This thesis presents a unified approach to artificial compressibility methods and local low Mach number preconditioning methods. A single preconditioner is derived that can act as the artificial compressibility or the local preconditioning under a given Mach number condition. The preconditioning matrix is applied to a single solver so that the solver can analyze problems ranging from incompressible fluid flows to supersonic flows with shocks.
Multiple well-known local Mach number preconditioners and artificial compressibility methods are introduced and examined. It is shown that both techniques can be considered to be modifications of the bulk thermal expansion coefficient and isothermal compressibility. To unify the techniques, preconditioners that can be defined in terms of the bulk thermal expansion coefficient and isothermal compressibility are preferable.
A single matrix, called the unified preconditioning matrix, that includes several preconditioners and artificial compressibility methods is proposed. The modifications of the bulk thermal expansion coefficient and isothermal compressibility that yield each method are provided in tabular form. Additionally, a new artificial compressibility method that is proper for the unified system is proposed.
Then a formulation is presented which gives a seamless transition between incompressible and compressible systems. It is shown that a proper formulation of enthalpy is the crucial step towards the unification of these systems. Such a formulation enables the development of a single solver using the unified preconditioner.
The unified preconditioner is applied to Inha Aerodynamic Analysis and Design Laboratory's in-house unstructured solver, UMSAPv. In particular, the unified preconditioner is applied to the upwind approach, Roe's approximate Riemann solver, and the JST artificial dissipation approach. Also, a Roe average which can be applied to the unified systems in a single form is newly derived. It is shown that the average satisfies Roe's three mathematical conditions; hyperbolicity, consistency and property U.
Both incompressible and compressible analyses are performed for laminar and turbulent flows around various geometries. In particular, the proposed artificial compressibility method of the unified preconditioner is thoroughly validated. Parametric studies are conducted to verify the proper ranges for the parameters of the artificial compressibility method. Also, error analyses are conducted for the proposed method. In addition, it is shown that in extreme cases, low Mach number flows can differ from incompressible fluid flows. Numerical results of the artificial compressibility method and the local low Mach number preconditioner are compared. Additionally, transonic flows accompanying shocks are analyzed to validate the solver in the compressible regions.