A hybrid Lagrangian–Eulerian particle‐level set method for numerical simulations of two‐fluid turbulent flows

A coupled Lagrangian interface‐tracking and Eulerian level set (LS) method is developed and implemented for numerical simulations of two‐fluid flows. In this method, the interface is identified based on the locations of notional particles and the geometrical information concerning the interface and fluid properties, such as density and viscosity, are obtained from the LS function. The LS function maintains a signed distance function without an auxiliary equation via the particle‐based Lagrangian re‐initialization technique. To assess the new hybrid method, numerical simulations of several ‘standard interface‐moving’ problems and two‐fluid laminar and turbulent flows are conducted. The numerical results are evaluated by monitoring the mass conservation, the turbulence energy spectral density function and the consistency between Eulerian and Lagrangian components. The results of our analysis indicate that the hybrid particle‐level set method can handle interfaces with complex shape change, and can accurately predict the interface values without any significant (unphysical) mass loss or gain, even in a turbulent flow. The results obtained for isotropic turbulence by the new particle‐level set method are validated by comparison with those obtained by the ‘zero Mach number’, variable‐density method. For the cases with small thermal/mass diffusivity, both methods are found to generate similar results. Analysis of the vorticity and energy equations indicates that the destabilization effect of turbulence and the stability effect of surface tension on the interface motion are strongly dependent on the density and viscosity ratios of the fluids. Copyright © 2007 John Wiley & Sons, Ltd.     

3D two phase flows numerical simulations by SL‐VOF method

This paper describes the development of a semi‐Lagrangian computational method for simulating complex 3D two phase flows. The Navier–Stokes equations are solved separately in both fluids using a robust pseudo‐compressibility method able to deal with high density ratio. The interface tracking is achieved by the segment Lagrangian volume of fluid (SL‐VOF) method. The 2D SL‐VOF method using the concepts of VOF, piecewise linear interface calculation (PLIC) and Lagrangian advection of the interface is herein extended to 3D flows. Three different test cases of SL‐VOF 3D are presented for validation and comparison either with 2D flows or with other numerical methods. A good agreement is observed in each case. Copyright © 2004 John Wiley & Sons, Ltd.     

Critical evaluation of CFD codes for interfacial simulation of bubble‐train flow in a narrow channel

Computational fluid dynamics (CFD) codes that are able to describe in detail the dynamic evolution of the deformable interface in gas–liquid or liquid–liquid flows may be a valuable tool to explore the potential of multi‐fluid flow in narrow channels for process intensification. In the present paper, a computational exercise for co‐current bubble‐train flow in a square vertical mini‐channel is performed to investigate the performance of well‐known CFD codes for this type of flows. The computations are based on the volume‐of‐fluid method (VOF) where the transport equation for the liquid volumetric fraction is solved either by the methods involving a geometrical reconstruction of the interface or by the methods that use higher‐order difference schemes instead. The codes contributing to the present code‐to‐code comparison are an in‐house code and the commercial CFD packages CFX, FLUENT and STAR‐CD. Results are presented for two basic cases. In the first one, the flow is driven by buoyancy only, while in the second case the flow is additionally forced by an external pressure gradient. The results of the code‐to‐code comparison show that only the VOF method with interface reconstruction leads to physically sound and consistent results, whereas the use of difference schemes for the volume fraction equation shows some deficiencies. Copyright © 2007 John Wiley & Sons, Ltd.     

Algebraic coupled level set-volume of fluid method for surface tension dominant two-phase flows

In this study, an Algebraic Coupled Level Set-Volume of Fluid (A-CLSVOF) method is proposed for the simulation of interfacial capillary flows. The Volume of Fluid (VOF) method is utilized to incorporate one-way coupling with the Level Set (LS) function in order to improve the accuracy of the surface tension force calculation and to reduce the presence of parasitic currents. In this method, both VOF and LS functions are transported where the new volume fraction determines the interface seed position utilized by the reinitialization procedure for the LS field. Computational efficiency is enhanced through the use of the advected LS field, serving as an initial condition for the reinitialization procedure. The Hamilton–Godunov function is used with a second order (in space and time) discretization scheme to produce a signed distance function. In order to evaluate the performance of the methodology implemented here for capillary dominant flows, four different cases were considered: 1. static droplet; 2. capillary wave relaxation; 3. Rayleigh–Taylor instability and 4. droplet impact on a liquid pool. The simulations demonstrated reduction in spurious currents, accurately predicted capillary pressure and demonstrated overall improvements in efficiency and error reduction owing to the addition of LS advection. The present methodology is tested against previously published experimental results for a droplet impact on a deep liquid pool. Simulation results demonstrate excellent agreement with measured interface height up to and beyond the formation of a Rayleigh–Jet and break-up formation of a secondary daughter drop.     

