Time-averaged Force Research Articles

Dynamics of an acoustically levitated fluid droplet captured by a low-order immersed boundary method

We present a variant of the immersed boundary (IB) method that implements acoustic perturbation theory to model acoustically levitated fluid droplets. Instead of resolving sound waves numerically, our hybrid method solves acoustic scattering semi-analytically to obtain the corresponding time-averaged acoustic forces on the droplet. This framework allows the droplet to be simulated on inertial timescales of interest, and therefore works with much larger time steps than traditional compressible flow solvers. To benchmark this technique and demonstrate its utility, we implement the hybrid IB method for a single droplet in a standing wave. Simulated droplet shape deformations and streaming profiles agree with available theoretical predictions. Our simulations also yield insights into the streaming profiles for elliptical droplets, for which a comprehensive analytic solution does not yet exist.

The effect of Reynolds number on the separated flow over a low-aspect-ratio wing

At high incidence, low-aspect-ratio wings present a unique set of aerodynamic characteristics, including flow separation, vortex shedding and unsteady force production. Furthermore, low-aspect-ratio wings exhibit a highly impactful tip vortex, which introduces strong spanwise gradients into an already complex flow. In this work, we explore the interaction between leading-edge flow separation and a strong, persistent tip vortex over a Reynolds number range of $600 \leq Re \leq 10{\,}000$ . In performing this study, we aim to bridge the insight gained from existing low-Reynolds-number studies of separated flow on finite wings ( $Re \approx 10^2$ ) and turbulent flows at higher Reynolds numbers ( $Re \approx 10^4$ ). Our study suggests two primary effects of the Reynolds number. First, we observe a break from periodicity, along with a dramatic increase in the intensity and concentration of small-scale eddies, as we shift from $Re = 600$ to $Re = 2500$ . Second, we observe that many of our flow diagnostics, including the time-averaged aerodynamic force, exhibit reduced sensitivity to Reynolds number beyond $Re = 2500$ , an observation attributed to the stabilising impact of the wing tip vortex. This latter point illustrates the manner by which the tip vortex drives flow over low-aspect-ratio wings, and provides insight into how our existing understanding of this flow field may be adjusted for higher-Reynolds-number applications.

Evaluating impedance boundary conditions to model interfacial dynamics in acoustofluidics

We present a numerical study to investigate the efficacy of impedance boundary conditions in capturing the interfacial dynamics of a particle subjected to an acoustic field and study the concomitant time-averaged acoustic streaming and radiation force fields. While impedance boundary conditions have been utilized to represent fluid–solid interface in acoustofluidics, such models assume the solid material to be locally reactive to the acoustic waves. However, there is a limited understanding of when this assumption holds true, raising concerns about the suitability of impedance boundary conditions. Here, we systematically investigate the applicability of impedance boundary conditions by comparing the predictions of an impedance boundary approach against a fully coupled fluid–solid model. We contrast the oscillation profiles of the fluid–solid interface predicted by the two models. We consider different scatterer materials to identify the extent to which the differences in interfacial dynamics impact the time-averaged fields and highlight the divergence within the predictions of the two models. Our findings indicate that, although impedance boundary conditions can yield qualitatively similar results to the full model in certain situations, the predictions from the two models generally differ both qualitatively and quantitatively. These results underscore the importance of exercising caution when applying these boundary conditions to model general acoustofluidic systems.

A time-averaged method to analyze slender rods moving in tubes

The motion of slender rods in narrow tubes involves the coupling of axial movements and transverse vibrations, in which the transverse vibration exhibits high frequency and chaotic characteristics. These characteristics render the computational cost of the time integration to the motions of rods and tubes expensive. To solve this problem, this paper proposes a novel time-averaged method, which contains two innovations: 1) By time averaging the traditional dynamic equations, the time-averaged dynamic equations describing the rod-tube system are established based on the time-averaged concept used in turbulence problems; 2) Two time-averaged contact models are developed, by fitting the time-averaged sample set using the polishing function and the machine learning, respectively, to construct the relationship between the time-averaged contact gaps and the time-averaged contact forces. The time-averaged method can use a large time step, which is far larger than that required for the time integration of traditional methods, thus reducing the computational cost. Inclined and rolling experiments of the rod-tube system, as well as numerical experiments of the control rod drop in a nuclear reactor, are conducted to validate the accuracy of the time-averaged method. Comparative analyses are carried out between the time-averaged method and six continuous contact models, demonstrating the high computational efficiency of the proposed method. Moreover, the experimental results can provide benchmark data for algorithm verification for scholars studying dynamic contact algorithms.

