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.     

A direct numerical simulation of axisymmetric liquid–gas two‐phase laminar flows in a pipe

An accurate finite‐volume‐based numerical method for the simulation of an isothermal two‐phase flow, consisting of a liquid slug translating in a non‐reacting gas in a circular pipe is presented. This method is built on a sharp interface concept and developed on an Eulerian Cartesian fixed‐grid with a cut‐cell scheme and marker points to track the moving interface. The unsteady, axisymmetric Navier–Stokes equations in both liquid and gas phases are solved separately. The mass continuity and momentum flux conditions are explicitly matched at the true surface phase boundary to determine the interface shape and movement. A quadratic curve fitting algorithm with marker points is used to yield smooth and accurate information of the interface curvatures. It is uniquely demonstrated for the first time with the current method that conservation of mass is strictly enforced for continuous infusion of flow into the domain of computation. The method has been used to compute the velocity and pressure fields and the deformation of the liquid core. It is also shown that the current method is capable of producing accurate results for a wide range of Reynolds number, Re, Weber number, We, and large property jumps at the interface. Copyright © 2010 John Wiley & Sons, Ltd.     

A high‐precision unstructured adaptive mesh technique for gas–liquid two‐phase flows

Adaptive mesh techniques are used widely in the numerical simulations of fluid flows, and the simulation results with high accuracies are obtained by appropriate mesh adaptations. However, gas–liquid two‐phase flows are still difficult to be simulated on adaptive meshes, especially on unstructured adaptive meshes, because the physical phenomena near gas–liquid interfaces are highly complicated and in general, not modeled appropriately on adaptive meshes. In this paper, a high‐precision unstructured adaptive mesh technique for gas–liquid two‐phase flows is developed and verified/validated. In the unstructured adaptive mesh technique, the PLIC algorithm is employed to simulate interfacial dynamic behaviors and, therefore, the reconstruction method for the interfaces in refined cells is developed, which satisfies the gas and liquid volume conservations and geometrical conservations of interfaces. In addition, the physics‐based consideration is performed on the momentum calculations near interfaces, and the calculation method with gas and liquid momentum conservations is developed. For verification, the slotted‐disk revolution problem is solved. As a result, the unstructured adaptive mesh technique succeeds in reproducing the slotted‐disk shape accurately and well maintaining the shape after one full‐revolution. The dam‐break problem is also simulated and the momentum conservative calculation method succeeds in providing physically appropriate results, which show good agreements with experimental data. Therefore, it is confirmed that the developed unstructured adaptive mesh technique is very efficient to simulate gas–liquid two‐phase flows accurately. Copyright © 2010 John Wiley & Sons, Ltd.     

Discretizing two‐dimensional complex fractured fields for incompressible two‐phase flow

A method is introduced to discretize irregular and complex two‐dimensional fractured media. The geometry of the fractured media is first analysed by searching and treating the complex configurations. Based on that, the method generated a good mesh quality and allows for including finer grids. An incompressible two‐phase flow problem is solved to compare the developed method and a public method based on the approximation of a 1D fracture by the edges of a 2D finite element grid of the porous media. The comparison showed that the developed method (i) represents better the fractured domain by maintaining the geometric integrity of input surfaces and geologic data, (ii) provides, for sample and complex fractured domains, excellent and more accurate results, and (iii) is much less sensitive to the grid sizes. Furthermore, the method has to be more efficient than the other methods for transport problems and has to provide better predictable results; this is mainly based on point (ii) and because the method produces optimal triangular grids. Copyright © 2009 John Wiley & Sons, Ltd.     

Preconditioned Krylov subspace methods used in solving two‐dimensional transient two‐phase flows

This paper investigates the performance of preconditioned Krylov subspace methods used in a previously presented two‐fluid model developed for the simulation of separated and intermittent gas–liquid flows. The two‐fluid model has momentum and mass balances for each phase. The equations comprising this model are solved numerically by applying a two‐step semi‐implicit time integration procedure. A finite difference numerical scheme with a staggered mesh is used. Previously, the resulting linear algebraic equations were solved by a Gaussian band solver. In this study, these algebraic equations are also solved using the generalized minimum residual (GMRES) and the biconjugate gradient stabilized (Bi‐CGSTAB) Krylov subspace iterative methods preconditioned with incomplete LU factorization using the ILUT(p, τ) algorithm. The decrease in the computational time using the iterative solvers instead of the Gaussian band solver is shown to be considerable. Copyright © 1999 John Wiley & Sons, Ltd.     

