Approximate basis and a priori closure models for energy-conserving reduced order modeling of fluid flows

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Model order reduction (MOR) has been a field of active research in the past twenty years, more recently also in fluid dynamics. The main advantage of MOR is computational cost reduction, which, along with equally important accuracy, constitute main objective in the MOR community.
A main ongoing issue is that data volumes in fluid flow simulations (such as turbulent flows) are usually very large, hence processing is costly. An example of a high-cost operation in MOR is the construction of reduced basis (RB) via singular value decomposition (SVD) of snapshot data of turbulent flows. The present research, in its first objective, aims at tackling this problem by applying and adapting an incremental SVD algorithm (iSVD). The procedure does not require simultaneous access to the entire snapshot matrix, but
the price to pay is that accurate approximations of RB via iSVD are obtained only for low-index. ROMs require exactly those, therefore application of iSVD is plausible. The algorithm is tested on high-fidelity data representing transitional and turbulent flow solutions, obtained with an energy-conserving code (INS3D). Important iSVD paramters are identified and their influence on key properties of RB: orthogonality, zero-divergence and fidelity w.r.t. conventional SVD basis is examined.

The second objective concerns closure modeling. MOR by definition neglects a part of information.
Hence inaccuracies and/or instabilities often develop in the reduced order model (ROM) solution. The applied ROM framework is energy-conserving (EC-ROM), thereby ensuring non-linear stability. Accuracy is not guaranteed, therefore a correction is desirable. In ROM context several strategies exist. In the present research one such strategy, dissipation via a closure term, is examined in an `a priori' test. Based on the full order model (FOM) data and projection of it onto the reduced space, exact expression for missing information (exact closure term) is derived. Subsequently, an eddy viscosity (EV) ansatz is applied, whereby also high-fidelity data is used to compute EV. The turbulence model is of mixing-length type.
The related turbulent diffusive term with variable EV is regressed on the exact closure term.

It is concluded that iSVD is a feasible algorithm in MOR applications, particularly in combination with EC-ROM, provided that parameters of iSVD: increment size, maximum dimension of RB and threshold are far from their lower bounds. EV mixing length model is considered inadequate as a means of improving accuracy of EC-ROM in periodic shear-layer.