Browsing by Author "Tyson, William Conrad"

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    Application of r-Adaptation Techniques for Discretization Error Improvement in CFD
    Tyson, William Conrad (Virginia Tech, 2015-12-08)
    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.
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    The Effects of Pressure Gradient and Roughness on Pressure Fluctuations Beneath High Reynolds Number Boundary Layers
    Fritsch, Daniel James (Virginia Tech, 2022-09-16)
    High Reynolds number turbulent boundary layers over both smooth and rough surfaces subjected to a systematically defined family of continually varying, bi-directional pressure gradient distributions are investigated in both wind tunnel experiments and steady 2D and 3D Reynolds Averaged-Navier-Stokes (RANS) computations. The effects of pressure gradient, pressure gradient history, roughness, combined roughness and pressure gradient, and combined roughness and pressure gradient history on boundary growth and the behavior of the underlying surface pressure spectrum are examined. Special attention is paid to how said pressure spectra may be effectively modeled and predicted by assessing existing empirical and analytical modeling formulations, proposing updates to those formulations, and assessing RANS flow modeling as it pertains to successful generation of spectral model inputs. It is found that the effect of pressure gradient on smooth wall boundary layers is strongly non-local. The boundary layer velocity profile, turbulence profiles, and associated parameters and local skin friction at a point that has seen non-constant upstream pressure gradient history will be dependent both on the local Reynolds number and pressure gradient as well as the Reynolds number and pressure gradient history. This shows itself most readily in observable downstream lagging in key observed behaviors. Steady RANS solutions are capable of predicting this out-of-equilibrium behavior if the pressure gradient distribution is captured correctly, however, capturing the correct pressure gradient is not as straightforward as may have previously been thought. Wind tunnel flows are three-dimensional, internal problems dominated by blockage effects that are in a state of non-equilibrium due to the presence of corner and juncture flows. Modeling a 3D tunnel flow is difficult with the standard eddy viscosity models, and requires the Quadratic Constitutive Relation for all practical simulations. Modeling in 2D is similarly complex, for, although 3D effects can be ignored, the absence of two walls worth of boundary layer and other interaction flows causes the pressure gradient to be captured incorrectly. These effects can be accounted for through careful setup of meshed geometry. Pressure gradient and history effects on the pressure spectra beneath smooth wall boundary layers show similar non-locality, in addition to exhibiting varying effects across different spectral regions. In general, adverse pressure gradient steepens the slope of the mid-frequency region while favorable shallows it, while the high frequency region shows self-similarity under viscous normalization independent of pressure gradient. The outer region is dominated by history effects. Modeling of such spectra is not straightforward; empirical models fail to incorporate the subtle changes in spectral shape as coherent functions of flow variables without becoming overly-defined and producing non-physical spectral shapes. Adopting an analytical formulation based on the pressure Poisson equation solves this issue, but brings into play model inputs that are difficult to predict from RANS. New modeling protocols are proposed that marry the assumptions and limitations of RANS results to the analytical spectral modeling. Rough surfaces subjected to pressure gradients show simplifications over their smooth wall relatives, including the validity of Townsend's outer-layer-Reynolds-number-similarity Hypothesis and shortened history effects. The underlying pressure spectra are also significantly simplified, scaling fully on a single outer variable scaling and showing no mid-frequency slope pressure gradient dependence. This enables the development of a robust and accurate empirical model for the pressure spectra beneath rough wall flows. Despite simplifications in the flow physics, modeling rough wall flows in a steady RANS environment is a challenge, due to a lack of understanding of the relationship between the rough wall physics and the RANS model turbulence parameters; there is no true physical basis for a steady RANS roughness boundary condition. Improvements can been made, however, by tuning a shifted wall distance, which also factors heavily into the mathematical character of the pressure spectrum and enables adaptations to the analytical model formulations that accurately predict rough wall pressure spectra. This work was sponsored by the Office of Naval Research, in particular Drs. Peter Chang and Julie Young under grants N00014-18-1-2455, N00014-19-1-2109, and N00014-20-2821. This work was also sponsored by the Department of Defense Science, Mathematics, and Research for Transformation (SMART) Fellowship Program and the Naval Air Warfare Center Aircraft Division (NAWCAD), in particular Mr. Frank Taverna and Dr. Phil Knowles.
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    On Numerical Error Estimation for the Finite-Volume Method with an Application to Computational Fluid Dynamics
    Tyson, William Conrad (Virginia Tech, 2018-11-29)
    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.