The vorticity equation for two-dimensional incompressible viscous flows is formulated within the GENERIC formalism for non-equilibrium thermodynamics. The laws of conservation of energy and increasing entropy derived from the GENERIC formulation are properly inherited by the finite difference equations obtained by invoking the discrete variational derivative method. The law of increasing entropy corresponds to the dissipation of enstrophy for the vorticity equation. Some numerical experiments have been done to examine the usefulness of the proposed method.
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