extension of finite volume compressible flow solvers to multi dimensional variable density zero mach number flows when attempting to compute unsteady variable density flows at very small or zero mach number using a standard finite volume compressible flow solver one faces at least the following difficulties i spatial pressure variations vanish as the mach numberm

consistent projection methods for variable density incompressible navier stokes equations with continuous surface forces on a rectangular collocated mesh two consistent projection methods of second order temporal and spatial accuracy have been developed on a rectangular collocated mesh for variable density navier stokes equations with a continuous surface force instead of the original projection methods denoted as algorithms i and ii in this paper in which the updated cell center velocity from the intermediate velocity and the pressure gradient is not guaranteed solenoidal the consistent projection methods denoted as algorithms iii and iv obtain the cell center velocity based on an interpolation from a conservative fluxes with velocity unit on surrounding cell faces dependent on treatment of the continuous surface force the pressure gradient in algorithm iii or the sum of the pressure gradient and the surface force in algorithm iv at a cell center is then conducted from the difference between the updated velocity and the intermediate velocity in a consistent projection method a non viscous 3d static drop with serials of density ratios is numerically simulated using the consistent projection methods the spurious currents can be greatly reduced and the pressure jump across the interface can be accurately captured without oscillations the developed consistent projection method are also applied for simulation of interface evolution of an initial ellipse driven by the surface tension and of an initial sphere bubble driven by the buoyancy with good accuracy and good resolution

three dimensional volume conserving immersed boundary model for two phase fluid flows abstract we present a volume preserving scheme for two phase immiscible incompressible flows using an immersed boundary method ibm in a three dimensional space the two phase ibm employs a mixture of eulerian and lagrangian variables where the fluid interface is represented by discrete lagrangian markers exerting surface tension forces to the eulerian fluid domain and the markers are advected by the fluid velocity the interactions between the lagrangian markers and the fluid variables are linked by the discretized dirac delta function the present study extends the previous two dimensional research li et al volume preserving immersed boundary methods for two phase fluid flows int j numer meth fluids 69 2012 842 858 to the three dimensional space the key idea of the proposed method is relocating surface points along the normal directions to conserve the total volume we perform a number of numerical experiments to show the efficiency and accuracy of the proposed method

computations of multi fluid flows abstract full numerical simulations of three dimensional flows of two or more immiscible fluids of different densities and viscosities separated by a sharp interface with finite surface tension are discussed the method used is based on a finite difference approximation of the full navier stokes equations and explicit tracking of the interface between the fluids preliminary simulations of the rayleigh taylor instability and the motion of bubbles are shown

efficient pseudospectral flow simulations in moderately complex geometries a computationally efficient pseudospectral method is developed for incompressible flow simulations in two dimensional geometries involving periodicity in one direction and significant surface deformations a pseudoconformal mapping is used to map the flow domain into a rectangle thereby establishing an orthogonal curvilinear coordinate system within which the governing equations are formulated the time integration of the spectrally discretized two dimensional momentum equations is performed by a second order mixed explicit implicit time integration scheme the satisfaction of the continuity equation is obtained through the solution of a poisson equation for the pressure and the use of the influence matrix technique a highly efficient iterative solver has been developed for the solution of a generalized stokes problem at each time step based on a spectrally preconditioned biconjugate gradient algorithm which exhibits almost linear scalability requiring an ordernlog2nnumber of operations wherenis the number of unknowns numerical results are presented for two dimensional steady oscillatory and peristaltic flows within an undulating channel which agree well with previous results that have appeared in the literature

high order surface tension vof model for 3d bubble flows with high density ratio an improved volume of fluid vof method is presented which is applicable to high density ratio 3d flows for a large range of bubble reynolds number re the method is based on the navier stokes equations for incompressible multi phase flows which are discretized on a cartesian staggered grid the multi grid technique together with the pressure velocity coupling scheme for multi phase flows have resulted in an efficient solver which nearly exponentially converge with the number of iterations the convergence speed also shows negligible dependence on density ratio viscosity ratio and re a second order accurate non diffusive mass conservative phase transport model is presented which does not suffer from unphysical over or under shoots of the phase variable the high accurate normal curvature and surface tension force model in combination with the high order defect correction scheme for multi phase flows shows second order global accuracy when applied to the transient bubble rise where the viscosity ratio is equal to one in contrast the commonly used viscosity model for vof introduces a first order error for the same problem the vof method has been tested for different types of bubble flows at low re and for path oscillating and wobbling air bubbles in water with a diameter range of 1 82 6 mm the numerical results agree quantitatively with the available experimental data the investigations show that the proposed high accurate surface tension model can be used successfully for wobbling flows with bubble deformation while maintaining the mass of the phases the error in mass conservation is directly proportional to the residual in solving the discrete problem

