Advertisement

On the Sensitivity to the Filtering Radius in Leray Models of Incompressible Flow

  • Luca Bertagna
  • Annalisa QuainiEmail author
  • Leo G. Rebholz
  • Alessandro Veneziani
Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 47)

Abstract

One critical aspect of Leray models for the Large Eddy Simulation (LES) of incompressible flows at moderately large Reynolds number (in the range of few thousands) is the selection of the filter radius. This drives the effective regularization of the filtering procedure, and its selection is a trade-off between stability (the larger, the better) and accuracy (the smaller, the better). In this paper, we consider the classical Leray-\(\alpha \) and a recently introduced (by one of the authors) Leray model with a deconvolution-based indicator function, called Leray-\(\alpha \)-NL. We investigate the sensitivity of the solutions to the filter radius by introducing the sensitivity systems, analyzing them at the continuous and discrete levels, and numerically testing them on two benchmark problems.

Notes

Acknowledgements

The research presented in this work was carried out during LR’s visit at the Department of Mathematics and Computer Science at Emory University in the fall semester 2014. This support is gratefully acknowledged. This research has been supported in part by the NSF under grants DMS-1620384/DMS-1620406 (Quaini and Veneziani), DMS-1262385 (Quaini), Emory URC Grant 2015 Numerical Methods for Flows at Moderate Reynolds Numbers in LVAD (Veneziani), and DMS-1522191 (Rebholz).

