I'm wondering about the energy/flow of water in a pipe. The continuity equation states that u1A1=u2A2. Say for instance, a turbine is placed in the pipe to extract energy from the flow; by the conservation of mass, neither the cross-sectional area or the velocity of the flow can change if the pipe diameter is constant both upstream and downstream of the turbine. However, surely the turbine must extract energy from the flow, such that it has less energy on the downstream side? How is this reflected in the flow, when the volumetric flow is unchanged?

If your question is about energy, perhaps you should look at the equation that describes the energy of a fluid.

The pressure changes meanwhile. The turbine can harvest PV at most. In a turbine, the transformation into shaft power involves steps where pressure converts into speed - which does mean a change of section, locally where the blades are and the flow direction changes.