a quadratic spline based interface quasi reconstruction algorithm for accurate tracking of two phase flows a new quadratic spline based interface quasi reconstruction algorithm is presented which provides an accurate and continuous representation of the interface in a multiphase domain and facilitates the direct estimation of local interfacial curvature the fluid interface in each of the mixed cells is represented by piecewise parabolic curves and an initial discontinuous plic approximation of the interface is progressively converted into a smooth quadratic spline made of these parabolic curves the conversion is achieved by a sequence of predictor corrector operations enforcing function c 0 and derivative c 1 continuity at the cell boundaries using simple analytical expressions for the continuity requirements the efficacy and accuracy of the current algorithm has been demonstrated using standard test cases involving reconstruction of known static interface shapes and dynamically evolving interfaces in prescribed flow situations these benchmark studies illustrate that the present algorithm performs excellently as compared to the other interface reconstruction methods available in literature quadratic rate of error reduction with respect to grid size has been observed in all the cases with curved interface shapes only in situations where the interface geometry is primarily flat the rate of convergence becomes linear with the mesh size the flow algorithm implemented in the current work is designed to accurately balance the pressure gradients with the surface tension force at any location as a consequence it is able to minimize spurious flow currents arising from imperfect normal stress balance at the interface this has been demonstrated through the standard test problem of an inviscid droplet placed in a quiescent medium finally the direct curvature estimation ability of the current algorithm is illustrated through the coupled multiphase flow problem of a deformable air bubble rising through a column of water

an unsplit higher order godunov method for scalar conservation laws in multiple dimensions abstract in this paper we develop an unsplit higher order godunov method for scalar conservation laws in two dimensions the method represents an extension for the special case being considered of methods developed by colella and by van leer in our method we begin with a piecewise bilinear representation of the solution on each grid cell a new piecewise bilinear representation at the next time level is then obtained from a two step procedure in the first step a conservative predictor corrector scheme derived from the integral form of the differential equation and characteristic considerations is used to obtain average values of the solution over grid cells at the new time level next these new average values are then used to construct a limited piecewise bilinear profile for each cell at the new time level the resulting method is shown to satisfy a maximum principle for constant coefficient linear advection computational results are presented comparing the new method to colella s method for linear advection the method is also applied to two model problems from porous media flow miscible and immiscible displacement the new scheme provides accurate resolution of sharp fronts without any significant distortion

f quantum spaces and their dynamics abstract f quantum space has been suggested by riecan as an alternative model for quantum mechanics in this paper we study the dynamics of f quantum spaces

an interface capturing method with a continuous function the thinc method with multi dimensional reconstruction an interface capturing method with a continuous function is proposed within the framework of the volume of fluid vof method being different from the traditional vof methods that require a geometrical reconstruction and identify the interface by a discontinuous heaviside function the present method makes use of the hyperbolic tangent function known as one of the sigmoid type functions in the tangent of hyperbola interface capturing thinc method f xiao y honma k kono a simple algebraic interface capturing scheme using hyperbolic tangent function int j numer methods fluids 48 2005 1023 1040 to retrieve the interface in an algebraic way from the volume fraction data of multi component materials instead of the 1d reconstruction in the original thinc method a multi dimensional hyperbolic tangent function is employed in the present new approach the present scheme resolves moving interface with geometric faithfulness and compact thickness and has at least the following advantages 1 the geometric reconstruction is not required in constructing piecewise approximate functions 2 besides a piecewise linear interface curved quadratic surface can be easily constructed as well and 3 the continuous multi dimensional hyperbolic tangent function allows the direct calculations of derivatives and normal vectors numerical benchmark tests including transport of moving interface and incompressible interfacial flows are presented to validate the numerical accuracy for interface capturing and to show the capability for practical problems such as a stationary circular droplet a drop oscillation a shear induced drop deformation and a rising bubble

crystal growth and dendritic solidification we present a numerical method which computes the motion of complex solid liquid boundaries in crystal growth the model we solve includes physical effects such as crystalline anisotropy surface tension molecular kinetics and undercooling the method is based on two ideas first the equations of motion are recast as a single history dependent boundary integral equation on the solid liquid boundary a fast algorithm is used to solve the integral equation efficiently second the boundary is moved by solving a hamiltonjacobi type equation on a fixed domain formulated by osher and sethian for a function in which the boundary is a particular level set this equation is solved by finite difference schemes borrowed from the technology of hyperbolic conservation laws the two ideas are combined by constructing a smooth extension of the normal velocity off the moving boundary in a way suggested by the physics of the problem our numerical experiments show the evolution of complex crystalline shapes development of large spikes and corners dendrite formation and side branching and pieces of solid merging and breaking off freely

