[samycfd]

1)

the RHS is written by setting dP/dx|i-1/2= (axd-axc)i-1/2=0 at wall.

Why (axd-axc)i-1/2=0 too ? The condition at the wall is just on pressure? Oh but the velocity (u and v ) at the wall in the lid cavity is zero ? that's the reason ?

[FMDenaro]

Originally Posted by samycfd

1) Why (axd-axc)i-1/2=0 too ? The condition at the wall is just on pressure? Oh but the velocity (u and v ) at the wall in the lid cavity is zero ? that's the reason ?

if you fix dP/dn|i-1/2=0, the momentum equation says that you must fix also (axd-axc)i-1/2=0...it's a consequence of the equation, nothing to do with the fact that the velocity has zero value at wall

[FMDenaro]

just to let you see again the previous equation,

dP/dn = n.(ad-ac)

at section i-1/2 (supposed a wall) both RHS and LHS are zero. Note that does not depend on the condition v=0 at wall but is consequence of the fact you set homogeneous Neumann condition for the pressure

[samycfd]

Yes of course. Sorry. Now I got it.

I could also let dP/dn = n.(ad-ac) at walls and that's all. Why setting dp/dn=0 ?

[FMDenaro]

Originally Posted by samycfd

Yes of course. Sorry. Now I got it.

I could also let dP/dn = n.(ad-ac) at walls and that's all. Why setting dp/dn=0 ?

It could sound strange, but it is mathematically equivalent....just use the non-homogeneous Neumann condition

dP/dn|i-1/2 = (axd-axc)|i-1/2

substitute it in the equation

[(dP/dx)i+1/2 - (dP/dx)i-1/2]/dx = [(axd-axc)i+1/2 - (axd-axc)i-1/2]/dx

and you will se exactly the same.

The difference appears only at the end when you want, after computing the pressure in the inner nodes, to compute the pressure at walls to have the full pressure field.

[samycfd]

Oh yes it's kind of hidden but obvious now..

So I have to find a way of integrating this condition in the FE method.

[FMDenaro]

Originally Posted by samycfd

Oh yes it's kind of hidden but obvious now..

So I have to find a way of integrating this condition in the FE method.

yes, I think you have to work in the correct construction of the source term taking into account for the congruent BC.s

[samycfd]

Hi,

I was wondering, which BCs I have to set when I am computing the derivatives of velocity in the RHS ?

[FMDenaro]

Originally Posted by samycfd

Hi,

I was wondering, which BCs I have to set when I am computing the derivatives of velocity in the RHS ?

hello,
the velocities at walls are zero (for not moving wall), you compute the terms (axc-axd) and (ayc -ayd) in section i-1/2,i+1/2 and j-1/2, j+1/2 for the interior nodes.
When a section lies on a wall you procees as I wrote before

[samycfd]

yes I know that but still I get some different derivatives of velocities compare to the ones I exported from Fluent.

So do you agree with :
ac = rho*( V . grad ) V and axc = rho*(uu_x+vu_y) and ayc =rho*(uv_x+vv_y)
AND
ad = mu*(lap(V)) and axd = mu*(u_xx+u_yy) and ayd= mu*(v_xx + v_yy)

[FMDenaro]

Originally Posted by samycfd

yes I know that but still I get some different derivatives of velocities compare to the ones I exported from Fluent.

So do you agree with :
ac = rho*( V . grad ) V and axc = rho*(uu_x+vu_y) and ayc =rho*(uv_x+vv_y)
AND
ad = mu*(lap(V)) and axd = mu*(u_xx+u_yy) and ayd= mu*(v_xx + v_yy)

the terms written in quasi-linear form are correct (even if I would work with the conservative form).
Why do you want to compare the value with Fluent??

[samycfd]

Okay ! I would like to know which spatial schemes, they use in Fluent, forward, backward central differences or other because maybe the schemes could cause the solution not to converge!

[samycfd]

But if I assume rho an mu to be constant what the differences between the conservative and non conservative form ? Isn't it mathematically equivalent ?

