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.     

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.     

Experiences with two detached eddy simulation approaches through use of a high-order finite volume solver

In the present study, two advanced detached eddy simulation (DES) approaches, shear-layer-adapted delayed DES and zonal DES in mode II, which are known to help transition from RANS to LES mode, are employed in various flow problems in conjunction with a high-order finite volume solver. The numerical scheme, being only applicable on structured grids, has low-dissipation and low-dispersion features. Such features benefit mostly in the LES mode, minimizing the interference of numerical diffusion with subgrid eddy viscosity. First, corresponding subgrid models are validated via decaying homogeneous turbulence benchmark case. Then, a channel flow problem is chosen to examine these models in attached flow situations. Finally, flow around an airfoil at low Reynolds number is solved using the shear-layer-adapted delayed DES approach only, in an aim to obtain trailing-edge noise spectrum at an observer location. Despite some log-layer mismatch over turbulent boundary layers, which is typical of most DES methods, the combined application of high-resolution numerical method and advanced DES approaches, which are implemented on a stabilized Spalart-Allmaras turbulence model, shows merit in resolution of turbulence in regions of interest.     

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.     

Vortex mechanism of heat transfer enhancement in a channel with spherical and oval dimples

Vortex mechanism of heat transfer enhancement in a narrow channel with dimples has been investigated numerically using LES and URANS methods. The flow separation results in a formation of vortex structures which significantly enhance heat transfer on dimpled surfaces leading to a small increase in pressure loss. The heat transfer can be significantly increased by rounding the dimple edge and use of oval dimples. To get a deep insight into flow physics LES is performed for single phase flow in a channel with a spherical dimple. The instantaneous vortex formation and separation are investigated in and around the dimple area. Considered are Reynolds numbers (based on dimple print diameter) Re_D = 20,000 and Re_D = 40,000 the depth to print diameter ratio of Δ = 0.26. Frequency analysis of LES data revealed the presence of dominating frequencies in unsteady flow oscillations. Direct analysis of the flow field revealed the presence of coherent vortex structure inclined to the mean flow. The structure changes its orientation in time causing the long period oscillations with opposite-of-phase motion. Three dimensional proper orthogonal decomposition (POD) analysis is carried out on LES pressure and velocity fields to identify spatio-temporal structures hidden in the random fluctuations. Tornado-like spatial POD structures have been determined inside dimples.     

用基于M-SST模型的DES数值模拟喷流流场

脱体涡数值模拟方法(dettached eddy simulation,DES)是把雷诺平均Navier-Stokes方程(RANS)方法及大涡模拟方法(LES)结合起来模拟有脱体涡的湍流流场的数值模拟方法,其主要思想是在物面附近解雷诺平均Navier-Stokes方程、在其他区域采用Smagorinski大涡模拟方法。本文在剪切应力传输(SST)湍流模型的基础上用DES及混合非结构网格数值模拟具有横向喷流的湍流流场,算法采用Osher逆风格式,利用该套程序(包括网格生成及算法),对导弹在不同马赫数下的喷流流场进行了数值模拟,并与同时开展的实验研究的结果进行了对比,结果表明用该方法处理这类问题是较准确的。     

Proper orthogonal decomposition analysis of coherent motions in a turbulent annular jet

A three-dimensional incompressible annular jet is simulated by the large eddy simulation(LES) method at a Reynolds number Re = 8 500. The time-averaged velocity field shows an asymmetric wake behind the central bluff-body although the flow geometry is symmetric. The proper orthogonal decomposition(POD) analysis of the velocity fluctuation vectors is conducted to study the flow dynamics of the wake flow.The distribution of turbulent kinetic energy across the three-dimensional POD modes shows that the first four eigenmodes each capture more than 1% of the turbulent kinetic energy, and hence their impact on the wake dynamics is studied. The results demonstrate that the asymmetric mean flow in the near-field of the annular jet is related to the first two POD modes which correspond to a radial shift of the stagnation point. The modes 3 and 4 involve the stretching or squeezing effects of the recirculation region in the radial direction. In addition, the spatial structure of these four POD eigenmodes also shows the counter-rotating vortices in the streamwise direction downstream of the flow reversal region.     

