Skip to main content
Search
Search
Quick Search anywhere
Quick Search anywhere
Quick Search anywhere
Quick Search anywhere
Quick Search anywhere
Quick Search anywhere
Advanced Search
Skip main navigation
No Access

A low-dimensional tool for predicting force decomposition coefficients for varying inflow conditions

Published Online:October 14, 2013pp 368-381https://doi.org/10.1504/PCFD.2013.057101

Abstract

We develop a low-dimensional tool to predict the effects of unsteadiness in the inflow on force coefficients acting on a circular cylinder using proper orthogonal decomposition (POD) modes from steady flow simulations. The approach is based on combining POD and linear stochastic estimator (LSE) techniques. We use POD to derive a reduced-order model (ROM) to reconstruct the velocity field. To overcome the difficulty of developing a ROM using Poisson’s equation, we relate the pressure field to the velocity field through a mapping function based on LSE. The use of this approach to derive force decomposition coefficients (FDCs) under unsteady mean flow from basis functions of the steady flow is illustrated. For both steady and unsteady cases, the final outcome is a representation of the lift and drag coefficients in terms of velocity and pressure temporal coefficients. Such a representation could serve as the basis for implementing control strategies or conducting uncertainty quantification.

Keywords

reduced-order modelling, proper orthogonal decomposition, POD, linear stochastic estimator, LSE, force decomposition coefficient, FDC, unsteady inflow

