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

Question

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 1

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.

Answer 2

I think your guess of donburi is correct. The first sense line under 丼{どんぶり} in Shogakukan's 国語大辞典:

1. 厚手で深い陶製の食物を盛る鉢。どんぶりばち。
   "A thick and deep ceramic bowl which food is served in. [...]"

The food sense is the last sense given for this term:

4. 「どんぶりもの（丼物）」の略。
   "An abbreviation of donburimono".

 

UPDATE:

I think naruto is probably right: this may be a 茶碗{ちゃわん} instead. I hadn't considered the scale, and I don't think I've ever seen an earthenware chawan with an earthenware lid before. But given the size, this would be small for a 丼{どんぶり}.

UPDATE 2:

At 15cm, this could be a smaller 丼{どんぶり}. And regarding the term 丼{どんぶり} itself, this can refer to both the bowl and the foods traditionally served in this kind of bowl. The food is more specifically referred to as 丼{どんぶり}物{もの}.