Coherent structures and wavepackets in subsonic transitional turbulent jets

A large eddy simulation (LES) is performed for two subsonic jets with a Reynolds number of Re = 105, which have different core temperatures, i.e., the cold and hot jet. The far-field overall sound pressure levels (OASPL) and noise spectra are well validated against previous exper-imental results. It is found that the OASPL is raised by heating at shallow angles. The most energetic coherent struc-tures are extracted with specified frequencies using the filter based on the frequency domain variant of the snapshot method of proper orthogonal decomposition (POD). The m = 0, 1 modes have high coherence of near-field pres-sure for both jets, while the coherence of m = 0 modes is enhanced greatly by heating. Based on the coherent struc-tures, spatial wavepackets are educed and the characteristics of growth, saturation and decay are analyzed and compared between the two jets in detail. The results show that heat-ing would enhance the linear growth rate for high frequency components, and nonlinear growth rates for low frequency components in general, which are responsible for higher OASPL in the hot jet. The far-field sound generated by wavepackets is computed using the Kirchhoff extrapolation, which matches well with that of LES at shallow angles. This indicates that the wavepackets associated with coherent structures are dominant sound sources in forced transitional turbulent jets. Additionally, the present POD method is proven to be a robust tool to extract the salient features of the wavepackets in turbulent flows.     

Extended Proper Orthogonal Decomposition: Application to Jet/Vortex Interaction

The Proper Orthogonal Decomposition (POD) is used in the present work to study the interactions between different regions of a flow. The standard analysis would select structures that are best correlated with the entire fluctuating velocity field. It is therefore not helpful if one flow region S of interest contains only a small percentage of the total kinetic energy. Using POD modes computed in the sub-domain S only, extended modes are introduced using the method of snapshots. We demonstrate that they provide a decomposition of the velocity field in the whole domain and that the extended mode number p provides the only local contribution to the velocity field correlated with the projection of the velocity field on POD mode p in S. This method is general and can be applied to either experimental or numerical velocity fields. As an example, it is applied to the analysis of an internal turbulent flow in a model engine cylinder with tumble. Data are obtained at a given phase with Particle Image Velocimetry. We focus our analysis on the middle of the intake stroke when the energy containing intake jet rolls up to feed a large vortex structure. Preferred directions of the jet/vortex interaction are clearly identified. This revised version was published online in July 2006 with corrections to the Cover Date.     

Detached eddy simulation of flow around two wall-mounted cubes in tandem

We investigate the performance of unsteady Reynolds-averaged Navier–Stokes (URANS) computation and various versions of detached eddy simulation (DES) in resolving coherent structures in turbulent flow around two cubes mounted in tandem on a flat plate at Reynolds number (Re) of 22,000 and for a thin incoming boundary layer. Calculations are carried out using four different coherent structure resolving turbulence models: (1) URANS with the Spalart–Allmaras model; (2) the standard DES [Spalart, P.R., Jou, W.H., Strelets, M., Allmaras, S.R., 1997. Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. In: Liu, C., Liu, Z., (Eds.), Advances in DNS/LES. Greyden Press, Columbus, OH]; (3) the Delayed DES (DDES); and (4) the DES with a low-Re modification (DES-LR) [Spalart, P., Deck, S., Shur, M., Squires, K., Strelets, M., Travin, A., 2006. A new version of detached eddy simulation, resistant to ambiguous grid densities. Theor. Comput. Fluid Dyn. 20 (3), 181–195]. The grid sensitivity of the computed solutions is examined by carrying out simulations on two successively refined grids. The computed results for all cases are compared with the experimental measurements of Martinuzzi and Havel [Martinuzzi, R., Havel, B., 2000. Turbulent flow around two interfering surface-mounted cubic obstacles in tandem arrangement. ASME J. Fluids Eng. 122, 24–31] for two different cube spacings. All turbulence models reproduce essentially identical separation of the approach thin boundary layer and yield an unsteady horseshoe vortex system consisting of multiple vortices in the leading edge region of the upstream cube. Significant discrepancies between the URANS and all DES solutions are observed, however, in other regions of interest such as the shear layers emanating from the cubes, the inter-cube gap and the downstream wake. Regardless of the grid refinement, URANS fails to capture key features of the mean flow, including the second horseshoe vortex in the upstream junction and recirculating flow on the top surface of the downstream cube for the large cube spacing, and underestimates significantly turbulence statistics in most regions of the flow for both cases. On the coarse mesh, all three DES approaches appear to yield very similar results and fail to reproduce the second horseshoe vortex. The standard DES and DDES solutions obtained on the fine meshes are essentially identical and both suffer from premature switching to unresolved DNS, due to the mis-interpretation of grid refinement as wall proximity, which leads to spurious vortices in the inter-cube region. Numerical solutions show that the low-Re modification (DES-LR) is critical prerequisite in DES on the ambiguously fine – not fine enough for full LES – mesh to prevent excessive nonlinear drop of the subgrid eddy viscosity in low cell-Re regions like in the inter-obstacle gap. Mean flow quantities and turbulence statistics obtained with DES-LR on the fine mesh are in good overall agreement with the measurements in most regions of interest for both cases.     

