Discrete Bubble Model Research Articles

Eulerian–Lagrangian multiscale numerical analysis of multimodal partial shedding dynamics

The objective of this paper is to investigate the multimodal partial shedding dynamics from a multiscale perspective of cloud cavitating flows under two distinct cavity shedding mechanisms, namely the re-entrant jet mechanism and the shock wave propagation mechanism. A two-way Eulerian–Lagrangian coupling algorithm is applied to capture the multiscale vapor topologies from microbubble to large-scale cavities. The large-scale cavity evolution is solved through large eddy simulations (LES) with the volume of fraction (VOF) method in Eulerian frame. The sub-grid microbubbles are tracked in Lagrangian frame based on the discrete bubble model (DBM) method. The predictions agree well with experimental observation of the periodical cavity evolution and microbubble dynamics under both the re-entrant jet mechanism and shock wave mechanism around a NACA66 hydrofoil. The numerical simulation provides detailed analysis of the cavitating turbulent flow on the microbubble behavior with emphasis on the spatial-temporal distribution characteristics of microbubbles. The results show that the number and mean size of microbubbles in the cavitation region increase gradually with the growth of attached sheet cavity, development of re-entrant jet and collapse of largescale cavity for both cavitation patterns. Meanwhile, microbubbles are mainly distributed on the largescale interfaces where have high value of vorticity and turbulent kinetic energy under the effect of re-entrant jet and vortex structures. And the probability density functions (PDFs) of microbubble exhibit gamma distributions with a dominant peak at approximately 50 μm for both shedding mechanisms. However, the shock wave formation and propagation process only occurs in the final stage of cavitating flow under shock wave mechanism causing the condensation of vapor and the decrease of the number and mean size of microbubbles. Moreover, the microbubbles are uniformly distributed along the streamwise and vertical directions behind shock wave front.

Numerical investigation of partial cavitation in a Venturi tube by Eulerian-Lagrangian multiscale modelling

The cavitating flow in a Venturi tube has a complex flow structure. In this paper, the partial cavitation in an axisymmetric Venturi tube, dominated by bubbly shock, is investigated by utilizing Eulerian-Lagrangian multiscale modelling. The unsteady partial cavitation is simulated by Large Eddy Simulation coupled with the Volume of Fluid and the Discrete Bubble Model. The numerical results show a well agreement with the high-speed photography. Through a comprehensive analysis of the development and transportation of macro cavities and micro bubbles, a detailed explanation of the cavity shedding process triggered by bubbly shock is provided. The results highlight the precise capability of the multi-scale method in elucidating the intricate flow field induced by partial cavitation. The findings may pave the way for the further investigations on the underlying mechanisms of partial cavitation, fostering a deeper understanding of bubbly shock dominated cavitating flow.

Numerical erosion prediction of aluminum induced by cavitating jet using an Eulerian–Lagrangian method

Cavitation-induced erosion in pump machinery is a significant issue that leads to material loss and increased operating costs. This study develops a numerical method based on an energy balance approach to address the risk of cavitation erosion. Compared with other published methods, three improvements are made. Firstly, it assumes that the local instantaneous pressure is the driving force behind cavity collapse, rather than the far-away ambient pressure field. Secondly, it counts erosion only at the moment that the released shock wave is strong enough to cause surface damage. Thirdly, an Eulerian–Lagrangian method is introduced to simulate multiscale cavitation, in which the volume of fluid (VOF) method and a discrete bubble model (DBM) are combined to reproduce resolvable water-vapor interfaces and unresolvable discrete bubbles, respectively. The developed numerical modeling framework is validated by predicting the erosion of pure aluminum surface caused by a cavitating jet, and it is shown that the simulated erosion region fits well against the experimental results. Additionally, appropriate model coefficients of the erosion prediction method are introduced to achieve quantitative prediction of material mass loss, and the detailed erosion behavior is discussed.

