Apparatuses and methods for slicing products. Such an apparatus has an annular-shaped cutting head equipped with knife assemblies located at a circumference of the cutting head, and interior surfaces each located in a circumferential direction of the cutting head relative to one of the knife assemblies such that each interior surface leads one of the knife assemblies and trails a different one of the knife assemblies. Each knife assembly has a flat knife that extends radially inward to define a gate opening. Each interior surface has continuous flow paths defined therein that extend across at least a majority of the interior surface from adjacent a leading edge thereof through a trailing edge thereof and are fluidically connected to one of the gate openings between the interior surface and the knife that trails the interior surface.