Predicting rocker shoe parameters for reducing plantar pressure using machine learning methods

We investigate experimentally the effect of aspect ratio ( Open image in new window ) on the time-varying, three-dimensional flow structure of flat-plate wings rotating from rest at 45° angle of attack. Plates of Open image in new window = 2 and 4 are tested in a 50 % by mass glycerin–water mixture, with a total rotation of ϕ = 120° and a matched tip Reynolds number of 5,000. The time-varying, three-component volumetric velocity field is reconstructed using phase-locked, phase-averaged stereoscopic digital particle image velocimetry in multiple, closely-spaced chordwise planes. The vortex structure is analyzed using the \(\mathcal{Q}\)-criterion, helicity density, and spanwise quantities. For both Open image in new window s, the flow initially consists of a connected and coherent leading-edge vortex (LEV), tip vortex (TV), and trailing-edge vortex (TEV) loop; the LEV increases in size with span and tilts aft. Smaller, discrete vortices are present in the separated shear layers at the trailing and tip edges, which wrap around the primary TEV and TV. After about ϕ = 20°, the outboard-span LEV lifts off the plate and becomes arch-like. A second, smaller LEV and the formation of corner vortex structures follow. For Open image in new window = 4, the outboard LEV moves farther aft, multiple LEVs form ahead of it, and after about ϕ = 50° a breakdown of the lifted-off LEV and the TV occurs. However, for Open image in new window = 2, the outboard LEV lift-off is not progressive, and the overall LEV-TV flow remains more coherent and closer to the plate, with evidence of breakdown late in the motion. Inboard of about 50 % span, the Open image in new window = 4 LEV is stable for the motion duration. Up to approximately 60 % span, the Open image in new window = 2 LEV is distinct from the TV and is similarly stable. The Open image in new window = 2 LEV exhibits substantially higher spanwise vorticity and velocity. The latter possesses a “four-lobed” distribution at the periphery of the LEV core having adjacent positive (outboard) and negative (inboard) components, corresponding to a helical streamline structure. Both Open image in new window s show substantial root-to-tip velocity aft of the stable LEV, which drives outboard spanwise vorticity flux; flux toward the root is also present in the front portion of the LEV. For Open image in new window = 2, there is a strong flux of spanwise vorticity from the outboard LEV to the tip, which may mitigate LEV lift-off and is not found for Open image in new window = 4. The TV circulation for each Open image in new window is similar in magnitude and growth when plotted versus the chord lengths travelled by the tip, prior to breakdown. Streamwise vorticity due to the TV induces high spanwise velocity, and for Open image in new window = 2, the tilted LEV creates further streamwise vorticity which corresponds well to spanwise-elongated regions of spanwise velocity. For Open image in new window = 2, the TV influences a relatively greater portion of the span and is more coherent at later times, which coupled with the tilted LEV strongly contributes to the higher overall spanwise velocity and vorticity flux.

A synthetic jet issuing into a cross-flow influences the local velocity of the cross-flow. At the jet exit the jet is oriented in the wall-normal direction while the cross-flow is oriented in the streamwise direction, leading to a momentum transfer between the jet and the cross-flow. Streamwise momentum transferred from the cross-flow to the jet accelerates the pulses created by the jet. This momentum transfer continuous up to some point downstream where these pulses have the same velocity as the surrounding flow and are no longer blocking the cross-flow. The momentum transfer from the cross-flow to the jet leads to a momentum deficit in the cross-flow far downstream of the viscous near field of the jet. In the literature this momentum-flux deficit is often attributed to viscous blockage or to up-wash of low-momentum fluid. The present paper proposes and quantifies a third source of momentum deficit: a velocity induced opposite to the cross-flow by the vortical structures created by the synthetic jet. These vortical structures are reconstructed from measured data and their induced velocity is calculated using the Biot–Savart law. The three-dimensional three-component induced velocity fields show great similarity to the measured velocity fields, suggesting that this induced velocity is the main contributor to the velocity field around the synthetic jet and viscous effects have only a small influence. The momentum-flux deficit induced by the vortical structures is compared to the measured momentum-flux deficit, showing that the main part of this deficit is caused by the induced velocity. Variations with Strouhal number (frequency of the jet) and velocity ratio (velocity of the jet) are observed and discussed. An inviscid-flow model is developed, which represents the downstream evolution of the jet in cross-flow. Using the measured data as an input, this model is able to predict the deformation, (wall-normal) evolution and qualitative velocity field of the jet. The present study presents evidence that the velocity induced by the vortical structures forming a synthetic jet plays an important role in the development of and the velocity field around the jet.

