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Galerkin-based reduced-order models (G-ROMs) have provided efficient andaccurate approximations of laminar flows. In order to capture the complexdynamics of the turbulent flows, standard G-ROMs require a relatively largenumber of reduced basis functions (on the order of hundreds and eventhousands). Although the resulting G-ROM is still relatively low-dimensionalcompared to the full-order model (FOM), its computational cost becomesprohibitive due to the 3rd-order convection tensor contraction. The tensorrequires storage of $N^3$ entries with a corresponding work of $2N^3$operations per timestep, which makes such ROMs impossible to use in realisticapplications, such as control of turbulent flows. In this paper, we focus onthe scenario where the G-ROM requires large $N$ values and propose a novelapproach that utilizes the CANDECOMC/PARAFAC decomposition (CPD), a tensordecomposition technique, to accelerate the G-ROM by approximating the 3rd-orderconvection tensor by a sum of $R$ rank-1 tensors. In addition, we show that thetensor is partially skew-symmetric and derive two conditions for the CPdecomposition for preserving the skew-symmetry. Moreover, we investigate theG-ROM with the singular value decomposition (SVD). The G-ROM with CPdecomposition is investigated in several flow configurations from 2D periodicflow to 3D turbulent flows. Our numerical investigation shows CPD-ROM achievesat least a factor of 10 speedup. Additionally, the skew-symmetry preservingCPD-ROM is more stable and allows the usage of smaller rank $R$. Moreover, fromthe singular value behavior, the advection tensor formed using the $H^1_0$-PODbasis has a low-rank structure, and is preserved even in higher Reynoldsnumbers. Furthermore, for a given level of accuracy, the CP decomposition ismore efficient in size and cost than the SVD.

Accelerating the Galerkin Reduced-Order Model with the Tensor Decomposition for Turbulent Flows