People use different nomenclature for discussing fingerings. My preference, having played sax and clarinet that are often written in "instrument key" rather than in "concert key," would be to refer to the p'whistle with all 6 holes closed as "D." regardless of the "key" of the whistle.

"Blowing to pitch" as I described above should be sufficient for the first two octaves, uncovering one hole at a time bottom to top. Personal characteristics (the size of one's mouth) may affect the range of whistles one can play most easily. I have more difficulty with low notes on an F/Bb whistle than a D since it's difficult for me with my small mouth (you all believe me, I assume) to "open up" enough to blow to lower pitches on the long whistles. The top note or two on a shorter whistle might be a problem for someone with a "big mouth" due to difficulty in "squeezing down" enough to "whistle the note into the whistle."

The d' should be easy with all holes closed, then opening the top hole. The top hole works as a "trip hole" that creates a "disturbance" in the air column "about a quarter of the tube length down" which causes the air column inside to move to the 4x harmonic (2 octaves up) of the D it would otherwise play.

In this case, if you're already blowing to the "octave of D" (d), "doubling up" kicks you fairly easily to the next octave (d') note. (The top hole is about 1/4 of the way down the tube, which works out about right for the 4x harmonic of the tube tonic.)

On more complex instruments like saxophones, clarinets, etc, an "octave key," usually pressed with the left thumb, opens a "trip hole" but below about the 5th of the low end tone the hole is down among the other keys. Once the "G" key is opened the trip hole goes to someplace higher up on the horn, and on a sax the "upper trip hole" is way up on the top of the gooseneck. This all, of course, requires links and levers and stuff you don't have on a simple whistle.

A straight tube, with some sort of device to "make a distubance at one end" will have a particular frequency, since the compression wave in the air is "terminated" at the open end, so the time for a wave to travel down the tube and back determines the pitch.

A "leak" near the lower end of the tube, or "half-holing" the bottom hole, will "suck the end" of the air column up into the tube, although not necessarily all the way up to the "leaky hole," so false fingerings that open, or "leak," a particular hole somewhere at or just above the lowest closed hole can produce a "sharp" by shortening the effective length of the air column. (Writing the equations for where the effective end of the air column is, relative to the easily measured length of the tube, is really messy. The air movement doesn't just "chop off" at any particular place.)

On most simple instruments, such "false fingerings" can produce fairly accurate chromatic scales, and various "fingering charts" provide ways of playing the "notes in between" the diatonic scale notes. Details of the individual instrument may affect the accuracy of the "false notes," so some experimenting may be needed to find a "best set" of fingerings for a specific instrument. The difference between an open hole and a "leaky hole" can make a difference, but controlling a leak between "open" and "half hole" fingering can be difficult. Intonation - perfectly tuned pitch - for the "in betweens" is unlikely to be really accurate against a meter, but can be close enough to not spoil the music.

Slightly more complex instruments, like "Recorders," may have actual "half holes" drilled in to facilitate some accidentals. Covering/opening both of a pair of half-holes is like a whole hole, while covering/opening one of the two can produce a more accurate "leak" (half-hole). A few makers have even made the two adjacent half-holes in a pair slightly different in diameter so you can choose which one works best, but I think that's rather rare.

For most players, figuring out the right one or two accidentals may significantly expand versatility (esp. when playing with groups), to allow adding one sharp or one flat to the key signature; but trying to make a simple whistle into a fully chromatic instrument is likely to take more practice than really worthwhile. It's usually a lot easier to have two or three whistles to handle key changes.

John