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.     

Simulation of flow‐flexible body interactions with large deformation

A modified front‐tracking method was proposed for the simulation of fluid‐flexible body interactions with large deformations. A large deformable body was modeled by restructuring the body using a grid adaptation. Discontinuities in the viscosity at the fluid‐structure interface were incorporated by distributing the viscosity across the interface using an indicator function. A viscosity gradient field was created near the interface, and a smooth transition occurred between the structure and the fluid. The fluid motion was defined on the Eulerian domain and was solved using the fractional step method on a staggered Cartesian grid system. The solid motion was described by Lagrangian variables and was solved by the finite element method on an unstructured triangular mesh. The fluid motion and the structure motion were independently solved, and their interaction force was calculated using a feedback law. The interaction force was the restoring force of a stiff spring with damping, and spread from the Lagrangian coordinates to the Eulerian grid by a smoothed approximation of the Dirac delta function. In the numerical simulations, we validated the effect of the grid adaptation on the solid solver using a vibrating circular ring. The effects of the viscosity gradient field were verified by solving the deformation of a circular disk in a linear shear flow, including an elastic ring moving through a channel with constriction, deformation of a suspended catenary, and a swimming jellyfish. A comparison of the numerical results with the theoretical solutions was presented. Copyright © 2011 John Wiley & Sons, Ltd.     

A direct reinitialization approach of level‐set/splitting finite element method for simulating incompressible two‐phase flows

Computation of a moving interface by the level‐set (LS) method typically requires reinitialization of LS function. An inaccurate execution of reinitialization results in incorrect free surface capturing and thus errors such as mass gain/loss so that an accurate and robust reinitialization process in the LS method is essential for the simulation of free surface flows. In the present study, we pursue further development of the reinitialization process, which directly corrects the LS function after advection is carried out by using the normal vector to the interface instead of solving the reinitialization equation of hyperbolic type. The Taylor–Galerkin method is adopted to discretize the advection equation of the LS function and the P1P1 splitting finite element method is applied to solve the Navier–Stokes equation. The proposed algorithm is validated with the well‐known benchmark problems, i.e. stretching of a circular fluid element, time‐reversed single‐vortex, solitary wave propagation, broken dam flow and filling of a container. The simulation results of these flows are in good agreement with previously existing experimental and numerical results. Copyright © 2010 John Wiley & Sons, Ltd.     

Cahn–Hilliard modeling of particles suspended in two‐phase flows

In this paper, we present a model for the dynamics of particles suspended in two‐phase flows by coupling the Cahn–Hilliard theory with the extended finite element method (XFEM). In the Cahn–Hilliard model the interface is considered to have a small but finite thickness, which circumvents explicit tracking of the interface. For the direct numerical simulation of particle‐suspended flows, we incorporate an XFEM, in which the particle domain is decoupled from the fluid domain. To cope with the movement of the particles, a temporary ALE scheme is used for the mapping of field variables at the previous time levels onto the computational mesh at the current time level. By combining the Cahn–Hilliard model with the XFEM, the particle motion at an interface can be simulated on a fixed Eulerian mesh without any need of re‐meshing. The model is general, but to demonstrate and validate the technique, here the dynamics of a single particle at a fluid–fluid interface is studied. First, we apply a small disturbance on a particle resting at an interface between two fluids, and investigate the particle movement towards its equilibrium position. In particular, we are interested in the effect of interfacial thickness, surface tension, particle size and viscosity ratio of two fluids on the particle movement towards its equilibrium position. Finally, we show the movement of a particle passing through multiple layers of fluids. Copyright © 2011 John Wiley & Sons, Ltd.     

