A flume is a deep, narrow channel that is used to transport water from some source - such as a river - to the top of a waterwheel. This is used to direct water flow onto the blades of the wheel, allowing it to move.
The phrase flume is only used to describe the water channel in an overshot waterwheel. For undershot or breastshot waterwheels where no man made channel is necessary to direct water onto the wheel, the flume tends to be called the mill race.
Other special names for a flume include a leat, lade, or goit.