Dynamical force measurements for contacting soft surfaces upon steady sliding: Fixed-depth tribology.

The tribology between surfaces can have a profound impact on the response of a mechanical system, such as how granular particles are driven to flow. In this work, we perform experiments that time resolve the tangential and normal components of the force between two semicylindrical polydimethylsiloxane samples immersed in fluid, as they slide against each other in a range of controlled speeds. The time-averaged friction force shows a nonmonotonic dependence on the sliding speed over four decades, which is consistent with the paradigmatic Stribeck diagram and three dynamical regimes associated with it. Our specially designed fixed-depth setup allows us to study the fluctuation of force that exhibits strong stick-slip patterns in one of the regimes. Data from repetitive experiments reveal that both the "onset speed" for the stick-slip patterns and its spatial location along the sample change gradually during the course of our experiments, indicating changes on the sample surfaces. In addition, we conduct counterpart experiments by using spherical samples rubbing against each other, to make a direct connection of the interparticle tribology to the granular flow reported in our previous work [J.-C. Tsai et al., Phys. Rev. Lett. 126, 128001 (2021)10.1103/PhysRevLett.126.128001].

Connections between propulsive efficiency and wake structure via modal decomposition

We present experiments on oscillating hydrofoils undergoing combined heaving and pitching motions, paying particular attention to connections between propulsive efficiency and coherent wake features extracted using modal analysis. Time-averaged forces and particle image velocimetry measurements of the flow field downstream of the foil are presented for a Reynolds number of $Re=11\times 10^3$ and Strouhal numbers in the range $St=0.16\unicode{x2013}0.35$ . These conditions produce 2S and 2P wake patterns, as well as a near-momentumless wake structure. A triple decomposition using the optimized dynamic mode decomposition method is employed to identify dominant modal components (or coherent structures) in the wake. These structures can be connected to wake instabilities predicted using spatial stability analyses. Examining the modal components of the wake provides insightful explanations into the transition from drag to thrust production, and conditions that lead to peak propulsive efficiency. In particular, we find modes that correspond to the primary vortex development in the wakes. Other modal components capture elements of bluff body shedding at Strouhal numbers below the optimum for peak propulsive efficiency and characteristics of separation for Strouhal numbers higher than the optimum.

Continuous motion of particles attached to cavitation bubbles

Microbubble-mediated therapeutic gene or drug delivery is a promising strategy for various cardiovascular diseases (CVDs), but the efficiency and precision need to be improved. Here, we propose a cavitation bubble-driven drug delivery strategy that can be applied to CVDs. A bubble-pulse-driving theory was proposed, and the formula of time-averaged thrust driven by bubble pulses was derived. The continuous motion of particles propelled by cavitation bubbles in the ultrasonic field is investigated experimentally by high-speed photography. The cavitation bubbles grow and collapse continuously, and generate periodic pulse thrust to drive the particles to move in the liquid. Particles attached to bubbles will move in various ways, such as ejection, collision, translation, rotation, attitude variation, and circular motion. The cavity attached to the particle is a relatively large cavitation bubble, which does not collapse to the particle surface, but to the axis of the bubble perpendicular to the particle surface. The cavitation bubble expands spherically and collapses asymmetrically, which makes the push on the particle generated by the bubble expansion greater than the pull on the particle generated by the bubble collapse. The time-averaged force of the cavitation bubble during its growth and collapse is the cavitation-bubble-driven force that propels the particle. Both the cavitation-bubble-driven force and the primary Bjerknes force act in the same position on the particle surface, but in different directions. In addition to the above two forces, particles are also affected by the mass force acting on the center of mass and the motion resistance acting on the surface, so the complex motion of particles can be explained.

