Proper Orthogonal Decomposition Algorithm Research Articles

A Streamline Upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations Under Optimal Control

Abstract In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Péclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov–Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.

Flow fields prediction for data-driven model of 5 × 5 fuel rod bundles based on POD-RBFNN surrogate model

The fuel rod bundles are a crucial component of nuclear reactors. Research the flow characteristics of the fuel rod bundles can be computationally expensive, despite some progress made in the past few decades. In this paper, the fast prediction of flow fields for 5 × 5 fuel rod bundles is implemented based on a data-driven algorithm. First, a refined model of the 5 × 5 fuel rod bundles with spacer grid is established, and meshes are generated using a hybrid meshes technology. Then, the flow characteristics of the 5 × 5 fuel rod bundles are extensively analyzed using Computational Fluid Dynamics (CFD) methods. A comparison between the CFD results and experimental data reveals a strong agreement, thereby validating our research findings. Subsequently, a reduced-order model (ROM) called the proper orthogonal decomposition (POD)-radial basis function neural network (RBFNN) surrogate model is proposed. the POD algorithm is utilized to extracting the dominant flow modes, and the RBFNN is employed to predict the POD mode coefficients of non-sample points. By linearly combining the predicted POD mode coefficients with the POD modes, rapid prediction of flow fields for non-sample points can be achieved. Compared to the full-scale CFD simulations, it finds that the proposed POD-RBFNN surrogate model not only significantly improved efficiency, but also maintained a high level of predictive accuracy. It is believed that the research results hold important value for CFD calculations within nuclear reactor cores.

Data-driven identification and pressure fields prediction for parallel twin cylinders based on POD and DMD method

The mode analysis of parallel twin cylinders is conducted in this paper using two data-driven methods: proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). First, a high-fidelity computational fluid dynamics (CFD) model of parallel twin cylinders is established, and numerical simulations of the model are carried out. Subsequently, the fundamental principles of the POD and DMD algorithms are systematically introduced. Utilizing snapshots obtained from the high-fidelity CFD model, the POD and DMD methods are employed to extract the dominant flow structures. Furthermore, a comparison between the two data-driven methods is conducted by analyzing modal frequencies, pressure distribution, and the reconstruction errors of pressure fields. Finally, the pressure fields of non-sample points are predicted based on the POD–backpropagation neural network (BPNN) surrogate model and the DMD method, and the predicted results are compared with the CFD simulation results. It found that (i) the DMD method is capable of extracting the main coherent structures of the pressure fields, directly obtaining flow modes and their corresponding frequencies, and assessing the stability of flow modes; (ii) the DMD method can capture the main flow features of the pressure fields in both spatial and temporal dimensions, while the POD method is primarily efficient at capturing the spatial features of the pressure fields; (iii) in contrast to the frequency-ranked DMD method, the energy-ranked POD method can reconstruct the pressure fields using a smaller number of modes, indicating that the POD method has an advantage in terms of mode reduction; (iv) in contrast to the energy-ranked POD method, the frequency-ranked DMD method has a wider applicability to the range of flow types and has more advantages in stability analysis of complex dynamic systems; (v) the predicted pressure fields around the cylinder using the first five-order POD modes or DMD modes closely align with CFD calculation results. Additionally, the evolution of pressure fields predicted by the POD–BPNN surrogate model with the first five-order POD modes or the DMD method with the first 200-order DMD modes significantly agrees with CFD simulation results; (vi) the combined use of the POD–BPNN surrogate model and DMD methods allows efficient interpolation and extrapolation of samples, delivering exceptional predictive performance. This study offers insight into the coherent structures in parallel twin cylinders.

Flow fields prediction for data-driven model of parallel twin cylinders based on POD-RBFNN and POD-BPNN surrogate models