Direct numerical simulation of turbulent flow with evaporating droplets at high temperature

Direct numerical simulation (DNS) is used to investigate turbulent flows with evaporating fuel droplets. For the solution of the carrier gas fluid, the Eulerian method is employed, while for fuel droplets, the Lagrangian method is used. The two-way coupling interactions between the carrier fluid and the fuel droplets are described by the mass, momentum and energy transfers. Direct numerical simulation is performed by a compressible code, named S3D. In this paper, the effects of evaporating and non-evaporating droplets on isotropic turbulent flows are investigated. From the simulations it is found that for the case without evaporation, the inclusion of small droplets suppresses the turbulence, while evaporation usually enhances turbulence at later times for higher mass-loading ratios.     

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 arbitrary Lagrangian–Eulerian finite element approach to non‐steady state turbulent fluid flow with application to mould filling in casting

This paper presents a two‐dimensional Lagrangian–Eulerian finite element approach of non‐steady state turbulent fluid flows with free surfaces. The proposed model is based on a velocity–pressure finite element Navier–Stokes solver, including an augmented Lagrangian technique and an iterative resolution of Uzawa type. Turbulent effects are taken into account with the k–ε two‐equation statistical model. Mesh updating is carried out through an arbitrary Lagrangian–Eulerian (ALE) method in order to describe properly the free surface evolution. Three comparisons between experimental and numerical results illustrate the efficiency of the method. The first one is turbulent flow in an academic geometry, the second one is a mould filling in effective casting conditions and the third one is a precise confrontation to a water model. Copyright © 2000 John Wiley & Sons, Ltd.     

Evaluation of erosion in a two‐way coupled fluid–particle system

In this study, the effects of flow turbulence intensity, temperature, particle sizes and impinging velocity on erosion by particle impact are demonstrated numerically. Underlying turbulent flow on an Eulerian frame is described by the compressible Reynolds averaged Navier–Stokes equations with a RNG k–ε turbulence model. The particle trajectories and particle–wall interactions are evaluated by a Eulerian–Lagrangian approach in a two‐way coupling system. An erosion model considering material weight removal from surfaces is used to predict erosive wear. Computational validation against measured data is demonstrated satisfactorily. The analysis of erosion shows that the prevention of erosion is enhanced by increasing the effects of flow temperature and turbulence intensity and reducing particle inertial momentum. Copyright © 2001 John Wiley & Sons, Ltd.     

Identification of local extinction topology in axisymmetric bluff‐body diffusion flames with a reactedness‐mixture fraction presumed probability density function model

The effects of finite‐rate chemistry, such as partial extinctions and re‐ignitions, are investigated in turbulent non‐pre‐mixed reacting flows stabilized in the wake of an axisymmetric bluff‐body burner. A two‐dimensional large‐eddy simulation procedure is employed that uses a partial equilibrium/two‐scalar reactedness mixture fraction probability density function (PDF) combustion sub‐model, which is applied at the sub‐grid scale (SGS) level. An anisotropic sub‐grid eddy–viscosity and two equations for the SGS turbulence kinetic and scalar energies complete the SGS closure model. The scalar covariances required in the joint PDF formulation are obtained from an extended scale‐similarity assumption between the resolved and the sub‐grid fluctuations. Extinction due to strong turbulence/chemistry interactions is recognized with the help of a ‘critical’, locally variable, turbulent Damkohler number criterion, while transient localized extinctions and re‐ignitions are treated with a Lagrangian transport equation for a reactedness progress variable. Comparisons with available experimental data suggested that the formulated approach was capable of identifying the effects of large‐scale vortex structure activity, which were inherent in the reacting wake and dominant in the counterpart isothermal flows that otherwise would have been obscured if a standard time‐averaged procedure had been used. Additionally, the post‐extinction and re‐ignition behaviour and its time‐varying interaction with the large‐scale structure dynamics were more appropriately addressed within the context of the present time‐dependent method. Copyright © 2001 John Wiley & Sons, Ltd.     

