Data-driven closure of projection-based reduced order models for unsteady compressible flows

Abstract

A data-driven closure modeling based on proper orthogonal decomposition (POD) temporal modes is used to obtain stable and accurate reduced order models (ROMs) of unsteady compressible flows. Model reduction is obtained via Galerkin and Petrov–Galerkin projection of the non-conservative compressible Navier–Stokes equations. The latter approach is implemented using the least-squares Petrov–Galerkin (LSPG) technique and the present methodology allows pre-computation (i.e., not requiring hyper-reduction) of both Galerkin and LSPG coefficients. Closure is performed by adding linear and non-linear coefficients to the original ROMs and minimizing the error with respect to the POD temporal modes. In order to further reduce the computational cost of the ROMs, an accelerated greedy missing point estimation (MPE) hyper-reduction method is employed. A canonical compressible cylinder flow is first analyzed and serves as a benchmark. The second problem studied consists of the turbulent flow over a plunging airfoil undergoing deep dynamic stall. For both cases, regularization is required and an iterative Tikhonov methodology is proposed. For the first case, linear and non-linear closure coefficients are both low in intrusiveness, capable of providing results in excellent agreement with the full order model. Regularization of calibrated models is also straightforward for this case. On the other hand, the dynamic stall flow is significantly more challenging, specially when only linear coefficients are used. Results show that non-linear calibration coefficients outperform their linear counterparts when a POD basis with fewer modes is used in the reconstruction. However, determining a correct level of regularization is more complicated with non-linear coefficients. Hyper-reduced models show good results when combined with non-linear calibration and an appropriate sized POD basis.