quiz_2_sol - 2.29: Numerical Fluid Mechanics Solution of...

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Unformatted text preview: 2.29: Numerical Fluid Mechanics Solution of Quiz 2 MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.29 NUMERICAL FLUID MECHANICS— SPRING 2007 Solution of Quiz 2 Takehome 48 hours, Totally 25 points Due Thursday 4 p.m. 05/17/07, Focused on Lecture 12 to 25 (Last Lecture) Note that you are not allowed to collaborate or share your thoughts about the problems. Please state your assumptions and write down clearly what you think about the problems even if you cannot solve them to the endpoint. Furthermore, note that we do not need you to attach your codes and we will not look through them to find what you have done, instead explain your method. Problem 1 (5 points): The boundary layer equation for the self-similar incompressible flow over a flat plate can be cast as the following equation and boundary conditions set: f ''' + 1 2 f '' f = 0, where f ''' = d 3 f dx 3 , f '' = d 2 f dx 2 f (0) = f '(0) = f '( ! ) = 1 " # $ % $ 1. Solve the equation with your method of choice and plot the f(x) curve. 2. Find at which “x” the f ' value becomes equal to “0.99” . A minimum accuracy equal to 0.1% is expected. Solution: Note that this is a nonlinear differential equation and unfortunately most methods that we have studied (like finite difference and finite elements) cannot be applied routinely to this problem. However, while solution methods of BVP’s (BVP: boundary value problem) are usually limited to linear equations, we can solve arbitrary complicated ODEs (ODE: ordinary differential equations). 2.29: Numerical Fluid Mechanics Solution of Quiz 2 The key trick here is to transform the BVP into an ODE and it is accomplished by shooting method. Basically we choose an arbitrary value for f ''(0) and solve the below ODE set. Then we look at asymptotic behavior of our function at rather a large value, where we see a stationary value for f '( ! ) . Next we update the f ''(0) by trial and error (or more advanced techniques of root finding applied to f '( ! ) " 1 ) until we achieve our required accuracy. Y ( x ) = y 1 y 2 y 3 ! " # # # $ % & & & = f f ' f " ! " # # # $ % & & & dY dx = f ' f '' f "' ! " # # # $ % & & & = y 2 y 3 ’ 1 2 y 1 y 3 ! " # # # # # $ % & & & & & The whole process is done in attached “C2p29_Quiz2_1.m” file and here we have plots for some initial guesses on solution. 2.29: Numerical Fluid Mechanics Solution of Quiz 2 By playing with program we can find the right value of f ''(0) ! .33206 . This value can be deducted by following different runs summarized below (in each set the last value is based on linear approximation of 2 nd and 3 rd value). Note that “ode” default settings are based on a relative and absolute tolerance about 10 ! 3 and 10 ! 6 . On the other hand we have set the relative and absolute tolerance of error to conservative value of 10 ! 10 and 10 ! 12 to ensure the reliability of our printed value to 9 significant digits....
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.29 taught by Professor Henrikschmidt during the Spring '07 term at MIT.

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quiz_2_sol - 2.29: Numerical Fluid Mechanics Solution of...

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