Numerical Analysis of Slag–Steel–Air Four-Phase Flow in Steel Continuous Casting Model Using CFD-DBM-VOF Model

Argon injection is usually applied in the continuous casting mold to prevent submerged entry nozzle (SEN) clogging. However, the stability of the slag–steel interface is affected by the injected gas, even leading to the formation of the slag eye. A computational fluid dynamics–discrete bubble model–volume of fluid (CFD-DBM-VOF) model is established to predict the argon–slag–steel–air four-phase flow in the continuous casting mold. The bubble behavior is treated with the Lagrangian approach considering bubble coalescence and breakup. The movement behavior of the slag–steel interface is analyzed with and without argon blowing, validated with the water model. The results show that the large bubble tends to float up into the slag–steel interface near the SEN with argon injection, resulting in fluctuations in the slag–steel interface near the SEN. The bubble distribution, flow field, fluctuation height of the slag–steel interface and configuration of the slag eye in the mold are analyzed. Furthermore, the effect on the casting speed, gas flow rate and thickness of the slag layer is obtained based on the result. The mathematical prediction results showcase a combination of well-established phenomena and newly generated predictions.

Degassing Dissolved Oxygen through Bubbling: The Contribution and Control of Vapor Bubbles

An innovative yet sustainable approach for industrial deaeration is proposed, with demonstrated results and analyses, to contribute to finding solutions to improve energy efficiency in this field. Vacuum bubbling deaeration, sharing the same working principles of solubility control and the mass diffusion through vapor (or steam) with conventional thermal deaeration processes, works, however, at lower vacuum pressures. It neither resorts to heating nor requires any third-party materials such as membranes or gases, achieving orders of magnitude of reduction in the expected energy consumption in a simple and concrete way. In this study, the mechanisms of vapor bubble generation and retention were discussed by employing a vacuum bubbling model based on the experimental apparatus at Kongju National University, which uses a venturi-nozzle bubbler. The four parameters influencing vapor bubble generation and retention were identified as vessel pressure p1, nozzle depth Δh, nozzle performance p4−p3, and water temperature Tw. A series of deaeration experiments using the present approach for a tap water sample of 360~400 L were conducted under four different conditions to investigate the effects of the water temperature, vessel pressure, and bubbler nozzle depth. Final dissolved oxygen (DO) concentrations close to zero could be achieved with a vessel pressure of p1=1 kPa, with different bubbling times to reach a zero mg/L reading of DO concentration (case 2 and 3), which demonstrates the vital roles of the vapor bubble generation condition of (psat−p3) and retention condition of (p4−psat) in achieving the lowest DO concentration. Analysis of the test results, based on the discrete-bubble model with the measured DO concentrations and degassing rates, showed promising results in reproducing the experimental data. Though the potential of vacuum bubbling deaeration is demonstrated, for the first time, to its full extent, further research efforts are encouraged in many areas, including more case-specific validation test cases with optimum operating conditions along with the study of more detailed modeling for performance prediction, including energy analysis.

Large eddy simulations of bubbly flows and breaking waves with smoothed particle hydrodynamics

For turbulent bubbly flows, multi-phase simulations resolving both the liquid and bubbles are prohibitively expensive in the context of different natural phenomena. One example is breaking waves, where bubbles strongly influence wave impact loads, acoustic emissions and atmospheric-ocean transfer, but detailed simulations in all but the simplest settings are infeasible. An alternative approach is to resolve only large scales, and model small-scale bubbles adopting sub-resolution closures. Here, we introduce a large eddy simulation smoothed particle hydrodynamics (SPH) scheme for simulations of bubbly flows. The continuous liquid phase is resolved with a semi-implicit isothermally compressible SPH framework. This is coupled with a discrete Lagrangian bubble model. Bubbles and liquid interact via exchanges of volume and momentum, through turbulent closures, bubble breakup and entrainment, and free-surface interaction models. By representing bubbles as individual particles, they can be tracked over their lifetimes, allowing closure models for sub-resolution fluctuations, bubble deformation, breakup and free-surface interaction in integral form, accounting for the finite time scales over which these events occur. We investigate two flows: bubble plumes and breaking waves, and find close quantitative agreement with published experimental and numerical data. In particular, for plunging breaking waves, our framework accurately predicts the Hinze scale, bubble size distribution, and growth rate of the entrained bubble population. This is the first coupling of an SPH framework with a discrete bubble model, with potential for cost-effective simulations of wave–structure interactions and more accurate predictions of wave impact loads.

