[hamed.majeed]

Grid spacing near the wall or first node near the wall for laminar flow is also a requirement. In the attached picture taken from Fluent User guide 1997, it is discussed.

Q. What is the theoretical explanation of this? How did they end up with value of the factor mentioned in eq (A1) <=1.0

I know they did mention the blasius flat plate solution, which gives the boundary layer thickness delta(x).sqrt(U/(Mu.x))=5 (approx.) for u=0.99U.

[hamed.majeed]

How did they end up with delta(n)/R <=0.1 and delta(n)/H<=0.05 for laminar pipe and laminar parallel plate flow respectively.

[LuckyTran]

These are merely guidelines, they're not an exact science. I recommend you take them as they are and don't try to look too deeply into it. However, I would strongly urge you to stop thinking in terms of grid spacing near walls and first node near wall, and think in much simpler terms of x points across a boundary layer. That there is a wall is not important, the fact that there is a boundary layer is! Thinking this way, also lets you understand meshing requirements for free shear flows or jets or whatever flows: i.e. the non, wall-bounded flows.

The answer is clear (to me) in the last paragraph, basically someone tried it and got within 10% of the wall shear stress. Depending on what quantity you are trying to model, your needs may be slightly different. I've seen people numerically integrate the blasius equation using as few as 4 or 5 points and gotten the 99% boundary layer thickness to within a few percent.

To me, these are the common sense and should be intuitive grid requirements. If you are trying to resolve any curve, you need x number of points to be able to reproduce that curve. In the fluid dynamics and CFD sense, that curve we are trying to resolve is the boundary layer thickness. Hence there should be x points across a boundary layer. This can be the velocity boundary layer, density boundary layer, temperature boundary layer, etc.

"With four parameters you can fit an elephant to a curve, with five you can make him wiggle his trunk"

The recommendation is basically that there be 10 points across the pipe radius (or 10 points across the boundary layer thickness) and 20 points across the channel height (or again 10 points across the pipe half-height, or 10 pts across the boundary layer thickness). For non wall-bounded flows, people do DNS using similar requirements (although 16 pts and 32 pts are more popular because they are nice powers of 2). Again, try to imagine any flow feature, and just make sure you have x points to resolve that feature. The Fluent manual has basically chosen 10.

The difficulty is when you get to turbulence modelling of wall-bounded flows, where this stupid thing called wall y+ comes into play. The wall y+ requirement is an order of magnitude more restrictive than the x number of points across delta requirement. This leads to the infamous 80% of my grid is needed to resolve 20% of the flow or 90% of my computation is wasted solving 10% of the problem. This y+ requirement is a result of needing to either model the turbulence or explicitly trying to resolve the small length scales associated with wall-bounded flows (using LES or DNS). Without these two problems, we would never say the 1st cell needs to be any size.

[hamed.majeed]

Thanks Tran, That was informative.

But isn't the 1st node or y+ an important thing?? If yes, then simply adding 10 nodes won't do the job, it must mean adding 10 nodes what with a specific distribution!! Like clustered closer to the wall, and when we move away from the wall they should disperse.

[hamed.majeed]

This is what fluent used to explain near wall grid.

[LuckyTran]

Originally Posted by hamed.majeed

This is what fluent used to explain near wall grid.

This picture does a good job. Again, this falls into my category of x points over a boundary layer. You need to make sure you can fit the elephant.

Again, what are you trying to resolve? The wall shear stress? If so then you need to get the correct slope near the wall, which requires getting (two points) close enough to the wall that you get the correct slope there. If you want just the velocity profile and don't care about the wall shears stress, then your grid does not explicitly require you to have the correct wall shear stress (exactly). When you want the near-wall slope, you can place the 1st grid point as close as you want, but it's the 2nd point, that really determines where the slope is.

Or you can think of an alternative situation Say you have the 2nd grid point at an arbitrary location, say infinitely far away. Then no matter where you place the 1st grid point, you'll always have a biased slope. To get the wall shear stress, you need TWO points in the correct positions. From this logic is where the x points over a boundary layer condition arises. If I have say 10 points over delta, then I am fairly confident I will have 2 points in the wall. Remember, you never know (until you've solved the problem) what the velocity profile is so all you are doing is guessing.

You are correct to say that it is beneficial to have points clustered next to the wall and sparse far from the wall. But what is the logic behind that? You are here playing an optimization game and trying to maximize the information you can obtain at minimum cost. Near the wall there are large changes so you intuitively want more grid points there. That is, you are avoiding having a uniformly spaced grid. This is a separate intuition and a separate problem altogether. However, for a fixed number of cells, clustering them near the wall means you are giving up the information in the far-field (there's no free lunch).

