Data-based, reduced-order, dynamic estimator for reconstruction of nonlinear flows exhibiting limit-cycle oscillations

Abstract

We apply a data-based, linear dynamic estimator to reconstruct the velocity field from measurements at a single sensor point in the wake of an aerofoil. In particular, we consider a NACA0012 airfoil at Re = 600 and 16 • angle of attack. Under these conditions, the flow exhibits a vortex shedding limit cycle. A reduced order model (ROM) of the flow field is extracted using proper orthogonal decomposition (POD). Subsequently, a subspace system identification algorithm (N4SID) is applied to extract directly the estimator matrices from the reduced output of the system (the POD coefficients). We explore systematically the effect of the number of states of the estimator, the sensor location, the type of sensor measurements (one or both velocity components), and the number of POD modes to be recovered. When the signal of a single velocity component (in the stream wise or cross stream directions) is measured, the reconstruction of the first two dominant POD modes strongly depends on the sensor location. We explore this behaviour and provide a physical explanation based on the non-linear mode interaction and the spatial distribution of the modes. When however, both components are measured, the performance is very robust, and is almost independent of the sensor location when the optimal number of estimator states is used. Reconstruction of the less energetic modes is more difficult, but still possible.

Reconstruction of large‐scale flow structures in a stirred tank from limited sensor data

“…The location of the probe point is important for the quality of reconstruction 19 . In the present work, this point was chosen to be at the position of maximum turbulent kinetic energy and is indicated with a black dot (Point 1) in Figure 5.…”

“…where matrix A arises from the projection of the linear terms on φ i while the forcing term F(t) arises from the non-linear terms that appear in the right hand side of (2a) as well as the fact that we retain only a finite number of POD modes (for more details on the derivation of ( 8) see [19]). This equation represents a reduced-order model of the fluctuations.…”

“…We now define the state vector of the temporal POD coefficients and obtain its evolution equation by performing Galerkin projection, that is, by substituting the expansion () into the fluctuating equation set () and taking into account that the basis is divergence‐free and orthonormal. The resulting system takes the form where matrix arises from the projection of the linear terms on ϕ i while the forcing term F ( t ) arises from the nonlinear terms that appear in the right‐hand side of (Equation ) as well as the fact that we retain only a finite number of POD modes (for more details on the derivation of Equation () see Reference 19). This equation represents a reduced‐order model of the fluctuations.…”

“…The perturbations were assumed to have very small amplitude, so their evolution was governed by the Navier–Stokes equations, linearized around the Blasius velocity profile. The assumption of small perturbations was relaxed in Reference 19, where the method was applied to reconstruct nonlinear flows exhibiting limit‐cycle oscillations. In particular, the strongly periodic flow around a 2D aerofoil was considered.…”

Data-driven order reduction and velocity field reconstruction using neural networks: The case of a turbulent boundary layer

“…It is generally necessary to use an order reduction to derive some information that could be handled either to model or control the targeted flow. Data-driven methods are nowadays becoming more and more efficient and reliable even for fluid mechanics research 9,17 . Among successful applications, one can cite statistical learning 16 or machine learning 14,35 algorithms.…”