I have Helmholtz's book in front of me so I know what he intended his resonators to be used for. If they had audible overtones there would have been no point in making them. (Helmholtz just says any other tones produced by a resonator are "very distant" from the prime tone, he doesn't analyze them at all).

Can you construct anything that makes those 10x overtones perceptible? Are you saying they're musically relevant?

I can't imagine why you would want to invoke Helmholtz resonators as components of a flute. These days anybody who wants to get detailed answers can just simulate the flute directly, doing numerical solution of the relevant partial differential equations. Computers have been fast enough for that to work for about ten years now.