mixed stabilized finite element methods based on backward difference adams bashforth scheme for the time dependent variable density incompressible flows in this paper we consider a second order mixed stabilized finite element method based on pressure projection method for variable density incompressible flows the originality of the proposed approach is to use a stabilized method based on the difference between a consistent and an under integrated mass matrix of the pressure for variable density incompressible flows approximated by the lowest equal order finite element pairs a second order backward difference bdf for the temporal term and a second order adams bashforth ab for the explicit treatment of the nonlinear term lead to the presented second order bdf ab scheme the stability of the method was proved and the performance of the method is numerically illustrated finally comparison with some established methods a series of numerical experiments are given to show that this method has better stability and accuracy

a face oriented stabilized nitsche type extended variational multiscale method for incompressible two phase flow a novel extended variational multiscale method for incompressible two phase flow is proposed in this approach the level set method which allows for accurately representing complex interface evolutions is combined with an extended finite element method for the fluid field sharp representation of the discontinuities at the interface related to surface tension effects and large material parameter ratios are enabled by this approach to capture the discontinuities jump enrichments are applied for both velocity and pressure field nitsche s method is then used to weakly impose the continuity of the velocity field for a stable formulation on the entire domain residual based variational multiscale terms are supported by appropriate face oriented ghost penalty and fluid stabilization terms in the region of enriched elements both face oriented terms as well as interfacial terms related to nitsche s method are introduced such that it is accounted for viscous and convection dominated transient flows as a result stability and wellconditioned systems are guaranteed independent of the interface location the proposed method is applied to four numerical examples of increasing complexity two dimensional rayleigh taylor instabilities a twodimensional collapsing water column three dimensional rising bubbles as well as a three dimensional bubble coalescence excellent agreement with either analytical solutions or numerical and experimental reference data as well as robustness for all flow regimes is demonstrated for all examples

discrete singular convolution finite subdomain method for the solution of incompressible viscous flows this paper proposes a discrete singular convolution finite subdomain method dsc fsm for the analysis of incompressible viscous flows in multiply connected complex geometries the dsc algorithm has its foundation in the theory of distributions a block structured grid of fictitious overlapping interfaces is designed to decompose a complex computational geometry into a finite number of subdomains in each subdomain the governing navier stokes equations are discretized by using the dsc algorithm in space and a third order runge kutta scheme in time information exchange between fictitious overlapping zones is realized by using the dsc interpolating algorithm the taylor problem with decaying vortices could be solved to machine precision with an excellent comparison against the exact solution the reliability of the proposed method is tested by simulating the flow in a lid driven cavity the utility of the dsc fsm approach is further illustrated by two other benchmark problems viz the flow over a backward facing step and the laminar flow past a square prism the present results compare well with the numerical and experimental data available in the literature

multidimensional upwind methods for hyperbolic conservation laws abstract we present a class of second order conservative finite difference algorithms for solving numerically time dependent problems for hyperbolic conservation laws in several space variables these methods are upwind and multidimensional in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge and that all fluxes are evaluated and differenced at the same time in particular operator splitting is not used correct behavior at discontinuities is obtained by the use of solutions to the riemann problem and by limiting some of the second order terms numerical results are presented which show that the methods described here yield the same high resolution as the corresponding operator split methods

modelling merging and fragmentation in multiphase flows with surfer we introduce a new numerical method called surfer for the simulation of two and three dimensional flows with several fluid phases and free interfaces between them we consider incompressible fluids obeying the navier stokes equation with newtonian viscosity in the bulk of each phase capillary forces are taken into account even when interfaces merge or break up fluid interfaces are advanced in time using an exactly volume conserving variant of the volume of fluid algorithm thus allowing for full symmetry between fluid phases the navier stokes equation is solved using staggered finite differences on a mac grid and a split explicit time differencing scheme while incompressibility is enforced using an iterative multigrid poisson solver capillary effects are represented as a stress tensor computed from gradients of the volume fraction function this formulation is completely independent of the topology of interfaces and relatively easy to implement in 3d it also allows exact momentum conservation in the discretized algorithm numerical spurious effects or parasite currents are noticed and compared to similar effects in boltzmann lattice gas methods for immiscible fluids simulations of droplets pairs colliding in 2d and in 3d are shown interface reconnection is performed easily despite the large value of capillary forces during reconnection

computing interface motion in compressible gas dynamics a hamilton jacobi level set formulation of the equations of motion for propagating interfaces has been introduced recently by osher and sethian this formulation allows fronts to self intersect develop singularities and change topology the numerical algorithms based on this approach handle topological merging and breaking naturally work in any number of space dimensions and do not require that the moving front be written as a function instead the moving front is embedded as a particular level set in a fixed domain partial differential equation numerical techniques borrowed from hyperbolic conservation laws are then used to accurately capture the complicated surface motion that satisfies the global entropy condition for propagating fronts given by sethian in this paper we analyze the coupling of this level set formulation to a system of conservation laws for compressible gas dynamics we study both conservative and non conservative differencing of the level set function and compare the two approaches to illustrate the capability of the method we study the compressible rayleigh taylor and kelvin helmholtz instabilities for air air and air helium boundaries we perform numerical convergence studies of the method over a range of parameters and analyze the accuracy of this approach applied to these problems

