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.     

Advanced Proper Orthogonal Decomposition Tools: Using Reduced Order Models to Identify Normal Modes of Vibration and Slow Invariant Manifolds in the Dynamics of Planar Nonlinear Rods

Reduced order models for the dynamics of geometrically exact planar rods are derived by projecting the nonlinear equations of motion onto a subspace spanned by a set of proper orthogonal modes. These optimal modes are identified by a proper orthogonal decomposition processing of high-resolution finite element dynamics. A three-degree-of-freedom reduced system is derived to study distinct categories of motions dominated by a single POD mode. The modal analysis of the reduced system characterizes in a unique fashion for these motions, since its linear natural frequencies are near to the natural frequencies of the full-order system. For free motions characterized by a single POD mode, the eigen-vector matrix of the derived reduced system coincides with the principal POD-directions. This property reflects the existence of a normal mode of vibration, which appears to be close to a slow invariant manifold. Its shape is captured by that of the dominant POD mode. The modal analysis of the POD-based reduced order system provides a potentially valuable tool to characterize the spatio-temporal complexity of the dynamics in order to elucidate connections between proper orthogonal modes and nonlinear normal modes of vibration.     

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.     

A reduced‐order approach for optimal control of fluids using proper orthogonal decomposition

In this article, a reduced‐order modeling approach, suitable for active control of fluid dynamical systems, based on proper orthogonal decomposition (POD) is presented. The rationale behind the reduced‐order modeling is that numerical simulation of Navier–Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. The possibility of obtaining reduced‐order models that reduce the computational complexity associated with the Navier–Stokes equations is examined while capturing the essential dynamics by using the POD. The POD allows the extraction of a reduced set of basis functions, perhaps just a few, from a computational or experimental database through an eigenvalue analysis. The solution is then obtained as a linear combination of this reduced set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations (PDEs). It is used here in active control of fluid flows governed by the Navier–Stokes equations. In particular, flow over a backward‐facing step is considered. Reduced‐order models/low‐dimensional dynamical models for this system are obtained using POD basis functions (global) from the finite element discretizations of the Navier–Stokes equations. Their effectiveness in flow control applications is shown on a recirculation control problem using blowing on the channel boundary. Implementational issues are discussed and numerical experiments are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

A high‐fidelity low‐cost aerodynamic model using proper orthogonal decomposition

In this paper a high‐fidelity aerodynamic model is presented for use in parametric studies of weapon aerodynamics. The method employs a reduced‐order model obtained from the proper orthogonal decomposition (POD) of an ensemble of computational fluid dynamics (CFD) solutions with varying parameters. This decomposition produces an optimal linear set of orthogonal basis functions that best describe the ensemble of numerical solutions. These solutions are then projected onto this set of basis functions to provide a finite set of scalar coefficients that represent the solutions. A pseudo‐continuous representation of these projection coefficients is constructed, which allows predictions to be made of parameter combinations not in the original set of observations. The paper explores the performance of a few design‐of‐experiment approaches for the generation of the initial ensemble of computational experiments. Response surface construction methods based on parametric and non‐parametric models for the pseudo‐continuous representation of the projection coefficients are also evaluated. The model has been applied to two‐flow problems related to high‐speed weapon aerodynamics, inviscid flow around a flare‐stabilized hypersonic projectile and supersonic turbulent flow around a fin‐stabilized projectile with drooping nose control. Comparisons of model predictions with high‐fidelity CFD simulations suggest that the POD provides a reliable and robust approach to the construction of reduced‐order models. The practicality of the model is shown to be sensitive to the technique used to generate the ensemble of observations from which the model is constructed, while the accuracy of the approach depends on the pseudo‐continuous representation of the projection coefficients. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Modal decomposition‐based global stability analysis for reduced order modeling of 2D and 3D wake flows

The method for computation of stability modes for two‐ and three‐dimensional flows is presented. The method is based on the dynamic mode decomposition of the data resulting from DNS of the flow in the regime close to stable flow (fixed‐point dynamics, small perturbations about steady flow). The proposed approach is demonstrated on the wake flows past a 2D, circular cylinder, and a sphere. The resulting modes resemble the eigenmodes computed conventionally from global stability analysis and are used in model order reduction of the flow. The designed low‐dimensional Galerkin model uses continuous mode interpolation between dynamic mode decomposition mode bases and reproduces the dynamics of Navier–Stokes equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Closed-loop control of a free shear flow: a framework using the parabolized stability equations