A pressure correction‐volume of fluid method for simulation of two‐phase flows

A pressure correction method coupled with the volume of fluid (VOF) method is developed to simulate two‐phase flows. A volume fraction function is introduced in the VOF method and is governed by an advection equation. A modified monotone upwind scheme for a conservation law (modified MUSCL) is used to solve the solution of the advection equation. To keep the initial sharpness of an interface, a slope modification scheme is introduced. The continuum surface tension (CST) model is used to calculate the surface tension force. Three schemes, central‐upwind, Parker–Youngs, and mixed schemes, are introduced to compute the interface normal vector and the gradient of the volume fraction function. Moreover, a height function technique is applied to compute the local curvature of the interface. Several basic test problems are performed to check the order of accuracy of the present numerical schemes for computing the interface normal vector and the gradient of the volume fraction function. Three physical problems, two‐dimensional broken dam problem, static drop, and spurious currents, and three‐dimensional rising bubble, are performed to demonstrate the efficiency and accuracy of the pressure correction method. Copyright © 2010 John Wiley & Sons, Ltd.     

Finite‐volume‐type VOF method on dynamically adaptive quadtree grids

This paper describes a finite‐volume volume‐of‐fluid (VOF) method for simulating viscous free surface flows on dynamically adaptive quadtree grids. The scheme is computationally efficient in that it provides relatively fine grid resolution at the gas–liquid interface and coarse grid density in regions where flow variable gradients are small. Special interpolations are used to ensure volume flux conservation where differently sized neighbour cells occur. The numerical model is validated for advection of dyed fluid in unidirectional and rotating flows, and for two‐dimensional viscous sloshing in a rectangular tank. Copyright © 2004 John Wiley & Sons, Ltd.     

A Flexible Coupled Level Set and Volume of Fluid (flexCLV) method to simulate microscale two-phase flow in non-uniform and unstructured meshes

An innovative Flexible Coupled Level Set (LS) and Volume of Fluid (VOF) algorithm (flexCLV) to simulate two-phase flows at the microscale on unstructured and non-uniform meshes is proposed. The method combines the advantages of the VOF method in terms of mass conservation and the LS method in terms of accuracy of the surface tension implementation and can handle both 2D and 3D domains discretized by either structured hexaedra or unstructured tetrahedral grids with high aspect ratio elements, thus guaranteeing flexibility and robustness. The method is implemented within the VOF-based OpenFOAM’s solver interFoam, which is retained as the base algorithm for the interface advection, while the surface tension force is calculated by using the level set function reconstructed from the VOF’s fraction. The method is first validated in static flow conditions by simulating a circular bubble at equilibrium and then in dynamic flow conditions by studying a freely bubble rising in both 2D and 3D domains discretized by both structured and unstructured meshes. The proposed flexCLV algorithm is then used to simulate the dynamics of confined bubbles in circular microchannels in the low capillary number regime. 2D and 3D mesh grids with high aspect ratio elements are utilized to discretized the liquid film at the tube’s walls. The numerical results are compared with the available literature and simulations performed with the original interFoam solver in terms of bubble shape and velocity, thickness of the liquid film and amplitude of the bubble tail oscillations. Results compare very well with the experimental measurements and demonstrate the superior accuracy of the coupled flexCLV method with respect to the original VOF method when surface tension and accurate interface representation play a fundamental role. Importantly, the present study also provides a precious insight on the time-dependent patterns appearing on the bubble surface in the visco-inertial regime, which could be here investigated in detail.     

