Geometrically transformed spectral methods to solve partial differential equations in circular geometries, application for multi-phase flow

Circular geometries are ubiquitously encountered in science and technology, and the polar coordinate provides the natural way to analyze them; However, its application is limited to symmetric cases, and it cannot be applied to segments that are formed in multiphase flow problems in pipes. To address that, spectral discretization of circular geometries via orthogonal collocation technique is developed using geometrical mapping. Two analytical mappings between the circle and square geometries, namely, elliptical and horizontally squelched mappings, are employed. Accordingly, numerical algorithms are developed for solving PDEs in circular geometries with different boundary conditions for both steady state and transient problems. Various implementation issues are thoroughly discussed, including vectorization and strategies to avoid solving the differential–algebraic system of equations. Moreover, several case studies for symmetric and asymmetric Poisson equations with different boundary conditions are performed to evaluate several aspects of these techniques, such as error properties, condition number, and computational time. For both steady state and transient solvers, it was revealed that the computation time scales quadratically with respect to the grid size for both mapping and polar discretization techniques. However, due to the presence of the second mixed derivative, mapping techniques are more computationally costly. Finally, the squelched mapping was successfully employed to discretize the two, and three-phase gravity flows in sloped pipes.