Alireza Jalali

Accuracy Assessment of Finite Volume Discretizations of Convective Fluxes on Unstructured Meshes

The results of the 3 rd AIAA Drag Prediction Workshop showed that numerical errors are comparable in magnitude to physical modeling errors. One route to reducing numerical errors is to improve discretization accuracy on a fixed mesh. This paper presents novel techniques for analysis of truncation error for finite-volume discretizations on unstructured meshes. We apply these techniques to compare the truncation error of discretization schemes commonly used for convective flux approximation in cell-centered finite volume solvers. For that purpose, two classes of tests are considered. Analytical tests on topologically regular meshes are done to find the general form of truncation error for both linear and non-linear convection problems. Given the results of the analytic tests, a truncation error metric is defined based on the coefficients associated with the spatial derivatives in the series expansion of the truncation error. More complex numerical tests are conducted to extend the accuracy assessment to general unstructured meshes consisting of both isotropic and anisotropic triangles. We found that the choice of discretization does not change the truncation error of convective fluxes considerably on both isotropic and anisotropic meshes. Also, adding the artificial dissipation term to central discretization does not deteriorate the global accuracy of the flux integral associated with convection problems.

An efficient implicit unstructured finite volume solver for generalised Newtonian fluids

ABSTRACT An implicit finite volume solver is developed for the steady-state solution of generalised Newtonian fluids on unstructured meshes in 2D. The pseudo-compressibility technique is employed to couple the continuity and momentum equations by transforming the governing equations into a hyperbolic system. A second-order accurate spatial discretisation is provided by performing a least-squares gradient reconstruction within each control volume of unstructured meshes. A central flux function is used for the convective terms and a solution jump term is added to the averaged component for the viscous terms. Global implicit time-stepping using successive evolution–relaxation is utilised to accelerate the convergence to steady-state solutions. The performance of our flow solver is examined for power-law and Carreau–Yasuda non-Newtonian fluids in different geometries. The effects of model parameters and Reynolds number are studied on the convergence rate and flow features. Our results verify second-order accuracy of the discretisation and also fast and efficient convergence to the steady-state solution for a wide range of flow variables.

Higher-order Unstructured Finite Volume Methods for Turbulent Aerodynamic Flows

In this paper, we describe the steps for constructing a higher-order finite volume unstructured solver for turbulent aerodynamic flows. These include the strategies for curving the interior faces of a mesh, solution reconstruction on highly anisotropic meshes with curvature, robust implementation and coupling of a RANS turbulence model and efficient solution method. The solutions are verified by one of the verification test cases of the NASA Langley turbulence model resource. Also, the solutions and convergence behaviors are presented for fully turbulent flow over a flat plate and subsonic flow over the NACA 0012 airfoil. Our results show fast and efficient convergence for second-, thirdand fourthorder solutions for the flat plate test case and secondand third-order solutions for the airfoil. In addition, the accuracy of the solution is reasonable on meshes with sufficient degrees of freedom in which the turbulence features are captured appropriately.

Truncation and Discretization Error for Diffusion Schemes on Unstructured Meshes

The accuracy and reliability of CFD simulations depend on the ability to reduce and quantify physical modeling and numerical errors. The fact that numerical errors are at least as large as physical modeling errors was highlighted in the results of the 3rd AIAA Drag Prediction workshop, which also showed the increased severity of this issue for unstructured ow solvers. Furthermore, the varied local shape and connectivity of unstructured meshes make it di cult to quantify numerical error. The 5th AIAA Drag Prediction workshop focussed on reducing grid-related errors even further, where a grid re nement study was performed on a common grid sequence derived from a multiblock structured grid. The study had six di erent levels of grid re nement ranging from 136× 10 cells to 0.64× 10 cells, a much larger range than is typically seen, with structured overset and hexahedral, prismatic, tetrahedral, and hybrid unstructured grid formats. The results of the grid re nement study indicated that there was no clear advantage of any one grid type in terms of a reduced scatter in solution. Moreover, there were no clear breakouts with grid type or turbulence model. The conclusion was that discretization errors and turbulence modeling errors are both still major contributors to error in solution. The impact on solution accuracy by the interactions between mesh quality (cell size, shape, and anisotropy) and discretization schemes is not well understood and demands further investigation. The di erence between the discrete operator and the continuous PDE applied to the solution is referred to as the truncation error, while the di erence between the numerically approximated solution and the exact solution is the discretization error. The truncation error can be expressed in terms of the derivatives of an underlying smooth solution at the points of the discrete domain and can be used to estimate the discretization errors that occur during the approximate numerical solution of PDEs. It can be shown that for the special case of a linear di erential operator L, the truncation error τ can be used to calculate the discretization error e, as