Proper Orthogonal Decomposition Technique Research Articles

A novel dimensionality reduction iterative method for the unknown coefficient vectors in TGFECN solutions of unsaturated soil water flow problem

The main purpose of this paper is to reduce the dimensionality of unknown coefficient vectors in finite element (FE) solutions of two-grid FE Crank-Nicolson (CN) (TGFECN) format for the unsaturated soil water flow (USWF) problem with two strong nonlinear terms by using proper orthogonal decomposition (POD). For this purpose, our first step involves designing a time semi-discrete CN (TSDCN) scheme for the USWF problem and demonstrate the existence, boundedness, and error estimations of TSDCN solutions. Thereafter, we discretize the TSDCN scheme using the two-grid FE method to create a new TGFECN format and prove the existence, boundedness, and error estimations of TGFECN solutions. The primary focus should be on reducing the dimension of unknown coefficient vectors of TGFECN solutions through the utilization of the POD technique in creating a novel format, referred to as dimension reduction iterative TGFECN (DRITGFECN) format, while establishing the existence, boundedness, and error estimations for DRITGFECN solutions. Lastly, we use two sets of numerical tests to exhibit the advantage of the DRITGFECN format. Due to the presence of two highly nonlinear terms in the unsaturated soil flow problem, the development and analysis of DRITGFECN format pose greater challenges and necessitates more advanced technical skills compared to previous studies. However, the significance and broad applications of this research make it a valuable subject for investigation.

Effects of Liquid Film Instability on Spray Features of Liquid–Liquid Pintle Injectors

As one of the crucial structural parameters for the pintle injectors, the design of the skip distance ratio is still heavily empirical-dependent in engineering, and the intrinsic correlations between the skip distance ratio and spray characteristics of liquid–liquid pintle injectors are also unclear. Inspired by the observations that the instability of axial liquid film exerts significant influence on the downstream spray behaviors of a liquid–liquid pintle injector, the influence of the skip distance ratio on the instability of the axial liquid film and the subsequent spray characteristics of the pintle injector was experimentally investigated. The spray steadiness of the pintle injector was evaluated based on the instantaneous images using the proper orthogonal decomposition technique, and the effects of the skip distance ratio and the momentum ratio on the spray features were analyzed. Furthermore, a proactive strategy was proposed to adjust the liquid film instability by modifying the outer wall morphology of the pintle post, and the verification experiments were carried out based on three different wall morphologies. It was discovered that the rough wall enhances the axial liquid film instability and promotes the liquid sheet breakup and atomization of the pintle injector. These results provide new insights and valuable references for the optimization design of pintle injectors.

Three-dimensional wake transition of rectangular cylinders and temporal prediction of flow patterns based on a machine learning algorithm

This study investigates the three-dimensional (3D) wake transition in unconfined flows over rectangular cylinders using direct numerical simulation (DNS). Two different cross-sectional aspect ratios (AR) and Reynolds numbers (Re) are scrutinized: AR = 0.5 at Re = 200 and AR = 3 at Re = 600. The investigation focuses on characterizing the flow patterns and forecasting their temporal evolution utilizing the proper orthogonal decomposition (POD) technique coupled with a long short-term memory (LSTM) network. The DNS results reveal the emergence of an ordered mode A for AR = 3, attributed to the stabilizing effect of the elongated AR. On the other hand, the case with a smaller AR (= 0.5) exhibits a mode-swapping regime characterized by modes A and B's distinct and simultaneous manifestation. The spanwise wavelengths of mode A and mode B are approximately 4.7 and 1.2 D for AR = 0.5, while the spanwise wavelength of mode A is 3.5 D for AR = 3. The POD serves as a dimensionality reduction technique, and LSTM facilitates temporal prediction. This algorithm demonstrates satisfactory performance in predicting the flow patterns, including the instabilities of modes A and B, across both transverse and spanwise directions. The employed algorithm adeptly predicts the pressure time series surrounding the cylinders. The duration for training the algorithm is only about 0.5% of the time required for DNS computations. This research, for the first time, demonstrates the effectiveness of the POD–LSTM algorithm in predicting complex 3D instantaneous wake transition patterns for flow past rectangular cylinders.

