On the criteria for using DNS. Is there a formula (or rule of thumb) to use DNS in determining the number of grids necessary for its use? I have heard of the Kolmogorof's formula. What is it and How is it used?

In the DNS you resolve all the turbulence scale. The smallest eddies have dimension of the order of the Kolmogorov microscales. The K. microscale given by a dimensional analysis (see Fluid Mechanics Landau-Lifshitz) and the ratio between the microscale and macroscale is found to be Re^(-3/4). This means that if l is the dimension scale of your domain ( the distance between the walls in a plane pouseille flow) then the smallest vortex has a dimension in the order of l*Re^(-3/4). If your grid has this spacing then you can follow the whole energy cascade and you don't need any model. Hope it helps

With existing machine, DNS is used to solve for fluid flow with Re in the order of 1000 based on mean flow and integral length scale.

Hi. With DNS, the danger of not having a small enough grid in areas of small turbulent scales is to have energy accumulating at your local cutoff-scale (roughly the size of your local cell). One way to see if you're not doing that is by doing an FFT (fast Fourier transform) of your fluctuating velocity field and see if you don't have a peak of energy accumulation near that cutoff scale. Hope this helps.