Spectral holographic trapping: Creating dynamic force landscapes with polyphonic waves.

Acoustic trapping uses forces exerted by sound waves to transport small objects along specified trajectories in three dimensions. The structure of the time-averaged acoustic force landscape acting on an object is determined by the amplitude and phase profiles of the sound's pressure wave. These profiles typically are sculpted by deliberately selecting the amplitude and relative phase of the sound projected by each transducer in large arrays of transducers, all operating at the same carrier frequency. This approach leverages a powerful analogy with holographic optical trapping at the cost of considerable technical complexity. Acoustic force fields also can be shaped by the spectral content of the component sound waves in a manner that is not feasible with light. The same theoretical framework that predicts the time-averaged structure of monotone acoustic force landscapes can be applied to spectrally rich sound fields in the quasistatic approximation, creating opportunities for dexterous control using comparatively simple hardware. We demonstrate this approach to spectral holographic acoustic trapping by projecting acoustic conveyor beams that move millimeter-scale objects along prescribed paths. Spectral control of reflections provides yet another opportunity for controlling the structure and dynamics of an acoustic force landscape. We use this approach to realize two variations on the theme of a wave-driven oscillator, a deceptively simple dynamical system with surprisingly complex phenomenology.

Swing and reverse swing of a cricket ball: laminar separation bubble, secondary vortex and wing-tip-like vortices

Large eddy simulation of flow past a cricket ball with its seam at $30^\circ$ to the free stream is carried out for $5 \times 10^4 \le Re \le 4.5 \times 10^5$ . Three regimes of flow are identified on the basis of the time-averaged swing force coefficient ( $\bar {C}_Z$ ) – no swing (NS), conventional swing (CS, $\bar {C}_Z>0$ ) and reverse swing (RS, $\bar {C}_Z<0$ ). The effect of seam on the boundary layer is investigated. Contrary to the popular belief, the boundary layer does not transition to a turbulent state in the initial stages of CS. The seam energizes the laminar boundary layer and delays its separation. The delay is significantly larger in a region near the poles, whose extent increases with an increase in $Re$ causing $\bar {C}_Z$ to increase. Here $\bar {C}_Z$ assumes a near constant value in the later stage of CS. The boundary layer transitions to a turbulent state via formation of a laminar separation bubble (LSB) in the equatorial region and directly, without a LSB, in the polar region. The extent of the LSB shrinks while the region of direct transition near the poles increases with an increase in $Re$ . A LSB forms on the non-seam side of the ball in the RS regime. A secondary vortex is observed in the wake bubble. While it exists on the non-seam side for the entire range of $Re$ considered, the mixing in the flow introduced by the seam causes it to disappear beyond a certain $Re$ on the seam side. The pressure difference between the seam and non-seam sides sets up wing-tip-like vortices. Their polarity reverses with the switch from the CS to RS regime.

A numerical study on the aerodynamic effects of dynamic twisting on forward flight flapping wings

To better understand the secret of natural flying vertebrates such as how humming-birds twist their wings to achieve superb flight ability, we presented a numerical investigation of dynamic twisting based on a hummingbird-like flapping wing model. Computational fluid dynamic simulations were performed to examine the effects of dynamic twisting on the unsteady flow field, the generation of instantaneous aerodynamic forces, and the time-averaged aerodynamic performance. This research reveals the details of leading-edge vortices (LEVs) and the underlying mechanisms behind the positive effects of wing torsion. The results demonstrated that wing torsion can effectively maintain the favorable distribution of effective angle of attack along the wing spanwise, resulting in a higher time-averaged thrust and vertical force. Further, the proper parameters of dynamic twisting can also improve the propulsive efficiency in forward flight. Dynamic twisting also showed a superior ability in controlling the airflow separation over the wing surface and maintaining the stability of the LEV. The amplitudes of effective angle of attack associated with the highest peak thrust and the maximum thrust-to-power at different advanced ratios were also explored, and it was found that the amplitudes decrease with increasing advanced ratio. To improve the efficiency during larger advanced ratio, specific modifications to the pitching of the wing were proposed in this work. The research in this paper has promising implications for the bio-inspired flapping wing.

