Sampling and resolution characteristics in reduced order models of shallow water equations: Intrusive vs nonintrusive

We investigate the sensitivity of reduced order models (ROMs) to training data spatial resolution as well as sampling rate. In particular, we consider proper orthogonal decomposition (POD), coupled with Galerkin projection (POD-GP), as an intrusive ROM technique. For nonintrusive ROMs, we consider two frameworks. The first is using dynamic mode decomposition (DMD), and the second is based on artificial neural networks (ANNs). For ANN, we utilized a residual deep neural network, and for DMD we have studied two versions of DMD approaches; one with hard thresholding and the other with sorted bases selection. Also, we highlight the differences between mean subtracting the data (centering) and using the data without mean subtraction. We tested these ROMs using a system of 2D shallow water equations for four different numerical experiments, adopting combinations of sampling rates and spatial resolutions. For these cases, we found that the DMD basis obtained with hard thresholding is sensitive to sampling rate. The sorted DMD algorithm helps to mitigate this problem and yields more stabilized and converging solution. Furthermore, we demonstrate that both DMD approaches without mean subtraction provide significantly more accurate results than DMD with mean subtracting the data. On the other hand, POD is relatively insensitive to sampling rate and yields better representation of the flow field. Meanwhile, spatial resolution has little effect on both POD and DMD performances. Numerical results reveal that an ANN on POD subspace (POD-ANN) performs remarkably better than POD-GP and DMD in capturing system dynamics, even with a small number of modes.     

Efficiency of randomised dynamic mode decomposition for reduced order modelling

ABSTRACT

The purpose of this paper is the identification of a reduced order model (ROM) from numerical code output by non-intrusive techniques (i.e. not requiring projecting of the governing equations onto the reduced basis modes). In this paper, we perform a comparison between two methods of model order reduction based on dynamic mode decomposition (DMD). The first method is a deterministic (classic) DMD technique endowed with a dynamic filtering criterion of selection of modes used in the ROM model. The second method is an adaptive randomised DMD algorithm (ARDMD) based on a randomised singular value decomposition. This produced an accelerating algorithm, which is endowed with a few additional advantages. In addition, the reduced order model is guaranteed to satisfy the boundary conditions of the full model, which is crucial for surrogate modelling. For numerical illustration, we use the shallow water equations model.     

On linear and nonlinear aspects of dynamic mode decomposition

The approximation of reduced linear evolution operator (propagator) via dynamic mode decomposition (DMD) is addressed for both linear and nonlinear events. The 2D unsteady supersonic underexpanded jet, impinging the flat plate in nonlinear oscillating mode, is used as the first test problem for both modes. Large memory savings for the propagator approximation are demonstrated. Corresponding prospects for the estimation of receptivity and singular vectors are discussed. The shallow water equations are used as the second large‐scale test problem. Excellent results are obtained for the proposed optimized DMD method of the shallow water equations when compared with recent POD‐based/discrete empirical interpolation‐based model reduction results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Spatiotemporal representation of the dynamic modes in turbulent cavity flows

Proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) were used to extract the coherent structures in turbulent cavity flows. The spatiotemporal representation of the modes was achieved by performing the circular convolution of a change of basis on the data sequence, wherein the transformation function was extracted from the POD or DMD. The spatiotemporal representation of the modes provided significant insight into the evolutionary behavior of the structures. Self-sustained oscillations arise in turbulent cavity flows due to unsteady separation at the leading edge. The turbulent cavity flow at Re_D = 12,000 and a length to depth ratio L/D = 2 was analyzed. The dynamic modes extracted from the data clarified the presence of self-sustained oscillations. The spatiotemporal representation of the POD and DMD modes that caused self-sustained oscillations revealed the prevalent dynamics and evolutionary behavior of the coherent structures from their formation at the leading edge to their impingement at the trailing edge. A local minimum in the mode amplitude representing the energy contributions to the flow was observed upon the impingement of coherent structure at the trailing edge. The modal energy associated with the periodic formation of organized coherent structures followed by their dissipation upon impingement revealed the oscillatory behavior over time.     