An unfitted interior penalty discontinuous Galerkin method for incompressible Navier–Stokes two‐phase flow

A discontinuous Galerkin method for the solution of the immiscible and incompressible two‐phase flow problem based on the nonsymmetric interior penalty method is presented. Therefore, the incompressible Navier–Stokes equation is solved for a domain decomposed into two subdomains with different values of viscosity and density as well as a singular surface tension force. On the basis of a piecewise linear approximation of the interface, meshes for both phases are cut out of a structured mesh. The discontinuous finite elements are defined on the resulting Cartesian cut‐cell mesh and may therefore approximate the discontinuities of the pressure and the velocity derivatives across the interface with high accuracy. As the mesh resolves the interface, regularization of the density and viscosity jumps across the interface is not required. This preserves the local conservation property of the velocity field even in the vicinity of the interface and constitutes a significant advantage compared with standard methods that require regularization of these discontinuities and cannot represent the jumps and kinks in pressure and velocity. A powerful subtessellation algorithm is incorporated to allow the usage of standard time integrators (such as Crank–Nicholson) on the time‐dependent mesh. The presented discretization is applicable to both the two‐dimensional and three‐dimensional cases. The performance of our approach is demonstrated by application to a two‐dimensional benchmark problem, allowing for a thorough comparison with other numerical methods. Copyright © 2012 John Wiley & Sons, Ltd.     

A further work on multi‐phase two‐fluid approach for compressible multi‐phase flows

This paper is to continue our previous work Niu (Int. J. Numer. Meth. Fluids 2001; 36 :351–371) on solving a two‐fluid model for compressible liquid–gas flows using the AUSMDV scheme. We first propose a pressure–velocity‐based diffusion term originally derived from AUSMDV scheme Wada and Liou (SIAM J. Sci. Comput. 1997; 18 (3):633—657) to enhance its robustness. The scheme can be applied to gas and liquid fluids universally. We then employ the stratified flow model Chang and Liou (J. Comput. Physics 2007; 225 :240–873) for spatial discretization. By defining the fluids in different regions and introducing inter‐phasic force on cell boundary, the stratified flow model allows the conservation laws to be applied on each phase, and therefore, it is able to capture fluid discontinuities, such as the fluid interfaces and shock waves, accurately. Several benchmark tests are studied, including the Ransom's Faucet problem, 1D air–water shock tube problems, 2D shock‐water column and 2D shock‐bubble interaction problems. The results indicate that the incorporation of the new dissipation into AUSM⁺‐up scheme and the stratified flow model is simple, accurate and robust enough for the compressible multi‐phase flows. Copyright © 2008 John Wiley & Sons, Ltd.     

Simulation of single‐ and two‐phase flows on sliding unstructured meshes using finite volume method

The paper presents an efficient finite volume method for unstructured grids with rotating sliding parts composed of arbitrary polyhedral elements for both single‐ and two‐phase flows. Mathematical model used in computations is based on the ensemble averaged conservation equations. These equations are solved for each phase and in case of single‐phase flow reduce to the transient Reynolds‐averaged Navier–Stokes (TRANS) equations. Transient flow induced by rotating impellers is thus resolved in time. The use of unstructured grids allows an easy and flexible meshing for the entire flow domain. Polyhedral cell volumes are created on the arbitrary mesh interface placed between rotating and static parts. Cells within the rotating parts move each time step and the new faces are created on the arbitrary interfaces only, while the rest of the domain remain ‘topologically’ unchanged. Implicit discretization scheme allows a wide range of time‐step sizes, which further reduce the computational effort. Special attention is given to the interpolation practices used for the reconstruction of the face quantities. Mass fluxes are recalculated at the beginning of each time step by using an interpolation scheme, which enhances the coupling between the pressure and velocity fields. The model has been implemented into the commercially available CFD code AVL SWIFT (AVL AST, SWIFT Manual 3.1, AVL List GmbH, Graz, Austria, 2002). Single‐phase flow in a mixing vessel stirred by a six‐bladed Rushton‐type turbine and two‐phase flow in aerated stirred vessel with the four‐blade Rushton impeller are simulated. The results are compared with the available experimental data, and good agreement is observed. The proposed algorithm is proved to be both stable and accurate for single‐phase as well as for the two‐phase flows calculations. Copyright 2004 John Wiley & Sons, Ltd.     

