Three–Dimensional Turbulent RANS Adjoint–Based Error Correction

Engineering problems commonly require functional outputs of computational fluid dynamics (CFD) simulations with specified accuracy. These simulations are performed with limited computational resources. Computable error estimates offer the possibility of quantifying accuracy on a given mesh and predicting a fine grid functional on a coarser mesh. Such an estimate can be computed by solving the flow equations and the associated adjoint problem for the functional of interest. An adjoint-based error correction procedure is demonstrated for transonic inviscid and subsonic laminar and turbulent flow. A mesh adaptation procedure is formulated to target uncertainty in the corrected functional and terminate when error remaining in the calculation is less than a user-specified error tolerance. This adaptation scheme is shown to yield anisotropic meshes with corrected functionals that are more accurate for a given number of grid points then isotropic adapted and uniformly refined grids. Introduction Engineering problems commonly require computational fluid dynamics (CFD) solutions with functional outputs of specified accuracy. The computational resources available for these solutions are limited in practice, and errors in solutions and outputs are unknown. CFD solutions may be computed with an unnecessarily large number of grid points (and associated high cost) to ensure that the outputs are within a required accuracy. A method which estimates the error present in a computed functional offers the possibility to avoid the use of overly refined grids in order to guarantee accuracy. Unstructured grid technology promises easier initial grid generation for novel complex three-dimensional (3D) configurations compared with structured grid techniques. The use of unstructured grid technology for CFD simulations allows more freedom in adapting the discretization of the meshes to improve the fidelity of the simulation. Many previous efforts attempted to tailor the discretizations of unstructured meshes to increase solution accuracy while reducing computational cost. ∗Research Scientist, Computational Modeling and Simulation Branch, [email protected] Most of these adaptive methods focus on modifying discretizations to reduce local equation errors. These local errors are not guaranteed to directly impact error in global output functions. These methods, often referred to as feature-based adaptation, focus on resolving discontinuities or strong gradients in the flow field. Unfortunately, flow features (e.g., shocks) can be in the incorrect location due to errors elsewhere in the flow field. Also, resolving the flow in a location with large local error may have a minimal effect on the output function (e.g., a downstream shock). However, locations with small local errors (e.g., along a stagnation streamline) may have a large effect on forces. If the flow equations are linearized about the existing primal solution, a linear dual problem can yield a direct measure of the impact of local primal (flow equation) residual on a selected functional output. The combination of the primal and dual problems can also yield a correction to a specified functional on a given mesh. There are many examples of these techniques in the finite element communities. Pierce and Giles applied these methods to finite-volume discretizations. Venditti and Darmofal demonstrated these methods for compressible two-dimensional (2D) inviscid and viscous flow solutions. Müller and Giles also presented results for a similar approach. Park applied the methods of Venditti and Darmofal to 3D inviscid problems. Adaptation can reduce error in a output function by refining a discretization in locations where the local equation error weighted with the dual solution is large. This methodology targets refinement where it will have the most impact on reducing the error in an output functional. An alternative method is to refine the mesh to reduce uncertainty in the dual correction term because this correction can be computed to high accuracy. Adaptation to reduce the error prediction uncertainty is a robust and effective method for tuning a discretization to efficiently calculate a specific functional. Park demonstrated 3D isotropic adaptation coupled to a computer-aided design (CAD) description of the model for transonic, subsonic, and incompressible inviscid flow. The current study focuses on defining the requirements for extending Park’s methodology to anisotropic adaptation for turbulent flow.