[digger]

Dear Friends,

I have started to play with the implementation of some numerical procedures. I’ve just accomplished my finite volume solver for 2D advection-diffusion equation. I started to think about numerical implementation of steady-state incompressible Navier-Stokes model on a staggered grid. I have no very strong background in numerics, I’ve just read the book of Ferziger and Peric and the one of Moukalled et al., however the 2D staggered grid arrangement, for me is not too well documented therein (or maybe I just don’t understand it).

I would like to ask for your help. I just want to start out with one question but probably I will be refreshing this thread a couple of times during implementation. I would be kindly and sincerely grateful for your help.

This is what I’ve accomplished till now. I started with the derivation of discretized form of the u-momentum equation for some location in the „bulk” of my computational domain. This was not too complicated step (hoping I did it correctly).


As can be already seen for discretization of the viscous stress component (more precisely the derivative du/dy) on left boundary of (i,J) node I am using the velocity value at u(i-1,J+1), indicated by the green circle. But then I encountered the first conceptual problem when I started to think about the u-velocity node indicated by the green circle. Ferziger suggests that the shear stress on “east” face on u-momentum element should be calculated using east face of the v-momentum element. But for the u-velocity element I don’t have a “full” v-momentum element. How should I evaluate the du/dy derivative for this element indicated by the green circle?

[FMDenaro]

First of all, your momentum equations are written in quasi-linear form, therefore you can use only a FD formulation, not a FV one.


Then, consider the index i,j addressing the node xi,yj of the structured grid. A staggered notation could be this:


u(i,j)=u(xi,yj-dy/2)
v(i,j)=v(xi-dx/2,yj)
p(i,j)=p(xi-dx/2,yj-dy/2)


addressing the discretization of the divergence-free constraint in the pressure node. You see that some interpolation for the non linear terms is required.

[digger]

Thank you Sir,



the addressing is something that I didn’t think about for now, but I certainly consider your suggestion.


About the FD/FV formulation. I call it FV (maybe wrongly) as I obtained the discretized equation using approach typical for FV method, i.e., I integrated the differential equation, applied the divergence theorem and sum up the fluxes around the control volume. This is the discretised equation which I obtained finally.


About the non-linearity. I want to apply the Picard iteration, in which one of the velocity values is obtained from the previous inner iteration.


But I am not sure how to obtain du/dy discretized form on the “east” face of the u-momentum element (to evaluate tau_xy), which is placed in the top of the domain (the green circle).


Kind regards

[FMDenaro]

Originally Posted by digger

Thank you Sir,



the addressing is something that I didn’t think about for now, but I certainly consider your suggestion.


About the FD/FV formulation. I call it FV (maybe wrongly) as I obtained the discretized equation using approach typical for FV method, i.e., I integrated the differential equation, applied the divergence theorem and sum up the fluxes around the control volume. This is the discretised equation which I obtained finally.


About the non-linearity. I want to apply the Picard iteration, in which one of the velocity values is obtained from the previous inner iteration.


But I am not sure how to obtain du/dy discretized form on the “east” face of the u-momentum element (to evaluate tau_xy), which is placed in the top of the domain (the green circle).


Kind regards

you are doing a wrong approach... again, the quasi-linear form you are using is not suitable for the FV formulation. When you integrate the non linear flux and apply Gauss, you have the surface integral of the diadic normal component. There is no derivative at all.

[digger]

Dear Sir,



Please excuse me if I don’t understand. I have just started to learn this material. But there is no derivative in non-linear discretized convective term. I want to calculate the derivative in the stress tensor component:



https://en.wikipedia.org/wiki/Discre...okes_equations

[FMDenaro]

Ok, so let us forget the PDE you have written that is not the one you discretize.
Consider the node I,J in your figure (let me call i,j) and use this staggering definition


u(i,j)=u(xi,yj-dy/2)
v(i,j)=v(xi-dx/2,yj)
p(i,j)=p(xi-dx/2,yj-dy/2)


Now consider the balance for u(i,j), you have the x-derivatives (u(i+1,j)-u(i,j))/dx at the face est and (u(i,j)-u(i-1,j))/dx at the face west. In a similar way you have the y-derivative.

[digger]

I very much appreciate your help. Yes, the PDE was misleading I wrote it earlier to remind myself the terms in the governing equation.



Yes, your suggestion and addressing appears clear to me, however the particular problem I have is with derivative in y-direction. I would like to draw your attention to the green circle from my drawing, let me name it u(i,j) node, for a moment (its north face touches the top boundary of the domain). Now, let say I need to discretise du/dy on the west face. Which values of u’s should I use?

[FMDenaro]

Originally Posted by digger

I very much appreciate your help. Yes, the PDE was misleading I wrote it earlier to remind myself the terms in the governing equation.



Yes, your suggestion and addressing appears clear to me, however the particular problem I have is with derivative in y-direction. I would like to draw your attention to the green circle from my drawing, let me name it u(i,j) node, for a moment (its north face touches the top boundary of the domain). Now, let say I need to discretise du/dy on the west face. Which values of u’s should I use?

For incompressible flows there are not cross derivatives in the viscous terms. The diffusive term in the x-momentum equation is Integral[S] du/dn dS, therefore you need only to compute du/dx at w and e faces and du/dy at n and s faces.