An immersed boundary method for incompressible flows using volume of body function

A simple and effective immersed boundary method using volume of body (VOB) function is implemented on unstructured Cartesian meshes. The flow solver is a second‐order accurate implicit pressure‐correction method for the incompressible Navier–Stokes equations. The domain inside the immersed body is viewed as being occupied by the same fluid as outside with a prescribed divergence‐free velocity field. Under this view a fluid–body interface is similar to a fluid–fluid interface encountered in the volume of fluid (VOF) method for the two‐fluid flow problems. The body can thus be identified by the VOB function similar to the VOF function. In fluid–body interface cells the velocity is obtained by a volume‐averaged mixture of body and fluid velocities. The pressure inside the immersed body satisfies the same pressure Poisson equation as outside. To enhance stability and convergence, multigrid methods are developed to solve the difference equations for both pressure and velocity. Various steady and unsteady flows with stationary and moving bodies are computed to validate and to demonstrate the capability of the current method. Copyright © 2005 John Wiley & Sons, Ltd.     

An efficient method for capturing free boundaries in multi‐fluid simulations

An easy‐to‐use front capturing method is devised by directly solving the transport equation for a volume of fluid (VOF) function. The key to this method is a semi‐Lagrangian conservative scheme, namely CIP_CSL3, recently proposed by the author. In the CIP_CSL3 scheme, the first‐order derivative of the interpolation polynomial at each cell centre is used to control the shape of the reconstructed profile. We show in the present paper that the first‐order derivative, which plays a crucial role in reconstructing the interpolation profile, can also be used to eliminate numerical diffusion. The resulting algorithm can be directly used to compute the VOF‐like function and retain the compact thickness of the moving interface in multi‐fluid simulations. No surface reconstruction based on the value of VOF function is required in the method, which makes it quite economical and easy to use. The presented method has been tested with various interfacial flows including pure rotation, vortex shearing, multi‐vortex deformation and the moving boundaries in real fluid as well. The method gives promising results to all computed problems. Copyright © 2003 John Wiley & Sons, Ltd.     

Simulating bubble dynamics in a buoyant system

Multiphase flows are critical components of many physical systems; however, numerical models of multiphase flows with large parameter gradients can be challenging. Here, two different numerical methods, volume of fluid (VOF) and smoothed particle hydrodynamics (SPH), are used to model the buoyant rise of isolated gas bubbles through quiescent fluids for a range of Bond and Reynolds numbers. The VOF is an Eulerian grid–based method, whereas the SPH is Lagrangian and mesh free. Each method has unique strengths and weaknesses, and a comparison of the two approaches as applied to multiphase phenomena has not previously been performed. The VOF and SPH simulations are compared, verified, and validated. Results using two-dimensional VOF and SPH simulations are similar to each other and are able to reproduce numerical benchmarks and experimental results for sufficiently large Morton and Reynolds numbers. It is also shown that at low Reynolds numbers, the two methods, SPH and VOF, diverge in the transient regime of the bubble rise. Regimes that require simulations capable of representing three-dimensional drag are identified as well as regimes in which results from VOF and SPH diverge.     

Multi‐material incompressible flow simulation using the moment‐of‐fluid method

This paper compares the numerical performance of the moment‐of‐fluid (MOF) interface reconstruction technique with Youngs, LVIRA, power diagram (PD), and Swartz interface reconstruction techniques in the context of a volume‐of‐fluid (VOF) based finite element projection method for the numerical simulation of variable‐density incompressible viscous flows. In pure advection tests with multiple materials MOF shows dramatic improvements in accuracy compared with the other methods. In incompressible flows where density differences determine the flow evolution, all the methods perform similarly for two material flows on structured grids. On unstructured grids, the second‐order MOF, LVIRA, and Swartz methods perform similarly and show improvement over the first‐order Youngs' and PD methods. For flow simulations with more than two materials, MOF shows increased accuracy in interface positions on coarse meshes. In most cases, the convergence and accuracy of the computed flow solution was not strongly affected by interface reconstruction method. Published in 2009 by John Wiley & Sons, Ltd.     

