,z are:,（35）,Basis of Fluid Dynamics,14,压强和密度场表示为：,式（32）中x，y，z是流体质点在 t 时刻的运动坐标，即是时间变量 t 的函数。因此，根据复合函数求导法则，并考虑到,可得加速度在空间坐标x，y，z方向的分量为,（35）,流体动力学基础,15,The vector expression is,（35a）,In it,When using Eulers method to query variance ratio of other motion factors of fluid particle changing with time the normal formula is,（36）,is called total derivative，is called local derivative，is called migratory derivative.,Basis of Fluid Dynamics,16,矢量式为,（35a）,其中,用欧拉法求流体质点其它运动要素对时间变化率的一般式子为,（36）,称 为全导数，为当地导数，为迁移导数。,流体动力学基础,17,3-3 Basic Concepts of Fluid Motion,1.Stationary flow and nonstationary flow,Definition：,In factual engineering problems,motion factors of quite a few un steady flow changing with time very slowly which can be treated as steady flow problems approximatively.,Or else it is called nonstationary flow.,If all motion factors of each space point on fluid field dont change with time,this kind of flow is called steady flow.that is:,Basis of Fluid Dynamics,18,3-3 流体运动的基本概念,一、定常流动与非定常流动,定义：,在实际工程问题中，不少非定常流动问题的运动要素随时间变化非常缓慢，可近似地作为定常流动来处理。,否则，称为非定常流动。,若流场中各空间点上的一切运动要素都不随时间变化，这种流动称为定常流动。即,流体动力学基础,19,2.Trace and Streamline,Definition：,Figure 31 Trace,According to the differential equation of trace line is,（37）,When using Lagrange method to describe fluid motion the concept of trace line is introduced,(1).Trace,On special situation(x,y,z)the track of a certain fluid particle moveing with time is shown in Figure 3-1.,Basis of Fluid Dynamics,20,二、迹线和流线,定义：,根据 迹线微分方程为,（37）,流体动力学基础,用拉格朗日法描述流体运动引进迹线概念。,1、迹线,特定位置（x，y，z）处某流体质点随时间推移所走的轨 迹。如图31所示。,21,Basis of Fluid Dynamics,22,2、流线,定义：,流线的微分方程：,设流线上一点的速度矢量为流线上的微元线段矢量 根据流线定义，可得用矢量表示的微分方程为,（38）,若写成投影形式，则为,（38a）,流体动力学基础,用欧拉法形象地对流场进行几何描述，引进了流线的概念。,某一瞬时在流场中绘出的曲线，在这条曲线上所有质点的速度矢量都和该曲线相切，则此曲线称为流线。如图32。,23,example 31 Given that the velocity filed is,In it,k is constant,try to query the streamline equation.,Basis of Fluid Dynamics,24,例题31已知速度场为,其中k为常数，试求流线方程。,由式（38a）有,积分上式的流线方程为,如图33所示，该流动的流线为一族等角双曲线。,流线的性质：,解根据 及 可知流体运动仅限于 的上半平面。,流体动力学基础,（1）一般情况下，流线不能相交，且只能是一条光滑曲线；（2）在定常流动条件下，流线的形状、位置不随时间变化，且流线与迹线重合。,25,3.Stream tube,stream flow and cross section of flow,Definition:,(1).Stream tube,Take a random close curve C on fluid field,draw streamlines via every points on C,the pipe surrounded by these streamlines is called stream tube.As shown in Figure 34.,Because streamlines cant intersect fluid particles only can flow in the stream tube or via the surface of flow pipe on each time but cant go through the stream tube.so the stream tube just likes a really tube.,Basis of Fluid Dynamics,26,三、流管、流束与过流断面,定义：,由于流线不能相交，所以各个时刻，流体质点只能在流管 内部或沿流管表面流动，而不能穿越流管，故流管仿佛就是一 根真实的管子。,流体动力学基础,1、流管,在流场中取任意封闭曲线C，经过曲线C的每一点作流线，由这些流线所围成的管称为流管。如图34所示。,27,Basis of Fluid Dynamics,28,2、流束,3、过流断面,当组成流束的所有流线互相平行时，过流断面是平面；否则，过流断面是曲面。,流体动力学基础,流管内所有流线的总和称为流束。断面无穷小的流束称为微小流束，（元流）如图35中断面为 的流束。无数微小流束的总和称为总流。,定义：,与流束中所有流线正交的横断面称为过流断面。如图36所示。