Interface Tracking Research Articles

Volume-of-Fluid Based Finite-Volume Computational Simulations of Three-Phase Nanoparticle-Liquid-Gas Boiling Problems in Vertical Rectangular Channels

This study develops robust numerical algorithms for the simulation of three-phase (solid-liquid-gas) boiling and bubble formation problems in rectangular channels. The numerical algorithms are based on the Finite Volume Methods (FVM) and implement both the volume-of-fluid (VOF) methods for liquid-gas interface tracking as well as the volume-fraction methods to account for the concentration of embedded solid nano-particles in the liquid phase. Water is used as the base-liquid and the solid phase is modelled via metallic nano-particles (both aluminium oxide and titanium oxide nano-particles are considered) that are homogeneously mixed within the liquid phase. The gas phase is considered as a vapour arising from the bolling processes of the liquid-phase. The finite volume methodology is implemented on the OpenFOAM software platform, specifically by careful modification and manipulation of existing OpenFOAM solvers. The governing fluid dynamical equations, for the three-phase boiling problem, take into account the thermal conductivity effects of the solid (nano-particle), the momentum and energy equations for both the liquid-phase and the gas-phase, and finally the decoupled mass conservation equations for the liquid- and gas- phases. The decoupled mass conservation equations are specifically used to model the phase change between the liquid- and gas- phases. In addition to the FVM and VOF numerical methodologies for the discretization of the governing equations, the pressure-velocity coupling is resolved via the PIMPLE algorithm, a combination of the Pressure Implicit with Splitting of Operator (PISO) and the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithms. The computational results are presented graphically with respect to variations in time as well as in the nano-particle volume fractions. The simulations and results accurately capture the formation of vapour bubbles in the two-phase (particle-free) liquid-gas flow and additionally the computational algorithms are similarly demonstrated to accurately illustrate and capture simulated boiling processes. The presence of the nano-particles is demonstrated to enhance the heat-transfer, boiling, and bubble formation processes.

Unified modeling of cavitating sprays using a three-component volume of fluid method accounting for phase change and phase miscibility

In this study, a new three-fluid volume of fluid framework is presented in order to be used as a tool for providing physical insight—in a unified manner—to cavitating sprays and other complex multi-fluid, multiphase fluid flows. The framework accounts for phase change across a sharp interface between two fluids (gas and liquid) including miscibility between the fluid generated due to phase change (liquid vapor due to cavitation in the investigated cases) and one of the existent fluids (gaseous air). Systematic validation of the framework was performed over three cases. The first case is a bubble rising test case that an analytical solution for a two-phase system is available. Comparisons based on previous results from other interface tracking solvers and against the analytical solution are presented. This test case was then expanded by the authors so that a third non-condensable gas phase with a free-surface interface over a rising bubble was present. This second test case was used to further validate the three-fluid system behavior. Finally, experimental comparisons were made with a more realistic orthogonal spray geometry that captures different cavitation characteristics over a range of flow intensities. The predicted link between cavitation and the subsequent jet formation was highlighted. This is the first study, to our knowledge, that presents the performance of a three-fluid framework over a range of realistic injection conditions covering turbulent two-phase flows (no cavitation) to supercavitation.

A Fourth-Order Unfitted Characteristic Finite Element Method for Solving the Advection-Diffusion Equation on Time-Varying Domains

We propose a fourth-order unfitted characteristic finite element method to solve the advection-diffusion equation on time-varying domains. Based on a characteristic-Galerkin formulation, our method combines the cubic MARS method for interface tracking, the fourth-order backward differentiation formula for temporal integration, and an unfitted finite element method for spatial discretization. Our convergence analysis includes errors of discretely representing the moving boundary, tracing boundary markers, and the spatial discretization and the temporal integration of the governing equation. Numerical experiments are performed on a rotating domain and a severely deformed domain to verify our theoretical results and to demonstrate the optimal convergence of the proposed method.

