Snapshot Proper Orthogonal Decomposition Research Articles

Aeroacoustic Coupling in Rectangular Deep Cavities: Passive Control and Flow Dynamics

Deep cavity configurations are common in various industrial applications, including automotive windows, sunroofs, and many other applications in aerospace engineering. Flows over such a geometry can result in aeroacoustic coupling between the cavity shear layer oscillations and the surrounding acoustic modes. This phenomenon can result in a resonance that can lead to significant noise and may cause damage to mechanical structures. Flow control methods are usually used to reduce or eliminate the aeroacoustic resonance. An experimental set up was developed to study the effectiveness of both a cylinder and a profiled cylinder positioned upstream from the cavity in reducing the flow resonance. The cavity flow and the acoustic signals were obtained using particle image velocimetry (PIV) and unsteady pressure sensors, respectively. A decrease of up to 36 dB was obtained in the sound pressure levels (SPL) using the passive control methods. The profiled cylinder showed a similar efficacy in reducing the resonance despite the absence of a high-frequency forcing. Time-space cross-correlation maps along the cavity shear layer showed the suppression of the feedback mechanism for both control methods. A snapshot proper orthogonal decomposition (POD) showed interesting differences between the cylinder and profiled cylinder control methods in terms of kinetic energy content and the vortex dynamics behavior. Furthermore, the interaction of the wake of the control device with the cavity shear layer and its impact on the aeroacoustic coupling was investigated using the POD analysis.

Large eddy simulation of round jets with mild temperature difference

Understanding the behaviour of hot jets is crucial for various engineering and environmental applications. The present work studies the influence of heat transfer on the dynamics of horizontal round hot jets through Large Eddy Simulations (LES). Our focus lies on trajectory development, large-scale coherent structures, and turbulent kinetic budget analysis in the near-field and intermediate-field regions. LES of two horizontal round hot jets with Reynolds numbers (3934 and 5100) and corresponding Froude numbers (32.98 and 17.07) were carried out using buoyantPimpleFoam solver in OpenFOAM, and the simulation on an isothermal jet was also performed as a baseline for comparison. The results reveal that the jet core temperature decays faster in the streamwise direction but more slowly in the radial direction, indicating a wider temperature spread than velocity, and the maximum difference between the temperature and velocity spread is about 0.5D. Moreover, the energy associated with the large-scale coherent structure decreases with increasing initial jet temperature. The energy of the first two modes of snapshot Proper Orthogonal Decomposition (POD) and extended POD dropped by 12% and 14%, respectively. The coherent motion with the greatest correlation between the temperature and velocity fluctuations is identified as four pairs of Q1 and Q3 events, which are Reynolds shear stress dominant events. Furthermore, compared with the isothermal jet, the turbulent kinetic energy budgets of the hot jets indicate that the diffusion and generation terms are both reduced by approximately 50%, suggesting a transfer of more kinetic energy into potential energy rather than turbulence. The finding highlights the potential of heightened temperatures to mitigate instabilities associated with large-scale motions in hot jets. This study fills the gap on a comprehensive analysis of heat transfer effects on jet dynamics, and quantitative insights into the large-scale coherent structures are provided, contributing to a better understanding of hot jet behaviour.

Data-driven modal analysis of turbulent momentum exchange and heat transfer in composite porous fluid systems

