Computational fluid dynamics (CFD) is a tool that is widely used in most industries today. It is important to have rigorous techniques to estimate the error produced when using CFD. This thesis develops techniques to estimate discretization error for unsteady flows using the unsteady error transport equation (ETE) as well as defect correction. A framework to obtain exact truncation error and estimated truncation error is also presented. The technique and results for the steady-state cases are given and the algorithm used for the steady case is extended for the unsteady case. Numerical results are presented for the steady viscous Burgers' equation, unsteady viscous Burgers' equation, steady quasi-1D Euler equations, and unsteady 1D Euler equations when applied to a shock tube. Cases using either defect correction or ETE are shown to give higher orders of accuracy for the corrected discretization error estimates when compared to the discretization error of the primal solution. / Master of Science / Computational fluid dynamics (CFD) is a tool that is widely used in most industries today. It is used to understand complex flows that are difficult to replicate using experimental techniques or by theoretical methods. It is important to have rigorous techniques to estimate the error produced when using CFD even when the exact solution is not available for comparison. This paper develops techniques to estimate discretization error for unsteady flows. Discretization error has one of the largest error magnitudes in CFD solutions. The exact physics dictates the use of continuous equations but to apply CFD techniques, the continuous equations have to be converted to discrete equations. Truncation error is, the error obtained when converting the continuous equations to discrete equations. This truncation error is in turn, the local source term for discretization error. To reduce the discretization error in the discrete equations, the exact or estimated truncation error is either added as a source term to the discrete equations or is used along with the error transport equation to get a better estimate of the solutions. A framework to obtain exact truncation error and estimated truncation error is also presented. The framework is first applied to the steady equations and is verified with results from previous studies and is then extended to the unsteady flows.

Computational fluid dynamics (CFD) has proven to be an invaluable tool for both engineering design and analysis. As the performance of engineering devices become more reliant upon the accuracy of CFD simulations, it is necessary to not only quantify and but also to reduce the numerical error present in a solution. Discretization error is often the primary source of numerical error. Discretization error is introduced locally into the solution by truncation error. Truncation error represents the higher order terms in an infinite series which are truncated during the discretization of the continuous governing equations of a model. Discretization error can be reduced through uniform grid refinement but is often impractical for typical engineering problems. Grid adaptation provides an efficient means for improving solution accuracy without the exponential increase in computational time associated with uniform grid refinement. Solution accuracy can be improved through local grid refinement, often referred to as h-adaptation, or by node relocation in the computational domain, often referred to as r-adaptation. The goal of this work is to examine the effectiveness of several r-adaptation techniques for reducing discretization error. A framework for geometry preservation is presented, and truncation error is used to drive adaptation. Sample problems include both subsonic and supersonic inviscid flows. Discretization error reductions of up to an order of magnitude are achieved on adapted grids. / Master of Science

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.

The primary goal of the thesis is to provide some answers to the questions related to the key steps in the process of adaptive modeling of plates. Since the adaptivity depends on reliable error estimates, a large part of the thesis is related to the derivation of computational procedures for discretization error estimates as well as model error estimates. A practical comparison of some of the established discretization error estimates is made. Special attention is paid to what is called equilibrated residuum method, which has a potential to be used both for discretization error and model error estimates. It should be emphasized that the model error estimates are quite hard to obtain, in contrast to the discretization error estimates. The concept of model adaptivity for plates is in this work implemented on the basis of equilibrated residuum method and hierarchic family of plate finite element models.The finite elements used in the thesis range from thin plate finite elements to thick plate finite elements. The latter are based on a newly derived higher order plate theory, which includes through the thickness stretching. The model error is estimated by local element-wise computations. As all the finite elements, representing the chosen plate mathematical models, are re-derived in order to share the same interpolation bases, the difference between the local computations can be attributed mainly to the model error. This choice of finite elements enables effective computation of the model error estimate and improves the robustness of the adaptive modeling. Thus the discretization error can be computed by an independent procedure.Many numerical examples are provided as an illustration of performance of the derived plate elements, the derived discretization error procedures and the derived modeling error procedure. Since the basic goal of modeling in engineering is to produce an effective model, which will produce the most accurate results with the minimum input data, the need for the adaptive modeling will always be present. In this view, the present work is a contribution to the final goal of the finite element modeling of plate structures: a fully automatic adaptive procedure for the construction of an optimal computational model (an optimal finite element mesh and an optimal choice of a plate model for each element of the mesh) for a given plate structure.

