Hello! Here is my problem.

When solving INSEs using nonstaggered grid, I am confused by several methods.

In general, nonstagger grid difinition will cause pressure oscillation because of odd-even decoupling. Most people add high order terms explicitly or implicitly to smooth it, especially when it is in SIMPLE algorithms.

Abdallah(1987,1992,1997), however, used an different method to avoid it, either by proper boundary condition for pressure Poisson Eq., or by high order discretization for gradient for pressure.

What's the difference between the above two? Why does the latter work?

If I use a compact scheme with all derivatives 4-order despite the boundary condition, does the same effect can be reached?

Note, an other important issue is to keep delation free. Most high order methed cannot satisfy the constraint.

If you tie the two decoupling solutions together at the boundary they can (and generally will) still drift apart away from the boundary - hence it is not a general solution to the problem. In order to stop the decoupling, as you rightly state, all schemes add a measure of "smoothing" and this is most often 4th order in the absence of very strong pressure gradients. However, unlike the compressible community which is usually upfront about the smoothing that is being added, a number of incompressible schemes do not discuss the behaviour of their schemes in terms of the added smoothing relative to the underlying non-smoothed "centred" scheme.

Concerning a particular differencing scheme, the answer is yes if it includes a measure of smoothing. In order to maintain conservation one generally has to include one extra unknown in the differencing scheme (or drop an order in accuracy depending on how you look at it). For example, the QUICK scheme which used to be quite popular.

Thank you for your remark.

Sometimes it be considered be trade-off between the covenience and accuracy. Therefore, problem still exists with use nonstaggered grid in order to avoid the checkboard.

Zhi-Xing Yu

I am getting the feeling that we still don't know how to solve the incompressible flow problems. At the incompressible end, the pressure field is derived from the velocity field ( once the velocity field is known). In other word, there is only one set of independent solution, that is , the velocity field. In the pressure-based primitive variable approach, you are solving for the pressure field and the velocity field. Instead of one unknown field, you are making it two un-known field. You are solving these two unknown fields and hoping that they both will converge and become uniquely related. I think this is the source of the problem. So, if you try to decouple the pressure field from the governing equation first, and solve the velocity field, the job will be easier and straightforward. Trying to deal with two identical twins at the same time sure will create a lot of confusion. (or you will be an artist to do that.)

As you rightly state, for an incompressible flow the pressure field (apart from an additive constant) can be determined from the gradients of the velocity field. However, the primary role of solving the "pressure" related equation (or equations in some cases) is to drive the solution of the momentum equation towards the one (or one of the ones - it is a non-linear set of equations!) which also satisfies mass conservation. It is almost always the case that mass conservation is satisfied in an exact numerical sense at the expense of "accuracy" in the pressure field. For example, examine the predicted fields for a uniform plane impinging flow in which the pressure is known analytically.

It is quite straightforward to eliminate the pressure as an unknown in a number of ways both exact and "numerically good enough". The exact approach generally gives 2 equations in 2D instead of 3 but 6 equations in 3D instead of 4. The main reason this approach was largely abandoned in the early/mid 70s was because of the problems of imposing realistic boundary conditions. A number of time dependent codes in the LES area and some FE codes remove pressure as an unknown and retain a primitive variables formulation. However, they still impose mass conservation and the approach is close to "pressure correction" but without the introduction of something directly labelled pressure.

I think in the pressure-based approach, you can somehow let the pressure field to absorbe the in-accuracy part of the solution. I mean in this approach, you can obtain reasonable flow field solution with relatively coarse mesh.(if you try to integrate the flow field to give you the pressure field, you will get poor pressure field). But sometimes, it is very important to get a very accurate pressure field for certain applications. In those applications, you are forced to use very fine mesh so that you can get accurate pressure field back. On the other hand, in the decoupled formulation, you will be dealing with the flow field gradient quantities all the time ( such as the vorticity), you are forced to use fine mesh at the begining in order to obtain accurate velocity field. Base on my experience of using the pressure-based approach , there is always uncertainty about the accuracy of the solution. on the other hand, since you are forced to use fine mesh with the decoupled formulation , the results are always more reliable. ( both velocity and pressure field). My feeling is: the ability to obtain reasonable velocity field in the pressure-based formulation with coarse mesh, sometimes is missleading in the case when the pressure field is very important to your applications.(you get incorrect pressure field results.)

How about the projection method? Since the pressure is function of velocity, it seems reasonable to solve a intermediate velocity field and then project it to get the divergence-free velocity field. Then the pressure Poisson equation can be solved to get pressure field.

When you create a second-order pressure equation, it is very weak and must depend on the accurate formulation of the boundary conditions. For problems where the pressure boundary condition is known at the inlet only, the convergence of the pressure equation will be extremly "slow". You get the floating solution because of the weak pressure boundary condition. For external flows, where the presure can be assumed known over large part of the computational domain, the problem is not as severe as the internal flow problems.