[Rachit]

Hi everyone,

I have been trying to write a MATLAB code for 2D flow of the SIMPLE algorithm for incompressible flows on a collocated grid. I do understand that the Rhie Chow interpolation is used against the normal interpolation to calculate the velocity on the faces, in order to avoid checkerboarding effect. However, from what I gathered from the literature, there is no reference to where is the Rhie Chow interpolation used. In the calculation face velocities in the discretization of the continuity equation? Or is it used to calculate the face velocities in the calculation of convective flux in the momentum equation as well? If that is so then what is the point of using schemes like Upwind, Central Difference, QUICK, etc?

I would be extremely grateful, if someone could help me out on this.

Thanks in advance,
Rachit

[dsanthosh2]

you have to use Rhie and Chow interpolation in the continuity equation. Not in the momemtume equation.

[RodriguezFatz]

Originally Posted by dsanthosh2

you have to use Rhie and Chow interpolation in the continuity equation. Not in the momemtume equation.

But how do you get the flux at the cell faces for the convective term in the momentum euqation?

Originally Posted by Rachit

Or is it used to calculate the face velocities in the calculation of convective flux in the momentum equation as well? If that is so then what is the point of using schemes like Upwind, Central Difference, QUICK, etc?

I think it is used in both cases, continuity and momentum. Remember, that you don't really directly solve the continuity equation in schemes like SIMPLE...
The momentum term is not linear. Thus, you have to interpolate the flux on the cell face (as the "transporting" medium) and you have to interpolate the velocity on the cell faces (as the "independent variable"). If you have "rho*u*u", then "rho*u" is the flux through the face, and one single "u" is the independent variable. That's the one, you create your linear system of equations for. And this is also the one, you can decide the way of interpolation for (as you described).

If you have "rho*v*u" and you are creating the momentum equation for "u", then "rho*v" is the flux through the face, and "u" is the one you solve with your matrix...

Did that become clearer?

[andy_]

Originally Posted by Rachit

Hi everyone,

I have been trying to write a MATLAB code for 2D flow of the SIMPLE algorithm for incompressible flows on a collocated grid. I do understand that the Rhie Chow interpolation is used against the normal interpolation to calculate the velocity on the faces, in order to avoid checkerboarding effect. However, from what I gathered from the literature, there is no reference to where is the Rhie Chow interpolation used. In the calculation face velocities in the discretization of the continuity equation? Or is it used to calculate the face velocities in the calculation of convective flux in the momentum equation as well? If that is so then what is the point of using schemes like Upwind, Central Difference, QUICK, etc?

I would be extremely grateful, if someone could help me out on this.

Thanks in advance,
Rachit

Calling the addition of pressure gradient terms to the mass flux "interpolation" is rather misleading because there are no pressure terms in the mass flux. Nonetheless adding pressure smoothing terms to the mass flux is a requirement for many schemes.

The pressure smoothing terms are usually needed in the mass fluxes in the continuity equation to oppose pressure/velocity decoupling. For the mass fluxes in the other transport equations the pressure smoothing terms are optional but are usually included so that what is a mass flux is consistent and various numerical fixes that rely on the mass fluxes in a cell summing to zero hold.

The flux of momentum involves squared velocities. One velocity can be involved in determining the mass flux and the other velocity that is being transported by the mass flux can be evaluated using upwind, central difference, QUICK or whatever.

[RodriguezFatz]

Originally Posted by andy_

One velocity can be involved in determining the mass flux and the other velocity that is being transported by the mass flux can be evaluated using upwind, central difference, QUICK or whatever.

That's what I was trying to explain. Better read andy's post - it's easier to understand.

[andy_]

Originally Posted by RodriguezFatz

That's what I was trying to explain. Better read andy's post - it's easier to understand.

Had I seen that you had posted I would not have replied. My technique of putting posts to reply to in tabs and then getting on with other things while replying perhaps needs modifying.

[Rachit]

Dear Santo, Rodriguez and Andy,

Thank you for your replies. I shall work it out this way in my code. Thanks once again.

Regards,
Rachit

[Rachit]

Hi,

Looks like my confusion pertaining the Rhie Chow interpolation hasn't ended yet. From what's there in Versteeg and Malalasakera, to calculate the velocity at face e, I will need to the pressure at EE. What if my EE is outside the domain? Is there any way find the value of PEE without using the concept of ghost cells?

Also, to calculate the pressure smoothing term I need to aP which is the coefficient of variable u at point P. However, the coefficient is itself made up of convective flux terms (something for which I am using the Rhie Chow interpolation to calculate). What do I do then? Do I take the flux values from the previous step or one has to do something else?

Thanks in advance,
Rachit

[andy_]

> What if my EE is outside the domain?

What to do about the pressure smoothing on the boundary is up to you. For example, on a solid boundary the mass flow is zero but the pressure gradients will generally be non zero. So do you use a zero mass flux or include the pressure smoothing term? If the former, global mass conservation will be physically reasonable but there will be a jump caused by smoothing on one face but not the other and in the presence of strong pressure gradients from body forces like swirl it can be large and unreasonable. If the latter, you will have unreasonable mass fluxes on the solution boundary and issues such as coming up with ways to evaluate pressure gradients at the boundary.

> Do I take the flux values from the previous step or one has to do something else?

Storing the mass fluxes along with the solution variables is a common thing to do for schemes using Rhie and Chow type pressure smoothing.

[FMDenaro]

Justo to give you my opinion ...

First, start considering everywhere in the domain the Hodge decomposition

Vn+1 + Grad phi = V*

1) in the interior, after computing V* somehow, you will search for a "pressure" gradient such that it ensures Div Vn+1 = 0 -> Div Grad phi = Div V*
2) on the boundaries, you only have to prescribe something about Vn+1 and use the Hodge decomposition to ensures the correct mass flux by solving the modified pressure equation.

On co-located grids, you can use the approximate projection method (APM) to avoid checkerboard modes in the pressure, the RC interpolation can be applied on the flux defined by the Hodge decomposition.
However, the RC interpolation has some degrading in the accuracy...

If you want, some more deatils about possible remedy for checkerboard modes are addressed at:
http://onlinelibrary.wiley.com/doi/1....1368/abstract
http://onlinelibrary.wiley.com/doi/1....1368/abstract