On the utilisation of the coupled-oscillator approach as a surrogate model for sound propagation in the vicinity of bubble groups

The presence of bubble curtains have been shown to reduce the level of underwater noise propagation. This is due to the change in density and acoustic-impedance between the group of bubbles and the water. In addition, sound waves can be absorbed and scattered when they interact with bubble curtains. In this paper, it is shown that a mathematical model based on the discrete bubble model (DBM) and utilising the self-consistent coupled-oscillator theory is able to correctly predict the bubble collective resonance frequencies of 1D line, 2-D planar, and 3-D complex bubble cloud configurations. This framework can also accommodate polydisperse-sized bubbles. We use the DBM and conduct modal analysis on common 2-D and 3-D bubble curtain configurations and show that the fundamental modes in both configurations dominated the vibration of bubble curtains when subjected to an external acoustic source. In addition, it is demonstrated that the performance of the bubble curtain is most affected by the changes in the bubble size whereas the variations in inter-bubble spacing had no significant influence.

Assessment of cavitation erosion risk by Eulerian–Lagrangian multiscale modeling

Cavitation erosion in fluid machineries is a complicated physicochemical and instantaneous process, leading to material loss and reduced efficiency. As the cavitation and the consequent erosion are difficult to be measured and modeled, the present work aims to incorporate an improved energy balance approach into a new Eulerian–Lagrangian multiscale model for addressing the risk of cavitation erosion. The volume of fluid (VOF) and discrete bubble model (DBM) are coupled by a transforming algorithm to develop the multiscale cavitation model. With using the large eddy simulation (LES) approach, the multiscale cavitation features including the morphological evolution of large-cavities and characteristics of microbubbles are well reproduced. To assess the cavitation erosion, a modified aggressive indicator using a modified moving time-averaged pressure field as the driving pressure is proposed to consider the evolution of local pressure and vapor volume fraction. Furthermore, the erosion powers produced from large cavities and microbubbles are completely taken into account. The results show that the VOF-DBM multiscale model can better simulate the cavitation features around a twisted hydrofoil including the secondary shedding and the horse-shoe vapor structure, compared with the VOF method using the same mesh resolution. From the results of discrete bubbles, it is found that the primary shedding leads to the rapid growth of the number of discrete bubbles, which decreases during the transportation of the bubble cloud generated from the primary shedding. By taking the collapsing power of discrete bubbles into account, the prediction of cavitation erosion distribution is quite consistent with the experimental one, whereas the VOF method fails to predict the erosion region near the trailing edge.

Using a Lagrangian-Lagrangian approach for studying flow behavior inside a bubble column

Bubble columns are widely encountered in several industries, especially in the field of nuclear safety. The Eulerian-Eulerian and the Eulerian-Lagrangian methods are commonly used to investigate bubble columns. Eulerian approaches require additional tasks such as strict volume conservation at the interface and a predefined well-structured grid. In contrast, the Lagrangian approach can be easily implemented. Hence, we introduce a fully Lagrangian approach for the simulation of bubble columns using the discrete bubble model (DBM) and moving particle semi-implicit (MPS) methods. Additionally, we propose a rigorous method to estimate the volume fraction accurately, and verified it through experimental data and analytical results. The MPS method was compared with the experimental data of Dambreak. The DBM was verified by analyzing the terminal velocity of a single bubble for each bubble size. It agreed with the analytical results for each of the four drag correlations. Additionally, the improved method for calculating the volume fraction showed agreement with the Ergun equation for the pressure drop in a packed bed. The implemented MPS-DBM was used to simulate the bubble column, and the results were compared with the experimental results. We demonstrated that the MPS-DBM was in quantitative agreement with the experimental data.