This paper investigates the dynamics governing turbulent momentum exchange and heat transfer between pore flow within porous media and the turbulent flow passing over it. Employing high-fidelity pore-scale large eddy simulation, our investigation explores the fundamental mechanisms driving these phenomena. Modal analysis based on snapshot proper orthogonal decomposition (POD) is employed to quantify the modes of interaction between porous and non-porous regions, providing a comprehensive understanding of the underlying processes. Spatial and temporal modes reveal the existence of localized flow structures at the pore scale, contributing to time-varying patterns of information exchange. At the commencement of the porous block, the mean flow (Mode = 0) from the porous to non-porous region is the dominant mechanism in momentum exchange and heat transfer. This mode facilitates convective heat transfer from the porous to the non-porous region through upward and forward flow movements, showcasing positive flow leakage. In addition to the mean flow, the turbulent flux inherent in alternate POD modes (Mode ≠ 0) plays a substantial role in information propagation, influencing diverse directions. Spatial modes, complemented by statistical analysis, uncover a significant likelihood of observing negative vertical velocity values in the wake of the porous ligaments at the porous-fluid interface, indicative of negative flow leakage. This negative flow leakage precisely corresponds to the local penetration of fluid from the non-porous region into the porous region. Furthermore, our study reveals that information exchange via turbulence fluctuations manifests through complex outward and inward interactions in regions characterized by substantial positive flow leakage. Notably, these regions exhibit a distinct tendency for high-momentum streamwise-oriented flow to migrate outward from the porous region into the non-porous region (outward interactions). Conversely, inward interactions arise in these regions when the instantaneous magnitude of positive flow leakage is smaller than the mean value of positive flow leakage, emphasizing the pulsating nature of positive flow leakage. Finally, the distribution of the Nusselt number highlights that more than 60% of total heat transfer occurs within the initial one-third of the porous block length. Significantly, a notable portion of the porous ligaments experiences insufficient cooling due to positive flow leakage, underlining the critical implications of these findings for the understanding of turbulent momentum exchange and heat transfer in a composite porous-fluid system.

Impact of Transverse Acoustic Excitation and Air Swirl on a Confined Hollow Cone Sheet

Abstract This study investigates the effect of imposing symmetric transverse acoustic excitation and counter-rotating air swirl flow on a confined hollow cone spray sheet. We study the impact of these forcing conditions on two unstable modes—sheet flapping and sinuous wave growth modes—and their phasic relationship. We used the shadowgraphy technique, and only the sheet edges were analyzed using the method of snapshot proper orthogonal decomposition (POD). In this study, we used postprocessing techniques to estimate the nondimensional breakup length, phase averaged cone angle, and continuous wavelet transform (CWT) frequency. Results show that imposing acoustic excitation alone causes the sheet edges to flap, axisymmetrically, at the pressure antinode, while at the pressure node and base flow condition, the asymmetric sinuous wave mode is dominant. At the pressure intermediary, we find the possible dominance of both modes. While the dominant effect of air swirl alone was difficult to ascertain, results from CWT analysis show that POD mode 1 and mode 3 have very similar spectral content, with mode 3 showing a flapping mode. Finally, air swirl flow has a more dominant effect than acoustics in causing sheet breakup.

Flow structure and impinging interactions of two confined turbulent converging jets in crossflow

In this work, time-resolved particle image velocimetry (TR-PIV) measurements are carried out to study the flow structure and impinging interactions of two confined turbulent converging jets in crossflow, with the upstream and downstream jets inclined at 20° and 170° with respect to the oncoming crossflow direction. Experiments are conducted for fixed dimensionless nozzle separation distance of s=L/D=3, fixed dimensionless nozzle-to-target surface distance of H/D=6.73, fixed duct Reynolds number of Re=3500 and two jets' Reynolds numbers of Rej=3000 and 6000 that correspond to jets'-to-crossflow velocity ratios of rj=3.45 and 6.91, respectively. Flow visualization images showing ensemble-averaged and instantaneous flow distributions and turbulent characteristics including Reynolds stresses and turbulence intensities for equal and non-equal discharge velocities of the jets are presented on several orthogonal planes of the test section. The swirling strength criterion has been employed to identify coherent structures, and cross spectral analysis of the fluctuating velocity signals detecting subharmonic and harmonic sideband frequencies identify the merging and breakup interactions of the evolving coherent structures. The snapshot proper orthogonal decomposition technique (POD) has been applied to extract the dominant vortical structures and to quantify their relative and cumulative energy contributions to the total kinetic energy fluctuation. The spatial structure of the main four POD modes and the temporal projections of these POD coefficients have been explored to understand the spatio-temporal characteristics of the dominant turbulent fluctuations and their spectral characteristics. Our results show the effect of the velocity ratios on the jets' trajectories, penetration, spreading characteristics, crossflow entrainment through the inner and outer shear layers, transverse velocity decay, size of the converging region and location of the combining point. In addition, the dimensionless shedding frequencies of the oscillating combined jet, the developing wall jets and wake vortices have been obtained. Furthermore, it has been shown that for all values of the jets' velocity ratios employed in this study, a recirculation region does not develop in the converging region.

