[FMDenaro]

In your problem you used the Prandtl equations, that is parabolic x-component momentum equation (x is a time-like direction) coupled with the hyperbolic divergence-free equation.

You need to start from the BC at x=0 (trailing edge) and integrate the equations, for each position x, along the y direction. But that must be done only in the interior nodes. At x=L (last position) the velocity profile is computed as same as for the previous nodes, indeed your flat plate extends up to x->Inf.



You have to study the integration for a parabolic equation like the heat equation (but in your case the equation is not linear) but also to be aware how to manage the divergence-free constraint.

[researcher]

Originally Posted by FMDenaro

In your problem you used the Prandtl equations, that is parabolic x-component momentum equation (x is a time-like direction) coupled with the hyperbolic divergence-free equation.

You need to start from the BC at x=0 (trailing edge) and integrate the equations, for each position x, along the y direction. But that must be done only in the interior nodes. At x=L (last position) the velocity profile is computed as same as for the previous nodes, indeed your flat plate extends up to x->Inf.



You have to study the integration for a parabolic equation like the heat equation (but in your case the equation is not linear) but also to be aware how to manage the divergence-free constraint.

Thanks for the reply. I would say I have a finite length of plate, 0.05 m in length. What i have been doing is start from the interior nodes with the two given equations and reach column by column to the exit column. Here, I tried backward scheme and applied ghost node for the 2nd differential term. I am not getting a converged result. That is becoming troublesome.
Yes it is nonlinear but I am, for example, using u(i,j,k)*u(i,j,k-1) to linearize it in the momentum equation.
Any guidance is welcome/appreciated.

[LuckyTran]

Is there a reason you're trying to use a backward difference scheme in the y-direction? That's a very poor choice. Just use central differencing.

Or is that the purpose of this exercise?


If you still want to do backward diffferencing then it is:

And you need to retrieve u' on the wall. Now it becomes a question of how do you choose to evaluate the gradient du/dy? Also backward or...?

[FMDenaro]

Originally Posted by researcher

Thanks for the reply. I would say I have a finite length of plate, 0.05 m in length. What i have been doing is start from the interior nodes with the two given equations and reach column by column to the exit column. Here, I tried backward scheme and applied ghost node for the 2nd differential term. I am not getting a converged result. That is becoming troublesome.
Yes it is nonlinear but I am, for example, using u(i,j,k)*u(i,j,k-1) to linearize it in the momentum equation.
Any guidance is welcome/appreciated.

You have a 3D case? Write here the discretization of the full system of equations

[researcher]

Originally Posted by LuckyTran

Is there a reason you're trying to use a backward difference scheme in the y-direction? That's a very poor choice. Just use central differencing.

Or is that the purpose of this exercise?


If you still want to do backward diffferencing then it is:

And you need to retrieve u' on the wall. Now it becomes a question of how did you discretize the gradient du/dy? Also backward or...?

If i use central differencing in y direction then how about the continuity equation on the same outlet boundary? backward differencing? is it ok to use CD in momentum equation and backward diff in continuity?

[researcher]

Originally Posted by FMDenaro

You have a 3D case? Write here the discretization of the full system of equations

ok. here are the equations. pardon me for rather them being crude here.

inner nodes continuity:
u(i+1)-u(i-1)/(2*delx)+ v(i,j+1)-v(i,j-1)/(2*dely) = 0

inner nodes momentum:
u(i,j)(u(i+1)-u(i-1)/(2*delx) + v(i,j) (v(i,j+1)-v(i,j-1)/dely =mu/rho * (u(i,j-1)-2*u(i,j)+u(i,j+1))/dely^2

For outlet:
v(i,j)=v(i,j-1)-dely/delx (u(i,j)-u(i-1,j))
and using momentum equation
u(i,j)(u(i,j)-u(i-1))/delx + v(i,j)(u(i,j)-u(i,j-1))/dely = mu/rho* ( u(i,j-2) -2*u(i,j-1) +u(i,j))/dely^2

[researcher]

Originally Posted by researcher

If i use central differencing in y direction then how about the continuity equation on the same outlet boundary? backward differencing? is it ok to use CD in momentum equation and backward diff in continuity?

In addition see the situation:
If i apply central differencing, then wouldnt I be using inconsistent schemes in the momentum equation? like for example, in first term u velocity is out of the outlet to the right. How to address that ? or should backward diff work here in the same equation.

[LuckyTran]

There is no d2u/dy2 in the continuity equation... What is inconsistent with what?


You most likely have the Neumann condition at the outlet: d(anything)/dx=0. That's why you just duplicate the values from the interior node onto the outlet node. I'll say it again, there is no governing equation on a boundary node. Only boundary conditions.... Stop asking how and stop trying to discretize the governing equation at the outlet.... Even if you don't have a Neumann condition but some rather complex outlet BC, then you get the values per whatever is the BC.

For example... How do you determine the velocity at the inlet nodes? You use the boundary conditions. How do you determine the velocity at the upper boundary? It's U infinity. The boundary condition. How do you determine the velocity at the wall nodes? It's zero because of no-slip. No governing equation involved. Only boundary conditions.

