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Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier–Stokes equations

Journal Article · · Journal of Computational Physics
 [1];  [2];  [3]
  1. Univ. of Illinois at Urbana-Champaign, Urbana, IL (United States)
  2. Sandia National Lab. (SNL-CA), Livermore, CA (United States)
  3. Duke Univ., Durham, NC (United States)
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For a projection-based reduced order model (ROM) of a fluid flow to be stable and accurate, the dynamics of the truncated subspace must be taken into account. This paper proposes an approach for stabilizing and enhancing projection-based fluid ROMs in which truncated modes are accounted for a priori via a minimal rotation of the projection subspace. Attention is focused on the full non-linear compressible Navier–Stokes equations in specific volume form as a step toward a more general formulation for problems with generic non-linearities. Unlike traditional approaches, no empirical turbulence modeling terms are required, and consistency between the ROM and the Navier–Stokes equation from which the ROM is derived is maintained. Mathematically, the approach is formulated as a trace minimization problem on the Stiefel manifold. As a result, the reproductive as well as predictive capabilities of the method are evaluated on several compressible flow problems, including a problem involving laminar flow over an airfoil with a high angle of attack, and a channel-driven cavity flow problem.

Research Organization:
Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA)
Grant/Contract Number:
AC04-94AL85000
OSTI ID:
1333869
Report Number(s):
SAND--2015-3378J; PII: S0021999116301826
Journal Information:
Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: C Vol. 321; ISSN 0021-9991
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English

References (48)

Stabilization of projection-based reduced-order models: STABILIZATION OF PROJECTION-BASED REDUCED-ORDER MODELS journal February 2012
Stability Properties of POD-Galerkin Approximations for the Compressible Navier-Stokes Equations journal March 2000
Stabilization of projection-based reduced order models of the Navier–Stokes journal August 2012
On minimization on Stiefel manifolds journal December 2002
On self-sustained oscillations in two-dimensional compressible flow over rectangular cavities journal March 2002
Low-order modelling of laminar flow regimes past a confined square cylinder journal January 1999
Dynamic mode decomposition of numerical and experimental data journal July 2010
Dynamics of three-dimensional coherent structures in a flat-plate boundary layer journal September 1994
Proper orthogonal decomposition and low-dimensional models for driven cavity flows journal July 1998
A Survey of Model Reduction by Balanced Truncation and Some New Results journal May 2004
Turbulence and the dynamics of coherent structures. III. Dynamics and scaling journal January 1987
Symmetric Gauge Functions and Unitarily Invariant Norms journal January 1960
Balanced model reduction via the proper orthogonal decomposition conference August 2001
Reduced Basis Approximation and Error Bounds for Potential Flows in Parametrized Geometries journal January 2011
Reduction of Nonlinear Embedded Boundary Models for Problems with Evolving Interfaces text January 2014
Optimal Nonlinear Eddy Viscosity in Galerkin Models of Turbulent Flows text January 2014
Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basisa posteriori error bounds journal January 2005
Variational multiscale proper orthogonal decomposition: Navier-stokes equations: VARIATIONAL MULTISCALE POD journal December 2013
Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten journal December 1935
The approximation of one matrix by another of lower rank journal September 1936
libMesh : a C++ library for parallel adaptive mesh refinement/coarsening simulations journal November 2006
On minimization on Stiefel manifolds journal December 2002
Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison journal September 2012
Stabilization of projection-based reduced order models for linear time-invariant systems via optimization-based eigenvalue reassignment journal April 2014
Studies of shock/turbulent shear layer interaction using Large-Eddy Simulation journal May 2010
Goal-oriented, model-constrained optimization for reduction of large-scale systems journal June 2007
Stable Galerkin reduced order models for linearized compressible flow journal April 2009
Reliable reduced-order models for time-dependent linearized Euler equations journal June 2012
The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows journal June 2013
Reduction of nonlinear embedded boundary models for problems with evolving interfaces journal October 2014
Model reduction for compressible flows using POD and Galerkin projection journal February 2004
LES of scalar mixing in supersonic mixing layers journal January 2005
On self-sustained oscillations in two-dimensional compressible flow over rectangular cavities journal March 2002
Low-order modelling of laminar flow regimes past a confined square cylinder journal January 1999
Spectral analysis of nonlinear flows journal November 2009
Dynamic mode decomposition of numerical and experimental data journal July 2010
The dynamics of coherent structures in the wall region of a turbulent boundary layer journal July 1988
Dynamics of three-dimensional coherent structures in a flat-plate boundary layer journal September 1994
Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation journal July 2013
On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body journal April 2014
Principal component analysis in linear systems: Controllability, observability, and model reduction journal February 1981
A matrix inequality associated with bounds on solutions of algebraic Riccati and Lyapunov equations journal August 1987
Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics journal January 2015
The MATLAB ODE Suite journal January 1997
Model Reduction for Fluids, Using Balanced Proper Orthogonal Decomposition journal March 2005
An overview of the Trilinos project journal September 2005
A Finite-Time Thermodynamics of Unsteady Fluid Flows journal January 2008
Balanced Model Reduction via the Proper Orthogonal Decomposition journal November 2002

Cited By (8)

Physically constrained data‐driven correction for reduced‐order modeling of fluid flows journal October 2018
A hybrid stabilization approach for reduced‐order models of compressible flows with shock‐vortex interaction journal December 2019
Energy balance of reduced order models for unsteady flows using sparse coding journal August 2018
Construction of reduced-order models for fluid flows using deep feedforward neural networks journal June 2019
Data-driven construction of a reduced-order model for supersonic boundary layer transition journal July 2019
Assessment of reduced-order modeling strategies for convective heat transfer journal January 2020
Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations journal April 2019
Construction of Reduced Order Models for Fluid Flows Using Deep Feedforward Neural Networks text January 2019

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Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
Proper Orthogonal Decomposition (POD)
Stiefel manifold
compressible flow
projection-based reduced order model (ROM)
stabilization
trace minimization