On the stability and extension of reduced-order Galerkin models in incompressible flows |
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Authors: | Imran Akhtar Ali H Nayfeh Calvin J Ribbens |
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Institution: | 1. Department of Engineering Science and Mechanics, MC 0219, Virginia Tech, Blacksburg, VA, 24061, USA 2. Interdisciplinary Center for Applied Mathematics, MC 0531, Virginia Tech, Blacksburg, VA, 24061, USA 3. Department of Computer Science, Virginia Tech, Blacksburg, VA, 24061, USA
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Abstract: | 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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