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Optimization of an Implicit Large-Eddy Simulation Method for Underresolved Incompressible Flow Simulations

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In engineering applications, resolution is often low. In these underresolved regions, the truncation error of the underlying numerical schemes strongly affects the solution. If the truncation error functions as a physically consistent subgrid-scale model (that is, it models the evolution of otherwise resolved scales), resolution may remain low. Thereby, computational efficiency is improved. The sixth-order adaptive central-upwind weighted essentially nonoscillatory scheme with implicit scale separation, denoted as WENO-CU6-M1, potentially allows for physically consistent implicit subgrid-scale modeling, when shaped accordingly. In this work, finding an optimal formulation of WENO-CU6-M1 is considered within a deterministic design optimization framework. Possible surrogate modeling and sampling strategies are considered. Design optimization is based on evaluating the potential of a WENO-CU6-M1 scheme formulation to reproduce Kolmogorov scaling for a Taylor–Green vortex in its quasi-isotropic state. As in the absence of physical viscosity, kinetic energy dissipates exclusively due to the subgrid-scales, the Reynolds number is infinite, and the evolution of the flow is determined by proper subgrid-scale modeling. To complete the work, the effective numerical dissipation rate of the WENO-CU6-M1 model optimized for artificially compressible fluid flows is quantified, and it is compared to the original one. Not only is the zero viscosity limit considered, but the model behavior is benchmarked offdesign, for low to high Reynolds numbers. A comparison to an alternative explicit and implicit subgrid-scale model demonstrates its superior behavior for the chosen test flow.


  • [1] Margolin L. G. and Rider W. J., “A Rationale for Implicit Turbulence Modelling,” International Journal for Numerical Methods in Fluids, Vol. 39, No. 9, 2002, pp. 821–841. doi: IJNFDW 0271-2091 CrossrefGoogle Scholar

  • [2] Adams N., Hickel S. and Franz S., “Implicit Subgrid-Scale Modeling by Adaptive Deconvolution,” Journal of Computational Physics, Vol. 200, No. 2, 2004, pp. 412–431. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [3] Grinstein F., Margolin L. and Rider W., Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics, Cambridge Univ Press, Cambridge, England, U.K., 2007. CrossrefGoogle Scholar

  • [4] Balsara D. S. and Shu C.-W., “Monotonicity Preserving Weighted Essentially Non-Oscillatory Schemes with Increasingly High Order of Accuracy,” Journal of Computational Physics, Vol. 160, No. 2, 2000, pp. 405–452. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [5] Hickel S., Adams N. and Domaradzki J., “An Adaptive Local Deconvolution Method for Implicit LES,” Journal of Computational Physics, Vol. 213, No. 1, 2006, pp. 413–436. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [6] Hu X. Y., Wang Q. and Adams N. A., “An Adaptive Central-Upwind Weighted Essentially Non-Oscillatory Scheme,” Journal of Computational Physics, Vol. 229, No. 23, 2010, pp. 8952–8965. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [7] Jiang G. and Shu C.-W., “Efficient Implementation of Weighted ENO Schemes,” Journal of Computational Physics, Vol. 126, No. 1, 1996, pp. 202–228. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [8] Hu X. Y. and Adams N. A., “Scale Separation for Implicit Large Eddy Simulation,” Journal of Computational Physics, Vol. 230, No. 19, 2011, pp. 7240–7249. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [9] Schranner F. S., Hu X. and Adams N. A., “A Physically Consistent Weakly Compressible High-Resolution Approach to Underresolved Simulations of Incompressible Flows,” Computers and Fluids, Vol. 86, Nov. 2013, pp. 109–124. doi: CrossrefGoogle Scholar

  • [10] Schranner F. S., Domaradzki J. A., Hickel S. and Adams N. A., “Assessing the Numerical Dissipation Rate and Viscosity in Numerical Simulations of Fluid Flows,” Computers and Fluids, Vol. 114, July 2015, pp. 84–97. doi: CrossrefGoogle Scholar

  • [11] Taylor G. I. and Green A., “Mechanism of the Production of Small Eddies from Larger Ones,” Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, Vol. 158, No. 895, 1937, pp. 499–521. CrossrefGoogle Scholar

  • [12] Brachet M., Meiron D., Orszag S., Nickel B., Morf R. and Frisch U., “The Taylor–Green Vortex and Fully Developed Turbulence,” Journal of Statistical Physics, Vol. 34, Nos. 5–6, 1984, pp. 1049–1063. doi: JSTPBS 0022-4715 CrossrefGoogle Scholar

