Computational Fluid Dynamics (CFD) has been widely used as a tool to analyze physical phenomena involving fluids. To perform a CFD simulation, the governing equations are discretized to formulate a set of nonlinear algebraic equations. Typical spatial discretization schemes include finite-difference methods, finite-volume methods, and finite-element methods. Error introduced in the discretization process is called discretization error and defined as the difference between the exact solution to the discrete equations and the exact solution to the partial differential or integral equations. For most CFD simulations, discretization error accounts for the largest portion of the numerical error in the simulation. Discretization error has a complicated nonlinear relationship with the computational grid and the discretization scheme, which makes it especially difficult to estimate. Thus, it is important to study the discretization error to characterize numerical errors in a CFD simulation.
Discretization error estimation is performed using the Error Transport Equations (ETE) in this work. The original nonlinear form of the ETE can be linearized to formulate the linearized ETE. Results of the two types of the ETE are compared. This work implements the ETE for finite-volume methods and Discontinuous Galerkin (DG) finite-element methods. For finite volume methods, discretization error estimates are obtained for both steady state problems and unsteady problems. The work on steady-state problems focuses on turbulent flow modelled by the Spalart-Allmaras (SA) model and Menter's $k-omega$ SST model. Higher-order discretization error estimates are obtained for both the mean variables and the turbulence working variables. The type of pseudo temporal discretization used for the steady-state problems does not have too much influence on the final converged solution. However, the temporal discretization scheme makes a significant difference for unsteady problems. Different temporal discretizations also impact the ETE implementation. This work discusses the implementation of the ETE for the 2-step Backward Difference Formula (BDF2) and the Singly Diagonally Implicit Runge-Kutta (SDIRK) methods. Most existing work on the ETE focuses on finite-volume methods. This work also extends ETE to work with the DG methods and tests the implementation with steady state inviscid test cases. The discretization error estimates for smooth test cases achieve the expected order of accuracy. When the test case is non-smooth, the truncation error estimation scheme fails to generate an accurate truncation error estimate. This causes a reduction of the discretization error estimate to first-order accuracy. Discussions are made on how accurate truncation error estimates can be found for non-smooth test cases. / Doctor of Philosophy / For a general practical fluid flow problem, the governing equations can not be solved analytically. Computational Fluid Dynamics (CFD) approximates the governing equations by a set of algebraic equations that can be solved directly by the computer. Compared to experiments, CFD has certain advantages. The cost for running a CFD simulation is typically much lower than performing an experiment. Changing the conditions and geometry is usually easier for a CFD simulation than for an experiment. A CFD simulation can obtain information of the entire flow field for all field variables, which is nearly impossible for a single experiment setup. However, numerical errors are inherently persistent in CFD simulations due to the approximations made in CFD and finite precision arithmetic of the computer. Without proper characterization of errors, the accuracy of the CFD simulation can not be guaranteed. Numerical errors can even result in false flow features in the CFD solution. Thus, numerical errors need to be carefully studied so that the CFD simulation can provide useful information for the chosen application.
The focus of this work is on numerical error estimation for the finite-volume method and the Discontinuous Galerkin (DG) finite-element method. In general, discretization error makes the most significant contribution to the numerical error of a CFD simulation. This work estimates discretization error by solving a set of auxiliary equations derived for the discretization error of a CFD solution. Accurate discretization error estimates are obtained for different test cases. The work on the finite-volume method focus on discretization error estimation for steady state turbulent test cases and unsteady test cases. To the best of the author's knowledge, the implementation of the current discretization error estimation scheme has only been applied as an intermediate step for the error estimation of functionals for the DG method in the literature. Results for steady-state inviscid test cases for the DG method are presented.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/112792 |
| Date | 11 June 2021 |
| Creators | Wang, Hongyu |
| Contributors | Aerospace and Ocean Engineering, Roy, Christopher John, Wang, Kevin Guanyuan, Srinivasan, Bhuvana, Borggaard, Jeffrey T. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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