What’s the meaning of “hit him round the head”?

Dudley was sniffling in the back seat; his father had hit him round the head for holding them up while he tried to pack his television, VCR, and computer in his sports bag. (Harry Potter 1, Scholastic Paperbacks p41)

My guess about the meaning of this sentence was his father hit his son’s head with his palm once.

However, the more I think about it, the more ideas I get from it. I’d like to know if the following guesses of mine are correct or not.

1. hitting times

Dudley's father hit Dudley only one time because the writer uses the verb “hit” instead of “beat, etc.” and because “round” doesn’t add some information, like times, to the verb, since it’s not an adverb but a preposition.

2. Why does the writer use “round” instead of other prepositions? -- because

1). the head is cylindrical (or spherical), not flat. In other words, the shape of the target goes well with “round.”

2). the father hit his son’s head not with his fist, but with his palm which is along the head for a short moment. That is, the easy-to-bend quality of the hitting tool agrees with “round.”

3). the target area for the father to aim at is vague, not pinpoint. So, the hitter’s intention requires “round.”

4). as the father swings widely his hand (fist or palm whichever he likes), the path from the launching point to the head describes a circle. That means the shape of the orbit needs “round.”

5). Nope. 1-4 don't matter. British English sometimes uses ”round” for “on”, so there’s virtually no difference from “hit him on the head”. Take it easy.

Other ideas are always welcome. I’d appreciated it if you could help me.

"Hit/clout/slap round the head" is a UK idiom. It doesn't imply once or more than once.

Examples from the BNC: "clouted it very hard round the head with his stick"; "clouting him round the head"; "getting a tap round the head"; "who beats you round the head"; "Because you can't wallop them round the head can you?" and so on.

There's a more specific form "round the ear": this is nearly always "clip round the ear" (17 times in the BNC) but also "smack", "clout" and "belt" (6 instances in total). ("Clip" meaning a blow only occurs in this one phrase, as far as I know, but as a verb it can be used to mean "make an accidental, glancing contact", as "I clipped it with my mirror as I went past".)

Quite why we say "round" is not clear: it is certainly an aphetic form of the preposition "around". It doesn't seem to occur with other body parts than heads and ears (there are a few possible instances in the BNC, but they all might be interpreted as a literal "around"). It doesn't seem to imply any particular part of the head, or multiple blows "all round the head", as one might have supposed.