A discontinuous Galerkin‐based sharp‐interface method to simulate three‐dimensional compressible two‐phase flow

A numerical method for the simulation of compressible two‐phase flows is presented in this paper. The sharp‐interface approach consists of several components: a discontinuous Galerkin solver for compressible fluid flow, a level‐set tracking algorithm to follow the movement of the interface and a coupling of both by a ghost‐fluid approach with use of a local Riemann solver at the interface. There are several novel techniques used: the discontinuous Galerkin scheme allows locally a subcell resolution to enhance the interface resolution and an interior finite volume Total Variation Diminishing (TVD) approximation at the interface. The level‐set equation is solved by the same discontinuous Galerkin scheme. To obtain a very good approximation of the interface curvature, the accuracy of the level‐set field is improved and smoothed by an additional P_NP_M‐reconstruction. The capabilities of the method for the simulation of compressible two‐phase flow are demonstrated for a droplet at equilibrium, an oscillating ellipsoidal droplet, and a shock‐droplet interaction problem at Mach 3. Copyright © 2015 John Wiley & Sons, Ltd.     

Collision between high and low viscosity droplets: Direct Numerical Simulations and experiments

Binary droplet collisions are of importance in a variety of practical applications comprising dispersed two-phase flows. In the present work we focus on the collision of miscible droplets, where one droplet is composed of a high viscous liquid and the other one is of lower viscosity. This kind of collisions take place in, for instance, spray drying processes when droplets with different solid content collide in recirculation zones. The aim of this paper is to investigate the details of the flow inside the colliding droplets. For this purpose, two prototype cases are considered, namely the collision of equal sized droplets and the collision between a small and highly viscous droplet and a bigger low viscous droplet. A new experimental method has been developed in order to visualize the penetration and mixing process of two colliding droplets, where a fluorescence marker is added to one liquid and the droplets are excited by a laser. The results show a delay in the coalescence which takes place during the initial stage of a collision of droplets with different viscosities. Direct Numerical Simulations based on the Volume-of-Fluid method are used to study these collisions and to allow for a more detailed inspection of the mixing process. The method is extended to consider a second liquid with a different viscosity. In order to reproduce the delay of coalescence, an algorithm for the temporal suppression of the coalescence is applied. A predictive simulation of the delay is not possible, because the extremely thin air gap separating the droplets cannot be resolved by the numerics. This approach is validated by comparison with experimental data. The results provide local field data of the flow inside the collision complex, showing in particular a pressure jump at the liquid–liquid interface although no surface tension is present. The detailed analysis of the terms in the momentum balance show that the pressure jump results from the viscosity jump at the liquid–liquid interface.     

DROPLET COLLISION AND COALESCENCE MODEL

A new droplet collision and coalescence model was presented,a quick-sort method for locating collision partners was also devised and based on theoretical and experimental results,further advancement was made to the droplet collision outcome. The advantages of the two implementations of smoothed particle hydrodynamics (SPH) method were used to limit the collision of droplets to a given number of nearest droplets and define the probability of coalescence,numerical simulations were carried out for model validation.Results show that the model presented is mesh-independent and less time consuming,it can not only maintains the system momentum conservation perfectly,but not susceptible to initial droplet size distribution as well.     

On dual-grid level-set method for contact line modeling during impact of a droplet on hydrophobic and superhydrophobic surfaces

In this paper, a numerical methodology for modeling contact line motion in a dual-grid level-set method (DGLSM) – solved on a uniform grid for interface which is twice that for the flow equations – is presented. A quasi-dynamic contact angle model – based on experimental inputs – is implemented to model the dynamic wetting of a droplet, impacting on a hydrophobic or a superhydrophobic surface. High-speed visualization experiments are also presented for the impact of a water droplet on hydrophobic surfaces, with non-bouncing at smaller and bouncing at larger impact velocity. The experimental results for temporal variation of the droplet shapes, wetted-diameter and maximum height of the droplet match very well with the DGLSM based numerical results. The validation of the numerical results is also presented with already published experimental results, for the non-bouncing on a hydrophobic and bouncing on a superhydrophobic surface, at a constant impact velocity. Finally, a qualitative as well as quantitative performance of the DGLSM as compared to the traditional level set method (LSM) is presented by considering our experimental results. The accuracy of the partially refined DGLSM is close to that of the fine-grid based LSM, at a computation cost which is close to that of the coarse-grid based LSM. The DGLSM is demonstrated as an improved LSM for the computational multi-fluid dynamics (CMFD) simulations involving contact line motion.     