Influence of time-varying flow on dynamic flame characteristics in VCE: Numerical and experimental study

Afterburner shows obvious transient characteristics in variable cycle engines (VCEs), herein, a compact flame holder and injection device with well performance coordinating a rear variable bypass injector (RVABI) were designed and investigated under variable inlet conditions. Inlet aerodynamic parameters were dynamically simulated and adjusted from small to large to investigate the matching law of bypass ratio and guiding angle, flow structure, and combustion characteristics. A typical case was experimentally verified well, and the coherent modes of the transient velocity field were identified using snapshot proper orthogonal decomposition(POD). It is found from POD that the first modes contributing to the total fluctuation energy are decreased by 15% when the guiding angle increased to 31.92°. The inlet conditions have little effect on the vortex structure near the pilot, while the red-blue alternating vortex structures in the rear of the strut are mainly influenced by the coupling of back pressure between the strut and the pilot. The air entrainment strength guided by RVABI will affect the three-dimensional vortex system built by the flame holder, and cause flow separation or local flameout when the velocity distortion is out of 0.5–1.2. Increased air-entrainment strength generates high turbulent energy, which can improve flow separation and increase radial flame development's instability. Moreover, a guiding angle of nearly 5° has the same effect on air entrainment strength as a bypass ratio of 0.2.

Experimental investigation on effects of herringbone riblets on shock wave/boundary layer interactions of a compression ramp at Mach 3

In this paper, the effects of herringbone riblets (HRs) on a turbulent boundary layer and the shock wave/boundary layer interactions (SWBLIs) are experimentally investigated at the Mach number of 3. An array of three varied lengths of HRs strips are applied upstream of the separation zone of SWBLIs on a compression ramp model. High-speed schlieren, oil-flow visualization, and the planar laser scattering technique are used to examine the shock pattern and boundary layer developing over the surface of the model. The snapshot proper orthogonal decomposition technique and the Fast Fourier Transform method are applied to study the impact of HRs on the interaction between the shock wave and the boundary layer. The experiments provide convincing evidence that HRs make the separation line wavy and shrink the separation zone by about −39.54% for the longest HRs in the present experiments. Furthermore, it is also revealed that these microscale HRs induce large-scale streamwise vortical structures within the boundary layer as found in incompressible flows. It is believed that these vortices promote momentum transfer within the boundary layer hence providing the dominant mechanism for suppressing flow separation.