In this study the parabolized stability equations (PSE) are used to build reduced-order-models (ROMs) given in terms of frequency and time-domain transfer functions (TFs) for application in closed-loop control. The control law is defined in two steps; first it is necessary to estimate the open-loop behaviour of the system from measurements, and subsequently the response of the flow to an actuation signal is determined. The theoretically derived PSE TFs are used to account for both of these effects. Besides its capability to derive simplified models of the flow dynamics, we explore the use of the TFs to provide an a priori determination of adequate positions for efficiently forcing along the direction transverse to the mean flow. The PSE TFs are also used to account for the relative position between sensors and actuators which defines two schemes, feedback and feedforward, the former presenting a lower effectiveness. Differences are understood in terms of the evaluation of the causality of the resulting gain, which is made without the need to perform computationally demanding simulations for each configuration. The ROMs are applied to a direct numerical simulation of a convectively unstable 2D mixing layer. The derived feedforward control law is shown to lead to a reduction in the mean square values of the objective fluctuation of more than one order of magnitude, at the output position, in the nonlinear simulation, which is accompanied by a significant delay in the vortex pairing and roll-up. A study of the robustness of the control law demonstrates that it is fairly insensitive to the amplitude of inflow perturbations and model uncertainties given in terms of Reynolds number variations.     

Model reduction for supersonic cavity flow using proper orthogonal decomposition (POD) and Galerkin projection

The reduced-order model(ROM) for the two-dimensional supersonic cavity flow based on proper orthogonal decomposition(POD) and Galerkin projection is investigated. Presently, popular ROMs in cavity flows are based on an isentropic assumption,valid only for flows at low or moderate Mach numbers. A new ROM is constructed involving primitive variables of the fully compressible Navier-Stokes(N-S) equations, which is suitable for flows at high Mach numbers. Compared with the direct numerical simulation(DNS) results, the proposed model predicts flow dynamics(e.g., dominant frequency and amplitude) accurately for supersonic cavity flows, and is robust. The comparison between the present transient flow fields and those of the DNS shows that the proposed ROM can capture self-sustained oscillations of a shear layer. In addition, the present model reduction method can be easily extended to other supersonic flows.     

Large-eddy Simulation of Motored Flow in a Two-valve Piston Engine: POD Analysis and Cycle-to-cycle Variations

Large-eddy simulation (LES) has been performed for a single-cylinder, two-valve, four-stroke-cycle piston engine through 70 consecutive motored cycles. Initial comparisons of ensemble-averaged velocity fields have been made between LES and experiment, and proper orthogonal decomposition (POD) has been used to analyze the complex in-cylinder turbulent flows. Convergence of POD modes has been quantified, several POD variants have been explored, and sensitivity of results to analyzing different subsets of engine cycles has been studied. In general, it has been found that conclusions that were drawn earlier from POD analysis of a simplified non-compressing piston-cylinder assembly with a fixed valve carry over to the much more complex flow in this motored four-stroke-cycle engine. For the cases that have been examined, the first POD mode essentially corresponds to the ensemble-averaged mean velocity. The number of engine cycles required to extract converged POD modes increases with mode number, and varies with phase (piston position). There is little change in the lower-order phase-invariant POD modes when as few as 24 phases per cycle (30° between samples) are used, and complex 3-D time-dependent in-cylinder velocity fields through full engine cycles can be reconstructed using a relatively small number of POD modes. Quantification of cycle-to-cycle variations and insight into in-cylinder flow dynamics can be extracted through analysis of phase-invariant POD modes and coefficients.     

Study of linear response identification techniques and reduced‐order model generation for a 2D CFD scheme

Reduced‐Order Models (ROMs) have been the focus of research in various engineering situations, but it is only relatively recently that such techniques have begun to be introduced into the CFD field. The purpose of generating such models is to capture the dominant dynamics of the full set of CFD equations, but at much lower cost. One method that has been successfully implemented in the field of fluid flows is based on the calculation of the linear pulse responses of the CFD scheme coupled with an Eigensystem Realization algorithm (ERA), resulting in a compact aerodynamic model. The key to the models is the identification of the linear responses of the non‐linear CFD code. Two different methods have been developed and reported in literature for linear response identification; the first method linearizes the CFD code and the second method uses Volterra theory and the non‐linear code. As these methods were developed independently they have not previously been brought together and compared. This paper first explains the subtle, but fundamental differences between the two methods. In addition, a series of test cases are shown to demonstrate the performance and drawbacks of the ROMs derived from the different linear responses. The conclusions of this study provide useful guidance for the implementation of either of the two approaches to obtain the linear responses of an existing CFD code. Copyright © 2006 John Wiley & Sons, Ltd.     