Synthetic jets (SJs) enable tunable flow control across diverse applications by dynamically modulating the periodic formation and interaction of coherent vortical structures, such as vortex rings. This study experimentally investigates the effects of dynamic modulation—specifically actuation frequency and duty cycle (DC)—on the evolution of these structures. The modulation impacts on key SJ parameters, including jet strength, are carefully analyzed. Time-resolved particle image velocimetry, flow pathline visualization, and phase-averaged analysis were employed to characterize the spatiotemporal features of the unsteady SJ flows. Additionally, spectral proper orthogonal decomposition was used to extract frequency-dependent modal characteristics. Three distinct regimes of vortical behavior were identified depending on actuation conditions: (1) mutually isolated vortex rings with minimal interaction, (2) weakly interacting rings merging in the far-field, and (3) early merged, axially symmetric vortical structures near the orifice. These regimes correspond to transitions from vortex-dominated quasi-steady jets to phase-independent continuous jets. At low frequencies (3–5 Hz), SJs exhibit isolated vortex rings with extended travel due to long blowing phases and large stroke lengths, though the time-averaged jet strength is limited. At mid-range frequencies (10 Hz), SJs show strong transitional behavior sensitive to DC, producing distinct rings at low DC and merged rings at higher DC, with the strongest jets observed. At high frequencies (25 Hz), incomplete vortex ring formation leads to continuous jets insensitive to DC modulation. These results offer deeper insight into fine-tuning SJ dynamics and optimizing actuator designs for targeted flow control applications.

The interaction of a finite-span synthetic jet with a finite span low aspect ratio cylinder is explored experimentally using Stereoscopic Particle Image Velocimetry (SPIV) measurements in the near wake and along the surface of the cylinder in order to understand how local small changes to the near surface flow field result in large alterations to the wake structure. A single synthetic jet located at mid-span and azimuthally at  = 90˚ with respect to the freestream velocity was oriented such that the long dimension of its orifice was parallel to the freestream direction. Without flow control, downwash from the free end dominates the near wake of the cylinder, resulting in suppression of the von-Karman vortex street and the creation of a symmetrically-shedding arch-type vortex. Once the synthetic jet is actuated, the interaction between the synthetic jet and the local near-surface flow induces large-scale global changes to the flow field, altering the circulation and a narrowed wake, while also creating a vortex dislocation along one leg of the arch-type vortex outboard of the synthetic jet orifice. Above the jet orifice, a pair of counter-rotating streamwise vortices are formed, which entrain flow from the separated region inboard of the jet orifice and delay separation outboard of the jet orifice, creating the narrowed wake and vortex dislocation seen in the near wake measurements.

This paper presents a parametric study on the interaction of twin circular synthetic jets (SJs) that are in line with a crossflow over a flat plate. The resulting vortex structures under different actuation, and flow conditions are investigated using two-plane dye visualization in a water tunnel. The influence of four independent nondimensional parameters, i.e., the Reynolds number (ReL), Strouhal number (St), velocity ratio (VR), and phase difference (Δϕ), on the behavior of the twin SJs is studied. It is found that the increase of Reynolds number causes the SJ-induced vortex structures more turbulent, making the twin SJ interaction less organized. The increase of velocity ratio pushes the occurrence of interaction further away from the wall and makes the resulting vortex structures more sustainable. The St has no obvious influence on the interaction. And three types of vortex structures are observed under different phase differences: one combined vortex, two completely separated vortices, and partially interacting vortex structures. Based on this parametric study, a simple model is proposed to predict the resulting vortex pattern for the twin SJ interaction.