Phase proper orthogonal decomposition of non-stationary turbulent flow

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

Time-resolved PIV measurements inside and downstream of a 180° bend

The flow in a 180° bend with a radius of curvature 1.5 time the pipe diameter (Rc/D of 1.5) was studied for a Reynolds number of 34,800 to characterize the development of the mean velocity and Reynolds stresses and the unsteady motions that may affect the heat or mass transfer in the latter part of the bend. Two-component, time-resolved measurements were performed on the axial symmetry plane and on cross sectional planes in the bend and up to 1D downstream of the bend using refractive index matched Particle Image Velocimetry. The unsteady flow structures were examined using the Snapshot Proper Orthogonal Decomposition (POD) technique. Frequency spectra of the POD coefficients, transients of the coefficients and low-order reconstructions were used to characterize the time scales and dynamics of the motions. The results showed that the initial development of the flow was similar to that in a 90° bend but the development of the mean secondary flow, and the low-speed region associated with this feature, appeared delayed relative to the 90° bend in the middle section of the 180° bend. The results in the second half of the 180° bend showed that the mean longitudinal velocity near the inner wall was larger than that near the outer wall in the latter part of the bend. Changes in the unsteady interaction of the flow with the outer wall in the middle of the 180° bend also appeared to cause significant unsteadiness in the flow near the inner wall in the latter part of the bend. The mean secondary flow towards the end of the 180° bend was significantly diminished relative to the flow in the middle of the bend and largely suppressed by 1D downstream of the bend. Reconstructions of the instantaneous secondary flow showed occurrences of large secondary velocities along the sidewalls up to 170°, but these were substantively reduced even at 0.25D into the downstream bend.

Analysis of tip leakage flow unsteadiness in a transonic turbine cascade using data-driven modal decomposition methods

The evolution of tip leakage flow and subsequent vortical structures is accompanied by inherent unsteadiness. This paper presents a novel characterization for the unsteady dynamics of turbine tip flow using data-driven, i.e., equation-free modal decomposition methods, which are applied to the hybrid Reynolds-averaged Navier–Stokes/large eddy simulation data at a transonic condition. By combining these techniques, the identified eigen-tuples (eigenvalues, eigenvectors, and time evolution) are well determined, and the differences between the obtained patterns (modes) are also pronounced. The snapshot proper orthogonal decomposition (POD) analysis can isolate the large-scale fluctuating structures that populate the rear part of the suction-side, which is mainly attributed to the shock-induced vortex instability. Similar to the turbulence cascade, macro-scale coherent structures that correspond to the tip leakage vortex shedding phenomena and the subsequently transitional and noisy parts closely related to the dissipation can be well derived by a quadruple reconstruction. Three dynamic mode decomposition (DMD) variants including the amplitude selecting-DMD method, the DMD with criterion method, and the sparsity promoting (SP)-DMD method are also compared in extracting dominant modes from the periodic tip flow, and the SP-DMD method which can distill modes of broadband frequencies and low dissipation is proved to be more conducive to representing and reconstructing the complex tip flow. Additional spectral-kernel-based POD (SPOD) analysis that can identify the similar primary unsteadiness frequencies as the DMD method is also encompassed in this study. Specifically, although it manifests that a physical resemblance of the pattern of pressure fluctuations to tip eddy unsteadiness can be captured by all these approaches, the behavior of small-scale vortical interaction downstream of the trailing edge can be clearly isolated with the intrinsic Karman-type vortex layer shedding process via DMD and SPOD approaches, which also demonstrates that these two techniques are more favorable to decomposing the complex tip flows into uncoupled single-frequency coherent structures compared to the conventional POD method. On this basis, resulting modes of velocity components have been accounted for verifying their contributions to the turbulent kinetic energy fields. The ensuing observations can offer a glimpse of the complex dynamics in the tip region, which also sheds light on features previously masked by conventional analysis approaches.

Three-dimensional density field of a screeching under-expanded jet in helical mode using multi-view digital holographic interferometry

A synchronised multi-axis digital holographic interferometry set-up is presented for the study of 3-D flow fields with large density gradients. This optical configuration provides instantaneous interferograms with fine spatial resolution in six directions of projection. A regularised tomographic approach taking into account the presence of possible shock waves is furthermore considered to reconstruct 3-D density fields. Applied to a screeching under-expanded supersonic jet with helical dynamics, this set-up is used to provide dense optical phase measurements in the initial region of the jet. The jet mean density field is shown to be satisfactorily estimated with sharply resolved density gradients. In addition, an approach based on azimuthal Fourier transform and snapshot proper orthogonal decomposition (POD) applied to the instantaneous flow observations is proposed to study the main coherent dynamics of the jet. Relying on a cluster analysis of the azimuthal POD mode coefficients, a reduced dynamical model in the POD mode phase space is used as an approximation of the two observed limit cycles. A clear 3-D representation of the density field of a helical instability associated with screech mode C is then evidenced, with two equally probable directions of rotation. Switching between the two directions is reported, highlighting intermittency in the feedback loop. This helical structure is particularly seen to extend to the jet core, driving its internal dynamics and inducing out-of-phase density fluctuations between the outer and inner shear layers. These out-of-phase motions are related to the non-uniform radial distribution of fluctuation phase associated with the outer-layer Kelvin–Helmholtz instability wave.

