Dominant Proper Orthogonal Decomposition Modes Research Articles

Investigation of pulsed direct-current plasma jets in a turbulent boundary layer

Characteristics of the plasma jet produced by a pulsed direct-current (pulsed-DC) dielectric barrier discharge (DBD) and its interaction with a turbulent boundary layer (TBL) are investigated in detail using stereo particle imaging velocimetry. Quiescent-flow characterization results show that a positive starting vortex and a negative near-wall jet structure are induced by the pulsed-DC DBD plasma actuator. With increasing pulse width and discharge frequency, the jet velocity magnitude increases monotonously, as a direct result of the extension of fluid particle acceleration time. During the interaction with a cross-flow TBL, two streamwise vortices with opposite signs are observed at the two sides of the electrode junction, which essentially originate from the starting vortex and negative jet in quiescent air. The skin-friction drag variations are dominated by the cross-stream momentum transportation of streamwise vortices, with drag reduction in the vortex upwash zone and drag increase in the downwash zone. Compared with the conventional alternating-current DBD plasma actuators, the turbulent fluctuations produced by pulsed-DC DBD are much higher, which also affects the skin-friction drag. Further proper orthogonal decomposition (POD) analysis reveals that two distinctly different flow patterns are produced by pulsed-DC DBD working at small and large pulse widths. The dominant POD modes causing the most velocity fluctuation are the spanwise translation and deformation of plasma-induced streamwise vortices. These results provide insights into the basic phenomenon of pulsed-DC plasma jets in cross flow, which recently has demonstrated its promising applications in turbulent skin-friction reduction.

Experimental investigation of characteristics and influence of tip leakage vortex wandering in an axial compressor cascade

Vortex wandering is one of the most basic unsteady flow characteristics of the tip leakage vortex (TLV) in compressors. In this study, stereo particle image velocimetry (stereo-PIV) has been conducted in compressor cascades with various tip clearances to investigate the characteristics and influence of TLV wandering. The most effective vortex identification method for stereo-PIV data has been clarified. The wandering characteristics of the TLV are statistically analyzed, and the relationship between the vortex wandering and the dominant proper orthogonal decomposition mode is identified. The results reveal that TLV preferentially wanders along the pitchwise direction in the blade channel and gradually loses the dominant wandering direction downstream of the blade. The large displacement of the vortex center around its mean location is characterized by higher probabilities at small tip clearance size (1% chord length C). The spatial distribution characteristics demonstrate similarly concentric isocontour around the mean vortex center location for large tip clearances (3% and 5% chord length C). The effect of TLV wandering on the secondary velocity distribution, the tip flow blockage, and the distribution of Reynolds stress is explored using the vortex wandering corrected technique. The analysis verifies that the elevated turbulence kinetic energy in the core of the time-averaged uncorrected TLV is caused by vortex wandering rather than vortex deformation. The discussions of the current paper will enhance our knowledge of TLV wandering. Regardless of the reference to TLV, the interpretation of other swirling flows can benefit from the discussions presented here.

Reservoir computing reduced-order model based on particle image velocimetry data of post-stall flow

The present study proposes a reservoir computing reduced-order model (RCROM) of the post-stall flow around the National Advisory Committee for Aeronautics 0015 airfoil based on the time series velocity field, and the estimation accuracy of the RCROM is evaluated compared to that of a linear reduced-order model (LROM). The data were experimentally obtained by particle image velocimetry at a chord Reynolds number of 6.4 × 104 and an angle of attack of 18°. The low-dimensional description of the velocity field can be obtained by decomposing the velocity field with a proper orthogonal decomposition (POD) technique and by employing the leading POD mode coefficients as temporal variables of the data instead of the velocity field. Reservoir computing (RC) is adopted as a nonlinear function that predicts several steps ahead of the leading POD mode coefficients. The hyperparameters of RC are tuned by Bayesian optimization, and the optimized RCROM outperforms the LROM in terms of estimation accuracy. The estimation accuracy of the RCROM can be investigated under different numbers of the predicted dominant POD modes and prediction step conditions. As a result, the RCROM shows higher estimation accuracy than the LROM.

