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The Reynolds number is defined as $Re = \frac{\rho u L}{\mu}$, with the fluid's density $\rho$ and its viscosity $\mu$. The constants $u$ and $L$ are according to wikipedia the "velocity of the fluid with respect to the object" and a "characteristic linear dimension". Now I am a little bit confused by this terms: when one observes a flowing fluid with no obstacle/object, how can one define these properties?

If you look at a flow of a fluid through a channel in which an obstacle is placed, these constants can be set without any problem. But what happens, if we remove the obstacle? What is now a characteristic length scale?

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The characteristic length scale always is present in real life flow systems. It doesn't necessarily have to be the interfering object. In your channel example it would be the distance between the edges of the channel or the height of the channel. In a case with a flow through a pipe it will be the pipe diameter.

For example in a flow over a plate there is a boundary layer (layer that is affected by the presence of the plate). The height of this layer is the characteristic length scale in this case and it grows as the fluid flows further over the plate. Thus also the Reynolds number grows and eventually the flow will turn from laminar to turbulent at some point. An illustration of the situation below is from here.

enter image description here

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    $\begingroup$ The choice of the length scale seems to be kind of arbitrary in my eyes. Is it correct, that the classification of flows based on different Reynolds numbers is not very strict, as the number changes strongly because of the different definitions of the length scale? $\endgroup$
    – zimmerrol
    Nov 29, 2017 at 15:08
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    $\begingroup$ @FlashTek The scales are somewhat arbitrary, but not too much so. If you are studying say flow in a rectangular channel, it is foolish to take earth's diameter as your length scale for that problem. Whatever length scale you choose must be such that the flow depends strongly on it, for example channel height or channel width or its wetted perimeter. Sometimes the length scale you choose depends on a particular aspect of flow that you wish to study. For e.g. if you wish to study boundary layers then you may take its width as the length scale and then simplify the governing equations. $\endgroup$
    – Deep
    Nov 30, 2017 at 5:07

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