BasisofFhid Dynamic 2. Euler’ s method Definition 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 Euler's method(fluid field method) When we use euler 's 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 u,r,y,z ln=v,(x,y,2,) (32) ur=u(,y,z, t)
11 2. Euler’s method Definition： When we use Euler’s 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 Euler’s variables. So the velocity field can be expressed by the following formulas: ( ) ( ) u u (x y z t) u u x y z t u u x y z t z z y y x x , , , , , , , , , = = = （3—2） 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 Euler’s method (fluid field method)
滤动之学基础 、欧拉法 定义: 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)。 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: u,=u(x,y,z, 4) u,=u, x,y,z, 1) (3-2) u =ux,y, 2 12
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： ( ) ( ) u u (x y z t) u u x y z t u u x y z t z z y y x x , , , , , , , , , = = = （3—2） 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）
BasisofFhid Dynamics Pressure field and density field can be expressed as p=p(x,y,2 (3—3) p=pl,y (3-4) In the formula (3-2)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: dx_,dy_,,dz The acceleration components in direction of space coordinates of x, y, Zare du au u C x dt +u +u +u at du au (3-5) +. +u dt at O C.三 +L.-+ 13 dt at ax Ov 0z
13 Pressure field and density field can be expressed as: ( ) (x y z t) p p x y z t , , , , , , = = （3—3） （3—4） In the formula（3—2）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: x y uz dt dz u dt dy u dt dx = , = , = The acceleration components in direction of space coordinates of x, y, z are: z u u y u u x u u t u dt du a z u u y u u x u u t u dt du a z u u y u u x u u t u dt du a z z z y z x z z z y z y y y x y y y x z x y x x x x x + + + = = + + + = = + + + = = （3—5）
滤动之学基础 压强和密度场表示为: p=p,y (3—3) p=pl,y,2 (3-4) 式(3-2)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 x dz = 2 一=L dt dt 可得加速度在空间坐标x,y,z方向的分量为 tu +u xtu-x dt at +u ouy tuy ay (3-5) +u dt at ax dt a==+.+L +u 14 dt at L OX
14 压强和密度场表示为： ( ) (x y z t) p p x y z t , , , , , , = = （3—3） （3—4） 式（3—2）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 x y uz dt dz u dt dy u dt dx = , = , = 可得加速度在空间坐标x，y，z方向的分量为 z u u y u u x u u t u dt du a z u u y u u x u u t u dt du a z u u y u u x u u t u dt du a z z z y z x z z z y z y y y x y y y x z x y x x x x x + + + = = + + + = = + + + = = （3—5）
BasisofFhid Dynamics The vector expression is a- u ou +(uV)(3-5a) ot In it v- +J-+k ay az Local accelerate which shows the variety of velocity of fluid particles through fixed space points changing with Accelerate is time consisted by Migratory accelerate. Vhi which showsvariance ratio of velocity brought by the change of space situation of fluid particles When using Euler's method to query variance ratio of other motion factors of fluid particle changing with time the normal formula is +uV) (36) dt at dt is called total derivative,o is called local derivative at (l V)is called migratory derivative 15
15 The vector expression is (u )u t u dt du a + = = （3—5a） In it z k y j x i + + = Accelerate is consisted by Local accelerate: which shows the variety of velocity of fluid particles through fixed space points changing with time. Migratory accelerate which shows variance ratio of velocity brought by the change of space situation of fluid particles. t u (u )u When using Euler’s method to query variance ratio of other motion factors of fluid particle changing with time the normal formula is + ( ) = u dt t d （3—6） is called total derivative ， is called local derivative， is called migratory derivative. dt d t (u )