Multi-scale characteristics investigation of sheet/cloud cavitation

Sheet and cloud cavitation involve the formation of large-scale cavities and microbubbles across a wide range of length scales. In this study, a two-way coupling Eulerian-Lagrangian algorithm was utilized to simulate the cavitating flow around a NACA66 hydrofoil at an incidence of 8 degrees and a cavitation number of 1.2. The large eddy simulation (LES) and volume of fraction (VOF) methods are used to capture the large scale vapor structures in Eulerian frame. Meanwhile, the discrete bubble model (DBM) and simplified Rayleigh-Plesset equation are implied to solve the dynamic of Lagrangian microbubbles smaller than local grid scale. The results indicate that large-scale cavities are periodically shed downstream, influenced alternatively by the re-entrant jet and shock wave mechanisms, resulting in periodic variations of both the number and Sauter mean diameter of microscale bubbles. During the quasi-periodic evolution process of sheet/cloud cavitation, the microbubbles scatter around the large scale cavities where have strong turbulence intensity and high vorticity, and the probability density function of discrete bubble diameter similarly conforms to Gamma distribution with the dominant bubble size between 30 and 70μm. During the growth of an attached cavity, the number density of bubbles follows a power-law scaling with a value of -5/3 on radius. For the other stages, the bubble size spectrum of smaller microbubbles exhibits a -2/3 power-law scaling when the diameter of the microbubble is less than 200 μm. For larger bubbles with a diameter greater than 300 μm, the bubble density is proportional to the bubble radius to the power of -6.

Numerical assessment of cavitation erosion risk on the Delft twisted hydrofoil using a hybrid Eulerian-Lagrangian strategy

Cavitation damage is a major threat to the life span of fluid machinery and has been a hot research issue in engineering for a long time. The current work aims to show the ability to assess cavitation erosion risk on the twisted hydrofoil using a hybrid Eulerian-Lagrangian solver. The volume of fluid method is used for the liquid-vapor interface of resolved vapor structures, while the discrete bubble model is utilized to track micro-scale unresolvable bubbles. A two-way coupling approach is introduced to enable the transition between resolved cavities and bubbles based on their volume, relative location, and shape. A Lagrangian erosion model considering the effect of asymmetric bubble collapse is proposed to predict cavitation erosion risk. Compared with the experimental result, the current model can accurately identify the high erosion risk regions on the hydrofoil surface. The distribution and intensity of the high impact pressures emitted from bubbles is quantitatively evaluated. In addition, the erosion sensitive zones at different stages of the one cloud cavitation cycle are determined and the hydrodynamic mechanisms of aggressive collapse events are analyzed in detail. The results reveal that the potential erosion risk is highest in areas where primary shedding occurs, which causes the macroscopic cavity to roll up. Bubbles with high impact pressures are mainly focused around the edges of the shedding cavity. The secondary shedding contributes to erosive structure in a limited middle angle of attack area, such that when the primary shedding U-shaped structure evolves and begins to collapse, it becomes less erosive, and only isolated points of erosion is found. Further investigation demonstrates that this is due to a transformation from U-shaped vortex to O-shaped vortex, moving the bubbles gradually away from the hydrofoil surface, thereby reducing the cavitation erosion risk.

Modelling underwater noise mitigation of a bubble curtain using a coupled-oscillator model

Bubbles can inhibit underwater sound transmission, owing to density and acoustic-impedance mismatches and associated reflection and absorption of sound waves, and also scattering due to bubble-acoustic resonances. Thus, bubble curtains have been used to reduce underwater noise propagation. For the present study, a discrete bubble model (DBM) utilising self-consistent coupled-oscillator theory was developed to model noise mitigation by bubble curtains. The DBM is inherently able to handle polydisperse-sized and anisotropically-distributed bubbles. The DBM was shown to accurately predict the bubble collective resonance frequencies of 1D line, 2D planar and 3D complex bubble cloud configurations. Subsequently, the model demonstrated that the collective resonance frequency decreases as the inter-bubble spacing was reduced in 2D planar bubble screens and 3D cylindrical bubble curtains. Modal analysis indicated that the fundamental mode in both configurations dominated the vibration of bubble clouds when subjected to an external acoustic source. The broadband acoustic pulse levels were able to be reduced by approximately 17 dB using realistic cylindrical bubble curtain configurations with a diameter of 5 m and two different gas volume fraction values of 1% and 2%. Variations in the sizes of bubbles within a curtain were shown to have a significant impact on curtain performance, whereas variations in inter-bubble spacing had no significant impact, implying that bubble-curtain operators should focus on controlling the size of the bubbles.