POD-based study of turbulent plane Poiseuille flow: comparing structure and dynamics between quasi-linear simulations and DNS

Turbulence in the restricted nonlinear (RNL) dynamics is analysed and compared with direct numerical simulations (DNS) of Poiseuille turbulence at Reynolds number $R=1650$ . The structures are obtained by proper orthogonal decomposition (POD) analysis of the two components of the flow partition used in RNL dynamics: the streamwise mean flow and fluctuations. POD analysis of the streamwise mean flow indicates that the dominant POD modes, in both DNS and RNL dynamics, are roll-streaks harmonic in the spanwise direction. However, we conclude that these POD modes do not occur in isolation but rather are Fourier components of a coherent roll-streak structure. POD analysis of the fluctuations in DNS and RNL dynamics reveals similar complex structures consisting in part of oblique waves collocated with the streak. The origin of these structures is identified by their correspondence to POD modes predicted using a stochastic turbulence model (STM). These predicted POD modes are dominated by the optimally growing structures on the streak, which the STM predicts correctly to be of sinuous oblique wave structure. This close correspondence between the roll-streak structure and the associated fluctuations in DNS, RNL dynamics and the STM implies that the self-sustaining mechanism operating in DNS is essentially the same as that in RNL dynamics, which has been associated previously with optimal perturbation growth on the streak.

Proper orthogonal decomposition analysis of the cavitating flow around a hydrofoil with an insight on the kinetic characteristics

In this research, the cavitating flow around a NACA0015 (National Advisory Committee for Aeronautics) hydrofoil obtained by the large-eddy simulation method is analyzed using the proper orthogonal decomposition (POD) theory. Various fundamental mechanisms have been investigated thoroughly, including the reentrant jet behavior, pressure gradient mechanism, vortex dynamics, and dynamic properties of the hydrofoil. The influence of the vortex dynamics, pressure mechanism, and temporal/spatial evolution is revealed. The POD decomposition indicates that the first four dominant POD modes occupy 97.4% of the entire energy. Based on the vortex force field extracted from the first four single POD modes, it is found that the lift-and-drag characteristics in the cavitating flow are determined by the specific spatial distribution of mode vortex structures. In addition, the coupling of velocity pulsations and pressure fluctuations is carried out to obtain the POD modal pressure gradient field, which reveals that the pressure gradient has a close connection with the cavity evolution. Furthermore, the vortex force and pressure gradient are reconstructed using the first four modes, 17 modes, and 160 modes, which indicates that the low-order POD modes without the impact of small-scale structures and noise can clearly capture the fundamental aspects of the flow field.

Modal analysis of the liquid sheet breakup with and without acoustic forcing

• The coupling between the acoustic field and the liquid sheet atomization process is studied. • Modal decomposition techniques like Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) are employed. • Frequency response of the liquid sheet is found to be dependent on the laminar or turbulent nature of the impacting jets. • Liquid sheet breakup gets coupled with the external forcing when the acoustic energy goes into low frequency modes. • Decoupling occurs when the acoustic energy goes into the dominant modes which are higher ones having a higher frequency. The objective of the present study is to understand the coupling between the acoustics and the sheet atomization process. The phenomenon is studied through modal decompositions of time-resolved images of acoustically perturbed liquid sheets by employing the proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) techniques. The injector operating conditions are selected in order to allow the transition of a smooth liquid sheet to a highly turbulent sheet or fully developed spray. The sheet is destabilized by the traveling acoustic waves of known frequency and sound pressure level. POD modes of the smooth sheets and DMD modes of the turbulent sheets clearly highlight distinct effects of the acoustic perturbations on the liquid sheets. The frequency response of the liquid sheet is found to be dependent on the laminar or turbulent nature of the impacting jets. When the impinging jets are laminar, interestingly the dominant POD mode frequencies are either equal to or harmonics of the external acoustic excitation frequency which confirms the coupling of the atomization process with the acoustic perturbations. Below the lower cutoff frequency, the POD mode having frequency equal to the forcing frequency always dominates other modes. Also, for laminar jets, the DMD growth rate is maximum for the mode having frequency equal to the external forcing. However, for the turbulent impinging jets, the first DMD mode frequency is not always equal to the external forcing frequency but one of the mode in the DMD frequency spectrum show the frequency which is either close to the forcing frequency or close to its integer multiple.

