Orthogonal Decomposition Method Research Articles

Real-time reconstruction of 3D transient non-uniform temperature field for thermal protection system based on machine learning

Reconstructed temperature field will provide critical input for monitoring the aerodynamic heating on the thermal protection system (TPS). In this paper, a novel data-driven method is proposed that allows real-time reconstruction of the 3D transient non-uniform temperature field in a TPS ceramic tile. The measured temperature is obtained by embedding sparse sensors at different layers of the tile. The spatial feature bases and time projection coefficients of the non-uniform temperature fields are extracted using the multi-parameter proper orthogonal decomposition (POD) method. The Fully-connected Long Short-Term Memory (FC-LSTM) networks are used to establish the prediction relationship between the measurements and the full-field temperatures. The problem to estimate the projection coefficients of the spatial feature bases is solved by optimizing the loss function using Adam algorithm. The numerical examples show an approximation error less than 5% and the reconstruction time-consuming less than 1 s in 3D temperature field reconstruction (TFR). The proposed reconstruction method is robust at different measurement noise levels and tested point arrangements.

Investigation on the Reduced-Order Model for the Hydrofoil of the Blended-Wing-Body Underwater Glider Flow Control with Steady-Stream Suction and Jets Based on the POD Method

The rapid acquisition of flow field characterization information is crucial for closed-loop active flow control. The proper orthogonal decomposition (POD) method is a widely used flow field downscaling modeling method to obtain flow characteristics effectively. Based on the POD method, a flow field reduced-order model (ROM) is constructed in this paper for the flow field control of a hydrofoil of a blended-wing-body underwater glider (BWB-UG) with stabilized suction and blowing forces. Compared with the computational fluid dynamics (CFD) simulation, the computational time required to predict the target flow field using the established POD-ROM is only about 0.1 s, which is significantly less than the CFD simulation time. The average relative error of the predicted surface pressure is not more than 6.9%. These results confirm the accuracy and efficiency of the POD-ROM in reconstructing flow characteristics. The timeliness problem of fast flow field prediction in BWB-UG active flow control is solved by establishing a fast prediction model in an innovative way.

Geometrically parametrised reduced order models for studying the hysteresis of the Coanda effect in finite element-based incompressible fluid dynamics

This article presents a general reduced order model (ROM) framework for addressing fluid dynamics problems involving time-dependent geometric parametrisations. The framework integrates Proper Orthogonal Decomposition (POD) and Empirical Cubature Method (ECM) hyper-reduction techniques to effectively approximate incompressible computational fluid dynamics simulations. To demonstrate the applicability of this framework, we investigate the behaviour of a planar contraction-expansion channel geometry exhibiting bifurcating solutions known as the Coanda effect. By introducing time-dependent deformations to the channel geometry, we observe hysteresis phenomena in the solution.The paper provides a detailed formulation of the framework, including the stabilised finite elements full order model (FOM) and ROM, with a particular focus on the considerations related to geometric parametrisation. Subsequently, we present the results obtained from the simulations, analysing the solution behaviour in a phase space for the fluid velocity at a probe point, considered as the Quantity of Interest (QoI). Through qualitative and quantitative evaluations of the ROMs and hyper-reduced order models (HROMs), we demonstrate their ability to accurately reproduce the complete solution field and the QoI.While HROMs offer significant computational speedup, enabling efficient simulations, they do exhibit some errors, particularly for testing trajectories. However, their value lies in applications where the detection of the Coanda effect holds paramount importance, even if the selected bifurcation branch is incorrect. Alternatively, for more precise results, HROMs with lower speedups can be employed.

