Gradient Force Research Articles

Dynamics of currents in the Qiongzhou strait during spring and summer based on a numerical simulation

A high-precision three-dimensional hydrodynamic model based on the Semi-implicit Cross-scale Hydroscience Integrated System Model (SCHISM) was constructed to investigate the characteristics of residual currents in the Qiongzhou Strait during spring and summer. Results reveal an overall westward transport in the strait during spring and summer. However, the results also support the potential occurrence of transient eastward Eulerian residual currents, which should be evident predominantly at the surface in the central and eastern sections of the strait and mainly characterised by subtidal oscillations. The direction and intensity of these residual currents indicate a pronounced influence of pressure gradient forces on both sides of the strait. The eastward flow observed in spring may represent a compensatory mechanism resulting from the relaxation period of long-term easterly winds, whereas the generation of eastward residual currents in summer is influenced by wind stress curl differences in both the western and eastern areas of the strait. The direct impact of the summer monsoon on the flow field in the strait is relatively weak. Additionally, the direction and intensity of the residual currents in the strait exhibit potential interannual variations that are directly associated with interannual wind variations. This study provides valuable insights in relation to the ongoing discourse on the dynamics of residual currents in the Qiongzhou Strait.

Theoretical and simulation study of dispersive electron cooling

In electron cooling, the transverse cooling rate is usually smaller than the longitudinal rate, especially at high energies. By introducing dispersive cooling, it is possible to redistribute the cooling rate between longitudinal and transverse planes. Theoretically, achieving dispersive electron cooling requires an ion dispersion and a transverse gradient of longitudinal friction force. The latter depends on many factors such as the relative momentum offset, transverse displacement, e-beam density distribution, and space charge effect. Therefore, several methods can be employed to achieve dispersive electron cooling based on these factors. Based on the dc electron beam, these factors and their respective impacts on the cooling rate are discussed and analyzed. For the first time, we propose a new mechanism to achieve dispersive cooling for a uniform electron beam by placing part of the ion beam outside of the electron beam. Based on a linear friction force model, we propose a simple formula to numerically estimate the cooling rate redistribution effect of these methods. The analytical results are in good agreement with Monte Carlo calculation and numerical simulation. Published by the American Physical Society 2024

Comparison of focusing property and radiation force between autofocusing Bessel beams and focused Gaussian beams.

As abruptly autofocusing beams, autofocusing Bessel beams (ABBs) have been proven to be a class solution for the Helmholtz equation [Opt. Express31, 33228 (2023)10.1364/OE.500383]. In this paper, we use the Fresnel number as the basic parameter and accurately compare the focusing property and radiation force of ABBs versus focused Gaussian beams (FGBs) under the same Fresnel number. Unlike FGBs, ABBs can achieve autofocusing without the need for an initial focusing phase. Our analysis of the beam width defined by power in the bucket, revealed that FGBs exhibit uniform focusing along the straight line, whereas ABBs demonstrate accelerated focusing along the elliptic curve. At the same Fresnel number, FGBs exhibit a higher peak intensity in the focal plane, yet ABBs excel in gradient force on particles. In comparison to FGBs, ABBs exhibit smaller potential well widths, allowing for stable and precise trapping of high refractive index particles at the focal point. While FGBs are considered suitable for laser processing and ablation due to their high peak power density, ABBs possess significant advantages in optical manipulation due to their great gradient force. Furthermore, we conduct a comparative analysis between ABBs and circular Airy beams (CABs). The peak intensity and gradient force exhibited by CABs are slightly lesser than those of ABBs. CABs are appropriate for multi-point trapping along the axis, whereas ABBs are more suited for precise single-point trapping.

Corner states and particle trapping in waterborne acoustic crystals

Higher-order topological insulators exhibit intriguing capacity to confine energy on lower-dimensional boundaries owing to the unique bulk-boundary correspondence. Many potential practical applications of the higher-order states in classical wave systems have been proposed and achieved. In this work, we implement a second-order topological insulator in a waterborne acoustic crystal by drilling and grooving a copper plate based on a two-dimensional Su–Schrieffer–Heeger lattice with different intracell and intercell couplings. The far-field transmission spectrum and near-field pressure field distributions verify the existence of the one-dimensional edge states and zero-dimensional corner states in the bulk bandgap. Due to the highly localized edge and corner states, the polystyrene particles are trapped at the edges and corners by the acoustic gradient force. Our findings provide a good platform to manipulate underwater acoustic wave and may inspire topological acoustic applications.

