A method for removing a deposit from at least one component of an assembled, on-wing aircraft gas turbine engine, the gas turbine engine comprising a core gas turbine engine positioned downstream of a fan section, the core gas turbine engine having an absence of powered rotation during the removal of the deposit, the method comprising: operably coupling a delivery assembly to an annular inlet of the core gas turbine engine, the delivery assembly being coupled to a control unit and a storage vessel containing a cleaning fluid, wherein the delivery assembly is configured to develop atomized droplets having a median diameter of less than or equal to 50 microns; atomizing a portion of the cleaning fluid with the delivery assembly to develop a cleaning mist, the cleaning mist comprising a plurality of atomized droplets; suspending the atomized droplets of the cleaning mist within an airflow within at least one flow path of the core gas turbine engine, wherein the cleaning mist occupies a cross-sectional area of the at least one flow path to establish a simultaneous cross-sectional contact therein, wherein at least a portion of the atomized droplets remains suspended within the at least one flow path from the annular inlet to an axial position downstream of a high-pressure compressor of the core gas turbine engine; dissolving at least a portion of the deposit on the at least one component, the method characterized by impacting or precipitating a portion of the cleaning mist onto the at least one component so as to wet at least 80% of an exposed, inlet-facing surface of the at least one component; and in that the atomized droplets have a median diameter of less than or equal to 50 microns and the cleaning mist has a fluid-to-air mass ratio of at least 0.05 and less than or equal to 5.0 kilograms of cleaning fluid to kilograms of air. operably coupling a delivery assembly to an annular inlet of the core gas turbine engine, the delivery assembly being coupled to a control unit and a storage vessel containing a cleaning fluid, wherein the delivery assembly is configured to develop atomized droplets having a median diameter of less than or equal to 50 microns; atomizing a portion of the cleaning fluid with the delivery assembly to develop a cleaning mist, the cleaning mist comprising a plurality of atomized droplets; suspending the atomized droplets of the cleaning mist within an airflow within at least one flow path of the core gas turbine engine, wherein the cleaning mist occupies a cross-sectional area of the at least one flow path to establish a simultaneous cross-sectional contact therein, wherein at least a portion of the atomized droplets remains suspended within the at least one flow path from the annular inlet to an axial position downstream of a high-pressure compressor of the core gas turbine engine; dissolving at least a portion of the deposit on the at least one component, the method characterized by impacting or precipitating a portion of the cleaning mist onto the at least one component so as to wet at least 80% of an exposed, inlet-facing surface of the at least one component; and in that the atomized droplets have a median diameter of less than or equal to 50 microns and the cleaning mist has a fluid-to-air mass ratio of at least 0.05 and less than or equal to 5.0 kilograms of cleaning fluid to kilograms of air.