POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder

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Abstract

Vortex shedding around circular cylinders is a well known and studied phenomenon that appears in many engineering fields. A Reduced Order Model (ROM) of the incompressible ow around a circular cylinder is presented in this work. The ROM is built performing a Galerkin projection of the governing equations onto a lower dimensional space. The reduced basis space is generated using a Proper Orthogonal Decomposition (POD) approach. In particular the focus is into (i) the correct reproduction of the pres- sure field, that in case of the vortex shedding phenomenon, is of primary importance for the calculation of the drag and lift coefficients; (ii) the projection of the Governing equations (momentum equation and Poisson equation for pressure) performed onto different reduced basis space for velocity and pressure, respectively; (iii) all the relevant modifications necessary to adapt standard finite element POD-Galerkin methods to a finite volume framework. The accuracy of the reduced order model is assessed against full order results.

1. M. P. Païdoussis, Fluid-Structure Interactions. Slender Structures and Axial Flow. Volume 1. Academic Press, first ed., 1998.

2. M. P. Païdoussis, Fluid-Structure Interactions. Slender Structures and Axial Flow. Volume 2. Academic Press, first ed., 2003.

3. V. Strouhal, Über eine besondere Art der Tonerregung, Annalen der Physik, vol. 241, no. 10, pp. 216-251, 1878.

4. M. M. Zdravkovich, Flow around Circular Cylinders: Volume 2: Appli- cations, vol. 2. Oxford University Press, 2003.

5. M. M. Zdravkovich, Flow around Circular Cylinders: Volume 1: Funda- mentals, vol. 350. Cambridge University Press, 1997.

6. R. T. Hartlen and I. G. Currie, Lift-oscillator model of vortex-induced vibration, Journal of the Engineering Mechanics Division, vol. 96, no. 5, pp. 577-591, 1970.

7. M. Facchinetti, E. de Langre, and F. Biolley, Coupling of structure and wake oscillators in vortex-induced vibrations, Journal of Fluids and Structures, vol. 19, no. 2, pp. 123 - 140, 2004.

8. G. Stabile, H. G. Matthies, and C. Borri, A novel reduced order model for vortex induced vibrations of long exible cylinders, Submitted to Journal of Ocean Engineering, 2016.

9. J. S. Hesthaven, G. Rozza, and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer International Publishing, 2016.

10. A. Quarteroni, A. Manzoni, and F. Negri, Reduced Basis Methods for Partial Differential Equations. Springer International Publishing, 2016.

11. B. R. Noack and H. Eckelmann, A low-dimensional Galerkin method for the three-dimensional ow around a circular cylinder, Physics of Fluids, vol. 6, no. 1, pp. 124-143, 1994.

12. I. Akhtar, A. H. Nayfeh, and C. J. Ribbens, On the stability and extension of reduced-order Galerkin models in incompressible ows, The- oretical and Computational Fluid Dynamics, vol. 23, no. 3, pp. 213-237, 2009.

13. S. Lorenzi, A. Cammi, L. Luzzi, and G. Rozza, POD-Galerkin method for finite volume approximation of Navier-Stokes and RANS equations, Computer Methods in Applied Mechanics and Engineering, vol. 311, pp. 151 - 179, 2016.

14. M. Bergmann, C.-H. Bruneau, and A. Iollo, Enablers for robust POD models, Journal of Computational Physics, vol. 228, no. 2, pp. 516-538, 2009.

15. K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in uid dynamics, SIAM Journal on Numerical Analysis, vol. 40, no. 2, pp. 492-515, 2002.

16. J. Burkardt, M. Gunzburger, and H.-C. Lee, POD and CVT-based reduced-order modeling of Navier-Stokes ows, Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 1-3, pp. 337-355, 2006.

17. J. Baiges, R. Codina, and S. Idelsohn, Reduced-order modelling strategies for the finite element approximation of the incompressible Navier- Stokes equations, Computational Methods in Applied Sciences, vol. 33, pp. 189-216, 2014.

18. H. K. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics. The Finite Volume Method. London: Longman Group Ltd., 1995.

19. F. Moukalled, L. Mangani, and M. Darwish, The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM and Matlab. Springer Publishing Company, Incorporated, 1st ed., 2015.

20. H. G.Weller, G. Tabor, H. Jasak, and C. Fureby, A tensorial approach to computational continuum mechanics using object-oriented techniques, Computers in physics, vol. 12, no. 6, pp. 620-631, 1998.

21. H. Jasak, Error analysis and estimation for the finite volume method with applications to uid ows. PhD thesis, Imperial College, University of London, 1996.

22. R. Issa, Solution of the implicitly discretised uid ow equations by operator-splitting, Journal of Computational Physics, vol. 62, no. 1, pp. 40-65, 1986.

23. S. Patankar and D. Spalding, A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic ows, International Journal of Heat and Mass Transfer, vol. 15, no. 10, pp. 1787 - 1806, 1972.

