Combined Feature-Driven Richardson-Based Adaptive Mesh Refinement for Unsteady Vortical Flows

A DAPTIVE mesh refinement (AMR) is a useful approach for computational fluid dynamic (CFD) simulations that contain isolated relevant features like shocks or tip vortices, which are small in size with respect to the surface geometry but have a profound impact on the resulting flowfield. The fixed-wing aerodynamics community has used adaptively refined grids for high-fidelity solutions of transonic and supersonic flows [1–3]. In addition to shocks, trailing tip vortices occur in fixed-wing flight but are of greater interest for rotorcraft flight, in which the vortices shed from the blade tips can dominate the unsteady dynamics of the turbulent wake and can significantly impact vehicle performance, vibration, and noise. Accurate wake resolution can therefore lead to improvements in the prediction of rotor performance metrics [4], such as the figure of merit, a nondimensional parameter that represents the efficiency of a rotor in hover. Additionally, wake modeling is important because rotorcraft fly in their own wake, which may become entrained during hover and interact with the fuselage during forward flight [5]. However, despite the need to accurately resolve tip vortices, the rotorcraft community has not exercised AMR to the degree that the fixed-wing community has to model shocks, mainly due to the complexities in the unsteadiness of rotary-wing problems. Similar to shock modeling for fixed-wing cases, the spatial scales of trailing vortices are relatively smaller than the chord distance, thereby requiring relatively fine meshes and making the use of uniformly fine grids largely impractical [5]. Therefore, in this work, we develop an unsteady AMR strategy that targets vortical features with the goal of enhancing the resolution for both fixedand rotarywing problems. In particular for rotorcraft, complexities involving the inherently unsteady flowfield and the relative motion between the rotor and the fuselage make the implementation of efficient adaptive schemes especially challenging. Moreover, high-fidelity rotorcraft CFD are highly unsteady and require time-accurate simulations. Adjointbased AMR has shown promise for steady CFD applications [1,6,7], but time accurate solutions require the adjoint problem to be fully solved backwards in time, which is intractable for large-scale rotorcraft simulations that can involve 10 to 10 time-steps. Therefore, in this work we seek an alternative error-based refinement approach that specifically targets the vortex cores in a local manner without having to solve the full adjoint. Our approach first identifies the vortex cores using feature detection, and then the level of mesh resolution is set according to local solution error. In our earlier work [8,9], we developed four locally normalized methods that appropriately guided the AMR process based upon popular methods by the feature detection community [10–13]. A major goal of this development was to eliminate the parameter tuning that is required for common (dimensional) approaches, e.g., vorticity-based. Whereas dimensional approaches require highly tuned thresholds to select regions for refinement, the normalized schemes are able to mark regions with key vortical features using a fixed threshold, regardless of vortical strength, size, and/or resolution. However, while these nondimensional schemes effectively deal with the issue of identifying regions for refinement, the degree of mesh resolution still needs to be specified by the user. To reduce user dependency and improve computational efficiency, in this study, we examine a method of automatically setting the degree of mesh resolution by using the solution error as a guide. The objective of the current paper is to develop a solution-based error estimator that can be coupled with the nondimensional featurebased AMR. The error estimator is used to limit the amount of applied grid resolution so that additional refinement will be halted once the solution error is sufficiently low. Similar to a global functional, which is commonly used by adjoint approaches, our approach uses a local functional that is based upon quantities of interest to vortical motion. In effect, we aim to reduce the local error estimate through additional mesh refinement. Moreover, the Richardson estimator is quite practical because it is relatively simple to implement and efficient to execute. The remainder of the paper is organized as follows. A description of the adaptive overset grid-based CFD approach used for the present work is presented first in Sec. II. Thereafter, Sec. III briefly reviews the nondimensional feature-based approach, which is used to identify candidate regions for mesh refinement. Section IV offers a theoretical analysis of the Richardson error estimator, along with spatial accuracy validation tests. Then, the coupled AMR strategy that combines the feature identification with the Richardson Presented at the 49th AIAA Aerosciences Conference, Orlando, FL, January 4–7, 2011; received 14 October 2011; revision received 30 April 2012; accepted for publication 7 May 2012. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/12 and $10.00 in correspondence with the CCC. ∗Post-Doctoral Researcher; skamkar@merlin.arc.nasa.gov. AIAAMember. Aerospace Engineer; andrew.m.wissink@us.army.mil. AIAA Member. Aerospace Engineer; vsankaran@merlin.arc.nasa.gov. AIAA Member. Professor; jameson@baboon.stanford.edu. AIAA Fellow. AIAA JOURNAL Vol. 50, No. 12, December 2012