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.     

Experimental and numerical POD study of the Coanda effect used to reduce self-sustained tones

The two-dimensional flow along an inclined plate may be detached or reattached to the plate by Coanda effect. Experimentally, we explore the influence of the inclination angle and of the Reynolds number on the attachment and detachment phenomena, and on the hysteresis loop. A proper orthogonal decomposition (POD) of the flow is applied to a LES simulation resulting data. A low-dimensional dynamical model is obtained using by Galerkin projection of the Navier–Stokes equations upon the POD basis functions. We show that this model represents qualitatively the characteristics of the flow.     

Dynamic mode decomposition and reconstruction of transient cavitating flows around a Clark-Y hydrofoil

The transient cavitating flow around the Clark-Y hydrofoil is numerically investigated by the dynamic mode decomposition with criterion. Based on the ranking dominant modes, frequencies of the first four modes are in good accordance with those obtained by fast Fourier transform. Furthermore, the cavitating flow field is reconstructed by the first four modes, and the dominant flow features are well captured with the reconstructed error below 12% when compared to the simulated flow field. This paper offers a reference for observing and reconstructing the flow fields, and gives a novel insight into the transient cavitating flow features.     

A spatiotemporal decomposition of a fully inhomogeneous turbulent flow field

The three-dimensional orthogonal spatial modes and their temporal counterparts have been extracted from a large-eddy simulation of turbulent flow over a surface-mounted cube, using a space-time symmetric version of proper orthogonal decomposition (POD), proposed by Aubry et al. (1991). A relatively small domain of interest, located immediately above the top face of the flow obstacle, has been selected for the application of POD. Within that volume of interest, time records of the velocity field have been sampled at 6000 locations simultaneously. The space-time duality of POD can be demonstrated by deriving two alternative eigenvalue problems for either the orthogonal spatial modes or the orthogonal temporal modes. For a particular case, the choice between the two alternatives can be done on the basis of computational convenience and of data-storage requirements. The results show that the first spatiotemporal mode can be identified with the mean flow. The second spatiotemporal mode is dominated by the alternating vortex shedding from the side edges of the flow obstacle. A Fourier analysis of the second temporal mode leads to a Strouhal number of S=0.125 which corresponds to the measured Strouhal number for the vortex shedding (Martinuzzi, 1992). The third and the fourth spatiotemporal modes are connected with the rolls created at the horizontal leading edge of the cube. For the flow field investigated, the dual space-time point of view of POD is rather realistic in the sense that the first four spatiotemporal modes can actually be observed in the flow.This work is currently supported by the German Research Society (DFG), Priority Research Program, Project No. We 705/3 (Wengle /Römer). We also gratefully acknowledge the support by the Universität der Bundeswehr München (UniBwM) and by the Leibniz Computing Center of the Bavarian Academy of Sciences (LRZ).     