24. A. Caiazzo, T. Iliescu, V. John, and S. Schyschlowa, A numerical investigation of velocity-pressure reduced order models for incompressible ows, Journal of Computational Physics, vol. 259, pp. 598 - 616, 2014.

25. G. Rozza, D. Huynh, and A. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics, Archives of Computational Methods in Engineering, vol. 15, no. 3, pp. 229-275, 2008.

26. F. Chinesta, A. Huerta, G. Rozza, and K. Willcox, Model Order Reduction, Encyclopedia of Computational Mechanics, In Press, 2017.

27. F. Chinesta, P. Ladeveze, and E. Cueto, A Short Review on Model Order Reduction Based on Proper Generalized Decomposition, Archives of Computational Methods in Engineering, vol. 18, no. 4, p. 395, 2011.

28. A. Dumon, C. Allery, and A. Ammar, Proper general decomposition (PGD) for the resolution of Navier-Stokes equations, Journal of Com- putational Physics, vol. 230, no. 4, pp. 1387-1407, 2011.

29. L. Sirovich, Turbulence and the Dynamics of Coherent Structures part I: Coherent Structures, Quarterly of Applied Mathematics, vol. 45, no. 3, pp. 561-571, 1987.

30. A. Quarteroni and G. Rozza, Numerical solution of parametrized Navier-Stokes equations by reduced basis methods, Numerical Meth- ods for Partial Differential Equations, vol. 23, no. 4, pp. 923-948, 2007.

31. G. Rozza, Reduced basis methods for Stokes equations in domains with non-affine parameter dependence, Computing and Visualization in Sci- ence, vol. 12, no. 1, pp. 23-35, 2009.

32. D. Xiao, F. Fang, A. Buchan, C. Pain, I. Navon, J. Du, and G. Hu, Non linear model reduction for the navier stokes equations using residual deim method, Journal of Computational Physics, vol. 263, pp. 1 - 18, 2014.

33. M. Barrault, Y. Maday, N. C. Nguyen, and A. T. Patera, An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations, Comptes Rendus Mathematique, vol. 339, no. 9, pp. 667 - 672, 2004.

34. K. Carlberg, C. Farhat, J. Cortial, and D. Amsallem, The GNAT method for nonlinear model reduction: Effective implementation and application to computational uid dynamics and turbulent ows, Jour- nal of Computational Physics, vol. 242, pp. 623 - 647, 2013.

35. B. R. Noack, P. Papas, and P. A. Monkewitz, The need for a pressureterm representation in empirical Galerkin models of incompressible shear ows, Journal of Fluid Mechanics, vol. 523, pp. 339-365, 01 2005.

36. A. E. Deane, I. G. Kevrekidis, G. E. Karniadakis, and S. A. Orszag, Lowdimensional models for complex geometry ows: Application to grooved channels and circular cylinders, Physics of Fluids A: Fluid Dynamics, vol. 3, no. 10, pp. 2337-2354, 1991.

37. X. Ma and G. Karniadakis, A low-dimensional model for simulating three-dimensional cylinder ow, Journal of Fluid Mechanics, vol. 458, pp. 181-190, 2002.

38. F. Ballarin, A. Manzoni, A. Quarteroni, and G. Rozza, Supremizer stabilization of POD-Galerkin approximation of parametrized steady incompressible Navier-Stokes equations, International Journal for Nu- merical Methods in Engineering, vol. 102, no. 5, pp. 1136-1161, 2015.

39. G. Rozza and K. Veroy, On the stability of the reduced basis method for Stokes equations in parametrized domains, Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 7, pp. 1244 - 1260, 2007.

40. G. Rozza, D. B. P. Huynh, and A. Manzoni, Reduced basis approximation and a posteriori error estimation for Stokes ows in parametrized geometries: Roles of the inf-sup stability constants, Numerische Math- ematik, vol. 125, no. 1, pp. 115-152, 2013.

41. I. Kalashnikova and M. F. Barone, On the stability and convergence of a Galerkin reduced order model (ROM) of compressible ow with solid wall and far-field boundary treatment, International Journal for Numerical Methods in Engineering, vol. 83, no. 10, pp. 1345-1375, 2010.

42. S. Sirisup and G. Karniadakis, Stability and accuracy of periodic ow solutions obtained by a POD-penalty method, Physica D: Nonlinear Phenomena, vol. 202, no. 3-4, pp. 218 - 237, 2005.

43. W. R. Graham, J. Peraire, and K. Y. Tang, Optimal control of vortex shedding using low-order models. Part I:open-loop model development, International Journal for Numerical Methods in Engineering, vol. 44, no. 7, pp. 945-972, 1999.

44. S. Makridakis, Accuracy measures: theoretical and practical concerns, International Journal of Forecasting, vol. 9, no. 4, pp. 527 - 529, 1993.

45. G. Stabile and G. Rozza, Stabilized Reduced order POD-Galerkin techniques for finite volume approximation of the parametrized Navier- Stokes equations, submitted, 2017.

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