Mode Decomposition on Surface-Mounted Cube

In this paper, the flow around the surface-mounted cube is decomposed into modes using Proper Orthogonal Decomposition (POD) and Koopman mode decomposition, respectively. The objective of the paper is twofold. Firstly, a comparison of the two decomposition methods for a highly separated flow is performed. Secondly, an evaluation of Detached Eddy Simulation (DES) for simulating a time-accurate flow, to be used as input data for the two mode decomposition methods, is accomplished. The knowledge on the accuracy and usefulness of the modes computed with from DES flow fields can then be the foundation for other studies for applied geometries in vehicle aerodynamics. The flow is simulated using DES, which enables time-accurate simulations on flows around realistic vehicle geometries. Most of the first eight modes computed with DES in a reference domain can also be found among the first eight computed with LES in reference work. Since the POD modes computed with DES resemble those computed with LES, the conclusion is that DES is suitable to use for mode decomposition. When comparing the POD and Koopman modes, many similarities can be found in both the spatial and temporal modes. For this case, where the flow contains a broad band of frequencies, it is concluded that the advantage of using Koopman modes, decomposing by frequency, cannot be fully utilized, and Koopman modes are very similar to the POD modes.     

Analysis of time-resolved PIV measurements of a confined turbulent jet using POD and Koopman modes

We present a comparative analysis of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) computed from experimental data of a turbulent, quasi 2-D, confined jet with co-flow (Re?=?11,500, co-flow ratio inner-to-outer flow ≈2:1). The experimental data come from high-speed 2-D particle image velocimetry. The flow is fully turbulent, and it contains geometry-dependent large-scale coherent structures; thus, it provides an interesting benchmark case for the comparison between POD and DMD. In this work, we address issues related to snapshot selections (1), convergence (2) and the physical interpretation (3) of both POD and DMD modes. We found that the convergence of POD modes follows the criteria of statistical convergence of the autocovariance matrix. For the computation of DMD modes, we suggest a methodology based on two criteria: the analysis of the residuals to optimize the sampling parameters of the snapshots, and a time-shifting procedure that allows us to identify the spurious modes and retain the modes that consistently appear in the spectrum. These modes are found to be the ones with nearly null growth rate. We then present the selected modes, and we discuss the way POD and DMD rank them. POD analysis reveals that the most energetic spatial structures are related to the large-scale oscillation of the inner jet (flapping); from the temporal analysis emerges that these modes are associated with a low-frequency peak at St?=?0.02. At this frequency, DMD identifies a similar mode, where oblique structures from the walls appear together with the flapping mode. The second most energetic group of modes identified is associated with shear-layer oscillations, and to a recirculation zone near the inner jet. Temporal analysis of these modes shows that the flapping of the inner jet might be sustained by the recirculation. In the DMD, the shear-layer modes are separated from the recirculation modes. These have large amplitudes in the DMD. In conclusion, the DMD modes with eigenvalues on the unit circle are found to be similar to the most energetic POD modes, although differences appear due to the fact that DMD isolates structures associated with one frequency only.     

On the use of sensitivity analysis in model reduction to predict flows for varying inflow conditions

The proper orthogonal decomposition (POD)‐based model reduction method is more and more successfully used in fluid flows. However, the main drawback of this methodology rests in the robustness of these reduced order models (ROMs) beyond the reference at which POD modes have been derived. Any variation in the flow or shape parameters within the ROM fails to predict the correct dynamics of the flow field. To broaden the spectrum of these models, the POD modes should have the global characteristics of the flow field over which the predictions are required. Mixing of snapshots with varying parameters is one way to improve the global nature of the modes but is computationally demanding because it requires full‐order solutions for a number of parameter values in order to assemble atextitrich enough database on which to perform POD. Instead, we have used sensitivity analysis (SA) to include the flow and shape parameters influence during the basis selection process to develop more robust ROMs for varying viscosity (Reynolds number), changing orientation and shape definition of bodies. This study aims at extending these ideas to inflow conditions to demonstrate the effectiveness of the proposed approach in capturing the effect of varying inflow on the dynamics of the flow over an elliptic cylinder. Numerical experiments show that the newly derived models allow for a more accurate representation of the flows when exploring the parameter space. Copyright © 2011 John Wiley & Sons, Ltd.     