固体火箭燃气射流驱动液柱过程的CFD分析

固体火箭燃气射流驱动液柱过程会产生一个复杂的非稳态多相流场,为了研究液柱对固体火箭发动机工作过程中射流流场的降温效果,并揭示燃气冲击液柱的流动演化和气水之间的相互作用,利用FLUENT软件中耦合了液态水汽化方程的VOF多相流计算模型对燃气与液柱之间的耦合流动及相变过程进行了数值模拟,并与无液柱情况下射流流场的计算结果进行了对比分析。计算结果表明,当有液柱平衡体时射流流场中的压力、温度、速度波动幅度均减小,减弱了射流流场中的湍流脉动强度;液柱与燃气之间的汽化以及液柱的阻碍作用减小了射流流场的轴向发展位移,尾管后的完全发展射流流场核心区域内的压力峰值降低了0.9 MPa,温度峰值降低了503 K,速度峰值降低了291 m/s,验证了实验中液柱对燃气射流流场的降温效果。     

竖直流道宽度对气泡运动行为影响的数值模拟

用数值方法模拟了竖直通道宽度对气泡在液体中的非定常运动、变形以及传热特性的影响。在这个模拟中,界面跟踪采用了VOF方法,并采用PL IC进行界面重构。主流场计算采用有限容积方法将控制方程离散,其中扩散项采用中心差分格式,对流项采用一阶迎风格式。用成熟的S IM PLE算法求解N-S方程的速度与压力的耦合问题。引入CSF模型处理运动界面的表面张力。利用所编制的程序计算了竖直流道中的单个气泡的形状、运动特性以及气泡内外流场与传热特性,并对竖直通道宽度在不同情况下,对气泡的形状、运动特性以及传热特性进行了进一步的研究。得到了一系列有价值的结果,并与实验结果比较。表明数值模拟结果与实验结果吻合的较好。     

Volume preserving immersed boundary methods for two‐phase fluid flows

In this article, we propose a simple area‐preserving correction scheme for two‐phase immiscible incompressible flows with an immersed boundary method (IBM). The IBM was originally developed to model blood flow in the heart and has been widely applied to biofluid dynamics problems with complex geometries and immersed elastic membranes. The main idea of the IBM is to use a regular Eulerian computational grid for the fluid mechanics along with a Lagrangian representation of the immersed boundary. Using the discrete Dirac delta function and the indicator function, we can include the surface tension force, variable viscosity and mass density, and gravitational force effects. The principal advantage of the IBM for two‐phase fluid flows is its inherent accuracy due in part to its ability to use a large number of interfacial marker points on the interface. However, because the interface between two fluids is moved in a discrete manner, this can result in a lack of volume conservation. The idea of an area preserving correction scheme is to correct the interface location normally to the interface so that the area remains constant. Various numerical experiments are presented to illustrate the efficiency and accuracy of the proposed conservative IBM for two‐phase fluid flows. Copyright © 2011 John Wiley & Sons, Ltd.     

Validation of the S‐CLSVOF method with the density‐scaled balanced continuum surface force model in multiphase systems coupled with thermocapillary flows

Three numerical methods, namely, volume of fluid (VOF), simple coupled volume of fluid with level set (S‐CLSVOF), and S‐CLSVOF with the density‐scaled balanced continuum surface force (CSF) model, have been incorporated into OpenFOAM source code and were validated for their accuracy for three cases: (i) an isothermal static case, (ii) isothermal dynamic cases, and (iii) non‐isothermal dynamic cases with thermocapillary flow including dynamic interface deformation. Results have shown that the S‐CLSVOF method gives accurate results in the test cases with mild computation conditions, and the S‐CLSVOF technique with the density‐scaled balanced CSF model leads to accurate results in the cases of large interface deformations and large density and viscosity ratios. These show that these high accuracy methods would be appropriate to obtain accurate predictions in multiphase flow systems with thermocapillary flows. Copyright © 2016 John Wiley & Sons, Ltd.     

