Proper orthogonal decomposition-based reduced order modeling is a technique that can be used to develop low dimensional models of fluid flow. In this technique, the Navier-Stokes equations are projected onto a finite number of POD basis functions resulting in a system of ODEs that model the system. The overarching goal of this work is to determine the best methods of applying this technique to generate reliable models of fluid flow. The first chapter investigates some basic characteristics of the proper orthogonal decomposition using the Burgers equation as a surrogate model problem. In applying the POD to this problem, we found that the eigenvalue spectrum is affected by machine precision and this leads to non-phsical negative eigenvalues in the POD. To avoid this, we introduced a new method called deflation that gives positive eigenvalues, but has the disadvantage that the orthogonality of the POD modes is more affected by numerical precision errors. To reduce the size of eigenproblem of POD process, the well-known snapshot method was tested. It was found that the number of snapshots required to obtain an accurate eigenvalue spectrum was determined by the smallest time scale of the phenomenon. After resolving this time scale, the errors in the eigenvalues and modes drop rapidly then converge with second-order accuracy. After obtaing POD modes, the ROM error was assessed using two errors, the error of projection of the problem onto the POD modes (the out-plane error) and the error of the ROM in the space spanned by POD modes (the in-plane error). The numerical results showed not only is the in-plane error bounded by the out-plane error (in agreement with theory) but it actually converges faster than the out-of-plane error. The second chapter is dedicated to building a robust POD-ROM for long term simulation of Navier-Stokes equation. The ability of the POD method to decompose the simulation and the capability of POD-ROM to simulate a low and high Reynolds flow over a NACA0015 airfoil was studied. We observed that POD can be applied for low Reynolds flows successfully if a proper stabilization method is used. For the high Reynolds case, the convergence of the eigenvalues spectrum with respect to duration of time window from we observed that the number of modes needed to simulate a certain time window increases almost linearly with the length of the time window. So, generating a POD-ROM for high Reynolds flow that reproduced the correct long-term limit cycle behavior needs many more modes than has been usually used in the literature. In the last chapter, we address the problem that the standard method of generating POD modes may be inaccurate when used "off-design" (at parameter values not used to generate the POD). We tested some of the popular methods developed to remedy that problem. The accuracy of these methods was in direct relation with the amount of data provided for those methods. So, in order to generate appropriate POD modes, very large POD problems must be solved. To avoid this, a new multi-level method, called recursive POD, for enriching the POD modes is introduced that mathematically provides optimal POD modes while reducing the computational size of problem to a manageable degrees. A low Reynolds flow over NACA 0015, actuated with constant suction/blowing of a fluidic jet located on top surface of airfoil is used as benchmark to test the technique. The flow is shifted from one periodic state to another periodic state due to fluidic jet effect. It was found that the modes extracted with the recursive POD method are as accurate as the modes of the best known method, global POD, while the computational effort is lower.
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