Does the Reynolds number uses the velocity of the fluid or the velocity of the object going through it?

Question

On the Wikipedia page for the Reynolds number, it is defined as:

$$ Re = {\frac {\rho uL}{\mu }} $$ where:

ρ is the density of the fluid (SI units: kg/m3)
u is the flow speed (m/s)
L is a characteristic linear dimension (m) (see the below sections of this article for examples)
μ is the dynamic viscosity of the fluid (Pa·s or N·s/m2 or kg/(m·s))

However, on the Drag page, the formula for the Reynolds number is the same but the variables mean something different.

• $ u $ is the speed of the object relative to the fluid
• $ L $ is the diameter of the object.

It seems that on one page the Reynolds number is determined completely by the properties of the fluid.

However, on the second page, the object crossing the fluid has an influence on the Reynolds number.

How can the variables in the Reynolds number have different interpretations?

Answer 1

Reynolds numbers aren't a single unified concept

The Reynolds number is not one unified thing, it does not have an unambiguous definition, and it is not comparable between different geometries as a result.

(We still do it, compare Reynolds numbers that are irreconcilibly different and should never be compared... But we know that we're not supposed to and we are mostly concerned with comparisons that are as course or coarser than one order of magnitude to try to absorb the geometric ambiguity. “They both have a Reynolds number around 10⁵” for example. Because of the geometric difference, two different geometries won't have the same levels at which they switch over from laminar to turbulent, nevertheless we might try to argue that both things are probably turbulent if the geometric differences are not huge and the Reynolds numbers are comparably large.)

The Reynolds number is fundamentally about how you construct a scale model. “I want to study the impact of drag forces on my pickup truck when the back gate is up or down, but driving it through the atmosphere I am getting a lot of noise because the wind isn't constant and I have to measure its gas consumption but that depends on traffic and tire temperature, yadda yadda, so I bought a hobbyist model pickup truck that is 10cm high and I want to put it into a ‘wind tunnel’ experiment, except that the best one I have does not use air but water.” That's the scenario you should picture. So I want to investigate roughly the same geometry, but I am changing out the fluid and the size, and I want to know what fluid velocity corresponds to driving this pickup truck at 100km/hr.

And the Reynolds number swoops in to say, “compare ME, make me the same between the two.” The Reynolds number says that the fluid has an intrisic momentum diffusion coefficient in m²/s, which should be divided by the characteristic length scale to get a characteristic velocity scale. Re = 100 means “things are moving at 100 diffusion-velocities for this fluid and size scale.”

Some different “geometries”

It helps to have some examples in mind, here's some:

1. A flow of fluid comes in from infinity at some fixed flow velocity, then travels around an object like a marble that is fixed in space. While the fluid has many different speeds as it gets closer to the object, we all agree to use the speed off at infinity upstream as our reference speed, and the diameter of the marble as the length.

2. A flow is moving through a pipe and there is no object. Velocity and length mean something very different now! Length is going to be something like the inner diameter of the pipe, and velocity is no longer expected to be uniform across the cross-section of the pipe, so you have to choose something and you probably choose the peak velocity in the very middle or an average or something.

3. Now fix a marble inside the pipe. You now have an additional dimensionless parameter to play around with, the diameter of the marble divided by the diameter of the pipe, $d/D$. And so you've got many choices of characteristic length you might use for Reynolds number calculation... $D,d, (d+D)/2, D-d, \sqrt{dD}, \sqrt{D^2-d^2},$ and so on... None of these is technically right or wrong because the scaling argument says “you have to preserve all of your dimensionless parameters as you scale: Re AND $d/D$,” which protects whichever choice you make. So you probably choose one based on further investigation, maybe $\sqrt{D^2-d^2}$ happens to offer some nice pretty linear-looking graph in the paper you are publishing, so you choose to use that.

4. A particle is moving through a stationary fluid, this geometry is not quite related to the geometry 1 by reference frame changes, because before we were holding the particle fixed, but here we are usually firing a particle into the fluid and letting it come to rest. So the definition is subtly different again, maybe the velocity is the initial velocity of the particle when it enters the fluid.

Four very different situations, defining very different numbers! No wonder I am saying that you cannot compare them directly! But they each satisfy their purpose, which is to aid us constructing scale models of each one of these phenomena.

How this answers your question

You are observing a discrepancy between definitions for example (1) and (4) above, and you are treating it as a real discrepancy, in other words you are expecting this to be a very precisely defined idea about where maybe laminar turns into turbulent or some such.

But the idea was never defined as precisely as you want it to be. It was always a scaling heuristic, which proved helpful in creating some pretty graphs for scientific papers. The transition from laminar to turbulent might be 10² in one geometry and 10⁴ in some other geometry, hinting perhaps that one of those is more streamlined and delicate than the other one, but not unambiguously so. (Perhaps instead the choice of what length to measure is somewhat anomalous, for example. So like: This hourglass shape is measured best with calipers on the thinnest part, but that boulder shape is measured best by a circumference of tape drawn around its largest part.)

Answer 2

Flow speed is how fast the fluid is moving relative to the object. It doesn't matter which of them is considered fixed in place. There is no fixed reference frame for constant linear motion anyway. It's always relative.