It is a physical reason.

Close to the wall, there is a thin sublayer with predominantly molecular diffusion. The sublayer has a substantial influence upon the remaining part of the flow. An adequate numerical resolution of a solution in the sublayer requires a very fine mesh because of sublayer thinness and high gradients of the solution.

Dear Dr Mounir and Mohammed,

Thank you for your reply. I am trying to relate the following mathematically: interpolation method, gradients and meshing size. There should be some sufficient condition for the mesh size near the wall which is a function with parameters like gradients and interpolation weights, etc. Because the term "Very Finer or Finer or Coarse mesh" is not well defined. Whatever you feel coarser may be finer for me. Whatever you feel finer may be very finer or coarser for me. So, there should be some condition like CFL to produce smooth solution.

But, in my case, uniform finer grid which satisfies CFL also produced damping solution. Finer mesh also produced a damping solution after a few time steps (greater time steps than coarser). The damping occurs near the wall boundary only. However, coarser mesh on the internal cells and finer mesh near the wall boundary produced smooth solution. It works for fine:coarse = 4:1. However, if I make 10:1, then it damps at fewer steps and disappear. I concluded that mesh grading has some relation with gradient and interpolation methods in FVM. I am not confident with my conclusion and I do not find any such relation in the literature. So, I am confused.

I need to redefine the problem with different velocity and different geometry scaling, and this can lead a computationally expensive research if I make trial and error on mesh grading.

If you like a more mathematical reason (previous answers were focused on good physical reasoning), you can think about the Nyquist theorem. The smallest wavelenght described by a grid of size h has lenght 2*h. Thenrefore, if you need to solve in the BL strong gradients you need to have 2*h comparable to the characteristic wavelenght in the BL.

However, this is for a resolved boundary layer, you can use wall-model BC.s

Due to the formation of boundary layer close to the wall, the velocity gradients are very large compared to free stream. So it is necessary to capture the velocity gradient near the wall, that's why finer mesh is used.

Previous answers already said the mathematical and physical reasons for using a fine mesh near the wall: small eddies and strong gradients. I don't know what method are you using - RANS? LES? - but, if you're solving the wall flow and not using the wall-functions mentioned by Mohammed Islam, you need your first grid point to be around y+=1.

This is an example of discussion related to physical issues

Article Grid-point requirements for large eddy simulation: Chapman’s...

when you have intense gradient in variables you need fine mesh to detect that.

near the wall for example when you have no slip boundary condition, you have steep gradient in velocity near the wall,in very small region velocity changes rapidly, and must have several meshes in this region.

Because there is large gradient in the boundary.

Pleas read this article

https://link.springer.com/journal/11470

It's only important to have really compact meshing near wall if you need to precisely break for the overflow bordering to the wall. The turbulent overflow near a flat wall can be divided into four regions. At the wall, the fluid fastness is zero, and in a thin tier above this, the overflow fastness is direct with distance from the wall. This region is called the ropy sublayer, or laminar sublayer. Further out from the wall is a region called the buffer tier. In the buffer region, turbulence stresses begin to dominate over ropy stresses and it ultimately connects to a region where the overflow is exhaustively turbulent and the norm overflow fastness is related to the log of the distance to the wall. This is known as the log- law region. Yea further out from the wall, the overflow transitions to the free- canal region. The ropy and buffer layers are really thin and if the distance to the end of the buffer tier is x, either the log- law region will extend about 100x out from the wall. It's possible to use a ropy CFD model to calculate the overflow field in all four of these regions. Notwithstanding, since the consistence of the buffer tier is so small, it can be beneficent to use an approximation in this region. One answer is to use"Wall functions"that ignore the overflow field in the buffer region and analytically calculate a nonzero fluid fastness at the wall. By using a wall function expression, you assume an consequent answer for the overflow in the ropy tier and the consequential models will have significantly lower computational necessaries. This is a really useful approach for multiple practical engineering plays. Notwithstanding, either you'll want to consider a turbulence model that solves the concentrated inpouring government, If you need a reach of perfection beyond what the wall function utterances give. For instance, you may want to figure lift and drag on an object or figure the heat transfer between the fluid and the wall.