coupling of the interface tracking and the two fluid models for the simulation of incompressible two phase flow abstract the volume of fluid vof method which uses an interface tracking algorithm for the simulation of the two phase flow is coupled with the two fluid model which is based on time and space averaged equations and cannot track the interface explicitly the idea of the present work is to use the vof method in the parts of the computational domain where the grid density allows surface tracking in the parts of the domain where the flow is too dispersed to be described by the interface tracking algorithms the two fluid model is used the equations of the two fluid model are less accurate than the vof model due to the empirical closures required in the averaged equations however in the case of the sufficiently dispersed flow the two fluid model results are still much closer to the real world than the results of the vof method which do not have any physical meaning when the grid becomes too coarse each model in the present work uses a separate set of equations suitable for description of two dimensional incompressible viscous two phase flow similar discretization techniques are used for both sets of equations and solved with the same numerical method coupling of both models is achieved via the volume fraction of one of the fluids which is used in both models a special criterion for the transition between the models is derived from the interface reconstruction function in the vof method an idealized vortical flow and the rayleigh taylor instability are used as tests of the coupling in both cases the time development causes mixing of the fluids and dispersion of the interface that is beyond the capabilities of the model based on the vof method therefore the two fluid model gradually replaces the interface tracking model in the final stages of the rayleigh taylor instability when both fluids are approaching their final positions and the tractable interface appears again the two fluid model is gradually replaced by the vof method

probability of fuzzy events defined as denumerable additivity measure abstract in this paper the probability of fuzzy events is defined as a denumerable additivity measure this definition is based on a non conventional approach of separativity between fuzzy subsets the presented measure fulfils all properties analogous to the properties of classical probability in the crisp case further the notion of conditional probability of fuzzy events complete fuzzy repartition and independent fuzzy events are defined by means of the probability measure considered here connections between all the above notions are presented in this paper too finally the bayes formula is proved for the fuzzy case

a 3d adaptive mesh refinement algorithm for multimaterial gas dynamics abstract adaptive mesh refinement amr in conjunction with high order upwind finite difference methods has been used effectively on a variety of problems in this paper we discuss an implementation of an amr finite difference method that solves the equations of gas dynamics with two material species in three dimensions an equation for the evolution of volume fractions augments the gas dynamics system the material interface is preserved and tracked from the volume fractions using a piecewise linear reconstruction technique

on the entropy of dynamical systems in product mv algebras recently the so called product mv algebras have been introduced and the entropy of partitions in these algebras have been studied in this paper the previous results are applied to dynamical systems

two phase viscoelastic jetting a coupled finite difference algorithm on rectangular grids is developed for viscoelastic ink ejection simulations the ink is modeled by the oldroyd b viscoelastic fluid model the coupled algorithm seamlessly incorporates several things 1 a coupled level set projection method for incompressible immiscible two phase fluid flows 2 a higher order godunov type algorithm for the convection terms in the momentum and level set equations 3 a simple first order upwind algorithm for the convection term in the viscoelastic stress equations 4 central difference approximations for viscosity surface tension and upper convected derivative terms and 5 an equivalent circuit model to calculate the inflow pressure or flow rate from dynamic voltage

the entropy of fuzzy dynamical systems general scheme and generators in this paper some fuzzy modifications of the kolmogorov sinaj entropy are studied we introduce the general scheme which includes the mentioned entropies finally the kolmogorov sinaj theorem on generators is proved for special cases

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

spectral theory zeta functions and the distribution of periodic points for collet eckmann maps we study unimodal interval mapst with negative schwarzian derivative satisfying the collet eckmann condition dtn tc k cn for some constantsk 0 and c 1 c is the critical point oft we prove exponential mixing properties of the unique invariant probability density oft describe the long term behaviour of typical in the sense of lebesgue measure trajectories by central limit and large deviations theorems for partial sum processes of the form s n sigma i 0 n 1 f t i x and study the distribution of typical periodic orbits also in the sense of a central limit theorem and a large deviations theorem

a numerical method for the incompressible navier stokes equations based on an approximate projection in this method we present a fractional step discretization of the time dependent incompressible navier stokes equations the method is based on a projection formulation in which we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields our treatment of the diffusion convection step uses a specialized second order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high reynolds number in contrast to conventional projection type discretizations that impose a discrete form of the divergence free constraint we only approximately impose the constraint i e the velocity field we compute is not exactly divergence free the approximate projection is computed using a conventional discretization of the laplacian and the resulting linear system is solved using conventional multigrid methods numerical examples are presented to validate the second order convergence of the method for euler finite reynolds number and stokes flow a second example illustrating the behavior of the algorithm on an unstable shear layer is also presented