,定义：,29,4.Discharge and average velocity of section,(1).Discharge,Definition：,The fluid quantity through a certain spatial curved surface in unit time is called Discharge.,In this formula is the cosine of inclination of velocity vector and the unit vector in normal orientation of infinitesimal area.,Basis of Fluid Dynamics,30,四、流量与断面平均速度,1、流量,定义：,两种表示方法：,流经任意曲面的流量,（310）,式中 为速度矢量与微元面积 法线方向单位矢量 的夹角余弦。,流体动力学基础,单位时间内通过某一特定空间曲面的流体量称为流量。,31,(2).Average velocity of section,5.One-,two-,and three dimensional flow,The discharge Q flowing across the cross section of flow is divided by area of cross section A.namely,definition：,The motion factor which is the function of a coordinate is called one-dimensional flow.The motion factor which is the function of two coordinates is called two-dimensional flow.The motion factor which is the function of three coordinates is called three-dimensional flow.,definition：,Basis of Fluid Dynamics,32,2、断面平均流速,五、一元流动、二元流动、与三元流动,流体动力学基础,定义：,运动要素是一个坐标的函数，称为一元流动。运动要素是二个坐标的函数，称为二元流动。运动要素是三个坐标的函数，称为三元流动。,定义：,33,3-4 Continuity Equation,Take a infinitesimal hexahedron on a random point in fluid field.as shown in Figure 37。The mass of it changes with space and time.,(1)Space change,for example：for the x orientation the mass flowing into the hexahedron in unit time is.the mass flowing out of it is the increased mass is Also,the increased mass of y and z orientation are separately and,Basis of Fluid Dynamics,34,3-4 连续方程式,在流场的任意点处取微元六面体，如图37。六面体中的质量随空间和时间变化。,（1）空间变化,例如：对于x轴方向，单位时间流入微元六面体的质量为流出的质量为其质量增加为同样y、z 轴方向的质量增加分别为,流体动力学基础,35,namely,（312）,physical meaning：,The increased quantity of mass in space should equal to the increased quantity of mass because of the density change.,Basis of Fluid Dynamics,36,（2）时间变化,根据质量守恒定律，流体运动的连续方程式为：,即,（312）,物理意义：,流体动力学基础,空间上质量的增加量应该等于由于密度变化而引起的质量增加量。,37,（1）Steady compressible fluid，then formula（312）turns into,（314）,In column coordinate system continuity equation is,（315）,（315a）,Basis of Fluid Dynamics,38,（1）恒定压缩性流体，则式（312）变为,（314）,在柱坐标系中，连续方程式为,（315）,式中 是速度 u 在 坐标上的分量。,在球坐标系中，连续方程式为,（315a）,流体动力学基础,（2）非压缩性流体，常数，则式（312）变为,39,3-5 Motion Differential Equation of Ideal Fluid,At last section the continuity equation was discussed.It reflects the conditions that velocity field of fluid motion must satisfy.It is a kinematics equation.Now let us analyze the kinematics relations between the stress and motion of fluid.That is to build the kinematics equation of ideal fluid,1.Motion Differential Equation of Ideal Fluid（Eulers equation）,Consider the infinitesimal right-angled hexahedron whose length of sides are,as shown in Figure 38.In it the coordinate of point A is，the outside forces act on this right-angled hexahedron are two kinds:surface pressure and quality strength,Basis of Fluid Dynamics,40,3-5 理想流体的运动微分方程,上节讨论了连续性方程，它反映了流体运动速度场必须满足的条件，这是一个运动学方程。现在我们分析流体受力及运动之间的动力学关系，即建立理想流体动力学方程。