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?
(source: mh-aerotools.de)
Best Answer
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.