Alternative and Explicit Derivation of the Lattice Boltzmann Equation for the Unsteady Incompressible Navier-Stokes Equation

A lattice Boltzmann equation for fully incompressible flows is derived through the utilization of appropriate ansatzes. The result is a singular equilibrium distribution function which clarifies the algorithm for general implementation, and ensures correct steady and unsteady behavior. Through the Chapman-Enskog expansion, the exact incompressible Navier-Stokes equations are recovered. With 2D and 3D canonical numerical simulations, the application, accuracy, and workable boundary conditions are shown. Several unique benefits over the standard equation and alternative forms presented in literature are found, including faster convergence rate and greater stability.