face offsetting a unified approach for explicit moving interfaces dynamic moving interfaces are central to many scientific engineering and graphics applications in this paper we introduce a novel method for moving surface meshes called the face offsetting method based on a generalized huygens principle our method operates directly on a lagrangian surface mesh without requiring an eulerian volume mesh unlike traditional lagrangian methods which move each vertex directly along an approximate normal or user specified direction our method propagates faces and then reconstructs vertices through an eigenvalue analysis locally at each vertex to resolve normal and tangential motion of the interface simultaneously the method also includes techniques for ensuring the integrity of the surface as it evolves face offsetting provides a unified framework for various dynamic interface problems and delivers accurate physical solutions even in the presence of singularities and large curvatures we present the theoretical foundation of our method and also demonstrate its accuracy efficiency and flexibility for a number of benchmark problems and a real world application

lattice boltzmann model for free surface flow and its application to filling process in casting a generalized lattice boltzmann model to simulate free surface is constructed in both two and three dimensions the proposed model satisfies the interfacial boundary conditions accurately a distinctive feature of the model is that the collision processes is carried out only on the points occupied partially or fully by the fluid to maintain a sharp interfacial front the method includes an anti diffusion algorithm the unknown distribution functions at the interfacial region are constructed according to the first order chapman enskog analysis the interfacial boundary conditions are satisfied exactly by the coefficients in the chapman enskog expansion the distribution functions are naturally expressed in the local interfacial coordinates the macroscopic quantities al the interface are extracted from the least square solutions of a locally linearized system obtained from the known distribution functions the proposed method does not require any geometric front construction and is robust for any interfacial topology simulation results of realistic filling process are presented rectangular cavity in two dimensions and hammer box campbell box sheffield box and motorblock in three dimensions to enhance the stability at high reynolds numbers various upwind type schemes are developed free slip and no slip boundary conditions are also discussed

numerical simulation of liquid gas phase flow during mold filling abstract a numerical model for simulation of liquid gas phase flow during mold filling is presented the incompressible navier stokes equations are discretized on a fixed cartesian mesh with finite difference method the fractional step scheme is employed for enforcing incompressibility constraint the free surface effects are calculated using the volume of fluid method based on the piecewise linear interface reconstruction and split lagrangian advection of volume fraction field adding limited compressibility to the gas phase led to improvement in convergence rate of poisson equation solver about 2 fold this new concept permits simulation of two phase incompressible free surface flow during mold filling without using outflow boundary condition numerical results are presented for several test cases which demonstrate validity capability and performance of the proposed method in contrast with the traditional fully incompressible approach

second order accurate volume of fluid algorithms for tracking material interfaces we introduce two new volume of fluid interface reconstruction algorithms and compare the accuracy of these algorithms to four other widely used volume of fluid interface reconstruction algorithms we find that when the interface is smooth e g continuous with two continuous derivatives the new methods are second order accurate and the other algorithms are first order accurate we propose a design criteria for a volume of fluid interface reconstruction algorithm to be second order accurate namely that it reproduce lines in two space dimensions or planes in three space dimensions exactly we also introduce a second order unsplit volume of fluid advection algorithm that is based on a second order finite difference method for scalar conservation laws due to bell dawson and shubin we test this advection algorithm by modeling several different interface shapes propagating in two simple incompressible flows and compare the results with the standard second order operator split advection algorithm although both methods are second order accurate when the interface is smooth we find that the unsplit algorithm exhibits noticeably better resolution in regions where the interface has discontinuous derivatives such as at corners

toward efficient and accurate interface capturing on arbitrary hybrid unstructured grids the thinc method with quadratic surface representation and gaussian quadrature abstract a novel interface capturing scheme is proposed to compute moving interface on arbitrary hybrid unstructured grids different from conventional volume of fluid vof schemes that require complicated geometric manipulations for interface reconstructions the present method can be viewed as a hybrid geometric algebraic type vof approach i e the interface is implictly retrieved from an algebraic function that effectively makes use of the geometrical information such as normal direction and curvature of the interface unlike previous versions of the thinc tangent of hyperbola interface capturing method the interface is represented by a quadratic surface for grid cells of arbitrary shapes in this scheme the gaussian quadrature is used to estimate the integration of the cell wise multi dimensional hyperbolic tangent reconstruction function which is then used to retrieve the interface from the volume fraction value of the target cell the gaussian quadrature is also used to compute the numerical fluxes from the reconstruction function numerical accuracy for reconstruction and flux computation can be effectively improved by increasing quadrature points the whole solution procedure follows the finite volume method for advection transport and is thus simple and easy to use for cells of arbitrary shapes as verified in the benchmark tests in this paper the presented scheme so called thinc qq thinc method with quadratic surface representation and gaussian quadrature scheme shows significantly improved geometrical fidelity of interface representation particularly for curved surface despite algorithmic simplicity the solution quality of thinc qq is comparable to other existing vof methods with plic geometrical interface reconstructions and thus an accurate and efficient vof scheme of great practical significance for unstructured grids