Examination of Haines Jump in Microfluidic Experiments via Evolution Graphs and Interface Tracking

This work examines a type of rapid pore-filling event in multiphase flow through permeable media that is better known as Haines Jump. While existing microfluidic experiments on Haines Jump mostly seek to maintain quasi-steady states through very low bulk flow rates over long periods of time, this work explores the combined use of a highly structured microscale transport network, high-speed fluorescent microscopy, displacement front segmentation algorithms, and a tracking algorithm to build evolution graphs that track displacement fronts as they evolve through high-speed video recording. The resulting evolution graph allows the segmentation of a high-speed recording in both space and time, potentially facilitating topology-cognitive computation on the transport network. Occurrences of Haines Jump are identified in the microfluidic displacement experiments and their significance in bulk flow rates is qualitatively analyzed. The bulk flow rate has little effect on the significance of Haines Jump during merging and splitting, but large bulk flow rates may obscure small bursts at the narrowest part of the throat.

Modal analysis and interface tracking of multiphase flows using Dynamic Mode Decomposition

Non-invasive visualization techniques for multiphase flows are critical to understanding primary atomization and sprays. We use back-lit imaging to identify the liquid–gas interface of two-phase flows at high temporal and spatial resolutions and employ Dynamic Mode Decomposition (DMD) to study the shape and frequency of instabilities of a liquid jet surrounded by a coaxial annular airblast atomizer. However, DMD is not suitable for interface tracking, so we develop a data-driven two-step approach using the optical sensor data. The method uses DMD on the optical flow field estimated from image snapshot pairs. We demonstrate our method to a representative toy problem of an oscillating drop and on the primary atomization of a numerical planar liquid jet. Finally, we apply our method to the experimental liquid jet from the coaxial airblast atomizer using back-lit imaging. Our method is able to accurately reconstruct and predict the flow and preserves the sharpness of the fluid interface.

Improved Volume-Of-Solid Formulations for Micro-Continuum Simulation of Mineral Dissolution at the Pore-Scale

We present two novel Volume-of-Solid (VoS) formulations for micro-continuum simulation of mineral dissolution at the pore-scale. The traditional VoS formulation (VoS-ψ) uses a diffuse interface localization function ψ to ensure stability and limit diffusion of the reactive surface. The main limitation of this formulation is that accuracy is strongly dependent on the choice of the localization function. Our first novel improved formulation (iVoS) uses the divergence of a reactive flux to localize the reaction at the fluid-solid interface, so no localization function is required. Our second novel formulation (VoS-ψ′) uses a localization function with a parameter that is fitted to ensure that the reactive surface area is conserved globally. Both novel methods are validated by comparison with experiments, numerical simulations using an interface tracking method based on the Arbitrary Eulerian Lagrangian (ALE) framework, and numerical simulations using the VoS-ψ. All numerical methods are implemented in GeoChemFoam, our reactive transport toolbox and three benchmark test cases in both synthetic and real pore geometries are considered: 1) dissolution of a calcite post by acid injection in a microchannel and experimental comparison, 2) dissolution in a 2D polydisperse disc micromodel at different dissolution regimes and 3) dissolution in a Ketton carbonate rock sample and comparison to in-situ micro-CT experiments. We find that the iVoS results match accurately experimental results and simulation results obtained with the ALE method, while the VoS-ψ method leads to inaccuracies that are mostly corrected by the VoS-ψ’ formulation. In addition, the VoS methods are significantly faster than the ALE method, with a speed-up factor of between 2 and 12.

Cell alternation algorithm for simulating bubble growth in boiling flows through volume of fluid (VOF) method in fluent

This study focuses on the numerical illustration of bubble growth and addresses the inherent interface deformation problem associated with phase change in the Volume of Fluid (VOF) interface tracking method. To avoid the interface deformation, this study proposes a cell alternation algorithm that could simulate bubble growth with a sharp interface. The proposed algorithm identifies two types of cells near the vicinity of the interface, which include the primary-interface-cell and small-interface-cell. This algorithm computes mass transfer associated with a phase change at the primary interface cell and relocates it to a nearby cell full of vapor. The predictive accuracy of the proposed method is assessed with a one-dimensional Stefan phase change problem. In the case of the Stefan phase change problem, the results of simulations agree well with the results of the theoretical solution, and the predictive accuracy increases as the grid are refined. In addition, the proposed method is verified with two engineering problems which include bubble growth on a constant temperature surface and bubble growth in a microchannel. The simulation results show a good agreement with the results of the experiment in terms of bubble growth rate, bubble shape, and other parameters associated with experiment. Also, an insight into the temperature and velocity fields is provided in simulation which cannot be achieved directly through experiment.

