Reynolds number is the ratio between inertial forces and viscous forces according to wikipedia. However, I still do not understand its derivation process and how to, for example, calculate the Reynolds Number for a cylinder in large pond of water with the water moving in a direction with speed v. 
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1$\begingroup$ Which bit have you got difficulty with? $\endgroup$– KoyovisCommented Jun 5, 2017 at 2:01
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$\begingroup$ @Koyovis: My question is mainly about how to apply this concept, so for a cylinder in a fluid, how would you calculate the number? $\endgroup$– Harry YangCommented Jun 5, 2017 at 3:31
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3$\begingroup$ You'd go to Physics SE and ask them. This is not an aviation question at all - your example is at best, a boat, and certainly not an aircraft. $\endgroup$– NijCommented Jun 5, 2017 at 5:54
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$\begingroup$ A way it's applied in aviation is to do accurate scale models -- see for example How are wind tunnels used with scale models?. $\endgroup$– ChrisWCommented Jun 7, 2017 at 0:34
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$\begingroup$ I don't like the derivation of $Re$. For most of the non-linear velocity profile $\frac{v}{L} \neq \frac{dv}{dx}$. Are we living in pre-calculus era? This may be used in the 19th century because of the lack of measuring techniques. It is still followed in the 21st century even after having CFD and PIV. Please correct me, if I'm wrong. $\endgroup$– AGNCommented Jan 2, 2020 at 8:27
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2 Answers
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You can derive the Reynolds number by non-dimensionalization of the Navier-Stokes momentum equation: Wiki link
The choice of reference length and velocity is somewhat arbitrary: usually they are chosen so that they represent "fundamental" dimensions of the study case, e.g.:
- For a cylinder in a flow: freestream velocity and cylinder diameter
- For a pipe: mean velocity and inner diameter
- For a flat plate: freestream velocity and distance from leading edge
In the end of the day, it is a matter of conventions, but what you choose must be physically relevant (e.g. you cannot choose the plate thickness) in a way that, increasing either of them, you go from laminar to turbulent flow. Once the choice of the above pair (length, velocity) is standardized, you can define a critical Reynolds number, which marks the boundary from laminar to turbulent flow (actually, a range of it because of the complex behaviour of fluids during laminar-turbulent transition).
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Experimentally it is found the larger is the ratio of inertial force to viscous force the more turbulent is the flow
The trouble is be able to get from the experimental conditions predictable results applicable to real condition
In the theoretical abstract model a volume is a cubic length, while an area is a square length, thus in the abstract we are not trying to determine to what reel volume this applies neither to what real area this applies, that’s why the volume does not show as L x l x H and the area does not show as L x l .
In order to go from the abstract to the real condition and since the Reynolds number is experimental and to standardize the matter it has been accepted to consider a characteristic dimension for every shape, such as for open large area, narrow section pipe, square pipes etc
As a conclusion in the derivation the characteristic dimension is given nominally as simple as possible that’s why it looks strange.