Characteristics and mechanisms of air-core vortex meandering in a free-surface intake flow

The air-core vortex generated at hydraulic intakes is usually associated with efficiency reduction and performance deterioration of the hydraulic machinery. Numerical simulations of the air-core vortex in a horizontal intake system are performed for various intake submergence depths and flow rates. Our study focus on the characteristics and mechanisms of the air-core vortex meandering phenomenon. Statistical analyses of the meandering coordinates of the vortex core are carried out, which characterize the meandering scope, dispersion radii and frequencies of the meandering coordinates of the air-core vortex. Parametric studies of the effects of the flow parameters on the air-core vortex meandering indicate that the dispersion radii increase as the intake flow rate increases or the intake submergence depth decreases, and the meandering frequencies decrease with increasing flow rate. Snapshot proper orthogonal decomposition (POD) analyses of the velocity and vorticity fields reveal that the large-scale energetic coherent structures in the first two modes cause streamwise and spanwise displacement of the air-core vortex, and the small-scale coherent structures with low energy fractions in the third to sixth modes contribute to elliptical deformation of the vortex core. Multiple relationships among the temporal frequencies of the decomposed POD modes and the meandering coordinates are discovered, which further proves that the large-scale energetic coherent structure in the turbulent flow field is the intrinsic reason for the meandering phenomenon of the air-core vortex.

Experimental investigation of the flow-spray field in a realistic concentric staged high-temperature-rise combustor

The present work investigated the flow and spray fields in a realistic concentric staged high-temperature-rise (HTR) combustor in isothermal conditions using high-speed (10 kHz) particle image velocimetry (PIV) and 10 Hz kerosene planar laser-induced fluorescence (kerosene-PLIF). The two stages of the combustor, including a pilot stage and a main stage, were operated both individually and jointly to investigate their corresponding flow fields and the interactions between the flow from the two stages. From the time-averaged flow field results, strong coupling between the pilot and main swirling flows has been identified, leading to the formation of a primary recirculation zone (PRZ) and a lip recirculation zone (LRZ). From the instantaneous flow fields, the presence of a precessing vortex core (PVC) was observed when the two stages were flowing jointly, and the dominant PVC unsteady motions located in the inner shear layer of the pilot swirling was identified using snapshot proper orthogonal decomposition (snapshot POD) with dominant frequencies at 1400 and 1515.4 Hz. Comparing the flow field with the spray field indicates that the unsteady flow motion, in addition to the mean bulk flow, plays a key role in the understanding of the spray distribution. Noticeably, it is shown in present study that the dominant unsteady flow motions properly captured from the POD analyses show good agreement with some primary features of the spray distribution, which can be valuable to unraveling the complex flow-spray coupling in the HTR combustor operated under realistic conditions.

Optimal Design for Suppressing Time Fluctuation Part of Two-Dimensional Jet in Crossflow

This paper proposes a shape optimisation method for suppressing a time fluctuation part of the two-dimensional jet-in-cross flow. Previously, the author has formulated a shape optimisation problem for suppressing transient fluids using Snapshot proper orthogonal decomposition (POD). This study extends it to a two-dimensional jet-in-cross flow. For this purpose, the eigenvalues of Snapshot POD are integrated to a cost function, where the constraint functions are the nonstationary Navier–Stokes equation and the eigenvalue equation of Snapshot POD. An objective functional is described using Lagrange multipliers and the finite element method as the sum of the cost and constraint functions. To verify the proposed optimisation strategy, using the gradient method for domain deformation, all triangles over a mesh are deformed, which causes the cost function to decrease. Finally, the eigenvalues of Snapshot POD in the initial and optimal domains are compared.