The vortex dynamics during the initial rotation of a rotating blade were investigated using two-component and stereo digital particle image velocimetry. The structure and dynamics of spanwise vorticity in the leading-edge vortex (LEV) was investigated on two rotating at plates of aspect ratios (AR), 2 and 4. Reynolds numbers of 4,000, 8,000, and 16,000, based on tip velocity, and angles of attack of 25 , 35 , and 45 were investigated at ve azimuthal locations ( = 90 , 180 , 235 , 270 , and 320 ), and two span locations (25% and 50%). Circulation measurements of the leading-edge vortex structure exhibit trends of increasing circulation with increasing Reynolds number and angle of attack, independently, for both aspect ratios. In the particular case of the aspect ratio 4 plate at the 25% spanwise position and the aspect ratio 2 plate at the 50% spanwise position, measurements demonstrate that the LEV of the smaller aspect ratio plate has lower circulation for a given angle of attack and Reynolds number (based on the local velocity). A strong region of counter-rotating vorticity was observed on the surface of the plate due to the interactions of the LEV and the body. It is proposed that the interactions between the LEV and counterrotating surface vorticity play an important role in governing the dynamics and strength of the LEV which may ultimately determine whether the LEV remains attached. An analysis of vorticity transport in the LEV was conducted to estimate the relative contributions of spanwise ux, tilting, the shear layer, and annihilation to the rate of change of circulation of the LEV. Stereoscopic particle image velocimetry (SPIV) was implemented at three parallel chordwise planes to obtain the three components of velocity and vorticity vectors needed for the analysis. Results of the vorticity transport equation indicate that annihilation of the LEV due to entrainment of the surface vorticity is an important factor governing the dynamics of the LEV.

An experimental investigation is conducted on the vortices induced by twin synthetic jets (SJs) in line with a laminar boundary layer flow over a flat plate. The twin SJs operating at four different phase differences, i.e., Δϕ = 0°, 90°, 180°, and 270°, are visualized using a stereoscopic color dye visualization system and measured using a two-dimensional particle image velocimetry (PIV) system. It is found that depending on the phase difference of twin SJs, three types of vortex structures are produced. At Δϕ = 90°, the two hairpin vortices interact in a very constructive way in terms of the vortex size, strength, and celerity, forming one combined vortex. At Δϕ = 270°, the two individual hairpin vortices do not have much interaction, forming two completely separated hairpin vortices that behave like doubling the frequency of the single SJ case. At Δϕ = 0° and 180°, the two hairpin vortices produced by the twin SJ actuators are close enough, with the head of one hairpin vortex coupled with the legs of the other, forming partially interacting vortex structures. Quantitative analysis of the twin SJs is conducted, including the time histories of vortex circulation in the mid-span plane as well as a selected spanwise-wall-normal plane, and the influence of the twin SJs on the boundary layer flow filed. In addition, dynamic mode decomposition analysis of the PIV data is conducted to extract representative coherent structures. Through this study, a better understanding in the vortex dynamics associated with the interaction of in-line twin SJs in laminar boundary layers is achieved, which provides useful information for future SJ-array applications.

This paper describes a detailed study on effects of both cavity and orifice height on flow field governing a single synthetic jet in a quiescent air. The synthetic jet, known as a zero net mass-flux jet, is generated by a vibrating diaphragm to oscillate flow through a small orifice. The tested synthetic jet actuator consists of the orifice with the rectangular slot and cavity installed the PZT disk of 31.2mm diameter. The orifice height is chosen from 0.5, 1.0, 2.0 and 4.0mm, and the cavity height from 0.5 to 8.9mm, respectively. The test is conducted in variable input voltages and frequencies to PZT disk. With both the displacement of PZT disk on sinusoidal voltages and jet velocities above the orifice, the synthetic jet depends on not only oscillation of the PZT disk but sound pressure in the actuator. The simple lumped elements modeling is tested for the various cavity and orifice height, which does not require the iterative calculation on 1 degree forced oscillating. The analytical and experimental results are in good agreement on the maximum synthetic jet velocities for the orifice height of 0.5, 1.0 and 2.0mm in the cavity height higher than 0.75mm. The actuator with the cavity height of 2.5mm gives the maximum synthetic jet velocities to a high level, optimizing actuator size have to be considered the effect of both the cavity and orifice height for a micro synthetic jet actuator.

