Classical Proper Orthogonal Decomposition Research Articles

Direct numerical simulations of an airfoil undergoing dynamic stall at different background disturbance levels

Thin airfoil dynamic stall at moderate Reynolds numbers is typically linked to the sudden bursting of a small laminar separation bubble close to the leading edge. Given the strong sensitivity of laminar separation bubbles to external disturbances, the onset of dynamic stall on a NACA0009 airfoil section subject to different levels of low-amplitude free stream disturbances is investigated using direct numerical simulations. The flow is practically indistinguishable from clean inflow simulations in the literature for turbulence intensities at the leading edge of ${Tu} = 0.02\,\%$ . At slightly higher turbulence intensities of ${Tu} = 0.05\,\%$ , the bursting process is found to be considerably less smooth and strong coherent vortex shedding from the laminar separation bubble is observed prior to the formation of the dynamic stall vortex (DSV). This phenomenon is considered in more detail by analysing its appearance in an ensemble of simulations comprising statistically independent realisations of the flow, thus proving its statistical relevance. In order to extract the transient dynamics of the vortex shedding, the classical proper orthogonal decomposition method is generalised to include time in the energy measure and applied to the time-resolved simulation data of incipient dynamic stall. Using this technique, the dominant transient spatiotemporally correlated features are distilled and the wave train of the vortex shedding prior to the emergence of the main DSV is reconstructed from the flow data exhibiting dynamics of large-scale coherent growth and decay within the turbulent boundary layer.

Model order reduction for the 1D Boltzmann-BGK equation: identifying intrinsic variables using neural networks

Kinetic equations are crucial for modeling non-equilibrium phenomena, but their computational complexity is a challenge. This paper presents a data-driven approach using reduced order models (ROM) to efficiently model non-equilibrium flows in kinetic equations by comparing two ROM approaches: proper orthogonal decomposition (POD) and autoencoder neural networks (AE). While AE initially demonstrate higher accuracy, POD’s precision improves as more modes are considered. Notably, our work recognizes that the classical POD model order reduction approach, although capable of accurately representing the non-linear solution manifold of the kinetic equation, may not provide a parsimonious model of the data due to the inherently non-linear nature of the data manifold. We demonstrate how AEs are used in finding the intrinsic dimension of a system and to allow correlating the intrinsic quantities with macroscopic quantities that have a physical interpretation.

Nonlinear model order reduction for problems with microstructure using mesh informed neural networks

Many applications in computational physics involve approximating problems with microstructure, characterized by multiple spatial scales in their data. However, these numerical solutions are often computationally expensive due to the need to capture fine details at small scales. As a result, simulating such phenomena becomes unaffordable for many-query applications, such as parametrized systems with multiple scale-dependent features. Traditional projection-based reduced order models (ROMs) fail to resolve these issues, even for second-order elliptic PDEs commonly found in engineering applications. To address this, we propose an alternative nonintrusive strategy to build a ROM, that combines classical proper orthogonal decomposition (POD) with a suitable neural network (NN) model to account for the small scales. Specifically, we employ sparse mesh-informed neural networks (MINNs), which handle both spatial dependencies in the solutions and model parameters simultaneously. We evaluate the performance of this strategy on benchmark problems and then apply it to approximate a real-life problem involving the impact of microcirculation in transport phenomena through the tissue microenvironment.

Fast Prediction of the Temperature Field Surrounding a Hot Oil Pipe Using the POD-BP Model

