Trying to understand math eq in a paper: extrapolating back the force to grid point?
 

JOURNAL OF COMPUTATIONAL PHYSICS 123, 450–465 (1996)
ARTICLE NO. 0036
Numerical Simulation of a Cylinder in Uniform Flow: Application of
a Virtual Boundary Method
E. M. SAIKI AND S. BIRINGEN

can any one explain the math given in eq 7...

extrapolating back the force to grid point ? <3 :):):):)

well, look at figure 4b as an example. lets say you are at x=4.12 and y=5.32 (roughly). here we have part of the cylinder surface which does not coincide with the grid points (shown here by the intersection of solid lines). you want to know how much force you need to add here so that the fluid surrounding this point (here called the virtual boundary point, i.e. point x=4.12 and y=5.32 for example) is zero (satisfies the no-slip condition). maybe the concept of adding forces is confusing? well, if you did not have any force acting on the fluid, the flow would simply pass through the cylinder (because the mesh is not aligned with the flow, as you can see, the cylinder geometry is just put on top of the mesh, as in figure 4a). so we need this force to slow down the flow and we want the flow to be zero at the boundary. with the method described in the paper, they can calculate the force necessary at any point x_s that would slow down the flow to zero at the boundary. to understand equation 7, you need to understand equation 1 first. you have the fluid velocity U and the body velocity v of the cylinder. for stationary cyclinders you have v=0. if the cylinder is moving in space (for example oscilating or simply translating in one coordinate direction) then v is not equal 0 and needs to be computed from v=dx_s / dt. so you have v, by choosing alpha and beta you can observe the effect of F on U. I have to admit at this point that this concept is a bit strange to me (but probably more so because I am not very familiar with the immersed boundary method) because I would assume U should equal v in order to get a no slip condition. However, I have seen seminars where they computed a virtual flow field inside the cylinder so as to match the zero velocity on the boundary so it might go in this direction. Anyhow, with U, v, alpha and beta you can calculate the force necessary to slow the flow down so as to simulate the presence of the cyclinder. You use that force in equation 7, along with a weight function, if you want to see it this way (D_i,j) to calculate the force (F_i,j) needed at the grid points i,j (which is not the same as in equation 1, remember that equation 1 calculates the forces at the virtual boundary x_s). So yeah, in equation 7 you really just extrapolating the information from x_s back to i,j i.e. the force at x_s to the points i,j which are the points that are available to you in your solver and where you solve the momentum equation on.

i know, it was a bit more detailed but i did not quit know where your problem was so i took some more steps inbetween, hoped that helped anyway.

I suggest a careful reading of the paper of Peskin in JCP 1977, it give much details about the ideas of immersed boundary method.
The key is that Cartesian and Lagrangian equations exchange each other some coupling term expressed by discrete Dirac representation.

hello Tom-Robin Teschner!
 

thanks for your reply... I got some helpful info <3 :)
but I think, I could not pose my question properly.... :(

I want to know how the extrapolation is derived in eq 7 ???

As I wrote, the paper of Peskin gives that idea: the force on a node (i,j) of a Cartesian grid is "extrapolated" by considering the portion of the body contour that has effect on a weighted area. That can be mathematically defined by an integral containing the Dirac function that modulates the distance of interaction. Peskin defined a discretization of such integral that provides the "extrapolation" to the Cartesian grid. You can then see the similarity of Eq.(7) you are asking for

thanks every one once again... my code worked!!!
 

here is the result :
CLICK HERE!

your comment and suggestion was quite helpful! :D :)

Seems good! :)

Isn't this difficult to say without knowledge of the mesh density and also some information about sensitive parameters such as drag and lift?

But I agree, it looks very nice :)

yes, such methods must be validated quantitatively by assessing that the normal velocity on a wall is really zero...bu the movie is nice :)

Yes Simbelmynë, FM you surely have a point of argument! :)

I just extended my LD cavity code with some extra module to see whether I can do it or not...

right now I am doing the validation part with flow over cylinder...

Thanks for your comment and suggestion... love this forum :D