  In this paper we compute the Ricci flow formulas for invariant metrics on
prinicpal $G$-bundles compatible with the connection. Our primary focus is on
torus bundles which we use to study a notion of Bakry-\'Emery Ricci flow as
well as Ricci flow on circle bundles over K\"ahler-Einstein manifolds. The
latter application gives us solutions to Ricci flows on Heisenberg groups and
implicit solutions to Ricci flows on Berger 3-spheres and several other
3-dimensional manifolds.
