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Solving Navier-Stokes Numerically (Discretization)

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Old   March 15, 2024, 17:24
Default Solving Navier-Stokes Numerically (Discretization)
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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.

\rho\frac{\partial u}{\partial t}=-\frac{\partial p}{\partial x}+\mu\frac1r\frac\partial{\partial r}\Big(r\frac{\partial u}{\partial r}\Big)

Boundary conditions for the problem are symmetry at the centre and no-slip on the walls. Initially, the fluid is at rest.

\begin{aligned}
& \frac{\partial u}{\partial r}(r=0, t)=0 \\
& u(r=R, t)=0 \\
& u(r, t=0)=0
\end{aligned}

Last edited by kokogu; March 19, 2024 at 15:39.
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Old   March 16, 2024, 00:30
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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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Old   March 16, 2024, 04:30
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Quote:
Originally Posted by LuckyTran View Post
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
Yup read it but, still, I cannot discretise the spatial term.

\frac\partial{\partial r}\left(r\frac{\partial u}{\partial r}\right)\approx\frac1{\Delta r}\left(r_{i+1/2}\frac{u_{i+2}^{j}-u_{i+1}^{j41}}{\Delta r}-r_{i}\frac{u_{i}^{j}-u_{i-1}^{j+1}}{\Delta r}\right)

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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Old   March 16, 2024, 06:01
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Quote:
Originally Posted by kokogu View Post
Yup read it but, still, I cannot discretise the spatial term.

\frac\partial{\partial r}\left(r\frac{\partial u}{\partial r}\right)\approx\frac1{\Delta r}\left(r_{i+\frac12}\frac{u_{i+1}^{j}-u_{i}^{j}}{\Delta r}-r_{i-\frac12}\frac{u_{i}^{j}-u_{i-1}^{j}}{\Delta r}\right)

Maybe like this? But I could not grasp why are we doing smth like this. Also I am gonna solve this with MATLAB



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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Old   March 16, 2024, 06:27
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Originally Posted by FMDenaro View Post
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.
Could you help me discretize this? For FTCS, BTCS, and Crank-Nicolson Method.
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Old   March 16, 2024, 06:38
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Originally Posted by kokogu View Post
Could you help me discretize this? For FTCS, BTCS, and Crank-Nicolson Method.

Since that seems a list of homeworks, start to code, when you have a specific question, ask here.
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Old   March 16, 2024, 15:19
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Quote:
Originally Posted by kokogu View Post
Maybe like this? But I could not grasp why are we doing smth like this. Also I am gonna solve this with MATLAB
If you don't discretize it, then it is a 2D PDE. Do you know how to solve a PDE? If you discretize it, then it becomes an algebraic formula (for FTCS) or it becomes a linear system (for BTCS and Crank-Nicolson). Do you know how to solve Ax=b? That's why you discretize it.


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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