Optimal flow control using a POD-based reduced-order model

ABSTRACT

The flow control constitutes a challenging research topic with numerous applications in aeronautics, aerodynamics, fluid mechanics, civil engineering, and so on. Due to the use of iterative algorithms, it is costly in both CPU time and memory requirements. In this paper, an optimal control strategy with reduced-order model built with proper orthogonal decomposition approach is proposed. The methodology is developed to control an anisothermal flow in a lid-driven cavity with one control parameter and then extended to at two control parameters. The control is realized in quasi-real time, which opens the way to promising practical applications.

Notes

¹The parameters can be Reynolds number, Richardson number, Grashof number, …

²ω₁ and ω₂ are penalty terms of the objective functional to regularize the optimization problem [47].

³The target temporal coefficient is associated to the target temperature field as:

(16)

It is obtained with the target control parameter α_tar.

⁴The detail of the calculations is not given here. The reader can refer to the works of Joslin et al. [48], in which the same methodology is employed on the momentum and mass conservation equations.

⁵The RIC is defined such as:

Table 1. Reconstruction POD error as a function of the POD mode number.

(29)

where λ are the eigenvalues, N^POD are the number of POD modes and N^snap the number of snapshots.

⁶For this, new simulations with Code_Saturne have been realized.

⁷The classical method is a classical POD–ROM; the mean velocity field is decomposed according to the control function method ([33, 34, 36, 37]):

(32)

u_s(x) corresponds to the solutions at large time of Navier–Stokes equations, i.e., the velocity snapshot with which the POD basis has been built and at the final time. Since , α and u_s(x) are known, it is straightforward to compute defined as .

⁸For this, news simulations with Code_Saturne have been realized.

⁹They must therefore be updated at each algorithm iteration.