A mass‐conserving Level‐Set method for modelling of multi‐phase flows

A mass‐conserving Level‐Set method to model bubbly flows is presented. The method can handle high density‐ratio flows with complex interface topologies, such as flows with simultaneous occurrence of bubbles and droplets. Aspects taken into account are: a sharp front (density changes abruptly), arbitrarily shaped interfaces, surface tension, buoyancy and coalescence of droplets/bubbles. Attention is paid to mass‐conservation and integrity of the interface. The proposed computational method is a Level‐Set method, where a Volume‐of‐Fluid function is used to conserve mass when the interface is advected. The aim of the method is to combine the advantages of the Level‐Set and Volume‐of‐Fluid methods without the disadvantages. The flow is computed with a pressure correction method with the Marker‐and‐Cell scheme. Interface conditions are satisfied by means of the continuous surface force methodology and the jump in the density field is maintained similar to the ghost fluid method for incompressible flows. Copyright © 2005 John Wiley & Sons, Ltd.     

Optimization of unsteady incompressible Navier–Stokes flows using variational level set method

This paper presents the optimization of unsteady Navier–Stokes flows using the variational level set method. The solid–liquid interface is expressed by the level set function implicitly, and the fluid velocity is constrained to be zero in the solid domain. An optimization problem, which is constrained by the Navier–Stokes equations and a fluid volume constraint, is analyzed by the Lagrangian multiplier based adjoint approach. The corresponding continuous adjoint equations and the shape sensitivity are derived. The level set function is evolved by solving the Hamilton–Jacobian equation with the upwind finite difference method. The optimization method can be used to design channels for flows with or without body forces. The numerical examples demonstrate the feasibility and robustness of this optimization method for unsteady Navier–Stokes flows.Copyright © 2012 John Wiley & Sons, Ltd.     

NUMERICAL PREDICTION OF TWO-PHASE FLOW IN BUBBLE COLUMNS

A numerical model is described for the prediction of turbulent continuum equations for two-phase gas–liquid flows in bubble columns. The mathematical formulation is based on the solution of each phase. The two-phase model incorporates interfacial models of momentum transfer to account for the effects of virtual mass, lift, drag and pressure discontinuities at the gas–liquid interface. Turbulence is represented by means of a two-equation k–ϵ model modified to account for bubble-induced turbulence production. The numerical discretization is based on a staggered finite-volume approach, and the coupled equations are solved in a segregated manner using the IPSA method. The model is implemented generally in the multipurpose PHOENICS computer code, although the present appllications are restricted to two-dimensional flows. The model is applied to simulate two bubble column geometries and the predictions are compared with the measured circulation patterns and void fraction distributions.     

A new volume of fluid advection algorithm: the defined donating region scheme

This paper presents a new volume of fluid (VOF) advection algorithm, termed the defined donating region (DDR) scheme. The algorithm uses a linear piecewise method of free surface reconstruction, coupled to a fully multi‐dimensional method of cell boundary flux integration. The performance of the new scheme has been compared with the performance of a number of alternative schemes using translation, rotation and shear advection tests. The DDR scheme is shown to be generally more accurate than linear constant and flux limited schemes, and comparable with an alternative linear piecewise scheme. The DDR scheme conserves fluid volume rigorously without local redistribution algorithms, and generates no fluid ‘flotsam’ or other debris, making it ideal in applications where stability of the free surface interface is paramount. Copyright © 2001 John Wiley & Sons, Ltd.     

A conservative local interface sharpening scheme for the constrained interpolation profile method

A conservative local interface sharpening scheme has been developed for the constrained interpolation profile method with the conservative semi‐Lagrangian scheme, because the conservative semi‐Lagrangian scheme does not feature a mechanism to control the interface thickness, thus causing an increase of numerical error with the advance of the time step. The proposed sharpening scheme is based on the conservative level set method proposed by Olsson and Kreiss. However, because their method can cause excessive deformation of the free‐surface in certain circumstances, we propose an improvement of the method by developing a local sharpening technique. Several advection tests are presented to assess the correctness of the advection and the improved interface sharpening scheme. This is followed by the validations of dam‐breaking flow and the rising bubble flows. The mass of the fluid is exactly conserved and the computed terminal velocity of the rising bubble agrees well with the experiments compared with other numerical methods such as the volume of fluid method (VOF), the front tracking method, and the level set method. Copyright © 2011 John Wiley & Sons, Ltd.