22nd AIAA Computational Fluid Dynamics Conference, Jun 18, 2015
In this paper, we perform a numerical estimation of discretization error using the error transpor... more In this paper, we perform a numerical estimation of discretization error using the error transport equation, derived from the primal PDE. The viability of using this method for obtaining higher order estimates for unstructured finite-volume discretizations of scalar linear and nonlinear scalar PDEs has previously been demonstrated, and here we examine how this extends to steady state solutions to the Euler equations, a nonlinear system of PDEs. Considerations for the error transport equation with and without linearization were made. Comparisons of results show that using the fully nonlinear form has verifiable properties as well as being superior in accuracy of the error estimate in some situations, although the Newton linearization can be adequate in others. The major results for 1D and 2D test cases were consistent with scalar problems. With arbitrary choices of discretization orders for the primal and error PDEs and residual source term, the error estimate obtained is in general not sharp and converges to the exact error at the same order as the primal discretization. However, using a discretization scheme where the source term for the error equation is the residual based on a reconstruction of the converged primal solution that is the same order as the error equation discretization leads to a sharp, high order estimate compared to other combinations. Therefore, we demonstrate that there are nominal accuracy combinations for discretizing the primal and error equations, and evaluating the residual source term, that require more computational work but are actually less accurate asymptotically in obtaining an estimate of error, which are choices that one should never make in practice. In addition, some results for the runtime costs are obtained for evaluating the feasibility of applying this error estimation approach compared to higher order primal discretizations.
The current state-of-the-practice technology for high-lift aerodynamic simulations is to solve th... more The current state-of-the-practice technology for high-lift aerodynamic simulations is to solve the Reynolds-averaged Navier–Stokes (RANS) equations on a fixed grid or a refinement sequence of fixed grids. The Fixed-Grid Reynolds-Averaged Navier–Stokes Technology Focus Group set out to determine meshing requirements and best practices, whether RANS can accurately predict the change in aerodynamic performance with changes in flap deflection, whether RANS modeling can produce accurate results near [Formula: see text], and the effects of underconvergence and solution strategy on computed results. Eighteen groups of participants submitted over 100 datasets. Challenges with grid convergence and iterative convergence made it impossible to definitively answer all the questions we had posed. Despite this, we can conclude that meshes with at least half a billion cells (more than one billion degrees of freedom) are required for grid convergence away from stall; that RANS simulations cannot currently be reliably used to predict aerodynamic coefficients near stall, nor changes in coefficients with changes in flap angle; that iterative underconvergence remains a significant source of uncertainty in outputs; and that solution initialization can have an important effect on solution behavior, including flow separation patterns.
The accuracy of flow simulations is a major concern in computational fluid dynamics (CFD) applica... more The accuracy of flow simulations is a major concern in computational fluid dynamics (CFD) applications. A possible outcome of inaccuracy in CFD results is missing a major feature in the flowfield. Many methods have been proposed to reduce numerical errors and increase overall accuracy, but these are not always efficient or even feasible. In this study, the principal component analysis (PCA) has been performed on compressible flow simulations around an airfoil to map the high-dimensional CFD data to a lower-dimensional PCA subspace. A machine learning classifier based on the extracted principal components has been developed to detect the simulations that miss the separation bubble behind the airfoil. The evaluative measures indicate that the model is able to detect most of the simulations where the separation region is poorly resolved. Moreover, a single mode responsible for the missing flow separation was uncovered that could be the subject of future studies. The results demonstrate that a machine learning model based on the principal components of the dataset is a promising tool for detecting possible missing flow features in CFD.
22nd AIAA Computational Fluid Dynamics Conference, Jun 18, 2015
In this paper, we perform a numerical estimation of discretization error using the error transpor... more In this paper, we perform a numerical estimation of discretization error using the error transport equation, derived from the primal PDE. The viability of using this method for obtaining higher order estimates for unstructured finite-volume discretizations of scalar linear and nonlinear scalar PDEs has previously been demonstrated, and here we examine how this extends to steady state solutions to the Euler equations, a nonlinear system of PDEs. Considerations for the error transport equation with and without linearization were made. Comparisons of results show that using the fully nonlinear form has verifiable properties as well as being superior in accuracy of the error estimate in some situations, although the Newton linearization can be adequate in others. The major results for 1D and 2D test cases were consistent with scalar problems. With arbitrary choices of discretization orders for the primal and error PDEs and residual source term, the error estimate obtained is in general not sharp and converges to the exact error at the same order as the primal discretization. However, using a discretization scheme where the source term for the error equation is the residual based on a reconstruction of the converged primal solution that is the same order as the error equation discretization leads to a sharp, high order estimate compared to other combinations. Therefore, we demonstrate that there are nominal accuracy combinations for discretizing the primal and error equations, and evaluating the residual source term, that require more computational work but are actually less accurate asymptotically in obtaining an estimate of error, which are choices that one should never make in practice. In addition, some results for the runtime costs are obtained for evaluating the feasibility of applying this error estimation approach compared to higher order primal discretizations.
The current state-of-the-practice technology for high-lift aerodynamic simulations is to solve th... more The current state-of-the-practice technology for high-lift aerodynamic simulations is to solve the Reynolds-averaged Navier–Stokes (RANS) equations on a fixed grid or a refinement sequence of fixed grids. The Fixed-Grid Reynolds-Averaged Navier–Stokes Technology Focus Group set out to determine meshing requirements and best practices, whether RANS can accurately predict the change in aerodynamic performance with changes in flap deflection, whether RANS modeling can produce accurate results near [Formula: see text], and the effects of underconvergence and solution strategy on computed results. Eighteen groups of participants submitted over 100 datasets. Challenges with grid convergence and iterative convergence made it impossible to definitively answer all the questions we had posed. Despite this, we can conclude that meshes with at least half a billion cells (more than one billion degrees of freedom) are required for grid convergence away from stall; that RANS simulations cannot currently be reliably used to predict aerodynamic coefficients near stall, nor changes in coefficients with changes in flap angle; that iterative underconvergence remains a significant source of uncertainty in outputs; and that solution initialization can have an important effect on solution behavior, including flow separation patterns.
The accuracy of flow simulations is a major concern in computational fluid dynamics (CFD) applica... more The accuracy of flow simulations is a major concern in computational fluid dynamics (CFD) applications. A possible outcome of inaccuracy in CFD results is missing a major feature in the flowfield. Many methods have been proposed to reduce numerical errors and increase overall accuracy, but these are not always efficient or even feasible. In this study, the principal component analysis (PCA) has been performed on compressible flow simulations around an airfoil to map the high-dimensional CFD data to a lower-dimensional PCA subspace. A machine learning classifier based on the extracted principal components has been developed to detect the simulations that miss the separation bubble behind the airfoil. The evaluative measures indicate that the model is able to detect most of the simulations where the separation region is poorly resolved. Moreover, a single mode responsible for the missing flow separation was uncovered that could be the subject of future studies. The results demonstrate that a machine learning model based on the principal components of the dataset is a promising tool for detecting possible missing flow features in CFD.
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Papers by Carl Ollivier-Gooch