A washing and drying appliance includes a tub having an air outlet. A drum rotationally operates within the tub. An air handling system draws process air from inside the tub and through a lint filter disposed within the air outlet that filters particulate from the process air. During a filter cleaning phase, a fluid delivery system delivers impulse amounts of wash fluid toward a sump of the tub. The impulse amounts of wash fluid are maintained between an outer surface of the drum and an inside surface of the tub. The drum operates to direct the wash fluid around the drum and along the inside surface of the tub and over an interior surface of the lint filter to remove particulate from the lint filter. A fluid pump is coupled with the sump to direct wash fluid and particulate to an external outlet.