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Let us assume we have circular flow in a bowl or a tea cup.

(1) If the velocity of the fluid in the bowl is low, we have laminar flow.

(2) However, to my understanding turbulent flow is defined as the curl of a velocity field, i.e., as soon as we have a vortex-like flow structure, we can speak of turbulence. Independent of the velocity of the flow in the bowl, we get a vortex.

How can I solve this contradiction?

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Turbulence is a much broader concept, it implies the existence of vorticity and some other features, while vorticity alone does not imply the existence of turbulence. I quote:

Every aspect of turbulence is controversial. Even the definition of fluid turbulence is a subject of disagreement. However, nearly everyone would agree with some elements of the following description:

(1.) Turbulence is associated with vorticity. In any case, the existence of vorticity is surely a prerequisite for turbulence in the sense that irrotational flow is smooth and steady to the extent that the boundary conditions permit.4

(2.) Turbulent flow has a very complex structure, involving a broad range of spaceand time-scales.

(3.) Turbulent flow fields exhibit a high degree of apparent randomness and disorder. However, close inspection often reveals the presence of orderly embedded flow structures (sometimes called coherent structures).

(4.) Turbulent flows are three-dimensional (unless constrained to be two-dimensional by strong rotation or stratification), and have a high rate of viscous energy dissipation.

(5.) Advected tracers are rapidly mixed by turbulent flow.

(6.) Turbulent flow fields often exhibit high levels of intermittency. (Roughly speaking, a flow is intermittent if its variability is dominated by infrequent large events.)

Source

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  • $\begingroup$ I thought turbulence dominated the regime $R \gg 1$ where dissipation due to viscosity is zero. Instead, dissipation occurs at the molecular level. Is this true? $\endgroup$ – Ryan Thorngren Sep 4 '18 at 20:44

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