On a symmetry-preserving unconditionally stable projection method on collocated unstructured grids for incompressible flows

This work establishes the conditions for having a symmetry-preserving unconditionally stable projection method for incompressible flows, such as PISO or Fractional Step Method (FSM), on collocated unstructured grids based on a compact stencil Poisson equation. Its formulation is based on preserving the underlying symmetries of the differential operators. In addition, a general theorem for these types of projection methods will be proven. To establish an unconditionally stable method, this theorem gives the mathematical requirements for the operators and the geometrical conditions a mesh must satisfy, even in cases where the mesh is highly distorted. This will be proven both theoretically and numerically. Conservation of (global) kinetic energy is also a key feature in simulations. Within this context, two canonical cases are tested, a turbulent channel flow at Re_r = 395 to show the robustness of the method, and an air-filled differentially heated cavity at Ra = 10^10 (based on the cavity height), to show that the (artificial) kinetic energy error introduced by the pressure is negative and very small compared with the physical dissipation.