  • [13] Brachet M., Meneguzzi M., Vincent A., Politano H. and Sulem P.-L., “Numerical Evidence of Smooth Self-Similar Dynamics and Possibility of Subsequent Collapse for Three-Dimensional Ideal Flow,” Physics of Fluids, Vol. 4, No. 12, 1992, pp. 2845–2854. doi: CrossrefGoogle Scholar

  • [14] Chorin A. J., “A Numerical Method for Solving Incompressible Viscous Flow Problems,” Journal of Computational Physics, Vol. 135, No. 2, 1997, pp. 118–125. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [15] Temam R., “Une Méthode d’Approximation des Solutions des Équations Navier-Stokes,” Bulletin de la Société Mathématique de France, Vol. 96, 1968, pp. 115–152. CrossrefGoogle Scholar

  • [16] Roe P. L., “Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes,” Journal of Computational Physics, Vol. 43, No. 2, 1981, pp. 357–372. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [17] Shu C.-W. and Osher S., “Efficient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes,” Journal of Computational Physics, Vol. 77, No. 2, 1988, pp. 439–471. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [18] Shu C.-W., “Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws,” Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Vol. 1697, edited by Quarteroni A., Lecture Notes in Mathematics, Springer, Berlin, 1998, pp. 325–432. CrossrefGoogle Scholar

  • [19] Borges R., Carmona M., Costa B. and Don W., “An Improved Weighted Essentially Non-Oscillatory Scheme for Hyperbolic Conservation Laws,” Journal of Computational Physics, Vol. 227, No. 6, 2008, pp. 3191–3211. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [20] Henrick A., Aslam T. and Powers J., “Mapped Weighted Essentially Non-Oscillatory Schemes: Achieving Optimal Order Near Critical Points,” Journal of Computational Physics, Vol. 207, No. 2, 2005, pp. 542–567. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [21] Gerolymos G., Sénéchal D. and Vallet I., “Very-High-Order Weno Schemes,” Journal of Computational Physics, Vol. 228, No. 23, 2009, pp. 8481–8524. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [22] Xu-Dong L. and Stanley Osher T. C., “Weighted Essentially Non-Oscillatory Schemes,” Journal of Computational Physics, Vol. 115, No. 1, 1994, pp. 200–212. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [23] Taylor E., Wu M. and Martin M., “Optimization of Nonlinear Error for Weighted Essentially Non-Oscillatory Methods in Direct Numerical Simulations of Compressible Turbulence,” Journal of Computational Physics, Vol. 223, No. 1, 2007, pp. 384–397. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [24] Fauconnier D., Langhe C. D. and Dick E., “Construction of Explicit and Implicit Dynamic Finite Difference Schemes and Application to the Large-Eddy Simulation of the Taylor–Green Vortex,” Journal of Computational Physics, Vol. 228, No. 21, 2009, pp. 8053–8084. doi: JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [25] Gram J., “Über die Entwicklung Reeler Funktionen in Reihen Mittels der Methode der Kleinsten Quadrate,” Journal für die Reine und Angewandte Mathematik, Vol. 94, 1883, pp. 41–73. Google Scholar

  • [26] Legendre A. M., Nouvelles Methodes pour la Determination des Orbites des Comtes, F. Didot, Paris, 1805, pp. 72–75. Google Scholar

  • [27] Venter G., Review of Optimization Techniques, Wiley, Hoboken, NJ, 2010, pp. 7–8. CrossrefGoogle Scholar

  • [28] Beasley D., Martin R. and Bull D., “An Overview of Genetic Algorithms: Part 1. Fundamentals,” University Computing, Vol. 15, No. 2, 1993, pp. 58–69. Google Scholar

  • [29] Zingg D. W., Nemec M. and Pulliam T. H., “A Comparative Evaluation of Genetic and Gradient-Based Algorithms Applied to Aerodynamic Optimization,” European Journal of Computational Mechanics, Vol. 17, Nos. 1–2, May 2008, pp. 103–126. doi: CrossrefGoogle Scholar

  • [30] Harder R. L. and Desmarais R., “Interpolation Using Surface Splines,” Journal of Aircraft, Vol. 9, No. 2, 1972, pp. 189–191. doi: LinkGoogle Scholar

  • [31] Wilson D. C. and Mair B. A., Applied and Numerical Harmonic Analysis, Springer Science+Business Media, New York, 2003, pp. 311–341, Chap. 12. Google Scholar

  • [32] Powell M. J. D., “A Thin Plate Spline Method for Mapping Curves into Curves in Two Dimensions,” Computational Techniques and Applications (CTAC95), edited by May R. L. and Easton A. K., World Scientific, Singapore, 1995, pp. 43–57. Google Scholar