DNS
The Direct Numerical Simulation is a method of simulation in Computational Fluid Dynamics that permits to solve directly the Navier-Stokes equations without using any modeling, averaging or approximation other than the numerical discretizations performed on these equations. Consequently, with this method, every instantaneous quantities of the flow have to be known and then, all of the fluid motions contained in the flow are considered to be resolved. Since the Direct Numerical Solution requires all significant turbulent structures (eddies) to be adequately captured, the mesh size must be smaller than the dissipation scale. That means an important number of mesh elements.
That’s why, with this method, you have extremely long calculus time. The more mesh points you have, the longer the calculus will be. And since if the speed of flow increases, the number of time steps increases, then similarly, the faster the flow is, the longer the calculus will be. So the cost of the Direct Numerical Simulation is very high.
Even if the capacity and the power of the tools of calculation keep improving, the use of this method remains limited to simple cases of application with reasonable speeds of flow. Indeed, most of the applications are still too demanding for the currently available computers.
Nevertheless, this method represents a good way of understanding turbulent flow behaviors in practical applications.

LES
The Large Eddy Simulation is a numerical method of resolution of the partial differential equations governing turbulent fluid flow. This technique forms a good alternative to Direct Numerical Simulation by solving only the large-scale motions and modeling the small-scale ones. A simulation that treats the large eddies but approximates the small eddies makes perfect sense because the large-scale motions are usually much more energetic than the small ones. In order to define the large-scale field, a filter is used. Kolmogorov’s scale permits to determine which