A structure-preserving geometric construction of a fully explicit discrete dynamical system equivalent to the Navier-Stokes equations based on conjugate Voronoi-Delaunay tessellations

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The motion of viscous fluids is governed by the Navier-Stokes equations. The inherent nonlinearity of the Navier-Stokes equations has historically led to the widespread use of implicit global solvers or explicit schemes with artificial stabilization. In this work, we present a fully explicit computational framework based on a structure-preserving geometric formulation defined on a conjugate Voronoi-Delaunay tessellation. Unlike conventional approaches derived from differential operator discretization, the proposed approach constructs a discrete dynamical system directly from geometric coupling relations, in which the governing conservation laws emerge as exact algebraic conservation identities of the discrete structure. Within this framework, conservation of mass and linear momentum is rigorously satisfied at the discrete level through algebraic cancellation based on the conjugate geometry of Voronoi and Delaunay cells. Furthermore, by defining the energy density on the Delaunay complex, the formulation preserves the total discrete energy exactly in the inviscid limit, establishing a discrete Hamiltonian dynamical structure. These fundamental invariants are preserved independently of mesh resolution and do not rely on asymptotic approximation or truncation arguments. This ensures the intrinsic physical and mathematical consistency of the discrete system. Thermodynamic consistency is ensured by determining pressure exclusively as a thermodynamic potential through the equation of state, rather than treating it as an independently transported variable. This eliminates the structural inconsistency arising from the mismatch between the independently transported pressure field and the pressure determined by the thermodynamic state. Such inconsistencies can act as a source of numerical instability by violating the intrinsic thermodynamic compatibility of the system. In addition, the formulation separates advective transport from force integration using a semi-Lagrangian transport mechanism combined with geometrically defined interaction forces. This separation provides a consistent and robust framework for fully explicit time integration while ensuring that numerical dissipation scales with the characteristic flow velocity rather than the acoustic wave speed, thereby preserving physically relevant flow structures across a wide range of regimes. A rigorous volumetric stability condition is derived, establishing a critical time-step constraint determined by the geometric volume scale of Voronoi cells. Numerical validation demonstrates high-order effective convergence behavior, consistent with the smoothness and structural consistency of the geometric formulation. By constructing a conservative and thermodynamically consistent discrete dynamical system whose properties follow directly from geometric structure, the present work provides a mathematically well-defined and physically faithful explicit formulation of viscous fluid dynamics. This establishes a unified structure-preserving computational framework for the discrete representation of Navier-Stokes dynamics, applicable across laminar, transitional, and turbulent flow regimes without empirical stabilization or turbulence modeling.