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Advertisement | Early bird registration is officially open for March Meeting 2023–one of the largest and most exciting conferences in physics! Witness groundbreaking physics research, network with potential employers, and prepare for career success at March Meeting 2023. Register today » | | | | | | Not an APS member? Join today to start connecting with a community of more than 50,000 physicists. | | | | EDITORIALS AND ANNOUNCEMENTS | Gokul Pathikonda and Tie Wei Phys. Rev. Fluids 7, 110001 (2022) – Published 7 November 2022 | | | Editors' Suggestion Dylan Reynolds, Gustavo M. Monteiro, and Sriram Ganeshan Phys. Rev. Fluids 7, 114201 (2022) – Published 16 November 2022  | We derive the governing equation of flow for a three-dimensional fluid with a parity-broken viscosity tensor when confined to a Hele-Shaw cell. When such a fluid is pushed through a channel, a transverse force is exerted on the walls, and when a bubble of air expands into a region of such fluid, a circulation develops in the far field. The Saffman-Taylor stability condition is also modified, with these terms tending to stabilize the two fluid interface. Such experiments can in principle facilitate the measurement of parity odd coefficients in both synthetic and natural active matter systems. | | | | | | Editors' Suggestion Jinyul Hwang and Jae Hwa Lee Phys. Rev. Fluids 7, 114603 (2022) – Published 14 November 2022  | In wall turbulence, meandering behaviors of large-scale structures observed in the logarithmic layer is a crucial spatial feature for understanding the spatial organization of these structures and improving the structure-based turbulence model. These structures extend from the near-wall region to the edge of boundary layers. Their meandering motions leave an imprint on the two-point turbulence statistics across the flow, especially in the logarithmic region. Here, we demonstrate the influence of the meandering motions of wall-attached structures on the two-point correlation and premultiplied two-dimensional spectra by analyzing direct numerical simulation data of the turbulent boundary layer. | | | | | | Virgile Thiévenaz and Alban Sauret Phys. Rev. Fluids 7, 110501 (2022) – Published 7 November 2022 | | | Arpit Mishra, Claire Bourquard, Arnab Roy, Rajaram Lakkaraju, Parthasarathi Ghosh, and Outi Supponen Phys. Rev. Fluids 7, 110502 (2022) – Published 7 November 2022 | | | Yorgos Stergiou, Marcus J. B. Hauser, Anne De Wit, Gábor Schuszter, Dezső Horváth, Kerstin Eckert, and Karin Schwarzenberger Phys. Rev. Fluids 7, 110503 (2022) – Published 7 November 2022 | | | Samar Alqatari, Thomas E. Videbæk, Sidney R. Nagel, Anette Hosoi, and Irmgard Bischofberger Phys. Rev. Fluids 7, 110504 (2022) – Published 7 November 2022 | | | Paul Lilin and Irmgard Bischofberger Phys. Rev. Fluids 7, 110505 (2022) – Published 7 November 2022 | | | Palas Kumar Farsoiya, Stéphane Popinet, and Luc Deike Phys. Rev. Fluids 7, 110506 (2022) – Published 7 November 2022 | | | Georgios Matheou Phys. Rev. Fluids 7, 110507 (2022) – Published 7 November 2022 | | | Hanul Hwang, Dokyun Kim, and Parviz Moin Phys. Rev. Fluids 7, 110508 (2022) – Published 7 November 2022 | | | Carola Seyfert and Alvaro Marin Phys. Rev. Fluids 7, 110509 (2022) – Published 7 November 2022 | | | A. Ceci, S. Pirozzoli, J. Romero, M. Fatica, R. Verzicco, and P. Orlandi Phys. Rev. Fluids 7, 110510 (2022) – Published 7 November 2022 | | | Ranjiangshang Ran, Quentin Brosseau, Brendan C. Blackwell, Boyang Qin, Rebecca L. Winter, and Paulo E. Arratia Phys. Rev. Fluids 7, 110511 (2022) – Published 7 November 2022 | | | S. Ravichandran and Rama Govindarajan Phys. Rev. Fluids 7, 110512 (2022) – Published 8 November 2022  | The formation of raindrops in clouds requires the collision and coalescence of water droplets. The rapidity of this process has long been a puzzle in atmospheric fluid dynamics. We review one possible explanation - inertial particle caustics around strong vortices. We apply this simple physical picture to data from direct numerical simulations of highly turbulent flow, and show that caustics around vortices can, at high Reynolds number, provide the seed for the rapid formation of larger droplets, and thus rain, in ice-free clouds. | | | | | | Suin Shim, Janine K. Nunes, Guang Chen, and Howard A. Stone Phys. Rev. Fluids 7, 110513 (2022) – Published 10 November 2022  | Diffusiophoretic motion of particles is influenced by the pH of surrounding liquid, due to the varying zeta potential of surfaces at different pH. By using the particles with an isoelectric point (pI), we study