Three-dimensional particle image velocimetry experimental study of transient motion characteristics of different scale vortices in gasoline engine

The paper adopted three-dimensional (3D) Particle Image Velocimetry (PIV) combined with high-speed measurement technology to study the transient motion characteristics of tumble flow induced by small and medium-sized vortices on an optical gasoline engine. Meanwhile, the Proper Orthogonal Decomposition (POD) method was introduced to analyze the evolution of transient vortices, revealing the three-dimensional motion principle of vortices with different spatial scales within the cylinder. The results showed that increasing the engine speed can cause boosted in-plane and out-of-plane velocity, especially for the in-plane velocity. However, the average velocity in the plane was higher than that out of the plane with the difference generally within an order of magnitude. The strong intake jet was the main reason for bringing turbulent kinetic energy in the cylinder. The average kinetic energy and turbulent kinetic energy in the cylinder are both significantly improved at high engine speed. From the results of this experiment, it was revealed that there exists isotropic turbulent pulsation characteristics. Overall, the corresponding flow characteristics of different decomposed modes were different. The first mode section had the lowest average velocity ratio, while the other two had the highest. Due to the dissipation of large vortices into small vortices or the shear action between flow layers, new vortices were generated, leading to the intermittent phenomenon of the motion path of the vortex cluster in the target plane. The average vortex size and velocity circularity corresponding to two varied engine speeds presented a downward trend. The average vortex size was smaller at high engine speed, while the vortex intensity was higher. In addition, increasing the engine speed would decrease the vortex scale but increase the vortex strength. This work provided new fundamental insights into analyzing the characteristics of micro-scaled vortices in turbulent flow, as well as the organization of intake in the gasoline engine.

Optimal transonic buffet aerodynamic noise PSD predictions with Random Forest: Modeling methods and feature selection

To address challenges such as long computational cycles and high experimental measurement costs in obtaining Power Spectral Density (PSD) of aerodynamic noise, this study aims to enable a rapid assessment of buffet frequency and aerodynamic noise levels. The paper conducts a comparative analysis of the impact of different modeling methods and input features on prediction accuracy, proposing the time domain model, the full frequency domain model, and the single frequency model. The research reveals that the frequency domain model has an advantage over the time domain model in predicting aerodynamic noise, emphasizing the importance of selecting appropriate modeling methods. Additionally, based on whether frequency information is used as input features, the study introduces the full frequency domain model and the single frequency model. Results indicate that the single frequency model can significantly reduce the maximum relative error and Root Mean Square Error of the full frequency domain model by approximately three orders of magnitude, lowering the reconstruction error of the Proper Orthogonal Decomposition method by 2–3 orders of magnitude. Furthermore, this model demonstrates generalization across Mach numbers, angles of attack, and spatial positions, ensuring that the maximum absolute error of discrete peak frequencies is controlled within 1 Hz, and the relative error of discrete narrowband peaks is maintained at around 1 %.

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.

Proper Orthogonal Decomposition Based Response Analysis of Inlet Distortion on a Waterjet Pump

This study addresses the challenge of performance degradation in waterjet pumps due to non-uniform suction flow. Utilizing the Proper Orthogonal Decomposition (POD) method, it decomposes and reconstructs the flow features within a waterjet pump under non-uniform inflow into a series of modes ranked in descending order of energy. By analyzing the modes with dominant energy, which contain complex information about the flow field, it is revealed that modes 1 and 2 predominantly represent the formation of a concentrated vortex, whereas modes 3 and 4 illustrate its spatial offset. Notably, in the hub section, mode 3 exhibits a delayed flow separation caused by the reduction of circumferential vortex (CV), with a consequent lift in blade loading at the leading edge and a higher head compared to mode 1. In the shroud section, the delayed flow separation in mode 3 suppressed reverse flow and the concentrated separation vortex (CSV) and then increased the blade loading, ultimately enhancing the pump head. The findings provide significant insights into optimizing waterjet pump performance by detailing the interactions between various flow structures and pump components, effectively filling a knowledge gap in applying dimensionality reduction techniques within the distorted flow fields of water jet pumps.

