A Cartesian grid method for two‐phase gel dynamics on an irregular domain

We develop a numerical method for simulating models of two‐phase gel dynamics in an irregular domain using a regular Cartesian grid. The models consist of transport equations for the volume fractions of the two phases, polymer network and solvent; coupled momentum equations for the two phases; and a volume‐averaged incompressibility constraint. Multigrid with Vanka‐type box relaxation scheme is used as a preconditioner for the Krylov subspace solver (GMRES) to solve the momentum and incompressibility equations. Ghost points are used to enforce no‐slip boundary conditions for the velocity field of each phase, and no‐flux boundary conditions for the volume fractions. The behavior of the new method, including its rate of convergence, is explored through numerical experiments for a problem in which strong phase separation develops from an initially (almost) homogeneous phase distribution. We also use the method to explore situations, motivated by biology, which show that imposed boundary velocities can cause substantial redistribution of network and solvent. Copyright © 2010 John Wiley & Sons, Ltd.     

A semi‐implicit finite difference method for non‐hydrostatic,free‐surface flows

In this paper a semi‐implicit finite difference model for non‐hydrostatic, free‐surface flows is analyzed and discussed. It is shown that the present algorithm is generally more accurate than recently developed models for quasi‐hydrostatic flows. The governing equations are the free‐surface Navier–Stokes equations defined on a general, irregular domain of arbitrary scale. The momentum equations, the incompressibility condition and the equation for the free‐surface are integrated by a semi‐implicit algorithm in such a fashion that the resulting numerical solution is mass conservative and unconditionally stable with respect to the gravity wave speed, wind stress, vertical viscosity and bottom friction. Copyright © 1999 John Wiley & Sons, Ltd.     

A high-resolution finite-difference method for simulating two-fluid,viscoelastic gel dynamics

An important class of gels are those composed of a polymer network and fluid solvent. The mechanical and rheological properties of these two-fluid gels can change dramatically in response to temperature, stress, and chemical stimulus. Because of their adaptivity, these gels are important in many biological systems, e.g. gels make up the cytoplasm of cells and the mucus in the respiratory and digestive systems, and they are involved in the formation of blood clots. In this study we consider a mathematical model for gels that treats the network phase as a viscoelastic fluid with spatially and temporally varying material parameters and treats the solvent phase as a viscous Newtonian fluid. The dynamics are governed by a coupled system of time-dependent partial differential equations which consist of transport equations for the two phases, constitutive equations for the viscoelastic stresses, two coupled momentum equations for the velocity fields of the two fluids, and a volume-averaged incompressibility constraint. We present a numerical method based on a staggered grid, second order finite-difference discretization of the momentum equations and a high-resolution unsplit Godunov method for the transport equations. The momentum and incompressibility equations are solved in a coupled manner with the Generalized Minimum Residual (GMRES) method using a multigrid preconditioner based on box-relaxation. We present results on the accuracy and robustness of the method together with an illustration of the interesting behavior of this gel model for the four-roll mill problem.     

Schwarz domain decomposition for the incompressible Navier–Stokes equations in general co‐ordinates

This paper describes a domain decomposition method for the incompressible Navier–Stokes equations in general co‐ordinates. Domain decomposition techniques are needed for solving flow problems in complicated geometries while retaining structured grids on each of the subdomains. This is the so‐called block‐structured approach. It enables the use of fast vectorized iterative methods on the subdomains. The Navier–Stokes equations are discretized on a staggered grid using finite volumes. The pressure‐correction technique is used to solve the momentum equations together with incompressibility conditions. Schwarz domain decomposition is used to solve the momentum and pressure equations on the composite domain. Convergence of domain decomposition is accelerated by a GMRES Krylov subspace method. Computations are presented for a variety of flows. Copyright © 2000 John Wiley & Sons, Ltd.     

