Proper Orthogonal Decomposition Technique Research Articles

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<!PCT!>, 72<!PCT!>, 52<!PCT!>, and 34<!PCT!> of the total wave energy of the four pores we examined is associated with the identified sausage modes and about 23<!PCT!>, 17<!PCT!>, 39<!PCT!>, and 49<!PCT!> with their kink modes, while the first pore also receives a contribution of about 11<!PCT!> 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.

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.

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.

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 behaviour 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 analyse via SPOD large dataset in turn prevents unlocking novel mechanisms and dynamical behaviours 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 analyse 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.

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.

Efficient Frequency and Time-Domain Simulations of Delayed PEEC Models With Proper Orthogonal Decomposition Techniques

Efficient Frequency and Time-Domain Simulations of Delayed PEEC Models With Proper Orthogonal Decomposition Techniques

Qualitative Investigation of Wake Composition in Offshore Wind Turbines: A Combined Computational and Statistical Analysis of Inner and Outer Blade Sections

High-fidelity numerical simulations are used to thoroughly analyze the evolution of the wake behind a megawatt-scale offshore wind turbine. The wake features are classified in terms of wake dynamics composition and the associated turbulence characteristics originating from the inner and outer sections of the blades. Understanding the wake is essential for developing compact layouts for future wind farms. We employed a transient Sliding Mesh Interface (SMI) technique to analyze the fully dynamic wake evolution of the offshore NREL 5MW full turbine. Our high-fidelity results have been validated against previously published results in the literature. We thoroughly investigated the dominant structures of the wake using Proper Orthogonal Decomposition (POD) techniques, which we applied to transient simulations of fully developed flows after five wind turbine revolutions over the snapshot data. Our findings show that the inner section of the blades, which is composed of airfoils with larger cross-sections, is responsible for the dominant components of the wake, while the contribution of the wake from the outer section of the blade is significantly lower. Therefore, designing more aerodynamic sections for the blade’s inner section can help reduce the dominant wake components and thus decrease the inter-turbine distance in future wind farms.

A predictive surrogate model for hemodynamics and structural prediction in abdominal aorta for different physiological conditions

Background and objectiveThis study investigates the application of a Predictive Surrogate Model (PSM) for the prediction of the fluid and solid variables in the abdominal aorta by integrating Proper Orthogonal Decomposition (POD) and Long Short-Term Memory (LSTM) techniques. MethodsThe Fluid-Structure Interaction (FSI) solver, which serves as the Full-Order Model (FOM), can capture the blood hemodynamics and structural mechanics precisely for a variety of physiological states, namely the rest and exercise conditions. ResultsDetailed analyses have been conducted on velocity components, pressure, Wall Shear Stress (WSS), and Oscillatory Shear Index (OSI) variables. Firstly, the reconstruction error has been derived based on a specific number of POD bases to assess the Reduced Order Model (ROM). Notably, the reconstruction error for velocity components in the rest condition is one order of magnitude higher than that in the exercise condition, yet both remained below 10%. This error for pressure is even more minimal, being less than 1%. ConclusionsThe PSM is evaluated against rest and exercise conditions, exhibiting promising results despite the inherent complexities of the physiological conditions. Despite the inherent complexities of phenomena in the aorta, the predictive model demonstrates consistent error magnitudes for velocity components and wall-related indices, while solid variables show slightly higher errors.

A Reduced-Dimension Extrapolating Method of Finite Element Solution Coefficient Vectors for Fractional Tricomi-Type Equation

We here employ a proper orthogonal decomposition (POD) to reduce the dimensionality of unknown coefficient vectors of finite element (FE) solutions for the fractional Tricomi-type equation and develop a reduced-dimension extrapolating FE (RDEFE) method for the fractional Tricomi-type equation. For this purpose, we first develop an FE method for the fractional Tricomi-type equation and provide the existence, unconditional stability, and error analysis for the FE solutions. We then develop the RDEFE method for the fractional Tricomi-type equation by means of the POD technique and analyze the existence, unconditional stability, and errors for the RDEFE solutions by using the matrix analysis. Lastly, we provide two numerical examples to verify our theoretical results and to explain the advantages of the RDEFE method.

An efficient compact difference-proper orthogonal decomposition algorithm for fractional viscoelastic plate vibration model

In this article, we aim to prove the solvability for the fractional viscoelastic plate vibration model through the Galerkin argument in Sobolev spaces as a compensation for the absence of its mathematical theory, then propose an efficient compact difference-proper orthogonal decomposition (CD-POD) algorithm. Rigorous numerical analyses are conducted to show that the CD-POD algorithm inherits the merits of the compact difference and the proper orthogonal decomposition technique, that is, it is unconditionally stable, possesses the spatial and temporal convergence rates of fourth order and second order, respectively, and reduces greatly the storage and computing cost, and thus may adapts for long time simulation for the plate vibration model. As a bridge for the plate vibration model and the CD-POD algorithm, a compact difference scheme is also proposed and analyzed previously. Numerical experiments, one of which is for resonance case, indicate that the CPU time of the CD-POD algorithm is at least 20 times less than that of the compact difference scheme, for resonance case or for non-resonance case, and thus verifies again that the CD-POD algorithm is a reasonable computation candidate for long time simulation of the viscoelastic plate vibration model.