[FMDenaro]

Originally Posted by samycfd

But if I assume rho an mu to be constant what the differences between the conservative and non conservative form ? Isn't it mathematically equivalent ?

Actually the mathematical equivalence exists also if rho is not constant (take the equation d rho/dt + Div (rho v) =0 and substitute into the momentum equation), but the numerical equivalence between the two formulations does not ..

[samycfd]

I don't understand how it could change something numerically ..

[FMDenaro]

Originally Posted by samycfd

I don't understand how it could change something numerically ..

let me give you a simple example for a term as uux and as (uu)x both discretized at second order centred FD:

u(i)*(u(i+1)-u(i-1))/2/dx

and

(u(i+1)^2-u(i-1)^2)/2/dx

you will easily see that they produce different local truncation errors.
Furthermore, in a FV formulation you cannot work with the quasi-linear form but you must use the conservative (divergent) form.

[samycfd]

Okay yes I got it. I have tried to fully implement the idea in 2D. And with the 2nd dimension I faced a new problem :

[(dP/dx)i+1/2 - (dP/dx)i-1/2]/dx = [(axd-axc)i+1/2 - (axd-axc)i-1/2]/dx

So in 2D we get :

[(dP/dx)i+1/2j - (dP/dx)i-1/2j]/dx + [(dP/dy)ij+1/2 - (dP/dy)ij-1/2]/dy= [(axd-axc)i+1/2j - (axd-axc)i-1/2j]/dx + [(axd-axc)ij+1/2 - (axd-axc)ij-1/2]/dy

My question is did we get :
dP/dx|i-1/2= (axd-axc)i-1/2=0 ( Like i 1D ) OK
dP/dy|i-1/2= (ayd-ayc)i-1/2=0 ( NEW condition because 2D case ) ?

from the BC : dP/dn = n.(ad-ac)

[FMDenaro]

Originally Posted by samycfd

Okay yes I got it. I have tried to fully implement the idea in 2D. And with the 2nd dimension I faced a new problem :



So in 2D we get :

[(dP/dx)i+1/2j - (dP/dx)i-1/2j]/dx + [(dP/dy)ij+1/2 - (dP/dy)ij-1/2]/dy= [(axd-axc)i+1/2j - (axd-axc)i-1/2j]/dx + [(axd-axc)ij+1/2 - (axd-axc)ij-1/2]/dy

My question is did we get :
dP/dx|i-1/2= (axd-axc)i-1/2=0 ( Like i 1D ) OK
dP/dy|i-1/2= (ayd-ayc)i-1/2=0 ( NEW condition because 2D case ) ?

from the BC : dP/dn = n.(ad-ac)

The expression dP/dy= (ayd-ayc) only where you are on a wall where n.Grad P = dP/dy (normal projetion to the wall that allows you to apply Neumann condition), that does not happen on a section i-1/2 where dP/dy is a tangential projection...

[samycfd]

i make a mistake :

[(dP/dx)i+1/2j - (dP/dx)i-1/2j]/dx + [(dP/dy)ij+1/2 - (dP/dy)ij-1/2]/dy= [(axd-axc)i+1/2j - (axd-axc)i-1/2j]/dx + [(ayd-ayc)ij+1/2 - (ayd-ayc)ij-1/2]/dy

I was thinking that too so boundary conditions are missing: If i-1/2 lying on a wall, then how I can deal with the term (ayd-ayc)ij-1/2

[FMDenaro]

Originally Posted by samycfd

i make a mistake :

[(dP/dx)i+1/2j - (dP/dx)i-1/2j]/dx + [(dP/dy)ij+1/2 - (dP/dy)ij-1/2]/dy= [(axd-axc)i+1/2j - (axd-axc)i-1/2j]/dx + [(ayd-ayc)ij+1/2 - (ayd-ayc)ij-1/2]/dy

I was thinking that too so boundary conditions are missing: If i-1/2 lying on a wall, then how I can deal with the term (ayd-ayc)ij-1/2

the Neumann condition must be applied on any section having j-1/2 (this section is parallel to the x-direction), therefore

(dP/dy)ij-1/2=(ayd-ayc)ij-1/2