Spacing Ratio Effects on the Evolution of the Flow Structure of Two Tandem Circular Cylinders in Proximity to a Wall

The flow around two tandem circular cylinders in proximity to a wall is investigated using particle image velocimetry (PIV) for Re = 2 × 103. The spacing ratios L/D are 1, 2, and 5, and the gap ratios G/D are 0.3, 0.6, and 1. The proper orthogonal decomposition (POD) method and λci vortex identification method are used to investigate the evolution of flow structure, and the influences of L/D and G/D on flow physics are shown. At L/D = 2 and G/D = 0.3, a “pairing” process occurs between the wall shear layer and the upstream cylinder’s lower shear layer, resulting in a small separation bubble behind the upstream cylinder. At L/D = 1, the Strouhal number (St) increases with decreasing G/D. At three gap ratios, the St gradually decreases as L/D increases. At G/D = 0.3, there is nearly a 49.98% decrease from St = 0.3295 at L/D = 1 to St = 0.1648 at L/D = 5, which is larger than the reductions in cases of G/D = 0.6 and G/D = 1. The effects of L/D on the evolution of flow structure at G/D = 0.6 are revealed in detail. At L/D = 1, the vortex shedding resembles that of the single cylinder. As L/D increases to 2, a squarish flow structure is formed between two cylinders, and a small secondary vortex is formed due to induction of the lower shear layer of the upstream cylinder. At L/D = 5, there is a vortex merging process between the upper wake vortices of the upstream and downstream cylinders, and the lower wake vortex of the upstream cylinder directly impinges the downstream cylinder. In addition, the shear layers and wake vortices of the upstream cylinder interact with the wake of the downstream cylinder as L/D increases, resulting in reductions in velocity fluctuations, and the production and turbulent diffusion of turbulent kinetic energy are decreased behind the downstream cylinder.

A preserving accuracy two-grid reduced-dimensional Crank-Nicolson mixed finite element method for nonlinear wave equation

In this paper, we mainly resort to a proper orthogonal decomposition (POD) method to study the reduced-dimension of unknown mixed finite element (MFE) solution coefficient vectors in the two-grid Crank-Nicolson MFE (TGCNMFE) method for the nonlinear wave equation and build a preserving accuracy reduced-dimension extrapolated TGCNMFE (RDETGCNMFE) method for the nonlinear wave equation. For this purpose, we first build a new TGCNMFE method for the nonlinear wave equation and demonstrate the existence, unconditional stability, and error estimates for the TGCNMFE solutions. From there, we build a preserving accuracy RDETGCNMFE method for the nonlinear wave equation by using the POD method to decrease the dimension of unknown TGCNMFE solution coefficient vectors and employ the matrix analysis to analyze the existence, unconditional stability, convergence, and errors for the RDETGCNMFE solutions. We finally make use of numerical experiments to verify our theoretical results and to show the advantages of RDETGCNMFE method.

A depth-variant seismic wavelet extraction method for basis pursuit inversion with an impedance trend constraint

The seismic images produced by prestack depth migration indicate more accurate subsurface structures than time images, resulting in a growing need for depth-domain inversion. However, due to the strong nonstationarity exhibited by depth-domain seismic data, time-domain inversion methods based on the convolutional model cannot be directly applied in the depth domain. To address this issue, we develop a method for extracting a depth-variant seismic wavelet, which is then combined with a nonstationary convolutional model to enable direct inversion of the depth-domain acoustic impedance (AI). First, we extend the Morlet wavelet to the depth domain and develop an orthogonal matching pursuit spectral decomposition method using the depth-domain Morlet wavelet. We then investigate the waveforms and wavenumber spectra similarities between the depth-domain Morlet wavelet and depth-domain Ricker wavelet and extract depth-variant Ricker wavelets from the depth-wavenumber spectrum. We add a depth-domain impedance trend constraint to the conventional basis pursuit inversion to enhance the lateral continuity of the inversion results. Then, we attain direct inversion of the depth-domain AI. Tests of synthetic and field data demonstrate that our method achieves high-accuracy inversion results while maintaining high computational efficiency, highlighting our approach’s effectiveness and strong reservoir characterization potential.

