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• I found this phrase: "come the heavy swell over somebody" in my dictionary which means "Trying to leave an impression on somebody's mind by saying or presenting or demonstrating oneself to be of something great." Could anyone please tell me whether it is a common phrase among natives? If not, what some other phrases I can use instead of it?
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I've never heard of it, and search results suggest that it barely exists. There are only two or three relevant hits in Google Books, one in a dictionary of idiomatic expressions. "swell" in this sense (i.e. an important person) is a dated word. An equivalent that could be used nowadays is "come the high and mighty with (someone)".

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HasibrahmanCould anyone please tell me whether it is a common phrase among natives?

No. I've never heard it. It sounds like it might have been used in the 1800s.

HasibrahmanIf not, what are some other phrases I can use instead of it?

brag, boast; swagger; lord it over; put on airs; act important; look pleased with oneself; get on one's high horse

Of course, it depends on the register and context.

CJ