<u> pronounced "ew"

Solution 1:

In most cases the pronunciation of u as /ju:/ is an indication that the word is connected with a French word. French u is regulary pronounced as /y/ as in

• French le duc /dyk/ - English duke /dju:/, AmE /du:k/
• F dû/due /dy/ - E due /dju:/
• F la vue /vy/ - E view /vju:/

Also German words with ü become words with /ju:/ in English

• G München - E Munich /'mju:nik/

The letter u has the pronunciation (PN) /ʌ/ as in to cut, and the PN /ju:/ in open syllables as in cu-te. An open syllable has the structure vowel (no consonant); here only the written form is relevant, not the spoken form. A closed syllable has the structure vowel+ consonant.

In American English /ju:/ is reduced to /u:/ after a certain group of consonants as in

• duke /du:k/
• new /nu:/

In his comment below Peter Shor has enumerated the special consonants after which /ju:/ is reduced to /u:/.

Solution 2:

I specUlate that the "ew" (/ju/) is more common in Southern Regional English. E.g. there are towns in the South spelled with "u" which have the /ju/ sound: Bude, Mississippi, is pronounced like its English namesake, Bude/Bewd, England. Could it be another of those archaic survivals from dialectical isolation in the rural South? (e.g., "Pierre" pronounced as "Per"; "person" as "poisson"; "roasting" as "rowsn"; and the aformentioned "new" as "gnew", a pronunciation quite common in my rural South of the past. Not sure if "shoe"/"shew" is related, for other such words ("hoe", "froe") have a simple long "o" sound.