Numerical modeling of the dynamics of a viscous-fluid droplet

A numerical model developed on the basis of the level set method is proposed. This makes it possible to describe both the nonlinear oscillations of a single viscous-fluid droplet and the fragmentation and coalescence processes. The Navier-Stokes equations written in “velocity-pressure” variables on a rectangular uniform grid in cylindrical coordinates are solved using the method of splitting into physical processes. Non-oscillating solutions for two-phase media with a characteristic density ratio of less than 10⁻³ and Re > 1000 are obtained. The possibilities of the approach proposed are demonstrated with the reference to the problem of a droplet falling from a capillary (detachment from the capillary, formation of a “Plato ball”, droplet motion, collision with a plane wall, droplet oscillations on the wall, and droplet spreading). A comparison of the numerical results with the known calculation models and experimental data shows satisfactory agreement with respect to both the phases and the shape of the droplet.     

Numerical simulation of droplet ejection of thermal inkjet printheads

In this study, we present a method to predict the droplet ejection in thermal inkjet printheads including the growth and collapse of a vapor bubble and refill of the firing chamber. The three‐dimensional Navier–Stokes equations are solved using a finite‐volume approach with a fixed Cartesian mesh. The piecewise‐linear interface calculation‐based volume‐of‐fluid method is employed to track and reconstruct the ink–air interface. A geometrical computation based on Lagrangian advection is used to compute the mass flux and advance the interface. A simple and efficient model for the bubble dynamics is employed to model the effect of ink vapor on the adjacent ink liquid. To solve the surface tension‐dominated flow accurately, a hierarchical curvature‐estimation method is proposed to adapt to the local grid resolution. The numerical methods mentioned earlier have been implemented in an internal simulation code, CFD3. The numerical examples presented in the study show good performance of CFD3 in prediction of surface tension‐dominated free‐surface flows, for example, droplet ejection in thermal inkjet printing. Currently, CFD3 is used extensively for printhead development within Hewlett‐Packard. Copyright © 2015 John Wiley & Sons, Ltd.     

A combined level set/ghost cell immersed boundary representation for floating body simulations

A six degrees of freedom (6DOF) algorithm is implemented in the open‐source CFD code REEF3D. The model solves the incompressible Navier–Stokes equations. Complex free surface dynamics are modeled with the level set method based on a two‐phase flow approach. The convection terms of the velocities and the level set method are treated with a high‐order weighted essentially non‐oscillatory discretization scheme. Together with the level set method for the free surface capturing, this algorithm can model the movement of rigid floating bodies and their interaction with the fluid. The 6DOF algorithm is implemented on a fixed grid. The solid‐fluid interface is represented with a combination of the level set method and ghost cell immersed boundary method. As a result, re‐meshing or overset grids are not necessary. The capability, accuracy, and numerical stability of the new algorithm is shown through benchmark applications for the fluid‐body interaction problem. Copyright © 2016 John Wiley & Sons, Ltd.     

A Coupled Level Set and Volume of Fluid method for automotive exterior water management applications

Motivated by the need for practical, high fidelity, simulation of water over surface features of road vehicles a Coupled Level Set Volume of Fluid (CLSVOF) method has been implemented into a general purpose CFD code. It has been implemented such that it can be used with unstructured and non-orthogonal meshes. The interface reconstruction step needed for CLSVOF has been implemented using an iterative ‘clipping and capping’ algorithm for arbitrary cell shapes and a re-initialisation algorithm suitable for unstructured meshes is also presented. Successful verification tests of interface capturing on orthogonal and tetrahedral meshes are presented. Two macroscopic contact angle models have been implemented and the method is seen to give very good agreement with experimental data for a droplet impinging on a flat plate for both orthogonal and non-orthogonal meshes. Finally the flow of a droplet over a round edged channel is simulated in order to demonstrate the ability of the method developed to simulate surface flows over the sort of curved geometry that makes the use of a non-orthogonal grid desirable.     

Simulations of free surface flows with implementation of surface tension and interface sharpening in the two-fluid model

Two-fluid model used for free surface flows with large characteristic scales is improved; the smeared interface is sharpened with conservative level set method and the surface tension force with wetting angle is implemented. Surface tension force is split between two phases with several models. Detailed analysis showed the splitting of surface tension force with volume averaging as the most appropriate. The improved two-fluid model with interface sharpening and implemented surface tension is validated on several test cases. The pressure jump over a droplet interface test case showed that the pressure jump in simulation converges with grid refinement to the analytical one. The parasitic currents in simulation are one order of magnitude larger than in simulation with volume of fluid model. In the oscillating droplet test case the time period of oscillating droplet with initially ellipsoid or square shape is similar to the analytical time period. In the rising bubble test case, the rising bubble position, terminal velocity, and circularity are similar to the one observed in simulations with level set model. The wetting angle is implemented in the two-fluid model with interface sharpening and surface tension force. Model is tested in the simulation of droplet in contact with wall with different wetting angles.     