Spontaneous motion of solid object on open channel

Spontaneous motion of a solid object floating on an unsymmetrical geometric open channel is investigated. The open channel was created by selectively fabricating hydrophilic patterns on a superhydrophobic substrate, confining water within the hydrophilic region, thus forming a distinct open channel. As a rectangular foam block was placed on a triangular open channel, the variation of the spanwise width of the wetted area leads to a gradient of surface tension force along the centerline that is able to drive the foam block directionally. What is more, two open channels could be parallelly arranged together to drive even larger object. A theoretical model was developed to explain the mechanism, which agrees well with the experimental results. The findings of this work extend the application of capillary force, which could be used in areas such as self-driven microfluidics, surface lubrication, and open microchannels.

Targeted observation for the climatology temperature in the Kuroshio region based on the CNOP approach

To reduce the cost of ocean observations and improve prediction accuracy of the Kuroshio region temperature, this study investigates the related targeted observation by using the conditional nonlinear optimal perturbation (CNOP) approach. Results show that the scheme of vertical-integrated energy is more suitable for the identification of sensitive area in the related targeted observation. By conducting a set of observation system simulation experiments (OSSEs), we discovered that the sensitive areas identified by the CNOP exert substantial influence on temperature predictions within the target area. The dynamic diagnosis further indicated that the pressure gradient and Coriolis force in the momentum equations greatly contribute the development of the prediction biases. These findings implied that the implement of CNOP-based targeted observation represents a cost-effective strategy for enhancing temperature predictions in the Kuroshio region.

Reducing a Tropical Cyclone Weak-Intensity Bias in a Global Numerical Weather Prediction System

Abstract The operational Canadian Global Deterministic Prediction System suffers from a weak-intensity bias for simulated tropical cyclones. The presence of this bias is confirmed in progressively simplified experiments using a hierarchical system development technique. Within a semi-idealized, simplified-physics framework, an unexpected insensitivity to the representation of relevant physical processes leads to investigation of the model’s semi-Lagrangian dynamical core. The root cause of the weak-intensity bias is identified as excessive numerical dissipation caused by substantial off-centering in the two time-level time integration scheme used to solve the governing equations. Any (semi)implicit semi-Lagrangian model that employs such off-centering to enhance numerical stability will be afflicted by a misalignment of the pressure gradient force in strong vortices. Although the associated drag is maximized in the tropical cyclone eyewall, the impact on storm intensity can be mitigated through an intercomparison-constrained adjustment of the model’s temporal discretization. The revised configuration is more sensitive to changes in physical parameterizations and simulated tropical cyclone intensities are improved at each step of increasing experimental complexity. Although some rebalancing of the operational system may be required to adapt to the increased effective resolution, significant reduction of the weak-intensity bias will improve the quality of Canadian guidance for global tropical cyclone forecasting. Significance Statement Global numerical weather prediction systems provide important guidance to forecasters about tropical cyclone development, motion, and intensity. Despite recent improvements in the Canadian operational model’s ability to predict tropical cyclone formation, the system systematically underpredicts the intensity of these storms. In this study, we use a set of increasingly simplified experiments to identify the source of this error, which lies in the numerical time-stepping scheme used to solve the model equations. By decreasing numerical drag on the tropical cyclone circulation, intensity predictions that resemble those of other global modeling systems are achieved. This will improve the quality of Canadian tropical cyclone guidance for forecasters around the world.

