In general, I think turbulence is resulted only from the viscosity term as in the Naiver-Stokes equation, and it dissipates energy in the flow. But the compressible Euler equations, which already ignored the viscosity, may generate shock waves that dissipate energy as well. So my question is, if the flow governed by the Euler equations can be a turbulent flow.

  • $\begingroup$ I don't know what your definition of turbolent flow is, but annals.math.princeton.edu/wp-content/uploads/… constructs compactly supported solutions, which both create energy from nothing and then dissipate it $\endgroup$ – Federico Nov 27 '18 at 17:50
  • $\begingroup$ I think that this question may be on-topic here, but you will probably have much better answers at Physics.SE. AFAIK, a fundamental mechanism in turbulence is the vortex stretching, which demands (I'm not sure...) viscosity to works. With Euler equations you can have hydrodynamic instability, which is the "first step" to turbulence, but I think that only N-S equations can predict turbulence in its full meaning. $\endgroup$ – rafa11111 Nov 27 '18 at 17:52
  • $\begingroup$ Complementing my previous comment, another fundamental mechanism (and related to vortex stretching) is the energy cascade, that is the transfer of energy from larger vortices to smaller ones, until the cascade ends at a scale in which all energy from its vortices is dissipated into thermal energy, due to viscosity. $\endgroup$ – rafa11111 Nov 27 '18 at 17:55
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    $\begingroup$ @rafa11111: Vortex stretching is independent of viscosity, it occurs whenever vorticity is aligned with the, well, the "stretching" determined by the velocity gradient (formally, whenever $(\omega\cdot\nabla)u>0$). My understanding is that the energy cascade is primarily driven by vortex stretching and other inertial effects, and it's only once the cascade has transferred energy down to the Kolmogorov microscales that viscosity then starts taking over. So I would expect turbulence to exist as long as there is vortex stretching, even if there is no viscosity. But, I am not a fluid dynamicist. $\endgroup$ – Rahul Nov 27 '18 at 18:27
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    $\begingroup$ @rafa11111: But I agree that the question is more on-topic for the physics site, and I've cast a vote to transfer it. $\endgroup$ – Rahul Nov 27 '18 at 18:29

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