The solution to partial differential equations generally requires approximations that result in numerical error in the final solution. Of the different types of numerical error in a solution, discretization error is the largest and most difficult error to estimate. In addition, the accuracy of the discretization error estimates relies on the solution (or multiple solutions used in the estimate) being in the asymptotic range. The asymptotic range is used to describe the convergence of a solution, where an asymptotic solution approaches the exact solution at a rate proportional to the change in mesh spacing to an exponent equal to the formal order of accuracy. A non-asymptotic solution can result in unpredictable convergence rates introducing uncertainty in discretization error estimates. To account for the additional uncertainty, various discretization uncertainty estimators have been developed. The goal of this work is to evaluation discretization error and discretization uncertainty estimators based on Richardson extrapolation for computational fluid dynamics problems. In order to evaluate the estimators, the exact solution should be known. A select set of solutions to the 2D Euler equations with known exact solutions are used to evaluate the estimators. Since exact solutions are only available for trivial cases, two applications are also used to evaluate the estimators which are solutions to the Navier-Stokes equations: a laminar flat plate and a turbulent flat plate using the k-Ï SST turbulence model. Since the exact solutions to the Navier-Stokes equations for these cases are unknown, numerical benchmarks are created which are solutions on significantly finer meshes than the solutions used to estimate the discretization error and uncertainty. Metrics are developed to evaluate the accuracy of the error and uncertainty estimates and to study the behavior of each estimator when the solutions are in, near, and far from the asymptotic range. Based on the results, general recommendations are made for the implementation of the error and uncertainty estimators. In addition, a new uncertainty estimator is proposed with the goal of combining the favorable attributes of the discretization error and uncertainty estimators evaluated. The new estimator is evaluated using numerical solutions which were not used for development and shows improved accuracy over the evaluated estimators. / Master of Science

To avoid the large amount of damage that floods can cause, spillway tunnels are used to control water levels. To ensure the safety of water transportation through spillway tunnels, the behaviour of the water throughout the tunnel is important to know. Physical experiments are time consuming and expensive, hence CFD simulations are a profitable option for investigating the performance of the spillway tunnel. In this project, simulations of water flow in a spillway tunnel were executed. A three dimensional model of the spillway tunnel in Zipingpu Water Conservancy Project was created in the software ANSYS Gambit. A coarse, middle and fine mesh with both hexahedral- and tetrahedral elements were also created for the model in ANSYS Gambit. The meshes were imported to ANSYS Fluent where the simulations, and a convergence analysis were made. The water flow was set to be described by the Reynolds-Averaged Navier-Stokes model, using the pressure solver, k-epsilon model and the VOF model. Physical experiments had previously been performed, and the simulated results were compared to these, in an attempt to find the parameters to replicate the experimental results to the greatest extent possible. The inlet velocity of the tunnel was known and the inlet boundary was set as a velocity inlet. The ceiling of the tunnel was set as a pressure inlet, the floor and walls were set as wall, and the outlet was set as pressure outlet. The simulated results showed similar behavior as the experimental results, but all differed from the experimental results. The grid convergence index, estimating the results' dependency on the mesh was 6.044 %. The flow was analyzed, and where the flow had unfavorable characteristics, such as a high cavitation number, the geometry of the spillway was altered in ANSYS Gambit to investigate if an improved geometry for the spillway tunnel could be found. The water flow in the revised geometry was simulated in ANSYS Fluent, and results showing flow with lower cavitation numbers was found.