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Effervescent atomization has been a topic of considerable investigation in the literature due to its important advantages over other atomization mechanisms. This work contributes to the development of both effervescent atomizers and also laser-based techniques for spray investigation In order to develop non-intrusive measurement techniques for spray applications, a procedure is suggested to characterize the shape of droplets using image-based droplet analyzers. Image discretization which is a major source of error in droplet shape measurement is evaluated using a simulation. The accuracy of StereoPIV system in conducting droplet velocity measurement in a spray field is also investigated. To assist in the design of effervescent atomizers, bubble formation during gas injection from a micro-tube into liquid cross-flow is investigated using a Shadow-PIV/PTV system. The generated spray fields of two effervescent atomizers which operate using a porous and a typical multi-hole air injector are compared using qualitative images and Shadow-PTV measurement.

The largest and most difficult numerical approximation error to estimate is discretization error. Residual-based discretization error estimation methods are a category of error estimators that use an estimate of the source of discretization error and information about the specific application to estimate the discretization error using only one grid level. The higher-order terms are truncated from the discretized equations and are the local source of discretization error. The accuracy of the resulting discretization error estimate depends solely on the accuracy of the estimated truncation error. Residual-based methods require only one grid level compared to the more commonly used Richardson extrapolation which requires at least two. Reducing the required number of grid levels reduces computational expense and, since only one grid level is required, can be applied to unstructured grids where multiple quality grid levels are difficult to produce. The two residual-based discretization error estimators of interest are defect correction and error transport equations. The focus of this work is the development, improvement, and evaluation of various truncation error estimation methods considering the accuracy of the truncation error estimate and the resulting discretization error estimates. The minimum requirements for accurate truncation error estimation is specified along with proper treatment for several boundary conditions. The methods are evaluated using various Euler and Navier-Stokes applications. The discretization error estimates are compared to Richardson extrapolation. The most accurate truncation error estimation method was found to be the k-exact method where the fine grid with a correction factor was considerably reliable. The single grid methods including the k-exact require that the continuous operator be modified at the boundary to be consistent with the implemented boundary conditions. Defect correction showed to be more accurate for areas of larger discretization error; however, the cost was substantial (although cheaper than the primal problem) compared to the cost of solving the ETEs which was essential free due to the linearization. Both methods showed significantly more accurate estimates compared to Richardson extrapolation especially for smooth problems. Reduced accuracy was apparent with the presence of stronger shocks and some possible modifications to adapt to singularies are proposed for future work. / Ph. D.

A detailed code verification study of an unstructured finite volume Computational Fluid Dynamics (CFD) code is performed. The Method of Manufactured Solutions is used to generate exact solutions for the Euler and Navier-Stokes equations to verify the correctness of the code through order of accuracy testing. The verification testing is performed on different mesh types which include triangular and quadrilateral elements in 2D and tetrahedral, prismatic, and hexahedral elements in 3D. The requirements of systematic mesh refinement are discussed, particularly in regards to unstructured meshes. Different code options verified include the baseline steady state governing equations, transport models, turbulence models, boundary conditions and unsteady flows. Coding mistakes, algorithm inconsistencies, and mesh quality sensitivities uncovered during the code verification are presented. In recent years, there has been significant work on the development of algorithms for the compressible Navier-Stokes equations on unstructured grids. One of the challenging tasks during the development of these algorithms is the formulation of consistent and accurate diffusion operators. The robustness and accuracy of diffusion operators depends on mesh quality. A survey of diffusion operators for compressible CFD solvers is conducted to understand different formulation procedures for diffusion fluxes. A patch-wise version of the Method of Manufactured Solutions is used to test the accuracy of selected diffusion operators. This testing of diffusion operators is limited to cell-centered finite volume methods which are formally second order accurate. These diffusion operators are tested and compared on different 2D mesh topologies to study the effect of mesh quality (stretching, aspect ratio, skewness, and curvature) on their numerical accuracy. Quantities examined include the numerical approximation errors and order of accuracy associated with face gradient reconstruction. From the analysis, defects in some of the numerical formulations are identified along with some robust and accurate diffusion operators. / Ph. D.