Investigation of cavitation noise using Eulerian-Lagrangian multiscale modeling

We have employed the large eddy simulation (LES) approach to investigate the cavitation noise characteristics of an unsteady cavitating flow around a NACA66 (National Advisory Committee for Aeronautics) hydrofoil by employing an Eulerian-Lagrangian based multiscale cavitation model. A volume of fluid (VOF) method simulates the large cavity, whereas a Lagrangian discrete bubble model (DBM) tracks the small bubbles. Meanwhile, noise is determined using the Ffowcs Williams-Hawkings equation (FW-H). Eulerian-Lagrangian analysis has shown that, in comparison to VOF, it is more effective in revealing microscopic characteristics of unsteady cavitating flows, including microscale bubbles, that are unresolvable around the cloud cavity, and their impact on the flow field. It is also evident that its evolution of cavitation features on the hydrofoil is more consistent with the experimental observations. The frequency of the maximum sound pressure level corresponds to the frequency of the main cavity shedding for the noise characteristics. Using the Eulerian-Lagrangian method to predict the noise signal, results show that the cavitation noise, generated by discrete bubbles due to their collapse, is mainly composed of high-frequency signals. In addition, the frequency of cavitation noise induced by discrete microbubbles is around 10 kHz. A typical characteristic of cavitation noise, including two intense pulses during the collapsing of the cloud cavity, is described, as well as the mechanisms that underlie these phenomena. The findings of this work provide for a fundamental understanding of cavitation and serve as a valuable reference for the design and intensification of hydrodynamic cavitation reactors.

Investigation on multiscale features of cavitating flow in convergent-divergent test section using Eulerian–Lagrangian method

Cavitating flows in fluid machineries always present complicated phase morphologies including a wide range of interface length scales. In the present work, an Eulerian–Lagrangian cavitation model is proposed for simulating the unsteady cavitating flow and investigating the multiscale cavitating flow features in a convergent-divergent test section. Most importantly, a two-way transition algorithm is newly developed mainly based on the shape of resolved cavities and the relative size of microscale bubbles compared to the cells, for taking the transition between microscale discrete bubbles and macroscale vapor cavities into consideration. The transient flow field of the two-phase system is described by the filtered Navier–Stokes equation and the sub-grid scale model. The volume of fluid (VOF) method and a discrete bubble model (DBM) are combined for simulating resolvable water-vapor interfaces and unresolvable discrete bubbles, respectively. Applying the proposed multiscale modeling framework, the dissipated bubbles caused by the collapse of shedding cavities that are unable to be represented by the VOF method are well predicted. The results show that the VOF-DBM model is less sensitive to the mesh resolution compared to the VOF method and it is found the dissipated bubbles are transporting around the vortex core region. Detailed bubble dynamics including DBM to VOF and VOF to DBM transitions and the interaction between macroscale cavities and microscale bubbles are well presented, indicating that the proposed VOF-DBM model is a promising approach to reveal the multiscale cavitation characteristics within limited computing resources.

Transport in turbulent, recurrent flows: Time-extrapolation and statistical symmetrization

We investigate bubble dynamics in a submerged double-jet as an example of weakly coupled, Lagrangian transport. The setup resembles continuous casting of steel at Reynolds numbers Re=136000 to Re=272000. Using short time series of flow fields obtained from large-eddy simulations (LES) with a discrete bubble model, we calculate a database of transport patterns and time-extrapolate them to long durations in the framework of recurrence CFD (rCFD). A dedicated averaging procedure along bubble trajectories allows us to study their transport with large steps at little numerical costs and monitor their spatial distribution. Besides time-extrapolation for fixed Re with a single database, we demonstrate how several time series can be combined to (i) enforce symmetries and (ii) approximately model conditions in between. We compare bubble volume fraction fields and total hold-up obtained from LES and from rCFD and find very good to satisfying agreement with speed up factors of more than 500.

Modeling of Bubbly Flow using a Combined Volume of Fluid and Discrete Bubble Model: Investigation on Interphase Forces

The gas-liquid two-phase flow with interfacial behaviors and bubble-liquid interactions is widely encountered in industrial processes such as that in gas-liquid reactors. The complicated phase structure makes it difficult to be modeled. The present work proposes a multi-scale mathematical model to simulate the bubbly flow in a square column. The volume of fluid (VOF) method is applied to treat the separated interface, and the discrete bubble model (DBM) is incorporated to handle the dynamics of dispersed bubbles. The hybrid model is validated against the benchmark experimental data to study the accuracy and suitability of the modeling framework for bubbly flows. And the influence of interphase forces on bubbly flow patterns and velocity profiles is investigated. It is found that the employment of both pressure gradient force and Ishii-Zuber drag model provides fairly good agreements with experimental data for velocity profiles.

