Analytical and Numerical Solutions for Emergence of Air Cavities in Ducts and Geysering

Resumen

This Thesis presents a continuous nite element model for the computation of two-phase ows with moving interfaces. The method is based on the Non-Oscillatory Finite Element (NFEM) algorithm and integrates the signpreserving ux correction methodology, predicting with high accuracy ow dynamics and interface motion. The procedure is composed of three main stages: transport of a bounded phase function to couple uids motion and the contact discontinuity, reinitialisation step to recover resolution of phase eld, and solution of equations of motion, where both incompressible and weakly compressible assumptions are considered. For nearly incompressible ows the continuity equation is modi ed to preserve mass conservation by considering the parametric de nition of density. Flux correction technique takes action on the three aforementioned steps. In phase function advection, limiting process that assures positivity of solution, incorporates a straightforward re nement to remove global mass residuals present in the earliest version of algorithm. Besides, new correction does not endanger e cacy of the original. To reconstruct phase function after transport, a novel non-linear (and conservative) streamlined di usion equation is proposed, with an anisotropic diffusivity comprising arti cial compression and di usive uxes oriented along interface displacements direction. Iterative procedure employed to solve this equation integrates ux correction techniques to keep phase function bounds. Finally, hydrodynamics resolution incorporates an improved bound estimation that includes interface information to substantially reduce nonphysical overshoots appearing along the contact discontinuity. On the other hand, stability of arti cial strati ed ows has been explored in problems involving Kelvin-Helmholtz instabilities. This study indicates that, to avoid nonphysical ampli cation of perturbations, thickness of numerical representation of interface should be reduced to some extent. Then, strategies to decrease transition thickness between both uids are examined, and interface re nement results the most suited. Consequently, a novel inexpensive nested-grid re nement is proposed. The algorithm is also founded in ux-correction principles, ensuring conservation and monotonicity of the variables during dynamical adaptation. E cacy of numerical model is assessed with stringent benchmark tests both for transport/reinitialisation and for two uids interface propagation. Second target of this work is to scrutinise dynamics of emergence and propagation of air cavities and resulting geysering events. Presented numerical model along with supplementary theoretical approaches have been used for this purpose. Analytical model, accomplished by a control volume analysis, is able to predict dynamics of single and consecutive elongated rising bubbles and takes into account gas expansion e ects and free surface position to determine impulsion of water above the bubble. This model also reveals conditions that trigger a sudden bubble decompression, and therefore a severe geysering event. In numerical experiments, weakly compressible uid assumption is essential for proper momentum transfer between phases in the aforementioned dynamics, particularly for bubble rising process. A rst series of simulations reproduces air cavities propagating in straight and inclined ducts. Results show a good agreement with existing laboratory outputs. A second set of simulations examines ow conditions for emergence of air pockets in ducts, giving rise to simple solutions that provide the required ow rate to avoid the intrusion of air. Finally, axisymmetric and complete three dimensional versions of the numerical model are used to perform rising Taylor bubbles in vertical ducts and geysering events. These outputs complement analytical results by giving precise ow details, in particular above the ground level.