References

  1. 1.
    Anitescu M, Layton WJ (2007) Sensitivities in large eddy simulation and improved estimates of turbulent flow functionals. SIAM J Sci Comput 29(4):1650–1667MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bertagna L, Quaini A, Veneziani A (2016) Deconvolution-based nonlinear filtering for incompressible flows at moderately large Reynolds numbers. Int J Numer Methods Fluids 81(8):463–488MathSciNetGoogle Scholar
  3. 3.
    Borggaard J, Burns J (1995) A sensitivity equation approach to shape optimization in fluid flows. In: Flow control (Minneapolis, MN, 1992), vol 68 of IMA Vol Math Appl, Springer, New York, pp 49–78Google Scholar
  4. 4.
    Borggaard J, Burns J (1997) A PDE sensitivity equation method for optimal aerodynamic design. J Comput Phys 136(2):366–384MathSciNetzbMATHGoogle Scholar
  5. 5.
    Borggaard J, Verma A (2000) On efficient solutions to the continuous sensitivity equation using automatic differentiation. SIAM J Sci Comput 22(1):39–62MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bowers AL, Rebholz LG (2012) Increasing accuracy and efficiency in FE computations of the Leray-deconvolution model. Numer Methods Partial Differ Equ 28(2):720–736MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bowers AL, Rebholz LG (2013) Numerical study of a regularization model for incompressible flow with deconvolution-based adaptive nonlinear filtering. Comput Methods Appl Mech Engrg 258:1–12MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bowers AL, Rebholz LG, Takhirov A, Trenchea C (2012) Improved accuracy in regularization models of incompressible flow via adaptive nonlinear filtering. Internat J Numer Methods Fluids 70(7):805–828MathSciNetGoogle Scholar
  9. 9.
    Brenner SC, Scott LR (2008) The mathematical theory of finite element methods. Springer, New YorkzbMATHGoogle Scholar
  10. 10.
    Cao Y, Titi ES (2009) On the rate of convergence of the two-dimensional \(\alpha \)-models of turbulence to the Navier-Stokes equations. Numer Funct Anal Optim 30(11–12):1231–1271MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cheskidov A, Holm DD, Olson E, Titi ES (2005) On a Leray-\(\alpha \) model of turbulence. Proc R Soc Lond Ser A Math Phys Eng Sci 461(2055):629–649MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dunca A, Epshteyn Y (2006) On the Stolz-Adams deconvolution model for the large-eddy simulation of turbulent flows. SIAM J Math Anal 37(6):1890–1902MathSciNetzbMATHGoogle Scholar
  13. 13.
    Geurts BJ, Holm DD (2002) Leray simulation of turbulent shear layers. In: Castro IP, Hancock PE, Thomas TG (eds), Advances in Turbulence IX: Proceedings of the Ninth European Turbulence Conference (Southampton, 2002), CIMNE, pp 337–340Google Scholar
  14. 14.
    Geurts BJ, Holm DD (2003) Regularization modeling for large-eddy simulation. Phys Fluids 15(1):L13–L16MathSciNetGoogle Scholar
  15. 15.
    Geurts BJ, Holm DD (2006) Leray and LANS-\(\alpha \) modelling of turbulent mixing. J Turbul 7(10):33MathSciNetzbMATHGoogle Scholar
  16. 16.
    Geurts BJ, Kuczaj AK, Titi ES (2008) Regularization modeling for large-eddy simulation of homogeneous isotropic decaying turbulence. J Phys A 41(34):344008 (29p)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Godfrey AG, Cliff EM (1998) Direct calculation of aerodynamic force derivatives: a sensitivity equation approach. In: 36th AIAA Aerospace Sciences Meeting and Exhibit. AIAA, Paper 98-0393, 12pGoogle Scholar
  18. 18.
    Graham JP, Holm DD, Mininni P, Pouquet A (2011) The effect of subfilter-scale physics on regularization models. J Sci Comput 49(1):21–34MathSciNetzbMATHGoogle Scholar
  19. 19.
    Graham JP, Holm DD, Mininni PD, Pouquet A (2008) Three regularization models of the Navier-Stokes equations. Phys Fluids 20:35107 (15p)zbMATHGoogle Scholar
  20. 20.
    Gunzburger MD (1989) Finite element methods for viscous incompressible flows: a guide to theory, practice, and algorithms. Academic Press, BostonGoogle Scholar
  21. 21.
    Gunzburger MD (1999) Sensitivities, adjoints and flow optimization. Int J Numer Methods Fluids 31(1):53–78MathSciNetzbMATHGoogle Scholar
  22. 22.
    Hecht F (2012) New development in FreeFem++. J Numer Math 20(3–4):251–265MathSciNetzbMATHGoogle Scholar
  23. 23.
    Hecht MW, Holm DD, Petersen MR, Wingate BA (2008) The LANS-\(\alpha \) and Leray turbulence parameterizations in primitive equation ocean modeling. J Phys A 41(34):344009 (23p)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Heywood JG, Rannacher R (1990) Finite-element approximation of the nonstationary Navier-Stokes problem. IV: Error analysis for second-order time discretization. SIAM J Numer Anal 27(2):353–384MathSciNetzbMATHGoogle Scholar
  25. 25.
    Heywood JG, Rannacher R, Turek S (1996) Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int J Numer Methods Fluids 22(5):325–352MathSciNetzbMATHGoogle Scholar
  26. 26.
    Layton W (2008) Introduction to the numerical analysis of incompressible viscous flows. SIAMGoogle Scholar
  27. 27.
    Layton W, Manica CC, Neda M, Rebholz LG (2008) Helicity and energy conservation and dissipation in approximate deconvolution LES models of turbulence. Adv Appl Fluid Mech 4(1):1–46MathSciNetzbMATHGoogle Scholar
  28. 28.
    Layton W, Mays N, Neda M, Trenchea C (2014) Numerical analysis of modular regularization methods for the BDF2 time discretization of the Navier-Stokes equations. ESAIM Math Model Numer Anal 48(3):765–793MathSciNetzbMATHGoogle Scholar
  29. 29.
    Layton W, Rebholz L (2012) Approximate deconvolution models of turbulence: analysis, phenomenology and numerical analysis. Springer, HeidelbergzbMATHGoogle Scholar
  30. 30.
    Layton W, Rebholz L, Trenchea C (2012) Modular nonlinear filter stabilization of methods for higher Reynolds numbers flow. J Math Fluid Mech 14(2):325–354MathSciNetzbMATHGoogle Scholar
  31. 31.
    Leray J (1934) Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math 63(1):193–248MathSciNetzbMATHGoogle Scholar
  32. 32.
    Liu Y, Tucker P, Kerr R (2008) Linear and nonlinear model large-eddy simulations of a plane jet. Comput Fluids 37(4):439–449zbMATHGoogle Scholar
  33. 33.
    Lunasin E, Kurien S, Titi ES (2008) Spectral scaling of the Leray-\(\alpha \) model for two-dimensional turbulence. J Phys A 41(34):344014 (10p)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Pahlevani F (2006) Sensitivity computations of eddy viscosity models with an application in drag computation. Int J Numer Methods Fluids 52(4):381–392MathSciNetzbMATHGoogle Scholar
  35. 35.
    Quarteroni A, Sacco R, Saleri F (2007) Numerical mathematics, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  36. 36.
    Sagaut P, Lê T (1997) Some investigations of the sensitivity of large eddy simulation. Technical report, ONERAGoogle Scholar
  37. 37.
    Sagaut P, Lê TH (1997) Some investigations on the sensitivity of large eddy simulation. In: Chollet J-P, Voke PR, Kleiser L (eds) Direct and Large-Eddy Simulation II: Proceedings of the ERCOFTAC Workshop (Grenoble, 1996). Springer, Dordrecht, pp 81–92Google Scholar
  38. 38.
    Stanley L, Stewart D (2002) Design sensitivity analysis: computational issues of sensitivity equation methods. Number 25 in Frontiers in Applied Mathematics. SIAM, Philadelphia, PAGoogle Scholar
  39. 39.
    Verstappen R (2008) On restraining the production of small scales of motion in a turbulent channel flow. Comput Fluids 37(7):887–897MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Luca Bertagna
    • 1
  • Annalisa Quaini
    • 2
    Email author
  • Leo G. Rebholz
    • 3
  • Alessandro Veneziani
    • 4
  1. 1.Department of Scientific ComputingFlorida State UniversityTallahasseeUSA
  2. 2.Department of MathematicsUniversity of HoustonHoustonUSA
  3. 3.Department of Mathematical SciencesClemson UniversityClemsonUSA
  4. 4.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

Personalised recommendations