Computational fluid dynamics (CFD) simulations can provide tremendous insight into complex physical processes and are often faster and more cost-effective to execute than experiments. However, each CFD result inherently contains numerical errors that can significantly degrade the accuracy of a simulation. Discretization error is typically the largest contributor to the overall numerical error in a given simulation. Discretization error can be very difficult to estimate since the generation, transport, and diffusion of these errors is a highly nonlinear function of the computational grid and discretization scheme. As CFD is increasingly used in engineering design and analysis, it is imperative that CFD practitioners be able to accurately quantify discretization errors to minimize risk and improve the performance of engineering systems. In this work, improvements are made to the accuracy and efficiency of existing error estimation techniques. Discretization error is estimated by deriving and solving an error transport equation (ETE) for the local discretization error everywhere in the computational domain. Truncation error is shown to act as the local source for discretization error in numerical solutions. An equivalence between adjoint methods and ETE methods for functional error estimation is presented. This adjoint/ETE equivalence is exploited to efficiently obtain error estimates for multiple output functionals and to extend the higher-order properties of adjoint methods to ETE methods. Higher-order discretization error estimates are obtained when truncation error estimates are sufficiently accurate. Truncation error estimates are demonstrated to deteriorate on grids with a non-smooth variation in grid metrics (e.g., unstructured grids) regardless of how smooth the underlying exact solution may be. The loss of accuracy is shown to stem from noise in the discrete solution on the order of discretization error. When using conventional least-squares reconstruction techniques, this noise is exactly captured and introduces a lower-order error into the truncation error estimate. A novel reconstruction method based on polyharmonic smoothing splines is developed to smoothly reconstruct the discrete solution and improve the accuracy of error estimates. Furthermore, a method for iteratively improving discretization error estimates is devised. Efficiency and robustness considerations are discussed. Results are presented for several inviscid and viscous flow problems. To facilitate the study of discretization error estimation, a new, higher-order finite-volume solver is developed. A detailed description of the code base is provided along with a discussion of best practices for CFD code design. / Ph. D. / Computational fluid dynamics (CFD) is a branch of computational physics at the intersection of fluid mechanics and scientific computing in which the governing equations of fluid motion, such as the Euler and Navier-Stokes equations, are solved numerically on a computer. CFD is utilized in numerous fields including biomedical engineering, meteorology, oceanography, and aerospace engineering. CFD simulations can provide tremendous insight into physical processes and are often preferred over experiments because they can be performed more quickly, are typically more cost-effective, and can provide data in regions where it may be difficult to measure. While CFD can be an extremely powerful tool, CFD simulations are inherently subject to numerical errors. These errors, which are generated when the governing equations of fluid motion are solved on a computer, can have a significant impact on the accuracy of a CFD solution. If numerical errors are not accurately quantified, ill-informed decision-making can lead to poor system performance, increased risk of injury, or even system failure. In this work, research efforts are focused on numerical error estimation for the finite -volume method, arguably the most widely used numerical algorithm for solving CFD problems. The error estimation techniques provided herein target discretization error, the largest contributor to the overall numerical error in a given simulation. Discretization error can be very difficult to estimate since these errors are generated, convected, and diffused by the same physical processes embedded in the governing equations. In this work, improvements are made to the accuracy and efficiency of existing discretization error estimation techniques. Results are presented for several inviscid and viscous flow problems. To facilitate the study of these error estimators, a new, higher-order finite -volume solver is developed. A detailed description of the code base is provided along with a discussion of best practices for CFD code design.

This work examines the Method of Nearby Problems as a way to generate analytical exact solutions to problems governed by partial differential equations (PDEs). The method involves generating a numerical solution to the original problem of interest, curve fitting the solution, and generating source terms by operating the governing PDEs upon the curve fit. Adding these source terms to the right-hand-side of the governing PDEs defines the nearby problem. In addition to its use for generating exact solutions the MNP can be extended for use as an error estimator. The nearby problem can be solved numerically on the same grid as the original problem. The nearby problem discretization error is calculated as the difference between its numerical solution and exact solution (curve fit). This is an estimate of the discretization error in the original problem of interest. The accuracy of the curve fits is quite important to this work. A method of curve fitting that takes local least squares fits and combines them together with weighting functions is used. This results in a piecewise fit with continuity at interface boundaries. A one-dimensional Burgers' equation case shows this to be a better approach then global curve fits. Six two-dimensional cases are investigated including solutions to the time-varying Burgers' equation and to the 2D steady Euler equations. The results show that the Method of Nearby Problems can be used to create realistic, analytical exact solutions to problems governed by PDEs. The resulting discretization error estimates are also shown to be reasonable for several cases examined. / Master of Science