High‐order implicit time integration for unsteady incompressible flows

The spatial discretization of unsteady incompressible Navier–Stokes equations is stated as a system of differential algebraic equations, corresponding to the conservation of momentum equation plus the constraint due to the incompressibility condition. Asymptotic stability of Runge–Kutta and Rosenbrock methods applied to the solution of the resulting index‐2 differential algebraic equations system is analyzed. A critical comparison of Rosenbrock, semi‐implicit, and fully implicit Runge–Kutta methods is performed in terms of order of convergence and stability. Numerical examples, considering a discontinuous Galerkin formulation with piecewise solenoidal approximation, demonstrate the applicability of the approaches and compare their performance with classical methods for incompressible flows. Copyright © 2011 John Wiley & Sons, Ltd.     

A 3D incompressible Navier–Stokes velocity–vorticity weak form finite element algorithm

The velocity–vorticity formulation is selected to develop a time‐accurate CFD finite element algorithm for the incompressible Navier–Stokes equations in three dimensions.The finite element implementation uses equal order trilinear finite elements on a non‐staggered hexahedral mesh. A second order vorticity kinematic boundary condition is derived for the no slip wall boundary condition which also enforces the incompressibility constraint. A biconjugate gradient stabilized (BiCGSTAB) sparse iterative solver is utilized to solve the fully coupled system of equations as a Newton algorithm. The solver yields an efficient parallel solution algorithm on distributed‐memory machines, such as the IBM SP2. Three dimensional laminar flow solutions for a square channel, a lid‐driven cavity, and a thermal cavity are established and compared with available benchmark solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

Convergence acceleration of segregated algorithms using dynamic tuning additive correction multigrid strategy

A convergence acceleration method based on an additive correction multigrid–SIMPLEC (ACM‐S) algorithm with dynamic tuning of the relaxation factors is presented. In the ACM‐S method, the coarse grid velocity correction components obtained from the mass conservation (velocity potential) correction equation are included into the fine grid momentum equations before the coarse grid momentum correction equations are formed using the additive correction methodology. Therefore, the coupling between the momentum and mass conservation equations is obtained on the coarse grid, while maintaining the segregated structure of the single grid algorithm. This allows the use of the same solver (smoother) on the coarse grid. For turbulent flows with heat transfer, additional scalar equations are solved outside of the momentum–mass conservation equations loop. The convergence of the additional scalar equations is accelerated using a dynamic tuning of the relaxation factors. Both a relative error (RE) scheme and a local Reynolds/Peclet (ER/P) relaxation scheme methods are used. These methodologies are tested for laminar isothermal flows and turbulent flows with heat transfer over geometrically complex two‐ and three‐dimensional configurations. Savings up to 57% in CPU time are obtained for complex geometric domains representative of practical engineering problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Unstructured finite volume discretization of two‐dimensional depth‐averaged shallow water equations with porosity

This paper deals with the numerical discretization of two‐dimensional depth‐averaged models with porosity. The equations solved by these models are similar to the classic shallow water equations, but include additional terms to account for the effect of small‐scale impervious obstructions which are not resolved by the numerical mesh because their size is smaller or similar to the average mesh size. These small‐scale obstructions diminish the available storage volume on a given region, reduce the effective cross section for the water to flow, and increase the head losses due to additional drag forces and turbulence. In shallow water models with porosity these effects are modelled introducing an effective porosity parameter in the mass and momentum conservation equations, and including an additional drag source term in the momentum equations. This paper presents and compares two different numerical discretizations for the two‐dimensional shallow water equations with porosity, both of them are high‐order schemes. The numerical schemes proposed are well‐balanced, in the sense that they preserve naturally the exact hydrostatic solution without the need of high‐order corrections in the source terms. At the same time they are able to deal accurately with regions of zero porosity, where the water cannot flow. Several numerical test cases are used in order to verify the properties of the discretization schemes proposed. Copyright © 2009 John Wiley & Sons, Ltd.     