pH-dependent diffusiophoresis under a pH gradient. Various particle behaviors in the absence and presence of wall diffusioosmosis are demonstrated using a dead-end pore geometry. | | | | | | Maziyar Jalaal, Borge ten Hagen, Hai le The, Christian Diddens, Detlef Lohse, and Alvaro Marin Phys. Rev. Fluids 7, 110514 (2022) – Published 17 November 2022  | Many living microorganisms experience an affinity to populate boundaries. The reasons for such affinity can be complex. Here we show that a simple synthetic microswimmer (Janus catalytic colloidal particles) tends to accumulate in the vicinity of liquid interfaces in sessile droplets. We show that the main mechanism is related to their active swimming motion, which is dominating even in the presence of evaporation-driven flows within the sessile droplet. | | | | | | Paulo E. Arratia Phys. Rev. Fluids 7, 110515 (2022) – Published 21 November 2022  | How do microorganisms move in fluids that contain particles and/or polymers? Such fluids often display nonlinear rheological behavior such as shear-rate dependent viscosity and viscoelasticity. The effects of such rheological behavior on the swimming behavior of microorganism at low Reynolds number are only recently being elucidated. This article highlights a few of the key developments in the field of swimming in complex fluids. | | | | | | Letter Omkar T. Patil, Michael A. Meehan, and Peter E. Hamlington Phys. Rev. Fluids 7, L111501 (2022) – Published 29 November 2022  | Buoyant plumes are known to oscillate at a characteristic frequency, but little is known about how this frequency changes when two plumes are in close proximity. In this work, we conduct a series of high-fidelity simulations varying the spacing and width between two-dimensional helium plumes. We find that the global trends are similar to that of interacting reacting plumes, and four distinct regimes of interaction can be identified, most notably a scaling regime when the plumes are at moderate distances apart. | | | | | | Interfacial Phenomena and Flows | Letter Hansol Wee, Christopher R. Anthony, and Osman A. Basaran Phys. Rev. Fluids 7, L112001 (2022) – Published 23 November 2022  | We examine the breakup of low-viscosity (μ) liquid filaments. When μ is small, the filament initially thins as if it were inviscid and its minimum radius hmin obeys a universal scaling law. Here, we use simulations to show that for fluids of sufficiently small μ, a coefficient value in the scaling law predicted from computations agrees with theory to three decimal places and inviscid power-law behavior can be observed over 2-3 decades in hmin as tb−t→0 where tb is the filament break up time. Transition from the inviscid regime to a viscous one is also demonstrated from simulations. | | | | | | Biological and Biomedical Flows | Paul M. Ryan and Charles W. Wolgemuth Phys. Rev. Fluids 7, 113101 (2022) – Published 21 November 2022  | Filaments rotating in fluids arise throughout nature and human engineering, from the thin helices that drive bacterial swimming to methods for laying deep sea cables. Previous work examined the instability that occurs when a filament of circular cross-section rotates in a viscous fluid. Here we extend that analysis to probe the dynamics beyond onset of the instability and how the behavior is affected by cross-sectional shape. Our analysis reveals a rich phase space and elucidates the past discrepancy between immersed boundary simulations and other analytic and computational approaches. | | | | | | Leonid Kagan, Peter V. Gordon, and Gregory Sivashinsky Phys. Rev. Fluids 7, 113501 (2022) – Published 9 November 2022  | Nonlinear Rayleigh-B ́enard convection in an infinite Prandtl number fluid layer between poorly conducting boundaries is considered as a model for convection in the Earth's upper mantle. It is shown that accounting for the generally neglected impact of viscous dissipation may lead to the development of large-scale spatiotemporal chaotic dynamics governed by the familiar Kuramoto-Sivashinsky (KS) equation, Φτ+∇4Φ+2∇2Φ−(∇Φ)2+αΦ=0, known to occur in various physical systems. The figure shows an irregular pattern of chaotically recombining cells developing at α=0.1. | | | | | | Drops, Bubbles, Capsules, and Vesicles | Nikos Pelekasis, Maria Vlachomitrou, and Alkmini Lytra Phys. Rev. Fluids 7, 113601 (2022) – Published 15 November 2022  | Compression only is a counterintuitive response pattern of lipid coated microbubbles that are strain softening by nature and normally exhibit a bias towards expansion during volume pulsation. It is a result of shell viscoelastic behavior and arises for shells