AFRL-AFOSR-VA-TR-2015-0403 The Development of High-Order Methods for Real World Applications

With increased computational power and progress in numerical methods over the past several decades, Computational Fluid Dynamics (CFD) is now used routinely as a powerful tool in the design of aircraft. Current production CFD codes used in the aerospace industry are usually second order accurate. High-order methods have the potential to achieve higher accuracy at less cost than low-order methods. This potential has been demonstrated conclusively for smooth problems in the latest International Workshops on High-Order Methods. For non-smooth problems, solution based hp-adaptation offers the best promise. Adjoint-based adaptive methods have the capability to dynamically distribute computing resources to areas most important for predicting engineering parameters, such as lift and drag coefficients. The primary objective of the present study is to develop robust and efficient high-order CFD methods and tools for the compressible Navier-Stokes equations that can provide engineering accuracy for real world industry problems. This report outlines progresses in the following areas. The flux reconstruction (FR) or the correction procedure via reconstruction (CPR) method used in this work is a high-order differential formulation. We develop a parallel adjoint-based adaptive CPR solver which can work with any element-based error estimate and handle arbitrary discretization order for mixed elements. First, a dual-consistent discrete form of the CPR method is derived. Then, an efficient and accurate adjoint-based error estimation for the CPR method is developed and its accuracy and effectiveness are verified for the linear and non-linear partial differential equations (PDE). The current method has been applied to aerodynamic problems. Numerical tests show that significant savings in the number of DOFs can be achieved through the adjoint-based adaptation. The usage of large-eddy simulation (LES) methods for the computation of turbulent flows has increased substantially in recent years. By resolving large energetic scales, LES has the potential to exhibit good performance especially for vortex dominated or massively separated flows. Due to the disparate length scales in LES, high-order methods are preferred for their high accuracy. Recently, models of the sub-grid scale (SGS) stress with the high-order FR/CPR method have been evaluated on 3D turbulent flows and the 1D Burgers’ equation. Preliminary studies show that implicit LES (ILES), which does not involve any SGS model, has the most efficient and accurate results. In addition, a mathematical analysis of the scale similarity is performed, revealing that the ratio of the resolved stress to the SGS stress is γ, where γ is the ratio of the second filter width to the first filter width, under the assumption of small filter width. For high-order methods to be effective, they must be paired with high-order meshes that include high-order representations of curved boundaries. A user-friendly, GUI-based software named meshCurve is developed to convert linear unstructured meshes to curved high-order meshes . Using the state-of-the-art algorithms, the software reconstructs the curved geometry from the linear surface mesh, while retaining sharp feature curves . The upgrade process is automatic and does not require a CAD file. meshCurve may be used by CFD practitioners and researchers to easily produce quality high-order meshes from existing low-order meshes. The CGNS standard is used as the file format for input and output. 2 DISTRIBUTION A: Distribution approved for public release