A numerical continuous model for the hydrodynamics of fluid particle systems

In order to understand the hydrodynamic interactions that can appear in a fluid particle motion, an original method based on the equations governing the motion of two immiscible fluids has been developed. These momentum equations are solved for both the fluid and solid phases. The solid phase is assumed to be a fluid phase with physical properties, such as its behaviour can be assimilated to that of pseudo‐rigid particles. The only unknowns are the velocity and the pressure defined in both phases. The unsteady two‐dimensional momentum equations are solved by using a staggered finite volume formulation and a projection method. The transport of each particle is solved by using a second‐order explicit scheme. The physical model and the numerical method are presented, and the method is validated through experimental measurements and numerical results concerning the flow around a circular cylinder. Good agreement is observed in most cases. The method is then applied to study the trajectory of one settling particle initially off‐centred between two parallel walls and the corresponding wake effects. Different particle trajectories related to particulate Reynolds numbers are presented and commented. A two‐body interaction problem is investigated too. This method allows the simulation of the transport of particles in a dilute suspension in reasonable time. One of the important features of this method is the computational cost that scales linearly with the number of particles. Copyright © 1999 John Wiley & Sons, Ltd.     

A parallel finite element sliding mesh technique for the simulation of viscous flows in agitated tanks

A parallel sliding mesh algorithm for the finite element simulation of viscous fluid flows in agitated tanks is presented. Lagrange multipliers are used at the sliding interfaces to enforce the continuity between the fixed and moving subdomains. The novelty of the method consists of the coupled solution of the resulting velocity–pressure‐Lagrange multipliers system of equations by an ILU(0)‐QMR solver. A penalty parameter is introduced for both the interface and the incompressibility constraints to avoid pivoting problems in the ILU(0) algorithm. To handle the convective term, both the Newton–Raphson scheme and the semi‐implicit linearization are tested. A penalty parameter is introduced for both the interface and the incompressibility constraints to avoid the failure of the ILU(0) algorithm due to the lack of pivoting. Furthermore, this approach is versatile enough so that it allows partitioning of sliding and fixed subdomains if parallelization is required. Although the sliding mesh technique is fairly common in CFD, the main advantage of the proposed approach is its low computational cost due to the inexpensive and parallelizable calculations that involve preconditioned sparse iterative solvers. The method is validated for Couette and coaxial stirred tanks. Copyright © 2011 John Wiley & Sons, Ltd.     

Comparisons of CIP,compact and CIP‐CSL2 schemes for reproducing internal solitary waves

Six different models were evaluated for reproducing internal solitary waves which occur and propagate in a stratified flow field with a sharp interface. Three stages were used to compute internal solitary waves in a stratified field: (1) first‐phase computation of momentum equations, (2) second‐phase computation of momentum equations, which corresponds to computing the Poisson's equation, and (3) density computation. The six models discussed in this paper consisted of combinations of four different schemes, a three‐point combined compact difference scheme (CCD), a normal central difference scheme (CDS), a cubic‐polynomial interpolation (CIP), and an exactly conservative semi‐Lagrangian scheme (CIP‐CSL2). The residual cutting method was used to solve the Poisson's equation. Three tests were used to confirm the validity of the computations using KdV theory; i.e. the incremental wave speed and amplitude of internal solitary waves, the maximum horizontal velocity and amplitude, and the wave form. In terms of the shape of an internal solitary wave, using CIP for momentum equations was found to provide better performance than CCD. These results suggest one of the most appropriate scheme for reproducing internal solitary waves may be one in which CIP is used for momentum equations and CCD to solve the Poisson's equation. Copyright © 2005 John Wiley & Sons, Ltd.     