An evolve‐then‐filter regularized reduced order model for convection‐dominated flows

In this paper, we propose a new evolve‐then‐filter reduced order model (EF‐ROM). This is a regularized ROM (Reg‐ROM), which aims to add numerical stabilization to proper orthogonal decomposition (POD) ROMs for convection‐dominated flows. We also consider the Leray ROM (L‐ROM). These two Reg‐ROMs use explicit ROM spatial filtering to smooth (regularize) various terms in the ROMs. Two spatial filters are used: a POD projection onto a POD subspace (Proj) and a POD differential filter (DF). The four Reg‐ROM/filter combinations are tested in the numerical simulation of the three‐dimensional flow past a circular cylinder at a Reynolds number Re=1000. Overall, the most accurate Reg‐ROM/filter combination is EF‐ROM‐DF. Furthermore, the spatial filter has a higher impact on the Reg‐ROM than the regularization used. Indeed, the DF generally yields better results than Proj for both the EF‐ROM and L‐ROM. Finally, the CPU times of the four Reg‐ROM/filter combinations are orders of magnitude lower than the CPU time of the DNS. Copyright © 2017 John Wiley & Sons, Ltd.     

Mode decomposition of a noise suppressed mixing layer

Noise is generated in atwo-dimensional mixing layer due to the growing of instability waves and vortex pairings. The adjoint-based control methodology has shown to be arobust tool to suppress noise radiation. The mode decomposition algorithms such as the compressible versionof proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are employed toanalyze thespatial/spatial-temporal coherent structures for a consecutive data sets of the controlled mixing layer and itsuncontrolled counterpart. The analyses of POD indicate that the y-direction body forcecontrol mainly modify themost energetic spatialstructures, and increase the uniformity of the flow. The analyses of DMD show us prevalent frequencies andcorresponding mode structures, and the stability characteristics of each mode can be obtained fromDMD-spectrum. The spectral signatures illustrate that a lot of neutral/slightly damping modesemerging in uncontrolled flow within the frequency range (ω < 0.4) are suppressed due to control, relevant spatial-temporal structures are also varied, which iscoincident with the change of far-field noise spectra. From the view of mode decomposition, the action of control redistribute the energy forfrequency components of ω < 0.4 by weakening nonlinearities and regularizing corresponding dynamicstructures in streamwise direction, and thus suppress the noise radiation. Moreover, the POD- and DMD-analysis in this studydemonstrate that DMD can serve as an important supplement for POD in analyzing a time-resolved physicalprocess.     

Data-driven feature identification and sparse representation of turbulent flows

Identifying coherent structures in fluid flows is of great importance for reduced order modelling and flow control. However, extracting such structures from experimental or numerical data obtained from a turbulent flow can be challenging. A number of modal decomposition algorithms have been proposed in recent years which decompose time-resolved snapshots of data into spatial modes, each associated with a single frequency and growth-rate. Most prominently among them is dynamic mode decomposition (DMD). However, DMD-like algorithms create an arbitrary number of modes. It is common practice to then choose a smaller subset of these modes, for the purpose of model reduction and analysis, based on some measure of significance. In this work, we present a method of post-processing DMD modes for extracting a small number of dynamically relevant modes. We achieve this through an iterative approach based on the graph-theoretic notion of maximal cliques to identify clusters of modes and representing each cluster with a single representative mode.     

Dynamical mode decomposition of Gurney flap wake flow

The present work uses dynamic mode decomposition (DMD) to analyze wake flow of NACA0015 airfoil with Gurney flap. The physics of DMD is first introduced. Then the PIV-measured wake flow velocity field is decomposed into dynamical modes. The vortex shedding pattern behind the trailing edge and its high-order harmonics have been captured with abundant information such as frequency, wavelength and convection speed. It is observed that high-order dynamic modes convect faster than low-order modes; moreover the wavelength of the dynamic modes scales with the corresponding frequency in power law.     

2D Burgers equation with large Reynolds number using POD/DEIM and calibration

Model order reduction of the two‐dimensional Burgers equation is investigated. The mathematical formulation of POD/discrete empirical interpolation method (DEIM)‐reduced order model (ROM) is derived based on the Galerkin projection and DEIM from the existing high fidelity‐implicit finite‐difference full model. For validation, we numerically compared the POD ROM, POD/DEIM, and the full model in two cases of Re = 100 and Re = 1000, respectively. We found that the POD/DEIM ROM leads to a speed‐up of CPU time by a factor of O(10). The computational stability of POD/DEIM ROM is maintained by means of a careful selection of POD modes and the DEIM interpolation points. The solution of POD/DEIM in the case of Re = 1000 has an accuracy with error O(10^?3) versus O(10^?4) in the case of Re = 100 when compared with the high fidelity model. For this turbulent flow, a closure model consisting of a Tikhonov regularization is carried out in order to recover the missing information and is developed to account for the small‐scale dissipation effect of the truncated POD modes. It is shown that the computational results of this calibrated ROM exhibit considerable agreement with the high fidelity model, which implies the efficiency of the closure model used. Copyright © 2016 John Wiley & Sons, Ltd.     