[FMDenaro]

First, do you use staggered or colocated arrangement?


Then there is no physical outlet in the flat plate theory, the length of the plate is infinite. You just have a computational finite domain but the theory of the characteristic says you cannot prescribe a velocity but it is a result fo the computation of the equations.

[researcher]

Originally Posted by LuckyTran

There is no d2u/dy2 in the continuity equation... What is inconsistent with what?


You most likely have the Neumann condition at the outlet: d(anything)/dx=0. That's why you just duplicate the values from the interior node onto the outlet node. I'll say it again, there is no governing equation on a boundary node. Only boundary conditions.... Stop asking how and stop trying to discretize the governing equation at the outlet.... Even if you don't have a Neumann condition but some rather complex outlet BC, then you get the values per whatever is the BC.

For example... How do you determine the velocity at the inlet nodes? You use the boundary conditions. How do you determine the velocity at the upper boundary? It's U infinity. The boundary condition. How do you determine the velocity at the wall nodes? It's zero because of no-slip. No governing equation involved. Only boundary conditions.

I agree with you 100%. However I would like you to convince me that I do not have to apply equations on the outlet boundary where both u and v are unknown. Just tell me how else would we find u and v on the outlet boundary. Is there any other method? inlet, wall and free stream uinf are clear, agreed.

[researcher]

There is no neumann condition on the outlet.

[researcher]

Originally Posted by fmdenaro

first, do you use staggered or colocated arrangement?


Then there is no physical outlet in the flat plate theory, the length of the plate is infinite. You just have a computational finite domain but the theory of the characteristic says you cannot prescribe a velocity but it is a result fo the computation of the equations.

it is finite difference formulation, akin to colocation. My only query is how to apply these equations to the outlet boundary.

[LuckyTran]

There is a boundary condition at the outlet, you just don't realize it yet.


I'm suggesting that it is a Neumann condition based on the fact that you haven't realized it yet, but it could be something more complicated like a characteristic BC or a non-reflecting BC or... lots of things. But it's not a governing equation. A boundary value problem is not defined until you have specified boundary conditions on ALL boundaries.

[FMDenaro]

Originally Posted by researcher

it is finite difference formulation, akin to colocation. My only query is how to apply these equations to the outlet boundary.

How do you compute the velocity at the node just before the "outlet" section? Repeat the procedure. You have an equation like




u(i+1,j)=u(i,j) + F(u,v)


See Sec.3.3
https://essay.utwente.nl/63314/1/BSc...Puttkammer.pdf

[researcher]

Originally Posted by LuckyTran

There is a boundary condition at the outlet, you just don't realize it yet.


I'm suggesting that it is a Neumann condition based on the fact that you haven't realized it yet, but it could be something more complicated like a characteristic BC or a non-reflecting BC or... lots of things. But it's not a governing equation. A boundary value problem is not defined until you have specified boundary conditions on ALL boundaries.

As i said, we just have zero gradient mentioned. Let me attach the problem at hand.

[FMDenaro]

Originally Posted by LuckyTran

There is a boundary condition at the outlet, you just don't realize it yet.


I'm suggesting that it is a Neumann condition based on the fact that you haven't realized it yet, but it could be something more complicated like a characteristic BC or a non-reflecting BC or... lots of things. But it's not a governing equation. A boundary value problem is not defined until you have specified boundary conditions on ALL boundaries.

Actually, no, there is no BC you can prescribe due to the parabolic/hyperbolic character of the equations: the domain of integration is open. The profile is determined by integrating the equation at every station x since the flat plate is infinite.

[LuckyTran]

Originally Posted by FMDenaro

Actually, no, there is no BC you can prescribe due to the parabolic/hyperbolic character of the equations: the domain of integration is open. The profile is determined by integrating the equation at every station x since the flat plate is infinite.

Actually, I agree. If this is indeed an infinite flat plate (and it should be since this is the Prandtl equation), it means the domain of integration is open on the right and there is no physical outlet and no need to discuss boundary nodes at the outlet. Then the calculation procedure for the right most cell is same as the procedure for the one just before it, recursively so.

[researcher]

Originally Posted by FMDenaro

How do you compute the velocity at the node just before the "outlet" section? Repeat the procedure. You have an equation like




u(i+1,j)=u(i,j) + F(u,v)


See Sec.3.3
https://essay.utwente.nl/63314/1/BSc...Puttkammer.pdf

very useful document. thanx

[researcher]

For the convective term in steady equation, is it correct to write:

u du/dx = u(i,j)* (u(i,j)-u(i-1,j))/delx ?

I am of the view that it is nonlinear. How should one proceed? Should one change the scheme? any guidance please.

[FMDenaro]

Originally Posted by researcher

For the convective term in steady equation, is it correct to write:

u du/dx = u(i,j)* (u(i,j)-u(i-1,j))/delx ?

I am of the view that it is nonlinear. How should one proceed? Should one change the scheme? any guidance please.

This is a first order upwind scheme, the backward derivative is correct for positive u, as is in your case.