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Solving Navier-Stokes Numerically (Discretization) |
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March 15, 2024, 17:24 |
Solving Navier-Stokes Numerically (Discretization)
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#1 |
New Member
Kokogu
Join Date: Mar 2024
Posts: 3
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Unsteady, incompressible, laminar and fully developed flow inside a circular pipe can be expressed by following the 1D parabolic PDE form of the Navier-Stokes equation where u(r,t) is to be calculated.
Boundary conditions for the problem are symmetry at the centre and no-slip on the walls. Initially, the fluid is at rest. Last edited by kokogu; March 19, 2024 at 15:39. |
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March 16, 2024, 00:30 |
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#2 |
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Lucky
Join Date: Apr 2011
Location: Orlando, FL USA
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https://en.wikipedia.org/wiki/Crank%...icolson_method
see also the table on forward euler and backward euler for a very obvious hint on how to do those cases as well |
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March 16, 2024, 04:30 |
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#3 | |
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Kokogu
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Quote:
Maybe like this? But I could not grasp why are we doing smth like this. Also I am gonna solve this with MATLAB Last edited by kokogu; March 19, 2024 at 15:40. |
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March 16, 2024, 06:01 |
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#4 | |
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Filippo Maria Denaro
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Quote:
This is just the formula obtained by the successive discretization from the outer to inner operator d/dx using a centred second order discretization on the step dr. This is done to avoid a larger stencil having the issue of the checkerboard modes. |
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March 16, 2024, 06:27 |
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#5 |
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Kokogu
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Could you help me discretize this? For FTCS, BTCS, and Crank-Nicolson Method.
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March 16, 2024, 06:38 |
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#6 |
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Filippo Maria Denaro
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March 16, 2024, 15:19 |
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#7 | |
Senior Member
Lucky
Join Date: Apr 2011
Location: Orlando, FL USA
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Quote:
Furthermore, the general N-S and other important PDE's are non-linear. In this example you start with the parabolised navier-stokes which is linear to begin with. But discretizing a non-linear PDE also results in the same algebraic equations (for FTCS) and linear system (for BTCS and CN). The reason you are being assigned parabolised navier-stokes, is the discretization is simple and it facilitates learning how to discretize PDE's. Clearly if students struggle with discretizing the parabolised navier-stokes equations, then the full navier-stokes is even more daunting and then students baby-rage and quit prematurely. |
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Tags |
btcs, discretisation, ftcs, navier stokes equation, navier-stokes solver |
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