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.
If the inline PDF is not rendering correctly, you can download the PDF file here.
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.