A new method for prediction of power coefficient and wake length of a horizontal axis wind turbine based on energy analysis

Power coefficient and wake length are important indexes for wind farm design and operational control. Accurate estimation of this coefficient is of great significance for the design of wind energy conversion system. In the present study, new methods for estimating rotor power coefficient (Cp) and wake length (WL) of wind turbine from an energy-based perspective are proposed, and the corresponding prediction models are established. Large eddy simulation (LES) coupled with the Actuator Line Method (ALM) are used to simulate the flow field around the wind turbine, and then transient snapshot Proper Orthogonal Decomposition (POD) method is adopted to analyze the energy contained in each plane of the velocity field. Results show that the new model for power coefficient Cp prediction is effective and the maximum error between the estimated value and that obtained with the conventional method is 1.7%. Meanwhile, the wake lengths predicted with the new model are all about 28.5D for the studied tip speed ratios, and the velocity contours perpendicular to the center of wind turbine verify the good accuracy of the model. The POD analysis results also show that the first-order POD mode characterizing the large-scale turbulence structures maintains more than 99.5% energy of the inflow plane, and offers primarily the energy extracted by the wind turbine. This method can be used in the conditions of the transient velocity field easily obtained, and will rich the method of rotor power coefficient and wake length estimation.

An LES study of secondary motion and wall shear stresses in a pipe bend

Large-eddy simulations (LES) of a single-phase, turbulent flow in a 90° pipe bend are performed at three Reynolds numbers (5300, 27 000, and 45 000) to investigate the correlation between secondary flow motion and wall shear stresses, which is suspected to be a potential mechanism responsible for material erosion. The isothermal flows are validated against available experimental and numerical data first. The snapshot proper orthogonal decomposition (POD) is applied for the medium and high Reynolds number flows to identify the secondary flow motions and the oscillation of the Dean vortices that are found to cause swirl-switching. Distinguished frequencies of the POD time coefficients at Strouhal numbers of 0.25 and 0.28 are identified for Reynolds numbers at 27 000 and 45 000, respectively. Moreover, shear stress on the pipe wall and the associated power spectral density are obtained and shown to have the same oscillating frequency as the swirl-switching.

Meandering of a wing-tip vortex in a grid-generated turbulent flow

The interaction of a National Advisory Committee for Aeronautics 0012 wing-tip vortex with a grid-generated turbulent flow was experimentally investigated. The experiments were conducted in the near- and mid-wake regions at three free-stream turbulence (FST) levels, viz., 0.5%, 3%, and 6%, at a Reynolds number, based on the wing chord length, cw, of 2×105. Stereoscopic particle image velocimetry measurements were carried out at four downstream positions: x/cw = 1.25, 3.25, 6.25, and 7.75. Streamwise vorticity contours showed that the wing-tip vortex decayed with increased FST and downstream distance. Turbulent surroundings were found to affect the meandering amplitude of the vortex, which increased with the dispersed scatter pattern of the vortex center motion, resulting in a meandering-induced turbulence. Meandering-corrected turbulent kinetic energy revealed the existence of a laminar core at the center of the vortex surrounded by low turbulence levels outside the core. This was attributed to the stabilizing Coriolis effects of the strong rotational motion inside the vortex core, which tend to re-laminarize the turbulent fluid crossing the periphery of the vortex. Snapshot proper orthogonal decomposition analysis on the coherent component of the velocity field revealed two dominant modes forming a helical dipole, consistent with the helical displacement of a Kelvin wave with an azimuthal wavenumber |m=1|. An analysis of the terms balancing the rate of decay of the mean enstrophy revealed that increasing FST increases the stretching of the mean enstrophy within the vortex core while it reduces both its transport and convection terms. Nonetheless, the latter contributions were larger in all cases studied acting as the main mechanism for mean enstrophy decay.