An optimisation–based domain–decomposition reduced order model for the incompressible Navier-Stokes equations

The aim of this work is to present a model reduction technique in the framework of optimal control problems for partial differential equations. We combine two approaches used for reducing the computational cost of the mathematical numerical models: domain–decomposition (DD) methods and reduced–order modelling (ROM). In particular, we consider an optimisation–based domain–decomposition algorithm for the parameter–dependent stationary incompressible Navier–Stokes equations. Firstly, the problem is described on the subdomains coupled at the interface and solved through an optimal control problem, which leads to the complete separation of the subdomain problems in the DD method. On top of that, a reduced model for the obtained optimal–control problem is built; the procedure is based on the Proper Orthogonal Decomposition technique and a further Galerkin projection. The presented methodology is tested on two fluid dynamics benchmarks: the stationary backward–facing step and lid-driven cavity flow. The numerical tests show a significant reduction of the computational costs in terms of both the problem dimensions and the number of optimisation iterations in the domain–decomposition algorithm.

Reynolds number effect correction of multi-fidelity aerodynamic distributions from wind tunnel and simulation data

Conventional wind tunnels cannot simulate actual flight Reynolds numbers because of testing condition constraints. The changes in aerodynamic characteristics caused by differences in the Reynolds number are especially sensitive on transonic supercritical airfoils and wings. Accordingly, this study proposes a correction method for the Reynolds number effect based on feature extraction of multi-fidelity aerodynamic distributions. Correcting wind tunnel test results at low Reynolds numbers with large amounts of numerical simulation data across various Reynolds numbers allows this method to achieve greater precision on airfoil pressure distribution predictions at high Reynolds numbers. The first step is to extract the primary features of the surface pressure distribution of the supercritical airfoil using the proper orthogonal decomposition technique. Next, a multi-fidelity neural network model is employed to relate the (low-fidelity) simulation data to the (high-fidelity) wind tunnel data acquired at low Reynolds numbers. Finally, the model is used to predict the airfoil pressure distributions at high Reynolds numbers, and the predictions are validated using wind tunnel test data collected from the RAE2822 airfoil. The results indicate that this approach can provide more accurate estimations than numerical simulations and single-fidelity models. Additionally, it can effectively reduce the impact of the Reynolds number difference on the transonic aerodynamic properties of supercritical airfoils.

A production term study of delayed detached eddy simulation for turbulent near wake based on proper orthogonal decomposition

In this study, we investigate the distribution of turbulent kinetic energy (TKE) in the near wake of a circular cylinder in a turbulent flow. Numerical calculations were performed using the delayed detached eddy simulation method, incorporating two different production terms for TKE: one in its original form and the other with the Kato–Launder correction term. Our results demonstrate that the turbulence model utilizing the Kato–Launder correction term exhibits a strong correlation between TKE and the vorticity field, which is related to the calculations of mean velocity, velocity fluctuation, and other parameters that are in closer agreement with direct numerical simulation and experimental values. By employing the proper orthogonal decomposition technique, we extract and reconstruct three significant modes within the flow: the shear layer mode, vortex shedding mode, and near-wake bubble mode. The findings reveal that the Kato–Launder correction term offers a more detailed portrayal closer to the real flow physics. Conversely, the original form of the TKE production term exhibits an uneven energy distribution among the three modes and affects the role of the diffusion term within the flow. This leads to a less accurate representation of the vortex shedding mode and an excessive diffusion effect in the near-wake bubble mode. Finally, possible modifications of the turbulence model in this problem are given to enhance the portrayal of these characteristics. This work presents an analytical framework that enables a comprehensive analysis of turbulence models, providing valuable physical insights and guidance for improvement.

Flow structure and impinging interactions of two confined turbulent converging jets in crossflow

In this work, time-resolved particle image velocimetry (TR-PIV) measurements are carried out to study the flow structure and impinging interactions of two confined turbulent converging jets in crossflow, with the upstream and downstream jets inclined at 20° and 170° with respect to the oncoming crossflow direction. Experiments are conducted for fixed dimensionless nozzle separation distance of s=L/D=3, fixed dimensionless nozzle-to-target surface distance of H/D=6.73, fixed duct Reynolds number of Re=3500 and two jets' Reynolds numbers of Rej=3000 and 6000 that correspond to jets'-to-crossflow velocity ratios of rj=3.45 and 6.91, respectively. Flow visualization images showing ensemble-averaged and instantaneous flow distributions and turbulent characteristics including Reynolds stresses and turbulence intensities for equal and non-equal discharge velocities of the jets are presented on several orthogonal planes of the test section. The swirling strength criterion has been employed to identify coherent structures, and cross spectral analysis of the fluctuating velocity signals detecting subharmonic and harmonic sideband frequencies identify the merging and breakup interactions of the evolving coherent structures. The snapshot proper orthogonal decomposition technique (POD) has been applied to extract the dominant vortical structures and to quantify their relative and cumulative energy contributions to the total kinetic energy fluctuation. The spatial structure of the main four POD modes and the temporal projections of these POD coefficients have been explored to understand the spatio-temporal characteristics of the dominant turbulent fluctuations and their spectral characteristics. Our results show the effect of the velocity ratios on the jets' trajectories, penetration, spreading characteristics, crossflow entrainment through the inner and outer shear layers, transverse velocity decay, size of the converging region and location of the combining point. In addition, the dimensionless shedding frequencies of the oscillating combined jet, the developing wall jets and wake vortices have been obtained. Furthermore, it has been shown that for all values of the jets' velocity ratios employed in this study, a recirculation region does not develop in the converging region.

On the Usefulness of the Proper Orthogonal Decomposition on the Description of the Highly Concentrated Sediment Release Phenomena Resulting from a Two-Phase Solid-Fluid Simulation: Effect of the Ambient Current

On the Usefulness of the Proper Orthogonal Decomposition on the Description of the Highly Concentrated Sediment Release Phenomena Resulting from a Two-Phase Solid-Fluid Simulation: Effect of the Ambient Current