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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.

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  • $\begingroup$ 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 $\endgroup$
    – Quillo
    Feb 27 at 23:47
  • $\begingroup$ @Quillo I assume that is because all the energy has cascaded down and ultimately dissipates due to the viscosity? $\endgroup$
    – CBBAM
    Feb 27 at 23:49
  • $\begingroup$ It’s a good question, but needs more focus. Can you give an example with your question? $\endgroup$
    – joseph h
    Feb 28 at 0:07
  • $\begingroup$ @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. $\endgroup$
    – CBBAM
    Feb 28 at 0:19
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    $\begingroup$ 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). $\endgroup$
    – Quillo
    Feb 28 at 9:20

1 Answer 1

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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.

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  • $\begingroup$ Thank you, this is precisely the kind of intuition I was looking for! $\endgroup$
    – CBBAM
    Feb 28 at 17:09

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