On the stability and extension of reduced-order Galerkin models in incompressible flows

Theoretical and Computational Fluid Dynamics - 2009
Imran Akhtar1,2, Ali H. Nayfeh1, Calvin J. Ribbens3
1Department of Engineering Science and Mechanics, MC 0219, Virginia Tech, Blacksburg, USA
2Interdisciplinary Center for Applied Mathematics, MC 0531, Virginia Tech, Blacksburg, USA
3Department of Computer Science Virginia Tech, Blacksburg, USA

Tóm tắt

Proper orthogonal decomposition (POD) has been used to develop a reduced-order model of the hydrodynamic forces acting on a circular cylinder. Direct numerical simulations of the incompressible Navier–Stokes equations have been performed using a parallel computational fluid dynamics (CFD) code to simulate the flow past a circular cylinder. Snapshots of the velocity and pressure fields are used to calculate the divergence-free velocity and pressure modes, respectively. We use the dominant of these velocity POD modes (a small number of eigenfunctions or modes) in a Galerkin procedure to project the Navier–Stokes equations onto a low-dimensional space, thereby reducing the distributed-parameter problem into a finite-dimensional nonlinear dynamical system in time. The solution of the reduced dynamical system is a limit cycle corresponding to vortex shedding. We investigate the stability of the limit cycle by using long-time integration and propose to use a shooting technique to home on the system limit cycle. We obtain the pressure-Poisson equation by taking the divergence of the Navier–Stokes equation and then projecting it onto the pressure POD modes. The pressure is then decomposed into lift and drag components and compared with the CFD results.

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Theoretical and Computational Fluid Dynamics

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