Flow around cylinders is an important phenomenon in many different engineering fields. In this paper, the fast prediction of the pressure fields of parallel twin cylinders is implemented based on a data-driven algorithm. Firstly, the pressure fields of parallel twin cylinders with a low Reynolds number are obtained through the Computational Fluid Dynamics (CFD) method. The pressure fields at different time steps are collected to form a snapshot matrix. The Proper Orthogonal Decomposition (POD) algorithm is then applied to obtain the POD basis vectors of the snapshot matrix, enabling the reconstruing the pressure fields. Subsequently, two reduced-order models (ROM) called the POD-RBFNN and POD-BPNN surrogate models are proposed in this paper. The POD-RBFNN surrogate model uses the Radial Basis Function Neural Network (RBFNN) to train the POD mode coefficients obtained from the POD algorithm, while the POD-BPNN surrogate model uses the Backpropagation Neural Network (BPNN) for the same purpose. Linearly combining the POD mode coefficients predicted by the POD-RBFNN or POD-BPNN surrogate models with the POD basis vectors obtained from the POD algorithm enables fast and efficient prediction of pressure fields for non-sample points. Finally, comparisons are made between the predicted pressure fields obtained from these two surrogate models and the actual values obtained through CFD simulations. It is found that both the POD-RBFNN and POD-BPNN surrogate models proposed in this paper not only significantly improve efficiency but also maintain a high level of accuracy. However, the training time of the POD-RBFNN surrogate model is significantly shorter than that of the POD-BPNN surrogate model. Additionally, the POD-RBFNN surrogate model exhibits smaller Root Mean Square Errors (RMSE) and Mean Absolute Error (MAE). For the data-driven model of parallel twin cylinders described in this paper, the POD-RBFNN surrogate model is more suitable to predict the pressure fields. The research results in this paper are believed to hold significant value for CFD calculations of parallel twin cylinder models, offering essential guidance for a deeper understanding of cylinder flow problems and optimizing engineering design.

Intelligent flow field reconstruction based on proper orthogonal decomposition dimensionality reduction and improved multi-branch convolution fusion

The rapid and accurate reconstruction of the supersonic combustor flow field is of great significance for sensing and predicting the combustion state. Existing deep learning methods pay less attention to the convergence speed of flow field reconstruction, which results in longer training and prediction times for the models. This study proposes a method for reconstructing the flow field in supersonic combustor by combining a reduced-order model based on proper orthogonal decomposition (POD) with a multi-branch convolutional neural network. This method first analyzes the effectiveness of POD reconstruction. Then, based on the wall pressure data of the supersonic engine combustor, it performs flow field image reconstruction. Finally, through error calculation and gradient updating with low-resolution principal component flow field shadow images obtained from the POD algorithm, the high-precision and efficient prediction of flow field images is achieved. Different equivalence ratio hydrogen fuel combustion experiments were conducted in a pulsed combustion wind tunnel with an incoming flow Mach number of 2.5. The learning model was trained and tested using the dataset obtained from these experiments. Numerous experiments demonstrated that the model can effectively reconstruct the wave structures of complex flow fields. Multiple evaluation indicators indicated that the reconstructed flow field of the combustor shows good agreement with that obtained from ground wind tunnel testing. Furthermore, after introducing the POD dimensionality reduction model, the training time was reduced by 32.03%, effectively improving the training time complexity of the model.

Imperfection-induced internal resonance in nanotube resonators

Through molecular dynamics simulations, we demonstrate the possibility of internal resonances in single-walled carbon nanotubes. The resonant condition is engineered with a lack of symmetry in the boundary condition and activated by increasing the energy exchange with a coupled thermal bath. The critical temperature threshold for initiating modal interaction is found to be chirality-dependent. By applying the proper orthogonal decomposition algorithm to molecular dynamics time responses, we show how the thermal fluctuations influence the vibrational behaviour of the nanotube leading to both flexural–flexural and flexural–longitudinal resonances. Understanding the interaction between nanotube resonators and the thermal bath is crucial for designing and optimizing their performance for various nanoscale sensing, actuation, and signal processing applications.

Phase proper orthogonal decomposition of non-stationary turbulent flow

A phase proper orthogonal decomposition (phase POD) method is demonstrated utilizing phase averaging for the decomposition of spatiotemporal behavior of statistically non-stationary turbulent flows in an optimized manner. The proposed phase POD method is herein applied to a periodically forced statistically non-stationary lid-driven cavity flow, implemented using the snapshot proper orthogonal decomposition algorithm. Space-phase modes are extracted to describe the dynamics of the chaotic flow, in which four central flow patterns are identified for describing the evolution of the energetic structures as a function of phase. The modal building blocks of the energy transport equation are demonstrated as a function of the phase. The triadic interaction term can here be interpreted as the convective transport of bi-modal interactions. Non-local energy transfer is observed as a result of the non-stationarity of the dynamical processes inducing triadic interactions spanning across a wide range of mode numbers.

Machine learning and reduced order modelling for the simulation of braided stent deployment.

