A novel Gappy reduced order method to capture non-parameterized geometrical variation in fluid dynamics problems

Nissrine Akkari (1) , Fabien Casenave (2) , David Ryckelynck (3)

1 CDM - Centre des Matériaux
2 LAREG - LAboratoire de Recherche en Géodésie
3 LMSP - Laboratoire de Mécanique des Systèmes et des Procédés

In this work, we propose a new Gappy reduced order method to fill the gap within an incomplete turbulent and incompressible data field in such a way to satisfy the physical and topological changes of the fluid flow after a non-parameterized geometrical variation in the fluid domain 1. A single baseline simulation is assumed to be performed prior geometrical variations. The proposed method is an enhancement of the Gappy-POD method proposed by Everson and Sirovich in 1995, in the case where the given set of empirical eigenfunctions is not sufficient and is not interpolant for the recovering of the modal coefficients for each Gappy snapshot by a least squares procedure. This happens when the available data cannot be written as an interpolation of the baseline POD modes. This is typically the case when we introduce non-parameterized geometrical modifications in the fluid domain. Here, after the baseline simulation, additional solutions of the incompressible Navier-Stokes equations are solely performed over a restricted fluid domain, that contains the geometrical modifications. These local Large Eddy Similations that we will call hybrid simulations are performed by using the immersed boundary technique, where the latter is a fluid boundary and is defined by the baseline velocity field. Then, we propose to repair the POD modes using a local modification of the baseline POD modes in the restricted fluid domain. The modal coefficients of the least squares optimization of the Gappy-POD technique are now well recovered thanks to these updated modes, i.e. the residual of the Gappy-POD technique in the restricted fluid domain is now equal to zero. Furthermore, we will propose a physical correction of the latter enhanced Gappy-POD modal coefficients thanks to a Galerkin projection of the full Navier-Stokes equations upon the new compression modes of the available data. This repairing procedure of the global velocity reconstruction by the physical constraint was tested on a 3D semi-industrial test case of a typical aeronautical injection system. The speed-up relative to this new technique is equal to 100, which allows us to perform an exploration of two new designs of the aeronautical injection system.

Mots clés: Proper Orthogonal Decomposition (POD); novel Gappy reduced order method; Gappy-POD; Non-parameterized geometrical variation; hybrid approach; Locally available data; Galerkin projection; dynamic extrapolation; Navier-Stokes equations; Large Eddy Simulation (LES); eeronautical injection system; design exploration in industry; efficiency; robustness

Domaines: Mécanique des fluides [physics.class-ph]; Mathématiques [math]

Nissrine Akkari, Fabien Casenave, David Ryckelynck. A novel Gappy reduced order method to capture non-parameterized geometrical variation in fluid dynamics problems. 2019. ⟨hal-02344342v1⟩