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1、1 Prandtle BlasiusPrandtle Blasius SolutionSolution PrandtlePrandtle used boundary layer concept and imposed used boundary layer concept and imposed approximation valid for large Reynolds number flows approximation valid for large Reynolds number flows to simplify the governing to simplify the gover
2、ning NavierNavier Stokes equations H Stokes equations H BlasiusBlasius 1883 1883 1970 one of 1970 one of PrandtlPrandtl s s students solved students solved these simplified equations these simplified equations 2 從NavierNavierNavierNavier Stokes Equations Stokes Equations Stokes Equations Stokes Equa
3、tions 開始開始 開始開始 3 The The NavierNavier Stokes EquationsStokes Equations UnderUnder incompressible flow with constant viscosity incompressible flow with constant viscosity conditionsconditions the the NavierNavier Stokes equations are reduced to Stokes equations are reduced to 2 2 2 2 2 2 z 2 2 2 2 2
4、 2 y 2 2 2 2 2 2 x z w y w x w g z p z w w y w v x w u t w z v y v x v g y p z v w y v v x v u t v z u y u x u g x p z u w y u v x u u t u 這是已經過 不可壓縮 且 黏度是constant 假設 後 所得到的簡化後的Navier Stokes equations 這是已經過 不可壓縮 且 黏度是constant 假設 後 所得到的簡化後的Navier Stokes equations 4 Prandtle BlasiusPrandtle Blasius So
5、lution Solution 1 10 1 10 The details of viscous incompressible flow past any object can bThe details of viscous incompressible flow past any object can be e obtained by solving the governing obtained by solving the governing NavierNavier Stokes equation Stokes equation ForFor steadysteady two dimen
6、sionaltwo dimensional laminar flow with laminar flow with negligible negligible gravitational effectsgravitational effects these equations reduced to the following these equations reduced to the following In addition the conservation of massIn addition the conservation of mass No analytical solution
7、 2 2 2 2 2 2 2 2 y v x v y p1 y v v x v u y u x u x p1 y u v x u u 0 y v x u Prandtle進一步簡化 請注意簡化過程的假設 綜合假設 綜合假設 不可壓縮 2D 重 效應忽 黏度 constant steady 注意每一假設後 方程式會 如何改變 不可壓縮 2D 重 效應忽 黏度 constant steady 注意每一假設後 方程式會 如何改變 三個未知數 需要三個方程式 5 Prandtle BlasiusPrandtle Blasius Solution Solution 2 10 2 10 Simplific
8、ationSimplification Since the boundary layer is thin it is expected that the coSince the boundary layer is thin it is expected that the component of mponent of velocity normal to the plate is much smaller than the parallel tvelocity normal to the plate is much smaller than the parallel to the o the
9、plate and that the rate of change of any parameter across the plate and that the rate of change of any parameter across the boundary layer should be much greater than that along the flow boundary layer should be much greater than that along the flow direction That isdirection That is yx 0 y v x u 2
10、2 y u y u v x u u 再進一步簡化 1 垂直板的速度遠低於平行板的速度 2 邊界層很 薄 y方向梯度當然遠大於x方向梯度 仍無解析解仍無解析解 得再加得再加no pressure no pressure no pressure no pressure variationsvariationsvariationsvariations 剩下兩個未知數兩個方程式 v v u and u and u and u and 6 Prandtle BlasiusPrandtle Blasius Solution Solution 3 10 3 10 Governing equationsGov
11、erning equations Boundary conditionsBoundary conditions Solution Solution are extremely difficult to obtain are extremely difficult to obtain Second order partial Second order partial differential equationsdifferential equations 0 y u Uu y 0v 0u 0y 0 y v x u 2 2 y u y u v x u u 看似簡單 但 總結簡化過程後的結果總結簡化
12、過程後的結果 7 Prandtle BlasiusPrandtle Blasius Solution Solution 4 10 4 10 Blasius reduced the partial differential equations to an ordinary differential equation The velocity profile u U should be similar for all values of x Thus the velocity profile is of the dimensionless form where Is an unknown func
13、tion to be determined Is an unknown function to be determined y g U u 徒弟出手 將偏微分轉成常微分徒弟出手 將偏微分轉成常微分 設定無因次速度分佈曲線u U 與無因 次板垂直向距離y u U僅與y 有 關 關係不明 也是我們想要探討的 見下頁 8 Prandtle BlasiusPrandtle Blasius Solution Solution 5 10 5 10 Set a dimensionless similarity variableSet a dimensionless similarity variable T
14、he velocity componentThe velocity component Is an unknown function Is an unknown function to be determined to be determined and the stream functiony x U 2 1 f f x U 2 1 f x U 2 1 x f xU f x U 2 1 x f xU x v 2 1 Uf fxU yy u x U fU x 1 2 設定一個無因次變數與stream function設定一個無因次變數與stream function 代入方程式代入方程式 2
15、2 y u y u v x u u 0 y v x u 9 Prandtle BlasiusPrandtle Blasius Solution Solution 6 10 6 10 2 2 y u y u v x u u 0 y v x u 0 ff f2 d fd f d fd 2 2 2 3 3 2 2 d fd x2 U x u 2 2 d fd x UU y u 3 32 2 2 d fd x U y u 原來要解u與v 原來要解u與v 現在要解f f f 現在要解f f f 10 Prandtle BlasiusPrandtle Blasius Solution Solution 7
16、 10 7 10 With boundary conditionsWith boundary conditions f f 0 at 0 f 1 at Nonlinear thirdNonlinear third order order ordinary differential equationordinary differential equation Solution No analytical solution Solution No analytical solution Easy to integrate to obtain numerical solutionEasy to integrate to obtain numerical solution BlasiusBlasius solved it using a power series expansion aboutsolved it using a power series expansion about 0 0 BlasiusBlasius solutionsolution 0 ff f2 d fd f d fd