Flow dynamics in the wake of a rectangular cylinder near a moving wall

Dynamics of the three-dimensional flow past a rectangular cylinder placed near a moving wall is studied for different values of the aspect-ratio (0.5 ≤ AR ≤ 2.0) for a constant value of the gap-ratio (G/D = 0.5) and Reynolds number (Re = 160). Spatially localized energetic coherent structures and single frequency oscillatory structures are obtained by employing proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). The structures of Mode A, Mode B and Mode C vortices are captured by these POD and DMD modes. In comparison to the streamwise and transverse velocity POD modes, energy is distributed more evenly among the spanwise velocity POD modes. The wave speed of the travelling modes has been obtained using POD. The streamwise wave speed of the dominant POD mode first increases and then decreases with an increase in AR. Energy transfer among the POD modes is faster compared to energy fluctuation in the flow. Strong nonlinearity in the flow is observed at AR = 0.5. Compared to higher values of AR, at AR = 0.5 energy is more evenly distributed among the DMD modes. The DMD modes with higher frequency have a lower spatial wavelength, which reflects in their spatial structures.

Mode vortex and turbulence in ventilated cavitation over hydrofoils

In this paper, the complex interaction of gas-vortex-turbulence in ventilated cavitation over a NACA0015 hydrofoil is investigated by using large eddy simulation. The proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are used for analyzing the three-dimensional fluctuation modes of gas-liquid interfaces induced by turbulent coherent structures in ventilated cavitation. The results show that among the first four dominant POD modes possessing 99.5% modal kinetic energy in the ventilated cavitating flow, and the turbulent POD mode 2 contains the multi-scale coherent structures and acts as a bridge for the energy scale transition from mode 1 to mode 3 through the gas-liquid interface oscillations. The DMD modal structures are reconstructed from the roller-shaped energetic structure captured by POD mode 2, which further reveal the process of breaking down near the hydrofoil leading edge, changing into the hairpin shape and finally stretching into the horseshoe shape above the hydrofoil trailing edge for the turbulent coherent structures on the cavity interfaces. Such low-frequency wake structures are energetic but sustained by means of the high-frequency turbulent fluctuations, with the interfacial oscillation amplitude changed through the formation and transition of the multi-scale coherent structures. Further, the modal turbulent kinetic energy transport theory is introduced to describe the ventilated coherent structures. It is worth noting that the large-scale energetic structures in POD mode 2 are mainly drawing energy from mean flows through the Reynolds stress, and the modal turbulent energy stored by shedding vortex and air cavities in the ventilated wake flow is redistributed and balanced due to the turbulent and pressure diffusion effects. Besides, the uneven spatial distribution of turbulent energy caused by the vortex structures of different modal scales is proved to promote the spatial oscillation level of the ventilated cavity interfaces. This study is helpful to understand the interaction mechanism between turbulence and gas-liquid interfacial fluctuations for the ventilated wake flow.

