Rhie and Chow should applied in equation of "pressure correction" not in momentum eqs.

@ Javad Rostami

As per I have studied in text and looked on web, it is applied for interpolating velocities at cell faces (like u_e=(u_E+u_P)/2 +other terms of pressure etc.

Once the equation is formed for all cell centers and pressures at particular node say P then we will have pressure at node P also included (unlike only linear interpolation where the values are skipped)

After substituting this in continue, we get pressure correction equation.

Kindly correct if I am wrong anywhere.

Realy, I didn't study in Rhie and chow deeply. I found it's formula in some literatures and applied to my code. I applied it in momentum and pressure correction eqs. but when I remove it from momentum eqs. I didn't find any change in my results. Then I applied it just in pressure correction eq. (I did it because I didn't find in literature that where I should applied it in momentum eqs. or in pressure correction eq.)

Rhie and Chow interpolation is a procedure to calculate mass fluxes at cell faces. These mass fluxes are used in the discretisation of the convective terms - momentum, turbulence model, temperature, etc. and are the fluxes used in the pressure correction equation. With the pressure-correction equation, you want these fluxes to satisfy mass-conservation.

Regarding your first question, discretising the x-momentum equation at cel i+0.5,j is done with staggered grids, not with a collocated grid. With a collocated grid you discretise all equation at cell i,j. I think there is a bit of confusion here.

If you are writing your own code, I would advise you to use the Finite Volume approach, that assures you some basic conservation properties for the discretised equations, not only when grid space goes to zero. This does not mean that you won't use Finite Differences, it is just a different starting point to discretise the equations. Also, I would advise you to look at some sample codes. You can find one here: ftp://ftp.springer.de/pub/technik/peric/ . It is the implementation of the algorithm described in the book Computational Methods for Fluid Dynamics, from Ferziger and Perić.

Dear Lopes,

Reg my first query, expanding about i,j in case of collocated grids, how would the momentum terms such as del(rho*u*u)/del(x) be discretized. Had the case been FVM, I would have written easily (as expression is readily available) for u_e which ultimately brings pressure at i,j node into picture.

However, in case of FDM, I am not able to get how to implement this.

Secondly, reg using FVM I too had plans to use the same as I have studied it in dept but the issue is with geometry I am dealing, trapezoidal shape. I am not able to get how to create mesh and go ahead with it.

With FDM I had planned to use mapping (and mesh metrics) to achieve the same.

Can you help me out in this regard.

Thanks

I really think you should have a look at the Ferziger and Perić book, because it has the answers to your questions.

Briefly, discretisation of del(rho*u*u)/del(x) is something like (rho_e * u_RCe * u_e - rho_w * u_RCw * u_w) / (delta x). u_RCe is what you calculate with Rhie and Chow procedure. rho_e and u_e are interpolations between nodes i,j and i+1,j (w is interpolation between i-1,j and i,j). However, while you probably can use linear interpolation for rho_e, most likely you will need something that is 'upwind biased' to get u_e (QUICK is a good choice). Here, I'm assuming you are not going to do DNS or LES. You are right that the discretised pressure gradient it will be (p(i+1,j) - p(i-1,j)), so no p(i,j). That is the reason why you can get oscillations with a collocated grid!

You can deal with a non-orthogonal mesh with FVM - that's what I do. You don't need metrics, instead you need the face-surface vectors of the control volumes. Again, you can get this in the book I mentioned.

Dear Sharma,

If I understood your problem correctly you are trying to pass something that has been developed in the FVM framework (integration) to the FDM framework (finite differentiation).

In the FVM framework, the continuity equation would become, after integration over the control volume:

(uA)_e - (uA)_w + (vA)_n + (vA)_s = 0

(d rho / dt = 0 and e, w, n and s with the usual meanings).

In the FDM framework it would be, using middle point values to approximate the derivatives:

(u_{i+1/2, j} - u_{i-1/2, j} ) / dx + (v_{i, j+1/2} - v_{i, j-1/2} ) / dy = 0

So, you have to evaluate, for example, u_{i+1/2, j}, which is in the exact same position u_e is in the FVM framework. This value can be evaluated with the very same RC expression. (Corrections to compensate the effect of relaxation and inertia factors can be found in the paper of Majumdar.("Role of Underrelaxation in Momentum Interpolation for Calculation of Flow with Nonstaggered Grid", in Numerical Heat Transfer, vol. 113, pp. 125-132, 1988))

I see no changes in the procedure to get the values of the velocity components at the faces, you would only multiply them by areas in the FVM and use them directly in the finite difference in the FDM.

Best regards and good luck.

Helio

Dear Deewakar,

I agree with Alexandre Lopes that it would be a good idea to read the book by Ferziger and Peric. It describes very well the finite volume method, which is not much different than the finite difference method but it has more potential.

As again Alexandre Lopes explained, in collocated grids the problem is that the discrete expressions for pressure force (FVM) or pressure gradient (FDM) allow the existence of pressure oscillations from node to node, because they relate pressures at two nodes apart, not at neighbouring nodes, i.e. p(i+1,j) - p(i-1,j) does not involve p(i,j). Therefore you get a mean pressure field that is correct, but it has spurious oscillations which spoil the aesthetics and can cause other serious convergence problems. The "momentum interpolation" remedy adds artificial pressure terms to the continuity equation which do relate directly the pressures at neighbouring nodes: p(i+1,j) - p(i,j). These terms, since they are artificial (the continuity equation doesn't really directly involve pressure), must be very small and tend to zero with grid refinement, which is ideed the case. However, their inclusion in the continuity equation means that if the continuity equation is satisfied then no spurious pressure oscillations can exist.

In finite volume methods, these artificial momentum interpolation terms are introduced in the definition of the mass fluxes, but in finite difference methods where mass fluxes are not defined you get the same effect by introducing analogous terms in the continuity equation, in the definition of the velocities.

The classic form of momentum interpolation has the drawback that it relies on the use of the SIMPLE algorithm. You can find a variant of momentum interpolation which is independent of SIMPLE, together with a discussion on momentum interpolation and references in this paper:

A. Syrakos & A. Goulas "Estimate of the truncation error of finite volume discretisation of the Navier-Stokes equations on colocated grids". Int. J. Numer. Methods Fluids, vol. 50, p. 103-130

But again, the book by Ferziger & Peric is highly recommended. Hope this helps!

Alexandros