Large eddy simulation of cavitating flow around a twist hydrofoil and investigation on force element evolution using a multiscale cavitation model

Cavitating flows always include micro-bubbles and macro-cavities over a wide range of length scale, and understanding the vorticity force evolution induced by cavitation is also a challenging issue. In the present work, a hybrid multiscale cavitation model incorporated with the large eddy simulation approach is applied for comprehensively revealing the unsteady cavitating flow features. The macroscale cavities are captured by the volume of fluid (VOF) method while the microscale bubbles are modeled using a discrete bubble model (DBM) in the Lagrangian way. Simulations of the cavitating flow on a twist hydrofoil are performed and validated against the experimental result. Good agreement in terms of cavity morphology and shedding frequency is obtained. Applying the multiscale cavitation model, the microscale cavitation bubbles that are unable to be predicted by the VOF method are well represented especially in the region near the trailing edge. To investigate the vorticity force evolution, the force element method is adopted to quantitatively analyze the change of force elements along with the evolution of the cavity. The overall effect of cavitation on the lift and drag forces is also investigated. Finally, the superiority of the multiscale VOF-DBM model is attested compared with the original VOF method.

Hybrid Front-Tracking and Discrete Bubble Model Approach for Bubbly Flows with a Free Surface

Hybrid Front-Tracking and Discrete Bubble Model Approach for Bubbly Flows with a Free Surface

Large eddy simulation of tip-leakage cavitating flow using a multiscale cavitation model and investigation on model parameters

For understanding tip-leakage cavitating flow features, the present work aims to implement a multiscale model to comprehensively reproduce the complicated phase structure. The volume of fluid (VOF) interface capturing method is applied to simulate macroscale cavities, while a discrete bubble model using the Lagrangian formulation is newly developed to take the microscale bubbles into account. The Schnerr–Sauer cavitation model is incorporated into the VOF model to calculate the mass transfer rate between phases from the macroscale point of view. For microscale bubbles, the simplified Rayleigh–Plesset equation is adopted to simulate the bubble growing and collapsing stages. An algorithm for coupling the approaches simulating macroscale cavities and microscale bubbles is also implemented to achieve multiscale simulation. Unsteady flow features are simulated using the large eddy simulation approach. The results show that an anti-diffusive compression scheme for the spatial discretization of volume fraction equation is relatively accurate for simulating the tip-leakage cavitating flow. Applying the multiscale model, the tip-leakage cavitating flow features with multiple time and space scales including the formation of glass cavity tube and the transport of bubble clouds can be revealed. Suitable model parameters including the coefficient of saturated pressure, and the bubble evaporation and condensation coefficients are studied.

Multiscale modeling of tip-leakage cavitating flows by a combined volume of fluid and discrete bubble model

For the tip-leakage cavitating flow, the existence of both interface and microbubbles at scales separated by orders of magnitude makes it difficult to be comprehensively reproduced by numerical modeling. This work aims to develop a multiscale model that directly resolves the large-scale cavities and models small discrete bubbles. The volume of fluid (VOF) method and the Schnerr–Sauer cavitation model are first adopted to represent the phase field of water and vapor from a macroscale point of view. A discrete bubble model (DBM) based on the Lagrangian formulation is then developed to simulate the microscale cavitation bubbles that are smoothed in the macroscale model due to the limited mesh resolution. The transition between DBM and VOF is also implemented to achieve the multiscale simulation. For modeling of the turbulence, the scale adaptive simulation approach is used. The tip-leakage cavitating flows induced by the NACA0009 hydrofoil under the conditions with different gaps according to the reported experiment are simulated. Results show that, using the proposed multiscale model, better agreements can be obtained compared with the macroscale model, and specific phenomena can be well revealed including the bubble nucleation, growth and collapse, the interaction between discrete bubbles and large-scale cavities.