Laminar-turbulent transition and Reynolds number

[curiously new]

TL;DR

I'd like to derive an equation for Reynolds number as a function of pressure for a pinhole leak in a pressurized gas line. The line is pressurized to 1 atmosphere and is leaking into vacuum.

Hi all!

This is my first post here, so hopefully I am not in violation of any rules or etiquette. I'm looking to derive an equation for Reynolds number as function of pressure for a pinhole leak in a pressurized gas line. The line is regular air and is pressurized to 1 atmosphere and is leaking into vacuum.

I know that Reynolds number can be defined by the equation

$Re=\frac{\rho u L}{\mu}$ ,
​

where:

$\rho$ is the density of the air,​

$u$ is the fluid flow speed,​

$L$ is the characteristic dimension, and​

$\mu$ is the dynamic viscosity of the air.​

​

If I constrain the system so that the air does no work and no work is done on the air, and that there is no expansion/compression of the gas, I believe that I can apply Bernoulli's principle to the air and consider it an incompressible fluid. Then, for a rough approximation, I can use Bernoulli's equation for an incompressible fluid

$\frac{u^2}{2}+gz+\frac{p}{\rho}=\mbox{constant}$ ,
​

where:

$u$ is the fluid flow speed,​

$g$ is the acceleration due to gravity,​

$z$ is the distance from the reference plane,​

$p$ is the pressure at the point of interest, and​

$\rho$ is the density of the air.​

​

Now, if I place my plane of reference at the pinhole leak, then $z=0$ and the middle term vanishes. It was suggested to me to assume that the gas inside of the line is not moving so that the fluid flow speed in Bernoulli's equation is zero, $u=0$, so that

$C=\frac{p}{\rho}$ ,​

where $C$ is the arbitrary energy constant.

Outside of the line is a vacuum and so $p=0$ and we have

$C=\frac{u^2}{2}$ ,​

and since energy is conserved, we can equate the two and solve for the fluid flow speed, such that

$u=\sqrt{2\frac{p}{\rho}}$ .​

Substituting this into the equation for Reynolds number, I get

$Re=\frac{\rho u L}{\mu}=\frac{\rho}{\mu}\sqrt{2\frac{p}{\rho}} L=\frac{L}{\mu}\sqrt{2\rho p}$ .​

Finally, at STP this site gives $\mu=1.825\times10^{-5}\ \mbox{kg}\cdot\mbox{m}^{-1}\cdot\mbox{s}^{-1}$, and $\rho=1.204\ \mbox{kg}\cdot\mbox{m}^{-3}$, and if I just wing a pinhole size with a diameter of 10 microns, then $L=10\times10^{-6}\ \mbox{m}$, then I can write

$Re(p)=0.85\sqrt{p}\ (\mbox{s}\cdot\sqrt{\frac{\mbox{m}}{\mbox{kg}}})$ .​

So at 1 atmosphere, I get a Reynolds number of $Re(1\ \mbox{atm})\approx270$. According to the Reynolds number Wikipedia page, the laminar-turbulent transition will occur at a Reynolds number $Re_x\approx5\times10^5$, where $x$ is the distance from the leak. If I keep the pressure at 1 atmosphere, I find that I need a pinhole diameter of about 18mm to achieve that sort of Reynolds number.

I'm just looking for a rough approximation, and was wondering how confident that I could be in my derivation above. But my real question concerns the suggested Reynolds number for the laminar-turbulent transition boundary. How can I calculate Reynolds number at varying distances away from the leak?

Thanks in advance for any insights!

 

[hutchphd]

I am not expert in this field but the assumption that gas leaking into vacuum is an incompressible fluid seems to me a very bad starting point.
There are folks here who actually know what they are talking about so hopefully they will comment.