Vortex Shedding from a Hemisphere in a Turbulent Boundary Layer

Supercritical turbulent boundary layer flow over a hemisphere with a rough surface (Re= 150000) has been simulated using Large Eddy Simulation (LES) and analyzed using the Karhunen--Loève expansion (“Proper Orthogonal Decomposition,” POD). The time-dependent inflow condition is provided from a separate LES of a boundary layer developing behind a barrier fence and a set of vorticity generators. LES results using significantly different grid resolutions are compared with a corresponding wind tunnel experiment to demonstrate the reliability of the simulation. The separation processes are analyzed by inspecting second-order moments, time spectra, and instantaneous velocity distributions. Applying POD, a detailed study of the spatiotemporal structure of the separation processes has been carried out. From this analysis it can be concluded that the major event in the separated flow behind the obstacle is the shedding of “von Kármán”-type vortices, which can be represented by the first three energetically dominant modes. Received 23 January 1997 and accepted 19 February 1998     

Coherent structures in the boundary layer of a flat thick plate

We use POD and EPOD (extended POD) analysis to extract the main features of the flow over a thick flat plate simulated with an LES. Our goal is to better understand the coupling between the velocity field and the surface pressure field. We find that POD modes based on the full velocity and energy fields contain both flapping and shedding frequencies. Pressure modes are found to be uniform in the spanwise direction and the most intense variations take place at the mean reattachment point. Velocity modes educed from the pressure modes with EPOD are seen to correspond to eddies shed by the recirculation bubble.     

A two‐phase adaptive finite element method for solid–fluid coupling in complex geometries

In this paper we present a method to solve the Navier–Stokes equations in complex geometries, such as porous sands, using a finite‐element solver but without the complexity of meshing the porous space. The method is based on treating the solid boundaries as a second fluid and solving a set of equations similar to those used for multi‐fluid flow. When combined with anisotropic mesh adaptivity, it is possible to resolve complex geometries starting with an arbitrary coarse mesh. The approach is validated by comparing simulation results with available data in three test cases. In the first we simulate the flow past a cylinder. The second test case compares the pressure drop in flow through random packs of spheres with the Ergun equation. In the last case simulation results are compared with experimental data on the flow past a simplified vehicle model (Ahmed body) at high Reynolds number using large‐eddy simulation (LES). Results are in good agreement with all three reference models. Copyright © 2010 John Wiley & Sons, Ltd.     

Spectral and modal analysis of the flow in a helical coil steam generator experiment with large eddy simulation

This paper presents an analysis of flow in the shell side of a 24-degree cut helical coil steam generator (HCSG) with large eddy simulation (LES). In the study, the open source, high-order spectral element CFD code Nek5000 was employed for numerical simulation. LES was performed at a Reynolds number of 9000 based on the inlet velocity and tube diameter. Spectral analysis was carried out along the tube bundle to investigate vortex shedding and to reveal shedding frequencies in HCSGs. Wavelet analysis was carried out to determine the time-frequency information of the fluctuating velocities in the domain. Flow was decomposed into modes with proper orthogonal decomposition, which revealed coherent structures in the flow and evolution of coherent structures in time. For the validation of Nek5000 in complex geometries, results from the numerical simulation were compared with experiments conducted at Texas A&M University. Overall, the LES results showed good agreement with experimental data.     

Meandering jets in shallow rectangular reservoirs: POD analysis and identification of coherent structures

The effect of the shallowness on meandering jets in a shallow rectangular reservoir is investigated. Four meandering flows were investigated in an experimental shallow rectangular reservoir. Their boundary conditions were chosen to cover a large range of friction numbers (defined with the sudden expansion width). Due to the unsteady characteristics of the flows, a proper orthogonal decomposition (POD) of the fluctuating part of the surface velocity fields measured using Large-Scale Particle Image Velocity was used for discriminating the flow structures responsible for the meandering of the jet. Less than 1 % of the calculated POD modes significantly contribute to the meandering of the jet, and two types of instability are in competition in such a flow configuration. The sinuous mode is the dominant mode in the flow, and it induces the meandering of the flow, while the varicose mode is a source of local mixing and weakly participates to the flow. The fluctuating velocity fields were then reconstructed using the POD modes corresponding to 80 % of the total mean fluctuating kinetic energy, and the coherent structures were identified using the residual vorticity, their centres being localised using a topology algorithm. The trajectories of the structures centres emphasise that at high friction number the coherent structures are small and laterally paired in the near, middle and far fields of the jet, while with decreasing friction number, the structures merge into large horizontal vortices in the far field of the jet, their trajectories showing more variability in space and time. The analysis of the stability regime finally reveals that the sinuous mode is convectively unstable and may become absolutely unstable at the end of the reservoir when the friction number is small.