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High-Order Accurate, Low Numerical Diffusion Methods for Aerodynamics

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High-Order Accurate, Low Numerical Diffusion Methods for Aerodynamics ARTICLE IN PRESS Progress in Aerospace Sciences 41 (2005) 192–300 www.elsevier.com/locate/paerosci High-order accurate, lownumerical diffusion methods for aerodynamics John A. Ekaterinaris Foundation for Research and Technology FORTH, Institute of Applied and Computational Mathematics IACM, FORTH/IACM, 71110 Heraklion, Crete, Greece Abstract In recent years numerical methods have been widely used to effectively resolve complex flow features of aerodynamics flows with meshes that are reasonable for today’s computers. High-order numerical methods were used mainly in direct numerical simulations and aeroacoustics. For many aeronautical applications, accurate computation of vortex- dominated flows is important because the vorticity in the flow field and the wake of swept wings at an incidence and rotor blades largely determines the distribution of loading. The main deficiency of widely available, second-order accurate methods for the accurate computation of these flows is the numerical diffusion of vorticity to unacceptable levels. Application of high-order accurate, low-diffusion numerical methods can significantly alleviate this deficiency of traditional second order methods. Furthermore, higher-order space discretizations have the potential to improve detached eddy simulation predictions of separated flows with significant unsteadiness. Recently developed high-order accurate finite-difference, finite-volume, and finite-element methods are reviewed. These methods can be used as an attractive alternative of traditional low-order central and upwind computational fluid dynamics methods for improved predictions of vortical and other complex, separated, unsteady flows. The main features of these methods are summarized, from a practical user’s point of view, their applicability and relative strength is indicated, and examples from recent applications are presented to illustrate their performance on selected problems. r 2005 Elsevier Ltd. All rights reserved. Contents 1. Introduction . 194 1.1. Prior work on high-order numerical methods . 196 2. Governing equations . 198 2.1. Conservative form of the NS equations . 198 2.2. Entropy splitting . 199 2.3. Time integration of the Euler and NS equations . 200 2.3.1. Explicit temporal schemes. 201 2.3.2. Implicit schemes. 202 2.3.3. Time accurate solutions with multigrid. 202 E-mail address: [email protected]. 0376-0421/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.paerosci.2005.03.003 ARTICLE IN PRESS J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 193 3. High-order finite-difference schemes . 203 3.1. Centered compact schemes . 203 3.2. Boundary closures for high-order schemes . 204 3.3. Simultaneous evaluation of the first and second derivative with compact schemes . 205 3.4. Modified high-order finite-difference schemes . 206 3.4.1. Upwind high-order schemes . 206 3.4.2. Dispersion-relation-preserving (DPR) scheme . 207 3.5. Spectral-type filters. 207 3.6. Characteristic-based (ACM) filters . 208 3.6.1. ENO and WENO ACM filters . 210 3.7. Finite-volume compact difference-based schemes. 211 3.8. Results with ACM filters . 212 3.9. Results with compact schemes and filters . 218 4. ENO and WENO schemes . 227 4.1. High-order reconstruction . 228 4.2. Approximation in one dimension. 228 4.3. One-dimensional conservative approximation of the derivative . 229 4.4. ENO reconstruction . 230 4.5. 1D finite-volume ENO scheme . 231 4.6. 1D finite-difference ENO scheme . 232 4.7. WENO approximation . 232 4.8. Multidimensional ENO and WENO reconstruction. 234 4.8.1. Finite-volume reconstruction for Cartesian mesh. 234 4.8.2. Two-dimensional reconstruction for triangles . 235 4.8.3. Multidimensional finite-difference ENO . 236 4.9. Optimization of WENO schemes . 236 4.10. Compact WENO approximation . 237 4.11. Hybrid compact-WENO scheme for the Euler equations . 239 4.12. Applications of ENO and WENO . 240 5. The discontinuous Galerkin (DG) method . 256 5.1. DG space discretization . 257 5.2. Element bases . 258 5.3. Arbitrary high-order DG schemes . 260 5.4. Analysis of the DG method for wave propagation . 260 5.5. Dissipative and dispersive behavior of high-order DG method . 261 5.6. Limiting of DG expansions . 261 5.6.1. Rectangular elements . 262 5.6.2. Triangular elements . 262 5.6.3. Component-wise limiters. 263 5.7. DG stabilization operator . 264 5.8. DG space discretization of the NS equations . 264 5.9. The DG variational multiscale (VMS) method . 265 5.10. Implicit time marching of DG discretizations . 269 5.11. p-type multigrid for DG . 270 5.12. Results with the DG method. 270 6. The spectral volume (SV) method . 276 6.1. Fundamentals of the SV method . 277 6.2. SV method in one dimension. 279 6.2.1. Reconstruction for SV in one dimension . 279 6.3. Polynomial reconstruction in triangular SVs . 281 6.4. Linear, quadratic, and cubic SV reconstructions . 283 6.5. Spectral volume partitions. 284 6.6. Multidimensional limiters . 284 6.7. The SV method for two-dimensional systems . 285 6.8. Multidimensional TVD and TVB limiters.
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