1、 Fluid Mechanics12Chapter 3 Basis of Fluid Dynamics 34 Continuity Equation31 Preface32 Methods to Describe Fluid Motion33 Basic Concepts of Fluid Motion 35 Motion Differential Equation of Ideal Fluid36 Bernoulli Equation and Its Application37 System and Control Volume38 Momentum Equation39 Moment of
2、 Momentum Equation Exercises of Chapter 33 第三章第三章 流体动力学基础流体动力学基础 34 连续方程式连续方程式31 引言引言32 描述流体运动的方法描述流体运动的方法33 流体运动的基本概念流体运动的基本概念35 理想流体的运动微分方程理想流体的运动微分方程36 伯努利方程及其应用伯努利方程及其应用37 系统与控制体系统与控制体38 动量方程动量方程39 动量矩方程动量矩方程 第三章第三章 习习 题题4Chapter 3 Basis of Fluid Dynamics3-1 3-1 Preface The backgrounds,fundament
3、als 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.5第三章第三章 流体动力学基础流体动力学基础 3-1 3-1 引言引言 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。63-23-2 Methods to Describe the Fluid
4、MotionMethods 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 bei
5、ng researched objects.The essential of lagrangian method is a method of particle coordinates.73-23-2 描述流体运动的方法描述流体运动的方法描述流体运动的方法：描述流体运动的方法：一、一、拉格朗日法拉格朗日法 定义：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。8 when we use lagranges method to describe the fluid motion th
6、e 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 us
7、ing 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.9 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
8、多采用欧拉法。102.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）
9、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
10、 point changing with time which is called Eulers method(fluid field method).11二、欧拉法二、欧拉法 定义：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）。12Pressure field and density
11、field can be expressed as:（33）（34）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 dir
12、ection of space coordinates of x,y,z are:（35）13压强和密度场表示为：（33）（34）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）14The vector expression is（35a）In it Accelerate is consisted by Local accelerate:which shows the variety of velocity of fluid particles through fixed s
13、pace points changing with time.Migratory accelerate which shows variance ratio of velocity brought by the change of space situation of fluid particles.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 tota
14、l derivative，is called local derivative，is called migratory derivative.15矢量式为（35a）其中加速度的组成当地加速度 。表示通过固定空间点的流体质点速度随时间的变化。迁移加速度 。表示流体质点所在空间位置的变化所引起的速度变化率。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。163-3-3 Basic Concepts of Fluid Motion 1.Stationary flow and nonstationary flow Definition：In fact
15、ual 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 i
16、s called steady flow.that is:173-3 流体运动的基本概念一、定常流动与非定常流动一、定常流动与非定常流动 定义：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 182.Trace and Streamline Definition：Figure 31 TraceAccording to the differential equation of trace line is（37）When using Lagrange
17、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 3-1.19二、迹线和流线二、迹线和流线 定义：定义：图 31 迹 线 根据 迹线微分迹线微分方程方程为（37）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。2
18、0 Figure 32 streamline Definition：(2).Streamline When using Eulers method to describe fluid motion vividly the concept of Streamline is introduced A streamline is a curve which is drawed on fluid field in a certain instant.On this curve velocity vector of all particles are tangent with the curve.Jus
19、t as shown in Figure 32。If the formula(3-8)is expressed by projection form，then it is The differential equation of streamline:Suppose the velocity vector of a certain point on srteamline is the micro unit segment vector on streamline is ,According to the definition of streamline the differential equ
20、ation expressed by vector is（38）（38a）21 图 32 流 线2、流线 定义：定义：流线的微分方程：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）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。22 example 31 Given that the velocity filed is In it,k is constant,try to query the st
21、reamline equation.from formula（38a）we can get integral of it is solution According to and we can obtain that the fluid motion is only limit to the upper half plane of .Just as shown in Figure 33,the flowing streamlines are a group of equiangular hyperbolas.Figure 33 hyperbolic streamline (1)On norma
22、l circumstance streamlines cant intersect ,moreover it must be smoothed curves.(2)On the condition of steady flow the shape and situation cant change with time.characters of streamline:23 例题例题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：流线的性质：解解根据 及 可知流体运动仅限于 的上半平面。图33双曲流线 （1）
23、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。243.Stream tube,stream flow and cross section of flow Definition:Figure 34 stream tubeFigure 35 stream flow and whole streamFigure 36 cross section of flow(1).Stream tube Take a random close curve C on fluid field,draw streamlines via every
24、 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 reall
25、y tube.25三、流管、流束与过流断面三、流管、流束与过流断面 定义：定义：图 34 流管图35流束和总流图 36 过 流 断 面 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。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。26(2)Stream flow The summation of all streamlines in stream tube is called stream flow.The stream whose sections is in
26、finitesimal is called elementary flow.As in Figure 3-5 the stream tube whose section is .The summation of countless elementary flow is called whole stream.Definition：(3)Cross section of flow When all the streamlines which consist the streamline tube keep parallel the cross section is a plane or else
27、 the Cross section is a curve surface.The transects which keep orthogonal with all the streamlines in the streamline tube are called cross section of flow.As shown in Figure 3-6.Definition：272、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