Endoluminal reconstruction using flow diverters represents a novel paradigm for the minimally invasive treatment of intracranial aneurysms. The configuration assumed by these very dense braided stents once deployed within the parent vessel is not easily predictable and medical volumetric images alone may be insufficient to plan the treatment satisfactorily. Therefore, here we propose a fast and accurate machine learning and reduced order modelling framework, based on finite element simulations, to assist practitioners in the planning and interventional stages. It consists of a first classification step to determine a priori whether a simulation will be successful (good conformity between stent and vessel) or not from a clinical perspective, followed by a regression step that provides an approximated solution of the deployed stent configuration. The latter is achieved using a non-intrusive reduced order modelling scheme that combines the proper orthogonal decomposition algorithm and Gaussian process regression. The workflow was validated on an idealized intracranial artery with a saccular aneurysm and the effect of six geometrical and surgical parameters on the outcome of stent deployment was studied. We trained six machine learning models on a dataset of varying size and obtained classifiers with up to 95% accuracy in predicting the deployment outcome. The support vector machine model outperformed the others when considering a small dataset of 50 training cases, with an accuracy of 93% and a specificity of 97%. On the other hand, real-time predictions of the stent deployed configuration were achieved with an average validation error between predicted and high-fidelity results never greater than the spatial resolution of 3D rotational angiography, the imaging technique with the best spatial resolution (0.15mm). Such accurate predictions can be reached even with a small database of 47 simulations: by increasing the training simulations to 147, the average prediction error is reduced to 0.07mm. These results are promising as they demonstrate the ability of these techniques to achieve simulations within a few milliseconds while retaining the mechanical realism and predictability of the stent deployed configuration.

A constrained proper orthogonal decomposition model for upscaling permeability

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

Equivalent-Input-Disturbance-Based Dynamic Tracking Control for Soft Robots via Reduced-Order Finite-Element Models

This article develops a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">systematic</i> framework for dynamic tracking control of soft robots. To this end, we propose a new projector for the proper orthogonal decomposition algorithm to significantly reduce the large-scale robot models, obtained from finite-element methods (FEM), while preserving their structure and stability properties. Such a property preservation enables an effective equivalent-input-disturbance-based scheme for dynamic tracking control of elastic soft robots with various geometries and materials. The proposed control scheme is composed of three key components, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> , feedforward control, disturbance-estimator control, and feedback control. To account for the trajectory reference, the feedforward action is designed from the dynamic FEM reduced-order robot model. The disturbance-estimator control action is obtained from an unknown input observer, which also provides the estimates of the reduced states for the feedback control design. The feedback gains of the observer-based controller are computed from an optimization problem under linear matrix inequality constraints. The closed-loop tracking properties are guaranteed using the Lyapunov stability theory. The effectiveness of the proposed dynamic control framework has been demonstrated via both high-fidelity SOFA simulations and experimental validations, performed on two soft robots with different natures. In particular, comparative studies with state-of-the-art control methods have been also carried out to highlight the interests of the new soft robot control results. This article is complemented with a video: <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://bit.ly/2VVwtLn</uri> .

Transient characteristics of a typical vacuum ejector—An experimental study

Using simultaneous measurements of unsteady pressures in conjunction with time-resolved Schlieren images and oil flow visualization, we investigate the characteristics of a typical vacuum ejector's starting transient, steady-state, and shut-down transient. With varying primary jet chamber pressure, the pressure evolution in the secondary chamber shows smooth, perturbed, rapid, and steady evacuation stages, as well as hysteresis and rapid filling stages. It is noticed that the evacuation in the secondary chamber is improved during stopping transient just before the unstart event. By using oil flow images, we illustrate the separation bubble characteristics during each stage of the vacuum ejector operation and their influence on the pressure evolution. Through cross correlation, it has been determined that the primary jet flapping during the starting transient causes the jet to attach to one of the diffuser walls. We also demonstrate that the primary jet undergoes both longitudinal and lateral oscillations in the starting transient, the former having a major effect on unsteadiness in the secondary chamber using proper orthogonal decomposition and spectral proper orthogonal decomposition algorithms and power spectral density (PSD). Simultaneous acquisition of unsteady pressures and high-speed Schlieren images allow us to correlate the frequency peaks (PSD spectra) in the flow. Using magnitude-squared coherence and cross correlation analyses, we confirm communication of unsteadiness and its direction of propagation between the secondary chamber and the diffuser. In this study, we demonstrate that a high ramping rate of primary jet chamber pressure reduces the unsteadiness in the secondary chamber during the transient starting phase.

