In Fantastic Beasts: The Crimes of Grindelwald, Dumbledore says:
"Well, I've always felt an affinity with the great magical birds. There's a story in my family that a phoenix will come to any Dumbledore who is in desperate need."
Fantastic Beasts: The Crimes of Grindelwald - Original Screenplay - Page 43<
When Ariana Dumbledore was attacked by the Muggle boys, why didn't a phoenix come to her aid, but Fawkes came to Harry (who is not a Dumbledore) in Chamber of Secrets?.