Resonance mechanism of flapping wing based on fluid structure interaction simulation

Certain insect species have been observed to exploit the resonance mechanism of their wings. In order to achieve resonance and optimize aerodynamic performance, the conventional approach is to set the flapping frequency of flexible wings based on the Traditional Structural Modal (TSM) analysis. However, there exists controversy among researchers regarding the relationship between frequency and aerodynamic performance. Recognizing that the structural response of wings can be influenced by the surrounding air vibrations, an analysis known as Acoustic Structure Interaction Modal (ASIM) is introduced to calculate the resonant frequency. In this study, Fluid Structure Interaction (FSI) simulations are employed to investigate the aerodynamic performance of flapping wings at modal frequencies derived from both TSM and ASIM analyses. The performance is evaluated for various mass ratios and frequency ratios, and the findings indicate that the deformation and changes in vortex structure exhibit similarities at mass ratios that yield the highest aerodynamic performance. Notably, the flapping frequency associated with the maximum time-averaged vertical force coefficient at each mass ratio closely aligns with the ASIM frequency, as does the frequency corresponding to maximum efficiency. Thus, the ASIM analysis can provide an effective means for predicting the optimal flapping frequency for flexible wings. Furthermore, it enables the prediction that flexible wings with varying mass ratios will exhibit similar deformation and vortex structure changes. This paper offers a fresh perspective on the ongoing debate concerning the resonance mechanism of Flexible Flapping Wings (FFWs) and proposes an effective methodology for predicting their aerodynamic performance.

Role of the deviation motion on the aerodynamic performance of a mosquito wing in hover

The deviation angles during wing flapping motions are generally small for many insects and are known to have insignificant or even detrimental effects on the flight performance. Unlike many other insects, a mosquito hovers with a small sweeping amplitude but relatively large deviation angles. Thus, we investigate the effect of a deviation motion (‘figure-eight’ motion measured by Bomphrey et al. (2017)) on the aerodynamic performance of a mosquito wing in hover by comparing it with that without deviation motion (called in-plane motion). During the stroke, the deviation motion greatly changes the angle of attack and wing moving direction with respect to the stroke plane, but little changes the wing speed. These changes by the deviation motion significantly influence the instantaneous force and power generation. In particular, with the deviation motion, a large drag force is generated at early phase of the downstroke due to the increased angle of attack, and contributes to the vertical force generation. As results, the time-averaged vertical force is increased by the deviation motion, while the time-averaged power required is only slightly increased. Therefore, the mosquito wing generates a larger vertical force more efficiently with the deviation motion, suggesting that a deviation motion may be beneficial for an insect flight with a small sweeping amplitude.

Wall-bounded periodic snap-through and contact of a buckled sheet

Fluid flow passing a post-buckled sheet placed between two close confining walls induces periodic snap-through oscillations and contacts that can be employed for triboelectric energy harvesting. The responses of a two-dimensional sheet to a uniform flow and wall confinement in both equilibrium and post-equilibrium states are numerically investigated by varying the distance between the two ends of the sheet, gap distance between the confining walls and flow velocity. Cases with strong interactions between the sheet and walls are of most interest for examining how contact with the walls affects the dynamics of the sheet and flow structure. At equilibrium, contact with the wall displaces the sheet to form a nadir on its front part, yielding a lower critical flow velocity for the transition to snap-through oscillations. However, reducing the gap distance between the walls below a certain threshold distinctly shifts the shape of the sheet, alters the pressure distribution and eventually leads to a notable delay in the instability. The contact between the oscillating sheet and the walls at post-equilibrium is divided into several distinct modes, changing from sliding/rolling contact to bouncing contact with increasing flow velocity. During this transition, the time-averaged contact force exerted on the sheet decreases with the flow velocity. The vortices generated at the extrema of the oscillating sheet are annihilated by direct contact with the walls and merging with the shear layers formed by the walls, resulting in a wake structure dominated by the unstable shear layers.