a balanced force refined level set grid method for two phase flows on unstructured flow solver grids a balanced force refined level set grid method for two phase flows on structured and unstructured flow solver grids is presented to accurately track the phase interface location an auxiliary high resolution equidistant cartesian grid is introduced in conjunction with a dual layer narrow band approach this refined level set grid method allows for parallel efficient grid convergence and error estimation studies of the interface tracking method the navier stokes equations are solved on an unstructured flow solver grid with a novel balanced force algorithm for level set methods based on the recently proposed method by francois et al m m francois s j cummins e d dendy d b kothe j m sicilian m w williams a balanced force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework j comput phys 213 2006 141 173 for volume of fluid methods on structured grids to minimize spurious currents a second order converging curvature evaluation technique for level set methods is presented the results of several different test cases demonstrate the effectiveness of the proposed method showing good mass conservation properties and second order converging spurious current magnitudes

a volume of fluid method based on multidimensional advection and spline interface reconstruction a new volume of fluid method for tracking two dimensional interfaces is presented the method involves a multidimensional advection algorithm based on the use of edge matched flux polygons to integrate the volume fraction evolution equation and a spline based reconstruction algorithm the accuracy and efficiency of the proposed method are analyzed using different tests and the results are compared with those obtained recently by other authors despite its simplicity the proposed method represents a significant improvement and compares favorably with other volume of fluid methods as regards the accuracy and efficiency of both the advection and reconstruction steps

reconstructing volume tracking a new algorithm for the volume tracking of interfaces in two dimensions is presented the algorithm is based upon a well defined second order geometric solution of a volume evolution equation the method utilizes local discrete material volume and velocity data to track interfaces of arbitrarily complex topology a linearity preserving piecewise linear interface geometry approximation ensures that solutions generated retain second order spatial accuracy second order temporal accuracy is achieved by virtue of a multidimensional unsplit time integration scheme we detail our geometrically based solution method in which material volume fluxes are computed systematically with a set of simple geometric tasks we then interrogate the method by testing its ability to track interfaces through large controlled topology changes whereby an initially simple interface configuration is subjected to vortical flows numerical results for these strenuous test problems provide evidence for the algorithm s improved solution quality and accuracy

a surface marker algorithm coupled to an area preserving marker redistribution method for three dimensional interface tracking in this work we present a new interface tracking algorithm based on surface markers to reconstruct and advect interfaces in three dimensional space the algorithm is coupled to a marker redistribution method which is area preserving along the interface the interface is described by a set of closed lines which define a coarse lagrangian quadrangular mesh in the fixed computational grid fixed markers are located where the lines cross and are maintained during the whole simulation at each time step the interface is first reconstructed and then all markers are advected by following streamlines in the reconstruction step new markers are determined by computing the intersections of the interface lines with the grid cell faces and by adding area conservation markers between fixed and grid intersection markers intersection and conservation markers defined at the previous time step are discarded the method maintains a smooth geometrical description of the interface for both two dimensional and three dimensional tests with accurate volume conservation even in very challenging situations where the fluid bodies progressively deform and stretch developing localized regions with very high curvature and thin fluid filaments the method compares favorably with front capturing methods such as volume of fluid and level set and other hybrid techniques such as the particle level set method

piecewise invertible dynamical systems the aim of the paper is the investigation of piecewise monotonic maps t of an interval x the main tool is an isomorphism of x t with a topological markov chain with countable state space which is described by a 0 1 transition matrix m the behavior of the orbits of points in x under t is very similar to the behavior of the paths of the markov chain every irreducible submatrix of m gives rise to a t invariant subset l of x such that l is the set x of all limit points of the orbit of an x x the topological entropy of l is the logarithm of the spectral radius of the irreducible submatrix which is a l 1 operator besides these sets l there are two t invariant sets y and p such that for all x x the set x is either contained in one of the sets l or in y or in p the set p is a union of periodic orbits and y is contained in a finite union of sets y with y x and has topological entropy zero this isomorphism of x t with a topological markov chain is also an important tool for the investigation of t invariant measures on x results in this direction which are published elsewhere are described at the end of the paper furthermore a part of the proofs in the paper is purely topological without using the order relation of the interval x so that some results hold for more general dynamical systems x t

zeta functions and transfer operators for piecewise monotone transformations given a piecewise monotone transformationt of the interval and a piecewise continuous complex weight functiong of bounded variation we prove that the ruelle zeta function z of t g extends meromorphically to z 1 where lim g tn 1 g tg 1 n and thatz is a pole of if and only ifz 1 is an eigenvalue of the corresponding transfer operator l we do not assume that l leaves a reference measure invariant

