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I would like to know how turbulence arises from the standard Navier-Stokes equations, both mathematically and also physically. At least I suspect this is the case as many of the "vanilla" Navier-Stokes simulations seem to develop turbulent.
I am a mathematician who does not have a strong physics background, and so far all the resources I have looked at either contain physics beyond my understanding or evade too much math.
The Navier Stokes equations are non-linear in velocities, and thus have multiple solutions, some of which are time-dependent turbulent solutions. In practice, the most stable solution prevails. In turbulent flow, the fluid velocity and pressure vary rapidly with time and spatial position, even for so-called "steady flows." In "steady" turbulent flows the time average velocities and pressure are constant, however. The time averages of velocity component products give rise to the so-called turbulent stresses, which are typically much larger than the viscous stresses based on the average velocities. These play a major role in determining the time-average flow and pressure.
