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I have a long duct with a fan in the middle of it. The fan causes a steady airflow in the pipe which is at low speeds (incompressible) the mean flow in the pipe is about 10 m/s. However the fan blades are spinning very fast with tip speeds up to 100 m/s and looking at the plots of normalised velocity over a segment of the fan blades (similar to the image below) one sees that the Mach number of the flow moving past the fan blades can attain appreciable mach number (i.e. Mach number > 0.3) so that compressible flow may result. It seems like the acceleration of a flow due to curvature over an obstacle can often induce large velocities even if the oncoming flow is incompressible. Is it then the case that most flows are locally compressible? Am I looking at this in the right way?

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(source: mh-aerotools.de)

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    $\begingroup$ Notice the caveat "at low speeds"? That's a hint that there is an approximation at work here. The next question to ask yourself is 'what sets the scale of "low"' and from that to deduce what physics is being neglected. $\endgroup$ – dmckee Aug 21 '14 at 14:39
  • $\begingroup$ Not exactly sure what you are hinting at. "What sets the scale of low?"- I guess that would be the fan speed and also the throttle opening position of the duct. $\endgroup$ – Dipole Aug 21 '14 at 15:09
  • $\begingroup$ No, Jack, he means "what terms in the complete equation can be deemed small enough to ignore and which flow rates?" $\endgroup$ – Carl Witthoft Aug 21 '14 at 16:53
  • $\begingroup$ There has been a minor comment edit to reduce abrasiveness. Try to remember to "Be nice." $\endgroup$ – dmckee Aug 21 '14 at 23:49
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I'll answer the easy question first -- you are looking at it the right way.

Now for the other question... it's really impossible to say that "most" flows are locally compressible. Although that's also a lie because every flow is compressible! Incompressibility is an approximation that makes the math easier, but even the slowest flows of air are technically compressible.

With that out of the way, the question you need to ask yourself is: does it matter? So in your ducted fan example, at the very tip the flow is around Mach 0.3. Engineers typically take $M < 0.3$ as incompressible. So you're right at this arbitrary limit. If it's only the tip of the fan, and it's just barely compressible, does it really affect the analysis of the flow so much more accurate to include compressibility that it's worth the added difficulty? Probably not.

But let's consider when the inflow is $M = 0.2$. Okay, so it's approximately incompressible inflow. But now the tip velocity may induce something like $M \approx 1.0$ if it's going fast enough and now you have strong compression waves or shocks forming at the tip. Now you really need to consider compressibility effects or you're missing major parts of the physics.

So basically all flows are compressible and saying they are incompressible is an approximation. One we are free to make whenever we think it's okay to make it. If you did the analysis assuming the flow was compressible, and did it assuming the flow was incompressible, you would likely see in your situation that the difference is really small. So it's okay to call it all incompressible.

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  • $\begingroup$ +1 thanks, and I agree with what you say. Im wondering what information we need to use the incompressible approximation justifiably. In this case, knowing that the tip speed is say $M = 0.1$ seems to motivate incompressibility, however if the blades are at aggressive incidence angles (within the range of not stalling the flow) maybe the flow on the suction side can attain $M = 0.8$ which would definitely not justify incompressibility. So my question is: does the max velocity always determine if we should regard the flow as incompressible or not? Hope that makes sense. $\endgroup$ – Dipole Aug 21 '14 at 16:15
  • $\begingroup$ @Jack Max velocity is part of it, but also how much of the flow is at that velocity. If only the tip is at Mach 0.3 then it's probably not a big deal. If the tip is at Mach 0.7 then most of the blade is probably above Mach 0.3 and it may matter. But it also depends on the analysis you're doing. If you are interested in blade dynamics/flutter then the compressibility may be important even at 0.1. If all you care about is the mean flow 10 feet downstream of the fan, then it may not matter. $\endgroup$ – tpg2114 Aug 21 '14 at 16:24
  • $\begingroup$ Ok great , got it! $\endgroup$ – Dipole Aug 21 '14 at 16:45

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