POD Analysis of the Wake of Two Tandem Square Cylinders

This study aims to investigate the wake of two tandem square cylinders based on the Proper Orthogonal Decomposition (POD) analyses of the PIV and hotwire data. The cylinder centre-to-centre spacing ratio L/w examined is from 1.2 to 4.2, covering the four flow regimes, i.e., extended body, reattachment, transition and co-shedding. The Reynolds number examined was 1.3 × 104. A novel Proper Orthogonal Decomposition (POD) technique (hereafter referred to as PODHW) is developed to analyse data from single point hotwire measurements, offering a new perspective compared to the conventional POD analysis (PODPIV) based on Particle Image Velocimetry (PIV) data. A key finding is the identification of two distinct states, reattachment and co-shedding, within the transition flow regime at L/w = 2.8, which PODPIV fails to capture due to the limited duration of the PIV data obtained. This study confirms, for the first time, the existence of these states as proposed by Zhou et al. (2024), highlighting the advantage of using PODHW for capturing intermittent flow phenomena. Furthermore, the analysis reveals how the predominant coherent structures contribute to the total fluctuating velocity energy in each individual regime. Other aspects of the flow are also discussed, including the Strouhal numbers, the contribution to the total fluctuating energy of the flow from the first four POD modes, and a comparison between different regimes.

Effects of serration angle on noise reduction mechanism of trailing edge serration

This study investigates the effects of serration angle on the three-dimensional flow field and noise reduction mechanism of a NACA6512-63 airfoil using large eddy simulation (LES) and spectral proper orthogonal decomposition (SPOD) techniques. The serration angle is defined as the ratio of serration wavelength (λ) to serration amplitude (h). The mean flow field analysis on suction side of serration reveals the outward flow towards the serration edges and the downwash flow between the valleys that generate the counter-rotating vortex pair. This flow alteration changes the serration effective angle of the serration, consequently affecting the noise reduction performance. Additionally, by comparing the general solution of Lyu’s semi-analytical equation with the SPOD mode results, the noise reduction mechanism is examined, and the frequency range in which noise reduction occurs is analyzed. The linearly distributed multipole-like pressure patterns, similar to those in the semi-analytical solution, are observed in the low-frequency range for all three cases, with the λ/h = 0.3 case exhibiting the most effective noise reduction. The findings suggest that the local deviation of the turbulence axis, caused by the strong outward flow near the serration edges, can hinder the formation of the optimal pressure field distribution required for effective noise reduction.

The prediction on non-Gaussian characteristics of wind pressure for the long-span roof in the mountainous area using proper orthogonal decomposition–deep learning framework

To make an accurate prediction of the non-Gaussian characteristics of wind pressure for the long-span roof, this study combines the proper orthogonal decomposition (POD) technique, convolutional neural network (CNN), and long short-term memory (LSTM) network to propose a novel POD-CNN-LSTM framework. Then, the proposed framework was well validated based on the wind tunnel testing of a long-span roof structure, and some error criteria, such as mean square root error and correlation coefficient, were adopted to evaluate the prediction accuracy of the non-Gaussian characteristics. Furthermore, two other methods, POD-CNN and POD-LSTM, were also used to conduct a comparative study. The obtained results illustrate that compared to POD-CNN and POD-LSTM, the proposed framework can achieve better performance on the pulsating wind pressure coefficient. For predictions of non-Gaussian characteristics, the output results of the proposed POD-CNN-LSTM show fewer errors, which means the predictions are close to the measured results, including skewness, kurtosis, and wind pressure probability density distributions. To summarize, the proposed POD-CNN-LSTM framework shows superiority over others, which means the proposed framework has good potential for the practical application of non-Gaussian prediction of the engineering structure.

Sausage, kink, and fluting magnetohydrodynamic wave modes identified in solar magnetic pores by Solar Orbiter/PHI

