This estimation is generally used for channel flows where you have that x and z are directions of flow periodicity (homogeneous flow). These spatial directions are discretized by uniform grid steps in the whole domain. Conversely, along the wall normal direction (y) the resolution of the BL requires at least 3-4 nodes within y+=1, then the grid is via via enlarged approaching the center of the channel.

I suggest to read this paper

https://www.google.it/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0ahUKEwj_pObA_tjWAhWGDsAKHcxhB10QFggwMAA&url=http%3A%2F%2Ftfc.snu.ac.kr%2Fjboard%2Ffile_down.php%3Ftid%3D11%26file_name%3D89.pdf%26data_file%3D617_1405313764.pdf&usg=AOvVaw1UqaQgPF9NuSODya4y_dW7

This estimation is generally used for channel flows where you have that x and z are directions of flow periodicity (homogeneous flow). These spatial directions are discretized by uniform grid steps in the whole domain. Conversely, along the wall normal direction (y) the resolution of the BL requires at least 3-4 nodes within y+=1, then the grid is via via enlarged approaching the center of the channel.

I suggest to read this paper

https://www.google.it/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0ahUKEwj_pObA_tjWAhWGDsAKHcxhB10QFggwMAA&url=http%3A%2F%2Ftfc.snu.ac.kr%2Fjboard%2Ffile_down.php%3Ftid%3D11%26file_name%3D89.pdf%26data_file%3D617_1405313764.pdf&usg=AOvVaw1UqaQgPF9NuSODya4y_dW7

Thank you Filippo.

Yes, I was following 'Chapman's revisited' paper to arrive at the mentioned grid resolutions.  Understood that we need to follow the specific grid resolution for wall bounded flows like channels, pipe etc. That (high computation expense) might be the reason, very few works are published for wall bounded flows' wall pressure fluctuations. 

But I could see some papers published with LES of high Re numbers of the order 10^6 - 10^7 for external flows. For eg: cylinder cross flows.

Q1. Does that mean, the grid estimates by Chapman & Moin is to be used for wall bounded flows only?

Q2. How do we arrive at grid resolution for external flow problems? 

Q3. Is there any guidelines available for bluffbody flows with high Re numbers and larger computational domain?

The requirement is expressed for resolving the boundary layer both for confined walls and for BL over bluff body. A further example is the mixing layer (no walls) between two or more flows. Of course, separated flow behind bluff body could also require a refined grid to solve the small structures of the flow.

But for Re=O(10^6-10^7) I suppose the papers you read use a wall-model BC, without resolving the BL. Check the grid resolution they adopt. Remember that evaluating the pressure distribution on a body requires less grid resolution than the tangential stresses. 

Thank you for the information that grid specs are to be followed for wall bounded flow as well as bluff body flows. For a wall bounded flow, computational domain is relatively smaller (pipe, channel etc.) than a bluff body domain (flow around a cylinder, aerofoil etc. ). Uniform grids with specific del_x_+, del_y_+ etc. is somewhat achievable, but for a bigger domain as of aerofoil, I think the no. of grids go out of proportion and one need enormous computation facilities for such simulations. OR is there a reasonable approach available for bluff body flows.

WMLES, Hybrid RANS/LES are the suggested options by Piomelli (2008) with a limitation of delta/15 for the grid sizes. Is this the only criteria generally used for WMLES grids or any coarse del_x_+, del_y_+ values are mentioned in any literatures ?

At present, DES is a hybrid methodology to be adopted for flow around airfoil at nominal Re number

Article Detached-Eddy Simulation

Good Read. I shall try DES than going for LES.

For turbulent boundary layer or channel flows, your estimations are correct except that dy_max+ should be smaller than 0.5, so that you can resolve the wall region.

Bare in mind that u_\tau changes in the downstream direction in the TBL case.

In the most general case (such as external flows), the requirements of grid resolution are best described by the kolmogorov length scale (\eta). For DNS, max(dx, dy, dz)/\eta < 4~10; for LES, max(dx, dy, dz)/\eta < 50~100, depending on the accuracy of the numerical algorithm. As for the requirement of grid at wall, dy+

I have written the resolutions in viscous wall units (based on friction velocity) and you have mentioned in terms of Kolmogrov length scales. Hope both are of the same order.

Gone through your paper and its a good work. Very high mesh density gives good results, but I cannot think of such fine grid as of now and I am looking for standard flow problems only (flat plate, pipes etc.)

Q 1. I understand that for internal flows, uniform grids are taken along streamwise and spanwise directions. y+ is maintained

There is no need to maintain uniform grid for an external flow, unless you are using Fourier spectral method.

That could depends on the level of resolution you want not only on the BL around the bluff body but also on the turbulent vortex shedding

Thank you both for your time.

To make myself more clear - Does that mean uniform grids and required LES grid resolution are a pre requisite for internal flows like channel, pipe etc. And it need not be strictly followed for external flows (outside b.layer).

My objective is to obtain turbulent wall pressure fluctuations using LES. Since velocity gradients contribute to pressure and are maximum near wall, fine grids are definitely needed near the wall. I am not sure of the contribution from larger eddies outside b.layer to the pressure on the surface (wall). Since isotropic turbulence is considered, it is important to have good resolution in the streamwise as well as spanwise direction also.

From the above discussions, LES grid resolution is calculated based on viscous wall units (del_x+, del_y+, del_z+ ), which are based on friction velocities. As the velocity gradient is maximum in the direction normal to wall, friction velocity component normal to the wall will be more than other directions.

Q1. For viscous wall unit calculations. do we need to calculate friction velocity based on the velocity gradient in that particular direction ? OR can we take a rms value of fric_velocity for the calculations ?

the friction velocity is defined by the stress at the wall, therefore depends on the velocity gradient

Yes. My question was about velocity gradient in which direction ? Whether we need to use friction velocity of each direction for calculating viscous wall unit of that particular cordinate ?

For eg: streamwise flow is in x axis. del_x+ =(del_x * u*)/nu;

spanwise flow in z axis . del_z+ =(del_z*u*)/nu;

u* - friction velocity

In each direction, do we have to use the same u* irrespective of its direction OR calculate individually ?

From a statistic point of view, the flow has no components in the spanwise direction where the forcing pressure gradient is zero

Thank you.

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After doing literature review on LES and doing some hands on CFD simulations, following are my findings for the question.

1. The grid resolution requirement depends on the problem you need to solve. For eg: if the problem is to estimate wall pressure fluctuations inside turbulent boundary layer (TBL), the grid needs to be neccesarily small than b.layer itself. Hence the above said viscous wall units have to be maintained.

Eg:2 - If your objective is to capture the flow features/aerodynamic noise in the wake region of a flow around a cylinder, you can make your grid coarse than the above. I cannot say the wall viscous units in numbers, as I am yet to do such a a simulation.

Above said unit/numbers are written/suggested by authors from their years long experience. They are to be trusted eventhough variations are allowed if it is backed by theory. Finally, the grid resolution depends on the eddy scales which needs to be resolved.

Above viscous wall units are meant for wall resolved simulations. Wall modelled resolutions will definitely be coarse than this.

2. Wall normal y+ needs to be kept