Solving Navier-Stokes Numerically (Discretization)

kokogu · March 15, 2024, 17:24

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.

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

LuckyTran · March 16, 2024, 00:30

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

kokogu · March 16, 2024, 04:30

Quote:

Originally Posted by LuckyTran

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

FMDenaro · March 16, 2024, 06:01

Quote:

Originally Posted by kokogu

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.

kokogu · March 16, 2024, 06:27

Quote:

Originally Posted by FMDenaro

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.

FMDenaro · March 16, 2024, 06:38

Quote:

Originally Posted by kokogu

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.

LuckyTran · March 16, 2024, 15:19

Quote:

Originally Posted by kokogu

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.

March 15, 2024, 17:24	Solving Navier-Stokes Numerically (Discretization)	#1
kokogu New Member Kokogu Join Date: Mar 2024 Posts: 3 Rep Power: 2	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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March 16, 2024, 00:30		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,762 Rep Power: 66	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