Efficient unsteady aerodynamic loads prediction based on nonlinear system identification and proper orthogonal decomposition

In the present work, an efficient surrogate-based framework is developed for the prediction of motion-induced surface pressure fluctuations and integral force and moment coefficients. The model construction is realized by performing forced-motion computational fluid dynamics (CFD) simulations, while the result is processed via the proper orthogonal decomposition (POD) to obtain the predominant flow modes. Subsequently, a nonlinear system identification is carried out with respect to the applied excitation and the resulting POD coefficients. For the input/output model identification task, a recurrent local linear neuro-fuzzy approach is employed in order to capture the linear and nonlinear characteristics of the dynamic system. Once the reduced-order model (ROM) is trained, it can substitute the flow solver within unsteady aerodynamic or aeroelastic simulation frameworks for a given configuration at fixed freestream conditions. For demonstration purposes, the ROM approach is applied to the LANN wing in high subsonic and transonic flow. Due to the characteristic lambda-shock system, the unsteady aerodynamic surface pressure distribution is dominated by nonlinear effects. Numerical investigations show a good correlation between the results obtained by the ROM methodology in comparison to the full-order CFD solution. In addition, the surrogate approach yields a significant speed-up regarding unsteady aerodynamic calculations, which is beneficial for multidisciplinary computations.     

International journal of computational fluid dynamics real-time prediction of unsteady flow based on POD reduced-order model and particle filter

An integrated method consisting of a proper orthogonal decomposition (POD)-based reduced-order model (ROM) and a particle filter (PF) is proposed for real-time prediction of an unsteady flow field. The proposed method is validated using identical twin experiments of an unsteady flow field around a circular cylinder for Reynolds numbers of 100 and 1000. In this study, a PF is employed (ROM-PF) to modify the temporal coefficient of the ROM based on observation data because the prediction capability of the ROM alone is limited due to the stability issue. The proposed method reproduces the unsteady flow field several orders faster than a reference numerical simulation based on Navier–Stokes equations. Furthermore, the effects of parameters, related to observation and simulation, on the prediction accuracy are studied. Most of the energy modes of the unsteady flow field are captured, and it is possible to stably predict the long-term evolution with ROM-PF.     

Principal interval decomposition framework for POD reduced‐order modeling of convective Boussinesq flows

A principal interval decomposition (PID) approach is presented for the reduced‐order modeling of unsteady Boussinesq equations. The PID method optimizes the lengths of the time windows over which proper orthogonal decomposition (POD) is performed and can be highly effective in building reduced‐order models for convective problems. The performance of these POD models with and without using the PID approach is investigated by applying these methods to the unsteady lock‐exchange flow problem. This benchmark problem exhibits a strong shear flow induced by a temperature jump and results in the Kelvin–Helmholtz instability. This problem is considered a challenging benchmark problem for the development of reduced‐order models. The reference solutions are obtained by direct numerical simulations of the vorticity and temperature transport equations using a compact fourth‐order‐accurate scheme. We compare the accuracy of reduced‐order models developed with different numbers of POD basis functions and different numbers of principal intervals. A linear interpolation model is constructed to obtain basis functions when varying physical parameters. The predictive performance of our models is then analyzed over a wide range of Reynolds numbers. It is shown that the PID approach provides a significant improvement in accuracy over the standard Galerkin POD reduced‐order model. This numerical assessment of the PID shows that it may represent a reliable model reduction tool for convection‐dominated, unsteady‐flow problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Reduced order modelling method via proper orthogonal decomposition (POD) for flow around an oscillating cylinder

This paper presents reduced order modelling (ROM) in fluid–structure interaction (FSI). The ROM via the proper orthogonal decomposition (POD) method has been chosen, due to its efficiency in the domain of fluid mechanics. POD-ROM is based on a low-order dynamical system obtained by projecting the nonlinear Navier–Stokes equations on a smaller number of POD modes. These POD modes are spatial and temporally independent. In FSI, the fluid and structure domains are moving, owing to which the POD method cannot be applied directly to reduce the equations of each domain. This article proposes to compute the POD modes for a global velocity field (fluid and solid), and then to construct a low-order dynamical system. The structure domain can be decomposed as a rigid domain, with a finite number of degrees of freedom. This low-order dynamical system is obtained by using a multiphase method similar to the fictitious domain method. This multiphase method extends the Navier–Stokes equations to the solid domain by using a penalisation method and a Lagrangian multiplier. By projecting these equations on the POD modes obtained for the global velocity field, a nonlinear low-order dynamical system is obtained and tested on a case of high Reynolds number.     