A Lagrangian–Eulerian model of particle dispersion in a turbulent plane mixing layer

A Lagrangian–Eulerian model for the dispersion of solid particles in a two‐dimensional, incompressible, turbulent flow is reported and validated. Prediction of the continuous phase is done by solving an Eulerian model using a control‐volume finite element method (CVFEM). A Lagrangian model is also applied, using a Runge–Kutta method to obtain the particle trajectories. The effect of fluid turbulence upon particle dispersion is taken into consideration through a simple stochastic approach. Validation tests are performed by comparing predictions for both phases in a particle‐laden, plane mixing layer airflow with corresponding measurements formerly reported by other authors. Even though some limitations are detected in the calculation of particle dispersion, on the whole the validation results are rather successful. Copyright © 2002 John Wiley & Sons, Ltd.     

Two-phase micro- and macro-time scales in particle-laden turbulent channel flows

The micro-and macro-time scales in two-phaseturbulent channel flows are investigated using the direct numerical simulation and the Lagrangian particle trajectorymethods for the fluid-and the particle-phases,respectively.Lagrangian and Eulerian time scales of both phases are calculated using velocity correlation functions.Due to flowanisotropy,micro-time scales are not the same with the theoretical estimations in large Reynolds number(isotropic) turbulence.Lagrangian macro-time scales of particle-phaseand of fluid-phase seen by particles are both dependent onparticle Stokes number.The fluid-phase Lagrangian integral time scales increase with distance from the wall,longerthan those time scales seen by particles.The Eulerian integral macro-time scales increase in near-wall regions but decrease in out-layer regions.The moving Eulerian time scalesare also investigated and compared with Lagrangian integraltime scales,and in good agreement with previous measurements and numerical predictions.For the fluid particles themicro Eulerian time scales are longer than the Lagrangianones in the near wall regions,while away from the walls themicro Lagrangian time scales are longer.The Lagrangianintegral time scales are longer than the Eulerian ones.Theresults are useful for further understanding two-phase flowphysics and especially for constructing accurate predictionmodels of inertial particle dispersion.     

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.     

Finite‐element/level‐set/operator‐splitting (FELSOS) approach for computing two‐fluid unsteady flows with free moving interfaces

The present work is devoted to the study on unsteady flows of two immiscible viscous fluids separated by free moving interface. Our goal is to elaborate a unified strategy for numerical modelling of two‐fluid interfacial flows, having in mind possible interface topology changes (like merger or break‐up) and realistically wide ranges for physical parameters of the problem. The proposed computational approach essentially relies on three basic components: the finite element method for spatial approximation, the operator‐splitting for temporal discretization and the level‐set method for interface representation. We show that the finite element implementation of the level‐set approach brings some additional benefits as compared to the standard, finite difference level‐set realizations. In particular, the use of finite elements permits to localize the interface precisely, without introducing any artificial parameters like the interface thickness; it also allows to maintain the second‐order accuracy of the interface normal, curvature and mass conservation. The operator‐splitting makes it possible to separate all major difficulties of the problem and enables us to implement the equal‐order interpolation for the velocity and pressure. Diverse numerical examples including simulations of bubble dynamics, bifurcating jet flow and Rayleigh–Taylor instability are presented to validate the computational method. Copyright © 2004 John Wiley & Sons, Ltd.     

Unified one‐fluid formulation for incompressible flexible solids and multiphase flows: Application to hydrodynamics using the immersed structural potential method (ISPM)

In this paper, we present a two‐dimensional computational framework for the simulation of fluid‐structure interaction problems involving incompressible flexible solids and multiphase flows, further extending the application range of classical immersed computational approaches to the context of hydrodynamics. The proposed method aims to overcome shortcomings such as the restriction of having to deal with similar density ratios among different phases or the restriction to solve single‐phase flows. First, a variation of classical immersed techniques, pioneered with the immersed boundary method (IBM), is presented by rearranging the governing equations, which define the behaviour of the multiple physics involved. The formulation is compatible with the “one‐fluid” formulation for two‐phase flows and can deal with large density ratios with the help of an anisotropic Poisson solver. Second, immersed deformable structures and fluid phases are modelled in an identical manner except for the computation of the deviatoric stresses. The numerical technique followed in this paper builds upon the immersed structural potential method developed by the authors, by adding a level set–based method for the capturing of the fluid‐fluid interfaces and an interface Lagrangian‐based meshless technique for the tracking of the fluid‐structure interface. The spatial discretisation is based on the standard marker‐and‐cell method used in conjunction with a fractional step approach for the pressure/velocity decoupling, a second‐order time integrator, and a fixed‐point iterative scheme. The paper presents a wide d range of two‐dimensional applications involving multiphase flows interacting with immersed deformable solids, including benchmarking against both experimental and alternative numerical schemes.     

