Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison

Wang, Zhu; Akhtar, Imran; Borggaard, Jeff; Iliescu, Traian

doi:10.1016/j.cma.2012.04.015

Cited by 289 publications

(341 citation statements)

References 71 publications

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“…964 × 10 4 , Pr = 712 × 10 −3 , and Gr = 7 . 369 × 10 7 , reasonable values in a quiet room. The simulation was run from zero velocity and temperature and snapshots were collected to t f = 78 [s] seconds.…”

Section: Boussinesq Equation Mes-based Pod Rom Stabilizationmentioning

confidence: 99%

Learning-based robust stabilization for reduced-order models of 2D and 3D Boussinesq equations

Benosman¹,

Borggaard²,

San³

et al. 2017

Applied Mathematical Modelling

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a b s t r a c tWe present some results on the stabilization of reduced-order models (ROMs) for thermal fluids. The stabilization is achieved using robust Lyapunov control theory to design a new closure model that is robust to parametric uncertainties. Furthermore, the free parameters in the proposed ROM stabilization method are optimized using a data-driven multiparametric extremum seeking (MES) algorithm. The 2D and 3D Boussinesq equations provide challenging numerical test cases that are used to demonstrate the advantages of the proposed method.

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Section: Boussinesq Equation Mes-based Pod Rom Stabilizationmentioning

confidence: 99%

Learning-based robust stabilization for reduced-order models of 2D and 3D Boussinesq equations

Benosman¹,

Borggaard²,

San³

et al. 2017

Applied Mathematical Modelling

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“…Recent works improving upon POD have dealt with the combination of Galerkin strategies with POD [3,8], stabilization techniques [1,7,29], and regularized/large eddy simulation POD models for turbulent flows [32,33].…”

Section: Introductionmentioning

confidence: 99%

An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier--Stokes Equations

Gunzburger¹,

Jiang²,

Schneier³

2017

SIAM J. Numer. Anal.

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Abstract. The definition of partial differential equation (PDE) models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., optimization, control, uncertainty quantification, and other settings, solutions are needed for many sets of parameter values. We consider the case of the time-dependent Navier-Stokes equations for which a recently developed ensemble-based method allows for the efficient determination of the multiple solutions corresponding to many parameter sets. The method uses the average of the multiple solutions at any time step to define a linear set of equations that determines the solutions at the next time step. To significantly further reduce the costs of determining multiple solutions of the Navier-Stokes equations, we incorporate a proper orthogonal decomposition (POD) reduced-order model into the ensemble-based method. The stability and convergence results for the ensemble-based method are extended to the ensemble-POD approach. Numerical experiments are provided that illustrate the accuracy and efficiency of computations determined using the new approach.Key words. Ensemble methods, proper orthogonal decomposition, reduced-order models, Navier-Stokes equations.1. Introduction. Computing an ensemble of solutions of fluid flow equations for a set of parameters or initial/boundary conditions for, e.g., quantifying uncertainty or sensitivity analyses or to make predictions, is a common procedure in many engineering and geophysical applications. One common problem faced in these calculations is the excessive cost in terms of both storage and computing time. Thanks to recent rapid advances in parallel computing as well as intensive research in ensemble-based data assimilation, it is now possible, in certain settings, to obtain reliable ensemble predictions using only a small set of realizations. Successful methods that are currently used to generate perturbations in initial conditions include the Bred-vector method, [30], the singular vector method, [5], and the ensemble transform Kalman filter, [4]. Despite all these efforts, the current level of available computing power is still insufficient to perform high-accuracy ensemble computations for applications that deal with large spatial scales such as numerical weather prediction. In such applications, spatial resolution is often sacrificed to reduce the total computational time. For these reasons the development of efficient methods that allow for fast calculation of flow ensembles at a sufficiently fine spatial resolution is of great practical interest and significance.Only recently, a first step was taken in [22,23] where a new algorithm was proposed for computing an ensemble of solutions of the time-dependent Navier-Stokes equations (NSE) with different initial condition and/or body forces. At each time step, the new method employs the same coefficient matrix for all ensemble members. This reduces the problem of solving multi...

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“…Noack et al (2011) derive a nonlinear eddy viscosity model, based on a Finite-Time Thermodynamics (FTT) closure . Nonlinear models based on the Galerkin projection of filtered Navier-Stokes equations have been pursued by Wang et al (2011Wang et al ( , 2012). An approach of completely different nature is suggested by Balajewicz et al (2013).…”

Section: Introductionmentioning

confidence: 99%

On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body

et al. 2014

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We investigate a hierarchy of eddy-viscosity terms in POD Galerkin models to account for a large fraction of unresolved fluctuation energy. These Galerkin methods are applied to Large Eddy Simulation data for a flow around the vehicle-like bluff body called Ahmed body. This flow has three challenges for any reduced-order model: a high Reynolds number, coherent structures with broadband frequency dynamics, and meta-stable asymmetric base flow states. The Galerkin models are found to be most accurate with modal eddy viscosities as proposed by Rempfer & Fasel (1994). Robustness of the model solution with respect to initial conditions, eddy viscosity values and model order is only achieved for state-dependent eddy viscosities as proposed by Noack, Morzyński & Tadmor (2011). Only the POD system with state-dependent modal eddy viscosities can address all challenges of the flow characteristics. All parameters are analytically derived from the Navier-Stokes based balance equations with the available data. We arrive at simple general guidelines for robust and accurate POD models which can be expected to hold for a large class of turbulent flows.

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Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison

Cited by 289 publications

References 71 publications

Learning-based robust stabilization for reduced-order models of 2D and 3D Boussinesq equations

Learning-based robust stabilization for reduced-order models of 2D and 3D Boussinesq equations

An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier--Stokes Equations

On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body

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