Is there any good approach to determining if an amount of air passing through a given slit or gap will cause a whistling sound? I'm unsure with what shapes whistling might become an issue, and also at what air velocities.
I don't know. But generally in dust collection (which is allowed to be loud) we try to stay below 5000 fpm through ducts as above that you tend to get lots of noise. Through an open orifice, I have no idea off the top of my head. It also depends on the angle of incidence between the air path and the orifice (think of a flute or soda bottle)
There isn't a good approach of which I am aware. To really figure it out you will likely need some aeroacoustic simulations. It's going to depend on the geometry of the orifice, slit, or gap, the geometry of the vessel through which the air is flowing, and the details of that air flow. The whistling is caused by the oscillating of the shear layer that forms at the orifice. The amplitude and frequency of those oscillations as well as whether or not the surroundings are capable of resonating with that frequency are going to determine how audible any whistle is going to be. Depending on the geometry in question, there are going to be a few rules of thumb to help roughly determine the frequency, but that won't give you much information about whether it is going to be audible.
It isn't restricted to commercial stuff. I've had cannulas (nasal prongs) for my oxygen equipment that whistled so loudly in my nose that I couldn't go to sleep. It seemed to be at their whim: sometimes they did and sometimes they didn't. When they did, I would bend and twist and beat them until they got down to a level that I could eventually ignore.