A methodology is presented for performing numerical aerodynamic shape optimization based on the threedimensional Reynolds-averaged Navier-Stokes (RANS) equations. An initial multiblock structured mesh is first fit with B-spline volumes that form the basis for a hybrid mesh movement scheme that is tightly integrated with the geometry parameterization based on B-spline surfaces. The RANS equations and the one-equation Spalart-Allmaras turbulence model are solved in a fully coupled manner using an efficient parallel Newton-Krylov algorithm with approximate-Schur preconditioning. Gradient evaluations are performed using the discrete-adjoint approach with analytical differentiation of the discrete flow and mesh movement equations. The overall methodology remains robust even in the presence of large shape changes. Several examples of lift-constrained drag minimization are provided, including a study of the common research model wing geometry, a wing-body-tail geometry with a prescribed spanwise load distribution, and a blended-wing-body configuration. An example is provided that demonstrates that a wing optimized based on the Euler equations exhibits substantially inferior performance when subsequently analyzed based on the RANS equations relative to a wing optimized based on the RANS equations.1555
This work demonstrates the performance of Jetstream, a high-fidelity aerodynamic shape optimization methodology for three-dimensional turbulent flows. The geometry parameterization and mesh movement is accomplished using B-spline volumes and linear elasticity mesh movement. The Euler or Reynolds-averaged Navier-Stokes (RANS) equations are solved at each iteration using a parallel Newton-Krylov-Schur method. The equations are discretized in space using summation-by-parts operators with simultaneous approximation terms to enforce boundary and block interface conditions. The gradients are evaluated using the discrete-adjoint method to allow for gradient-based optimization using a sequential quadratic programming algorithm. The goal of this work is to investigate the performance of Jetstream for three test problems. The first problem is the drag minimization of a two-dimensional symmetric airfoil in transonic inviscid flow, under a geometric constraint that the airfoil have a thickness greater than or equal to that of a NACA 0012 airfoil. Although the shock waves are not quite eliminated, they are substantially weakened, such that the drag coefficient is reduced by 86% compared to the NACA 0012 airfoil. The second problem is drag minimization through optimizing the twist distribution of a three-dimensional wing characterized by NACA 0012 sections in subsonic inviscid flow, subject to a lift constraint. A nearly elliptical spanwise lift distribution is achieved by the optimized twist distribution, leading to a span efficiency factor of 0.98. The third problem is drag minimization through optimizing the sections and twist distribution of the blunt-trailing-edge Common Research Model wing in transonic turbulent flow, subject to lift and pitching moment constraints. For this case the optimization is performed based on the solution of the RANS equations, with the Spalart-Allmaras turbulence model fully coupled and linearized. The drag coefficient is reduced by eleven counts, or 6%, when analyzed on a fairly fine mesh.
An efficient, high-fidelity numerical aerodynamic shape optimization tool is presented. The algorithm includes an integrated geometry parameterization and mesh movement scheme based on B-spline volumes, an efficient parallel Newton-Krylov-Schur algorithm for solving the three-dimensional Reynolds-Averaged Navier-Stokes (RANS) equations, a discrete-adjoint gradient evaluation, and a gradient-based optimizer which is capable of performing large-scale optimizations subject to linear and nonlinear constraints. Several cases are presented to demonstrate the performance of the algorithm. First, an optimization is performed for a rectangular wing that is initially fit with NACA0012 sections in order to demonstrate the robustness of the mesh movement and flow analysis given substantial changes in the geometry. The optimizer is able to achieve substantial drag reduction at the target lift by altering the camber and by increasing the sweep angle. We then present a study of the wing geometry extracted from the Common Research Model (CRM) wing-body geometry; we consider the CRM wing with a sharp trailing edge, as well as a wing with the same planform, but given NACA0012 sections. Given section and twist design variables, each initial geometry yields an optimized design that demonstrates improved drag compared to the initial shape. The optimizations of the planar wing with NACA0012 sections and the CRM wings were additionally run with an Euler-based algorithm; RANS analyses were performed on the Euler-optimized geometries such that they could be compared directly with the results of the RANS-based optimizations. In the case of the planar wing with NACA0012 sections, which specified a low target lift coefficient, the Euler-based optimizer produced a very similar design which yielded the same drag coefficient as the RANS-based optimization. However, the CRM study shows that the RANS-based optimizations result in designs with much lower drag compared to the Eulerbased optimizations. We conclude that, in general, viscous and turbulent effects should be taken into account when performing aerodynamic shape optimization.
The focus of this study is to understand the origin of the chiral recognition for a host-guest system containing complexes with different stoichiometries. Each enantiomer of 2-naphthyl-1-ethanol forms two different 1 : 1 complexes with β-cyclodextrin, leading to the formation of three different 2 : 2 complexes. One of these 2 : 2 complexes leads to excimer emission of the guest. Fluorescence studies were employed to determine the binding isotherms for the 1 : 1 and 2 : 2 complexes. No chiral discrimination was directly observed for the formation of the 1 : 1 complexes, while higher equilibrium constants (29% from binding isotherms and 40% from kinetic studies) were observed for the formation of the 2 : 2 complexes with (R)-2-naphthyl-1-ethanol when compared to the formation of the 2 : 2 complexes formed from (S)-2-naphthyl-1-ethanol. The relaxation kinetics was studied using stopped-flow experiments. The formation of the 2 : 2 complexes was followed by detecting the excimer emission from one of the 2 : 2 complexes. The relaxation kinetics was faster for (S)-2-naphthyl-1-ethanol, where a higher dissociation rate constant, by 47%, was observed, suggesting that the chiral discrimination occurs because the interaction between two cyclodextrins is more favorable for the complexes containing (R)-2-naphthyl-1-ethanol when compared to (S)-2-naphthyl-1-ethanol. The same overall equilibrium constants were observed for the 1 : 1 complexes with both enantiomers showing that at a given cyclodextrin concentration the sum of the two types of 1 : 1 complexes is the same for both enantiomers. However, analysis of the binding isotherms indicates that the ratio between the two different 1 : 1 complexes for each enantiomer was different for (R)- and (S)-2-naphthyl-1-ethanol.
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