Is this normal behaviour?

Hi group, looking for advice on problems integrating a new cat.

A week ago my older cat passed away (he was the alpha of my two neutered
males, the only two cats). I went to the rescue and adopted a 8 YO female
which was aparently used to other cats and friendly etc. For the first 2
days they studiously ignored each other aside from a few touch-noses but
this morning the new cat attacked Worf and has been hunting & chasing him
around the house (not just out of the immediate vicinity) to the point where
he cowers in fear and growls whenever she is in sight. I now have her shut
in a spare room with toys, bed, litter tray, food etc and Worf has gone
outside & not come back yet

Worf is a sociable cat and used to others, there are a number of cats in the
neighbourhood and they all come in and out of each others houses with no
trouble between them. As he was not the previous 'alpha' of the house, he is
generally passive in nature and therefore is not attacking back although he
is physically twice the size.

Would this behaviour be normal with a new cat asserting herself or does it
look like her nature has been mis-described as I was also told that she'd
been in the shelter for an extended period of time without finding a home
which is strange as she is a pretty little cat. She's also bitten my
neighbour who came in to see her, and after a couple of minutes of petting
also tried to claw & bite me. She's a Turkish Van if her breed gives any
clues.

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I.P.Freely