Mechanical & Aerospace Engineering
High Order Computational Fluid Dynamics
Major Professor: Dr. Aaron Katz. Computational Fluid Dynamics (CFD) has become a mature, practical, and useful tool for design of fluid dynamic problems in industry. Many commercial and open source products are available to solve a variety of design issues. Indeed, many integrated product suites with a multitude of solver strategies, turbulence models, automated meshing of geometries, and multiphysics capabilities are generally available. The techniques available to CFD practitioners today are generally second-order accurate in space and time. Second-order methods are generally adequate for many situations. However, there are other situations, (rotorcraft design and flapping wings, for example) for which the established practices produce too much numerical dissipation and cannot accurately resolve vortex-dominated flows. Recent advances in turbulence modeling, including Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS), have been shown to require reduced numerical dissipation. In order to obtain realistic answers without excessive grid refinement, methods with a higher order of accuracy (third or greater) are required. While many higher-order methods have been and are being actively developed in academia, they have made very little progress into the professional engineering world. This disappointing situation can be ascribed to a few reasons. First, high-order methods possess greater mathematical stiffness, requiring more sophisticated solution techniques. Second, high-order methods require complex techniques to handle strong discontinuities in the solution. It is well-known that high-order approximations suffer from Gibb's Phenomenon, or oscillations in the approximation that do not exist in the exact solution. This is especially true in high-gradient regions and shocks. Third, most high-order methods require a complete rewrite of a low-order software package code base. Many software packages in use today have been in development for many years, using code that has been tweaked, optimized, debugged and polished, encompassing many thousands of lines of code. Creating a new code base is a daunting proposition that requires enormous investments of time, energy, and capital.
My current research topic is called Flux Correction (FC), a method which shows much promise in overcoming these concerns. The FC method works by cancelling truncation error terms arising in a traditional node-centered Galerkin finite volume method, improving the accuracy to third-order. This is accomplished by adding a correction to the numerical flux that defines the scheme. The correction, which requires an additional gradient computation, is the major change to the traditional scheme. By using a traditional Finite Volume scheme as the basis for FC, much of a second-order code base may be reused. As the traditional method remains essentially the same, mature solution techniques and limiters can be applied to the method to reduce solution time and implement shock-capturing.