The criteria for a best grid has multiple objectives. The problem of how many grid points to use, and the distribution of those numbers of grid points are two separate problems. Yet another reason why you shouldn't take these guidelines too literally. Many of them are very intuitive and you should follow your intuition rather than trying to find the answer in some bible.

Remember, numerical results should be mesh independent (wherever this meaning applies anyway)! That is the minimum publishing criteria in all respectable journals. Don't waste time meshing! Meshing is not your problem, solving the problem is. Of course I may be wrong, and if you are interested in studying those problems, then feel free to explore.

Mesh something that makes sense, run it. Do your mesh dependency study if you are not convinced. Remesh if necessary. This is a much more systematic approach than to wonder whether you need 8,9, or 10 grid points or if your growth ratio should be 1.1,1.2, 1.3, 1.4, 1.5 or if they should be uniform, linear, parabolic, tanh, or whatever. The answer to all these depends so strongly on the specific problem that it is hard to give a general answer. Only fully developed flow in pipes is parabolic. Flat plate boundary layers are not parabolic, neither are developing flows in pipes. Turbulent boundary layers, depends who you ask but you can model them many many ways.

Excuse the lengthy post, but I just want to reassure you that: "you know what you're doing"

[hamed.majeed]

Thanks again for the response.
Yes you are right about the grid specific to what I am trying to resolve. And yeah....fine grid do give you exponentially increasing computational power and time.

For example, if we consider a pipe flow, lets say laminar pipe flow. Now we have a velocity inlet and outlet. The wall dictates the no slip boundary conditions to the computational grid. If I am not accurately resolving the boundary layer, how can I expect to get good results? I mean that my entrance length would be messed up, same as my velocity profile, maximum velocity (at center), and hence average velocity won't be accurate.
Yes, I do appreciate the fact that computational methods work within certain limits, and you got to be satisfied with this fact. Mesh independence test will surely tell you a lot about how refined your mesh could be....that is why probably in academic publications...you cannot go without doing the mesh independence test!!

The thing which you said about the 1st and 2nd node...is it because of the 2nd order numerical scheme and second order derivatives in the NS equations??

Thank you for clarification!

[LuckyTran]

Originally Posted by hamed.majeed

The thing which you said about the 1st and 2nd node...is it because of the 2nd order numerical scheme and second order derivatives in the NS equations??

For CFD, we are solving for the velocity field and not the slope field. So I have made the arbitrary claim that velocity is more important. For example, it doesn't matter (to me) how accurate your wall shear stress is if you can't tell me what the velocity field looks like. Under this framework, what I said has almost nothing to do with any numerical schemes and almost nothing to do with the equations being solved.

If you are solving for velocity and you want to calculate the wall shear stress at the wall you need two points to get a slope. This result is true no matter how accurate your scheme is, or regardless of what equations you are solving. I can't imagine you have any scheme that can get the slope with 1 point as long as you are in velocity space. Referring to the same picture again, suppose that profile is the solution and you already know it to be the solution (you have already solved the problem, either numerically or analytically). You now insert grid points to resample and hence discretize the curve. Depending on how many and where those points are, you may succeed or fail to produce a satisfactory curve. But it also depends on what variable you are interested in. With two points I cannot reproduce the parabolic arc, but I might be able to produce a shear stress. You can imagine that you have selected the only numerical scheme that produces the true result. You can also imagine, with equal validity, that all numerical schemes can accurately lead to that curve. You still have the same problem of picking your grid points to be in right place. No matter what, you need to fit the elephant! So the grid requirement is like 0th order in terms of necessity and the numerical schemes are higher order.

Where does the scheme come into play?

So in my example I have assumed that you somehow magically obtained the exact solution. Of course, you need the correct numerical schemes to get to that solution in the first place. When you consider numerical schemes, you need to add those as additional requirements. The schemes will add additional mesh restrictions, but these are new requirements that are different than the x cells over delta.

Then there is the question of how you get the slope at the wall. Obviously you need more than 1 point to get any information about the slope. You can imagine fitting a line in-between the points. You can vaguely imagine fitting a parabola through two points. Actually you need 3 points to fit a parabola, which is exactly the new requirement! Suppose you could fit a parabola (or any fancy curve you can think of). You need to position your grid points in the right positions to get these slopes.

So when am I wrong? Say you have a fancy solver that solves for only velocity gradients and not actual velocity. Or alternatively, you are solving everything in the frequency domain. In frequency domain you are solving amplitudes in spectral density which is a different beast than velocity(x,y,z,t). Over there, you tune your grid to make sure you have the right spectral content.