Efficient algorithm for proper orthogonal decomposition of block-structured adaptively refined numerical simulations

Adaptive mesh refinement (AMR) is increasingly being used to simulate fluid flows that have vastly different resolution requirements throughout the computational domain. Proper orthogonal decomposition (POD) is a common tool to extract coherent structures from flow data and build reduced order models, but current POD algorithms do not take advantage of potential efficiency gains enabled by multi-resolution data from AMR simulations. Here, we explore a new method for performing POD on AMR data that eliminates repeated operations arising from nearest-neighbor interpolation of multi-resolution data onto uniform grids. We first outline our approach to reduce the number of computations with examples and provide the complete algorithms in the appendix. We examine the computational acceleration of the new algorithms compared to the standard POD method using synthetically generated AMR data and operation counting. We then use CPU times and operation counting to analyze data from an AMR simulation of an axisymmetric buoyant plume, finding that we are able to reduce the computational time by a factor of approximately 2−5 when using three levels of grid refinement. The new POD algorithm is the first to eliminate redundant operations for matrix multiplications with repeated values in each matrix, making it ideal for POD of data from AMR simulations.

Data-driven reduced order modeling for parametrized time-dependent flow problems

This paper proposes a nonintrusive reduced basis (RB) method based on dynamic mode decomposition (DMD) for parameterized time-dependent flows. In the offline stage, the reduced basis functions are extracted by a two-step proper orthogonal decomposition algorithm. Then, a novel hybrid DMD regression model that combines windowed DMD and optimized DMD is introduced for the temporal evolution of the RB coefficients. To improve the stability of this method for complex nonlinear problems, we introduce a threshold value to modify the DMD eigenvalues and eigenvectors. Moreover, the interpolation of the coefficients in parameter space is conducted by a feedforward neural network or random forest algorithm. The prediction of the RB solution at a new time/parameter value can be recovered at a low computational cost in the online stage, which is completely decoupled from the high-fidelity dimension. We demonstrate the performance of the proposed model with two cases: (i) laminar flow past a two-dimensional cylinder and (ii) turbulent flow around a three-dimensional SD7003 airfoil. The results show reasonable efficiency and robustness of this novel reduced-order model.

Error analysis of proper orthogonal decomposition data assimilation schemes with grad–div stabilization for the Navier–Stokes equations

The error analysis of a proper orthogonal decomposition (POD) data assimilation (DA) scheme for the Navier–Stokes equations is carried out. A grad–div stabilization term is added to the formulation of the POD method. Error bounds with constants independent on inverse powers of the viscosity parameter are derived for the POD algorithm. No upper bounds in the nudging parameter of the data assimilation method are required. Numerical experiments show that, for large values of the nudging parameter, the proposed method rapidly converges to the real solution, and greatly improves the overall accuracy of standard POD schemes up to low viscosities over predictive time intervals.

A Data-Driven Model Reduction Method for Parabolic Inverse Source Problems and its Convergence Analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems efficiently. Our method consists of offline and online stages. In the off-line stage, we explore the low-dimensional structures in the solution space of the parabolic partial differential equations (PDEs) in the forward problem with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problem extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in the parabolic inverse source problems, which is referred to as the POD algorithm in this paper. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD algorithm in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD algorithm for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the POD algorithm provides considerable computational savings over the finite element method.

Adaptive proper orthogonal decomposition for large scale reliable soil moisture estimation

A major challenge in automatic irrigation of extensive agricultural fields is large scale soil moisture monitoring. Proper orthogonal decomposition (POD) is a widespread data-driven dimension reduction technique which can be combined with QR pivoting method for estimation of high-dimensional signals and optimal sensor placement. However, it requires computation of tailored basis functions which should be extracted from known training data. This is feasible for problems with constant features such as face recognition. However, using fixed bases (and probably fixed sensor selection) may not be an appropriate approach for estimation of signals with time-variant features. This paper demonstrates that the POD algorithm can be implemented adaptively in a reliable fashion to address this issue. The probability of the mean estimation error to remain within a predetermined threshold is studied and theoretical results on permitted update intervals are provided. The adaptive tailored basis and sensor selection increases suitability to large scale applications with time-variant features. The performance of the algorithm on a NASA soil moisture dataset is analyzed.