Accelerated particle beams in a 3D simulation of the quiet Sun

Context. Charged particles are constantly accelerated to non-thermal energies by the reconnecting magnetic field in the solar atmosphere. Our understanding of the interactions between the accelerated particles and their environment can benefit considerably from three-dimensional atmospheric simulations that account for non-thermal particle beam generation and propagation. In a previous publication, we presented the first results from such a simulation, which considers quiet Sun conditions. However, the original treatment of beam propagation ignores potentially important phenomena such as the magnetic gradient forces associated with a converging or diverging magnetic field. Aims. Here we present a more general beam propagation model incorporating magnetic gradient forces, the return current, acceleration by the ambient electric field, corrected collision rates due to the ambient temperature, and collisions with heavier elements than hydrogen and the free electrons they contribute. Neglecting collisional velocity randomisation makes the model sufficiently lightweight to simulate millions of beams. We investigate how each new physical effect in the model changes the non-thermal energy transport in a realistic three-dimensional atmosphere. Methods. We applied the method of characteristics to the steady-state continuity equation for electron flux to derive ordinary differential equations for the mean evolution of energy, pitch angle, and flux with distance. For each beam, we solved these numerically for a range of initial energies to obtain the evolving flux spectrum, from which we computed the energy deposited into the ambient plasma. Results. Magnetic gradient forces significantly influence the spatial distribution of deposited beam energy. The magnetic field converges strongly with depth in the corona above loop footpoints. This convergence leads to a small coronal peak in deposited energy followed by a heavy dip caused by the onset of magnetic mirroring. Magnetically reflected electrons carry away 5 to 10% of the injected beam energy on average. The remaining electrons are relatively energetic and produce a peak in deposited energy below the transition region a few hundred kilometres deeper than they would in a uniform magnetic field. A diverging magnetic field at the beginning of the trajectory, which is common in the simulation, enhances the subsequent impact of magnetic mirroring. The other new physical effects do not qualitatively alter the picture of non-thermal energy transport for the atmospheric conditions under consideration.

Magnetically driven capsules with multimodal response and multifunctionality for biomedical applications

Untethered capsules hold clinical potential for the diagnosis and treatment of gastrointestinal diseases. Although considerable progress has been achieved recently in this field, the constraints imposed by the narrow spatial structure of the capsule and complex gastrointestinal tract environment cause many open-ended problems, such as poor active motion and limited medical functions. In this work, we describe the development of small-scale magnetically driven capsules with a distinct magnetic soft valve made of dual-layer ferromagnetic soft composite films. A core technological advancement achieved is the flexible opening and closing of the magnetic soft valve by using the competitive interactions between magnetic gradient force and magnetic torque, laying the foundation for the functional integration of both drug release and sampling. Meanwhile, we propose a magnetic actuation strategy based on multi-frequency response control and demonstrate that it can achieve effective decoupled regulation of the capsule’s global motion and local responses. Finally, through a comprehensive approach encompassing ideal models, animal ex vivo models, and in vivo assessment, we demonstrate the versatility of the developed magnetic capsules and their multiple potential applications in the biomedical field, such as targeted drug delivery and sampling, selective dual-drug release, and light/thermal-assisted therapy.

Drifts effect on the divertor W leakage mechanisms under different dissipative divertor conditions of EAST

A new drift module has been developed in the DIVIMP code, enabling two-dimensional simulations of tungsten (W) transport in the edge plasma with full drifts included. By using the SOLPS-DIVIMP code package, the impact of drifts on W transport and screening has been investigated for various levels of dissipative divertor conditions and different divertor geometric configurations in EAST. Simulation results reveal that E⃗×B⃗ drifts can enhance the W leakage by more than one order of magnitude. Under the favorable Bt direction, W erosion mainly occurs on the outer divertor target, making the W leakage from the outer divertor region the dominator. A leakage path from the near-scrape-off layer (SOL) region is revealed by the modeling results. In the leakage path, both the ion temperature gradient force and the reversed poloidal E⃗×B⃗ drift are pointing upstream. With the radial E⃗×B⃗ drift pushing W ions from the well-screened far-SOL region to the near-SOL region, the leakage from the near-SOL region becomes significant. As the divertor condition varies from the low-recycling regime to the deep detachment regime, the decrease of the ion temperature gradient velocity and poloidal E⃗×B⃗ drift velocity narrows the width of the near-SOL leakage tunnel and thus enhances W screening. While under the unfavorable Bt , W erosion and leakage from the inner divertor target matters, the leakage mechanism especially the leakage path from the near-SOL region is similar as the favorable Bt cases. Furthermore, the effect of different divertor geometries on the W screening has been investigated. The configuration with the outer strike point (OSP) on the horizontal divertor plate is proved to narrow the near-SOL leakage tunnel, and thus the unreversed poloidal E⃗×B⃗ drift pointing to the divertor target dominates and helps to enhance the divertor W screening. For the same D 2 puffing rate, the W leakage ability of cases with the OSP on the horizontal target can be more than 10 times weaker than the cases with the OSP on the vertical target, especially when the divertor is detached.

