By variational analysis, a couple of variational principles for dam spillway problem were testified a couple of maximum principle and minimum principle, and by variational analysis, furthermore, this couple of maximum and minimum principles was proved to be a pair of complementary extremum principles. So it not only provides a more rigorous theoretical foundation for their application, but also contributes to find the FEM solutions of problems in fluid mechanics.

variational principle, complementary extremum principle, moment function, azimuthal angle function.