P-094 Fundamental understanding of sperm motion in viscoelastic media

Abstract Study question How does viscosity influence the flagellar beating behaviour of free-swimming bull, mouse, and human sperm? Summary answer Sperm flagellar beating behaviour exhibits a transition from an irregular three-dimensional (3D) beating at 5 mPa·s to an organized two-dimensional (2D) waveform at 20 mPa·s. What is known already Sperm migrate in a complex viscoelastic environment through the female reproductive tract. The viscoelastic properties of the oviductal fluid significantly influence the progressive motility of sperm, acting as one of the key guidance mechanisms in vivo. However, the biomechanics of sperm flagellar activity in response to varying viscosity of the oviductal fluid is poorly understood. Understanding sperm flagellar behaviour in physiologically relevant environments is crucial to understanding reproduction and may help to describe unknown causes of infertility. Lack of high-speed high-resolution imaging techniques and automated image-analysis capabilities have been the main barriers to fully describe the flagellar beating behaviour. Study design, size, duration We used a custom-built high-speed high-resolution dark-field microscopy platform to resolve the flagellar dynamics of human, bull, and mouse sperm near surfaces in viscoelastic media ranging in viscosity from 1 to 250 mPa·s. The imaging system includes an automated image analysis algorithm to quantify sperm flagellar waveform and motility characteristics by extracting the flagellar centreline, reconstructing the waveform and calculating tangent-angle profiles. 20 sperm from 3 different bull, mice and humans were analyzed. Participants/materials, setting, methods Bull, mouse, and human sperm were used in this study. In each experiment, a diluted sperm sample in a buffer supplemented with methylcellulose was used and free-swimming sperm were imaged using dark-field microscopy at 200 frames per second. An automated image analysis algorithm was used to extract sperm flagellar centreline and Proper Orthogonal Decomposition (POD) was then used to study sperm flagellar waveform. Statistical analysis was performed using one-way ANOVA. Main results and the role of chance The reconstructed flagellar beating pattern was different for sperm swimming in low and high-viscosity media. Bull sperm exhibited a lower flagellar beating amplitude along the end piece when swimming in a high-viscosity media, a potential energy-efficient strategy to navigate a high viscosity fluid. The first two dominant POD modes (shape modes) describe more than 90% of the beating pattern for all species. Bull sperm exhibited a transition mode with irregular loops in 5 mPa·s buffer, but the flagellar shape cycle created an organised repetitive circular cycle in 1 mPa.s (3D beating) buffer and at viscosities higher than 5 mPa·s (2D beating). Human sperm also indicated a similar behaviour but with the transition happening at higher viscosities. Mouse sperm in high-viscosity media had a lower flagellar beating amplitude across the principal piece and higher beating amplitude across the end piece. The flagellar shape cycle in mouse sperm showed a periodic flagellar beating behaviour at high-viscosities (>20 mPa·s), but a shape cycle with distorted loops at lower viscosities. Our results showed in quantitative detail that increasing viscosity alters sperm flagellar beating pattern, and how sperm migration behaviour in low viscosity media can be distinct from their swimming behaviour in vivo. Limitations, reasons for caution A more comprehensive study of sperm motility parameters such as curvilinear velocity, average path velocity and straight line velocity with a larger sample size is required to fully characterise sperm swimming behaviour as a function of viscosity. Wider implications of the findings The increasing viscosity of the oviductal fluid regulates the sperm flagellar beating behaviour to switch from a 3D swimming behaviour with irregular shape cycles at lower viscosities to a 2D slithering mode with repetitive circular shape cycles at higher viscosities to achieve a more energy-efficient beating pattern for navigation. Trial registration number Not applicable

Lower Order Description and Reconstruction of Sparse Scanning Lidar Measurements of Wind Turbine Inflow Using Proper Orthogonal Decomposition

Preview measurements of the inflow by turbine-mounted lidar systems can be used to optimise wind turbine performance or alleviate structural loads. However, nacelle-mounted lidars suffer data losses due to unfavourable environmental conditions and laser beam obstruction by the rotating blades. Here, we apply proper orthogonal decomposition (POD) to the simulated line-of-sight wind speed measurements of a turbine-mounted scanning lidar obtained from two large eddy simulations. This work aimed at identifying the dominant POD modes that can be used to subsequently derive a reduced-order representation of the turbine inflow. Secondly, we reconstructed the data points lost due to blade passage by using Gappy-POD. We found that only a few modes are required to capture the dynamics of the wind field parameters commonly used for lidar-assisted wind turbine control, such as the effective wind speed, vertical shear and directional misalignment. By evaluating turbine-relevant metrics in the time and frequency domain, we found that a ten-mode reconstruction could accurately describe most spatio-temporal variations in the inflow. Furthermore, a modal interpretation is presented by direct comparison with these wind field parameters. We found that the Gappy-POD method performs substantially better than spatial interpolation techniques, accurately reconstructing up to even 50% of missing data. A POD-based wind field reconstruction offers a trade-off between wind field reconstruction techniques requiring flow assumptions and more complex physics-based representations, offers dimensional reduction and can overcome the blade passage limitation of nacelle-mounted lidar systems.

Fast and Accurate Proper Orthogonal Decomposition using Efficient Sampling and Iterative Techniques for Singular Value Decomposition

In this article, we propose a computationally efficient iterative algorithm for proper orthogonal decomposition (POD) using random sampling based techniques. In this algorithm, additional rows and columns are sampled and a merging technique is used to update the dominant POD modes in each iteration. We derive bounds for the spectral norm of the error introduced by a series of merging operations. We use an existing theorem to get an approximate measure of the quality of subspaces obtained on convergence of the iteration. Results on various datasets indicate that the POD modes and/or the subspaces are approximated with excellent accuracy with a significant runtime improvement over computing the truncated SVD. We also propose a method to compute the POD modes of large matrices that do not fit in the RAM using this iterative sampling and merging algorithms.

