A doubt of stagged grid FVM(Finite Volume Method)?
Dear all,
Say channel flow, there are "M2" pcells from lower to upper boundary in vertical direction(y) at fixed streamwise(x) position. So we have to envalue u(streamwise), v(spanwise), p at "M2" different points respectively, but for w(vertical), there are only "M21" points unknown to be evalued. Obviously it is due to the stagged grid method. I just wonder if it will cause some mathematical problems. Thanks. Zhengtong 
Re: A doubt of stagged grid FVM(Finite Volume Meth
(1). I don't like the staggered grid approach, especially because u,v,w, and p are not defined at the same location. (2). So, if you don't understand it, why not try some other methods. I don't think people care about how the method is formulated, as long as they can get something from the code. (3). I would say that staggered grid method is ugly, and the collocated grid method is not reliable. And you really have to try it in order to know the exact problem. Math problem? I don't think so. The conceptual problem? yes. (when you are dealing with approximations)

I agree what do you use?
John:
Many of us may share your views regarding staggered grid etc., but now I invite you to tell us what you use? Sharing experience is the most powerful feature of this excellent forum. 
Re: I agree what do you use?
(1). For low speed flows, I have been using vorticityvelocitystream function approach, since early 70's. (2). It has been applied to laminar flows, turbulent flows, variable density flows, and arbitrary geometry through coordinate transformation for 2D and 3D problems. (3). For compressible flows, I have been using explicit MacCormack method for flow all the way to Mach 25. (4). It seems to me that the use of unstructured mesh comes mainly from the unwillingness to understand the physics of the flow and the geometry. It is hopeless to get the right answer using this type of mesh. (5). The use of the pressurebased approach will always create what I call the "double guessing" of the flow field and the pressure field at the same time. Since the low speed flow depends only on the flow field, introducing the pressure variable at the same time is not necessary. In other words, using the pressurebased approach requires more than twice the effort and suffer from the coupling effect of the two fields. (6). The difficulty of compressible flow comes from the existence of the shocks, which can only be captured by the use of the proper mesh and the first order scheme. Most socalled advanced methods are "artificial method" trying to eliminate the overshoot and undershoot, which tends to have bad secondary effects. (7). The only way to obtain the good solution of a turbulent flow is to model the flow for the individual case, because it has been known that there is no general turbulence model. Each case has to be modelled separately. (8). It seems to me that the problem in CFD today is "poor understanding of the physics of the problem, improper selection and use of the method and mesh, and no modeling effort for each individual case". The basic framwork developed over twenty years ago, is still good in numerical methods, transformation, mesh generation, and turbulence modeling. (9). I have made comment ten years ago that the unstructured tri/tet mesh was not appropriate to get accurate flow field solutions. The hybrid mesh concept which is being developed and used only recently, actually was already used at very high level when Prandtl developed his boundary layer theory many, many years ago. The accuracy obtained in the boundary layer solution is actually much higher than any unstructured methods used today. (10). In my opinion, the development of 90's is actually moving backward. In other words, we are not back to the early 80's. What I am trying to say is: most people are still trying to solve the square cavity flow, and we are still facing the modeling problem of separated flows. The negative impact is: more codes are being used to produce the unreliable solutions today, which will have severe consequence on the system and product design. (11). So, to make any progress, we will have to go back to the early 80's, and look at the individual problem separately. Looking for a general code with a general turbulence model and an automatic mesh generation is just like a dream, like the story of the " crouching dragon and hiding tiger".

error correction,(10)We are now back to...
(10). ...We are not back .. should be (10)...We are now back to...

