We combine the spectral element method with the smoothed profile method (SPM) to obtain an efficient method for flows with moving boundaries in complex geometries. SPM uses a fixed non-conforming computational mesh and represents the particles by indicator functions to construct a penalty force term in the Navier-Stokes equations. The method is similar to the immersed boundary method in that they both use a force distribution to effectively impose the constraints on the fluid motion to approximate the boundary conditions. However, for spectral element discretizations, the smooth profile of SPM leads to high order accuracy. While the original method employs a fully-explicit time-integration scheme, we develop a high-order semi-implicit splitting scheme to improve accuracy and stability. We first analyze the error of the hybrid method for several prototype flow problems. We show that the modeling error of SPM is a non-monotonic function of the time step size and the interface thickness of the smooth profile. The optimum time step size balances the thickness of the Stokes layer and that of the profile interface. Subsequently, we propose an extension of SPM to simulate {\em electrohydrodynamic flows} allowing for spatially varying electrical conductivities. In addition to the Navier-Stokes equations, the Poisson-Boltzmann and electric charge continuity equations are also cast into forms based on SPM. The method is verified by benchmark problems of electroosmotic flow in straight channels and electrophoresis of charged insulating cylinders. We also present simulations on the electrophoresis of charged microtubules, and show that the simulated electrophoretic mobility and anisotropy agree with the experimental results. In the last part we present numerical simulation results for engineering flows involving three-dimensional moving domains. In particular, the flow patterns around the vortex-induced vibrations of two tandem cylinders are successfully resolved, and results are also shown for a rotating three-bladed propeller in a duct which can be used in waterjet propulsion. Compared to the simulation results based on an arbitrary Lagrangian Euler formulation, SPM yields satisfactory agreement, yet it is computationally much cheaper allowing for fast three-dimensional simulations on a laptop.
Luo, Xian,
"A spectral element/smoothed profile method for complex-geometry flows"
(2009).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0VT1QBN