TR-PIV measurements of turbulent confined impinging twin-jets in crossflow

In this work, time-resolved particle image velocimetry (TR-PIV) measurements have been carried out to study the flow structure and flow dynamics of submerged transverse impinging twin-jet flows discharged into a confined laminar crossflow. A parametric study has been carried out for fixed spacing ratio between the jets of s=L/D=3, fixed jet-to-target spacing ratio of H/D=6.73, fixed crossflow Reynolds number of Re=3500, and two jet Reynolds numbers based on the outlet velocities and jet diameter D of Rej=8670 and 9933 that correspond to jet-to-cross-flow velocity ratios of r=vj/u0=10 and 11.46, respectively. Flow measurements in several different planes of the test section are acquired to assess the confinement effect of the bounded jets. Flow visualization images of the mean velocity, vorticity, strain rate distributions, turbulence kinetic energy, and Reynolds stresses have been obtained. The results reveal the dependence of Rej on the trajectory, penetration, crossflow entrainment and lateral spreading of the twin jets. Spectral analysis of the fluctuating velocity components shows the characteristic frequencies related to vortex pairing and vortex breakup, and the shedding frequencies of the upstream and downstream wall jets have also been obtained. The swirling strength criteria has been applied to identify coherent structures, and their development confirms that these structures undergo two or three stages of vortex pairing prior to the vortex breakdown activity. The experimental data has been analyzed using proper orthogonal decomposition (POD), and the most energetic modes, the relative and cumulative energy contributions, dominant POD modes, and temporal POD coefficients have been obtained. The results show that the for Rej=8670, the first two POD modes account for 16.50% of the total kinetic energy, and the cumulative energy of the first 20 POD modes is 55.10%. For Rej=9933, the first two POD modes account for 17.98% of the total energy, and the sum of energies of the first 20 POD modes is 57.13%.

Response of a plate in turbulent channel flow: Analysis of fluid–solid coupling

The paper performs simulation of a rectangular plate excited by turbulent channel flow at friction Reynolds numbers of 180 and 400. The fluid–structure interaction is assumed to be one-way coupled, i.e, the fluid affects the solid and not vice versa. We solve the incompressible Navier–Stokes equations using finite volume direct numerical simulation in the fluid domain. In the solid domain, we solve the dynamic linear elasticity equations using a time-domain finite element method. The obtained plate averaged displacement spectra collapse in the low frequency region in outer scaling. However, the high frequency spectral levels do not collapse in inner units. This spectral behavior is reasoned using theoretical arguments. The resonant vibration is stronger at the third natural frequency than at the first natural frequency. We explain this behavior by comparing the fluid and solid length scales. We further study the sources of plate excitation using a novel formulation. This formulation expresses the average displacement spectrum of the plate as an integrated contribution from the fluid sources within the channel. Analysis of the sources reveals that at the plate natural frequencies, the contribution of the fluid sources to the plate excitation peaks in the buffer layer. The corresponding wall-normal width is found to be ≈0.75δ. The integrated contribution of the overlap and outer regions together to the plate response is comparable to that from the buffer region for Reτ=180 and exceeds the buffer region contribution for Reτ=400. We analyze the decorrelated features of the sources using spectral Proper Orthogonal Decomposition (POD) of the net displacement source. We enforce the orthogonality of the modes in an inner product with a symmetric positive definite kernel. The dominant spectral POD mode contributes to the entire plate excitation. The contribution of the remaining modes from the different wall-normal regions undergo destructive interference resulting in zero net contribution. The envelope of the dominant mode further shows that the intensity of the sources peaks in the buffer region and the wall-normal width of the sources extend well into the outer region of the channel.