 

[boneh3ad]

If I constrain the system so that the air does no work and no work is done on the air, and that there is no expansion/compression of the gas, I believe that I can apply Bernoulli's principle to the air and consider it an incompressible fluid. Then, for a rough approximation, I can use Bernoulli's equation for an incompressible fluid

$\frac{u^2}{2}+gz+\frac{p}{\rho}=\mbox{constant}$ ,
​

where:

$u$ is the fluid flow speed,​

$g$ is the acceleration due to gravity,​

$z$ is the distance from the reference plane,​

$p$ is the pressure at the point of interest, and​

$\rho$ is the density of the air.​

As @hutchphd surmised, these are terrible assumptions for your application. In fact, what you have described is a highly compressible flow and treating it as incompressible will result in a wildly inaccurate answer. That doesn't necessarily make your goal here a fool's errand, but it does mean that Bernoulli won't apply.

As a result, temperature of the gas in the line will be important, as will determining whether your flow is choked or not (hint: it is) and whether your flow has any opportunity to become supersonic (it doesn't assuming the hole in question has constant diameter). Making assumptions such as isentropic flow will also ease your calculations and would be reasonable here.

So at 1 atmosphere, I get a Reynolds number of $Re(1\ \mbox{atm})\approx270$. According to the Reynolds number Wikipedia page, the laminar-turbulent transition will occur at a Reynolds number $Re_x\approx5\times10^5$, where $x$ is the distance from the leak. If I keep the pressure at 1 atmosphere, I find that I need a pinhole diameter of about 18mm to achieve that sort of Reynolds number.

When it comes to Wikipedia, caveat emptor always applies. In this case, there are several problems with this discussion. First, if you read the article you cite, that specifically applies to boundary-layer flow over a flat plate. Even if that was the situation here, it is still wrong in a general sense. If you actually follow the citation in the Wikipedia article, it comes from a heat transfer book by Incropera and Dewitt. I happen to have a copy of it with me, so here is what it says:

In calculating boundary layer behavior, it is frequently reasonable to assume that transition begins at some location $x_c$, as shown in Figure 6.6. This location is determined by the critical Reynolds number, $Re_{x,c}$. For flow over a flat plate, $Re_{x,c}$ is known to vary from approximately $10^5$ to $3\times 10^6$, depending on surface roughness and the turbulence level of the free stream. A representative value of
Re_{x,c} \equiv \dfrac{\rho u_{\infty} x_c}{\mu} = 5\times 10^5
is often assumed for boundary layer calculations and, unless otherwise noted, is used for the calculations of this text that involve a flat plate.

So let's talk about this for a moment.

• This applies only to boundary layers, which does not apply to your flow (as noted on Wikipedia).
• This is a massive simplification over what really happens (as noted in Incropera and Dewitt).
• Even for boundary layers, this is highly problematic (e.g. the transition Reynolds number on an airplane wing is often more like $10^7$).

The bottom line here is that you need to treat the flow as compressible and then choose a suitable transition criterion based on your actual flow. What you have is a free jet, which would have a Reynolds number based on the orifice diameter that would determine the behavior.

 

[osilmag]

​

Now, if I place my plane of reference at the pinhole leak, then z=0 and the middle term vanishes. It was suggested to me to assume that the gas inside of the line is not moving so that the fluid flow speed in Bernoulli's equation is zero, u=0

The speed of pressurized gas will not be zero especially if your reference is at an opening or nozzle.

 

[boneh3ad]

The speed of pressurized gas will not be zero especially if your reference is at an opening or nozzle.

Assuming the speed of the pressurized gas to be zero can actually be a very reasonable assumption under some circumstances.

 

[curiously new]

Thank you for your kind replies! I'm understanding that my initial constraint of the air so that it can be treated as an incompressible fluid is a large misstep. After reading boneh3ad's reply (apologies, I can't seem to work out how to link your profile yet), I've also investigated the boundary layer and realized that I had assumed that it was any boundary layer, and I have learned it's true meaning.

I'm a recent grad, and my boss (a late career Ph.D.) has given me this assignment and offered a little direction, and one of his hints was to constrain the air as an incompressible fluid. My suspicion is that they really just wanted to introduce me to the topics of the project and that this will probably see a more rigorous treatment later. Your replies have given me a lot to think about and look into.

Thank you so much for your time and assistance!