Impact of Oxygen Content on Flame Dynamics in a Non-Premixed Gas Turbine Model Combustor

In this study, large eddy simulation (LES) was used to investigate the dynamic characteristics of diffusion flames in a swirl combustion chamber at an oxygen content of 11–23 wt% and temperature of 770 K. The proper orthogonal decomposition (POD) method was employed to obtain flame dynamic modes. The results indicate that oxygen content has a significant impact on the downstream flow and flame combustion characteristics of the swirl combustion chamber. With oxygen content increasing, the size of the recirculation zone is reduced, and the flame field fluctuations are intensified. The pressure and heat release fluctuations under different oxygen contents were analyzed using frequency spectrum analysis. Finally, the flame modes were analyzed using the POD method, and it was found that the coherent structures are asymmetric relative to the local coordinate system. At an oxygen content of 11 wt%, they exhibit larger coherent structures, while at an oxygen content of 23 wt%, they exhibit numerous turbulent structures.

Fluidic undulation effects on carangiform swimmers propelled by internal active bending moments

With different shapes and material properties, fish all achieve undulatory swimming gait under the action of internal active muscle stimulation and external fluid forces. Such locomotion can be decomposed into deformation affected by internal and external forces in the body frame and overall translation and rotation solely determined by fluid forces. In order to revisit the undulatory swimming gait, we investigate the hydrodynamic performance of two-dimensional flexible carangiform swimmers with varying stiffnesses and thicknesses, which are driven by the active internal bending moments, and employ the complex orthogonal decomposition and Fourier decomposition methods to quantitatively measure and analyze the proportion of undulation. It is found that standing wave deformation characteristics are prominently observed along fish-like bodies with high stiffness, whereas traveling wave characteristics are more evident in bodies with lower stiffness. The self-propelled fish body demonstrates lateral oscillation and rotation around its center of mass, namely, the heaving and pitching movement, particularly in specimens with high stiffness. The present analysis shows that the heaving and pitching locomotion induced by the fluid significantly increase the traveling wave proportion by modulating the amplitude and phase of the left and right traveling waves viewed in forward frame. We called it fluidic undulation effects (FUE), which is different from the undulation of body deformation. This effect is more pronounced for large stiffnesses and thin airfoils. The standing wave deformation observed with a large stiffness transforms into a traveling wave propulsion pattern, with its traveling wave index even slightly surpassing that of a small-stiffness pattern. Although the efficiency of the standing wave deformation is low, it facilitates a faster forward speed (body lengths per stroke). The positive impact of the FUE on the swimming performance is also confirmed by restricting the recoil motions of the lateral translation and rotation of the body. Furthermore, we observe that there is no undulatory swimming gait that has both the highest energy efficiency and the highest speed.

Nonlinear reduction using the extended group finite element method

In this paper, we develop a nonlinear reduction framework based on our recently introduced extended group finite element method. By interpolating nonlinearities onto approximation spaces defined with the help of finite elements, the extended group finite element formulation achieves a noticeable reduction in the computational overhead associated with nonlinear finite element problems. However, the problem’s size still leads to long solution times in most applications. Aiming to make real-time and/or many-query applications viable, we consider model order reduction and complexity reduction techniques to reduce the problem size and efficiently handle the reduced nonlinear terms, respectively. For proof of concept, we focus on the proper orthogonal decomposition and discrete empirical interpolation methods. While similar approaches based on the group finite element method only focus on semilinear problems, our proposed framework is also compatible with quasilinear problems. Compared to existing methods, our reduced models prove to be superior in many different aspects as demonstrated in three numerical benchmark problems.