Modeling of binary droplet collisions for application to inter-impingement sprays

In this paper, we focused on modeling the collision phenomenon between two liquid droplets for application in spray simulations. It has been known that the existing O’Rourke collision model widely used in CFD codes is inaccurate in determining collision outcomes and droplet behavior. In addition, since the collision probability of the model follows a statistical approach involving computational cell geometry, the prediction results should be strongly dependent on the cell size. As a result, to more accurately calculate droplet collisions, the technique for predicting the droplet velocity and its direction after collision must be extended for use in spray modeling. Further, it is also necessary to consider all the possible collision outcomes, such as bouncing, stretching separation, reflexive separation and coalescence. Therefore, this paper describes the appropriateness of a composite concept for modeling collision outcomes and the implementation of deterministic collision algorithms into a multidimensional CFD code for the calculation of post-collisional droplet movements. Furthermore, the existing model does not consider the formation of satellite droplets. For this reason, our present modeling concept includes a fragmenting droplet collision model. Using the present model, we have validated the collision interactions between liquid droplets under high Weber number conditions by comparing our calculations with experimental results from a binary droplet collision. This paper also deals with the application of the model to inter-impingement sprays by analyzing the atomization characteristics, such as mean droplet size and velocity, spray tip penetrations and spray-shapes of the impinging spray using the suggested collision algorithms and then comparing the results with available experimental data.     

Numerical prediction of droplet dynamics in turbulent flow,using the level set method

In the present article, the droplet dynamics in turbulent flow is numerically predicted. The modelling is based on an interfacial marker-level set (IMLS) method, coupled with the Reynolds-averaged Navier–Stokes (RANS) equations to predict the dynamics of turbulent two-phase flow. The governing equations for time-dependent, two-dimensional and incompressible two-phase flow are described in both phases and solved separately using a control volume approach on structured cell-centred collocated grids. The topological changes of the interface are predicted by applying the level set approach. The kinematic and dynamic conditions on the interface separating the two phases are satisfied. The numerical method proposed is validated against a well-known computational fluid dynamics problem. Further, the deformation and breakup of a single droplet either suddenly moved in air or exposed to turbulent stream are numerically investigated. In general, the developed numerical method demonstrates remarkable capability in predicting the characteristics of complex turbulent two-phase flows.     

Coupled ghost fluid/two‐phase level set method for curvilinear body‐fitted grids

A coupled ghost fluid/two‐phase level set method to simulate air/water turbulent flow for complex geometries using curvilinear body‐fitted grids is presented. The proposed method is intended to treat ship hydrodynamics problems. The original level set method for moving interface flows was based on Heaviside functions to smooth all fluid properties across the interface. We call this the Heaviside function method (HFM). The HFM requires fine grids across the interface. The ghost fluid method (GFM) has been designed to explicitly enforce the interfacial jump conditions, but the implementation of the jump conditions in curvilinear grids is intricate. To overcome these difficulties a coupled GFM/HFM method was developed in which approximate jump conditions are derived for piezometric pressure and velocity and pressure gradients based on exact continuous velocity and stress and jump in momentum conditions with the jump in density maintained but continuity of the molecular and turbulent viscosities imposed. The implementation of the ghost points is such that no duplication of memory storage is necessary. The level set method is adopted to locate the air/water interface, and a fast marching method was implemented in curvilinear grids to reinitialize the level set function. Validations are performed for three tests: super‐ and sub‐critical flow without wave breaking and an impulsive plunging wave breaking over 2D submerged bumps, and the flow around surface combatant model DTMB 5512. Comparisons are made against experimental data, HFM and single‐phase level set computations. The proposed method performed very well and shows great potential to treat complicated turbulent flows related to ship flows. Copyright © 2007 John Wiley & Sons, Ltd.     

An improved diffuse interface method for three-dimensional multiphase flows with complex interface deformation

An improved diffuse interface (DI) method is proposed for accurately capturing complex interface deformation in simulations of three-dimensional (3D) multiphase flows. In original DI methods, the unphysical phenomenon of interface thickening or blurring can affect the accuracy of numerical simulations, especially for flows with large density ratio and high Reynolds number. To remove this drawback, in this article, an interface-compression term is introduced into the Cahn-Hilliard equation to suppress the interface dispersion. The additional term only takes effect in the interface region and works normal to the interface. The difference of the current method from the previous work is that the compression rate can be adjusted synchronously according to the magnitude of local vorticity, which is strongly correlated to the interface dispersion and changes with the computational time and interface position. Numerical validations of the proposed method are implemented by simulating problems of Laplace law, Rayleigh-Taylor instability, bubble rising in a channel, and binary droplet collision. The obtained results agree well with the analytical solutions and published data. The numerical results show that the phenomenon of interface dispersion is suppressed effectively and the tiny interfacial structures in flow field can be captured accurately.     