Re: A doubt of stagged grid FVM(Finite Volume Meth
hi friend,
I agree with you. The staggered grid will cause the number of CV's to be different for Scalar variables than for vectors. It all depends upon how you choose to stagger. patankar suggested staggering in X direction. one can as well stagger in Y direction. What's the harm? If not used properly you can run in to not satisfying consistency which may lead to divergence. One headache with staggered grids is repeated interpolations. To avoid the trouble u can consider half a control volume at boundaries then you will have M2 X M2 for all variables at interior points. However i disagree with Jhon that staggered grids have problem with solution accuracy. My experience shows accuracy with staggered grids is far better than non staggered. especially div(rhoU) the mass conservation, which is far better (10e8 for indivudial cells) than non staggered (10e5). Methods for nonstag. grids like Peric's PWIM ultimately evaluate cell face velocities and store them for use in next iteration, as good as staggered approach. abhijit 
Re: A doubt of stagged grid FVM(Finite Volume Meth
No, quite the opposite  it is a SOLUTION to a mathematical problem that exists when all variables are located in the same place.

Re: A doubt of stagged grid FVM(Finite Volume Meth
There are mixed methods now which define cartesian velocities at the cell centers and the contravariant fluxes at the cell faces (Shyy and Mittal [JCP, 00], Zang et al. [JCP, 94/95] etc.). I think the outstanding and difficult of the issues with pressurevelocity coupling remains the prescription of boundary condition.
In my experience, boundary conditions seem to be a problem for the implicit timeintegration scheme I am using. I would like to hear about the implementation of any implicit time integration scheme for nonstaggered (or mixed) grid layouts. I need a scheme that can integrate at CFL > 1020. Thanks. 
Re: A doubt of stagged grid FVM(Finite Volume Meth
Hi all,
I just wanted to ballance out some of the staggeredgrid bashing. There is no doubt from a geometrical point of view (and the resulting coding difficulties) the staggered grid is not elegant. However, the whole idea in a FV context of closing the integral relations at the integration points in terms of nodal values is very elegantly solved by putting the required nodal variables at the integration points (ie. staggering the grid). In fact the terminology of writing a set of "closure equations" for the integration point variables only evolved when we went back to the colocated grid and said: what do we need here to make this work?...well really this is a closure problem and was trivial with a stagered approach. In addition, the stagered grid satisfies strong conservation of kinetic energy (not independant once conservation of momentum is specified in a numerical method) whereas the collocated grid with Rhie and Chow interpolation does not. See the discussion by Ferziger and Peric in their text. I think that future methods should look to capture some of the best qualities of both methods...who knows what is on the way! Regards, Duane 
Its "Crouching Tiger, Hidden Dragon" :)
not the other way round, although still a dream....
Eulerian/Eulerian multiphase flows add extra problems for describing everything as "vorticity" in part because the description of a dispersed phase as a continuum is very rough, and to abstract that one step more really begs the questions as to what the appropriate boundary conditions are for dispersed phase and the interphase drag. However there are so few good analog experiments against to which to compare most of it is numerical fantasy, and I indulge freely. 
Re: Its "Crouching Tiger, Hidden Dragon" :)
(1). I am not sure about the name of the movie. All I know is that it is not real. It is more or less a dream. (2). I think, in the multiphase area, the problem is mainly the modeling of the physics. Without the proper modeling, the computational part is hard to carry out. What I am saying is, a lot need to be done in the basic modeling of the multiphase flow, before one can apply the computational part to it. In other words, we need a good description of the flow first.(in mathematical terms or equations. In terms of vorticity? I don't know.)(3). The CFD seems to give people the impression that solving fluid dynamics problem is like what's going on in the movie. It is hard to know whether it is real or not. Colorful fluid dynamics is quite similar to the special effects in the movie. Even the story is quite artificial. I think, it is a good movie, if you look at it from the entertainment point of view. (4). Anyway, it is hard to rule out the possibility that someone might use CFD to study the aerodynamics of a man flying in the air or walking up the wall.

Re: A doubt of stagged grid FVM(Finite Volume Meth
(1). Yes, "who knows what is on the way!" (2). It is likely that "nothing" is on the way. (3). The discussion is really on the need to develop improved methods or new ideas. (4). It is easy to follow the past, and it is also easy to say what was wrong in the past. (5).The new ideas will not come from the use of existing codes. The discussion is one way to encourage the new development of ideas.

