# Reynolds number, turbulence regime, and drag force

I am trying to model a system in which cubes of about 2 cm in size are floating in a circular water thank of about 30 cm in diameter. The cubes move around under the influence of the fluid flow induced by four inlets that point toward the center of the tank, and are located at the positions $0$, $\pi/2$, $\pi$, and $3\pi/2$. The flow velocity ranges from 0 to 10 cm/s, with an average velocity around 6 cm/s.

My questions are the following:

• What would be the Reynolds number of the system? In particular, should I take as characteristic length the size of the cubes, or that of the tank?
• For such a system, what is the limit Reynolds number for the turbulent regime?
• What would be the correct form of the drag force, and do you intuitively think that the orientation of the blocks is negligible from a drag coefficient point of view?

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## 2 Answers

I'd say that you have several regimes that are well defined:

• The behavior of the fluid as it exits in inlet jets and enters the bulk without interference from the cubes. [Length scale set by the exit aperture?]
• Flow of the fluid around isolated cubes when far from the edges of the tank (far being several times the characteristic size of the cube). [Length scale set by the side of the cube.]
• Flow of the fluid toward, along and away from the sides of the tank away from the jets and without interference from the cubes. [Length scale set by the boundary behavior?]

which is the good news, unfortunately you also have all the cases that mix and match the various length scales:

• case with cubes interacting with the jet near the aperture
• case with cubes in motion near the walls
• case with cubes in close proximity to one another

You can probably find existing treatments for all the former cases, but the latter ones are going to be tricky, and you'll note that they feature at least two length scales.

Yuck.

This must be part of why they say CFD is hard.

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Well, the answer is more complicated than I hoped, but that's interesting, thanks. – Greg Mar 2 '11 at 7:32

The characteristic length can be whatever you want it to be. Characteristic lengths are only prescribed for a limited range of "standard" problems. For that matter, what do you select as your velocity?

Likewise, the dependence of flow characteristics on Reynolds number depends on how you define your Reynolds number. To my knowledge, unlike, say, flow between parallel plates, pipe flow, or flow over an airfoil, this is not a well-studied problem, so there really is no answer anyone can give to your second question. You'll have to physically do the experiment to see.

If you are referring to the drag force on a specific block, I would ask: does the coefficient of drag for an airfoil change with angle of attack? There's your answer.

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