Solar pores are intense concentrations of magnetic flux that emerge through the solar photosphere. When compared to sunspots, they are much smaller in diameter and can therefore be affected and buffeted by neighbouring granular activity to generate significant magnetohydrodynamic (MHD) wave energy flux within their confines. However, observations of solar pores from ground-based telescope facilities may struggle to capture subtle motions that are synonymous with higher-order MHD wave signatures because of the seeing effects that are produced in the Earth’s atmosphere. Hence, we exploited timely seeing-free and high-quality observations of four small magnetic pores from the High Resolution Telescope (HRT) of the Polarimetric and Helioseismic Imager (PHI) on board the Solar Orbiter spacecraft during its first close perihelion passage in March 2022 (at a distance of 0.5 au from the Sun). Through acquisition of data under stable observing conditions, we were able to measure the area fluctuations and horizontal displacements of the solar pores. Cross correlations between perturbations in intensity, area, line-of-sight velocity, and magnetic fields, coupled with the first-time application of novel proper orthogonal decomposition techniques on the boundary oscillations, provided a comprehensive diagnosis of the embedded MHD waves as sausage and kink modes. Additionally, the previously elusive m = 2 fluting mode is identified in the most magnetically isolated of the four pores. An important consideration lies in how the identified wave modes contribute to the transfer of energy into the upper solar atmosphere. Approximately 56%, 72%, 52%, and 34% of the total wave energy of the four pores we examined is associated with the identified sausage modes and about 23%, 17%, 39%, and 49% with their kink modes, while the first pore also receives a contribution of about 11% linked to the fluting mode. This study reports the first-time identification of concurrent sausage, kink, and fluting MHD wave modes in solar magnetic pores.

A reduced-order two-grid method based on POD technique for the semilinear parabolic equation

In the conventional two-grid (TG) method, the nonlinear system on the fine grid is transformed into a nonlinear subsystem on the coarse grid and a linear subsystem on the fine grid to reduce computational costs. It has been successfully applied in various fields. Nonetheless, its computational efficiency remains relatively low. For this, we develop a novel reduced-order two-grid (ROTG) method with less degrees of freedom for solving the semilinear parabolic equation. For the two subsystems mentioned, the proper orthogonal decomposition (POD) technique is utilized to substantially reduce degrees of freedom. An a priori error estimate for the ROTG scheme is derived. Finally, we conduct several numerical tests to observe the ROTG method's behavior and verify the theoretical analysis.

Estimation of wind dynamic load linear transfer matrix using proper orthogonal decomposition method and genetic algorithm

ABSTRACT Due to the advantages of Laplace domain in both time and frequency domains, the dynamic analysis and control tools in engineering software like MATLAB and Python have been designed based on it. Dynamic analysis of structures requires determination of the overall transfer matrix, which is obtained by multiplying the structure's transfer matrix (TM) with the load TM. With the input vector of unit intensity white-noise processes and the overall TM, the structural responses can be obtained. Furthermore, having the load TM is crucial for implementation of active control systems. In this article, the TM of along-wind loads was estimated for a tall building using the proper orthogonal decomposition (POD) method and the genetic algorithm (GA). To evaluate the effectiveness of the proposed approach, an example of a tall building was provided. The wind load cross-power spectral matrix (XPSD) was decomposed using POD technique, followed by the truncation of higher loading modes. Subsequently, the loading TM was derived by estimating the target non-linear functions with linear transfer functions through GA technique. Then, the responses of the structure were computed using the loading TM. The results demonstrated the efficacy of the proposed method in acquiring the loading TM and predicting wind-induced responses.

A Reduced-Dimension Weighted Explicit Finite Difference Method Based on the Proper Orthogonal Decomposition Technique for the Space-Fractional Diffusion Equation

A kind of reduced-dimension method based on a weighted explicit finite difference scheme and the proper orthogonal decomposition (POD) technique for diffusion equations with Riemann–Liouville fractional derivatives in space are discussed. The constructed approximation method written in matrix form can not only ensure a sufficient accuracy order but also reduce the degrees of freedom, decrease storage requirements, and accelerate the computation rate. Uniqueness, stabilization, and error estimation are demonstrated by matrix analysis. The procedural steps of the POD algorithm, which reduces dimensionality, are outlined. Numerical simulations to assess the viability and effectiveness of the reduced-dimension weighted explicit finite difference method are given. A comparison between the reduced-dimension method and the classical weighted explicit finite difference scheme is presented, including the error in the L2 norm, the accuracy order, and the CPU time.

