Thanks again everyone, for your comments and links. I swear I had a Quena-style flute from an importer about 10 years ago, but it was a cheapy ($12?), nobody at the store had any idea how it worked. I worked at it quite some time and realized I had to block the end of the flute with my lower lip/chin to form a partial seal, and also allow a space to remain between the lip and the blade area. Unfortunately the one I had was probably made as a toy, was high pitched, the intonation was off and the blade area didn’t look very well made. It was bamboo. But I liked the type of tone I heard.
I’ve seen several labia that are ‘flat’ in that they slope in the same direction on the top as on the bottom.
Look at how a basic clarke is made-- the metal tube is folded in to create the labium. The inside edge of the labium slopes toward the inside of the tube, and the outside edge of the labium also slopes in toward the center of the tube. I’ve made a few 3d printed whistle heads with a labium profile so that toward the window, the top edge slopes inward and the inside edge slopes outward. They’re breathy and quiet. I would like to try and make some more that are sloped more like a folded tube whistle.
The feadog whistle I have, as well as the clarke sweetones and generations, follow the slope pattern you’d see in a folded metal tube whistle like a classic clarke. Recorders are often built with the same pattern. The stream is split but both directions of air travel tend to push the air toward the outside of the tube. I suspect this is key to generate more outward air pressure and volume. This inward-sloped labium profile is impossible to make on a bamboo endblown flute without generous carving, but a transverse flute has a similar relation to your breath as directed by your lips. A quena is much easier to make undercut, and I remember seeing a plastic recorder with an undercut inside edge once I sawed the beak off.