Tobias Leicht

Adjoint-based error estimation and adaptive mesh refinement for the RANS and k–ω turbulence model equations

Abstract In this article we present the extension of the a posteriori error estimation and goal-oriented mesh refinement approach from laminar to turbulent flows, which are governed by the Reynolds-averaged Navier–Stokes and k – ω turbulence model (RANS- kω ) equations. In particular, we consider a discontinuous Galerkin discretization of the RANS- kω equations and use it within an adjoint-based error estimation and adaptive mesh refinement algorithm that targets the reduction of the discretization error in single as well as in multiple aerodynamic force coefficients. The accuracy of the error estimation and the performance of the goal-oriented mesh refinement algorithm is demonstrated for various test cases, including a two-dimensional turbulent flow around a three-element high lift configuration and a three-dimensional turbulent flow around a wing-body configuration.

Error estimation and anisotropic mesh refinement for 3d laminar aerodynamic flow simulations

This article considers a posteriori error estimation and anisotropic mesh refinement for three-dimensional laminar aerodynamic flow simulations. The optimal order symmetric interior penalty discontinuous Galerkin discretization which has previously been developed for the compressible Navier-Stokes equations in two dimensions is extended to three dimensions. Symmetry boundary conditions are given which allow to discretize and compute symmetric flows on the half model resulting in exactly the same flow solutions as if computed on the full model. Using duality arguments, an error estimation is derived for estimating the discretization error with respect to the aerodynamic force coefficients. Furthermore, residual-based indicators as well as adjoint-based indicators for goal-oriented refinement are derived. These refinement indicators are combined with anisotropy indicators which are particularly suited to the discontinuous Galerkin (DG) discretization. Two different approaches based on either a heuristic criterion or an anisotropic extension of the adjoint-based error estimation are presented. The performance of the proposed discretization, error estimation and adaptive mesh refinement algorithms is demonstrated for 3d aerodynamic flows.

Error estimation and adaptive mesh refinement for aerodynamic flows

We consider the adjoint-based error estimation and goal-oriented mesh refinement for single and multiple aerodynamic force coefficients as well as residual-based mesh refinement applied to various three-dimensional laminar and turbulent aerodynamic test cases defined in the ADIGMA project.

Generalized adjoint consistent treatment of wall boundary conditions for compressible flows

Using numerical fluxes on boundary faces like on interior faces increases stability.Extension of the adjoint consistency analysis to any consistent wall boundary flux.Adjoint consistent discretization of integral quantities like force coefficients.Associated treatment of local quantities like surface pressure and skin friction. In this article, we revisit the adjoint consistency analysis of Discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equations with application to the Reynolds-averaged Navier-Stokes and k - ω turbulence equations. Here, particular emphasis is laid on the discretization of wall boundary conditions. While previously only one specific combination of discretizations of wall boundary conditions and of aerodynamic force coefficients has been shown to give an adjoint consistent discretization, in this article we generalize this analysis and provide a discretization of the force coefficients for any consistent discretization of wall boundary conditions. Furthermore, we demonstrate that a related evaluation of the c p - and c f -distributions is required. The freedom gained in choosing the discretization of boundary conditions without loosing adjoint consistency is used to devise a new adjoint consistent discretization including numerical fluxes on the wall boundary which is more robust than the adjoint consistent discretization known up to now.While this work is presented in the framework of Discontinuous Galerkin discretizations, the insight gained is also applicable to (and thus valuable for) other discretization schemes. In particular, the discretization of integral quantities, like the drag, lift and moment coefficients, as well as the discretization of local quantities at the wall like surface pressure and skin friction should follow as closely as possible the discretization of the flow equations and boundary conditions at the wall boundary.

High-order unstructured grid generation and Discontinuous Galerkin discretization applied to a 3D high-lift configuration

Discontinuous Galerkin (DG) methods allow high-order flow solutions on unstructured or locally refined meshes by increasing the polynomial degree and using curved instead of straight-sided elements. However, one of the currently largest obstacles to applying these methods to aerodynamic configurations of medium to high complexity is the availability of appropriate higher order curved meshes. In this article, we describe a complete chain of higher order unstructured grid generation and higher order Discontinuous Galerkin flow solution applied to a turbulent flow around a 3D high-lift configuration. This includes (i) the generation of an appropriately coarse straight-sided mesh, (ii) the evaluation of additional points on the CAD geometry of the curved wall boundary for defining a piecewise polynomial boundary representation, (iii) a higher order mesh deformation to translate the curvature from the wall boundary into the interior of the computational domain, and (iv) the description of a Discontinuous Galerkin discretization which is sufficiently stable to allow a flow computation on the resulting curved mesh. Finally, a fourth order flow solution of the RANS and k-omega turbulence model equations computed on a fourth order unstructured hybrid mesh around a 3D high-lift configuration will be compared against wind-tunnel measurements of the SWING project.

