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A Reduced Basis Technique for Long-Time Unsteady Turbulent Flows
Numerical Analysis · Submitted 10 Oct 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We illustrate our approach and we demonstrate its effectiveness by studying the dependence of a two-dimensional turbulent lid-driven cavity flow on the Reynolds number.
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The abstract, sentence by sentence
We present a reduced basis technique for long-time integration of parametrized incompressible turbulent flows. The new contributions are threefold. First, we propose a constrained Galerkin formulation that corrects the standard Galerkin statement by incorporating prior information about the long-time attractor. For explicit and semi-implicit time discretizations, our statement reads as a constrained quadratic programming problem where the objective function is the Euclidean norm of the error in the reduced Galerkin (algebraic) formulation, while the constraints correspond to bounds for the maximum and minimum value of the coefficients of the $N$-term expansion. Second, we propose an \emph{a posteriori} error indicator, which corresponds to the dual norm of the residual associated with the time-averaged momentum equation. We demonstrate that the error indicator is highly-correlated with the error in mean flow prediction, and can be efficiently computed through an offline/online strategy. Third, we propose a Greedy algorithm for the construction of an approximation space/procedure valid over a range of parameters; the Greedy is informed by the \emph{a posteriori} error indicator developed in this paper. We illustrate our approach and we demonstrate its effectiveness by studying the dependence of a two-dimensional turbulent lid-driven cavity flow on the Reynolds number.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present a reduced basis technique for long-time integration of parametrized incompressible turbulent flows. | contribution 0.97 | 0.89 | 0.15 | asserted 0.99 |
| s1 | The new contributions are threefold. | contribution 0.99 | 0.54 | 0.03 | asserted 0.85 |
| s2 | First, we propose a constrained Galerkin formulation that corrects the standard Galerkin statement by incorporating prior information about the long-time attractor. | contribution 0.98 | 0.94 | 0.09 | asserted 1.00 |
| s3 | For explicit and semi-implicit time discretizations, our statement reads as a constrained quadratic programming problem where the objective function is the Euclidean norm of the error in the reduced Galerkin (algebraic) formulation, while the constraints correspond to bounds for the maximum and minimum value of the coefficients of the $N$-term expansion. | method 0.67 | 0.92 | 0.05 | asserted 1.00 |
| s4 | Second, we propose an \emph{a posteriori} error indicator, which corresponds to the dual norm of the residual associated with the time-averaged momentum equation. | contribution 0.99 | 0.93 | 0.07 | asserted 0.97 |
| s5 | We demonstrate that the error indicator is highly-correlated with the error in mean flow prediction, and can be efficiently computed through an offline/online strategy. | result 0.95 | 0.95 | 0.17 | asserted 1.00 |
| s6 | Third, we propose a Greedy algorithm for the construction of an approximation space/procedure valid over a range of parameters; the Greedy is informed by the \emph{a posteriori} error indicator developed in this paper. | contribution 0.98 | 0.95 | 0.09 | asserted 1.00 |
| s7 | We illustrate our approach and we demonstrate its effectiveness by studying the dependence of a two-dimensional turbulent lid-driven cavity flow on the Reynolds number. | contribution 0.59 | 0.96 | 0.37 | asserted 1.00 |
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Original abstract
We present a reduced basis technique for long-time integration of parametrized incompressible turbulent flows. The new contributions are threefold. First, we propose a constrained Galerkin formulation that corrects the standard Galerkin statement by incorporating prior information about the long-time attractor. For explicit and semi-implicit time discretizations, our statement reads as a constrained quadratic programming problem where the objective function is the Euclidean norm of the error in the reduced Galerkin (algebraic) formulation, while the constraints correspond to bounds for the maximum and minimum value of the coefficients of the $N$-term expansion. Second, we propose an \emph{a posteriori} error indicator, which corresponds to the dual norm of the residual associated with the time-averaged momentum equation. We demonstrate that the error indicator is highly-correlated with the error in mean flow prediction, and can be efficiently computed through an offline/online strategy. Third, we propose a Greedy algorithm for the construction of an approximation space/procedure valid over a range of parameters; the Greedy is informed by the \emph{a posteriori} error indicator developed in this paper. We illustrate our approach and we demonstrate its effectiveness by studying the dependence of a two-dimensional turbulent lid-driven cavity flow on the Reynolds number.
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How the literature uses it
8 arXiv papers cite this in 10 sentences
Papers citing it · 5 sentences without a confident label
Cites it as background 4 papers
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Thus, many stabilized ROMs have been proposed over the years; see, e.g., [ref] for recent work on this topic.
Nonintrusive Stabilization of Reduced Order Models for Uncertainty Quantification of Time-Dependent Convection-Dominated Flows · Stabilized ROMs for Convection-Dominated Flows · 0.85
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These include stabilizing inner products applied to finite-difference [ref] and finite-element discretizations [ref] ; introducing dissipation via closure models [ref] or numerical dissipation [ref] ; performing nonlinear Galerkin projection based on approximate inertial manifolds [ref] ; including a pressure-term representation [ref] ; modifying the POD basis by including many modes (such that dissipative modes are captured), changing the norm [ref] , enabling adaptivity [ref] , or including basis functions that resolve a range of scales [ref] or respect the attractor's power balance [ref] ; modifying the projection by adopting a constrained Galerkin [ref] , constrained Petrov–Galerkin [ref] , or $L^1$-norm minimizing projection [ref] ; developing approaches tailored to the incompressible Navier–Stokes equations by introducing stabilizations based on supremizer-enriched velocity spaces
Conservative model reduction for finite-volume models · Introduction · 0.93
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Parametric ROMs can be used for evaluating solutions on new sets of parameter values or for time evolution that are different from those of the original simulations [ref] .
Development of a coupling between a system thermal-hydraulic code and a reduced order CFD model · Introduction · 0.88
What it cites · 3 sentences without a confident label
Compares with 1 paper
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We observe that a constrained formulation has been recently proposed in [ref] in the context of steady-state data assimilation: as in our work, the constraints in [ref] provide further information about the solution manifold; however, while in our work the constraints are designed to compensate for the effect of the unmodelled dynamics, in [ref] the constraints are designed to limit the effect of experimental noise.
Stabilization of (G)EIM in presence of measurement noise: application to nuclear reactor physics · Introduction · 0.98
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