with low bending resistance and shear viscosity. During sonication such microbubbles buckle when the shell is prestressed, leading to oscillations around deformed shapes of lower volume. The latter are energetically favored and arise due to parametric mode excitation. Lower values of shear shell viscosity enhance this pattern that can be used to improve protocols for the acoustic characterization of shell mechanical properties. | | | | | | F. J. Higuera Phys. Rev. Fluids 7, 113602 (2022) – Published 17 November 2022  | A simple model of hydrogen evolution at a horizontal cathode at the bottom of an acidic aqueous solution is proposed. The model accounts for convection due to suction of liquid through the electrode, as well as diffusion and migration of the charged species. The rate of growth of hydrogen bubbles and their spacing on the electrode are computed as functions of the mean current density and the suction velocity. | | | | | | Javier Sierra-Ausin, Paul Bonnefis, Antonia Tirri, David Fabre, and Jacques Magnaudet Phys. Rev. Fluids 7, 113603 (2022) – Published 28 November 2022  | The dynamics of a gas bubble suspended in an axisymmetric straining flow is examined with modern tools of global stability analysis. The complete bifurcation diagram of a bubble constrained to remain trapped at the stagnation point is determined. It is made of a stable branch, along which the bubble oscillates, and an unstable branch along which it develops non-oscillating waisted shapes prefiguring breakup. Two unstable non-oscillating modes also allow a free bubble to drift away from the stagnation point, not only along the elongation axis but also within the compressional plane thanks to a specific self-propulsion mechanism. | | | | | | Yu Tian, Zihan Peng, Yanchu Liu, Linsen Di, Ziyi Zhan, Dong Ye, Yin Guan, Xinping Zhou, Weiwei Deng, and YongAn Huang Phys. Rev. Fluids 7, 113604 (2022) – Published 29 November 2022  | Complex electrohydrodynamic behavior occurs during drop impact onto a solid surface under an external electric field. The process can be divided into three stages: (1) deformation of the drop in the electric field prior to contact; (2) initial contact of the drop with the substrate; and (3) the rich postcontact phenomena. Herein we provide a complete physical picture of the entire process of a drop impacting onto the solid surface under an external electric field. The various drop deposition modes are summarized in a phase diagram which sheds light on identifying appropriate electric fields for high-quality drop depositions without air bubble entrapments or jettings. | | | | | | A. Chahine, J. Sebilleau, R. Mathis, and D. Legendre Phys. Rev. Fluids 7, 113605 (2022) – Published 30 November 2022  | We conduct an experimental investigation of droplet sliding under the influence of a laminar or turbulent airflow for water and glycerin droplets. The onset of sliding is described with a critical Weber number that depends on the Reynolds number through the drag coefficient. During sliding, various shapes (oval, corner, and rivulet) are observed, and transitions are predicted using a capillary number. | | | | | | Interfacial Phenomena and Flows | Claudiu Patrascu and Ioana Rasuceanu Phys. Rev. Fluids 7, 114001 (2022) – Published 10 November 2022  | When water meets sunflower oil in a horizontal capillary tube, a water droplet can spontaneously form. The position of the breakup event can be manipulated by simply adjusting the oil content that prefills the capillary. For short length scales, the system advances with a constant velocity making it a good predictive tool for engineering applications. | | | | | | Vignesh Thammanna Gurumurthy, Molly Baumhauer, Aditya Khair, Ilia V. Roisman, Cameron Tropea, and Stephen Garoff Phys. Rev. Fluids 7, 114002 (2022) – Published 21 November 2022  | Corners formed between any two surfaces enhance the capillary flow along the corners resulting in the rise of thin liquid streams known as rivulets. In this work, we investigate the rivulet dynamics inside a square capillary under forced wetting. The continuously rising rivulets start decreasing in length linearly when the force is first applied and eventually reach a finite length above the central region of fluid in the capillary. We explain our observations using a simple model based on capillary rise. | | | | | | M. A. Herrada, A. Ponce-Torres, P. R. Kaneelil, A. A. Pahlavan, H. A. Stone, and J. M. Montanero Phys. Rev. Fluids 7, 114003 (2022) – Published 28 November 2022  | We study numerically the effect of a soluble surfactant on the stability of two-phase flow in a finite-length