A numerical study of constitutive models endowed with Pom-Pom molecular attributes

The branched polymer melts are modeled respectively in this investigation by the existing XPP and PTT–XPP models, along with the proposed S-MDCPP (Single/Simplified Modified Double Convected Pom-Pom) model developed on the basis of the existing MDCPP model. A pressure stabilized mass equation is formulated with the finite increment calculus (FIC) process to restrain and further eliminate spurious oscillations of pressure field due to the incompressibility of fluids. The discrete elastic viscous stress splitting (DEVSS) technique is employed, in order to retain an elliptic contribution in the weak form of the momentum equation. An inconsistent streamline-upwind (SU) method is applied to spatially discretize the constitutive equations. The mass, momentum conservation and constitutive equations are discretized and solved by the iterative stabilized fractional step algorithm along with the Crank–Nicolson implicit difference scheme. Thus the finite elements with equal low-order interpolation approximations for velocity–pressure–stress variables can be devised to numerically simulate the viscoelastic contraction flows for branched LDPE melts. The influences of the three viscoelastic constitutive models and the branched arms at the end of the Pom-Pom molecule on the rheological behaviors occurring in this complex flow are discussed. The numerical results demonstrate that the proposed S-MDCPP model is capable of reproducing some properties similar to those predicted by the XPP model in high shear flow and, on the other hand, reproducing some properties similar to those predicted by the PTT–XPP model in high elongational flow. Furthermore, the proposed S-MDCPP model is capable of well identifying the macromolecule topological structures of branched polymer melts.     

A numerical scheme for Euler–Lagrange simulation of bubbly flows in complex systems

An Eulerian–Lagrangian approach is developed for the simulation of turbulent bubbly flows in complex systems. The liquid phase is treated as a continuum and the Navier–Stokes equations are solved in an unstructured grid, finite volume framework for turbulent flows. The dynamics of the disperse phase is modeled in a Lagrangian frame and includes models for the motion of each individual bubble, bubble size variations due to the local pressure changes, and interactions among the bubbles and with boundaries. The bubble growth/collapse is modeled by the Rayleigh–Plesset (RP) equation. Three modeling approaches are considered: (a) one‐way coupling, where the influence of the bubble on the fluid flow is neglected, (b) two‐way coupling, where the momentum‐exchange between the fluid and the bubbles is modeled, and (c) volumetric coupling, where the volumetric displacement of the fluid by the bubble motion and the momentum‐exchange are modeled. A novel adaptive time‐stepping scheme based on stability‐analysis of the non‐linear bubble dynamics equations is developed. The numerical approach is verified for various single bubble test cases to show second‐order accuracy. Interactions of multiple bubbles with vortical flows are simulated to study the effectiveness of the volumetric coupling approach in predicting the flow features observed experimentally. Finally, the numerical approach is used to perform a large‐eddy simulation in two configurations: (i) flow over a cavity to predict small‐scale cavitation and inception and (ii) a rising dense bubble plume in a stationary water column. The results show good predictive capability of the numerical algorithm in capturing complex flow features. Copyright © 2010 John Wiley & Sons, Ltd.     

A projection method for the spectral solution of non‐homogeneous and incompressible Navier–Stokes equations

This paper is devoted to the development of a parallel, spectral and second‐order time‐accurate method for solving the incompressible and variable density Navier–Stokes equations. The method is well suited for finite thickness density layers and is very efficient, especially for three‐dimensional computations. It is based on an exact projection technique. To enforce incompressibility, for a non‐homogeneous fluid, the pressure is computed using an iterative algorithm. A complete study of the convergence properties of this algorithm is done for different density variations. Numerical simulations showing, qualitatively, the capabilities of the developed Navier–Stokes solver for many realistic problems are presented. The numerical procedure is also validated quantitatively by reproducing growth rates from the linear instability theory in a three‐dimensional direct numerical simulation of an unstable, non‐homogeneous, flow configuration. It is also shown that, even in a turbulent flow, the spectral accuracy is recovered. Copyright © 2012 John Wiley & Sons, Ltd.     