Experiments were performed within the Rutgers University Mach 3.4 supersonic wind tunnel to quantify the turbulent boundary layer in the test section, using stereoscopic particle image velocimetry (SPIV). Seeding of the tunnel was accomplished using atomized mineral oil, resulting in a Stokes number of 0.09. The Reynolds number based on momentum thickness and the friction-velocity Reynolds number were 64,000 and 4320, respectively, marking this as a high Reynolds number boundary layer. SPIV measurements were capable of resolving the mean velocity profile, along with turbulence quantities, permitting comparison against the known literature. Furthermore, the convection of vorticity concentrations representing hairpin vortices was observed within instantaneous vector fields. The resulting Reynolds stresses demonstrate that the boundary layer had typical values of anisotropy, while comparing well with data obtained within a slightly adverse pressure gradient. This was particularly noticeable from the trends in the Reynolds stresses, which exhibited higher-than-expected peak levels. These data provide a unique measurement of a supersonic turbulent boundary layer at Reynolds numbers not often encountered within existing data sets.

The structure and dynamics of the leading-edge vortex (LEV) were investigated for plunging nite-aspect-ratio wings at a chord Reynolds number of 10,000 while varying aspect ratio and root boundary condition. Stereoscopic particle image velocimetry (SPIV) measurements were used to characterize LEV dynamics and interactions with the plate in multiple chordwise planes. The relationship between the vorticity eld and the spanwise ow eld over the wing, and the inuence of root boundary conditions on these quantities has been investigated. The viscous symmetry plane was found to inuence this ow eld, in comparison to other studies, 1{3 by inuencing tilting of the LEV near the symmetry wall, and introducing a corewise root-to-tip ow near the symmetry plane. Modications in the root boundary conditions were found to signicantly aect this. LEV circulations for the dierent aspect ratio plates were also compared. At the bottom of the downstroke, the maximum circulation was found at the middle of the semi-span in each case. The circulation of the sAR = 2 wing was found to signicantly exceed that of the sAR = 1 wing and, surprisingly, the maximum circulation value was found to be independent of root boundary conditions for the sAR = 2 case and also closely matched that of the quasi-2D case.

Although macro-scale synthetic jets are well studied, our knowledge of the behaviour of micro-scale synthetic jets is still extremely limited due to the difficulty in performing detailed measurements with a desirable spatial solution. In the work presented in this paper, a PIV study of the characteristics of synthetic jets issued into quiescent air from orifices of 5mm and 0·5mm diameter respectively is undertaken. It is found that the vortex rollup is much weaker for the 0·5mm jets due to their low Stokes number which results in a near parabolic exit velocity profile. To ensure an appreciable vortex rollup that is desirable for effective flow control, the actuator has to be operated at much higher frequencies to ensure that the Stokes number is greater than a certain threshold value. Furthermore the study shows that the characteristics of synthetic jets of different scales are identical when the dimensionless stroke length (L), Stokes number (S) and Reynolds number (ReL) are the same. On the basis of these scaling parameters, the finding acquired from the studies on macro-scale actuators can be applicable to micro-scale actuators, which are more difficult to measure. Finally, it is also found that although the linear relationships between (L and ReL) and actuator operating conditions observed for macro-scale synthetic jets are no longer valid for micro-scale synthetic jets, the linear relationships between the dimensionless jet performance parameters and (L and ReL) still exist for micro-scale synthetic jets.

Synthetic jets are created by periodically ejecting and injecting fluid from an orifice or channel. Despite delivering no net mass flow per cycle, a synthetic jet delivers flow with net positive momentum. Small, compact synthetic jet actuators can be fabricated to operate in the subaudible acoustic range and can be packaged in orientations that allow them to deliver cooling air flow to electronic devices. The most promising orientation is one that delivers the jet flow in a direction normal to the heated surface such that it impinges on the surface as a periodic jet. In previous studies, numerical simulations have been performed by the authors, utilizing a canonical geometry, with the purpose of eliminating actuator artifacts from the fundamental physics that drive the problem. The present paper reports on laboratory experiments that have been performed in order to nearly replicate the idealized synthetic jet geometry and thus allow comparison to the previous numerical investigations. The periodic volume change in an upstream plenum required to produce the synthetic jet is accomplished with an acoustic speaker operated at low frequencies. The amplitude and the frequency at which the jet is actuated determine the Reynolds and Strouhal numbers, which are the dominant non-dimensional groups that control the behavior of the impinging synthetic jet. By maintaining the Re and the St in the laboratory experiments to match those of the small scale actuators, the laboratory experiments have been geometrically scaled up to allow highly resolved measurements of the unsteady velocity field and the local time-dependent Nusselt number on the target heated surface. Experiments were performed at variable jet Re, frequencies, and height from the target surface. The dependence of the surface averaged Nu to jet parameters generally agrees with the computational results. However, discrepancies found between numerical and empirical local data are under revision.