Multigrid Solver Algorithms for DG Methods and Applications to Aerodynamic Flows

In this chapter we collect results obtained within the IDIHOM project on the development of Discontinuous Galerkin (DG) methods and their application to aerodynamic flows. In particular, we present an application of multigrid algorithms to a higher order DG discretization of the Reynolds-averaged Navier-Stokes (RANS) equations in combination with the Spalart-Allmaras as well as the Wilcox-kω turbulence model. Based on either lower order discretizations or agglomerated coarse meshes the resulting solver algorithms are characterized as p- or h-multigrid, respectively. Linear and nonlinear multigrid algorithms are applied to IDIHOM test cases, namely theL1T2 high lift configuration and the deltawing of the second Vortex Flow Experiment (VFE-2) with rounded leading edge. All presented algorithms are compared to a strongly implicit single grid solver in terms of number of nonlinear iterations and computing time. Furthermore, higher order DG methods are combined with adaptive mesh refinement, in particular, with residual-based and adjoint-based mesh refinement. These adaptive methods are applied to a subsonic and transonic flow around the VFE-2 delta wing.

Higher Order Multigrid Algorithms for a Discontinuous Galerkin RANS Solver

A comparison of nonlinear and linear p- and h-multigrid algorithms will be presented with respect to both algorithmic convergence properties and run-time behavior. In addition to that a comparison with a single-level solver, namely a Backward--Euler method, will be presented as well. The algorithms will be used to solve the RANS-equations in combination with a k-w turbulence models for CFD applications in the area of compressible aerodynamic fows with high Reynolds numbers. The h-multigrid algorithms are formulated on agglomerated unstructured meshes. Results will be presented on both structured and unstructred meshes.

3D Application of Higher Order Multigrid Algorithms for a RANS-kω DG-Solver

We present an application of multigrid algorithms to a Discontinuous Galerkin discretization of the 3D Reynolds-averaged Navier–Stokes equations in combination with a kω turbulence model. Based on either lower order discretizations or agglomerated coarse meshes the resulting algorithms can be characterized as p- or h-multigrid, respectively. Linear and nonlinear multigrid algorithms are applied to a 3D numerical test case, namely the VFE-2 delta-wing with rounded leading edge. All presented algorithms are compared to a strongly implicit single grid solver in terms of run time behavior and nonlinear iterations.

Goal-oriented Error Estimation and hp-Adaptive Mesh Refinement for Aerodynamic Flows

We present adjoint-based techniques to estimate the error of a numerical flow solution with respect to a given target quantity like an aerodynamic force coefficient. This estimate can be used to judge the overall accuracy of a computation, to enhance the computed value of the target quantity and to drive a solution-adaptive mesh refinement process. The error estimation procedure is extended to multiple target functionals. The discontinuous ansatz spaces of the DG discretization allow for both element subdivision as well as a local increase of polynomial degrees for increasing the flow resolution. Targeting optimal rates of convergence, a smoothness estimation based on a truncated Legendre series expansion of the solution is employed to locally select the more promising strategy. Numerical examples for inviscid, laminar viscous and turbulent viscous flows including a three element airfoil high lift configuration and a wing-body aircraft configuration demonstrate the efficiency of the proposed algorithms.

Error estimation and anisotropic mesh refinement for aerodynamic flow simulations

Aerodynamic flow fields are dominated by anisotropic features at both boundary layers and shocks. Solution-adaptive local mesh refinement can be improved considerably by respecting those anisotropic features. Two types of anisotropy indicators are presented. Whereas the first one is based on polynomial approximation properties and needs the evaluation of second and higher order derivatives of the solution the second one exploits the inter-element jumps arising in discontinuous Galerkin methods and can easily be used with higher order discretizations and even hp-refinement. Examples for sub-, trans- and supersonic flows combining these anisotropic indicators with reliable residual or adjoint based error estimation techniques demonstrate the potential and limitations of this approach.