Projected data assimilation using sliding window proper orthogonal decomposition

Prediction of the state evolution of complex high-dimensional nonlinear systems is challenging due to the nonlinear sensitivity of the evolution to small inaccuracies in the model. Data Assimilation (DA) techniques improve state estimates by combining model simulations with real-time data. Few DA techniques can simultaneously handle nonlinear evolution, non-Gaussian uncertainty, and the high dimension of the state. We recently proposed addressing these challenges using a POD technique that projects the physical and data models into a reduced-dimensional subspace. Proper Orthogonal Decomposition (POD) is a tool to extract spatiotemporal patterns (modes) that dominate the observed data. We combined the POD-based projection operator, computed in an offline fashion, with a DA scheme that models non-Gaussian uncertainty in lower dimensional subspace. If the model parameters change significantly during time evolution, the offline computation of the projection operators ceases to be useful. We address this challenge using a Sliding Window POD (SW-POD), which recomputes the projection operator based on a sliding subset of snapshots from the entire evolution. The physical model projection is updated dynamically in terms of modes and number of modes, and the data model projection is also chosen to promote a sparse approximation. We test the efficacy of this technique on a modified Lorenz'96 model (L96) with a time-varying forcing and compare it with the time-invariant offline projected algorithm. In particular, dynamically determined physical and data model projections decrease the Root Mean Squared Error (RMSE) and the resampling rate.

Unlocking massively parallel spectral proper orthogonal decompositions in the PySPOD package

We propose a parallel (distributed) version of the spectral proper orthogonal decomposition (SPOD) technique. The parallel SPOD algorithm distributes the spatial dimension of the dataset preserving time. This approach is adopted to preserve the non-distributed fast Fourier transform of the data in time, thereby avoiding the associated bottlenecks. The parallel SPOD algorithm is implemented in the PySPOD library and makes use of the standard message passing interface (MPI) library, implemented in Python via mpi4py. An extensive performance evaluation of the parallel package is provided, including strong and weak scalability analyses. The open-source library allows the analysis of large datasets of interest across the scientific community. Here, we present applications in fluid dynamics and geophysics, that are extremely difficult (if not impossible) to achieve without a parallel algorithm. This work opens the path toward modal analyses of big quasi-stationary data, helping to uncover new unexplored spatio-temporal patterns. Program summaryProgram Title: PySPODCPC Library link to program files:https://doi.org/10.17632/jf5bf26jcj.1Developer's repository link:https://github.com/MathEXLab/PySPODLicensing provisions: MIT LicenseProgramming language: PythonNature of problem: Large spatio-temporal datasets may contain coherent patterns that can be leveraged to better understand, model, and possibly predict the behavior of complex dynamical systems. To this end, modal decomposition methods, such as the proper orthogonal decomposition (POD) and its spectral counterpart (SPOD), constitute powerful tools. The SPOD algorithm allows the systematic identification of space-time coherent patterns. This can be used to understand better the physics of the process of interest, and provide a path for mathematical modeling, including reduced order modeling. The SPOD algorithm has been successfully applied to fluid dynamics, geophysics and other domains. However, the existing open-source implementations are serial, and they prevent running on the increasingly large datasets that are becoming available, especially in computational physics. The inability to analyze via SPOD large dataset in turn prevents unlocking novel mechanisms and dynamical behaviors in complex systems.Solution method: We provide an open-source parallel (MPI distributed) code, namely PySPOD, that is able to run on large datasets (the ones considered in the present paper reach about 200 Terabytes). The code is built on the previous serial open-source code PySPOD that was published in https://joss.theoj.org/papers/10.21105/joss.02862.pdf. The new parallel implementation is able to scale on several nodes (we show both weak and strong scalability) and solve some of the bottlenecks that are commonly found at the I/O stage. The current parallel code allows running on datasets that was not easy or possible to analyze with serial SPOD algorithms, hence providing a path towards unlocking novel findings in computational physics.Additional comments including restrictions and unusual features: The code comes with a set of built-in postprocessing tools, for visualizing the results. It also comes with extensive continuous integration, documentation, and tutorials, as well as a dedicated website in addition to the associated GiHub repository. Within the package we also provide a parallel implementation of the proper orthogonal decomposition (POD), that leverages the I/O parallel capabilities of the SPOD algorithm.

A POD based extrapolation DG time stepping space-time FE method for parabolic problems

Based on the proper orthogonal decomposition technique, a new reduced-order discontinuous time stepping space-time finite element extrapolation iterative scheme is constructed to solve two-dimensional parabolic problems, which includes very few degrees of freedom but holds sufficiently high accuracy. A reduced space-time projection operator is defined, and its related error estimations are discussed. Then, the error estimate between the classical scheme solution and the reduced-basis scheme solution, and the a priori error estimate for the reduced-basis scheme are derived, respectively. These error estimates are analyzed by using the technique of combining Radau quadrature rule with finite element method, without considering any dual problems. The algorithm implementation of the reduced-order extrapolation iterative scheme is also provided. Finally, a numerical example is presented. The numerical results are consistent with theoretical ones. Moreover, it is shown that the new reduced-order extrapolation iterative scheme, by comparing to the standard approach, can reduce CPU time without loss of accuracy. Therefore, it is feasible and efficient for solving parabolic problems.