A numerical investigation of droplet impact on a heated wall in the film boiling regime

In this research, a novel approach is proposed to simulate the impingement of a water droplet on a superheated wall by solving conservation equations of mass, momentum and energy. Due to high temperature of the surface, a vapor layer is formed between droplet and surface. The vapor layer is captured using a very fine mesh near the surface. The level set method is applied for interface tracking and appropriate jump conditions are imposed at the interface by the ghost fluid method. The proposed algorithm is validated by comparing numerical results to the available experimental and analytical solutions. The effect of impact velocity, surfactant and polymer additives on the heat removal is studied. Simulations show that an increase in impact velocity enhances the heat removal. On the other hand, polymer additive has no significant effect while surfactant enhances dissipated heat by increasing the contact time or even by sticking the droplet on the surface.     

A coupled level set and volume-of-fluid method for sharp interface simulation of plunging breaking waves

A coupled level set and volume-of-fluid (CLSVOF) method is implemented for the numerical simulations of interfacial flows in ship hydrodynamics. The interface is reconstructed via a piecewise linear interface construction scheme and is advected using a Lagrangian method with a second-order Runge–Kutta scheme for time integration. The level set function is re-distanced based on the reconstructed interface with an efficient re-distance algorithm. This level set re-distance algorithm significantly simplifies the complicated geometric procedure and is especially efficient for three-dimensional (3D) cases. The CLSVOF scheme is incorporated into CFDShip-Iowa version 6, a sharp interface Cartesian grid solver for two-phase incompressible flows with the interface represented by the level set method and the interface jump conditions handled using a ghost fluid methodology. The performance of the CLSVOF method is first evaluated through the numerical benchmark tests with prescribed velocity fields, which shows superior mass conservation property over the level set method. With combination of the flow solver, a gas bubble rising in a viscous liquid and a water drop impact onto a deep water pool are modeled. The computed results are compared with the available numerical and experimental results, and good agreement is obtained. Wave breaking of a steep Stokes wave is also modeled and the results are very close to the available numerical results. Finally, plunging wave breaking over a submerged bump is simulated. The overall wave breaking process and major events are identified from the wave profiles of the simulations, which are qualitatively validated by the complementary experimental data. The flow structures are also compared with the experimental data, and similar flow trends have been observed.     

Modeling droplet collision and coalescence in an icing wind tunnel and the influence of these processes on droplet size distribution

A theoretical model of a two-phase air/dispersed water spray flow in an icing wind tunnel is presented here. The mutual interactions taking place within the dispersed phase known as binary droplet collisions, as well as gravitational sedimentation are considered. Where large droplets and low air stream velocities are concerned, the effect of gravity on droplet dynamics is considerable. Gravity causes the vertical deflection of droplet trajectories and an increase in liquid water content (LWC) in the bottom half of the wind tunnel. Droplet collision tends to influence the size, trajectory and velocity of droplets thus affecting the characteristics of the flow and, thereby, the formation of ice on the object placed in the wind tunnel. The present model simulates droplet motion and droplet collision in an icing wind tunnel, where it may be observed that bouncing, stable coalescence, or coalescence followed by separation are the possible outcomes of collision. In the theoretical examination, firstly, the effect of gravity on the vertical deflection of droplet trajectories and on the vertical distribution of the LWC near the icing object are taken into account, while droplet collision is disregarded. Then both factors are considered and collision outcome is determined together with the size and velocity of post-collision droplets. The initial droplet size distribution (DSD), as it occurs at the nozzle outlet, is estimated by a curve in accordance with previous experimental results. The DSD is determined theoretically near the icing object, which makes it possible to calculate the median volume diameter and the LWC of the aerosol cloud. The simulation results with regard to the LWC are compared to the experimental results obtained in this research and a satisfactory qualitative coincidence is to be found between them.     