Large eddy simulation of non-premixed flame behavior and NOx emission characteristics in a staged combustor with multi-swirling flows

This study utilizes Large Eddy Simulation (LES) to investigate flame characteristics and NOx emission generation in a concentric staged combustor with three distinct pilot swirler arrangements. We examine the effects of multi-swirling flow on the instantaneous flow field and flame characteristics using the Proper Orthogonal Decomposition (POD) method, and assess how variations in swirler structures influence thermoacoustic instability. Furthermore, a comparison of NOx emissions between two pilot swirler configurations, with similar mean flow field and flame characteristics, highlights the precessing vortex core's role in NOx emissions. Compared to a single-swirler setup, a double-swirler configuration produces a more distinct single precessing vortex core, thereby amplifying thermoacoustic instability. The extended venturi structure in the double-swirler setup mitigates the development of the swirl-induced vortex core, reducing thermoacoustic instability. However, the expansion section in this setup impedes initial fuel and air mixing, leading to an extended flame length, which in turn results in higher NOx emissions due to a more concentrated temperature distribution. With similar mean flow field and flame structure, the double-swirler combustor proves more effective in reducing NO emissions than the triple-swirler combustor.

Experimental study on quantitative characterization of cavitation internal structure based on the distribution of void fraction in pressure relief valve

Accurately quantify the complex structure of cavitation in pressure relief valve by experiment remains a challenging task. In this research, a quantitative method for assessing the vapor volume fraction in cavitation images by void fraction experimentally is proposed, this method is less affected by environmental interference and could achieve rapid and accurate characterization of cavitation. The results show that void fraction effectively represents changes in vapor volume fraction under different conditions. The influence of the grid division numbers on the vapor volume fraction is investigated, the optimal number of grid division is 93 in X direction, with a misjudgment number of 130 and a misjudgment rate of 1.62 %. The effects of grid numbers on vapor volume fraction under different cavitation image pixels is also discussed, the optimal number of grid division is 112 in flow direction, with a misjudgment number of 144 and a misjudgment rate of 1.21 %. Through processing the vapor volume fraction image using the Proper Orthogonal Decomposition (POD) method under different cavitation numbers, the results show that the cavitation number has an important influence on the proportion of re-entrant jet area and cavitation collapse area, when the cavitation number decreases to 0.047, the proportion of re-entrant jet area and cavitation collapse area achieve the maximum that 6.41 % and 8.68 %, respectively. Such quantitative analysis of the cavitation degrees under various working conditions can contribute to a better understanding of the impact of cavitation on valve dynamics mechanisms.

Binocular reconstruction of breaking ship bow waves in circulating water channel

This study conducts an experiment on the bow waves generated by a 1:80 KRISO container ship (KCS) model in a circulating water channel. The Froude numbers (characterized by the length of perpendiculars) reach high conditions of 0.377, 0.424, and 0.471. A binocular imaging system is employed to reconstruct three-dimensional configurations of the bow wave surface. The dynamic characteristics of bow wave fields are investigated by observing the standard deviation and employing the Proper Orthogonal Decomposition (POD) method. The power spectral density function presents a power-law relationship with frequency. A novel technique referred as Stereoscopic Foam Image Velocimetry (SFIV) is developed to extract the instantaneous three-dimensional velocity of the foam in breaking area. The velocity field obtained through SFIV defines the nominal time-averaged velocity and nominal turbulent energy. A correlation mapping from pixel coordinates to world coordinates is established to evaluate the actual physical uncertainty. The results demonstrate the superior spatiotemporal accuracy and resolution of binocular imaging technology in measuring the characteristics of bow waves. This research provides significant experimental data of surface three-dimensional configurations for the bow waves breaking issues at high Froude numbers.

Spectral feature extraction of rocket exhaust plume using spectral proper orthogonal decomposition