Aerodynamic damping investigations of light dynamic stall on a pitching airfoil via modal analysis

This study presents an investigation of the aerodynamic damping of light dynamic stall phenomena on a pitching NACA 0012 airfoil via Dynamic Modal Decomposition (DMD) and Proper Orthogonal Decomposition (POD) techniques. DMD analysis predicts a dominant mode having the same frequency of the pitching excitation and contributing all of the aerodynamic damping of the system. The temporal orthogonality of the DMD technique renders all other DMD modes orthogonal to both the dominant DMD mode as well as the pitching motion, resulting in zero aerodynamic work due to these DMD modes. However, the lack of temporal orthogonality of POD modes implies several modes contributing to the aerodynamic damping of the system. The leading-edge suction and the trailing-edge vortex were considered the most energetic features by the two most dominant POD modes and was also observed as the most significant spatial feature in the dominant DMD mode shape. A recently developed Hilbert transform-based methodology was extended here for computing chord-wise intra-cycle aerodynamic damping distribution. This method shows large variations of the intra-cycle aerodynamic damping distribution obtained from DDES surface pressure data over the cycle, which poses a challenge to extract meaningful information for possible flow control applications. However, the dominant DMD modal aerodynamic damping distribution shows no intra-cycle variations and predicts negative damping hot-spots near the leading and trailing edges of the chord. Such conclusions are significantly different than observed for an attached flow case or observed in a related study considering a deep dynamic stall regime at lower Reynolds number.

Analysis of wall-pressure fluctuation sources from direct numerical simulation of turbulent channel flow

The sources of wall-pressure fluctuations in turbulent channel flow are studied using a novel framework. The wall-pressure power spectral density (PSD) $(\phi_{pp}(\omega))$ is expressed as an integrated contribution from all wall-parallel plane pairs, $\phi_{pp}(\omega)=\int_{-\delta}^{+\delta}\int_{-\delta}^{+\delta}\Gamma(r,s,\omega)\,\mathrm{dr}\,\mathrm{ds}$, using the Green's function. Here, $\Gamma(r,s,\omega)$ is termed the net source cross spectral density (CSD) between two wall-parallel planes, $y=r$ and $y=s$ and $\delta$ is the half channel height. Direct Numerical Simulation (DNS) data at friction Reynolds number of $180$ and $400$ are used to compute $\Gamma(r,s,\omega)$. Analysis of the net source CSD, $\Gamma(r,s,\omega)$ reveals that the location of dominant sources responsible for the premultiplied peak in the power spectra at $\omega^+\approx 0.35$ and the wavenumber spectra at $\lambda^+\approx 200$ is in the buffer layer at $y^+\approx 16.5$ and $18.4$ for $Re_{\tau}=180$ and $400$, respectively. The contribution from a wall-parallel plane (located at distance $y^+$ from the wall) to wall-pressure PSD is log-normal in $y^+$ for $\omega^+>0.35$. A dominant inner-overlap region interaction of the sources is observed at low frequencies. Further, the decorrelated features of the wall-pressure fluctuation sources are analyzed using spectral Proper Orthogonal Decomposition (POD). We require the modes to be orthogonal in an inner product with a symmetric positive definite kernel. Spectral POD supports the case that the net source is composed of two decorrelated components - active (dominant mode) and inactive (remaining modes). The structure represented by the dominant POD mode at the premultiplied wall-pressure PSD peak inclines in the downstream direction. At the low-frequency linear PSD peak, the dominant mode resembles a large scale vertical pattern.

Uncertainty Quantification of Stochastic Impact Dynamic Oscillator Using a Proper Orthogonal Decomposition-Polynomial Chaos Expansion Technique

Abstract A proper orthogonal decomposition (POD)-based polynomial chaos expansion (PCE) is utilized in this article for the uncertainty quantification (UQ) of an impact dynamic oscillator. The time-dependent nonsmooth behavior and the uncertainties are decoupled using the POD approach. The uncertain response domain is reduced using the POD approach, and the dominant POD modes are utilized for the UQ of the response quantity. Furthermore, the PCE model is utilized for the propagation of the input uncertainties. Two different cases of impact oscillator are considered, namely, single impact and multiple impact. The contact between two bodies is modeled by Hertz’s law. For both the cases, UQ is performed on the projectile displacement, projectile velocity, and contact force. A highly nonsmooth behavior is noticed for the contact force. For that reason, most number of POD modes are required to assess the UQ of contact force. All the results are compared with the Monte Carlo simulation (MCS) and time domain PCE results. Very good accuracies are observed for the PCE and the POD-PCE predicted results using much less number of model evaluations compared to MCS. As the PCE coefficients are dependent on time, the PCE model is computed at each time step. On the contrary, for the POD-PCE model, the PCE coefficients are computed for the number of POD modes only: it is much less than the PCE model.