Thermotaxis of a deformable droplet in a confined Poiseuille flow

A droplet under a thermal gradient is known to migrate in a preferential direction, as governed by the variation of its interfacial tension with temperature. Contradicting the outcome of reported asymptotic analysis, here we show that the calculation of the droplet migration by considering the variation of interfacial tension with the imposed thermal field alone may be fundamentally incorrect. This error is attributed to the dynamically evolving interfacial temperature field due to a two-way coupling between the thermal field and the flow field, mediated by the droplet deformation and thermal diffusion. By directly capturing an inherent nonlinear coupling between the thermal field and the flow field using explicit interface tracking in a three-dimensional space, our boundary integral based analysis reveals that a linearly decreasing temperature profile imposed along the direction of a plane Poiseuille flow enhances the migration speed of the droplet in both the axial and cross-stream directions. This is in sharp contrast to a prediction of decelerated motion of the droplet under the same imposed thermal field, as obtained from asymptotic theory. We attribute this discrepancy to an alteration of the surface tension mediated interfacial stress due to the locally evolving temperature field, and a consequent concomitant alteration in the interfacial viscous stress to realize a tangential force balance at the interface. From scaling arguments, we show that the resulting change in the viscous drag force may occur over an order of magnitude, disrupting the outcome as compared to that obtained from asymptotic analysis. These results are likely to bear significant implications in controllable separation and sorting of deformable entities in confined fluidic media.

A novel ANN-CFD model for simulating flow in a vortex mixer

The study presents a novel ANN-CFD model for simulating flow in a vortex mixer (an unbaffled vessel stirred by a magnetic stirrer). Large eddy simulations (LES) are performed to simulate flow and turbulence considering presence of both air and liquid phases. The flow fields in air and liquid phases are coupled by applying suitable boundary conditions at the air–liquid interface represented by the vortex. The shape of the vortex is built-in in the computational domain and obtained by using an artificial neural network (ANN) model trained and validated with the data obtained from experiments carried out to capture the vortex shape under different parametric conditions. ANN-CFD model is validated with PIV data reported in literature. The ANN-CFD model reduces the computational time significantly by obviating the need of performing computationally very intensive interface tracking simulations. The model is used to understand hydrodynamics and turbulence characteristics of flow in the vortex mixer.

Effects of Gas Embolism on Pulsatile Flow Characteristics Within a Human Carotid Artery.

Atherosclerosis in carotid arteries depends mostly on hemodynamic parameters, and any disturbances in pulsatile flows may alter the hemodynamic parameters extensively. Gas emboli are one such source that can hinder and disturb standard blood flow patterns and potentially lead to occlusions and ischemia. To understand how gas embolism affects carotid artery hemodynamics, numerical simulation of coupled Newtonian two-phase laminar flow with interface tracking are performed in an anatomical image-based geometry with flow conditions closely approximating physiological flows. Bubble behavior and Pulsatile flow field changes are quantified. Significant deviation from flow without gas embolism is observed leading to nonstandard flow distributions. Results show that gas embolism promotes complex spatio-temporal variations in the carotid artery hemodynamics leading to higher time averaged shear stress values and greater regions of high oscillatory shear index, implying higher probability of atherosclerosis incidence. Depending on contact angle, gas emboli were found to be lodged in carotid sinus or exiting the carotid artery, which can potentially cause abnormalities in blood pressures, heart rates, and ischemia in downstream vasculature, respectively.

Application of Wireless Mesh Network in Interface Tracking Between Two Immiscible Fluids

The characteristics of the interaction between immiscible fluids are prominent because of their intriguing conduct in many situations in the environment and industry, such as lubrication process, oil extraction, physicochemical process, and polymers co-extrusion. We examine the time-dependent flow of two immiscible liquids via a horizontal channel in this paper. Due to external pressure, time, and other physical circumstances, the fluidfluid interface alters its location. It may collapse, even though it is non-slip and hyper-stick at the channel’s wall. Hence the single momentum equation by the volume of fluid (VOF) method is combined with Wireless Mesh Network’s to track the interface. The modeled coupled partial differential equation is analytically solved for fluid interface evolution, and using the proposed Algorithm, the coverage and uniformity between the node are calculated. Based on the calculations, the optimal node is selected to track the interface, resulting in discovering the lost information with the least distance. The advection and tracking of the interface with time through graphs. It is observed that the undulating sequence occurs faster for a considerable high time, and the proposed algorithms track the interface with optimal node selection.