Reduced order model analysis to identify possible aerodynamic noise sources of small axial fan: POD and CNN

This paper reports computational analysis of location and strength of sound source of the noise generated by a small axial fan widely used as an air-cooling system. High-fidelity Navier-Stokes simulations with high-resolution compact scheme are conducted with an implicit Large Eddy Simulation (LES) method on a HPC system and the resultant large-scale data confirms existence of unsteady vortex structures and their interactions around the impellers, boss and casing of the fan. To identify location and strength of the sound sources, reduced order model analysis is conducted for the distribution of pressure fluctuations in space and time. Snapshot POD (Proper Orthogonal Decomposition) analysis both in time and in circumferential direction, together with conventional FFT analysis, identifies location and strength of the sound sources. In addition, Convolutional Neural Network (CNN) is attempted, which shows more physical mode decomposition and separates some of the important features shown in the snapshot POD analysis. The study shows that the two data-mining techniques considered here identify possible aerodynamic noise sources of the axial fan clearly in comparison to those in the previous studies.

Cluster based nonuniform transformation field analysis: An efficient homogenization for inelastic heterogeneous materials

AbstractIn this article, the authors propose an efficient numerical homogenization method, the so‐called cluster based nonuniform transformation field analysis (NTFA), for inelastic heterogeneous material. The method enables to predict the effective properties directly from the local constitutive relations for microstructural phases of heterogeneous material, thus no requiring preliminary assumptions and any derivations for the effective constitutive relation as in the conventional NTFA. Evolution of plastic strain field is expressed by combination of nonuniform plastic strain modes evaluated from the nonlinear finite element analysis for training directions previously given and snapshot proper orthogonal decomposition. Eventually, microscopic stress and strain field are represented by scalar reduced variables and plastic strain modes. A simple maximization of minimum distance is proposed in order to choose uniformly distributed training directions on a unit sphere in the load space. Also, scalar reduced variables are directly determined from the integration of local constitutive relation, where material points are compressed into a number of clusters by clustering scheme. Thus, evolution of reduced variables is evaluated by the integration of local constitutive relation only in clusters. Numerical examples that consider material models followedJ2plasticity and pressure‐dependent non‐associated flow rule show that the current approach may predict the effective properties of inelastic heterogeneous material accurately and efficiently. It should be noted that the current approach can be available for material followed not only associated flow rule but also non‐associated flow one.

Suppressing unsteady motion of shock wave by high-frequency plasma synthetic jet

Three Plasma Synthetic Jet Actuators (PSJA) under the high-frequency actuation are used to control the Shock Wave Boundary Layer Interaction (SWBLI), a high-speed schlieren image processing method based on spatial Fourier transform as well as snapshot proper orthogonal decomposition were used to study the control effect of high-frequency plasma synthetic jet on low-frequency unsteadiness of SWBLI. The analysis of the base flow shows that the separated shock wave actually has both large- and small-amplitude vibrations at low frequency. And the results revealed that the PSJA with an operating frequency of 2 kHz has the ability to reduce the energy of low-frequency component of shock wave motion, indicating that the 2 kHz actuation can effectively suppress low-frequency unsteadiness of the separated wave. Compared with the actuation frequency of 2 kHz, the energy of low-frequency component of the shock wave is enhanced under the 8 kHz actuation, which aggravates the low-frequency unsteady motion of the shock wave. It is likely that the actuation frequency is too high, thus the intensity of the precursor shock wave induced by PSJA becomes weaker. Additionally, as the 4 kHz actuation is applied, the pulsation of the separation region was enhanced, it is speculated that the actuation frequency is coupled with the oscillation frequency of the separation region.