A Boundary Integral Method for Modeling Axisymmetric Flow Around a Rising Bubble in a Vertical Tube and Accurate Numerical Evaluation of Orthogonal Polynomials

Axisymmetric flow around a bubble rising in a 3D fluid has been studied intensively both numerically and experimentally over the years, but in the high Reynolds number regime, no matter for small oblate or elongated bubbles, current mathematical models face problems such as introducing an unnecessary singularity at the stagnation point, missing proper lower boundary conditions, and solving with low order finite difference methods, which leads to large numerical errors. Modeling the problem by both potential flow and viscous potential flow, we can represent the velocity potential using layer potentials and compute the bubble shape with spectral accuracy, with average numerical errors around $10^{-9}$. A dimensionless inviscid model is first built to study steady flow around a bubble in an infinitely long tube in the case of zero gravity. With a comprehensive discussion on parametrization, singular integrals, and numerical quadrature, we solve both oblate and elongated bubble shapes accurately and find different solution branches of bubble shapes characterized by the number of humps. These solution branches relating the Weber number and cross-section arc-length suggest that for any nonzero surface tension, there exists a countably infinite number of solutions. When there is gravity, kinematic viscosity is introduced to balance out the normal stress on the bubble surface. Different shapes of steadily rising bubbles under different Froude numbers are presented./nWe also study the corresponding time-dependent problem to illustrate the dynamics of unsteady bubbles, both small and elongated, and to confirm the non-existence of steadily rising bubbles in the inviscid model with nonzero gravity. The approximation of dipole density by spherical harmonics is computed to remove the hyper-singularity in the normal derivatives of the double layer potential on the bubble surface./nDue to the importance of accurate evaluation of orthogonal polynomials in the boundary integral method, we introduce a new way to evaluate orthogonal polynomials more accurately near the endpoints of the integration interval by evaluating a newly created set of associated orthogonal polynomials at corresponding points. Various ways to evaluate the associated orthogonal polynomials are discussed and implemented. Among them, the best method can achieve round-off error accuracy even for end-point evaluation of generic high-degree Jacobi polynomials and generalized Laguerre polynomials, which is 3 digits more accurate than the classic recurrence method in double precision when the degree is around 400. Based on the accurate evaluation, we perform one iteration of Newton's method to achieve more accurate quadrature abscissas near endpoints. We also calculate quadrature weights using the Christoffel formula based on our accurate evaluation. The resulting errors in abscissas and weights are compared to those of the Golub-Welsch algorithm, with our method being more accurate. Strategies on generic weight functions are given and tested on Maxwell polynomials of current interest in plasma physics.

Advisor: Wilkening, Jon