Writing conservaion equations with a control volume approach with staggered grids
I have a very fundamental question. Lets say that I have a medium (1D) and I want to write the momentum conservation equation and continuity (mass balance) equation for this medium. Based on CFD books, first I partition this medium with a grid (lets say uniform) which the boundaries of the girds coincide the left and right boundaries of the medium (the grid with solid lines). This is the gird that I solve pressure at the center point (grid for scalar properties) of it. Then I draw another grid but staggered half a gridblock and write the xmomentum on these grids (the grid with dashed lines). For writing the momentum and mass balance equations, I follow control volume approach which I evaluate the integral over surface boundary of v.n for each surface of gridblcoks. Everything looks fine and I solve for velocity at the boundary of the gridblock (as I need it to compute the convective flux) and I solve for pressure at the center of my original grid.
Now my question is that, DO I CONSERVE MOMENTUM ON THE whole MEDIUM? Because if you look at the grids, we see that there are half gridblcoks on the left and right sides close to the boundary that I don’t write momentum balance for these volumes (the volumes that are shown with a shaded area in the figure)!!! I understand that the effect of boundary is felt (lets say the boundary is wall and u_1=u_ghost and u_n=u_ghost ) but we don’t really write momentum blanace equation for these halfgrid volumes. Does anyone have an explanation for this? Please see the below figure for a better illustration. Thank you very much http://imageshack.us/photo/myimages/15/67923582.jpg/http://imageshack.us/photo/myimages/15/67923582.jpg/ http://imageshack.us/photo/myimages/15/67923582.jpg/ 
Quote:
Consider a medium going for x= 0 to L, uniformly discretized as xi=(i1)dx, i=1,N+1 The, x1 and xN+1 correspond to the physical boundaries of the medium. Velocity is colocated on such nodes, while pressure is staggered of halfsize such that p2 is colocated at x2dx/2 and so on. You solve pressure for i=2, N+1. You will solve the momentum equation going from x2 to xN and the "telescopic" property of the FVM allows the conservation. Then, all depends on the physical boundary conditions imposed at x1 and xN+1. You don't need ghost cells 
Quote:
Thank you very much Flippo for kindly responding to my question. I understand the gridding, but my confusion is about the control volumes that I consider for grids x2 to xN. These control volumes do not cover my whole physical region. So, I guess the heart of the answer is that ""telescopic" property of the FVM allows the conservation". Do you know about a good reference which I use to read about this property (with more kind of physical and intuitive discussion). My question arose, as I try to write the control volume equations for each defined control volumes for density and velocity. As I explained I come up with half control volumes in both ends close to the boundary that I dont know how to write the control volume equations for them. Note that its not about using already developed equations but I want to develop equations from the very beginning (integral forms) by myself. Thank you very much again for your help 
I understand your doubts. My opinion is that:
 FVM are automatically conservative (in discrete form) since they are derived from the integral form of the governing equation.  The staggered grid I considered before ensure the (local and global) conservation of mass, which in incompressible flow is mandatory. The momentum equation is locally conserved, also globally conserved in the domain that cover all the FVs (excluding the strips near the boundary). To ensure full conservation there are many techniques you can use, for example writing an equation on each volume adjacent to the boundary or using deformed volumes ore prescribing suitable BC. You can read Cap.9 of Versteeg H.K., Malalasekera W. Introduction to computational fluid dynamics, also the book of Peric and Ferziger can help. See also this paper of Morinishi http://www.google.it/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CEcQ FjAA&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fview doc%2Fdownload%3Fdoi%3D10.1.1.32.4898%26rep%3Drep1 %26type%3Dpdf&ei=KsojUI3ENab54QS4mYGAAw&usg=AFQjCN GbVO4FjeXrrVG9zyNJE3pKAf5Bew&sig2=MjhiX25ATWBPSbuQeryZg 
Thank you very much !

1 Attachment(s)
[QUOTE=Fabien;376236]
I understand the gridding, but my confusion is about the control volumes that I consider for grids x2 to xN. These control volumes do not cover my whole physical region. So, I guess the heart of the answer is that ""telescopic" property of the FVM allows the conservation". As I explained I come up with half control volumes in both ends close to the boundary that I dont know how to write the control volume equations for them. Taking the notation of Filipo and as he said it U(1) and U(NX+1) are boundary conditions. You discretize the momentum and solve it for nodes i=2 to i=NX. The drawback you mentioned (the fact that your control volumes do not cover the whole physical domain) comes from the cell centred configuration that you have implicitly chosen. If you take a node centred configuration your control volumes for momentum will cover your physical domain, but this time the control volumes for scalar will not. Check the sketch below. But whatever the configuration you chose it does not really matter. The integration remains the same. 
All times are GMT 4. The time now is 14:26. 