Mechanisms Accounting for the Formation of the Strong Winds that Caused the Tripping Incident of Transmission Line in Eastern Inner Mongolia

Meteorological disasters pose a serious threat to the State Grid Corporation of China, which covers ~ 88% of Chinese national territory. Of these, strong winds deserve a special attention, as they often induce windage yaw discharge of transmission lines and even toppling of transmission towers, resulting in serious economic losses. On 28 June 2023, a severe tripping incident of transmission line appears in Eastern Inner Mongolia due to strong winds. In this study, we conduct comprehensive analyses to clarify the favorable background conditions and governing mechanisms for producing the strong winds. Main results are shown as follows. Synoptic analysis indicates that, the favorable background environments for the event are characterized by a strong upper-level jet associated upper tropospheric divergence; an intense middle-level warm advection ahead of a shortwave trough; and a long-lived lower-tropospheric mesoscale vortex. The strong winds that cause the tripping incident mainly occur in the southeastern quadrant of the vortex. Vorticity budget presents that the period from the mesoscale-vortex’s formation to 4 h before is crucial to the mesoscale vortex, as cyclonic vorticity increases rapidly mainly due to the lower-level convergence-related vertical stretching. In contrast, the horizontal transport mainly results in a net export of cyclonic vorticity, which is the most detrimental factor. Kinetic energy (KE) budget shows that, after the mesoscale vortex forms, the strong winds within its southeastern quadrant enhance rapidly. Overall, the positive work done by the pressure gradient force associated with the mesoscale vortex dominates the enhancement of strong winds; the horizontal transport of KE is the second dominant factor, and the vertical transport of KE (i.e., the downward momentum transportation) shows the least contribution.

Dynamical Importance of the Trade Wind Inversion in Suppressing the Southeast Pacific ITCZ

AbstractSea surface temperature (SST) gradients are a primary driver of low‐level wind convergence in the east Pacific Inter‐Tropical Convergence Zone (ITCZ) through their hydrostatic relationship to the surface pressure gradient force (PGF). However, the surface PGF may not always align with SST gradients due to variations in boundary layer temperature gradients with height, that is, the boundary layer contribution to the surface PGF. In this study, we investigate the observed northern hemisphere position of the east Pacific ITCZ using a slab boundary layer model (SBLM) driven by different approximations of the boundary layer virtual temperature field. SBLM simulations using the entire boundary layer virtual temperature profile produce a realistic northern hemisphere ITCZ. However, SST‐only simulations produce excessive equatorial divergence and southern hemisphere convergence, resulting in a latitudinally confined double ITCZ‐like structure. Observed virtual temperature gradients highlight the importance of northward temperature gradients strengthening with height from the equator to 15°S below the trade wind inversion (TWI). Our interpretation is that the equatorial cold tongue induces relatively weak high surface pressure and double ITCZ‐like convergence because the resulting layer of cold air is shallow. Concurrently, relatively strong high surface pressure spreads out in the southern hemisphere due to interactions between stratocumulus clouds and the ocean surface. Together, the equatorial cold tongue and the TWI/stratocumulus clouds enable a more northern hemisphere dominant ITCZ. Thus, we provide evidence of a dynamical link between the equatorial cold tongue, low clouds, and double ITCZs, which continue to be problematic in Earth system models.

Activation energy with binary chemical reaction of convective radiative magnetized nanofluid flow with viscous dissipation and pressure gradient

ABSTRACT The present investigation deals with the study of activation energy with the binary chemical reaction on magnetohydrodynamic convective radiative heat and mass transfer of nanofluid flow with internal heat source/sink, viscous dissipation, pressure gradient force, and suction effects. The basic equations of conservation of momentum, energy, and mass diffusion are brought to a set of ordinary differential equations (nonlinear) using suitable similarity transformations, and these are then solved using the Homotopy Perturbation Method (HPM). The validity of the present results is established by comparing them with those obtained by the Runge-Kutta-Fehlberg method (RKFM). The results reveal that the temperature distribution increases with raising the value of the thermal radiation parameter. It is found that the concentration distribution increases by increasing the values of the activation energy parameter. In contrast, the reverse effect is observed by enhancing the chemical reaction parameter, Schmidt number, and pressure gradient parameter. Further, the skin-friction coefficient and Nusselt number decrement by enhancing the value of the Hartmann number, whereas the reverse trend is found by enhancing the pressure gradient and suction parameter values. Also, the Sherwood number retards with an enhancement in the Hartmann number, whereas it rises by increasing the pressure gradient and suction parameters.

