Having trouble when refining k-eps model over an aerofoil. Does K-eps struggle at when grid sizes are getting really small?

All equations struggle with a denser mesh. Allow more iterations and use relaxation factors. Look in the documentation for appropriate values.

If the residuals are unstable from the beginning of the solution instead of becoming unstable as the solution begins to converge then you probably have a problem with your model or you need the unsteady solver.

Phil