SPH modelling of depth‐limited turbulent open channel flows over rough boundaries

A numerical model based on the smoothed particle hydrodynamics method is developed to simulate depth‐limited turbulent open channel flows over hydraulically rough beds. The 2D Lagrangian form of the Navier–Stokes equations is solved, in which a drag‐based formulation is used based on an effective roughness zone near the bed to account for the roughness effect of bed spheres and an improved sub‐particle‐scale model is applied to account for the effect of turbulence. The sub‐particle‐scale model is constructed based on the mixing‐length assumption rather than the standard Smagorinsky approach to compute the eddy‐viscosity. A robust in/out‐flow boundary technique is also proposed to achieve stable uniform flow conditions at the inlet and outlet boundaries where the flow characteristics are unknown. The model is applied to simulate uniform open channel flows over a rough bed composed of regular spheres and validated by experimental velocity data. To investigate the influence of the bed roughness on different flow conditions, data from 12 experimental tests with different bed slopes and uniform water depths are simulated, and a good agreement has been observed between the model and experimental results of the streamwise velocity and turbulent shear stress. This shows that both the roughness effect and flow turbulence should be addressed in order to simulate the correct mechanisms of turbulent flow over a rough bed boundary and that the presented smoothed particle hydrodynamics model accomplishes this successfully. © 2016 The Authors International Journal for Numerical Methods in Fluids Published by John Wiley & Sons Ltd     

A two‐scale low‐Reynolds number turbulence model

In this study, a two‐scale low‐Reynolds number turbulence model is proposed. The Kolmogorov turbulence time scale, based on fluid kinematic viscosity and the dissipation rate of turbulent kinetic energy (ν, ε), is adopted to address the viscous effects and the rapid increasing of dissipation rate in the near‐wall region. As a wall is approached, the turbulence time scale transits smoothly from a turbulent kinetic energy based (k, ε) scale to a (ν, ε) scale. The damping functions of the low‐Reynolds number models can thus be simplified and the near‐wall turbulence characteristics, such as the ε distribution, are correctly reproduced. The proposed two‐scale low‐Reynolds number turbulence model is first examined in detail by predicting a two‐dimensional channel flow, and then it is applied to predict a backward‐facing step flow. Numerical results are compared with the direct numerical simulation (DNS) budgets, experimental data and the model results of Chien, and Lam and Bremhorst respectively. It is proved that the proposed two‐scale model indeed improves the predictions of the turbulent flows considered. Copyright © 2000 John Wiley & Sons, Ltd.     

Simulating surface tension with smoothed particle hydrodynamics

A method for simulating two‐phase flows including surface tension is presented. The approach is based upon smoothed particle hydrodynamics (SPH). The fully Lagrangian nature of SPH maintains sharp fluid–fluid interfaces without employing high‐order advection schemes or explicit interface reconstruction. Several possible implementations of surface tension force are suggested and compared. The numerical stability of the method is investigated and optimal choices for numerical parameters are identified. Comparisons with a grid‐based volume of fluid method for two‐dimensional flows are excellent. The methods presented here apply to problems involving interfaces of arbitrary shape undergoing fragmentation and coalescence within a two‐phase system and readily extend to three‐dimensional problems. Boundary conditions at a solid surface, high viscosity and density ratios, and the simulation of free‐surface flows are not addressed. Copyright © 2000 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.