The conventional teaching is that the vowels are AEIOU and sometimes Y.
Y is a consonant in Yet and Yoghurt. It is a vowel in City and Candy.
W is a consonant in Woman and Wine.
W is part of a diphthong in words like
PAWN, FAWN, FOWL, FEW
My ESL books based on phonics study teach the short vowel sound for 'Y", but W is not taught as an independent vowel letter. It modifies the preceeding vowel sound in some words to make a new vowel sound. There are similar lessons on R as in
FARM, BURN, TERN /ur/
Here are the current definitions:
VOWEL
1. Phonetics.
a. (in English articulation) a speech sound produced without occluding, diverting, or obstructing the flow of air from the lungs (opposed to consonant).
b. (in a syllable) the sound of greatest sonority, as i in grill. Compare consonant (def. 1b).
c. (in linguistic function) a concept empirically determined as a phonological element in structural contrast with consonant, as the (é) of be (bé), we (wé), and yeast (yést).
2. a letter representing or usually representing a vowel, as, in English, a, e, i, o, u, and sometimes w and y.
CONSONANT
1. Phonetics.
a. (in English articulation) a speech sound produced by occluding with or without releasing (p, b; t, d; k, g), diverting (m, n, ng), or obstructing (f, v; s, z, etc.) the flow of air from the lungs (opposed to vowel).
b. (in a syllable) any sound other than the sound of greatest sonority in the syllable, as b, r, and g in brig (opposed to sonant). Compare vowel (def. 1b).
c. (in linguistic function) a concept empirically determined as a phonological element in structural contrast with vowel, as the b of be, the w of we, the y, s, and t of yeast, etc.