SIAM Undergraduate Research Online

Volume 1, Issue 2


Basins of Attraction and Perturbed Numerical Solutions using Euler's Method

Published electronically September 2, 2008
DOI: 10.1137/08S010116

Author: Hendrik Orem (Harvey Mudd College)
Sponsor: Professor Rachel Levy (Harvey Mudd College)

Abstract: Small uncertainties in a dynamical system due to imperfect measurements or variations in the environment can dramatically impact the long term behavior of a trajectory. This phenomenon is studied in a population competition model by introducing a random error term into a numerical solver and investigating the effect on the behavior of solutions. Two methods for analyzing the impact of a random term are demonstrated.

Modeling the Fluid Flow around Airfoils Using Conformal Mapping

Published electronically October 6, 2008
DOI: 10.1137/08S010104

Authors: Nitin R. Kapania, Katherine Terracciano, Shannon Taylor (Franklin W. Olin College of Engineering)
Sponsor: Burt S. Tilley (Franklin W. Olin College of Engineering)

Abstract: The modeling of fluid interactions around airfoils is diffcult given the complicated, often non-symmetric geometries involved. The complex variable technique of conformal mapping is a useful intermediate step that allows for complicated airfoil flow problems to be solved as problems with simpler geometry. In this paper, we use the conformal mapping technique to model the fluid flow around the NACA 0012, 2215, and 4412 airfoils by using the Joukowsky transformation to link the flow solution for a cylinder to that of an airfoil. The flow around a cylinder was derived with the superposition of elementary potential flows using an inviscid, incompressible fluid model. Lift calculations as a function of angle of attack for each airfoil were obtained using the transformed flow solutions and fundamental theories of aerodynamics. These calculations are compared against lift calculations provided by the thin airfoil method. Lift calculations for the NACA 0012 airfoil match well with expected results, while there is a discrepancy at low angles of attack for the 2215 and 4412 airfoils.

Numerical Wave Scattering Taking Account of Energy Dissipation and Media Stiffness as Modeled by the Telegraph Equation

Published electronically December 9, 2008
DOI: 10.1137/08S010153

Authors: Sebastian Acosta and Pedro Acosta (Brigham Young University)
Sponsor: Dr. Vianey Villamizar (Brigham Young University)

Abstract: The telegraph equation is employed to model wave fields taking into account energy dissipation and media stiffness. The timeharmonic scattered waves generated by a line source incident upon cylindrical obstacles of arbitrary cross-section are studied. Solutions are found to depend strongly on the relative values of the frequency, damping, and stiffness coefficients. These coefficients are also found to have a significant effect on the far-field pattern. The analytical solution for a circular cylinder is reviewed. An approximate finite-difference solution is also obtained for the case of a two-dimensional scatterer with an arbitrary cross-section. Details are given for both soft and hard boundary conditions. The main feature of the numerical scheme is its computational efficiency based on the coupling between boundary conforming grids and a curvilinear coordinates version of the Dirichlet-to-Neumann non-reflecting boundary condition.