,一、理想流体运动微分方程（欧拉方程）,设在x，y，z轴方向上的单位质量力为 又设流体的密度为，加速度的三个分量为,流体动力学基础,考虑如图38所示的边长为 的微元直角六面体，其中角点A坐标为，作用在此直角六面体上的外力有两种：表面压力和质量力。,41,According to the newtons second law the motion equation on x orientation is,After simplifying the upper formula the result is,Basis of Fluid Dynamics,42,根据牛顿第二定律得x方向的运动方程式为,式中,上式简化后得,流体动力学基础,43,Substitute the formula(35）into the formula（316）the result is,the upper two formulas are motion differential equation of ideal fluid.They are also called Eulers motion differential equation.,In this formula x，y，z and t are four variables.are functions of x，y，z and t and are unknown quantity.are also functions of x，y and z，they are normally known.,Basis of Fluid Dynamics,（317）,44,将式（35）代入式（316）则得,上面二式即是理想流体运动的微分方程式，也叫做欧拉运动微分方程式。,式中x，y，z，t为四个变量，为x，y，z，t的函数，是未知量。也是x，y，z的函数，一般是已知的。,流体动力学基础,（317）,45,on column coordinate system Eulers motion differential equation is,Basis of Fluid Dynamics,46,在柱坐标系中，欧拉运动微分方程为,（318）,流体动力学基础,又式中 是速度 u 在 坐标轴上的分量。分别是单位质量的外力在 坐标轴上的分量。,47,3-6 Bernoulli Equation and Its Application,Bernoulli equation is the embodiment of the law of conversation and translation of energy in fluid mechanics.,1.Bernoulli equation of the ideal fluids,Multiply the formula（316）by separately and then summate all the end,so we can obtain,(319）,Under the steady conditions,Basis of Fluid Dynamics,48,3-6 伯努利方程及其应用,伯努利方程是能量守恒与转换定律在流体力学中的具体体现。,一、理想流体的伯努利方程,将式（316）中各式分别乘以。相加得,(319）,在稳定条件下,流体动力学基础,49,so,substitute it into the formula（319）,the end is,Basis of Fluid Dynamics,50,此外，稳定流时流线与迹线重合，质点沿流线运动，故流线上速度分量为,因此,代入式（319）,流体动力学基础,51,Basis of Fluid Dynamics,52,对于质量力只有重力的不可压缩流体，z 轴垂直向上为正，则上式可写成，积分上式得,(320）,式（320）就是单位质量不可压缩理想流体在稳定流条件下沿流线的伯努利方程式。,对于同一流线上任意两点，上式可写成,（321）,物理意义：,流体动力学基础,53,2.Bernoulli equation on the collection of stream,（322）,in it，after doing integral we obtain that the whole mechanical energy relationship through two cross sections of whole fluid is,Multiply each item of the formula（321）by，then the mechanical energy relationship of the whole fluid through two cross sections of flow of infinitesimal streamline tube in unit time is,In practice we often need to solve the whole fluid flowing problems.Such as the problems that fluids flow in pipes or channels.So we need extend it to the whole fluid by doing integral on cross section of flow additionally.,Basis of Fluid Dynamics,54,二、总流上的伯努利方程,（322）,其中，积分得通过总流两过流断面的总机械能之间的关系式为,将式（321）各项同乘以，则单位时间内通过微元流束两过流断面的全部流体的机械能关系式为,在工程实际中要求我们解决的往往是总流流动问题。如流体在管道、渠道中的流动问题，因此还需要通过在过流断面上积分把它推广到总流上去。,流体动力学基础,55,（333）,So if choose a area of waterway on gradual change sections,then,Basis of Fluid Dynamics,56,其中（1）它是单位时间内通过总流过流断面的流体位能和压能的总和。,在急变流断面上，各点的 不为常数，积分困难。在渐变流断面上，流体动压强近似地按静压强分布，各点的 为常数。,（333）,因此，若将过流断面取在渐变流断面上，则积分,流体动力学基础,57,（2）is the summation of fluid mechanical energy which go through the cross section of whole fluid in unit time.Because the velocity distribution on cross section is difficult to confirm the average velocity is often used to denote the factual kinetic energy on engineering.namely,（334）,On engineering calculation is often used.,Basis of Fluid Dynamics,58,（2）它是单位时间内通过总流过流断面的流体 动能的总和。