Turbulence
In 1932, the British physicist Sir Horace Lamb, in an address to the British Association for the Advancement of Science, reportedly said, "I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic."
Why then is the problem of turbulence so difficult? One reason is that the governing equations of turbulence are nonlinear partial differential equations, and appear to be insoluble. There are only partial proofs for the existence, uniqueness and regularity of solutions. What is more, these proofs correspond to simplified cases. It is not clear whether the equations themselves have some hidden randomness, or just the solutions. And if the latter, is it a consequence of the equations, or a consequence of the initial conditions?
DNS
solves fluid governing equations without any turbulence model.
RANS
solves the Reynolds averaging equations (the long-time average of a quantity or ensemble average) for fluid flow
LES
a practice to solve only for large eddies explicitly and to model the effect of the smaller and more universal eddies on the larger ones
