Hello what is the Pressure checkerboard problem ,in the solution of incompressible navier_stokes equations?

I would be very gratfull, if anyone send to me the name of articles, books or any other text about it.

Thanks you.

behzad

Read book by Patankar, "Numerical heat transfer and fluid flow". A checker board problem arises due to the nature of the central difference scheme when applied to the divergence operator and the pressure gradient operator.

Let me argue this from a finite element perspective. Unless you are using divergence-free finite elements, you must remove the divergence in each rectangle by subtracting off the gradient of an appropriate function. As you surely know, only the gradient of the correction appears in the governing equations, so the value of this function must be fixed some way, usually by assigning some value to it on some rectangle. It turns out that the divergence on all of the "red" squares are coupled where they meet at the corners, as are all of the "black" squares. But the red squares may not be not coupled to the black squares. Thus you must assign a reference value to the black squares and another to the red squares to avoid a singular system. If these assigned values are not compatible, the correction functions (but perhaps not their gradients) will be different and you get a "checkerboard." Not to worry though, often the velocity values are still good, even with the checkerboard.

If the functions you use for the divergence correction are not compatible with the functions for the velocity, things can get much worse.

do you have the electrical edition of this book ,i really appreciate if you can send a copy to me . thank you very much!

Dear egg

my thesis is about an overall investigation on all the algorithem , for the solution of incompressible navier stokes equations. I have several quations , for e.g. why me must use two distinct concept for velocity on collocated grid.( convecting and convected velocity)

please if you have experiences in related subjects , i would be very gratefull, to help me.

thank you

I'm sorry I do not know of convecting and convected velocity

Dear Behzad, If u use collocated grid, then two distinct concepts are needed for convecting and convected velocity. The reason , as I understand, is simple. In the collocated grid, the convected quantity should be taken in the center of the control volume, and the convecting quanatity must be taken on the wall. For example, some solute concentration is advected by convecting velocity. But when the advected variable is velocity itself, then keeping the same basic principle of collocated grid, we need to take the convecting velocity on the wall of the control volume containing the convected velocity in the center. As the position of the convecting and convected velocity is different in the grid, different concept is need. Hope this helps. Jaswant

Dear Jaswant

I think first and important reason for definning two distinct concept for velocity,have been checker board problem. but how this problem is solved?

basic quesion for me is the following question.

can we use collocated grid with out using two definition for velocity? and also cure the checker board problem?

thank you