An implicit two‐dimensional non‐hydrostatic model for free‐surface flows

An implicit finite volume model in sigma coordinate system is developed to simulate two‐dimensional (2D) vertical free surface flows, deploying a non‐hydrostatic pressure distribution. The algorithm is based on a projection method which solves the complete 2D Navier–Stokes equations in two steps. First the pressure term in the momentum equations is excluded and the resultant advection–diffusion equations are solved. In the second step the continuity and the momentum equation with only the pressure terms are solved to give a block tri‐diagonal system of equation with pressure as the unknown. This system can be solved by a direct matrix solver without iteration. A new implicit treatment of non‐hydrostatic pressure, similar to the lower layers is applied to the top layer which makes the model free of any hydrostatic pressure assumption all through the water column. This treatment enables the model to evaluate both free surface elevation and wave celerity more accurately. A series of numerical tests including free‐surface flows with significant vertical accelerations and nonlinear behaviour in shoaling zone are performed. Comparison between numerical results, analytical solutions and experimental data demonstrates a satisfactory performance. Copyright © 2006 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 Eulerian model for the motion of granular material with a large Stokes number in fluid flow

This study introduced a novel Euler–Euler approach for modeling granular multiphase flow. The motion of particles with a large Stokes number was investigated assuming that granular material has unilateral compressibility. Solid pressure in the momentum equations for granular multiphase flow was determined so that the unilateral incompressibility condition was satisfied. Using the continuity condition of the granular phase, the equation was rewritten in the optimal form to calculate the solid pressure. A discrete formulation of smoothed particle hydrodynamics was applied for the convective terms so that the discrete matrix was positive semidefinite for the convergence and the discretization for an unstructured mesh was allowed. Frictional stress was then determined from solid pressure and, by using the solid pressure and frictional stress, momentum equations for the granular phase were solved. The method was incorporated into ANSYS FLUENT by a UDF (user defined function). Model validation was performed through a comparison with two previous results, and efficacy of the proposed model was confirmed.     

A conservative semi‐implicit method for coupled surface–subsurface flows in regional scale

A semi‐implicit method for coupled surface–subsurface flows in regional scale is proposed and analyzed. The flow domain is assumed to have a small vertical scale as compared with the horizontal extents. Thus, after hydrostatic approximation, the simplified governing equations are derived from the Reynolds averaged Navier–Stokes equations for the surface flow and from the Darcy's law for the subsurface flow. A conservative free‐surface equation is derived from a vertical integral of the incompressibility condition and extends to the whole water column including both, the surface and the subsurface, wet domains. Numerically, the horizontal domain is covered by an unstructured orthogonal grid that may include subgrid specifications. Along the vertical direction a simple z‐layer discretization is adopted. Semi‐implicit finite difference equations for velocities and a finite volume approximation for the free‐surface equation are derived in such a fashion that, after simple manipulation, the resulting discrete free‐surface equation yields a single, well‐posed, mildly nonlinear system. This system is efficiently solved by a nested Newton‐type iterative method that yields simultaneously the pressure and a non‐negative fluid volume throughout the computational grid. The time‐step size is not restricted by stability conditions dictated by friction or surface wave speed. The resulting algorithm is simple, extremely efficient, and very accurate. Exact mass conservation is assured also in presence of wetting and drying dynamics, in pressurized flow conditions, and during free‐surface transition through the interface. A few examples illustrate the model applicability and demonstrate the effectiveness of the proposed algorithm. Copyright © 2015 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.     

Modified incompressible SPH method for simulating free surface problems

An incompressible smoothed particle hydrodynamics (I-SPH) formulation is presented to simulate free surface incompressible fluid problems. The governing equations are mass and momentum conservation that are solved in a Lagrangian form using a two-step fractional method. In the first step, velocity field is computed without enforcing incompressibility. In the second step, a Poisson equation of pressure is used to satisfy incompressibility condition. The source term in the Poisson equation for the pressure is approximated, based on the SPH continuity equation, by an interpolation summation involving the relative velocities between a reference particle and its neighboring particles. A new form of source term for the Poisson equation is proposed and also a modified Poisson equation of pressure is used to satisfy incompressibility condition of free surface particles. By employing these corrections, the stability and accuracy of SPH method are improved. In order to show the ability of SPH method to simulate fluid mechanical problems, this method is used to simulate four test problems such as 2-D dam-break and wave propagation.