Heat transfer enhanced surfaces for horizontal tube falling film evaporator characterized using laser interferometry

The falling film evaporators are a class of heat exchanger that is widely studied. The research for improving the efficiency is still of prime importance because of its widespread applications in many industries such as desalination, food processing, nuclear energy, refrigeration etc. Our motive in this work is to investigate the potential use of heat and mass transfer enhanced surfaces for horizontal tube falling film evaporator to scale up its effectiveness. Four different tube surfaces viz. bare copper tube, Al2O3 thermal spray coated tube, Al2O3 + TiO2 thermal spray coated tube and open cell copper metal foam wrapped tube were taken for the study. Scanning electron microscope, 3D surface profilometer, contact angle metre and energy-dispersive X-ray spectroscopy were utilized to study the surface texture, surface roughness, wettability and composition, respectively. Out of these four surfaces, a total of eight configurations were realized by varying the coating thickness, surface roughness and porosity. A new prototype of horizontal tube falling film evaporator was designed and fabricated with fully wetted flow. An in house developed novel method based on laser interferometry was utilized for the experimental investigation. This non-invasive technique was capable of measuring falling film thickness and falling film interface temperature simultaneously with unprecedented circumferential span. For the film thickness evaluation, an optical shadow method incorporating Otsu’s algorithm with precise interface tracking features was used. Mach–Zehnder Interferometer was employed to visualize the isotherm formation and to quantitatively infer the interface temperature. A 2D CFD study based on VOF model was carried out and found to be in good agreement with experimental as well as theoretical results. Further, the effect of flow rate, feed inlet water and body temperature were explored in detail and analysed statistically.

An incompressible smoothed particle hydrodynamics‐finite volume method coupling algorithm for interface tracking of two‐phase fluid flows

AbstractTwo‐phase flow involves complex interface evolution process such as the formation, development, pulsation, and rupture of phase interfaces. Numerical simulation is one of the important means to study two‐phase flow. The tracking and reconstruction of phase interface is the focus of two‐phase flow simulation. A two‐phase flow simulation algorithm based on coupled incompressible smoothed particle hydrodynamics (ISPH) method and finite volume method (FVM) is developed in this article. In present ISPH‐FVM coupling algorithm, one phase which has smaller volume is represented by SPH particles, while the other phase is defined on the FVM grids. The coupling of ISPH and FVM is achieved through the transfer and interaction of physical parameters at the overlapping area of the SPH particles and FVM grids. The continuous medium surface force model is also introduced into the ISPH‐FVM coupling algorithm to study the effect of surface tension on the two‐phase flow. Several numerical examples of two‐phase flow are adopted to verify the effectiveness of the ISPH‐FVM coupling method.

A lattice Boltzmann model for liquid-vapor-solid flow with thermal phase change

On the basis of phase-field theory, we develop a simple lattice Boltzmann (LB) model for liquid-vapor-solid flow with thermal phase change, which is able to deal with large fluid density contrasts. The conservative Allen-Cahn equation and Navier-Stokes equations are modified to take into account the phase change effects based on the Stefan boundary condition. Three distribution functions are used in the model, one of which is to track the interface among the fluids, one is employed for the fluid hydrodynamics, and the other one is utilized to solve the heat equation. The LB evolution equations are incorporated with proper source terms such that the modified interface tracking equation and hydrodynamic equations can be derived exactly. The proposed model for phase change is much simpler than the previous phase-field LB models. Combining with particle dynamics algorithms, the present model can further simulate liquid-vapor-solid systems with phase change. Several numerical tests, including the one-dimensional Stefan problem and droplet evaporation, have been performed to test the accuracy and stability of the present model, and the results obtained by this approach are in good agreement with the theory. Evaporation of a droplet wetting on a particle is simulated, it is found the fluid-solid interaction associated with the interface can be captured. Finally, as an application, the motion and growth of a rising vapor bubble with a particle through a superheated liquid is investigated.

