  In this paper, we present a general numerical platform for designing
accurate, efficient, and stable numerical algorithms for incompressible
hydrodynamic models that obeys the thermodynamical laws. The obtained numerical
schemes are automatically linear in time. It decouples the hydrodynamic
variable and other state variables such that only small-size linear problems
need to be solved at each time marching step. Furthermore, if the classical
velocity projection method is utilized, the velocity field and pressure field
can be decoupled. In the end, only a few elliptic-type equations shall be
solved in each time step. This strategy is made possible through a sequence of
model reformulations by fully exploring the models' thermodynamic structures.
The generalized Onsager principle directly guides these reformulation
procedures. In the reformulated but equivalent models, the reversible and
irreversible components can be identified, guiding the numerical platform to
decouple the reversible and irreversible dynamics. This eventually leads to
decoupled numerical algorithms, given that the coupling terms only involve
irreversible dynamics. To further demonstrate the numerical platform's power,
we apply it to several specific incompressible hydrodynamic models. The energy
stability of the proposed numerical schemes is shown in detail. The
second-order accuracy in time is verified numerically through time step
refinement tests. Several benchmark numerical examples are presented to further
illustrate the proposed numerical framework's accuracy, stability, and
efficiency.