28、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：定义：284.Discharge and average velocity of section(1).Discharge Definition：Two kinds of expressing methods:The method which is expressed by the fluid volume in unit time is called volumetric flow rate or discharge.That is The method which is expressed by the fluid ma
29、ss in unit time is called mass flow.That is .The discharge flowing via the random curved surface is（310）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 orientati
30、on of infinitesimal area .29四、流量与断面平均速度四、流量与断面平均速度1、流量 定义：定义：两种表示方法：两种表示方法：以单位时间通过的流体体积表示，称为体积流量（流量），记为以单位时间通过的流体质量表示，称为质量流量，记作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。单位时间内通过某一特定空间曲面的流体量称为流量。30(2).Average velocity of section5.One-,two-,and three dimensional flow（311）It is average velocity of sect
31、ion.The discharge Q flowing across the cross section of flow is divided by area of cross section A.namelydefinition：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 mot
32、ion factor which is the function of three coordinates is called three-dimensional flow.definition：312、断面平均流速 五、一元流动、二元流动、与三元流动五、一元流动、二元流动、与三元流动 （311）即为断面平均速度。流经过流断面的体积流量Q除以过流断面面积A，即定义：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：定义：323-4 Continuity Equation Take a infinitesimal
33、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 Figure 3 7for 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 o
34、f y and z orientation are separately and()()dzzdxdyudyydxdzuzy-rr 333-4 3-4 连续方程式连续方程式 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 图 3 7 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34namely（312）physical meaning：The increased quantity of mass in space should equal to the increased quantity
35、of mass because of the density change.（2）Time change Suppose the quality strength in the infinitesimal hexahedron is at any time.In unit time it turns into ,because the change of density the increased mass in the infinitesimal hexahedron in unit time isAccording to the law of mass conservation the c
36、ontinuity equation of fluid motion is：dxdydz+r()tdxdydzr35（2）时间变化 设任意时刻微元六面体内的质量力为 ，单位时间内变为 ，所以由于密度 的变化单位时间内微元六面体内增加的质量为 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：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。36（1）Steady compressible fluid ，then formula（312）turns into（313）（314）In column coordinate system continuity equation i
37、s（315）In it are components of velocity u on coordinates.In sphere coordinate system continuity equation is（315a）（2）incompressible fluid，is constant，then formula(312)turns into r37（1）恒定压缩性流体，则式（312）变为（313）（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）（2）非压缩性流体，常数，则式（312）变为 383-5 3-5 Mot
38、ion 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 t
39、he kinematics equation of ideal fluid1.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 ki
40、nds:surface pressure and quality strength Suppose the unit quality strengths on the x,y and z orientation are and the density of the fluid is ，then three components of acceleration are,zyxfff393-5 3-5 理想流体的运动微分方程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
41、理想流体运动微分方程（欧拉方程）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 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。表面压力和质量力。40According to the newtons second law the motion equation on x orientation is After simplifying the upper formula the result is（316）In a similar way Figure 3 8
42、dtdudxdydzdydzxpppdydzdxdydzfxrr=+-+In this formula()is pressure。zyxpp,=41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（316）同理 图 3 842Substitute 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 equa
43、tion.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.（317）43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。（317）44 on column coordinate sys
44、tem Eulers motion differential equation is（318）zuuruuruutuzpfruuzuuruuruuturpfruzuuruuruuturpfzzzzrzzrzrrzrrrrr+=-+=-+=-qrqqrqrqqqqqqJqqq2 in the formula,and are components of velocity u on coordinate axis .are components of outside force of unit mass on the coordinate axis of respectively.uruquz45
45、在柱坐标系中，欧拉运动微分方程为（318）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。463-6 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 fluidsMultiply the formula（316）by separately an
46、d then summate all the end,so we can obtain(319）Under the steady conditions473-6 3-6 伯努利方程及其应用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48sosubstitute it into the formula（319）,the end is in additional，when fluid keeps in steady flow the streamlines an
47、d traces are coincident and particles moves along streamlines,So the velocity components of streamlines are49 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）50 physical meaning：The kinetic energy of unit gravity fluid or is called specific kinetic energy.The pressure energy of unit gravity fluid or is ca
48、lled specific energy of pressure.The potential energy of unit gravity fluid or is called specific energy of position gugpz22r(320）The formula（320）is Bernoulli Equation of the unit quality incompressible fluid along streamlines under the steady flow conditions.For random two points on the same stream
49、line the upper formula can be rewrited as（321）the integral for it For the incompressible fluid whose quality strength is only the gravity the upper orientation of z axis is positive.So the upper formula can be rewrited as .51 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
50、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：物理意义：522.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 relati
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