# Solving The Prandtl Boundary Layer Equation in Fluid Dynamics Via Non-Linear Numerical Optimization

Fall 2019

## Degree Type

Restricted to Claremont Colleges Dissertation

## Degree Name

Engineering and Industrial Applied Mathematics Joint PhD with California State University Long Beach, PhD

## Program

Institute of Mathematical Sciences

Christiane Beyer

Marina Chugunova

Ortwin Ohtmer

## Keywords

Boundary Layer, Conformal Mapping, Optimization, Quasi-Newton

## Subject Categories

Engineering | Mathematics

## Abstract

The Finite Volume Method (FVM) is often applied to solve the non-linear Navier-Stokes differential equations. One of the main disadvantages of this method is that it requires a very fine mesh and a very large number of iterations for reasonable numerical solutions. In this research, a different approach is presented with columns of meshing for the boundary layer and a lower number of iterations with numerical results with a higher degree of accuracy. In a conventional Finite Volume Method (FVM), simple functions are used that have an accuracy of O(h2) with the mesh size h; this method uses only finite difference terms for the derivatives . The approach in this research is based on the Mehrstellen finite difference method in which more than one differential termsare used, resulting in an accuracy of O(h4) or O(h6).A powerful variation of the Quasi-Newton methods known as the BFGS Quasi-Newton iteration is applied with a quadratic convergence rate [41][43] while the conventional FVM converges linearly using the SIMPLE iteration approach. In this work, an Objective Function (or Penalty Function) and a gradient vector, as well as a Golden Section Search or Equality Algorithm, are specified in the thin boundary layer. The flow around profiles with sharp corners is mainly considered and applying conformal mappings. These sharp corners are opened for the calculation of a good initial solution as a starting point to reduce the number of iterations. Within the thin boundary layer, a fine mesh only in y-direction is generated, and then the BFGS Quasi-Newton method is applied. The velocity at the top of the boundary layer of any profile is calculated via conformal mapping. The Prandtl boundary layer equation is modified and solved via Mehrstellen Finite Difference formulas having only in finite difference terms. The differential term is replaced by from continuity equation in 2D space.

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