Three-dimensional vortex wake structure of a flapping-wing micro aerial vehicle in forward flight configuration

The unsteady flow field past a backward-facing step in a rectangular duct is investigated by adopting time-resolved particle image velocimetry (PIV) in the Reynolds number range of 2,640–9,880 based on step height and the inlet average velocity. The PIV realizations are subjected to post-processing techniques, namely, proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). At low Reynolds numbers, the second spatial POD modes indicate the presence of the shear layer mode, whereas this feature shifts to higher modes at higher Reynolds numbers. The corresponding temporal modes are Fourier-transformed to obtain the dominant frequency, whose Strouhal number corroborates the above observation. Short-time windows in the transverse velocity component along the shear layer are selected to investigate the temporal stability of the flow field by DMD to quantify the growth rate of the shear layer mode. The higher harmonics of this mode are also observed to grow, albeit at lesser rate. By relating to POD analysis, the most energetic structures were found to correspond to the unstable modes. The correlation between these unstable DMD modes and the Fourier-filtered flow fields for the same frequencies indicate better match for the lower operating Reynolds number case as compared to higher ones. The spatial stability analysis demonstrates the growth of the shear layer vortices, which is combined with the temporal stability analysis to evaluate the phase velocity of the identified shear layer structures. The calculated phase velocity magnitude of the shear layer is found to be reasonably below the local velocity as expected.     

Low-order modelling of shallow water equations for sensitivity analysis using proper orthogonal decomposition

This article presents a reduced-order model (ROM) of the shallow water equations (SWEs) for use in sensitivity analyses and Monte-Carlo type applications. Since, in the real world, some of the physical parameters and initial conditions embedded in free-surface flow problems are difficult to calibrate accurately in practice, the results from numerical hydraulic models are almost always corrupted with uncertainties. The main objective of this work is to derive a ROM that ensures appreciable accuracy and a considerable acceleration in the calculations so that it can be used as a surrogate model for stochastic and sensitivity analyses in real free-surface flow problems. The ROM is derived using the proper orthogonal decomposition (POD) method coupled with Galerkin projections of the SWEs, which are discretised through a finite-volume method. The main difficulty of deriving an efficient ROM is the treatment of the nonlinearities involved in SWEs. Suitable approximations that provide rapid online computations of the nonlinear terms are proposed. The proposed ROM is applied to the simulation of hypothetical flood flows in the Bordeaux breakwater, a portion of the ‘Rivière des Prairies' located near Laval (a suburb of Montreal, Quebec). A series of sensitivity analyses are performed by varying the Manning roughness coefficient and the inflow discharge. The results are satisfactorily compared to those obtained by the full-order finite volume model.     

Numerical analysis of characteristic features of shallow and deep cavity in supersonic flow

Flow past open cavities are numerically simulated at a Mach number of 1.5, and Reynolds number, based on initial momentum thickness at the front lip of cavity, of 3333 for variable depths (D) with constant length (L). The dominant frequency of oscillation shows a sudden jump when there is a transition from shallow (L/D > 1) to deep cavity (L/D < 1). The vorticity thickness displays two different growth rates along the length of cavity: (1) initial lower spreading rate, followed by (2) higher spreading rate. The lower spreading rate of shear layer is dictated by the type of cavity (either shallow or deep), while the higher spreading rate is directly related to the amplitude of oscillations. Proper orthogonal decomposition (POD) is implemented to visualise the coherent structures based on their energy content. The first two POD spatial structures in the shallow cavity represent vortex shedding, while in the deep cavity, they comprise vortex pairing interactions as in mixing layer. The higher POD modes contain coherent structures at mixed frequencies. The behaviour of coherent structures associated with a temporal frequency is further investigated using dynamic mode decomposition (DMD). The higher DMD modes confirm the dominance of mixing layer behaviour in the deep cavity.