A method for operating a rotating detonation engine, having a radially outer wall extending along an axis; a radially inner wall extending along the axis, wherein the radially inner wall is positioned within the radially outer wall to define an annular detonation chamber having an inlet and an outlet, wherein the method includes flowing liquid phase fuel along at least one wall of the radially inner wall and the radially outer wall in a direction from the outlet toward the inlet to cool the at least one wall and heat the liquid fuel to provide a heated liquid fuel; flowing the heated liquid fuel to a mixer at the inlet to reduce pressure of the heated liquid fuel, flash vaporize the heated liquid fuel and mix flash vaporized fuel with oxidant to produce a vaporized fuel-oxidant mixture; and detonating the mixture in the annular detonation chamber.