a boundary integral approach to unstable solidification we consider the supercooled stefan problem with a general anisotropic curvature and velocity dependent boundary condition on the moving interface we present numerical methods based on an integral equation formulation and including a new algorithm for moving curves with curvature dependent velocity these methods compute a periodic interface with o t accuracy where t is the time step previous work has been limited to short time spans and achieved slightly less than o t12 accuracy accurate numerical results are seen to agree with the predictions of linear stability theory this agreement has eluded previous authors because their numerical methods suffered from grid effects and their linear stability theory was incorrect we study the long time evolution of an unstable interface our computations exhibit the beginnings of a sidebranching instability when the boundary condition includes anisotropy and tip splitting in the isotropic case

cellwise conservative unsplit advection for the volume of fluid method we present a cellwise conservative unsplit ccu advection scheme for the volume of fluid method vof in 2d contrary to other schemes based on explicit calculations of the flux balances the ccu advection adopts a cellwise approach where the pre images of the control volumes are traced backwards through the flow map the donating regions of the fluxes are calculated via the streaklines of the grid intersections represented as polygonal chains whose vertices are determined by backward tracing of particles injected in the flow at different times high order accuracy is obtained from the fourth order runge kutta method where intermediate velocities along pathlines are determined with quadratic temporal and bicubic spatial interpolations the volumes of the donating regions are corrected in order to fulfill the discrete continuity of incompressible flows consequently the calculation produces non overlapping donating regions and pre images with conforming edges to their neighbors resulting in the conservativeness and the boundedness liquid volume fraction inside the interval 0 1 of the ccu advection scheme finally the update of the liquid volume fractions is computed from the intersections of the pre image polygons with the reconstructed interfaces the ccu scheme is tested on several benchmark tests for the vof advection together with the standard piecewise linear interface calculation plic the geometrical errors of the ccu compare favorably with other unsplit vof plic schemes finally potential improvements of the vof method with the use of more precise interface representation techniques and the future extension of the ccu scheme to 3d are discussed cellwise vof update through backward tracing of the cell s pre image accuracy enhanced via bicubic spatial interpolation and polygonal chain streakline the ccu advection scheme is fully conservative and bounded no limitation on the courant number geometrical errors dominated by plic interface representation errors

on the susceptibility function of piecewise expanding interval maps we study the susceptibility function psi z sum n 0 infty int z n x y rho 0 y frac partial partial y varphi f n y dy associated to the perturbation f t f tx circ f of a piecewise expanding interval map f and to an observable 1 is the formal derivative at t 0 of the average mathcal r t int varphi rho t dx of with respect to the srb measure of ft our analysis is based on a spectral description of transfer operators it gives in particular sufficient conditions on f x and which guarantee that z is holomorphic in a disc of larger than one or which ensure that a number may be associated to the possibly divergent series 1 we present examples of f x and so that mathcal r t is not lipschitz at 0 and we propose a new version of ruelle s conjecture

an investigation of interface sharpening schemes for multi phase mixture flows this work evaluates several approaches for sharp phase interface capturing in computations of multi phase mixture flows attention is focused on algebraic interface capturing strategies that fit directly within a finite volume muscl type framework in which dimension by dimension reconstruction of interface states based on extrapolated fluid properties is the norm in this scope linear sine wave and tangent hyperbola volume fraction reconstructions are examined for a range of problems including advection of a volume fraction discontinuity the rayleigh taylor instability a dam break problem an axisymmetric jet instability the rayleigh instability and flow within an aerated liquid injector an implicit dual time stepping approach applied directly to a preconditioned form of the governing equations is used for time advancement the results show that the sharpening strategies are successful in providing two to three cell capturing of volume fraction discontinuities

a second order projection method for variable density flows this paper describes a second order projection method for variabledensity incompressible flows the method is suitable for both finite amplitude density variations and for fluids that are modeled using a boussinesq approximation it is based on a second order fractional step scheme in which diffusion convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields the nonlinear convection terms are treated using a godunov type procedure that is second order for smooth flow and remain stable and non oscillatory for nonsmooth flows with low fluid viscosities the method is described for finite amplitude density variation and the simplifications for a boussinesq approximation are sketched numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems

a weakly compressible scheme with a diffuse interface method for low mach number two phase flows abstract a weakly compressible scheme for gas liquid two phase flows with low mach number is proposed for a fully explicit time integration the scheme is based on the fractional step method with the euler equation solver and time integration for viscous surface tension and gravity terms for the efficient solution of the euler equation we introduce the directional splitting method and the method of characteristics which is applicable to semi lagrangian methods a reduction of the speed of sound is also taken into consideration to enlarge the time step intervals the primitive flow variables are time integrated on the node center configuration to capture the gas liquid interface the conservative phase field equation is solved by using the multi moment method it is computed as a conservative form for the volume averaged value defined at the cell center in which the flow velocities are given on the node we discard the exact momentum conservation however the total mass of gas and liquid is conserved by introducing a phase field variable several benchmark problems are examined and the results of the explicit scheme are in good agreement with the experimental and computational references

