| The development of a general method for the direct solution of the Navier-Stokes equations, where no assumptions or modeling are required, with any equation of state, while maintaining thermodynamic equilibrium is the subject. This is accomplished through generalization of the Characteristic Based Split (CBS) method by removing isentropic assumptions and fully coupling the equation of state with the pressure and energy fields. The Modified CBS (MCBS) method is developed in rigor from first principles with the Navier-Stokes equations, where the equation of state is not required to be known or an analytical expression. Thermodynamic equilibrium, or thermodynamic consistency, where the pressure field from the equation of state, p(rho,T), is the same as the dynamic pressure field, is recovered through the implicit treatment of the temperature field during the solution of conservation of energy. Implicit treatment of both the pressure and temperature fields further enhances the MCBS method by permitting the integration over acoustic time scales if desired, achieving acoustic filtering without modification to the underlying governing equations.;The MCBS as implemented in a new Finite Element Method (FEM) code is applied to the study of compressible natural convection, where the entirety of Navier-Stokes equations is expressed, with several equations of state. Validation of the MCBS method for incompressible Boussinesq, incompressible thermodynamic Boussinesq, and compressible low-Mach natural convection in a cavity and near wall compressible thermal expansion waves is achieved with exceptional accuracy with the single MCBS implementation. Further, the solution of natural convection in a cavity using RefProp for the equation of state as well as all thermodynamic and transport properties was successfully achieved with the same implementation, providing real fluid results. The case of natural convection in a cavity is further pushed into higher Rayleigh numbers where the flow becomes time dependent in two dimensions and turbulent in three dimensions. In two dimensions the effect of very strong temperature differences and the equations of state is explored on the transient and time averaged steady state. In three dimensions steady state is achieved with a moderate Rayleigh number in a unit cube, and the turbulent transient captured directly without the use of a turbulence model for a high Rayleigh number. |
