"I can't imagine why you would want to invoke Helmholtz resonators as components of a flute."

Sadly you have now willingly joined the ranks of 'Conrad's Army of Incompetent Uneducated Sef Important Pseudo-Musicians'

Let tell you about the concept of "Lies To Children" - what you appear to think you know about Helmholtz Resonators is a very simplified idea based on Assumptions to ignore complexity that work for an Ocarina, but are critical in the wider reality!

I do wish people would do their own research!

Start at this University - Flute acoustics: an introduction

and follow all the links

After you have read the section on "The cork and the 'upstream space'"

I look forward to your public apology! "I can't imagine why you would want to invoke Helmholtz resonators as components of a flute." - indeed!

QUOTE
The cork and the 'upstream space'
Between the point where the embouchure riser meets the main bore of the flute and cork in the closed end of the instrument is a small volume of air. The cork is normally positioned to be about 17 mm from the centre of the embouchure hole (the exact value varies from player to player - see tuning wind instruments). Any very substantial variation seriously upsets the internal tuning of the flute. So how does this work?

Diagram of cross section showing cork, embouchure etc

This 'upstream air' acts like a spring - when you compress it, the pressure rises. The air in the embouchure riser tube can be considered as a mass. Together they can resonate like a mass bouncing on a spring (ie they form a Helmholtz resonator).This has a resonance over a broad range of frequencies, but centred at about 5 kHz. At much lower frequencies, which is to say over the playing range of the flute, it acts as an impedance in parallel with the main part of the bore, but an impedance whose magnitude decreases with frequency. The primary effect of this is good: with the cork correctly placed, it compensates for the frequency dependent end effects at the other end of the flute and so keeps the registers in tune with each other. On the other hand, it does reduce the variation in impedance with frequency when the frequency approaches the Helmholtz resonance, and so is one of the effects that limits the upper range of the instrument. If you push the cork in, as Charanga style players do, you can go further up into the fourth octave, but at the expense of having an instrument whose octaves are badly out of tune. If you want to know more about this effect, download our technical paper about it. To scale the highest reaches of the flute's range, search for 'high playability' fingerings on the virtual flute and the report on F#7 and G7
UNQUOTE

AARGGGGHHHHHH! said Charlie Brown!