Water collection and transportation on superhydrophilic/superhydrophobic bioinspired heterogeneous wettability surface

In recent years, the ability of superhydrophilic/superhydrophobic bioinspired heterogeneous wettability (SSBHW) surfaces to collect freshwater has gained significant attention due to their potential to mitigate water scarcity. Inspired by nature’s creatures, this paper presents an SSBHW surface that can efficiently collect freshwater resources from the atmosphere, drawing inspiration from cactus needles and spider silk spindles. The superhydrophobic (SHO) surface was prepared by wet etching and low surface energy modification on copper surface. Based on the SHO surface, the superhydrophilic (SHI) patterns consisting of spindle-shaped and wedge-shaped were constructed by laser ablation. The prepared SHI pattern was not only trapped airborne droplets but also enabled the directional transport of droplets. To enhance the efficiency of water collection, the smaller spindle-shaped pattern spacing in the SHO region was designed, which help to transport the condensed water quickly towards the SHI region because of the surface energy gradient force. The synergistic effect between the advantages offered by the SHO and SHI surfaces results in a significant improvement in water collection efficiency. The water collection efficiency of the SSBHW surface is 870 mg cm−2 h−1, which is 3.32 and 2.2 times higher than that of the SHI and SHO surfaces, respectively. In addition, the mechanism of enhancement of the moisture harvesting behavior of inhomogeneous wettable surfaces was analyzed. This work provides a new strategy for the study of catchment surfaces, making it an effective solution for obtaining fresh water resources in arid regions.

Особливості впливу тренувальної програми у підготовчому періоді річного макроциклу на морфо-функціональні, силові та швидкісно-силові показники армрестлерів 14-15 років

Purpose: to define features of the influence of training means in the preparatory period of an annual macrocycle on morpho-functional, power and speed-power indicators of armwrestlers of 14-15 years old. Material and methods. The following methods were used in the research: theoretical analysis and generalization of scientific and methodical information; pedagogical observation; pedagogical experiment; mathematical and statistical methods of the research with the use of the licensed program STATISTICA 10. The study was conducted on the territory of Poltava region in Kobeliaky on the basis of the State PTUZ «Professional Agrarian Lyceum» from January to June 2023, in which 10 armwrestlers aged 14–15 years took part who trained according to a speed- strength program Results: according to the conducted research the results of the sequential experiment are presented, which state the increase of functional indicators of cardiovascular and respiratory systems at the end of the research, also there was an increase of power indicators in all ЄДИНОБОРСТВА EDINOBORSTVA MARTIAL ARTS 2024 №1(31) 13 © Безкоровайний Д.О., Камаєв О.І., Орлов А.А., Глядя С.О. test exercises, and the increase of power was reliable (t-criterion ranged from 2,25 to 4,21 at p<0,05). Speed and power indicators (time of reaching the maximum force, speed and power index, force gradient, force for 500 ms and time of reaching the force of 1 kg) tended to improve, but reliable changes took place only in force for 500 ms and time of reaching the force of 1 kg (t- criterion ranged from 2,93 to 3,96 at p<0,05), and indicators of the left hand significantly increased. Conclusions. The block program of trainings allowed to improve considerably indicators of heart rate, blood pressure, vital index, coefficient of endurance that helped to achieve the reliable improvement of adaptive potential of young armwrestlers. During the experiment the indicators of strength of 14-15-year-old sportsmen in all test exercises increased significantly, on the basis of these changes the tendency to improvement of time of achievement of maximum force, gradient of force, speed-force index was determined, and results of the indicator of force for 500 ms and time of achievement of force in 1 kg significantly improved. Keywords: armwrestling, young athletes, explosive power, competitive exercises, speed and power indices.

Generation and propagation of circular Airy derivative beams carrying rotationally-symmetric power-exponent-phase vortices.

In this paper, the circular Airy derivative beams carrying rotationally-symmetric power-exponent-phase vortices are proposed for the first time, whose evolutionary properties are explored by theoretical analysis as well as experimental verification. The intensity and phase distributions of this kind of beam can be flexibly modulated by controlling its parameters such as derivative order, topological charge, and power order. Intriguingly, the evolution of such beams with different fractional topological charges is also investigated in detail by means of phase singularities distribution. In addition, from the perspective of transverse gradient force, the proposed beam is capable of efficiently protecting the trapped Rayleigh particles located at the beam center from the impact by surrounding particles. Furthermore, the beam width, orbital angular momentum (OAM) density and spiral spectrum are also compared and analyzed under different beam parameters. The proposed beams are expected to be useful for extending applications of optical vortices and autofocusing beam, especially for multi-regional particle gathering and central particle protecting.