Effect of wing planform on plunging wing aerodynamics

The unsteady aerodynamics of three plunging wings with different planform shapes was investigated in water tunnel experiments by means of force/moment and three-dimensional volumetric velocity measurements at a post-stall angle of attack. In contrast with a rectangular wing, a tapered wing does not exhibit a local minimum of the time-averaged lift coefficient at a reduced frequency of k≈ 1.7. This is due to the oblique shedding of trailing-edge vortices, which prevents the coupling with leading-edge vortices. The wing with a round-tip exhibits the same double maxima that correspond to the subharmonic and the fundamental frequency of vortex shedding in steady freestream. However, combined leading-edge and tip vortex produce larger time-averaged lift force with increasing reduced frequency. The variation of the bending moment is similar to that of the lift force in general. However, the mean bending moment arm shifts outboard with increasing reduced frequency due to the increased three-dimensionality of the vortex filaments and the tip-vortex effect. Maximum lift and bending moment become more similar for all three wings at higher reduced frequencies as the added mass effect becomes more significant.

Numerical investigation of non-planarity and relative motion for bionic slotted wings

Bird wings have split primary feathers that extend out from the wing surface. This structure is called the wingtip slot, which is recognized as a product of bird evolution to improve flight performance. In this paper, numerical simulations based on RANS (Reynolds-averaged Navier–Stokes) equations are conducted to examine and understand the influence of wingtip slots on six wings at Re = 100 000. The overlapping grid method, driven by an in-house UDF (User Defined Function), is used to model the motion of the bionic slotted wings. The motion law of the winglets is improved based on the law extracted from a level-flying bald eagle. Then the aerodynamic force, pressure distribution, vorticity contours, wake stream, and other flow structures of the slotted wings with different layouts were compared and analyzed. The results show a significant increase in aerodynamic force when the slotted wingtips are employed. The maximum lift-to-drag ratio is also improved in our designed wing model with a non-planar wingtip by a maximum of 34% from the base wing. Each winglet works as a single wing due to the existence of slots, with a chordwise pressure distribution similar to that of the main wing. The vortex structures of slotted wings show expressive changes in the tip vortex as compared with the base wing. Additionally, an innovative bionic slotted wing is proposed with a dynamic wingtip that forms varying gaps between winglets. Due to the collective mechanism of aerodynamic interaction among multiple winglets for the innovative wing, it acquires the optimal time-averaged force during a flapping period. As expected, the slotted wingtip reduces the main wingtip vortex intensity and creates weaker vortices. The non-planarity and relative motion of the wingtip strengthen its weakening effect on the wingtip vortex and wake.

Comparison of reduced order models based on dynamic mode decomposition and deep learning for predicting chaotic flow in a random arrangement of cylinders

Three reduced order models are evaluated in their capacity to predict the future state of an unsteady chaotic flow field. A spatially fully developed flow generated in a random packing of cylinders at a solid fraction of 0.1 and a nominal Reynolds number of 50 is investigated. For deep learning (DL), convolutional autoencoders are used to reduce the high-dimensional data to lower dimensional latent space representations of size 16, which were then used for training the temporal architectures. To predict the future states, two DL based methods, long short-term memory and temporal convolutional neural networks, are used and compared to the linear dynamic mode decomposition (DMD). The predictions are tested in their capability to predict the spatiotemporal variations of velocity and pressure, flow statistics such as root mean squared values, and the capability to predict fluid forces on the cylinders. Relative errors between 15% and 20% are evident in predicting instantaneous velocities, chiefly resulting from phase differences between predictions and ground truth. The spatial distribution of statistical second moments is predicted to be within a maximum of 5%–10% of the ground truth with mean error in the range of 1%–2%. Using the predicted fields, instantaneous fluid drag force predictions on individual particles exhibit a mean relative error within 20%, time-averaged drag force predictions to within 5%, and total drag force over all particles to within 1% of the ground truth values. It is found that overall, the non-linear DL models are more accurate than the linear DMD algorithm for the prediction of future states.

Hydrodynamic Diversity of Jets Mediated by Giant and Non-Giant Axon Systems in Brief Squid.

