We conduct a long-time-scale ($5000\,$ d) 3D simulation of a common-envelope event with a $2\, {\rm M}_{\odot }$ red giant and a $1\, {\rm M}_{\odot }$ main-sequence companion, using the moving-mesh hydrodynamic solver manga. Starting with an orbital radius of $52\, \mathrm{ R}_{\odot }$, our binary shrinks to an orbital radius of $5\, \mathrm{ R}_{\odot }$ in $200\,$ d. We show that over a time-scale of about $1500\,$ d, the envelope is completely ejected, while 80 per cent is ejected in about $400\,$ d. The complete ejection of the envelope is solely powered by the orbital energy of the binary, without the need for late-time reheating from recombination or jets. Motivated by recent theoretical and observational results, we also find that the envelope enters a phase of homologous expansion about $550\, \rm d$ after the start of our simulation. We also run a simplified 1D model to show that heating from the central binary in the envelope at late times does not influence the ejection. This homologous expansion of the envelope would likely simplify calculations of the observational implications such as light curves.

%0Journal Article