Hilbert transform revisited – Proper orthogonal decomposition applied to analytical signals of flow fields

The modes delivered by proper orthogonal decomposition (POD) are uncorrelated as per definition; but interestingly, they are not necessarily independent in terms of spatio-temporal flow-pattern dynamics. For instance, periodic structures that travel as waves through a series of snapshots often consist of pairs of modes with harmonic functions shifted 90 degree in phase and/or a spatial offset by a quarter of the spatial wave length of the convective flow pattern. Identification of such pairs, however, largely builds upon experience, visual inspection and/or the analysis of the reconstructed coefficients in cyclograms (Lissajous figures). This effort becomes even more challenging if measurement noise or other spurious information contaminates the raw data under consideration. One possibility to automatically pair corresponding patterns with common POD algorithms is the immediate application of the POD method to complex data (see Pfeffer et al., 1990). As outlined by Horel (1984), the Hilbert transform is a well-known and straight forward means to obtain the required extension of the original signal with an appropriate 90 degrees phase shift, which is independent of the fundamental frequencies. The complex extension of the original (real) signal Xi and its (discrete) Hilbert transform HT{Xi} as the imaginary part Xi +iHT{Xi} with the imaginary unit i is commonly known as the so-called analytical signal.

Experimental investigation of turbulent coherent structures interacting with a porous airfoil

The flow field on solid and porous airfoils subjected to turbulence shed by an upstream cylindrical rod and the corresponding far-field noise radiations are studied through particle image velocimetry (PIV) and microphone measurements, respectively. Three different Reynolds numbers based on the rod diameter are considered in a range between $${2.7\times 10^4}$$ and $${5.4\times 10^4}$$ , and two porous airfoil models are tested to analyze the influence of the design elements of the permeable treatment. A standard proper orthogonal decomposition (POD) algorithm is employed to band filter the different length scales that characterize the turbulent flow, making it feasible to determine which turbulence scales are affected by porosity. The aeroacoustic results indicate that the porous treatment of the wing profile leads to a noise reduction at low frequencies and a noise regeneration at high frequencies due to surface roughness. The investigation on the flow field shows that the main effect of porosity is to mitigate the turbulent kinetic energy in the stagnation region, attenuating the distortion of turbulence interacting with the airfoil surface. The application of the POD algorithm indicates that this effect acts mainly on the largest scales of turbulence.

The Optimization Algorithm for Large-Scale In Situ Stress Field

In situ stress state is a predominant factor for the design and safe construction of geotechnical engineering. For a real construction site, the amount of calculation using a finite element method for in situ stress field increases dramatically with the increase of the calculation freedom due to large-scale uncertainties. In order to reduce the computing cost without losing the accuracy of the calculation, an optimization algorithm combined with a reduced order model, which is realized by the proper orthogonal decomposition algorithm (POD) for large-scale in situ stress field, is put forward in this paper. The POD algorithm produces a set of orthogonal bases through the extraction of the field variables, combining with the Galerkin finite element method to create a reduced order numerical model. The reduced order model is then calculated with a global optimization algorithm to inversely find the solution for the actual in situ stress field. In order to verify the accuracy and efficiency of the method, two examples are presented to simulate the inverse calculation of the in situ stress field. They showed that the computation time of the POD method could reach 1/10 of the ordinary computation time. Also, the results showed good accuracy with a minimum computational expense, which can provide a reference for inverse calculation of large-scale in situ stress field.

Self-similarity for statistical properties in low-order representations of a large-scale turbulent round jet based on the proper orthogonal decomposition

This study is an experimental investigation into the self-similarity behavior of first and second order statistical quantities derived from a jet flow based on a) the original data and b) its low-order representations derived from the Proper Orthogonal Decomposition (POD) and c) a comparison of both. The flow under investigation is an air-helium turbulent round jet with Re≈15400 emerging from a tube into an ambient containing identical gas mass fraction and temperature as the jet at a constant pressure. Instantaneous two-dimensional velocity field measurements were obtained for downstream distances of 5.5d to 17.4d in the plane of the axis of the jet, via Particle Image Velocimetry. The snapshot POD algorithm was then applied to this data set to generate low-order representations with rank approximations 1, 5, 10 and 50. These then serve as the basis to derive the respective (rank truncated) statistical properties. All properties are non-dimensionalized with a self-similar framework as obtained from the original jet data. It is found that the statistical properties obtained from the low-order representations a) resemble in shape the asymptotic outline of the original jet and b) that the maximum values (for a given low-order representation) exhibit asymptotic states with increasing downstream distances. This is a strong indication that i) self-similar behavior is equally found in the low-order representations and that ii) this finding is mainly controlled by the large-scale vortices. The sole exception is the axial velocity root-mean-square values, where a distinct dip in the center line of the flow is found. This dip is successively filled up by smaller-scale turbulence for higher order truncations. Additionally, a new criterion – based on the maximum cross-correlation obtained through successive time traces of the temporal POD modes – is suggested to distinguish physically relevant modes from the POD basis in a more quantitative and explicit manner compared to traditional energy-based criteria.