Residual distribution schemes for advection and advection-diffusion problems on quadrilateral and hybrid meshes

Abstract This thesis provides a thorough theoretical and numerical analysis of residual distribution schemes for the scalar advection and advection-diffusion equations, focusing on the difficulties arising when extending FS schemes from triangular to quadrilateral grids. The study has been performed using the Fourier and the truncation error analyses on a structured mesh, in order to provide a strategy for the design of second-order-accurate stable schemes, and the results have been verified by numerical tests. A generalized modified wavenumber is defined, which provides a general mathematical framework for the multidimensional analysis and comparison of schemes belonging to different classes. The analysis demonstrates that, for the advection equation, linearity preserving schemes for quadrilaterals provide lower diffusion with respect to their triangle-based counterparts and show low or no damping for high frequency Fourier modes on general grids. Therefore, such schemes can be employed for pure advection problems adding an artificial dissipation term to damp marginally stable modes. Furthermore, concern- ing advection-diffusion problems, using a hybrid approach, which employs an upwind residual distribution scheme for the convective fluctuation and any other scheme for the diffusion term, leads to a first-order-accurate method. On the other hand, distributing the entire residual by an upwind scheme provides second-order accuracy; but such an approach is unstable for diffusion dominated problems since residual distribution schemes are characterized by undamped modes associated with the discretization of the diffusive fluctuation. In this work, this problem is ad- dressed by defining the conditions for a stable hybrid approach to be second-order-accurate and an optimal scheme having minimum dispersion error on a nine-point stencil is provided. Preliminary results are shown for the solution of the Euler equations on some well-documented test-cases. The flow through a cascade of gas-turbine rotor blades has been solved on a hybrid mesh, to show the applicability of the method to industrial flows. Finally, it is shown how to use the Fourier analysis and the generalized modified wavenumber as a tool to design and optimize multidimensional centred/upwind compact schemes. This can be considered a good starting point for future research, particularly considering that compact schemes are very suitable for aeroacoustic and Large Eddy Simulation (LES) for their excellent dispersion properties. Key words: fluctuation splitting, Fourier analysis, truncation error analysis, compact schemes.