[researcher]

Dear CFD community,
I am analyzing a simple rectangular domain and applying 2D Navier Stokes equations (zero pressure gradient) to it. As I apply the diffusion term d2u/dy2 to the node just above the wall on the exit boundary and do backward differencing, I get a term u(i,j-2). Now, since there is no node below (i,j-1), how do I go about it. Urgent help needed, please.

[LuckyTran]

If you are doing FVM then the diffusion term is actually the divergence of fluxes. The flux is what needs to be discretized, not d2u/dy2. The boundary flux is either a boundary condition or needs to be obtained from the solution.

[researcher]

please see the picture attached to make the situation clear.

[researcher]

Originally Posted by LuckyTran

If you are doing FVM then the diffusion term is actually the divergence of fluxes. The flux is what needs to be discretized, not d2u/dy2. The boundary flux is either a boundary condition or needs to be obtained from the solution.

I am using finite difference method. Of course in FVM it is in terms of fluxes.

[researcher]

Please also make me understand (with reference to the figure I attached above), how to use ghost cells. What would be the values of, say, velocities at the ghost cells.

[LuckyTran]

In FD you directly have node values on nodes. What is your boundary condition there?

[researcher]

Its wall right in the bottom.At the top it is the free stream condition.

[FMDenaro]

Your sketch is not clear and the formula you wrote is wrong


The derivative in i,j is (f(i,j+1)-2f(i,j)-f(i,j-1))/dy^2, no point in j-2 is involved in the node just above the wall.


But what is not clear is that you are at the outflow section, what are you prescribing as BC?

[LuckyTran]

The node at i,j is on the boundary. At boundaries, you impose boundary conditions. You simply don't discretize the governing equation there. What is the discretization of the diffusion term for the node at i,j is never a question.

What is the constraint at i,j? Is it f(i,j)= some number of df/dx and df/dy= something.


Even if i,j is an interior node adjacent to a boundary node, the same applies for the node at j-1. It has boundary conditions there. The boundary conditions determine what is the value of the virtual cells.

[researcher]

Originally Posted by FMDenaro

Your sketch is not clear and the formula you wrote is wrong


The derivative in i,j is (f(i,j+1)-2f(i,j)-f(i,j-1))/dy^2, no point in j-2 is involved in the node just above the wall.


But what is not clear is that you are at the outflow section, what are you prescribing as BC?

I am reattaching the image.

[researcher]

please see the schematic.

[researcher]

Originally Posted by LuckyTran

The node at i,j is on the boundary. At boundaries, you impose boundary conditions. You simply don't discretize the governing equation there. What is the discretization of the diffusion term for the node at i,j is never a question.

What is the constraint at i,j? Is it f(i,j)= some number of df/dx and df/dy= something.


Even if i,j is an interior node adjacent to a boundary node, the same applies for the node at j-1. It has boundary conditions there. The boundary conditions determine what is the value of the virtual cells.

So you mean one should start boundary node calculations starting from top (where Neumann condition exists) next to the free stream nodes rather than starting from the wall?

[researcher]

I am not discretizing at the wall. i,j is at one node above the wall node. I am applying d2u/dx2 there. Maybe I should start from the free stream top and come down on nodes until I reach the node adjacent to the wall.Correct me please.

[FMDenaro]

Originally Posted by researcher

I am not discretizing at the wall. i,j is at one node above the wall node. I am applying d2u/dx2 there. Maybe I should start from the free stream top and come down on nodes until I reach the node adjacent to the wall.Correct me please.

I do not understand what you are doing!

You should specify your flow problem, equations, bcs and formulation to integrate the NSE.

You are just addressing a term in the x-direction momentum

[LuckyTran]

You have a boundary condition at the j-1 node. Either you have du/dy= some value and you are done or you have u(i,j-1)=some value you use this to derive the correct finite difference formula (which is not 1 -2 1, because that is the formula for an interior node). For example, for a uniform grid:

[researcher]

I am attaching the whole problem. Please see.

[LuckyTran]

This gets more and more confusing with every picture taken by your phone.

Boundary nodes don't have governing equations, there is no discretization there. You don't discretize anything at outlets. Boundaries (outlets) have boundary conditions and that's all.

[researcher]

Originally Posted by LuckyTran

This gets more and more confusing with every picture taken by your phone.

Boundary nodes don't have governing equations, there is no discretization there. You don't discretize anything at outlets. Boundaries (outlets) have boundary conditions and that's all.

Honestly, I am perplexed now. it is a rectangular domain. From left enters air at 10 m/s. There is no pressure gradient ( I have quoted the equation). If I were to find the U and V velocities at the outlet, how should these two eqns (continuity and momentum) apply.

[researcher]

Let me elaborate it further :
Now, it is desired that backward differencing be applied to both continuity and momentum equations, how should these two equations look. There has to be a ghost node outside the domain to account for the 2nd order partial term in momentum equations. Can these two equations be consistently discretized in the mentioned scheme at the outlet? I would be grateful if you could please explain.

[researcher]

my bad, forgot to tell, it is flow over a flat plate. if pressure gradient does not exist, what condition should be set at the outlet. obviously one has to investigate laminar boundary layer and velocity gradient therein.