The spectral characterization of flow-field parameters provides a new perspective for understanding the spatiotemporal evolution of unsteady supersonic exhaust plumes and for extracting typical structures. In this study, a large-eddy simulation is performed to calculate the three-dimensional unsteady supersonic plume flow field of rocket engines, and a spectral proper orthogonal decomposition (SPOD) method with a spatiotemporal separation is established. This approach is used to extract the coherent structural features of the unsteady exhaust plume flow field and analyze the mode space structure at different frequencies. The three-dimensional reconstruction and denoising of the exhaust plume flow-field parameters can be achieved via the frequency- and time-domain reconstructions of the SPOD algorithm and oblique projection method, respectively. The ground rocket exhaust plume of ballistic evaluation motor-II is analyzed. The results indicate that the SPOD method can effectively extract the single-frequency mode structure of the reactive supersonic flow field, and that low-order behavior appears in the m = 0 and m = 1 azimuth modes. The potential core exhibits a high-frequency wave-packet structure that is affected by shock waves and shear layers. Time-domain reconstruction based on the oblique projection method facilitates the capture of the dynamic characteristics of the flow field. For the first-order SPOD mode, the frequency- and time-domain reconstruction errors are 3.3% and 1.5%, respectively. The frequency-domain reconstruction method exhibits a 4% improvement in denoising ability compared to low-pass filters. This study provides a novel method for the spectral characterization and spatiotemporal feature extraction of supersonic exhaust plume flow fields.

A novel accelerated convergence method for solving adjoint equations based on modal reduction

The efficiency of adjoint-based aerodynamic shape optimization depends critically on the solution efficiency of adjoint equations. In this letter, we employ the Proper Orthogonal Decomposition (POD) method to analyze the adjoint field samples and project them from the physical space into a low-order modal space. Subsequently, the full-order adjoint equations are reduced to low-order equations using the POD modes. Thus, we can efficiently predict the initial values for pseudo-time marching, thereby accelerating the solution of adjoint equations. Results indicate that the high-order POD modes are crucial for constructing the low-dimensional system. Moreover, this method can be seamlessly integrated with our previously established Dynamic Mode Decomposition (DMD) acceleration method to form a POD+DMD acceleration approach. Application of this approach to the flow past a National Advisory Committee for Aeronautics 0012 airfoil demonstrates a noteworthy 80.9% reduction in iteration numbers when solving the adjoint equations. Even for the airfoil located on the upper boundary of sampling space, the number of iterations is still reduced by 72.6%. Therefore, we believe that the proposed method holds significant promise for improving the efficiency of adjoint-based aerodynamic shape optimization in future research.

Error analysis of a reduced order method for the Allen-Cahn equation⁎

In this paper we carry out an error analysis for a reduced order method for the Allen-Cahn equation. First, an ensemble of snapshots is formed from the numerical solutions at some time instances of the full order model, which is a time-space discretisation of the Allen-Cahn equation. The reduced order model is essentially a new spatial discretisation method by using low dimensional approximations to the original approximation space. The low dimensional approximation space is generated from the ensemble of snapshots by applying a proper orthogonal decomposition method. To determine the error between the exact solution and the solution of the reduced order model. We consider a time-space discretisation for which an error estimate of the full model solution is available. Specifically, the full discretisation is based on a stabilized auxiliary variable approach for the time stepping and a spectral Galerkin method for the spatial discretisation. The advantages of this full discretisation are its unconditional stability, the availability of error estimates and its ease of implementation. An estimate of the errors in the H1 seminorm is rigorously derived for both the full order model and the reduced order model, which is then verified by some numerical examples.

Proper orthogonal decomposition method of constructing a reduced-order model for solving partial differential equations with parametrized initial values

This paper proposes a novel proper orthogonal decomposition method for constructing a reduced-order model. This model effectively computes solutions for various initial conditions in time-dependent partial differential equations. The mode obtained from the proposed proper orthogonal decomposition captures both the fluctuating component and the mean-field of the time-dependent solution. In this mode, the eigenvalues representing the mean-field, are significantly larger than those representing the fluctuating component. Consequently, determining the number of modes required for representing a solution cannot rely solely on the cumulative contribution rate. Therefore, the proposed method gives a scalar weight to the mean-field and controls the magnitude of the mean-field energy. We propose a method that considers the magnitude of the scaler weight alongside the cumulative contribution ratio for determining the number of modes. This method was evaluated using the time-dependent Burgers equation with parametric initial values. Our proposed method selects the appropriate mode to represent the given dataset, unlike the conventional method. A reduced-order model based on the proposed method effectively computes the time evolutions of the solutions using datasets for several parameters by imposing a condition that the cumulative contribution ratio exceeds 98%.