An energy-conservative cut-cell method and advanced B-spline-based filtering method for flow simulation

This dissertation develops two computational methods to improve the accuracy and stability of numerical simulations of turbulent flows. The first method develops an energy stable cut-cell approach to the spatial discretization of domains for simulating incompressible flows. The second method develops a B-spline-based dissipative filter that dynamically adjusts to local under-resolution and is applied to compressible flow simulations. Each method is demonstrated on a series of relevant and increasingly complex problems. The cut-cell method addresses the challenge of stable and accurate discretization of complex geometries in the simulation of an incompressible flow. The method uses a staggered variable arrangement on regular Cartesian grids combined with computational geometry to achieve discrete conservation and a summation-by-parts (SBP) property to enable provable energy stability in the presence of arbitrary geometries. The development emphasizes the structure of the discrete operators, designed to mimic the properties of the continuous ones while retaining a nearest-neighbor stencil. For convective transport, different forms are proposed (divergence, advective and skew-symmetric), and shown to be equivalent when the discrete continuity equation is satisfied. For diffusive transport, conservative and symmetric operators are proposed for both Dirichlet and Neumann boundary conditions. The accuracy and robustness of the method is demonstrated with Taylor-Couette flow, Taylor-Green vortex, lid-driven cavity and flows past a circular cylinder. A B-spline-based dissipative filter is developed that is provable dissipative for bounded domain simulations. The spectral regularization algorithm can be described using the singular values of the filter operator with the amount of filtering set by a scalar- or vector-valued penalty parameter. The penalty parameter can also be chosen to minimize the generalized cross validation (GCV) function that measures fit between the pre-filtered discrete solution and the filtered solution. Efficient algorithms for the GCV optimization are developed for both the scalar and vector penalty parameters. The scheme is demonstrated for shock-like solutions of the Burgers' equation, decaying Burgers' turbulence and compressible turbulent channel flow, revealing the filter scheme's numerical stability and ability to narrowly target the high wavenumber components of numerical solutions