The Reynolds number is classically described in terms of pipe geometries but its use has also been usefully extended to other more complex surface geometries to predict transitional flow behavior. But is there a surface geometry of scale so small that Reynold's number ceases to have meaning?


The answer of user3823992 is correct: Reynolds number will cease to have meaning when the hypothesis of continuum mechanics will cease to be verified. To complete his answer, if the characteristic length scale of the studied geometry is close to the free mean path (i.e. Knudsen number close to 1), you can no longer consider the Navier-Stokes equations to solve your problem. And so Reynolds number is not definable.

To go further, even if you are unable to determine a characteristic length scale to your geometry, I would say the Reynolds number never cease to have meaning.

The sense of the Reynolds number is to quantify the ratio between inertial forces and viscous forces. The common definition is $Re=\frac{UL}{\nu}$ but it can be seen as $Re = \frac{(u.\nabla)u}{\nu \nabla^2 u} = \frac{\textrm{inertial forces}}{\textrm{viscous forces}}$. It will obviously give back the first formula by taking some characteristic length scale of your flow and replace it in the formula.

Also even if there is no characteristic length scale in your specific geometry, you should be able to quantify each force and find the Reynolds number of the flow. Depending on its value, the flow will be seen as laminar, transitional or turbulent.


To add to Lalylulelo's excellent answer, the Reynolds number only has meaning with respect to a particular flow geometry. That is, it's only useful in comparing two flows of the same configuration. A Reynolds number corresponding to laminar flow in a pipe geometry might correspond to unstable or turbulent flow in some other geometry (never mind the fact that the other geometry may not have an obvious scale length that corresponds to the pipe diameter).


But is there a surface geometry of scale so small that Reynold's number ceases to have meaning?

I want to give an example where the "scale" doesn't necessarily mean physical scale. To do this, I will consider increasing altitude in Earth's atmosphere, therefore decreasing density.

Consider increasing altitude at which a plane (or a rocket) is moving. We'll keep both velocity and temperature constant. These assumptions are poor, but they go in opposite directions so an accurate revised picture has mixed implications. I will use D as the characteristic linear dimension of the craft, $\rho$ is the atmosphere density at that altitude, and $d$ is the air molecular diameter.

$$ Re = \frac{ \rho V D }{ \mu } \\ Kn = \frac{ k_B T }{ \sqrt{2} \pi d^2 P D } $$

As we increase altitude, using an isothermal model, pressure and density will decrease. This will be dictated by the ideal gas law (again, a gross assumption, but let me continue), which I will write in the form below. Let $h$ be the altitude, which we will write pressure and density as functions of. Let $H\approx 8 km$ be the characteristic height of Earth's atmosphere. To a first approximation, the viscosity doesn't change. In ideal gas models, it is generally dependant on temperature, and we're taking that to be constant.

$$ P = R_{specific} \rho T \\ T = \text{ constant} \\ P(h) = P_0 e^{-h/H} \\ \rho(h) = \rho_0 e^{-h/H} $$

With this, we can easily revise the forms of the Reynolds and Knudsen numbers. Here, $Kn_0$ and $Re_0$ are the atmospheric values at sea-level.

$$ Kn = Kn_0 e^{h/H} \\ Re = Re_0 e^{-h/H} $$

Interestingly, the flow around a high-altitude plane or a rocket transitions to be more laminar, while at the same time, the molecular path length between collisions grows to larger than the scale of the craft itself.

These physics are relevant if you look at concepts for an orbital atmospheric scoop to collect gas for propellant. While the regime would be transitional (Re close to 100 or so) for a small sattelite, it doesn't matter because the molecules move in straight-line paths. The high orbital velocity then causes some substantial collimation, and you wind up with the best candidate geometries to be more like long straws than funnels because the traditional fluid mechanics rules just don't apply until the density can be increased.

This low-density situation is sure to come up in many other important scientific fields as well, like low density plasmas. So a high Knudsen number could be a matter of small physical scale, but low density can put it at unity at fully macroscopic scales.


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