food - What would be the specific name of this type of bowl?

The closest thing that I can find is Donburi? but that seems more like types of food not bowls.

ADDING:

This is about 15 and a half cm/6in

Answer

I think this is a chawan (茶碗【ちゃわん】) rather than a donburi, because it's small. A typical chawan in my home is about 12 cm in diameter, and a donburi is about 18 cm or larger. Chawan literally means "tea bowl", but it's usually used for serving cooked rice (See this).

A typical and cheap chawan doesn't have a lid, but some do. If you want to be specific, you can also say 蓋【ふた】付【つ】き茶碗 (futa-tsuki chawan, "chawan with a lid").

Alternatively you can use a more generic term, お椀 (owan, "bowl").

EDIT: A bowl of 15 cm can be called a 丼, and the name would partly depend on how this is intended to be used.