Simulation of two‐phase flow with moving immersed boundaries

A two‐dimensional multi‐phase model for immiscible binary fluid flow including moving immersed objects is presented. The fluid motion is described by the incompressible Navier–Stokes equation coupled with a phase‐field model based on van der Waals' free energy density and the Cahn–Hilliard equation. A new phase‐field boundary condition was implemented with minimization of the free energy in a direct way, to specifically improve the physical behavior of the contact line dynamics for moving immersed objects. Numerical stability and execution time were significantly improved by the use of the new boundary condition. Convergence toward the analytical solution was demonstrated for equilibrium contact angle, the Lucas–Washburn theory and Stefan's problem. The proposed model may be used for multi‐phase flow problems with moving boundaries of complex geometry, such as the penetration of fluid into a deformable, porous medium. Copyright © 2010 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.     

Numerical simulation of a single bubble by compressible two‐phase fluids

The present work deals with the numerical investigation of a collapsing bubble in a liquid–gas fluid, which is modeled as a single compressible medium. The medium is characterized by the stiffened gas law using different material parameters for the two phases. For the discretization of the stiffened gas model, the approach of Saurel and Abgrall is employed where the flow equations, here the Euler equations, for the conserved quantities are approximated by a finite volume scheme, and an upwind discretization is used for the non‐conservative transport equations of the pressure law coefficients. The original first‐order discretization is extended to higher order applying second‐order ENO reconstruction to the primitive variables. The derivation of the non‐conservative upwind discretization for the phase indicator, here the gas fraction, is presented for arbitrary unstructured grids. The efficiency of the numerical scheme is significantly improved by employing local grid adaptation. For this purpose, multiscale‐based grid adaptation is used in combination with a multilevel time stepping strategy to avoid small time steps for coarse cells. The resulting numerical scheme is then applied to the numerical investigation of the 2‐D axisymmetric collapse of a gas bubble in a free flow field and near to a rigid wall. The numerical investigation predicts physical features such as bubble collapse, bubble splitting and the formation of a liquid jet that can be observed in experiments with laser‐induced cavitation bubbles. Opposite to the experiments, the computations reveal insight to the state inside the bubble clearly indicating that these features are caused by the acceleration of the gas due to shock wave focusing and reflection as well as wave interaction processes. While incompressible models have been used to provide useful predictions on the change of the bubble shape of a collapsing bubble near a solid boundary, we wish to study the effects of shock wave emissions into the ambient liquid on the bubble collapse, a phenomenon that may not be captured using an incompressible fluid model. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical study of breaking waves by a two‐phase flow model

A two‐phase flow model, which solves the flow in the air and water simultaneously, is presented for modelling breaking waves in deep and shallow water, including wave pre‐breaking, overturning and post‐breaking processes. The model is based on the Reynolds‐averaged Navier–Stokes equations with the k ?ε turbulence model. The governing equations are solved by the finite volume method in a Cartesian staggered grid and the partial cell treatment is implemented to deal with complex geometries. The SIMPLE algorithm is utilised for the pressure‐velocity coupling and the air‐water interface is modelled by the interface capturing method via a high resolution volume of fluid scheme. The numerical model is validated by simulating overturning waves on a sloping beach and over a reef, and deep‐water breaking waves in a periodic domain, in which good agreement between numerical results and available experimental measurements for the water surface profiles during wave overturning is obtained. The overturning jet, air entrainment and splash‐up during wave breaking have been captured by the two‐phase flow model, which demonstrates the capability of the model to simulate free surface flow and wave breaking problems.Copyright © 2011 John Wiley & Sons, Ltd.     