The heat transfer assessment of a buried hot oil pipe is essential for the economical and safe transportation of the pipeline, where the basis is to determine the temperature field surrounding the pipe quickly. This work proposes a novel method to efficiently predict the temperature field surrounding a hot oil pipe, which combines the proper orthogonal decomposition (POD) method and the backpropagation (BP) neural network, named the POD-BP model. Specifically, the BP neural network is used to establish the mapping relationship between spectrum coefficients and the preset parameters of the sample. Compared with the classical POD reduced-order model, the POD-BP model avoids solving the system of reduced-order governing equations with spectrum coefficients as variables, thus improving the prediction speed. Another advantage is that it is easy to implement and does not require tremendous mathematical derivation of reduced-order governing equations. The POD-BP model is then used to predict the temperature field surrounding the hot oil pipe, and the sample matrix is obtained from the numerical results using the finite volume method (FVM). In validation cases, both steady and unsteady states are investigated, and multiple boundary conditions, thermal properties, and even geometry parameters (different buried depths and pipe diameters) are tested. The mean errors of steady and unsteady cases are 0.845~3.052% and 0.133~1.439%, respectively. Appealingly, almost no time, around 0.008 s, is consumed in predicting unsteady situations using the proposed POD-BP model, while the FVM requires a computational time of 70 s.

Modal Decomposition Study of the Non-Reactive Flow Field in a Dual-Swirl Combustor

The modal decomposition study of the non-reactive flow field in a dual-swirl combustor is investigated through the large eddy simulation. The formation mechanism and function of various recirculation zones are elaborated by analyzing the time-averaged and instantaneous velocity contours of the center section. The precessing vortex core (PVC) is first visualized by the pressure iso-surface, and the evolution process is presented. Different dimensionality reduction methods are adopted to identify the coherent structures from the flow field. The most energetic spatial structure corresponding to the PVC and its second-order harmonic structure is extracted by the classical proper orthogonal decomposition (POD). The coherent structures with high frequency have relatively low energy content. In addition, a spectral proper orthogonal decomposition (SPOD) method, which can implement spatial-temporal decomposition simultaneously, is introduced to obtain the energy-based spatial structures at all characteristic frequencies. A triple-helix with azimuth wave number m = 3 and a quadruple-helix with azimuth wave number m = 4 are discovered as the third-order and the fourth-order harmonics of single-helix, respectively.

A hybrid projection/data-driven reduced order model for the Navier-Stokes equations with nonlinear filtering stabilization

We develop a Reduced Order Model (ROM) for the Navier-Stokes equations with nonlinear filtering stabilization. Our approach, that can be interpreted as a Large Eddy Simulation model, combines a three-step algorithm called Evolve-Filter-Relax (EFR) with a computationally efficient finite volume method. The main novelty of our ROM lies in the use within the EFR algorithm of a nonlinear, deconvolution-based indicator function that identifies the regions of the domain where the flow needs regularization. The ROM we propose is a hybrid projection/data-driven strategy: a classical Proper Orthogonal Decomposition Galerkin projection approach for the reconstruction of the velocity and the pressure fields and a data-driven reduction method to approximate the indicator function used by the nonlinear differential filter. This data-driven technique is based on interpolation with Radial Basis Functions. We test the performance of our ROM approach on two benchmark problems: 2D and 3D unsteady flow past a cylinder at Reynolds number 0≤Re≤100. The accuracy of the ROM is assessed against results obtained with the full order model for velocity, pressure, indicator function and time evolution of the aerodynamics coefficients.

A constrained proper orthogonal decomposition model for upscaling permeability

AbstractReservoir modeling and simulation are vital components of modern reservoir management processes. Despite the advances in computing power and the advent of smart technologies, the implementation of model‐based operational/control strategies has been limited by the inherent complexity of reservoir models. Thus, reduce order models that not only reduce the cost of the implementation but also provide geological consistent prediction are essential. This article introduces reduced‐order models based on the proper orthogonal decomposition (POD) coupled with linear interpolation for upscaling. First, using POD‐based models, low rank approximate (LRA) are obtained by projecting the high dimensional permeability dataset to a low dimensional subspace spanned by its trajectories to decorrelate the dataset. Next, the LRA is integrated into the interpolation algorithm to generate upscaled values. This technique is highly scalable since low‐rank approximations are achieved by the variation in the number of modes used for reconstruction. To test the validity and reliability of the model, we show its application to the practical dataset from SPE10 benchmark case2. From statistics of the error analysis, the classical POD algorithm seems to be more preferred for LRA; however, since non‐negativity of the permeability data set is a priority, the constrained POD (non‐negative POD) algorithm described in this article is more appropriate. Finally, we compared the POD‐based models to a traditional industry‐standard upscaling technique (e.g., arithmetic mean) to highlight our model benefits/performance. Results show that the POD‐based models, particularly the non‐negative POD model, produce considerably less error than the arithmetic mean model in the upscaling process.

