Have a look at these simulations:
http://www.astro.virginia.edu/VITA/ATHENA/implosion.html
http://www.astro.virginia.edu/VITA/ATHENA/blast.html
(click on the right hand images to play the animations)
They are not exactly the situation you describe (given the simulations are in a rectangular box), but they do show how interactions between initially smooth compression waves lead to instabilities and turbulent mixing.
What I think would happen in your scenario is that the slight imperfections in the waves travelling towards and away from the centre would interact and become unstable.
Further, this instability and subsequent mixing would reduce the peak denisty capable to well below fusion requirements.
