If "meshless" means "unstructured grid" then the answer is of course yes. You always compute values at discrete points or in discrete cells, and thus you need discrete points. How to distribute them and to get a decent solution at these points is coupled, i.e. the solver's accuracy character depends on the spatial discretization, and it is possible to get good solutions with unordered points. For DNS of turbulence however the resolution must be high in shear layers like near walls, no matter whether the grid is structured or not. With high-accuracy/order methods, you need less grid points/cells but it does not get cheaper this way, only better.

I suppose that for "meshless method" is here addressed a numerical method that adopts only a cloud of points without any type of connection using edge.

In any case, DNS is by definition always a direct solver of the NSE without any turbulence modelling. But this is not sufficient to define correctly that the formulation is really a DNS: the cloud of point must ensure to be able in solving the smallest characteristic lenght scale of the flow problem.

For example, have a look here

http://etd.fcla.edu/CF/CFE0005733/Book_-_Andres_Vidal_-_Word_2007.pdf

Article Meshfree Methods: A Comprehensive Review of Applications

@Tapan

I fully agree but resolve point 2: Near the wall any numerics/method suffers from some accuracy loss at constant grid spacing due to the wall BCs and stencils necessarily different from the inner field. That is why the spacing must be lowered there somewhat. The moving vortices (ejection event for instance) can indeed be resolved by a slighty coarser resolution with a high-order method.

Dear Tapan,

our DNS, and the ones I know, get unstable with spacing close to the wall not refined somewhat compared to the field. Also, moving structures are smaller and physical dissipation (viscous terms) are larger near the wall. Vortices in the outer layer are captured relatively well by our scheme with the reference spacing that must be found this way.

Dear,

this is my opinion:

1) no matter of the type of method, any time that the spatial domain is "traslated" from the continuous to the discrete sampling (structured/unstructured/meshless), we introduce a (can be local) Nyquist cut-off frequency. To be really denoted as DNS formulation we must ensure that the discrete cloud of nodes ensure to have that the Nyquist lenght scale is of the order of the Kolmogorov lenght scale.

2) The above constraint is not limited to the wall resolution but apply everywhere in the flow where strong gradients must be resolved

3) The quality of the method (accuracy, conservation preserving, dispersion preserving) is relevant. Just an advantage we have using the correct grid resolution is that the numerical errors (that is the local truncation error analysed in the wavenumber space) act at the higher part of the resolved spectrum. Therefore, if the resolution is correct, the numerical errors are confined in the part of the spectrum where the physical viscosity (and energy dissipation) is more relevant. That likely avoids contamination of errors backward in the inertial energy range.

4) I agree that the adoption of structured grid in the BL is more a hystorical issue started from RANS modelling. A correct LES/DNS formulation and grid generation would allow to use pure unstructured grid also in the BL.

Tapan K. Sengupta

yes, totally agree that SM has theoretically a spectral resolution not biased up to the Nyquist frequency. However, for non linear problems, SM practitioners adopts the de-aliasing technique that affects the last part of the resolved spectrum. I also agree that FV/FD produce a numerical smoothing (and sometimes dispersion) in the last portion of the resolved spectrum up to produce a zero-value of the numerical transfer function at the Nyquist frequency.

However, if you consider the DNS with the strict constraint of a grid size having everywhere the Nyquist frequency equal to the Kolmogorov frequency, then you see that is likely that the numerical smoothing acts in the range from the Taylor micro-scale to the Kolmogorov one. This way, the physical dissipation has a strongest role than the numerical errors and numerical errors are somehow prevented to affect the backward larger scales.

The reasining has value only for DNS, when we use LES we get into very different issues since the smoothing has to be considered part of the filter and the SGS model should adapt consequently. However, some years ag we developed a deconvolution-based FV with spectral-like resolution to have a built-in reconstruction of the resolved frequencies in the SGS model.

Concerning the last question, I am not a FEM user but I am not sure that structured grids are always used ....

Have a good day you, too.

I recommend the discussion between Filippo Maria Denaro and Tapan K. Sengupta , it is interesting conclusive and constructive.But,the question about mesh-less method is seeking for algorithm based on differential equations that solved with DNS without turbulence model instead of constructing grid(structured or unstructured ) i mean the type of differential equation is capable to sort out the difficulty of grid problems in particular for turbulent flow applications.

Ahmed Abdelhameed

An issue is about the fact the "meshless" is, in my opinion, a misunderstanding denomination. Any time we use a numerical solver we are involved in a "discrete" representation, that is we use finite dimensional vectors state. Now we can denote a "mesh" as an explicit discrete geometrical representation of the domain (as is used in FD/FV/FEM) but I think of "meshless" methods as an algebric discrete representation on a different type of mesh, not explicity represented. The key is that we work always with an algebric system of equations that are the "translation" of the continuous counterpart. In meshless methods the PDE are not the classical NSE so that it could be debated the fact that they undergo to the classical constraint of a DNS using the NSE. But I think that the idea in meshless methods of a scattered distribution of points has to take into account the characteristic physical lenght scales of the problem, hence we should always check the points distribution in terms of the Kolmogorov scale.

I am afraid, meshless method is not effective to do DNS. For most of meshes method, its accuracy is too low.

Jianming Liu Could you please explain it more? You mean that if we use meshless method and we have quite a lot of points, we cannot see the Kolmogorov length scale?