Adaptive POD-DEIM correction for Turing pattern approximation in reaction–diffusion PDE systems

Abstract We investigate a suitable application of Model Order Reduction (MOR) techniques for the numerical approximation of Turing patterns, that are stationary solutions of reaction–diffusion PDE (RD-PDE) systems. We show that solutions of surrogate models built by classical Proper Orthogonal Decomposition (POD) exhibit an unstable error behaviour over the dimension of the reduced space. To overcome this drawback, first of all, we propose a POD-DEIM technique with a correction term that includes missing information in the reduced models. To improve the computational efficiency, we propose an adaptive version of this algorithm in time that accounts for the peculiar dynamics of the RD-PDE in presence of Turing instability. We show the effectiveness of the proposed methods in terms of accuracy and computational cost for a selection of RD systems, i.e., FitzHugh–Nagumo, Schnakenberg and the morphochemical DIB models, with increasing degree of nonlinearity and more structured patterns.

Parametric model-order reduction for radiation transport using multi-resolution proper orthogonal decomposition

For parametric high-fidelity simulations, it is often desirable to utilize a reduced-order model (ROM) to emulate, at a reduced computational cost, parametric solutions of the governing partial differential equations (PDEs) for unseen parameter values. One commonly employed option is to utilize a data-driven, projection-based ROM supplemented with subspace identification via proper orthogonal decomposition (POD). POD discovers the ROM subspace by computing the singular value decomposition (SVD) of a set of training data from the full-order model (FOM). In streaming-dominated radiation transport simulations with localized sources, solutions often greatly vary over the spatial domain by many orders of magnitude. In such cases, machine-precision arithmetic can be insufficient to obtain an accurate SVD, resulting in a poorly performing ROM. We present a method called multiresolution POD (mrPOD) that mitigates these inaccuracies. The mrPOD method works by decomposing the spatial domain into regions and performing proper orthogonal decomposition on the training dataset separately in each region. mrPOD is tested on single energy group and multigroup atmospheric shielding transport problems and is shown to outperform classic POD.

Propulsion optimization of a jellyfish-inspired robot based on a nonintrusive reduced-order model with proper orthogonal decomposition

In this research, the propulsion of the proposed jellyfish-inspired mantle undulated propulsion robot (MUPRo) is optimized. To reliably predict the hydrodynamic forces acting on the robot, the proposed nonintrusive reduced-order model (NIROM) based on proper orthogonal decomposition (POD) additionally considers the POD basis that makes an important contribution to the features on the specified boundary. The proposed model establishes a mapping between the parameter-driven motion of the mantle and the evolution of the fluid characteristics around the swimmer. Moreover, to predict new cases where the input needs to be updated, the input of the proposed model is taken from the kinematics of the robot rather than extracted from full-order high-fidelity models. In this paper, it takes approximately 950 s to perform a simulation using the full-order high-fidelity model. However, the computational cost for one prediction with the proposed POD-NIROM is around 0.54 s, of which about 0.2 s is contributed by preprocessing. Compared with the NIROM based on the classic POD method, the proposed POD-NIROM can effectively update the input and reasonably predict the characteristics on the boundary. The analysis of the hydrodynamic performance of the MUPRo pinpoints that, over a certain period and with a certain undulation amplitude, the hydrodynamic force generated by the swinging-like mantle motion (k < 0.5) is greater, outperforming Aequorea victoria in startup acceleration. It is demonstrated that considering a certain power loss and a certain tail beat amplitude, the wave-like mantle motion (k > 0.5) can produce greater propulsion, which means higher propulsion efficiency.

