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Hello guys, I'm writing to get some help on an exercise I've been thinking but I can't get to solve.
I have to write the code for the Example 8.5 of the book White, Fluid Mechanics. Here is the problem and the solution I have to obtain.
It is about one duct that has three sections in which I have to obtain the value of the stream function in every node of the mesh. The PDE I have to solve is ##\dfrac{\partial^{2}\psi}{\partial x^{2}} + \dfrac{\partial^{2}\psi}{\partial y^{2}} = 0##, I have approximated it using the finite element difference (2n order) and with ##\Delta x = \Delta y = 0## I have obtained the approximation: ##\psi_{i,j} = \dfrac{1}{4} (\psi_{i+1,j} + \psi_{i-1,j} + \psi_{i,j+1} + \psi_{i,j-1})##, that is that I need the upper, lower, right and left values to obtain the ##\psi_{i,j}## value at any point of the mesh.
I know the boundary conditions which are:
Through the whole upper wall: ##\psi = 10##
Through the whole lower wall: ##\psi = 0##
Inlet: ##\psi(1,j) = 2\cdot (j-6)## from: ## j=7 ## to: ##j=10##
Outlet: ##\psi(16,j) = j-1 ## from: ##j=2## to: ##j=10##
Up to this point, everything is OK. But the problem is that I need to solve the mesh (inner nodes are unknown), and I only know boundary nodes. How can I compute the inner nodes values if I need 4 values (to solve: ##\psi_{i,j} = \dfrac{1}{4} (\psi_{i+1,j} + \psi_{i-1,j} + \psi_{i,j+1} + \psi_{i,j-1})##) ?
I need to compile a program with Matlab, but my problem is not coding the program... It is that I don't know how can I do it to use the last equation to obtain the inner nodes values...
Does anyone know how to do it? Any help is useful.
Thank you very much. I hope you guys understand which is my problem with this exercise.
I have to write the code for the Example 8.5 of the book White, Fluid Mechanics. Here is the problem and the solution I have to obtain.
It is about one duct that has three sections in which I have to obtain the value of the stream function in every node of the mesh. The PDE I have to solve is ##\dfrac{\partial^{2}\psi}{\partial x^{2}} + \dfrac{\partial^{2}\psi}{\partial y^{2}} = 0##, I have approximated it using the finite element difference (2n order) and with ##\Delta x = \Delta y = 0## I have obtained the approximation: ##\psi_{i,j} = \dfrac{1}{4} (\psi_{i+1,j} + \psi_{i-1,j} + \psi_{i,j+1} + \psi_{i,j-1})##, that is that I need the upper, lower, right and left values to obtain the ##\psi_{i,j}## value at any point of the mesh.
I know the boundary conditions which are:
Through the whole upper wall: ##\psi = 10##
Through the whole lower wall: ##\psi = 0##
Inlet: ##\psi(1,j) = 2\cdot (j-6)## from: ## j=7 ## to: ##j=10##
Outlet: ##\psi(16,j) = j-1 ## from: ##j=2## to: ##j=10##
Up to this point, everything is OK. But the problem is that I need to solve the mesh (inner nodes are unknown), and I only know boundary nodes. How can I compute the inner nodes values if I need 4 values (to solve: ##\psi_{i,j} = \dfrac{1}{4} (\psi_{i+1,j} + \psi_{i-1,j} + \psi_{i,j+1} + \psi_{i,j-1})##) ?
I need to compile a program with Matlab, but my problem is not coding the program... It is that I don't know how can I do it to use the last equation to obtain the inner nodes values...
Does anyone know how to do it? Any help is useful.
Thank you very much. I hope you guys understand which is my problem with this exercise.