Surface wave mechanism for directional motion of droplet on an obliquely vibrated substrate

The directional motion of a two-dimensional droplet on an obliquely vibrated substrate is studied numerically. The time dependent droplet profile is decomposed by using a proper orthogonal decomposition (POD) method. Two dominant POD modes of the capillary wave are identified. The first mode is quasi-harmonic, which leads to an apparent wetted area difference between uphill and downhill stages of the substrate vibration (ΔS). It plays a key role in the directional motion. The second mode is weak but contributes to ΔS subtly. The two modes qualitatively match the proposed “swaying” and “spreading” modes. Our decomposition directly reveals the connection between ΔS and the surface waves.

Efficient multi-objective shape optimization using proper orthogonal decomposition with variable fidelity concept

A variable fidelity concept is introduced in a re-parameterization approach based on the proper orthogonal decomposition (POD) to efficiently solve multi-objective aerodynamic shape optimization problems. The re-parameterization approach enables to extract dominant shape deformation modes from a database of good designs and to reduce the number of design variables. The present variable fidelity approach is proposed by utilizing low-fidelity functional evaluations to select the good designs. The proposed approach is investigated in two multi-objective aerodynamic shape optimization problems of 2D airfoil in which the combinations of viscous/inviscid simulations or fine/coarse grid simulations are treated as the high/low-fidelity evaluation methods. It can be confirmed that dominant POD modes obtained from low-fidelity evaluations are qualitatively equivalent with that obtained from high-fidelity evaluations. Non-dominated solutions obtained from a conventional optimization approach can be reproduced with smaller numbers of design variables using the dominant POD modes. The computational costs to solve the multi-objective aerodynamic shape optimization problems can be dramatically reduced by introducing the variable fidelity concept.

Galerkin-free model reduction for fluid-structure interaction using proper orthogonal decomposition

A Galerkin-free model reduction approach for fluid-structure interaction (FSI) is presented in this article. The reduced order model (ROM) is based on proper orthogonal decomposition (POD), where a reduced basis is formed using energy dominant POD modes. The reduced basis also consists of characteristics POD time modes that are derived from the POD time modes (coefficients) by using their periodicity. In addition to flow variables, the solution state vector comprises the mesh deformation, taking into account the structural deformation in FSI. A ROM solution is obtained by performing a linear interpolation of the reduced basis for changing operating/control parameters. The proposed Galerkin-free POD-ROM approach is demonstrated in terms of two test cases: a canonical case study of vortex-induced vibration (VIV) of a cylinder at Reynolds number Re=100, where simulations are performed for various structural-to-fluid mass ratios; and a shock wave boundary layer induced panel flutter. For the second case, we use previously computed high-fidelity simulations, considering only the effect of panel thickness on the aeroelastic coupling between the flexible panel and shock wave boundary layer interaction (SWBLI); the inflow is at Mach 2 and Reynolds number based on panel length Rea=50000. The presented Galerking-free ROM procedure is clean and robust for large mesh deformations, in addition to a significantly lower cost of computation compared to the FSI high-fidelity simulations.

A dynamic closure modeling framework for model order reduction of geophysical flows

In this paper, a dynamic closure modeling approach has been derived to stabilize the projection-based reduced order models in the long-term evolution of forced-dissipative dynamical systems. To simplify our derivation without losing generalizability, the proposed reduced order modeling (ROM) framework is first constructed by Galerkin projection of the single-layer quasigeostrophic equation, a standard prototype of large-scale general circulation models, onto a set of dominant proper orthogonal decomposition modes. We then propose an eddy viscosity closure approach to stabilize the resulting surrogate model considering the analogy between large eddy simulation (LES) and truncated modal projection. Our efforts, in particular, include the translation of the dynamic subgrid-scale model into our ROM setting by defining a test truncation similar to the test filtering in LES. The a posteriori analysis shows that our approach is remarkably accurate, allowing us to integrate simulations over long time intervals at a nominally small computational overhead.