References

  • 1. Akhtar, I. (2008). Parallel Simulations, Reduced-Order Modeling, and Feedback Control of Vortex Shedding using Fluidic Actuators. Blacksburg, VA:Virginia Tech , PhD thesis Google Scholar
  • 2. Akhtar, I. , Borggaard, J. , Hay, A. (2010). ‘Shape sensitivity analysis in flow models using a finite-difference approach’. Mathematical Problems in Engineering. 1-22, Article ID 209780, 10.1155/2010/209780 Google Scholar
  • 3. Akhtar, I. , Marzouk, O.A. , Nayfeh, A.H. (2009a). ‘A van der Pol-Duffing oscillator model of hydrodynamic forces on canonical structures’. Journal of Computational and Nonlinear Dynamics. 4, 4, 041006 Google Scholar
  • 4. Akhtar, I. , Nayfeh, A.H. (2010). ‘Model based control of vortex shedding using fluidic actuators’. Journal of Computational and Nonlinear Dynamics. 5, 4, 041015 Google Scholar
  • 5. Akhtar, I. , Nayfeh, A.H. , Ribbens, C.J. (2008). ‘A Galerkin model of the pressure field in incompressible flows’. in Proceedings of the 46th Aerospace Sciences Meeting and Exhibit. AIAA Paper No. 2008-611 Google Scholar
  • 6. Akhtar, I. , Nayfeh, A.H. , Ribbens, C.J. (2009b). ‘On the stability and extension of reduced-order Galerkin models in incompressible flows: a numerical study of vortex shedding’. Theoretical and Computational Fluid Dynamics. 23, 3, 213-237 Google Scholar
  • 7. Akhtar, I. , Wang, Z. , Borggaard, J. , Iliescu, T. (2012). ‘A new closure strategy for proper orthogonal decomposition reduced-order models’. Journal of Computational and Nonlinear Dynamics. 7, 3, 034503 Google Scholar
  • 8. Amsallem, D. , Farhat, C. (2008). ‘Interpolation method for adapting reduced-order models and application to aeroelasticity’. AIAA Journal. 46, 7, 1803-1813 Google Scholar
  • 9. Arian, E. , Fahl, M. , Sachs, E. (2000). ‘Trust-region proper orthogonal decomposition for optimal flow control’. Technical Report ICASE. Google Scholar
  • 10. Aubry, N. , Lian, W.Y. , Titi, E.S. (1993). ‘Preserving symmetries in the proper orthogonal decomposition’. SIAM Journal of Scientific Computing. 14, 2, 483-505 Google Scholar
  • 11. Ausseur, J.M. , Pinier, J.T. , Glausser, M.N. , Higuchi, H (2006). ‘Experimental development of a reduced-order model for flow separation control’. in Proceedings of the AIAA 30th Aerospace Sciences Meeting and Exhibit. AIAA Paper No. 2006-1251 Google Scholar
  • 12. Bakewell, H.P. , Lumley, J.L. (1967). ‘Viscous sublayer and adjacent wall region in turbulent pipe flow’. The Physics of Fluids. 10, 9, 1880-1889 Google Scholar
  • 13. Bergmann, M. , Cordier, L. , Brancher, J.P. (2005). ‘Optimal rotary control of the cylinder wake using POD reduced-order model’. Physics of Fluids. 17, 9, 097101-20 Google Scholar
  • 14. Berkooz, G. , Holmes, P. , Lumley, J.L. (1993). ‘The proper orthogonal decomposition in the analysis of turbulent flows’. Annual Review of Fluid Mechanics. 25, 539-575 Google Scholar
  • 15. Burkardt, J. , Du, Q. , Gunzburger, M.D. , Lee, H.C. (2002). ‘Reduced order modeling of complex systems’. in Biennal Conferences on Numerical Analysis, University of Dundee, Scotland, UK Google Scholar
  • 16. Deane, A.E. , Kevrekidis, I.G. , Karniadakis, G.E. , Orsag, S.A (1991). ‘Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinder’. Physics of Fluids A. 3, 10, 2337-2354 Google Scholar
  • 17. Dowell, E.H. , Hall, K.C. , Romanowski, M.C. (1998). ‘Eigenmode analysis in unsteady aerodynamics: reduced-order models’. Applied Mechanics Review. 50, 6, 371-385 Google Scholar
  • 18. Foias, C. , Jolly, M.S. , Kevrekidis, I.G. , Titi, E.S. (1991). ‘Dissipativity of the numerical schemes’. Nonlinearity. 4, 3, 591-613 Google Scholar
  • 19. Graham, W.R. , Peraire, J. , Tang, K.Y. (1999). ‘Optimal control of vortex shedding using low-order models. Part I – open-loop model development’. International Journal for Numerical Methods in Engineering. 44, 7, 945-972 Google Scholar
  • 20. Hall, K.C. , Thomas, J.P. , Dowell, E.H. (2000). ‘Proper orthogonal decomposition technique for transonic unsteady aerodynamic flows’. AIAA Journal. 38, 10, 1853-1862 Google Scholar
  • 21. Hay, A. , Akhtar, I. , Borggaard, J.T. (2012). ‘On the use of sensitivity analysis in model reduction to predict flows for varying inflow conditions’. International Journal for Numerical Methods in Fluids. 68, 1, 122-134 Google Scholar
  • 22. Hay, A. , Borggaard, J. , Akhtar, I. , Pelletier, D. (2010). ‘Reduced-order models for parameter dependent geometries based on shape sensitivity analysis’. Journal of Computational Physics. 229, 4, 1327-1352 Google Scholar
  • 23. Hay, A. , Borggaard, J.T. , Pelletier, D. (2009). ‘Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition’. Journal of Fluid Mechanics. 629, 41-72 Google Scholar
  • 24. Holmes, P. , Lumley, J.L. , Berkooz, G. (1996). Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge, UK:Cambridge University Press Google Scholar
  • 25. Leonard, B.P. (1979). ‘A stable and accurate convective modeling procedure based on quadratic upstream interpolation’. Computational Methods in Applied Mechanical Engineering. 19, 1, 59-98 Google Scholar
  • 26. Lie, T. , Farhat, C. (2007). ‘Adaptation of aeroelastic reduced-order models and application to an f-16 configuration’. AIAA Journal. 45, 6, 1244-1257 Google Scholar
  • 27. Ma, X. , Karniadakis, G. (2002). ‘A low-dimensional model for simulating three-dimensional cylinder flow’. Journal of Fluid Mechanics. 458, 181-190 Google Scholar
  • 28. Mittal, R. , Balachandar, S. (1996). ‘Direct numerical simulation of flow past elliptic cylinders’. Journal of Computational Physics. 124, 2, 351-367 Google Scholar
  • 29. Nayfeh, A.H. , Balachandran, B. (1995). Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods. New York, NY:Wiley , 449-454 Google Scholar
  • 30. Noack, B.R. , Papas, P. , Monkewitz, P.A. (2005). ‘The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows’. Journal of Fluid Mechanics. 523, 339-365 Google Scholar
  • 31. Siegel, S. , Cohen, K. , McLaughlin, T. (2006). ‘Numerical simulations of a feedback-controlled circular cylinder wake’. AIAA Journal. 44, 6, 1266-1276 Google Scholar
  • 32. Singh, S.N. , Myatt, J.H. , Addington, G.A. , Banda, S. , Hall, J.K. (2001). ‘Optimal feedback control of vortex shedding using proper orthogonal decomposition models’. Journal of Fluids Engineering. 123, 3, 612-618 Google Scholar
  • 33. Sirisup, S. , Karniadakis, G.E. (2004). ‘A spectral viscosity method for correcting the long-term behavior of pod models’. Journal of Computational Physics. 194, 1, 92-116 Google Scholar
  • 34. Sirovich, L. (1987). ‘Turbulence and the dynamics of coherent structures’. Quarterly of Applied Mathematics. 45, 561-590 Google Scholar
  • 35. Sirovich, L. , Kirby, M. (1987). ‘Low-dimensional procedure for the characterization of human faces’. Journal of Optical Society of America A. 4, 3, 529-524 Google Scholar
  • 36. Tadmor, E. (1989). ‘Convergence of spectral methods for nonlinear conservation laws’. SIAM Journal of Numerical Analysis. 26, 1, 30 Google Scholar
  • 37. Tang, D. , Kholodar, D. , Junag, J-N. , Dowell, E.H. (2001). ‘System identification and proper orthogonal decomposition method applied to unsteady aerodynamics’. AIAA Journal. 39, 8, 1569-1576 Google Scholar
  • 38. Taylor, J.A. , Glauser (2004). ‘Towards practical flow sensing and control via pod and LSE based low-dimensional tools’. Journal of Fluids Engineering. 126, 3, 337-345 Google Scholar
  • 39. Thomas, J.P. , Dowell, E.H. , Hall, K.C. (2003). ‘Three dimensional transonic aeroelasticity using proper orthogonal decomposition-based reduced-order models’. Journal of Aircraft. 40, 3, 544-551 Google Scholar
  • 40. Wang, Z. , Akhtar, I. , Borggaard, J. , Iliescu, T. (2011). ‘Two-level discretizations of nonlinear closure models for proper orthogonal decomposition’. Journal of Computational Physics. 230, 1, 126-146 Google Scholar
  • 41. Wang, Z. , Akhtar, I. , Borggaard, J. , Iliescu, T. (2012). ‘Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison’. Computer Methods in Applied Mechanics and Engineering. 237–240, 1, 10-26 Google Scholar
  • 42. Williamson, C.H.K. (1996). ‘Vortex dynamics in the cylinder wake’. Annual Review of Fluid Mechanics. 28, 477-539 Google Scholar
Previous article
Next article