scholarly journals Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations

2019 ◽  
Vol 24 (2) ◽  
pp. 45 ◽  
Author(s):  
Nissrine Akkari ◽  
Fabien Casenave ◽  
Vincent Moureau
Keyword(s):  
Stokes Equations ◽  
A Priori ◽  
Order Reduction ◽  
Reduced Order Modeling ◽  
Reduced Order Model ◽  
Navier Stokes ◽  
Reduced Basis ◽  
Order Model ◽  
Navier Stokes Equations ◽  
Reduced Order

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.

Using Functional Gains for Optimal Sensor Placement in Fluid-Structure Interaction

2009 ◽  
Author(s):  
Imran Akhtar ◽  
Jeff Borggaard ◽  
John A. Burns ◽  
Lizette Zietsman
Keyword(s):  
Differential Equations ◽  
Stokes Equations ◽  
Reduced Order Model ◽  
Navier Stokes ◽  
Cylinder Surface ◽  
Mean Flow ◽  
Order Model ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Functional Gains

Functional gains are integral kernels of the standard feedback operator and are useful in control of partial differential equations (PDEs). These functional gains provide physical insight into how the control mechanism is operating. In some cases, these functional gains can provide information about the optimal placement of actuators and sensors. The study is motivated by fluid flow control and focuses on the computation of these functions. However, for practical purposes, one must be able to compute these functions for a wide variety of PDEs. For higher dimensional systems, computing these gains is at least as challenging as the original simulation problem. To reduce the complexity of the governing equations, reduced-order models are often developed by reducing the PDEs to ordinary-differential equations (ODEs). In this study, we use proper orthogonal decomposition (POD)-Galerkin based approach and develop a reduced-order model of a bluff body wake. We solve the incompressible Navier-Stokes equations, simulate the flow past a circular cylinder, and record the snapshots of the flow field. We compute the POD eigenfunctions and project the Navier-Stokes equations onto these few of these eigenfunctions to develop a reduced-order model. Later, we modify the model by introducing a control function simulating suction actuation on the cylinder surface. We linearize the model about the mean flow and apply feedback control to suppress vortex shedding. We then compute the functional gains for the applied control. We identify these gains at various stations in the wake region and suggest optimum locations for the sensors.

Projection-based and neural-net reduced order model for nonlinear Navier–Stokes equations

2021 ◽  
Vol 89 ◽  
pp. 1294-1315
Author(s):  
My Ha Dao ◽  
Hoang Huy Nguyen ◽  
Chin Chun Ooi ◽  
Quang Tuyen Le
Keyword(s):  
Stokes Equations ◽  
Reduced Order Model ◽  
Navier Stokes ◽  
Neural Net ◽  
Order Model ◽  
Navier Stokes Equations ◽  
Reduced Order

scholarly journals Reduced order methods for nonlinear parametric problems with branching solutions

2018 ◽  
Author(s):  
Federico Pichi ◽  
Martin Wilfried Hess ◽  
Annalisa Quaini ◽  
Gianluigi Rozza
Keyword(s):  
Stokes Equations ◽  
Spectral Element ◽  
Navier Stokes ◽  
Reduced Basis ◽  
State Equations ◽  
Order Error ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Plate Equations ◽  
Basis Method

The aim of this work is to show the applicability of the reduced basis model reduction in nonlinear systems undergoing bifurcations. Bifurcation analysis, i.e., following the different bifurcating branches, as well as determining the bifurcation point itself, is a complex computational task. Reduced Order Models (ROM) can potentially reduce the computational burden by several orders of magnitude, in particular in conjunction with sampling techniques. In the first task we focus on nonlinear structural mechanics, and we deal with an application of ROM to Von Kármán plate equations, where the buckling effect arises, adopting reduced basis method. Moreover, in the search of the bifurcation points, it is crucial to supplement the full problem with a reduced generalized parametric eigenvalue problem, properly paired with state equations and also a reduced order error analysis. These studies are carried out in view of vibroacoustic applications (in collaboration with A.T. Patera at MIT). As second task we consider the incompressible Navier-Stokes equations, discretized with the spectral element method, in a channel and a cavity. Both system undergo bifurcations with increasing Reynolds - and Grashof - number, respectively. Applications of this model are contraction-expansion channels, found in many biological systems, such as the human heart, for instance, or crystal growth in cavities, used in semiconductor production processes. This last task is in collaboration with A. Alla and M. Gunzburger (Florida State University).

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