5动力学2MicrosoftPowerPoint演示文稿.ppt

1、Dynamics of Fluid Motion 一 Euler Equation of Motion 二 Bernoullis Equation 三 Bernoullis Equation for Streamtube 四 Momentum Equation 五 Moment of Momentum Equation,一 Euler Equation of Motion 1. Euler Equation * Consider a small element * Body force: * For ideal fluid: There are no shearing stresses. *

2、Application of Newtons second law in X direction Simplifying the equation above, we can obtain,in a similar manner, we have Euler Equation Note that: Euler equation must apply at every point throughout the fluid. It is valid only for an ideal fluid. It is applicable to both compressible and incompre

3、ssible flow.,2. Euler Equation in Sreamline Coordinates （流线坐标） * Streamline ( Natural) Coordinates Streamline coordinate is usually confined to flows that can be specified by two coordinate directions. The coordinate direction s is along a streamline and is taken as positive in the direction of flow

4、. The normal direction n is positive inward.,The advantage of this system is that there is no velocity component in n direction, the velocity must be tangent to the streamline. 的分量形式可表达为： 加速度：,其中,* Consider a small element which is cylindrical tube with cross section dA and length dl. (l: an arbitra

5、ry direction in flow field) Application of Newtons second law in l direction Now, pick the l direction as the streamline direction for a steady flow,Let the l direction be n direction of streamline coordinates: for incompressible flow 分析： 若流动发生在水平方向，得到： 上式说明：沿着压力降的方向上将有加速度产生，也就是说沿着等压线的方向上无加速度，如果有加速度

6、的话，只可能发生在与等压线垂直的方向上。, 记 piezometric head（测压水头） 对于不可压缩流体： We may immediately note that there can be no acceleration along a line of constant piezometric head. The acceleration in general must be perpendicular to lines of constant piezometric head and in the direction of decreasing values of h.,二 Bern

7、oullis Equation Daniel Bernoulli ( 1700-1782), Swiss, mathematician, published this equation in his Hydrodynamica (1738). 上式即为伯努利方程。 应用条件： 理想， 不可压， 定常， 重力流体， 同时 在一条流线上。,* 伯努利方程还可写作： 各项的物理意义： 三项分别是单位重量流体的压 力势能、位置势能和动能，或压力头（pressure head）、位置 头（elevation head）和速度头（velocity head），三项之和称 为 total head。 伯努利

8、方程的物理意义： The total head is a constant along a streamline in steady, incompressible, frictionless flows.,Example: Both tanks of Fig. 2 have a liquid depth z. Tank (a) discharges a jet of diameter D through the rounded orifice from whence it falls freely, whereas tank (b) has a pipe with rounded entra

9、nce connected to it, and after a length L, it discharges a jet also of diameter D. Which of the systems has the greater discharge through it ? Example: A siphon is used to draw water from the tank of Fig. 4. If the siphon line has a diameter of 6 cm and must rise 1m above the water level in the tank

10、, what is the maximum discharge that can be obtained without cavitation.,三 Bernoullis Equation for Streamtube （总流的伯努利方程） 为了从元流能量方程推出总流能量方程，需研究压力在过 流断面上的分布 * 缓变流（渐变流）、急变流 流线间夹角很小，流线的曲率半径很大，近乎平行直线流动。 在缓变流中，可导出在同一截面上任意点的 为常数(不 证)。,888,*使用上式时，应注意： 1、2两截面应取在管道的缓变流处 y1、y2常取截面形心处的值 V1、V2是平均速度 * 在工业上，计算得1.0

11、11.10之间，常取1.0，于是： * 对于圆截面，且速度分布是抛物线型， 2.0 * 流体流动越紊乱， 值越趋近于1.0 * 对于定常、不可压、无功交换，但有摩擦损失的流动有：,其中，hl为1截面到2截面之间的损失 * If the friction occurs, the total head must decrease in the flow direction. * Example A turbulent velocity distribution between parallel flat plates can be approximated by Calculate the ene

12、rgy correction factor,四 Momentum Equation 1. 推导过程 某系统的总动量： 系统的动量方程： 由雷诺输运定理： Since Reynolds transport theorem relates the system to the control volume at the instant the system passes through the control volume, the forces on the system are identical to those on the control volume. Hence,上式即为作用于控制体的

13、动量方程。 As applied to the control volume, the general momentum equation states that the net force is equal to the sum of the time rate of increase of momentum in the control volume and the net rate of flow of momentum across the control surface. 2. 定常流动动量方程的一元近似处理 定常流动下，上式变为： x方向的分量方程： 取控制体为流管：,设 V为有效

14、截面上的平均速度，U是平均速度V在x方向的分量， 工业上，比更接近于1，取1 又m1=m2，得出： 注意： 上式适用于定常流动及流管的条件下。 推导中没有对流体的种类加以限制，所以上式既适合于理想 流体 ，也适用于实际流体，既适合于不可压流体，也适合于可 压缩流体。, 外力包括作用在控制体上的全部质量力和控制面上的表面力。 动量方程只涉及控制面上的参数，不必考虑控制体内部的情况。 动量方程是矢量方程，所以，力矢量和速度投影到坐标轴上时 要注意正、负号。 Example ： A liquid jet strikes a curved plate as in Fig. 1 (1) If the 3

15、-in.-diameter jet with a velocity of 50 ft/s is deflected through an angle of 70when it hits the stationary vane, determine the horizontal and vertical components of the force of water on the vane. (2) Repeat part (a) with the one change that the vane no longer remains stationary but rather moves to

16、 the right at 15 ft/s. Also determine the actual velocity of the water leaving the vane.,Example Determine the force on the joint (at section A just upstream of the contracting nozzle ) if the discharge is 1 cfs. As shown in Fig. 2, the pipe is 4 in. in diameter and the nozzle discharges a 2-in. dia

17、meter jet of water. Assume that =1 at both sections A and B.,五 Moment of Momentum Equation （动量矩方程） * The Vector or Cross Product ( ) This product is a vector quantity defined as the product of the magnitudes of the two vectors and the sine of the smaller angle between them. The direction of is perpe

18、ndicular to the plane of and such that if , the vectors , and form a right-hand system. Also, if then C = AB sin 0 ,动量方程： 将两边都对同一轴取矩： The equation states that, relative to a common point, the algebraic sum of the moment of the forces (i. e., the torques) acting on the fluid equals the sum of the time rate of increase of of angular momentum within the control volume and the net rate of flow of angular momentum from the control volume. 在定常、流管条件下： 动量矩方程的实用形式： （去掉积分号，平均流速代替瞬时速度）,Example: Water at the rate of 2.5 l/s enters the hydraulic machine of Fig. 1 at the axis. It then