Adaptive Data-Driven Model Order Reduction for Unsteady Aerodynamics

A data-driven adaptive reduced order modelling approach is presented for the reconstruction of impulsively started and vortex-dominated flows. A residual-based error metric is presented for the first time in the framework of the adaptive approach. The residual-based adaptive Reduced Order Modelling selects locally in time the most accurate reduced model approach on the basis of the lowest residual produced by substituting the reconstructed flow field into a finite volume discretisation of the Navier–Stokes equations. A study of such an error metric was performed to assess the performance of the resulting residual-based adaptive framework with respect to a single-ROM approach based on the classical proper orthogonal decomposition, as the number of modes is varied. Two- and three-dimensional unsteady flows were considered to demonstrate the key features of the method and its performance.

Proper Orthogonal Decomposition for a Turbulent Boundary Layer with Aero-Optic Distortion

The velocity field associated with the deflection of a laser beam passing through a heated incompressible boundary layer in the wall-normal direction was studied using variants of the classical proper orthogonal decomposition (POD) technique. Due to the heating of the boundary layer, temperature, density, and index of refraction fields were variable in the boundary layer and were instantaneously affected by turbulent coherent structures. The time-varying index of refraction field led to a time-varying beam path of a laser traversing the boundary layer; the final angle of the laser beam was measured simultaneously with particle image velocimetry. The relationship between coherent structures in the velocity field and the final beam angle was studied using conditional POD and space-time POD. Conditional POD was performed on the wall-normal velocity field using conditions of large upstream and downstream laser deflections. The modes from conditional POD were compared to modes from POD without conditions. Physical features within some conditional POD modes were found to be shifted in the streamwise direction compared to unconditioned modes, suggesting that the streamwise position of velocity features was correlated to laser deflection. Space-time POD resulted in the same modes as the conditional POD and showed the modes convecting in the streamwise direction without a significant change in the size of the structures. This suggested a hypothesis that the laser deflects when coherent structures reach particular positions relative to the laser position as the structures convect, aligning the largest local velocity and density gradients with the laser beam path.

Data-driven surrogate modeling of aerodynamic forces on the superstructure of container vessels

The operation of fluid engineering systems is usually governed by a wide range of different parameters. Investigations of the entire parameter spectrum using classical, first-principle based CFD methods are costly with regards to CPU and wall-clock time. Therefore, a near real-time assessment of complex flows using CFD to support the operation is deemed unfeasible. The paper is concerned with methods for data-based surrogate models to predict the forces exerted by the aerodynamic pressure field on the superstructure of a full-scale container ship for different container loading conditions and wind directions. The strategy aims to assist a fuel-efficient operation and is based on a two-step approach. During an initial step, a reduced representation of pressure fields obtained from 3D Navier–Stokes simulations is compiled. To this extent, a classical proper orthogonal decomposition is compared with convolutional neural network autoencoders. A subsequent parameterization employs a feedforward neural network to link the reduced model with the operational parameters, i.e. the angle of attack and container loading condition, and to enable a rapid on board assessment. Both methods provide a similar agreement for the pressure fields, as well as the resulting forces, with the CNN-based surrogate model being significantly more compact.

Adaptive POD-Galerkin Technique for Reservoir Simulation and Optimization

In this work, a novel method with an adaptive functional basis for reduced order models (ROM) based on proper orthogonal decomposition (POD) is introduced. The method is intended to be applied in particular to hydrocarbon reservoir simulations, where a range of varying boundary conditions must be explored. The proposed method allows us to update the POD functional basis constructed for a specific problem setting in order to match varying boundary conditions, such as modified well locations and geometry, without the necessity to recalculate each time the whole set of basis functions. Such an adaptive technique allows us to significantly reduce the number of snapshots required to calculate the new basis, and hence reduce the computational cost of the simulations. The proposed method was applied to a two-dimensional immiscible displacement model, and simulations were performed using a high resolution model, a classical POD reduced model, and a reduced model whose POD basis was adapted to varying well location and geometry. Numerical simulations show that the proposed approach allows us to reduce the required number of model snapshots by a few orders of magnitude compared to a classical POD scheme, without noticeable loss of accuracy of calculated fluid production rates. Such an adaptive POD scheme can therefore provide a significant gain in computational efficiency for problems where multiple or iterative simulations with varying boundary conditions are required, such as optimization of well design or production optimization.

