# How does turbulence arise from Navier-Stokes?

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.

• Also Euler equation (i.e. Navier-Stokes with no viscosity) can have turbulent solutions. Navier Stokes, after an initial transient should relax to the stationary homogeneous state Feb 27 at 23:47
• @Quillo I assume that is because all the energy has cascaded down and ultimately dissipates due to the viscosity? Feb 27 at 23:49
• It’s a good question, but needs more focus. Can you give an example with your question? Feb 28 at 0:07
• @josephh I do not have a particular example in mind. I have been reading that all the information governing fluid flow is embedded within Navier-Stokes, including turbulent flow. I was wondering, in general, how this information arises from Navier-Stokes. Feb 28 at 0:19
• Exactly. To mantain turbulence in navier stokes the system need some external force. "Honey is less turbulent than water", in the sense that the energy cascade is faster and the dissipation length is bigger... very soon there is no turbulence (or even motion) at all. Turbulence without external forces is possible in perfect fluids (perfect = non viscous). Feb 28 at 9:20