Coupling of pore network modelling and volume of fluid methods for multiphase flow in fractured media

Modelling multiphase flow in fractured media is a challenging multi-disciplinary problem which is of particular importance for the production of hydrocarbons from unconventional geological formations. At the pore-scale, this can be studied using pore network models that are computationally efficient due to the employment of pore-space geometrical simplifications. There are two distinctive approaches to pore network modelling: quasi-static and dynamic. However, only dynamic pore network models allow capturing the transient nature of the multiphase flow. Nevertheless, the dynamic models have a number of limitations including numerical oscillations and the inability of interface tracking in a particular network element. In order to overcome these drawbacks, a hybrid numerical method which combines the Hagen–Poiseuille analytical solution of the Navier–Stokes equations and the volume of fluid advection scheme is developed and validated. The proposed multiphase model is benchmarked against the conventional volume of fluid implementation for various drainage and imbibition cases using two-dimensional demo cases. Besides, three-dimensional simulations are also performed to ensure the applicability of the proposed model to real fractured media. The single pressure algorithm is employed while the capillary pressure drop is explicitly introduced as an additional term in the Hagen–Poiseuille equation. The conducted validation and following analysis show the applicability of the developed model for a wide range of Reynolds and capillary numbers and viscosity ratio. Thus, the new model can be used to investigate transient immiscible multiphase flow phenomena in fractures by pore network modelling.

Isotherm-evolution-based interface tracking algorithm for modelling temperature-driven solid-liquid phase-change in multiphase flows

In this paper, an isotherm-evolution-based algorithm is proposed for effective tracking of the phase-change interface in temperature-driven solid-liquid phase-change process in multiphase flows (e.g., droplet freezing in the air). To fulfill the temperature condition on the interface, an interface indicator φ is defined, whose variation between specified maxima and minima marks the passing through the melting temperature. Previous approaches directly generate φ from the temperature field with explicit function and an artificial phase-change temperature interval, which could be affected by uneven temperature gradient and thus loses interfacial sharpness. The new algorithm devices an isotherm evolution equation to fulfill thermodynamic principles on the phase-change interface and maintain interface sharpness. Specifically, diffusive terms with explicit parameter are used to analytically ensure a finite interfacial width; and the migration term is introduced to cope with the phase-change temperature condition. The consistency of the proposed algorithm with the sharp interface limit is verified by comparing with analytical solutions of the Stefan problem. In numerical tests of the freezing droplet on substrate, the new algorithm presents better consistency of interfacial width throughout the computational domain and during the whole evolution process, as compared with the previous approach. The accuracy, convergence and robustness of the method as well as its capability in three-dimensional simulations are also validated through quantitative comparisons with reference results in the literature.

A pore-resolved interface tracking algorithm for simulating multiphase flow in arbitrarily structured porous media

We present an interface tracking algorithm that incorporates essential pore-scale invasion mechanisms for simulating multiphase flow in porous media in capillary dominated regime. The algorithm allows the capture of the actual shape of porous medium with complex geometrical features, including both convex and concave particles or irregular channels. It also incorporates the sharp edge pinning effect such that pinning of menisci at corners can be naturally taken into account. We firstly conduct simulations under imbibition and drainage conditions in a single junction micro-model and in disordered porous media for validation. It is shown that the invasion mechanisms at pore scale are accurately captured, and the macroscopic invasion morphology in porous media are reproduced. A series of simulations for fluid displacement processes are carried out across a wide range of intrinsic contact angles in a two-dimensional representation of a Berea sandstone. The transition between stable displacement and capillary fingering is characterized using displacement efficiency, fractal dimension, and fluid–fluid interfacial length. This work deepens the understanding of fluid–fluid displacement processes in porous media. Furthermore, the presented interface tracking algorithm offers a highly efficient tool for further exploration of effects of wettability, geometry, and topology on multiphase flow in realistic porous materials.

Novel Hybrid Brain-Computer Interface for Virtual Reality Applications Using Steady-State Visual-Evoked Potential-Based Brain-Computer Interface and Electrooculogram-Based Eye Tracking for Increased Information Transfer Rate.