Numerical modeling and experimental measurements of a high speed solid-cone water spray for use in fire suppression applications

Experimental measurements and numerical simulations of a high-speed water spray are presented. The numerical model is based on a stochastic separated flow technique that includes submodels for droplet dynamics, heat and mass transfer, and droplet–droplet collisions. Because the spray characteristics near the nozzle are difficult to ascertain, a new method for initialization of particle diameter size is developed that assumes a Rosin–Rammler distribution for droplet size, which correctly reproduces experimentally measured Sauter and arithmetic mean diameters. By relating the particle initialization to lower moments of the droplet statistics, it is possible to take advantage of measurements without substantial penalties associated with the greater experimental uncertainty of individual droplet measurements. Overall, very good agreement is observed in the comparisons of experimental measurements to computational predictions for the streamwise development of mean drop size and velocity. In addition, the importance of modeling droplet–droplet collisions is highlighted with comparison of selected droplet–droplet collision models.     

Level‐set based numerical simulation of a migrating and dissolving liquid drop in a cylindrical cavity

In the present paper the dissolution of a binary liquid drop having a miscibility gap and migrating due to thermo‐solutal capillary convection in a cylindrical cavity is studied numerically. The interest in studying this problem is twofold. From a side, in the absence of gravity, capillary migration is one of the main physical mechanisms to set into motion dispersed liquid phases and from the other side, phase equilibria of multi‐component liquid systems, ubiquitous in applications, often exhibit a miscibility gap. The drop capillary migration is due to an imposed temperature gradient between the cavity top and bottom walls. The drop dissolution is due to the fact that initial composition and volume values, and thermal boundary conditions are only compatible with a final single phase equilibrium state. In order to study the drop migration along the cavity and the coupling with dissolution, a previously developed planar two‐dimensional code is extended to treat axis‐symmetric geometries. The code is based on a finite volume formulation. A level‐set technique is used for describing the dynamics of the interface separating the different phases and for mollifying the interface discontinuities between them. The level‐set related tools of redistancing and off‐interface extension are used to enhance code resolution in the critical interface region. Migration speeds and volume variations are determined for different drop radii. Copyright © 2004 John Wiley & Sons, Ltd.     

含气泡油滴撞击油膜壁面时气泡的变形与破裂

油--气润滑过程中润滑油液滴受高速气流扰动易形成含气泡油滴,微气泡将对油滴撞击壁面时的运动过程以及壁面油膜 层的形成质量产生重要影响. 基于耦合的水平集--体积分数 方法,对含气泡油滴撞击油膜壁面行为进行数值模拟研究, 考察含气泡油滴撞击油膜壁面时气泡的变形运动过程,探讨气泡破裂的动力学机制,分析气泡大小、碰撞速度和液体黏度等因素对含气 泡油滴撞壁过程中气泡变形特征参数的影响规律. 研究表明:含气泡油滴撞击油膜壁面后气泡会发生变形,并破裂形成膜液滴;气泡随同 液滴运动过程中,气泡内外压力和速度梯度变化是使气泡发生破裂的主要诱因. 气泡大小对气泡破裂方式影响较大,气泡较小时发生单 点破裂,而气泡较大时更容易发生多处破裂. 不同大小气泡受力差异较大,气泡大小与破裂发生时刻没有明显相关性. 碰撞速度和液体 黏度对气泡的变形、破裂和破裂发生时刻都具有一定的影响. 碰撞速度越大,油滴动能越大,更容易产生气泡变形和破裂现象. 液体黏 度增大,在油滴撞壁运动前期促进气泡变形,而在运动后期可以阻延气泡破裂行为发生.      

Development of level set method with good area preservation to predict interface in two‐phase flows

A two‐step conservative level set method is proposed in this study to simulate the gas/water two‐phase flow. For the sake of accuracy, the spatial derivative terms in the equations of motion for an incompressible fluid flow are approximated by the coupled compact scheme. For accurately predicting the modified level set function, the dispersion‐relation‐preserving advection scheme is developed to preserve the theoretical dispersion relation for the first‐order derivative terms shown in the pure advection equation cast in conservative form. For the purpose of retaining its long‐time accurate Casimir functionals and Hamiltonian in the transport equation for the level set function, the time derivative term is discretized by the sixth‐order accurate symplectic Runge–Kutta scheme. To resolve contact discontinuity oscillations near interface, nonlinear compression flux term and artificial damping term are properly added to the second‐step equation of the modified level set method. For the verification of the proposed dispersion‐relation‐preserving scheme applied in non‐staggered grids for solving the incompressible flow equations, three benchmark problems have been chosen in this study. The conservative level set method with area‐preserving property proposed for capturing the interface in incompressible fluid flows is also verified by solving the dam‐break, Rayleigh–Taylor instability, bubble rising in water, and droplet falling in water problems. Good agreements with the referenced solutions are demonstrated in all the investigated problems. Copyright © 2010 John Wiley & Sons, Ltd.