microchannel. The two streams drag the surfactant molecules toward the downstream end of the interface against the action of the Marangoni stress. The sharp reduction of the interfacial tension at that end enhances the interface deformation and considerably destabilizes the system, translating into a sharp reduction of the critical capillary number even for very small surfactant volume concentrations. | | | | | | Laminar and Viscous Flows | Bradley Gibeau and Sina Ghaemi Phys. Rev. Fluids 7, 114101 (2022) – Published 2 November 2022  | The use of active surface deformations as an actuation strategy has been largely overlooked in the flow control literature. In this work, we investigate how active surface deformations applied over a range of frequencies and amplitudes affect a laminar boundary layer. We find that the strategy is capable of producing both high- and low-speed motions. These motions appear well-suited for flow control, especially at the lowest actuation frequencies. | | | | | | Karthik Menon, Sushrut Kumar, and Rajat Mittal Phys. Rev. Fluids 7, 114102 (2022) – Published 21 November 2022  | A novel force-partitioning method is used to quantify the lift induced on finite-span wings at low Reynolds numbers by streamwise and spanwise oriented vorticity over the wing and in its wake. Although spanwise vortices, such as the leading-edge vortex, are thought to be the dominant lift producing mechanism, we show that streamwise vorticity induces a higher net lift on the wing and spanwise vorticity in the wake can produce negative lift. We quantify the lift induced by vortex cores and their associated strain-dominated flow, and show that they exhibit very different vortex tilting and stretching dynamics that can diminish the influence of spanwise and augment streamwise vorticity. | | | | | | Editors' Suggestion Dylan Reynolds, Gustavo M. Monteiro, and Sriram Ganeshan Phys. Rev. Fluids 7, 114201 (2022) – Published 16 November 2022 | We derive the governing equation of flow for a three-dimensional fluid with a parity-broken viscosity tensor when confined to a Hele-Shaw cell. When such a fluid is pushed through a channel, a transverse force is exerted on the walls, and when a bubble of air expands into a region of such fluid, a circulation develops in the far field. The Saffman-Taylor stability condition is also modified, with these terms tending to stabilize the two fluid interface. Such experiments can in principle facilitate the measurement of parity odd coefficients in both synthetic and natural active matter systems. | | | | | | Qi-Lin Zhang, Rong-Yao Yang, Chun-Lei Wang, and Jun Hu Phys. Rev. Fluids 7, 114202 (2022) – Published 28 November 2022  | A terahertz electric field (TEF) is employed to stimulate an active pump for water transportation by a bias applied in a nanochannel under no external pressure gradient. The excellent pumping ability is attributed to the resonance coupling between the TEF and water molecules. This proposed TEF-driven pump design will offer a guide in polar molecule transport through artificial or biological nanochannels, particularly in a controllable, noncontact, and large-scale process. | | | | | | N. G. Hadjiconstantinou and M. M. Swisher Phys. Rev. Fluids 7, 114203 (2022) – Published 28 November 2022  | The hydrodynamic wall location at which the slip boundary condition is applied lies inside the fluid. Its distance from the fluid-solid interface is a strong function of the fluid state and system properties and cannot be neglected. Accounting for this distance holds the key to correct calculations of slip as, for example, when reconciling equilibrium and nonequilibrium measurements. | | | | | | S. Z. Hoque, K. Bhattacharyya, and A. K. Sen Phys. Rev. Fluids 7, 114204 (2022) – Published 29 November 2022  | We study the dynamics of an oblate shape micro-particle exposed to standing bulk acoustic waves in a microchannel. A parametric study examines the effects of initial orientation, aspect ratio, size, and initial location of the particle on the translational and rotational motion. We find that the particle undergoes rotation to minimize the acoustic radiation torque potential. The direction of rotation of the particle was found to change from anticlockwise to clockwise beyond a critical aspect ratio and larger particles are found to rotate faster and closer to the nodal plane. These results can be relevant to the dynamics of an elongated microorganism or biological cells in an acoustic field. | | | | | | Multiphase, Granular, and Particle-Laden Flows | Lorenzo Angelilli, Francisco E. Hernández Pérez, Hong G. Im, Pietro Paolo Ciottoli, and