Abstract
In the following paper, we consider the problem of constructing a time stable reduced order model of the 3D turbulent and incompressible Navier–Stokes equations. The lack of stability associated with the order reduction methods of the Navier–Stokes equations is a well-known problem and, in general, it is very difficult to account for different scales of a turbulent flow in the same reduced space. To remedy this problem, we propose a new stabilization technique based on an a priori enrichment of the classical proper orthogonal decomposition (POD) modes with dissipative modes associated with the gradient of the velocity fields. The main idea is to be able to do an a priori analysis of different modes in order to arrange a POD basis in a different way, which is defined by the enforcement of the energetic dissipative modes within the first orders of the reduced order basis. This enables us to model the production and the dissipation of the turbulent kinetic energy (TKE) in a separate fashion within the high ranked new velocity modes, hence to ensure good stability of the reduced order model. We show the importance of this a priori enrichment of the reduced basis, on a typical aeronautical injector with Reynolds number of 45,000. We demonstrate the capacity of this order reduction technique to recover large scale features for very long integration times (25 ms in our case). Moreover, the reduced order modeling (ROM) exhibits periodic fluctuations with a period of 2 . 2 ms corresponding to the time scale of the precessing vortex core (PVC) associated with this test case. We will end this paper by giving some prospects on the use of this stable reduced model in order to perform time extrapolation, that could be a strategy to study the limit cycle of the PVC.
Highlights
Reduced order modeling (ROM) of the complete Navier–Stokes equations by the projection of these equations upon a reduced order space, that is generated by a minimal number of spatial modes, is still an attractive research area especially when we consider turbulent flows such as the ones encountered in aeronautical engines and that feature a large range of scales
Reduced order modeling by proper orthogonal decomposition (POD) is a very good candidate for solving such problems
We are interested in the stabilization of POD-Galerkin reduced order modeling for the turbulent and incompressible Navier–Stokes equations and we propose a new stabilization and purely physical approach for the POD-Galerkin approximation of the turbulent and incompressible Navier–Stokes equations
Summary
Reduced order modeling (ROM) of the complete Navier–Stokes equations by the projection of these equations upon a reduced order space, that is generated by a minimal number of spatial modes, is still an attractive research area especially when we consider turbulent flows such as the ones encountered in aeronautical engines and that feature a large range of scales. This method was tested for computational fluid dynamics-based model of a linearized unsteady supersonic flow, the reduction of a computational structural dynamic system, and the stabilization of the reduction of a coupled computational fluid dynamics–computational structural dynamics model of a linearized aeroelastic system in the transonic flow regime Besides these reduced order techniques which are intrusive for the computational fluid dynamics physics, there are some new non-intrusive reduced order ones, as they rely only on the available snapshots data, without taking into account all the equations of physics as constraints, but rather some physical properties as the turbulent kinetic energy conservation by learning the orthogonal projection coefficients of the solutions over a POD basis using a metamodel or a neural network, see the work of Wang et al [14]. The enhanced ROM enables us to filter the PVC throughout the reduced order simulation even for large values of Q-criterion and for very long integration times
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