# Critical Reynolds number and characteristic length

• ntdiemai
In summary, the critical Reynolds number for fluid and characteristic length when a cylinder immerses and then rotates in the fluid (The fluid is initially static) is different depending on the reference you use.
ntdiemai
What are the critical Reynolds number for fluid and characteristic length when a cylinder immerses and then rotates in the fluid (The fluid is initially static)?
Please suggest the critical number for the transition from concentric flow to laminar and from laminar to turbulent flow.

ntdiemai said:
What are the critical Reynolds number for fluid and characteristic length when a cylinder immerses and then rotates in the fluid (The fluid is initially static)?
Please suggest the critical number for the transition from concentric flow to laminar and from laminar to turbulent flow.
Is this for schoolwork?

This is for my postgraduate research. I am working on a project which integrates material engineering and fluid dynamics.

Advanced schoolwork is sometimes allowed in the technical PF forums, but only when you show lots of effort and show us your work so far. I'll move this thread to the ME forum where it will get better responses, but please post your research and work so far on this question. Can you post links to the reading you've done so far? What are the Relevant Equations? What simulation software packages do you have available to work on this? Thanks.

Ok, I will do that :). Thank you!

This smacks of a situation where you've not really dug into the problem too deeply yet. Have you done anything to try to arrive at a solution on your own yet? I don't think most of us would feel particularly compelled to do your postgraduate research for you.

russ_watters and berkeman
Hi, I appreciate your time for reading and replying to my questions. As a postgraduate student, I am aware that it is the most important for myself to learn, understand and solve my problems by myself before I seek help.

At this moment, I am still reading to gain more knowledge about fluid dynamics. Most of the available materials mention flow in pipe or flow past a rotating cylinder, which are not relevant to my problem which is about a cylinder rotating in static fluid (static here means initially static before the cylinder rotates). Also, as I understand, the critical Reynolds numbers come from the combination of practical experiments and theory. So definitely, I need relevant references. So far, I could found only two references relevant to my case [1], [2].

The two references suggested the same characteristic length and formula (shown below) for calculating of Reynolds number. However, they suggested different critical Reynolds numbers to assess the flow pattern which leads to different conclusions in the flow pattern for my problem although the input data, e.g. cylinder radius, fluid viscosity, etc., are the same when I used those critical Reynolds numbers. I am not having a deep enough knowledge in fluid dynamics to decide which reference in the two ones I found is suitable for my problem. That is why I need help from an expert.

The formula for calculating of Reynolds number from the two references which are consistent to each other:

Re = (ρΩ(b)^2)/μ
where ρ fluid density
Ω fluid velocity
μ fluid viscosity

Critical Reynolds numbers from the references:
Reference [1] which was about a rotating disc in a static fluid. The suggested critical Reynolds number for the transition from concentric to laminar is 784, and from laminar to turbulent is 2x10^5.
Reference [2] which mentioned a rotating cylinder in a static fluid. The suggested critical Reynolds number for the transition from laminar to turbulent is 60.

References:
[1] R. I. Olivares, PhD thesis "The effect of sulfur on the dissolution of graphite and carbons in liquid iron-carbon alloys", The University of Newcastle, Australia, 1996.
[2] P. R. N. Childs and P. R. N. Childs, Chapter 6-Rotating Cylinders, Annuli, and Spheres. 2011.

Last edited by a moderator:
The rotating disc is not a great analogue here. Rotating disc have been extensively studied and are subject primarily to the crossflow instability, which dominates transition.

A rotating infinite cylinder, in contrast, is likely to be unstable primarily to Görtler vortices. It's a different physical mechanism so the correlations that work for one flow field won't work for the other.

ntdiemai
ntdiemai said:
the critical Reynolds number for fluid and characteristic length when a cylinder immerses and then rotates in the fluid
A problem for many.
here is a discussion of it.
https://engineering.stackexchange.c...-length-in-reynolds-number-calculations-in-gePipe flow - easy - everyone "knows" that the diameter( radius ) is the characteristic length.
Other configurations have to be looked up to see what has been used.
And other configurations have no look up, so you have to figure it out yourself, and that it seems is where you stand.

ntdiemai

berkeman
It's kind of odd to solve something like this so quickly. Generally, finding a simple correlation for transition Reynolds number is not possible. There are, of course, some examples that work out easily, like pipe flow, but this is more of an exception than the rule.

Really, it all depends on your goals here. You could probably concoct an experiment that does a good job of coming up with some decent results for your model. You'd have to take into account the cylinder size and the size of the fluid domain around it (if it becomes too much of an annulus, you'll get Taylor-Couette flow).

However, if you are trying to actually model transition in a situation like this, you'll end up with a pretty big project on your hands. Linear methods don't work well with flows like these because they produce substantial modification of the base flow, which then affects the stability characteristics of that flow. In other words, the problem rapidly becomes nonlinear.

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256bits and ntdiemai

## 1. What is the critical Reynolds number?

The critical Reynolds number is a dimensionless value used in fluid dynamics to determine the transition from laminar to turbulent flow. It is defined as the point at which the flow becomes unstable and turbulent eddies begin to form.

## 2. How is the critical Reynolds number calculated?

The critical Reynolds number is calculated using the formula Re_crit = ρVD/μ, where ρ is the fluid density, V is the flow velocity, D is the characteristic length, and μ is the dynamic viscosity of the fluid. The characteristic length is the characteristic dimension of the flow, such as the diameter of a pipe or the chord length of an airfoil.

## 3. What is the significance of the critical Reynolds number?

The critical Reynolds number is an important parameter in fluid mechanics as it determines the type of flow (laminar or turbulent) and affects the overall behavior and performance of the system. It is used to predict the onset of turbulence and to design systems for optimal flow conditions.

## 4. Can the critical Reynolds number vary for different fluids?

Yes, the critical Reynolds number is dependent on the properties of the fluid, such as density and viscosity. For example, a more viscous fluid will have a lower critical Reynolds number compared to a less viscous fluid. Additionally, the shape and surface roughness of the object in the flow also affect the critical Reynolds number.

## 5. How does the characteristic length affect the critical Reynolds number?

The characteristic length plays a crucial role in determining the critical Reynolds number. As the characteristic length increases, the critical Reynolds number also increases. This means that for larger objects or systems, the flow will remain laminar at higher velocities compared to smaller objects or systems with the same fluid properties.

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