Mauro Valorani Phys. Rev. Fluids 7, 114301 (2022) – Published 9 November 2022  | Liquid ammonia has been under investigation as a possible substitute of the current carbon-based fuels. However, its thermodynamical properties are responsible for a strong cooling effect of the surrounding environment and peculiar dispersion patterns that can affect the efficiency of a combustor. Through this computational study, it is highlighted how the rapid environmental cooling and change of carrier mixture make ammonia droplets condense, despite the ambient temperature being considerably higher than the saturation one, affecting the jet topology as well. | | | | | | Long Ju, Baochao Shan, and Zhaoli Guo Phys. Rev. Fluids 7, 114501 (2022) – Published 21 November 2022  | The present study investigates the effects of impurities on convective mixing processes in geological sequestration of CO2 at pore scale. We focus on the concentration expansion coefficient and diffusion rate of different impurities, the effects of which on the fingering phenomena, dissolution flux, as well as the onset time of convection ton are mainly analyzed. Based on the numerical results, the mathematical form of effective Rayleigh number Rae in a multicomponent system is fit, which could ensure that the scaling relations between ton and Rae are consistent with that in a pure system. | | | | | | J. Andrzej Domaradzki Phys. Rev. Fluids 7, 114601 (2022) – Published 8 November 2022  | Numerical simulations of turbulent flows at high Reynolds numbers rely on postulated turbulence models. We show that a model can be obtained by computing subgrid scale (SGS) energy transfer from the actual, resolved velocity fields in the course of simulations. This information, supplemented by asymptotic properties of the energy flux in the inertial range, allows self-contained simulations without postulating extraneous SGS models. For high Reynolds numbers such autonomous simulations lead to the inertial range k−5/3 spectrum with the correct value of the Kolmogorov constant. | | | | | | Emanuele Gallorini, Maurizio Quadrio, and Davide Gatti Phys. Rev. Fluids 7, 114602 (2022) – Published 14 November 2022  | We examine interaction mechanisms between the drag-reducing streamwise-traveling waves of spanwise wall velocity applied at the wall of a turbulent channel flow, and near-wall turbulent coherent structures. In particular, we focus our analysis on the drag-reduction-induced modifications of the quasi-streamwise vortices that populate the near-wall region of turbulent flows through a conditional averaging procedure. We observe that, while the streamwise-traveling waves share most of the drag-reducing mechanisms with the spanwise wall oscillations, some important differences related to the finite phase speed of the wave and the different properties of the generalized Stokes layer exist. | | | | | | Editors' Suggestion Jinyul Hwang and Jae Hwa Lee Phys. Rev. Fluids 7, 114603 (2022) – Published 14 November 2022 | In wall turbulence, meandering behaviors of large-scale structures observed in the logarithmic layer is a crucial spatial feature for understanding the spatial organization of these structures and improving the structure-based turbulence model. These structures extend from the near-wall region to the edge of boundary layers. Their meandering motions leave an imprint on the two-point turbulence statistics across the flow, especially in the logarithmic region. Here, we demonstrate the influence of the meandering motions of wall-attached structures on the two-point correlation and premultiplied two-dimensional spectra by analyzing direct numerical simulation data of the turbulent boundary layer. | | | | | | Cheng Cheng and Lin Fu Phys. Rev. Fluids 7, 114604 (2022) – Published 14 November 2022  | The present study investigates the scale characteristics of the log- and outer-region motions and structures in subsonic and supersonic wall turbulence. The energy distribution among the multiscale structures in the outer region is found to be dominated by the semilocal friction-Reynolds-number effects rather than the Mach-number effects. The geometrical characteristics of the self-similar structures populating the logarithmic region are also revealed by adopting a linear coherence spectrum. | | | | | | M. Obligado, S. Klein, and J. C. Vassilicos Phys. Rev. Fluids 7, 114606 (2022) – Published 18 November 2022  | The interaction between turbulent axisymmetric wakes plays an important role in many industrial applications, notably in the modeling of wind farms. This work proposes an experimental study on the interaction between