Hey all!

For my Ultra Agents series, I am going to need to make swirling portals. They can't be square, and should look something like a black-hole.
Because I possess only a rudimentary knowledge of visual effects, I'm not sure what to do. I have done a few tests already, but none have looked particularly good.
Any suggestions?
It should be noted that I don't currently own any high end visual effects programs, such as After Effects, but I will most definitely get one, if required.