由于过流断面上的速度分布一般难以确定，工程上常用断面平均速度 来表示实际动能，即,（334）,式中 为动能修正系数,（335）,工程计算中常取。,流体动力学基础,59,substitute the formulas（334）and（335）into（333）,simplify it when considering the stable flow，the end is,（336）,This is the Bernoulli equation of collection stream of ideal fluid.,（337）,where,So the factual Bernoullis equation of whole fluid is,The factual fluid has viscosity.Because the internal frictional resistances between the fluid layers do work a portion of mechanical energy is consumed and turns into heat energy.,The average energy loss of unit gravity fluid between two areas of waterway 1 and 2 on whole fluid.,2,1,-,h,Basis of Fluid Dynamics,60,将式（334）、（335）代入式（333）中考虑到稳定流动时，化简后得,（336）,这就是理想流体总流的伯努利方程。,（337）,式中,因此实际流体总流的伯努利方程为：,实际流体有粘性，由于流层间内摩擦阻力作功会消耗部分机械能转化为热能。,流体动力学基础,61,3.Applications of bernoulli equation,example32 The relative position of a fire fighting hose,nozzle and pump is expressed in Figure 39。The exit pressure of pump(the pressure on point A)is 2 atmosphere(gage pressure).The section diameter of discharge tube of pump is 50mm；The diameter of the nozzle exit is 20mm；the head loss of fire hose is supposed 0.5m；the water head loss of nozzle is 0.1m。Try to query the velocity of flow on nozzle exit,displacement of pump and the pressure on point B.,(1)Normal irrigation calculation,Basis of Fluid Dynamics,62,三、伯努利方程的应用,例题32一救火水龙带，喷嘴和泵的相对位置如图39。泵出口压力（A点压力）为2个大气压（表压），泵排出管断面直径为50mm；喷嘴出口C 的直径20mm；水龙带的水头损失设为0.5m；喷嘴水头损失为0.1m。试求喷嘴出口流速、泵的排量及B点压力。,1、一般水力计算,流体动力学基础,63,solution The energy equation for A and C sections are,The horizontal face across the point A is datum plane，so and(in air).The specific gravity of water is Acceleration of gravity is;the height of water column is,namely；,substituting each variable into the energy equation,we get,Basis of Fluid Dynamics,64,解 取A、C两断面写能量方程：,通过A点的水平面为基准面，则；（在大气中）；水的重度 重力加速度；水柱，即,将各量代入能量方程后，得,流体动力学基础,65,the velocity of flow of nozzle exit is,The displacement of pump is,in order to calculate the pressure on point B，choose B and C sections to calculate，namely,Do horizontal datum plane across point B.then,Substitute them into the equation,The pressure is,Basis of Fluid Dynamics,66,解得喷嘴出口流速为。,而泵的排量为,为计算B点压力，取B、C两断面计算，即,通过B点作水平面基准面，则,代入方程得,解得压力,流体动力学基础,67,(2)Throttle flowmeter,Now use venturi as an example to deduce the formula of calculating the flux.,Venturi is a kind of apparatus to measure the fluid flux in the conduit under pressure.It is consisted by three parts.They are slick constricted section、throat and expansion section.As shown in Figure 310.,The fluid section contracts as the liquid in the conduit flows via the throttle equipment.The increasing of velocity of flow and the falling of pressure on the contracting section bring the differential pressure on the forwar