Transitional Reynolds Number When Analysing Flow Through Orifices

[WhiteWolf98]

TL;DR Summary

I'm looking at laminar flow through an orifice, and a textbook I'm using has defined the transitional Reynolds number in a fashion I don't understand.

Greetings,

I need to work out the flow rate of of a flow through an orifice, where the size of the orifice and differential pressure are varied. The primary unknown in working out flow rate is the coefficient of discharge. A textbook I've been using to help me is "Hydraulic Control Systems" by Herbert E. Merritt. In it, he talks about both laminar and turbulent flow through orifices, and defines the coefficient of discharge for both.

From my understanding, the coefficient of discharge only varies when the flow is laminar, and is a constant for turbulent flow. What I'm confused about is a way in which the transitional Reynolds number is defined in the textbook, right at the end of the section on laminar flow.

It can be seen on page 45, the transitional Reynolds number is defined by 3-42. And it doesn't make any sense. Well, it does and it doesn't at the same time. How is it possible to have a transitional Reynolds number of 9, when the transitional Re number is generally between 2300 and 3500. Maybe I am being very dumb, but can someone else see what is going on here? Perhaps I am missing something crucial.

 

[jack action]

How is it possible to have a transitional Reynolds number of 9

By definition, a very tiny velocity in a very tiny orifice?

Also, it is specified that equation 3-38 was found by analyzing flows with $R \lt 10$, thus values in that region should be expected for equation 3-42 to be valid.

 

[boneh3ad]

TL;DR Summary: I'm looking at laminar flow through an orifice, and a textbook I'm using has defined the transitional Reynolds number in a fashion I don't understand.

Greetings,

I need to work out the flow rate of of a flow through an orifice, where the size of the orifice and differential pressure are varied. The primary unknown in working out flow rate is the coefficient of discharge. A textbook I've been using to help me is "Hydraulic Control Systems" by Herbert E. Merritt. In it, he talks about both laminar and turbulent flow through orifices, and defines the coefficient of discharge for both.

From my understanding, the coefficient of discharge only varies when the flow is laminar, and is a constant for turbulent flow. What I'm confused about is a way in which the transitional Reynolds number is defined in the textbook, right at the end of the section on laminar flow.

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It can be seen on page 45, the transitional Reynolds number is defined by 3-42. And it doesn't make any sense. Well, it does and it doesn't at the same time. How is it possible to have a transitional Reynolds number of 9, when the transitional Re number is generally between 2300 and 3500. Maybe I am being very dumb, but can someone else see what is going on here? Perhaps I am missing something crucial.

What constitutes a reasonable value for transition Reynolds number depends on the problem. The 2300 you cite is specifically for fully developed pipe flow. You are not currently studying fully developed pipe flow, so your scalings are all different.

For a boundary layer, a reasonable transition Reynolds number might be $10^6$. It's a failing of most mechanical engineering curricula that so many engineers finish fluid mechanics courses thinking that $Re_{tr}=2300$ is some kind of universal rule for transition.

 

[WhiteWolf98]

By definition, a very tiny velocity in a very tiny orifice?

Also, it is specified that equation 3-38 was found by analyzing flows with $R \lt 10$, thus values in that region should be expected for equation 3-42 to be valid.

Yes I saw this, that is a very tiny Reynolds number. The smallest Re number I'm dealing with is about 320. It's just bizarre to me to be talking about different flow regimes at such tiny values.

So depending on the conditions, are we saying that it's possible to get different flow regimes even at low Reynolds numbers?

 

[WhiteWolf98]

What constitutes a reasonable value for transition Reynolds number depends on the problem. The 2300 you cite is specifically for fully developed pipe flow. You are not currently studying fully developed pipe flow, so your scaling are all different.

For a boundary layer, a reasonable transition Reynolds number might be $10^6$. It's a failing of most mechanical engineering curricula that so many engineers finish fluid mechanics courses thinking that $Re_{tr}=2300$ is some kind of universal rule for transition.

Think you've answered my question there as I was typing it.

I'll be honest, I did think that. Thank you for your response