Conservative and nonconservative forces for Mie particles in acoustic trapping

A general acoustic force field can be decomposed into a conservative gradient force (GF) and a non-conservative scattering force (SF), which have very different physical and mathematical properties. However, the profiles of such forces for Mie particles are unknown, let alone their underlying physics. Here, by using a fast Fourier transform approach, we calculated the GF and SF for spherical particle of various sizes and various incident waves. For the same focused incident waves, the normalized GF and SF are similar for different particle sizes, while the total force can be quite different owing to the varying relative strength between the GF and SF. GF and SF possess symmetries that are not found in the incident waves, indicating that these physically and mathematically distinct forces have symmetries that are hidden from the beam profile. For a vortex beam carrying a well-defined topological charge, acoustic forces alone cannot trap particles.

Solute band transport in electroosmotic-pressure-driven flow of Carreau–Yasuda fluid over micropillar arrays

In this study, we numerically investigate the transport and dispersion characteristics of a neutral solute band in a fluid following generalized Carreau–Yasuda constitutive equations. The combined action of electroosmotic forcing and pressure gradient drives the flow through a microchannel embedded with micropillars. We analyze the band transport process by altering the flow-controlling parameters and fluid rheology. We observe an enhancement in band dispersion with increasing solute diffusivity. An augmentation in band dispersion is achieved using bluff micropillar designs compared to the streamlined micropillars. We also observe that the dispersion of solute band is less in shear-thickening fluids than in shear-thinning fluids. In a mixed electroosmotic-pressure-driven flow, the favorable pressure gradient conditions enhance the band dispersion phenomenon. Our results may prove valuable in understanding solute transport dynamics in non-Newtonian fluids, which has implications for lab-on-a-chip diagnostics and precise drug delivery.

Probing the surface electrical properties of clay minerals with electrostatic force microscopy and kelvin probe force microscopy

Many surface processes of clay minerals require in-depth understanding of their surface electrical properties, such as surface charge density, surface potential distribution, etc In this paper, electrostatic force microscopy (EFM) and Kelvin probe force microscopy (KPFM) were used to study the surface charge densities, surface potentials, electric field intensities, and electric field force gradients of three common clay minerals: kaolinite, montmorillonite, and illite. The properties were directly imaged, and the average surface permanent charge densities of kaolinite, montmorillonite, and illite were obtained to be −0.0060, −2.136, and −5.456 μC m−2, respectively. In addition, a good linear relationship was found between the surface charge densities obtained by KPFM and the layer charges calculated from the mineral chemical formulas of three clay minerals.

Strouhal number impact on propulsion efficiency in fully-active oscillating foils: Unpacking frequency, amplitude, and velocity relationships

Oscillating foils play a crucial role in marine and aerial organisms. Propulsion efficiency is gauged by the Strouhal number, combining frequency, amplitude, and velocity. This creates three parameters: reduced frequency, dimensionless amplitude, and Reynolds number. The hydrodynamics between these parameters and their effect on propulsion efficiency is complex. Using RANS simulations, this study shows that a large amplitude significantly increases propulsion efficiency. Flow analysis shows that increasing dimensionless amplitude enhances heaving velocity without affecting pitching velocity. However, an increase in reduced frequency increases both velocities, so there is greater inflow and horizontal force. Polynomial regression shows the horizontal force coefficient correlates with the square of the Strouhal number, and the power coefficient has a cubic relationship. A rise in reduced frequency increases the gradient of the horizontal force and power coefficient curves more than an increase in amplitude, suggesting decreasing propulsion efficiency. This clarifies why larger amplitude values are more efficient and why the optimal Strouhal number is between 0.3 and 0.6. Vortex analysis shows that lower reduced frequencies produce an undulatory wake pattern, not ideal for thrust. A higher amplitude, or frequency creates a leading-edge vortex pattern, reducing propulsion efficiency. A reverse Kármán vortex pattern represents optimal propulsion efficiency.