Reduced-order model of a reacting, turbulent supersonic jet based on proper orthogonal decomposition

Flow data are often decomposed using proper orthogonal decomposition (POD) of the space–time separated form, $\boldsymbol {q}'\left (\boldsymbol {x},t\right )=\sum _j a_j\left (t\right )\boldsymbol {\phi }_j\left (\boldsymbol {x}\right )$ , which targets spatially correlated flow structures in an optimal manner. This paper analyses permuted POD (PPOD), which decomposes data as $\boldsymbol {q}'\left (\boldsymbol {x},t\right )=\sum _j a_j\left (\boldsymbol {n}\right )\boldsymbol {\phi }_j\left (s,t\right )$ , where $\boldsymbol {x}=(s,\boldsymbol {n})$ is a general spatial coordinate system, $s$ is the coordinate along the bulk advection direction and $\boldsymbol {n}=(n_1,n_2)$ are along mutually orthogonal directions normal to the advection characteristic. This separation of variables is associated with a fundamentally different inner product space for which PPOD is optimal and targets correlations in $s,t$ space. This paper presents mathematical features of PPOD, followed by analysis of three experimental datasets from high-Reynolds-number, turbulent shear flows: a wake, a swirling annular jet and a jet in cross-flow. In the wake and swirling jet cases, the leading PPOD and space-only POD modes focus on similar features but differ in convergence rates and fidelity in capturing spatial and temporal information. In contrast, the leading PPOD and space-only POD modes for the jet in cross-flow capture completely different features – advecting shear layer structures and flapping of the jet column, respectively. This example demonstrates how the different inner product spaces, which order the PPOD and space-only POD modes according to different measures of variance, provide unique ‘lenses’ into features of advection-dominated flows, allowing complementary insights.

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“…Alomar et al. 2020). In a cylindrical coordinate system, radial velocity components directed away from the centreline are positive.…”

Section: Overview Of Space-only Pod and Permuted Pod Propertiesmentioning

confidence: 98%

“…For example, consider the transverse velocity component associated with planar data acquired through the centre of a round jet (as in e.g. Alomar et al 2020). In a cylindrical coordinate system, radial velocity components directed away from the centreline are positive.…”

Section: Coordinate Transformationsmentioning

confidence: 99%

Permuted proper orthogonal decomposition for analysis of advecting structures

Nair

Douglas

et al. 2021

J. Fluid Mech.

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“…ǫ t and ǫ m , respectively, assess the error due to truncation of the projection basis to p POD modes and the error due to prediction of the time-evolution of the p retained states of the model. In [30], it was shown that the total error ǫ total can be deduced from ǫ t and ǫ m following ǫ total = ǫ t + ǫ m (1 − ǫ t ). Finally, we introduce the ratio ǫ S between the Frobenius norm of the symmetric part N S of the nonlinear term N and the Frobenius norm of…”

Section: Quality Measures Of Reduced-order Modelsmentioning

confidence: 99%

“…It is the objective of this paper to compare the performance of POD-DEIM models with the more traditional Galerkin methods. More specifically, we will study the convergence of the models as the number of POD modes increases; we are, however, not interested in the calibration of models of very small size or in data-driven regression techniques [27], based on sparsitypromoting techniques [28] or on linear models [29,30]. We also propose a new reduction technique for the nonlinear terms that takes advantage of a supplementary POD basis for the representation of the nonlinear terms: however, instead of using interpolation to determine the coefficients (like in the DEIM technique), we proceed straightforwardly by projecting the nonlinear terms onto this additional basis.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear model reduction: a comparison between POD-Galerkin and POD-DEIM methods

Sipp¹,

Pando²,

Schmid³

2020

Preprint

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Several nonlinear model reduction techniques are compared for the three cases of the non-parallel version of the Kuramoto-Sivashinsky equation, the transient regime of flow past a cylinder at Re = 100 and fully developed flow past a cylinder at the same Reynolds number. The linear terms of the governing equations are reduced by Galerkin projection onto a POD basis of the flow state, while the reduced nonlinear convection terms are obtained either by a Galerkin projection onto the same state basis, by a Galerkin projection onto a POD basis representing the nonlinearities or by applying the Discrete Empirical Interpolation Method (DEIM) to a POD basis of the nonlinearities. The quality of the reduced order models is assessed as to their stability, accuracy and robustness, and appropriate quantitative measures are introduced and compared. In particular, the properties of the reduced linear terms are compared to those of the full-scale terms, and the structure of the nonlinear quadratic terms is analyzed as to the conservation of kinetic energy. It is shown that all three reduction techniques provide excellent and similar results for the cases of the Kuramoto-Sivashinsky equation and the limit-cycle cylinder flow. For the case of the transient regime of flow past a cylinder, only the pure Galerkin techniques are successful, while the DEIM technique produces reduced-order models that diverge in finite time.

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“…ROMs have been applied extensively across many fields of engineering. Recent developments include the application of proper orthogonal decomposition (POD) to transient heat conduction, 7 turbulent supersonic jets, 8 acoustic waves, 9 incompressible magnetohydrodynamics, 10 and many other problems. POD‐based methods have been combined with neural networks to model plasticity, 11 unsteady flows in a combustion problem, 12 the viscous Burgers equation, 13 and structural damage 14 .…”

Section: Introductionmentioning

confidence: 99%

An adaptive reduced order model for the angular discretization of the Boltzmann transport equation using independent basis sets over a partitioning of the space‐angle domain

Hughes

Buchan

2022

Numerical Meth Engineering

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This article presents a new reduced order model (ROM) for the angular discretization of the Boltzmann transport equation. The angular ROM is built over a partitioning of the space-angle phase-space, by generating independent, optimized angular basis function sets for each partition. The advantage is that each basis function set is optimized to represent the neutron flux distribution in a particular partition of space and angle, rather than being optimized for the entire domain. This serves to reduce the total number of basis functions required, and therefore the solve time. Two numerical examples are presented to demonstrate the efficacy of the methods-a dog-leg duct problem, involving particle streaming, and the Watanabe-Maynard problem, which includes significant particle scattering. In both cases, it is shown that the method reduces the angular flux error, for a given basis size or solve time, by around an order of magnitude in comparison to other, similar ROM methods. An adaptive version of the method is also presented, whereby the number of basis functions within each space-angle partition can vary independently. It is shown to potentially provide further significant reductions in error.

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Reduced-order model of a reacting, turbulent supersonic jet based on proper orthogonal decomposition

Cited by 7 publications

References 42 publications

Permuted proper orthogonal decomposition for analysis of advecting structures

Permuted proper orthogonal decomposition for analysis of advecting structures

Nonlinear model reduction: a comparison between POD-Galerkin and POD-DEIM methods

An adaptive reduced order model for the angular discretization of the Boltzmann transport equation using independent basis sets over a partitioning of the space‐angle domain

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