Re: I agree what do you use?
Thanks John,
I guess "the development of 90's is actually moving backward" means the RANS. Do you agree DNS and LES help to understand the machanism(physics) of turbulence ? Yes, the unstructured mesh seems too complicated to anlyse, I agree. People choose it just because they have no other good choice when they have to simulate complex boundary. Zhengtong 
Re: A doubt of stagged grid FVM(Finite Volume Meth
Hi Duane,
So you means just "is a closure problem and was trivial with a stagered approach"? I am afraid it maybe restrict vertical movement in channel flow coz it seems there is less freedom for vertical fluctuation. Zhengtong 
Re: A doubt of stagged grid FVM(Finite Volume Meth
Hi there,
Agree with you. I also don't like the repeated interpolations. Maybe I should try the half a control volume at boundaries later.Thanks 
Re: I agree what do you use?
(1). The reliable measurement is always the first step to validate the calculation and simulation. It provide the foundation to develop the turbulence model required for that particular category of problems. (2). We are not sure about DNS and LES whether the results can be used to guide the modeling of turbulent flows. In addition to the above uncertainty, DNS and LES are quite expensive and time consuming. (limited by the Reynolds number and mesh size) (3). The right approach should be: What is the physics of the problem? What would be the right turbulence model for it? And what is the most appropriate mesh to obtain the best resolution? (4). Currently, the focus is on the automatic mesh generation for arbitrary geometry. With this goal in mind, unstructured mesh approach is adapted. In the process, the automatic part of mesh generation is emphasized. A mesh generated in this way is not going to react to the accuracy requirement of the solution. Although the adaptive mesh refinement is possible, there is a lack of guidelines in the refinement process. (5). The need to use the unstructured automatic mesh generation, also limits the used of more accurate boundary condition treatment, and the implementation of turbulence models. In other words, the treatment of boundary conditions and turbulence models are being shaped and fitted into the unstructured mesh methodology.

conservation properties of the staggered gird
Hi there,
The staggered is not fully conservative unless uniform mesh are used. Please take a look at the two following publications: 1) Morinishi, Lund, Vasilyev and Moin. J. compt. Phys., vol 143, pp 90124, 1998. http://utacfdb.uta.edu/~felten/resume/5962a.pdf 2) Vasilyev, O.V. J. compt. Phys., vol 157, pp 746761, 2000. http://landau.mae.missouri.edu/~vasi...highorder.pdf I hope that, for all the participant of this discussion, the lecture of this two articles will help clarify some misconceptions about the collocated and staggered grid arrangement. sincerely, ** * Frederic Felten * * CFD Laboratory * * Box 19018 * * The University of Texas at Arlington * * Arlington, Texas 760190018, USA * * * * http://utacfdb.uta.edu/ * ** 
Re: conservation properties of the staggered gird
Mr. Felten, I beleive your statement applies to PDE and could be misleading.
1) It is well known that Finite Volume Methods are locally fully conservative. Even for staggered grid. As a matter a fact, if a Finite volume Method is not conservative, it won't converge (in theory). 2) Locally, The Finite Element method is not conservative. But, it is on the whole domain. Regards. 
Re: conservation properties of the staggered gird
Mr Perron,
The statment i previously made apllies to Finite difference method, and there is not much difference between the FDM and the FVM. when it comes to discretizing the equations. Talking about the convergence of the FVM, it still going to be stable (or "converge") if the error are dissipative (Sink). That's what Morinishi et al. have found and it's absolutly true. So please, take some time and read the two references i suggested earlier on this forum, and i'm sure that there won't be any misguidance or misunderstanding regarding the conservative properties of these two mesh arrangement. Sincerely, Frederic Felten. 
Re: conservation properties of the staggered gird
a little correction,
That's Vasilyev in the JCP 157, pp 746761, 2000. That stated the conservation error of the staggered grid on nonuniform mesh. But still the following is true: if the errors are dissipative (sink in the KE), then it still going to be stable and converge. Sincerely, Frederic Felten. 
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