Neural input is critical for establishing behavioral output, but understanding how neuromuscular signals give rise to behaviors remains a challenge. In squid, locomotion through jet propulsion underlies many key behaviors, and the jet is mediated by two parallel neural pathways, the giant and non-giant axon systems. Much work has been done on the impact of these two systems on jet kinematics, such as mantle muscle contraction and pressure-derived jet speed at the funnel aperture. However, little is known about any influence these neural pathways may have on the hydrodynamics of the jet after it leaves the squid and transfers momentum to the surrounding fluid for the animal to swim. To gain a more comprehensive view of squid jet propulsion, we made simultaneous measurements of neural activity, pressure inside the mantle cavity, and wake structure. By computing impulse and time-averaged forces from the wake structures of jets associated with giant or non-giant axon activity, we show that the influence of neural pathways on jet kinematics could extend to hydrodynamic impulse and force production. Specifically, the giant axon system produced jets with, on average, greater impulse magnitude than those of the non-giant system. However, non-giant impulse could exceed that of the giant system, evident by the graded range of its output in contrast to the stereotyped nature of the giant system. Our results suggest that the non-giant system offers flexibility in hydrodynamic output, while recruitment of giant axon activity can provide a reliable boost when necessary.

Acoustic radiation force on a spherical thermoviscous particle in a thermoviscous fluid including scattering and microstreaming.

We derive general analytical expressions for the time-averaged acoustic radiation force on a small spherical particle suspended in a fluid and located in an axisymmetric incident acoustic wave. We treat the cases of the particle being either an elastic solid or a fluid particle. The effects of particle vibrations, acoustic scattering, acoustic microstreaming, heat conduction, and temperature-dependent fluid viscosity are all included in the theory. Acoustic streaming inside the particle is also taken into account for the case of a fluid particle. No restrictions are placed on the widths of the viscous and thermal boundary layers relative to the particle radius. We compare the resulting acoustic radiation force with that obtained from previous theories in the literature, and we identify limits, where the theories agree, and specific cases of particle and fluid materials, where qualitative or significant quantitative deviations between the theories arise.

Effect of Wide-Spectrum Pulsed Magnetic Field on Solidification Structure of Pure Al at Constant Flow Velocity

The electromagnetic force generated by a pulsed magnetic field within a metal melt leads to changes in the internal temperature and flow fields of the molten metal, thus improving the solidification of the metal structure. Using the combination of a solidification test, experimental simulation and theoretical analysis, this study simulated the distribution of both electromagnetic force and the flow field in a metal melt under wide-spectrum pulse conditions, and studied the influence of a wide-spectrum pulsed magnetic field on the solidification structure of pure aluminium with a constant flow velocity. The results of this study show that the structural refinement of the solidification of pure aluminium can be different, in spite of equal flow velocity. Furthermore, this study shows that an applied time-averaged electromagnetic force causes crystal nuclei to pass through the solid–liquid interface boundary layer and promotes the growth of crystal grains. These grains flowed with the melt flow field to achieve both refinement and homogenization of the solidified structure.

Drag force on an oscillatory spherical bubble in shear‐thinning fluid

Power-law shear-thinning fluid motions induced by a translating spherical bubble with sinusoidal oscillation at a high frequency are numerically studied. We focus on reducing the time-averaged drag force $D$ on the bubble owing to the oscillation-enhanced shear-thinning effect. Under the assumption of negligible convection, the unsteady Stokes equation is directly solved in a finite-difference manner over a wide parameter space of the dimensionless oscillation amplitude $A$ (corresponding to the oscillatory-to-translational velocity ratio) and the power-law index $n$ of the viscosity. The results show that, for small amplitude ( $A\ll 1$ ), the drag reduction ratio $1-D/D_0$ (here, $D_0$ is $D$ with no oscillation) is proportional to $A^2$ . In contrast, for large amplitude ( $A \gg 1$ ), the drag ratio $D/D_0$ is proportional to $A^{n-1}$ , revealing a power-law behaviour. In the case of $A \gg 1$ for a strong shear-thinning fluid with small $n$ , the square of the vorticity over the entire domain is much smaller than that of the shear rate, and thus the bulk flow may be regarded as irrotational. To provide a fundamental perspective on the drag reduction mechanism, a theoretical model is proposed based on potential theory, and demonstrated to well capture the power-law relation between $D/D_0$ and $A$ .