A minimum residual projection to build coupled velocity-pressure POD-ROM for incompressible Navier-Stokes equations

Abstract The pressure term which appears in the ROM (reduced order model) associated to the incompressible Navier–Stokes equations, in particular for the shear flows, plays an important role on the velocity. The aim of this paper is to propose a Proper Orthogonal Decomposition based reduced order model (POD–ROM) to obtain both the velocity and pressure fields for incompressible flows. Two PODs are performed, one for the velocity and the other for the pressure. Contrary to existing projection methods available in the literature, the temporal velocity and pressure coefficients are sought by minimizing the residual of the momentum equation only, without the need of a Poisson equation. For the numerical test cases considered in this paper, the proposed minimum residual projection enables to obtain accurately the pressure field, and in turn to slightly improve the velocity one. The method is tested on two fluid flows: a transient mixed-convection flow and a periodic flow around a circular cylinder. In this last case, the drag, lift and pressure coefficients, as well as the Strouhal number are properly recovered compared to those of the full model.

[1]  D. Rempfer,et al.  Investigations of boundary layer transition via Galerkin projections on empirical eigenfunctions , 1996 .

[2]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[3]  John L. Lumley,et al.  A low-dimensional approach for the minimal flow unit of a turbulent channel flow , 1996 .

[4]  Cyrille Allery,et al.  Contribution à l'identification des bifurcations et à l'étude des écoulements fluides par des systèmes dynamiques d'ordre faible (P. O. D. ) , 2002 .

[5]  Jonathan C. Mattingly,et al.  Low-dimensional models of coherent structures in turbulence , 1997 .

[6]  Nadine Aubry,et al.  The dynamics of coherent structures in the wall region of a turbulent boundary layer , 1988, Journal of Fluid Mechanics.

[7]  Ali H. Nayfeh,et al.  On the stability and extension of reduced-order Galerkin models in incompressible flows , 2009 .

[8]  A. Hamdouni,et al.  On investigation of particle dispersion by a POD approach , 2008 .

[9]  Cédric Leblond,et al.  A priori space–time separated representation for the reduced order modeling of low Reynolds number flows , 2014 .

[10]  Roland W. Freund,et al.  Efficient linear circuit analysis by Pade´ approximation via the Lanczos process , 1994, EURO-DAC '94.

[11]  Traian Iliescu,et al.  A numerical investigation of velocity-pressure reduced order models for incompressible flows , 2014, J. Comput. Phys..

[12]  W.-H. Liu,et al.  Effect of temperature gradient orientation on the characteristics of mixed convection flow in a lid-driven square cavity , 2010 .

[13]  D. Ryckelynck,et al.  A priori hyperreduction method: an adaptive approach , 2005 .

[14]  G. Karniadakis,et al.  A spectral viscosity method for correcting the long-term behavior of POD models , 2004 .

[15]  Erwan Liberge,et al.  Reduced order modelling method via proper orthogonal decomposition (POD) for flow around an oscillating cylinder , 2010 .

[16]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[17]  Cédric Leblond,et al.  An optimal projection method for the reduced-order modeling of incompressible flows , 2011 .

[18]  Aziz Hamdouni,et al.  Numerical simulation of an oscillating cylinder in a cross-flow at low Reynolds number: Forced and free oscillations , 2009 .

[19]  S. Ravindran,et al.  A Reduced-Order Method for Simulation and Control of Fluid Flows , 1998 .

[20]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[21]  F. Archambeau,et al.  Code Saturne: A Finite Volume Code for the computation of turbulent incompressible flows - Industrial Applications , 2004 .

[22]  Max Gunzburger,et al.  POD and CVT-based reduced-order modeling of Navier-Stokes flows , 2006 .

[23]  Charles-Henri Bruneau,et al.  Enablers for robust POD models , 2009, J. Comput. Phys..

[24]  Janet S. Peterson,et al.  The Reduced Basis Method for Incompressible Viscous Flow Calculations , 1989 .

[25]  C. Norberg.  Flow around a Circular Cylinder: Aspects of Fluctuating Lift , 2001 .

[26]  Haecheon Choi,et al.  Suboptimal feedback control of vortex shedding at low Reynolds numbers , 1999, Journal of Fluid Mechanics.

[27]  Jean-Antoine Désidéri,et al.  Stability Properties of POD–Galerkin Approximations for the Compressible Navier–Stokes Equations , 2000 .

[28]  Aziz Hamdouni,et al.  Reduced‐order modelling for solving linear and non‐linear equations , 2011 .

[29]  M. Braza,et al.  Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder , 1986, Journal of Fluid Mechanics.

[30]  Bernd R. Noack,et al.  The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows , 2005, Journal of Fluid Mechanics.

[31]  Arthur Veldman,et al.  Proper orthogonal decomposition and low-dimensional models for driven cavity flows , 1998 .

[32]  R. Henderson.  Nonlinear dynamics and pattern formation in turbulent wake transition , 1997, Journal of Fluid Mechanics.

[33]  Charles-Henri Bruneau,et al.  Low-order modelling of laminar flow regimes past a confined square cylinder , 2004, Journal of Fluid Mechanics.

[34]  Eric James Grimme,et al.  Krylov Projection Methods for Model Reduction , 1997 .

[35]  S. S. Ravindran,et al.  Reduced-Order Adaptive Controllers for Fluid Flows Using POD , 2000, J. Sci. Comput..

[36]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[37]  P. Sagaut,et al.  Calibrated reduced-order POD-Galerkin system for fluid flow modelling , 2005 .