Identification of coexisting dynamics in boundary layer flows through proper orthogonal decomposition with weighting matrices

A different version of the classic proper orthogonal decomposition (POD) procedure introducing spatial and temporal weighting matrices is proposed. Furthermore, a newly defined non-Euclidean (NE) inner product that retain similarities with the POD is introduced in the paper. The aim is to emphasize fluctuation events localized in spatio-temporal regions with low kinetic energy magnitude, which are not highlighted by the classic POD. The different variants proposed in this work are applied to numerical and experimental data, highlighting analogies and differences with respect to the classic and other normalized variants of POD available in the literature. The numerical test case provides a noise-free environment of the strongly organized vortex shedding behind a cylinder. Conversely, experimental data describing transitional boundary layers are used to test the capability of the procedures in strongly not uniform flows. By-pass and separated flow transition processes developing with high free-stream disturbances have been considered. In both cases streaky structures are expected to interact with other vortical structures (i.e. free-stream vortices in the by-pass case and Kelvin–Helmholtz rolls in the separated type) that carry a significant different amount of energy. Modes obtained by the non-Euclidean POD (NE-POD) procedure (where weighted projections are considered) are shown to better extract low energy events sparse in time and space with respect to modes extracted by other variants. Moreover, NE-POD modes are further decomposed as a combination of Fourier transforms of the related temporal coefficients and the normalized data ensemble to isolate the frequency content of each mode.

Influence of flow field dynamics on soot evolution in an aero-engine model combustor

Large-Eddy Simulation of a sooting aero-engine model combustor is analyzed by means of proper orthogonal decomposition (POD) to investigate the influence of flow field dynamics on soot evolution. Since classical POD fails to extract the low-frequency intermittent dynamics of scalars such as soot volume fraction, the recently-introduced multiresolution POD (MRPOD) is deployed to treat the LES results instead. MRPOD essentially provides additional dynamic rankings of energetic POD modes via bandpass filters constructed using wavelet packet transform (WPT). In the investigated combustor major dynamics identified in both velocity and scalar fields include a precessing vortex core (PVC) at about 430 Hz and low-frequency dynamics (LFD) at below 100 Hz. Low-frequency dynamics of velocity are present in the whole combustor and affect the secondary air injection and recirculation. While both dynamics are found to influence soot volume fraction, the PVC impact is much smaller compared to the LFD impact.

Proper Orthogonal Decomposition of High-Speed Particle Image Velocimetry in an Open Cavity

Two-dimensional velocity data of flow over an open cavity acquired by pulse-burst particle image velocimetry at 37.5 kHz for freestream Mach numbers of 0.6, 0.8, and 0.94 have been analyzed using proper orthogonal decomposition (POD). The snapshot application of the POD resulted in modes consistent with previous studies on cavity flows under similar conditions. The time-resolved nature of velocity datasets also allowed for the application of the classical POD in both space and time which, for a stationary flow, is solved in the frequency domain. Eigenvalue spectra and convergence plots revealed that these POD modes are organized by capturing the maximum energy associated with dominant frequencies. The extracted eigenvectors revealed a strong dependency on frequency, with the most dominant modes corresponding to frequencies associated with Rossiter modes known to dominate open cavities. Mach number comparisons from the classical POD application reveal that their modes efficiently capture energy of dominant frequencies associated with the flow dynamics, regardless of the freestream conditions. Using a similarity parameter between spatial unitary basis functions, it was shown that the modes at different Mach numbers were quantitatively similar at the same Strouhal and POD mode numbers.