This study is focused on a new approach to control the mixing at the exhaust of a jet nozzle. Two dielectric barrier discharges are used to enhance the turbulent quantities in the jet wake. An axisymmetric air jet is equipped with a 22-degree-angle diffuser that houses two dielectric barrier discharge actuators, symmetrically placed along the lips of the diffuser. Streamwise, cross-stream planes, and turbulent spectra for free air jet velocities from 10 to 40 m/s are investigated by stereoscopic particle image velocimetry and laser Doppler velocimetry up to 8 jet diameters. The airflow is separated naturally along the diffuser wall and the results demonstrate that a full (at 10 m/s) or partial (at 20 m/s) flow reattachment can be performed, this process being associated with the shedding of coherent structures. The turbulent spectra suggest that the frequency of the vortex shedding is dictated by the electrical frequency applied to the actuators and that the flow structures can be manipulated up to 40 m/s. The time-averaged data confirm that a significant enhancement of the jet spreading, a jet core length reduction, and a turbulent kinetic energy increase are performed up to 30 m/s for a quasi-steady actuation. Unsteady actuations are also investigated and the results highlight that a Strouhal number from 0.25 to 0.32 is particularly effective to enhance the turbulent kinetic energy in the overall flowfield.

In the depths of the ocean, where minerals lie buried beneath layers of sediment, multi-stage pumps serve as the workhorses of deep-sea mining, lifting extracted materials through immense vertical distances. To ensure their efficiency and reliability, a comprehensive hydrodynamic analysis is required, focusing on vortex-induced vibrations and fluid–structure interactions. In this study, numerical simulations were conducted using computational fluid dynamics to investigate the internal flow characteristics of a multi-stage deep-sea pump. The distribution and formation mechanisms of vortices were analyzed using the omega vorticity identification method, which allowed for the capture of vortex structures and the evaluation of entropy production in the impeller and guide vanes across varying flow rates. The results revealed that at low flow rates, complex vortex structures, including flow separation and reflux vortices, were predominant at the impeller inlet. As the flow rate increased, these vortices gradually dissipated, leading to a more stable wake at the impeller outlet, while separation and reflux vortices became more pronounced at the guide vanes' inlet and outlet. A reduction in the blade passage vortex structure was also observed with increasing flow rates. Further analysis demonstrated that the stretching and twisting term played a crucial role in the formation, evolution, and dissipation of vortex structures at the guide vanes' inlet. Meanwhile, the Coriolis force was identified as a key factor in vortex generation at the impeller inlet and within the blade passage. These findings provide valuable insights into the vortex dynamics governing deep-sea pump hydraulics, offering a foundation for optimizing their design to enhance operational performance and ensure more efficient deep-sea resource extraction.

A computational fluid dynamics (CFD) validation workshop for synthetic jets and turbulent separation control (CFDVAL2004) was held in Williamsburg, Virginia in March 2004. Three cases were investigated: synthetic jet into quiescent air, synthetic jet into a turbulent boundary layer crossflow, and flow over a hump model with no-flow-control, steady suction, and oscillatory control. This paper is a summary of the CFD results from the workshop. Although some detailed results are shown, mostly a broad viewpoint is taken, and the CFD state-of-the-art for predicting these types of flows is evaluated from a general point of view. Overall, for synthetic jets, CFD can only qualitatively predict the flow physics, but there is some uncertainty regarding how to best model the unsteady boundary conditions from the experiment consistently. As a result, there is wide variation among CFD results. For the hump flow, CFD as a whole is capable of predicting many of the particulars of this flow provided that tunnel blockage is accounted for, but the length of the separated region compared to experimental results is consistently overpredicted.