Flow temporal reconstruction from non time-resolved data part II: practical implementation,methodology validation,and applications

This paper proposes a method to sort experimental snapshots of a periodic flow using information from the first three POD coefficients. Even in presence of turbulence, phase-average flow fields are reconstructed with this novel technique. The main objective is to identify and track traveling coherent structures in these pseudo periodic flows. This provides a tool for shedding light on flow dynamics and allows for dynamical contents comparison, instead of using mean statistics or traditional point-based correlation techniques. To evaluate the performance of the technique, apart from a laminar test on the relative strength of the POD modes, four additional tests have been performed. In the first of these tests, time-resolved PIV measurements of a turbulent flow with an externally forced main frequency allows to compare real phase-locked average data with reconstructed phase obtained using the technique proposed in the paper. The reconstruction technique is then applied to a set of non-forced, non time-resolved Stereo PIV measurements in an atmospheric burner, under combustion conditions. Besides checking that the reconstruction on different planes matches, there is no indication of the magnitude of the error for the proposed technique. In order to obtain some data regarding this aspect, two additional tests are performed on simulated non-externally forced laminar flows with the addition of a digital filter resembling turbulence (Klein et al. in J Comput Phys 186:652–665, 2003). With this information, the limitation of the technique applicability to periodic flows including turbulence or secondary frequency features is further discussed on the basis of the relative strength of the Proper Orthogonal Decomposition (POD) modes. The discussion offered indicates coherence between the reconstructed results and those obtained in the simulations. In addition, it allows defining a threshold parameter that indicates when the proposed technique is suitable or not. For those researchers interested on the background and possible generalizations of the technique, part I of this work (Legrand et al. in Exp Fluid (submitted in 2010) 2011) offers the mathematic fundamentals of the general space–time reconstruction technique using POD coefficients. Noteworthy, the involved computational time is relatively small: all the reconstructions have been performed in the order of minutes.     

1‐D numerical modelling of shallow flows with variable horizontal density

A1‐D numerical model is presented for vertically homogeneous shallow flows with variable horizontal density. The governing equations represent depth‐averaged mass and momentum conservation of a liquid–species mixture, and mass conservation of the species in the horizontal direction. Here, the term ‘species’ refers to material transported with the liquid flow. For example, when the species is taken to be suspended sediment, the model provides an idealized simulation of hyper‐concentrated sediment‐laden flows. The volumetric species concentration acts as an active scalar, allowing the species dynamics to modify the flow structure. A Godunov‐type finite volume scheme is implemented to solve the conservation laws written in a deviatoric, hyperbolic form. The model is verified for variable‐density flows, where analytical steady‐state solutions are derived. The agreement between the numerical predictions and benchmark test solutions illustrates the ability of the model to capture rapidly varying flow features over uniform and non‐uniform bed topography. A parameter study examines the effects of varying the initial density and depth in different regions. Copyright © 2009 John Wiley & Sons, Ltd.     

Continuation methods applied to the 2D Navier–Stokes equations at high Reynolds numbers

Nonlinearities arise in aerodynamic flows as a function of various parameters, such as angle of attack, Mach number and Reynolds number. These nonlinearities can cause the change from steady to unsteady flow or give rise to static hysteresis. Understanding these nonlinearities is important for safety validation and performance enhancement of modern aircraft. A continuation method has been developed to study nonlinear steady state solutions with respect to changes in parameters for two‐dimensional compressible turbulent flows at high Reynolds numbers. This is the first time that such flows have been analysed with this approach. Continuation methods allow the stable and unstable solutions to be traced as flow parameters are changed. Continuation has been carried out on two‐dimensional aerofoils for several parameters: angle of attack, Mach number, Reynolds number, aerofoil thickness and turbulent inflow as well as levels of dissipation applied to the models. A range of results are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

An improved algorithm for the shallow water equations model reduction: Dynamic Mode Decomposition vs POD