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.

Estimation of the fuel mixing of annular extruded fuel multi-jets in cavity flame holder at the supersonic combustion chamber via predictive surrogate model

Cavity flame holder is a well-organized method for fuel mixing inside the combustion chamber of supersonic vehicles. In this study, the combined machine learning technique of proper orthogonal decomposition (POD) combined with Long Short-Term Memory network (LSTM) is used for prediction of fuel jet penetration inside the cavity flame holder at free stream Mach=2.2 is investigated. Computational Fluid Dynamic is applied for the three-dimensional model of a cavity flame holder with extruded multi-nozzles. A comparison of single and extruded multi-nozzles for the fuel mixing at the combustor with cavity configuration is also done. The presented results show that the use of multi-jet efficiently increases the fuel mixing inside the cavity since the vortex structure is enhanced in this configuration. The usage of a shock creator pushes the shear layer into the cavity and limits the fuel penetration inside the cavity. Therefore, the shock creator is not recommended for the fuel mixing of extruded multi-jet injection systems.

Application of data-driven model reduction techniques in reactor neutron field calculations

High-order harmonic techniques can be used to recreate neutron flux distributions in reactor cores using the neutron diffusion equation. However, traditional source iteration and source correction iteration techniques have sluggish convergence rates and protracted calculation periods. The correctness of the implicitly restarted Arnoldi method (IRAM) in resolving the eigenvalue problems of the one-dimensional and two-dimensional neutron diffusion equations was confirmed by computing the benchmark problems SLAB_1D_1G and two-dimensional steady-state TWIGL using IRAM. By integrating Galerkin projection with Proper Orthogonal Decomposition (POD) techniques, a POD-Galerkin reduced-order model was developed and the IRAM model was used as the full-order model. For 14 macroscopic cross-section values, the TWIGL benchmark problem was perturbed within a 20% range. We extracted 100 sample points using the Latin hypercube sampling method, and 70% of the samples were used as the testing set to assess the performance of the reduced-order model The remaining 30% were utilized as the training set to develop the reduced-order model, which was employed to rebuild the TWIGL benchmark problem. The reduced-order model demonstrates good flexibility and can efficiently and accurately forecast the effective multiplication factor and neutron flux distribution in the core. The reduced-order model predicts keff and neutron flux distribution with a high degree of agreement compared to the full-order model. Additionally, the reduced-order model's computation time is only 10.18% of that required by the full-order model.The neutron flux distribution of the steady-state TWIGL benchmark was recreated using the reduced-order model. The obtained results indicate that the reduced-order model can accurately predict the keff and neutron flux distribution of the steady-state TWIGL benchmark.Overall, the proposed technique not only has the potential to accurately project neutron flux distributions in transient settings, but is also relevant for reconstructing neutron flux distributions in steady-state conditions; thus, its applicability is bound to increase in the future.

Gappy POD model for Structural Compliance of the Wing of a Civil Tilt-rotor

The selection of the most critical load conditions is an extremely important topic as it allows the design and structural verification to be limited to a limited subset of conditions that envelop the operational thousands of the Design Limit Load (DLL) book without necessarily having to apply a direct approach. Often, thousands of analyses are performed by a Detailed Finite Element Model (DFEM, including model loading, post-processing and full stress and buckling analysis) taking a long time with a higher probability of error. This work describes the effort to set up a surrogate model of a composite wing, which aims to supply, accurately and efficiently, the structural response of the wing to several solicitation loads that populate the DLL. The surrogate model is defined by employing the Proper Orthogonal Decomposition technique (POD), trained on partial knowledge of data. The snapshots collected are composed of load solicitations and, limited to a subset of wing elements, stress and strain states resulting from the structural response to the relative load solicitation in a numerical analysis. Gappy-POD algorithm is set up to deal with this lack of data in the querying stage when only the solicitations are known. The surrogate model training starts with a selection of loads from the DLL, while its reliability is increased by iteratively adding further FEM analyses based on error maps opportunely defined.