Bayesian identification of a projection-based reduced order model for computational fluid dynamics

In this paper we propose a Bayesian method as a numerical way to correct and stabilise projection-based reduced order models (ROM) in computational fluid dynamics problems. The approach is of hybrid type, and consists of the classical proper orthogonal decomposition driven Galerkin projection of the laminar part of the governing equations, and Bayesian identification of the correction term mimicking both the turbulence model and possible ROM-related instabilities given the full order data. In this manner the classical ROM approach is translated to the parameter identification problem on a set of nonlinear ordinary differential equations. Computationally the inverse problem is solved with the help of the Gauss-Markov-Kalman smoother in both ensemble and square-root polynomial chaos expansion forms. To reduce the dimension of the posterior space, a novel global variance based sensitivity analysis is proposed.

Component-level proper orthogonal decomposition for flexible multibody systems

Nonlinear finite element methods such as isogeometric analysis and absolute nodal coordinate formulation can be applied to modeling the flexible multibody systems (FMBS) subject to both large rotation and deformation. However, the computational expense is prohibitive for complex systems, especially for a parameterized FMBS. Proper orthogonal decomposition (POD), nonmodal model order reduction (MOR) technique, has been widely applied to nonlinear problems in order to reduce the computational expense of simulating the dynamic responses of FMBSs. This paper presents a component-level POD method, which applies the singular value decomposition process to snapshots of each component of an FMBS and utilizes constraint equations to satisfy the connecting conditions among the components. Furthermore, the paper provides a systematic study of the parameterized MOR (PMOR) for an FMBS, including the reduction of the constraint equations, energy conserving mesh sampling and weighting (ECSW), and the greedy-POD method. The ECSW method aims to further reducing the computational cost of matrix multiplications for evaluating the reduced stiffness matrix. Greedy-POD is applied to get a robust set of reduced order bases with parametric changes in simulation. Finally, three numerical examples validate the feasibility of the component-level POD as well as the whole PMOR process for an FMBS, thereby demonstrating that the component-level POD performs better numerical simulations in accuracy computational costs than the classical POD.

Single- and double-helix vortex breakdown as two dominant global modes in turbulent swirling jet flow

In this paper, we study the shape and dynamics of helical coherent structures found in the flow field of an annular swirling jet undergoing vortex breakdown. The flow field is studied by means of time-resolved tomographic particle image velocimetry measurements. The obtained flow fields are analysed using both classic and spectral proper orthogonal decomposition. Despite the simple geometrical set-up of the annular jet, the flow field is very complex. Two distinct large-scale helical flow structures are identified: a single and a double helix, both co-rotating with the swirl direction, and it is revealed that these structures are not higher harmonics of each other. The structures have a relatively low energy content which makes it hard to separate them from other dynamics of the flow field, notably turbulent motions. Because of this, classic proper orthogonal decomposition fails to identify both structures properly. Spectral proper orthogonal decomposition, on the other hand, allows them to be identified accurately when the filter size is set at around eight times the precession period. The precession frequencies of the single and double helices correspond to Strouhal numbers of 0.273 and , respectively. A global stability analysis of the mean flow field shows that these structures correspond to two separate global modes. The precessing frequencies obtained by the stability analysis and the related spatial structures match very well with the experimental observations. The current work extends our knowledge on turbulent vortex breakdown and on mean field global stability theory in general. It leads to the following conclusions. Firstly, single- and double-helix vortex breakdown are both manifestations of global modes. Previous studies have shown that both and modes can coexist in swirling jets. However, the mode has been identified as a second harmonic of the first mode, while this study identifies both as two independent global modes. Secondly, this work shows that the simultaneous occurrence of multiple helical global modes is possible within a turbulent flow and their shapes and frequencies are very well predicted by mean field stability analysis. The latter finding is of general interest as it applies to a wide class of fluid problems dominated by multiple oscillatory structures.