We propose an improved framework for dynamic mode decomposition (DMD) of 2‐D flows for problems originating from meteorology when a large time step acts like a filter in obtaining the significant Koopman modes, therefore, the classic DMD method is not effective. This study is motivated by the need to further clarify the connection between Koopman modes and proper orthogonal decomposition (POD) dynamic modes. We apply DMD and POD to derive reduced order models (ROM) of the shallow water equations. Key innovations for the DMD‐based ROM introduced in this paper are the use of the Moore–Penrose pseudoinverse in the DMD computation that produced an accurate result and a novel selection method for the DMD modes and associated amplitudes and Ritz values. A quantitative comparison of the spatial modes computed from the two decompositions is performed, and a rigorous error analysis for the ROM models obtained is presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Coherent structures and their influence on the dynamics of aeroelastic panels

A panel forced by a supersonic unsteady flow is numerically investigated using a finite difference method, a Galerkin approach, and proper orthogonal decomposition (POD). The aeroelastic model investigated is based on piston theory for modeling the flow-induced forces, and von Karman plate theory for modeling the panel. Structural non-linearity is considered, and it is due to the non-linear coupling between bending and stretching. Several novel facets of behavior are explored, and key aspects of using a Galerkin method for modeling the dynamics of the panel exhibiting limit cycle oscillations and chaos are investigated. It is shown that multiple limit cycles may co-exist, and they are both symmetric and asymmetric. Furthermore, the level of spatial coherence in the dynamics is estimated by means of POD. Reduced order models for the dynamics are constructed. The sensitivity to initial conditions of the non-linear aeroelastic system in the chaotic regime limits the capability of the reduced order models to identically model the time histories of the system. However, various global characteristics of the dynamics, such as the main attractor governing the dynamics, are accurately predicted by the reduced order models. For the case of limit cycle oscillations and stable buckling, the reduced order models are shown to be accurate and robust to parameter variations.     

Interpolation-based reduced-order modelling for steady transonic flows via manifold learning

This paper presents a parametric reduced-order model (ROM) based on manifold learning (ML) for use in steady transonic aerodynamic applications. The main objective of this work is to derive an efficient ROM that exploits the low-dimensional nonlinear solution manifold to ensure an improved treatment of the nonlinearities involved in varying the inflow conditions to obtain an accurate prediction of shocks. The reduced-order representation of the data is derived using the Isomap ML method, which is applied to a set of sampled computational fluid dynamics (CFD) data. In order to develop a ROM that has the ability to predict approximate CFD solutions at untried parameter combinations, Isomap is coupled with an interpolation method to capture the variations in parameters like the angle of attack or the Mach number. Furthermore, an approximate local inverse mapping from the reduced-order representation to the full CFD solution space is introduced. The proposed ROM, called Isomap+I, is applied to the two-dimensional NACA 64A010 airfoil and to the 3D LANN wing. The results are compared to those obtained by proper orthogonal decomposition plus interpolation (POD+I) and to the full-order CFD model.     

Assessment of global linear stability analysis using a time‐stepping approach for compressible flows

Global linear stability analysis combined with computational fluid dynamics (CFD) is considered useful for understanding the physics of fluid flows. However, the numerical techniques of global linear stability analysis for compressible flows have not been well established in comparison with those for incompressible flows. In this study, we develop and assess a set of appropriate numerical techniques required to conduct a global linear stability analysis for compressible flows. For the eigensystem analysis, the Arnoldi method combined with time integration is in effect to preserve the memory (RAM) size of the computer. The compact difference scheme is used for the CFD analysis from the viewpoints of computing accurate global modes and saving memory by reducing the number of grid points to obtain the necessary spatial resolution. To assess the proposed method, two‐dimensional compressible flow problems, including regularized cavity flow, flow around a square cylinder, and the compressible mixing layer, are analyzed, and it is confirmed that the proposed method can obtain accurate mode shapes, growth rate, and frequency of the corresponding global modes. In addition, influences and an appropriate formulation of the outflow boundary conditions are investigated. Results reveal that the outflow boundary causes spurious unstable modes in the global linear stability analysis, and the radiation and outflow boundary condition and the extension of the computational domain with grid stretching keep the spurious unstable modes to a minimum. Copyright © 2015 John Wiley & Sons, Ltd.     

缓变主流中三维气泡的非线性振动

空化现象和水下噪声机制与液体中气泡的动力学行为密切相关．在无粘势流的假定下，采用多参数摄动分析，研究了缓变主流中三维气泡的非线性体积模态振动．推导了关于缓变泡形展开的各阶扰动方程，获得了一阶振动的演化方程和一些特殊情况下的解析解；并采用高阶有限元离散的边界积分方程方法，对平面固壁和自由面附近三维气泡的固有频率进行了数值计算