two bluff bodies. We find that a simple mathematical expression for the wake interaction length based on nonequilibrium turbulence scalings can be used to collapse the streamwise developments of the second, third, and fourth moments of the streamwise fluctuating velocity. | | | | | | Enwei Zhang, Wangxia Wu, Qingquan Liu, and Xiaoliang Wang Phys. Rev. Fluids 7, 114607 (2022) – Published 21 November 2022  | Turbulent flow over a wavy wall represents many physical processes in natural environmental flow, such as aeolian sand or wind waves. We focus on the vortex generation and interaction in the two-dimensional wavy wall turbulent mass transfer. The centrifugal instability interprets the formation of the streamwise vortices. The spanwise vortices are verified to be generated through a Kelvin–Helmholtz instability deviating from the crest. A correlation of vorticity production with the statistics of the turbulent momentum (mass) flux demonstrates the likely mechanism of the vortex interaction and its effects on these statistics. | | | | | | Mahendra K. Verma and Soumyadeep Chatterjee Phys. Rev. Fluids 7, 114608 (2022) – Published 28 November 2022  | Typical isolated energy-conserving systems, including three-dimensional Euler turbulence, approach thermodynamic equilibrium, which is the most disordered state of the system. However, as shown in this paper, two-dimensional (2D) Euler turbulence remains out of equilibrium, with the asymptotic state more ordered than the intermediate state. We quantify the order of Euler turbulence using a unique measure called hydrodynamic entropy. The final state of 2D Euler turbulence exhibits a complex exchange of energy. | | | | | | Ghanesh Narasimhan, Dennice F. Gayme, and Charles Meneveau Phys. Rev. Fluids 7, 114609 (2022) – Published 28 November 2022  | Large Eddy Simulation (LES) is used to study the wind veer effects on the wake of a yawed wind turbine. The veer deflects the wake in the spanwise direction, and this effect can be captured in the wake model by adding a veer correction term. The counter-rotating vortex pair structures previously observed behind yawed turbines can also be recovered by subtracting the background veer vorticity. | | | | | | Sreethin Sreedharan Kallyadan and Priyanka Shukla Phys. Rev. Fluids 7, 114701 (2022) – Published 28 November 2022  | The motion of point vortices that preserves the initial geometry of the vortex arrangement is an important class of vortex motion associated with N-vortex systems, referred to as the self-similar motion. While the self-similar motion of three-point vortices is well understood, larger vortex systems still need to be explored. Here, we use simple concepts from linear algebra to numerically investigate the distribution of initial conditions that lead to the self-similar motion of point vortices. | | | | | | Wave Dynamics, Free Surface Flows, Stratified, and Rotating Flows | Sina Zendehroud, Roland R. Netz, and Julian Kappler Phys. Rev. Fluids 7, 114801 (2022) – Published 4 November 2022  | Many different types of waves that propagate along interfaces are found in nature. We derive the general dispersion relation for interfacial waves along a planar viscoelastic boundary that separates two viscoelastic bulk media, from which the known dispersion relations of different interfacial waves are recovered in the respective parameter limits. Our theory allows us to model acoustic wave-guiding phenomena at biological membranes, where viscoelastic properties are particularly relevant. | | | | | | Alain D. Gervais, Gordon E. Swaters, and Bruce R. Sutherland Phys. Rev. Fluids 7, 114802 (2022) – Published 14 November 2022  | Simulated fully localized internal gravity wave packets with moderately large initial amplitude exhibit finite transmission across a reflection level, due to the combined effects of the wave-induced shear locally cancelling the retrograde background shear, modulational instability, and the nonlinear generation and evolution of secondary wave packets. | | | | | | S. Liu, T. Waseda, and X. Zhang Phys. Rev. Fluids 7, 114803 (2022) – Published 22 November 2022  | We investigated the four-wave resonant and quasi-resonant interactions in a special degenerated case, wherein bichromatic mother waves are generated to give birth to a daughter wave. Through theoretical and numerical analyses, we first discover the threshold value of water depth when the four-wave resonance diminishes. Our work can shed light on the limiting shallow water depth where the energy exchange due to four wave resonance creases. | | | | | | | |
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