BasisofFhid Dynamic (2 ). Streamline When using Euler s method to describe fluid motion vividly the concept of Streamline is introduced Definition 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. Just as shown in Figure 3---2 The differential equation of streamline. ∠ Suppose the velocity vector of a certain point on srteamline is u=ui+uv+u. k, the micro unit segment ∠ vector on streamline is ds= dxi dyj +dck 3 ccordin g to the definiti n of streal ne the diffe ferential y equation expressed by vector is 花×ds=0 (38) If the formula(3-8)is expressed by projection form, then it is Figure 3---2 dz l(xy,=,)u,(x,y)2(x,y2:) 3--8a streamline
21 x y z o 1 2 3 4 Figure 3—2 streamline Definition： (2). Streamline When using Euler’s 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 . Just as shown in Figure 3—2。 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 equation expressed by vector is u u i u j u k, x y z = + + ds dxi dyj dzk = + + uds = 0 （3—8） ( ) ( ) u (x y z t) dz u x y z t dy u x y z t dx x y z , , , , , , , , , = = （3—8a）
滤动之学基础 2、流线 用欧拉法形象地对流场进行几何描述,引进了流线的概念。 定义 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。 流线的微分方程: 设流线上一点的速度矢量为=u21+y+uk,+z 流线上的微元线段矢量ds=cxi+y+k, 根据流线定义,可得用矢量表示的微分方 程为 i×ds=0 (38) y 若写成投影形式,则为 x dz n(y,.)1(x,y,)=n(x,y,2 (38)图32流线 22
22 x y z o 1 2 3 4 图 3—2 流 线 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 u u i u j u k, x y z = + + ds dxi dyj dzk , = + + uds = 0 （3—8） 若写成投影形式，则为 ( ) ( ) u (x y z t) dz u x y z t dy u x y z t dx x y z , , , , , , , , , = = （3—8a） 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3—2
BasisofFhid Dynamic example 3-1 Given that the velocity filed is u=kx uy==ky (20)In it, k is constant,try to query the streamline equation 0 [solution According to u,=0 and y20 we can obtain that the fluid motion is only limit to the upper half plane of xov from formula (3--8a) we can get dy kx一k integral of it is xy=C K (y) Just as shown in Figure 3-3, the flowing streamlines are a group of equiangular hyperbolas Figure 3-3 hyperbolic streamline characters of streamline (1 )On normal circumstance streamlines can'tintersect, moreover it must be smoothed curves (2 )On the condition of steady flow the shape and situation can' t change with tr Ime 23
23 [example 3—1] Given that the velocity filed is = 0 = − = z y x u u ky u kx (y 0) In it, k is constant ,try to query the streamline equation . from formula（3—8a）we can get ky dy kx dx − = integral of it is xy = c [solution] According to and we can obtain that the fluid motion is only limit to the upper half plane of . uz = 0 y 0 xoy Just as shown in Figure 3—3, the flowing streamlines are a group of equiangular hyperbolas . (x, y) x y o Figure 3—3 hyperbolic streamline (1)On normal circumstance streamlines can’t intersect ,moreover it must be smoothed curves . (2)On the condition of steady flow the shape and situation can’t change with time. characters of streamline:
滤动之学动 [例题3—1已知速度场为 u =kx y=-ky(y≥0)其中为常数,试求流线方程 u=0 解]根据u2=0及y≥0可知流体运动仅限于xoy的上半平面 由式(3-8a)有 kx k 积分上式的流线方程为xy=c 77777777 图3-3双曲流线 如图3-3所示,该流动的流线为一族等角双曲线。 流线的性质: (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 24
24 [例题3—1]已知速度场为 = 0 = − = z y x u u ky u kx (y 0) 其中k为常数，试求流线方程。 由式（3—8a）有 ky dy kx dx − = 积分上式的流线方程为 xy = c 如图3—3所示，该流动的流线为一族等角双曲线。 流线的性质： [解]根据 uz = 0 及 y 0 可知流体运动仅限于 xoy 的上半平面。 (x, y) x y o 图3—3双曲流线 （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
BasisofFhid Dynamics 3. Stream tube, stream flow and cross section of flow (1)Stream tube Definition 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 3-4. dA A2 Figure 3-4 Figure 3-5 stream Figure 3-6 cross section stream tube flow and whole stream of flOw Because streamlines cant intersect fluid particles only can flow in the stream tube or via the surface of flow pipe on each time but can t go through the stream tube. so the stream tube just likes a really tube 25
25 3. Stream tube , stream flow and cross section of flow Definition: L Figure 3—4 stream tube A1 A2 dA1 dA2 Figure 3—5 stream flow and whole stream Figure 3—6 cross section of flow (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 3—4. Because streamlines can’t intersect fluid particles only can flow in the stream tube or via the surface of flow pipe on each time but can’t go through the stream tube . so the stream tube just likes a really tube