Brain–computer interfaces (BCIs) based on electroencephalogram (EEG) have recently attracted increasing attention in virtual reality (VR) applications as a promising tool for controlling virtual objects or generating commands in a “hands-free” manner. Video-oculography (VOG) has been frequently used as a tool to improve BCI performance by identifying the gaze location on the screen, however, current VOG devices are generally too expensive to be embedded in practical low-cost VR head-mounted display (HMD) systems. In this study, we proposed a novel calibration-free hybrid BCI system combining steady-state visual-evoked potential (SSVEP)-based BCI and electrooculogram (EOG)-based eye tracking to increase the information transfer rate (ITR) of a nine-target SSVEP-based BCI in VR environment. Experiments were repeated on three different frequency configurations of pattern-reversal checkerboard stimuli arranged in a 3 × 3 matrix. When a user was staring at one of the nine visual stimuli, the column containing the target stimulus was first identified based on the user’s horizontal eye movement direction (left, middle, or right) classified using horizontal EOG recorded from a pair of electrodes that can be readily incorporated with any existing VR-HMD systems. Note that the EOG can be recorded using the same amplifier for recording SSVEP, unlike the VOG system. Then, the target visual stimulus was identified among the three visual stimuli vertically arranged in the selected column using the extension of multivariate synchronization index (EMSI) algorithm, one of the widely used SSVEP detection algorithms. In our experiments with 20 participants wearing a commercial VR-HMD system, it was shown that both the accuracy and ITR of the proposed hybrid BCI were significantly increased compared to those of the traditional SSVEP-based BCI in VR environment.

Transport Behavior of Commercial Anticancer Drug Protein-Bound Paclitaxel (Paclicad) in a Micron-Sized Channel.

Protein-bound paclitaxel has been developed clinically as one of the most successful chemotherapy drugs for the treatment of a wide variety of cancers. However, these medications, due to their nanoscale properties, may often induce capillary blocking while migrating through minute blood vessels. Considering the detrimental impact of this restriction, we investigated the transport of protein-bound paclitaxel, Paclicad, in a 7 μm microchannel mimicking the identical mechanical confinement of the blood capillaries. The drug was reported to migrate through a constricted microchannel without obstruction at a solution flow rate of 20-50 μL/h. The onset of an agglomeration site was observed at higher flow rates of 70-90 μL/h, while complete capillary obstruction was observed at 100 μL/h. The mobility of the particles was also calculated, and the results suggested that the presence of varying cross-sections affects the mobility of the drug particles. The trajectory of the particle migration was observed to be less tortuous at the higher flow rate, but the tortuous nature appeared to increase with the presence of agglomeration sites in the flow field. The experimental results were also compared with the computational model of the drug particle. The drug particle was modeled both as Newtonian and as an FENE-P viscoelastic drop. The drop interface tracking was done by the VOF method using the open source software Basilisk. The particle displacement was better estimated by both the FENE-P and Newtonian model at a flow rate of 30 μL/h, while deviation was observed at a flow rate of 50 μL/h. The FENE-P model was observed to show higher deformation than the Newtonian model at both flow rates. The experimental results provided better insight into the agglomeration tendency of Paclicad, migrating through a constricted microchannel at higher flow rates. The numerical model could be further employed to understand the more complex intravenous transport of drugs.

Numerical simulation of primary atomization process of liquid jet in subsonic crossflow

Studies of primary atomization of liquid transverse jets in subsonic crossflow consider surface waves as the main cause of jet fragmentation. The characteristics of surface waves during jet fragmentation have been initially analyzed, but there is a lack of understanding of the mechanism of surface wave generation and development. This paper investigates the primary atomization of liquid transverse jets in subsonic crossflow based on the numerical method of Large Eddy Simulation (LES) with the coupled level set and volume of fluid method (CLSVOF) interface tracking, combined with adaptive mesh encryption techniques, where the liquid phase medium is water. This paper focuses on the fine structure of the primary atomization of the jet column, including the evolution of the jet column and the process of shedding the surface of the jet column, to clarify the physical process of primary fragmentation of the jet. The near-field flow structure and gas–liquid interaction are also analyzed to investigate the mechanism of primary atomization of liquid transverse jets and the mechanism of surface wave generation and development in subsonic flows. The results show that the shear action of gas and liquid leads to the generation of Kelvin–Helmholtz (KH) unstable wave, and KH surface wave dominates the surface breaking of jet column. The extrusion of gas and liquid results in the unstable fluctuation of Rayleigh–Taylor (RT), and the RT surface wave dominates the jet column, and finally the cylindrical fracture occurs. The development trend of RT surface wave is to increase the wavelength along the jet direction. When the wavelength of the RT surface wave develops from 0.12 mm to 1.2 mm, the jet column will be columnar broken at the trough of the last RT wave.