On high‐resolution schemes for solving unsteady compressible two‐phase dilute viscous flows

A high‐resolution numerical scheme based on the MUSCL–Hancock approach is developed to solve unsteady compressible two‐phase dilute viscous flow. Numerical considerations for the development of the scheme are provided. Several solvers for the Godunov fluxes are tested and the results lead to the choice of an exact Riemann solver adapted for both gaseous and dispersed phases. The accuracy of the scheme is proven step by step through specific test cases. These simulations are for one‐phase viscous flows over a flat plate in subsonic and supersonic regimes, unsteady flows in a low‐pressure shock tube, two‐phase dilute viscous flows over a flat plate and, finally, two‐phase unsteady viscous flows in a shock tube. The results are compared with well‐established analytical and numerical solutions and very good agreement is achieved. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

Benchmark computations of diffuse interface models for two‐dimensional bubble dynamics

Diffuse interface models for incompressible two‐phase flow with large density ratios are tested on benchmark configurations for a two‐dimensional bubble rising in liquid columns. The benchmark quantities circularity, center of mass, and mean rise velocity are compared with reference solutions from Hysing et al. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical study of wave propagation in compressible two‐phase flow

We propose a new model and a solution method for two‐phase two‐fluid compressible flows. The model involves six equations obtained from conservation principles applied to a one‐dimensional flow of gas and liquid mixture completed by additional closure governing equations. The model is valid for pure fluids as well as for fluid mixtures. The system of partial differential equations with source terms is hyperbolic and has conservative form. Hyperbolicity is obtained using the principles of extended thermodynamics. Features of the model include the existence of real eigenvalues and a complete set of independent eigenvectors. Its numerical solution poses several difficulties. The model possesses a large number of acoustic and convective waves and it is not easy to upwind all of these accurately and simply. In this paper we use relatively modern shock‐capturing methods of a centred‐type such as the total variation diminishing (TVD) slope limiter centre (SLIC) scheme which solve these problems in a simple way and with good accuracy. Several numerical test problems are displayed in order to highlight the efficiency of the study we propose. The scheme provides reliable results, is able to compute strong shock waves and deals with complex equations of state. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical simulation of turbulent free surface flow with two‐equation k–ε eddy‐viscosity models

This paper presents a finite difference technique for solving incompressible turbulent free surface fluid flow problems. The closure of the time‐averaged Navier–Stokes equations is achieved by using the two‐equation eddy‐viscosity model: the high‐Reynolds k–ε (standard) model, with a time scale proposed by Durbin; and a low‐Reynolds number form of the standard k–ε model, similar to that proposed by Yang and Shih. In order to achieve an accurate discretization of the non‐linear terms, a second/third‐order upwinding technique is adopted. The computational method is validated by applying it to the flat plate boundary layer problem and to impinging jet flows. The method is then applied to a turbulent planar jet flow beneath and parallel to a free surface. Computations show that the high‐Reynolds k–ε model yields favourable predictions both of the zero‐pressure‐gradient turbulent boundary layer on a flat plate and jet impingement flows. However, the results using the low‐Reynolds number form of the k–ε model are somewhat unsatisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

Relaxation of the Navier–Stokes–Korteweg equations for compressible two‐phase flow with phase transition

The Navier–Stokes–Korteweg (NSK) system is a classical diffuse‐interface model for compressible two‐phase flow. However, the direct numerical simulation based on the NSK system is quite expensive and in some cases even not possible. We propose a lower‐order relaxation of the NSK system with hyperbolic first‐order part. This allows applying numerical methods for hyperbolic conservation laws and removing some of the difficulties of the original NSK system. To illustrate the new ansatz, we first present a local discontinuous Galerkin method in one and two spatial dimensions. It is shown that we can compute initial boundary value problems with realistic density ratios and perform stable computations for small interfacial widths. Second, we show that it is possible to construct a semi‐discrete finite‐volume scheme that satisfies a discrete entropy inequality. Copyright © 2015 John Wiley & Sons, Ltd.