Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations
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 probl...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2019-04-01
|
| Series: | Mathematical and Computational Applications |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2297-8747/24/2/45 |
| _version_ | 1811283504439230464 |
|---|---|
| author | Nissrine Akkari Fabien Casenave Vincent Moureau |
| author_facet | Nissrine Akkari Fabien Casenave Vincent Moureau |
| author_sort | Nissrine Akkari |
| collection | DOAJ |
| description | 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2.2</mn> </mrow> </semantics> </math> </inline-formula> 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. |
| first_indexed | 2024-04-13T02:13:49Z |
| format | Article |
| id | doaj.art-90c1ee84a6954d1585a03ce479afec1e |
| institution | Directory Open Access Journal |
| issn | 2297-8747 |
| language | English |
| last_indexed | 2024-04-13T02:13:49Z |
| publishDate | 2019-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematical and Computational Applications |
| spelling | doaj.art-90c1ee84a6954d1585a03ce479afec1e2022-12-22T03:07:13ZengMDPI AGMathematical and Computational Applications2297-87472019-04-012424510.3390/mca24020045mca24020045Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes EquationsNissrine Akkari0Fabien Casenave1Vincent Moureau2Safran Tech, Modelling and Simulation, Rue des Jeunes Bois, Châteaufort, 78114 Magny-Les-Hameaux, FranceSafran Tech, Modelling and Simulation, Rue des Jeunes Bois, Châteaufort, 78114 Magny-Les-Hameaux, FranceCORIA, CNRS UMR 6614, Université et INSA de Rouen, Campus Universitaire du Madrillet, Saint Etienne du Rouvray, 76800 Rouen, FranceIn 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2.2</mn> </mrow> </semantics> </math> </inline-formula> 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.https://www.mdpi.com/2297-8747/24/2/45reduced order modeling (ROM)proper orthogonal decomposition (POD)enhanced PODa priori enrichmentmodal analysisstabilizationdynamic extrapolation |
| spellingShingle | Nissrine Akkari Fabien Casenave Vincent Moureau Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations Mathematical and Computational Applications reduced order modeling (ROM) proper orthogonal decomposition (POD) enhanced POD a priori enrichment modal analysis stabilization dynamic extrapolation |
| title | Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations |
| title_full | Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations |
| title_fullStr | Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations |
| title_full_unstemmed | Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations |
| title_short | Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations |
| title_sort | time stable reduced order modeling by an enhanced reduced order basis of the turbulent and incompressible 3d navier stokes equations |
| topic | reduced order modeling (ROM) proper orthogonal decomposition (POD) enhanced POD a priori enrichment modal analysis stabilization dynamic extrapolation |
| url | https://www.mdpi.com/2297-8747/24/2/45 |
| work_keys_str_mv | AT nissrineakkari timestablereducedordermodelingbyanenhancedreducedorderbasisoftheturbulentandincompressible3dnavierstokesequations AT fabiencasenave timestablereducedordermodelingbyanenhancedreducedorderbasisoftheturbulentandincompressible3dnavierstokesequations AT vincentmoureau timestablereducedordermodelingbyanenhancedreducedorderbasisoftheturbulentandincompressible3dnavierstokesequations |