Extended Dynamic Mode Decomposition with Invertible Dictionary Learning

The Koopman operator has received attention for providing a potentially global linearization representation of the nonlinear dynamical system. To estimate or control the original system, the invertibility problem is introduced into the data-driven modeling, i.e., the observables are required to be reconstructed the original system’s states. Existing methods cannot solve this problem perfectly. Only linear or nonlinear but lossy reconstruction can be achieved. This paper proposed a novel data-driven modeling approach, denoted as the Extended Dynamic Mode Decomposition with Invertible Dictionary Learning (EDMD-IDL) to address this issue, which can be interpreted as a further extension of the classical Extended Dynamic Mode Decomposition (EDMD). The Invertible Neural Network (INN) is introduced in the proposed method, where its inverse process provides the explicit inverse on the dictionary functions, thus allowing the nonlinear and lossless reconstruction. An iterative algorithm is designed to solve the extended optimization problem defined by the Koopman operator and INN by combining the optimization algorithm based on the gradient descent and the classical EDMD method, making the method successfully obtain the finite-dimensional approximation of the Koopman operator. The method is tested on various canonical nonlinear dynamical systems and is shown that the predictions obtained in a linear fashion and the ground truth match well over the long-term, where only the initial status is provided. Comparison experiments highlight the superiority of the proposed method over the other EDMD-based methods. Notably, a typical example in fluid dynamics, cylinder wake, illustrates the potential of the method to be further extended to the high-dimensional system with tens of thousands of states. By combining the Proper Orthogonal Decomposition technique, nontrivial Kármán vortex sheet phenomenon is perfectly reconstructed. Our proposed method provides a new paradigm for solving the finite-dimensional approximation of the Koopman operator and applying it to data-driven modeling.

Numerical analysis of the flow over four side-by-side square cylinders with different gaps

This study conducts two-dimensional numerical simulations of the flow over four square cylinders arranged side by side at a low Reynolds number (Re) of 100. The investigation primarily centers on the influence of the gap to a square cylinder width ratio (g*) on the flow. The range of g* spans from 0.1 to 7.0. Within this parameter range, three distinct flow regimes emerge based on the inherent flow characteristics. These regimes are defined as follows: (1) single bluff body flow (g* ≤ 0.3), (2) flip-flopping flow (0.3 < g* < 2.0), and (3) modulated periodic flow (g* ≥ 2.0). Additionally, the modulated periodic flow is further categorized into three distinct flow patterns. Various aspects of these different flow regimes are examined, including vortex contours, velocity fields, and liquid force coefficients around the cylinders. Moreover, detailed illustrations are provided for the modulation behaviors in vortex structures and liquid force coefficients. Finally, the proper orthogonal decomposition technique is employed to identify and analyze the underlying spatial coherent structures in the flow field, offering further insights into the dynamic features of wakes.

Experimental investigation of flow-induced vibration and flow field characteristics of a flexible triangular cylinder

Flow-induced vibration (FIV) of a flexible cylinder with a triangular cross-section, allowed to oscillate in the cross-flow, inline and torsional direction, is studied experimentally through water tunnel tests. The dynamic response of the cylinder was studied for three different angles of attack ( $0^\circ, 30^\circ, 60^\circ$ ), at reduced velocities of 0.9–16.27, corresponding to Reynolds numbers of 364–3600. At the angle of attack of $0^\circ$ , vortex-induced vibration at low reduced velocity was observed, which transitioned to galloping at higher reduced velocities. At the angles of attack of $30^\circ$ and $60^\circ$ , galloping-type response was observed over the range of the reduced velocities tested. Our results show that the cylinder's torsional oscillation breaks the system's symmetry and affects the structural response at higher reduced velocities regardless of the angle of attack. The FIV response of the flexible triangular cylinder is distinct from that of a rigid flexibly mounted triangular cylinder due to torsional oscillation, spanwise flexibility and the two fixed boundary conditions limiting the amplitude of oscillation. Flow field analysis in the wake of the cylinder was done qualitatively and quantitatively using hydrogen bubble flow visualisation and time-resolved volumetric particle tracking velocimetry techniques, respectively. Our results show the existence of highly three-dimensional vortex structures in the wake of the cylinder. We studied the coupling between the vortex shedding modes in the wake of the cylinder and the structural vibration modes through the spatiotemporal mode analysis using the proper orthogonal decomposition technique to distinguish between different types of the FIV response observed.