Extension Ability of Reduced Order Model of Unsteady Incompressible Flows Using a Combination of POD and Fourier Modes

Document Type : Research Paper


Department of Mechanical Eng., School of Engineering, University of Qom, Iran


In this article, an improved reduced order modelling approach, based on the proper orthogonal decomposition (POD) method, is presented. After projecting the governing equations of flow dynamics along the POD modes, a dynamical system was obtained. Normally, the classical reduced order models do not predict accurate time variations of flow variables due to some reasons. The response of the dynamical system was improved using a calibration method based on a least-square optimization process. The calibration polynomial can be assumed as the pressure correction term which is vanished in projecting the Navier-Stokes equations along the POD modes. The above least- square procedure is a combination of POD method and the solution of an optimization problem. The obtained model can predict accurate time variations of flow field with high speed. For long time periods, the calibration term can be computed using a combined form of POD and Fourier modes. This extension is a totally new extension to this procedure which has recently been proposed by the authors. The results obtained from the calibrated reduced order model show close agreements to the benchmark DNS data, proving high accuracy of our model.


Main Subjects

[1] Couplet, M., Basdevant, C., Sagaut, P., Calibrated Reduced-order POD-Galerkin System for Fluid Flow Modeling, J. Comp. Physics 207, 2005, 192–220.
[2] Favier, J., Cordier, L., Kourta, A., Iollo, A., Calibrated POD Reduced-order Models of Massively Separated Flows in the Perspective of Their Control, ASME Joint U.S.-European Fluids Eng. Summer Meeting, Miami, Florida, USA, 2006.
[3] Galletti, B., Bruneau C.H., Zannetti L., Iollo, A., Low-order Modeling of Two-dimensional Flow Regimes Past a Confined Squared Cylinder, J. Fluid Mech. 503, 2004, 161-170.
[4] Holmes, P., Lumley, J.L., Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge Monographs on Mechanics, Cambridge University Press, 1996.
[5] Li, H., Luo, Z., Gao, J., A New Reduced-Order FVE Algorithm Based on POD Method for Viscoelastic Equations, Acta Mathematica Scientia, Vol. 33, Issue 4, 2013, 1076-1098.
[6] Chen, J., Han, D., Yu, B., Sun, B., Wei, J., POD-Galerkin reduced-order model for viscoelastic turbulent channel flow, Numerical Heat Transfer, Part B: Fundamentals, 72(3), 2017, 268-283.
[7] Moayyedi, M.K., Low-dimensional POD simulation of Unsteady Flow around Bodies with Arbitrary Shapes, PhD Dissertation, Sharif University of Technology, 2009.
[8] Moayyedi, M.K., Taeibi-Rahni, M., Sabetghadam, F., Accurate Low-dimensional Dynamical Model for Simulation of Unsteady Incompressible Flows, The 12th Fluid Dynamics Conference, Babol, Iran, 2009.
[9] Noack, B.R., Papas, P., Monkewitz, P.A., The Need for a Pressure-term Representation in Empirical Galerkin Models of Incompressible Shear Flows, J. Fluid Mech. 523, 2005, 339-365.
[10] Rowely, C.W., Model Reduction for Fluids, Using Balanced Proper Orthogonal Decomposition, Int. J. Bifurcation & Chaos 89, 2005, 110-119.
[11] Sabetghadam, F., Jafarpour, A., Ghaffari, S.A., α Regularization of the POD-Galerkin Dynamical System of the Kuramoto-Sivashinsky Equation, Applied Mathematics and Computation 218 (10), 2012, 6012-6026.
[12] Sabetghadam, F., Moayyedi, M.K., and Taeibi-Rahni, M., A Fast Approach for Temporal Calibration of Low-dimensional Dynamical Model for Simulation of Unsteady Incompressible Flows, The 9th Annual Conference of Iranian Aerospace Society, Science & Research Branch IAU, Tehran, Iran, 2010.
[13] Sirisup, S., Karnidakis, G.E., A Spectral Viscosity Method for Correcting the Long Term Behavior of POD Models, J. Comp. Physics 194, 2004, 92-116.
[14] Luo, Z., Li, H., Sun, P., Gao, J., A Reduced-order Finite Difference Extrapolation Algorithm based on POD Technique for the non-stationary Navier–Stokes Equations, Applied Mathematical Modelling, 37(7), 2013, 5464-5473.
[15] Luo, Z., Proper Orthogonal Decomposition-based Reduced-order Stabilized Mixed Finite Volume Element Extrapolating Model for the Non-stationary Incompressible Boussinesq equations, J. Mathematical Analysis and Applications, 425(1), 2015, 259-280.
[16] Moayyedi, M.K., Calibration of Reduced Order POD Model of Unsteady Incompressible Laminar Flow Using Pressure Representation as a Function of Velocity Field Modes, Tabriz University Journal of Mechanical Engineering, 48(2), 2018, 349-358.