a new fractional time stepping method for variable density incompressible flows this paper describes a new method for solving variable density incompressible viscous flows we have dealt with the momentum equation and the divergence free constraint in a new manner by rewriting the original equations the originality of the proposed approach is that we have used different numerical methods to evaluate the evolution of the velocity and pressure compared with some established methods the proposed approach is parameter free more flexible and can avoid the difficulties caused by the original equations the stability analysis of the method is performed to show that our method is stable finally numerical experiments are given to show the accuracy efficiency and validness of this method for variable density incompressible flows

volume of fluid interface tracking with smoothed surface stress methods for three dimensional flows motivated by the need for three dimensional methods for interface calculations that can deal with topology changes we describe a numerical scheme built from a volume of fluid interface tracking technique that uses a piecewise linear interface calculation in each cell momentum balance is computed using explicit finite volume finite differences on a regular cubic grid surface tension is implemented by the continuous surface stress or continuous surface force method examples and verifications of the method are given by comparing simulations to analytical results and experiments for sedimenting droplet arrays and capillary waves at finite reynolds number in the case of a pinching pendant drop both three dimensional and axisymmetric simulations are compared to experiments agreement is found both before and after the reconnections

a new volume of fluid advection algorithm a new vof advection algorithm is presented termed the stream scheme the algorithm uses a linear piecewise method for free surface reconstruction coupled to a unique fully multidimensional method of cell boundary flux integration the performance of the new algorithm is compared against other vof advection algorithms using a variety of standard advection tests in general the performance of the stream scheme appears superior to existing multidimensional and dimensionally split advection algorithms

numerical methods for tracking interfaces an overview is given of special numerical methods for tracking discontinuous fronts and interfaces these methods include surface tracking methods based on connected marker points along the interface volume tracking methods that track the volume occupied by the solution regions bounded by the interfaces and moving mesh methods where the underlying mesh is aligned and moved with the interface the pros and cons of the current methods are discussed and a new method is proposed that overcomes some of the difficulties encountered in approximating equations with multiple interacting interfaces

numerical study of flows of two immiscible liquids at low reynolds number we review our recent results on fingering bamboo waves and drop breakup in low reynolds number flows composed of two viscous liquids under shear

short note an interface capturing method with a continuous function the thinc method on unstructured triangular and tetrahedral meshes a novel interface capturing method is proposed to compute moving interfaces on unstructured grids with triangular 2d and tetrahedral 3d elements different from the conventional vof volume of fluid method which involves geometric reconstructions of the interface the present method is based on the algebraic reconstruction approach originally developed in the thinc tangent of hyperbola interface capturing scheme by xiao et al 2005 17 a continuous multidimensional hyperbolic tangent function is employed for retrieving the jump like distribution of the indicator function which avoids the explicit geometric representation of the interface and thus substantially reduces the algorithmic complexity in unstructured grids numerical diffusion and smearing are effectively eliminated and the compact thickness of the jump transition layer in the volume fraction is retained throughout the computation even for largely deformed interface the solution quality of the present scheme is comparable to the vof method with plic piecewise linear interface calculation algorithm

a geometrical area preserving volume of fluid advection method a new class of algorithms that preserve mass exactly for incompressible flows on a cartesian mesh are presented they amount to piecewise linear area preserving mappings of tessellations of the plane they are equivalent to volume of fluid vof advection methods which are decomposed into an eulerian implicit scheme in one direction followed by a lagrangian explicit step in the other one it is demonstrated that mass conservation is exact for incompressible flows and that there are no undershoots or overshoots of the volume fraction which thus always remains constrained between 0 and 1

a level set formulation of eulerian interface capturing methods for incompressible fluid flows a level set formulation is derived for incompressible immiscible navier stokes equations separated by a free surface the interface is identified as the zero level set of a smooth function eulerian finite difference methods based on this level set formulation are proposed these methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection numerical experiments are presented to demonstrate the effectiveness of the methods

an eulerian projection method for quasi static elastoplasticity a well established numerical approach to solve the navier stokes equations for incompressible fluids is chorin s projection method 1 whereby the fluid velocity is explicitly updated and then an elliptic problem for the pressure is solved which is used to orthogonally project the velocity field to maintain the incompressibility constraint in this paper we develop a mathematical correspondence between newtonian fluids in the incompressible limit and hypo elastoplastic solids in the slow quasi static limit using this correspondence we formulate a new fixed grid eulerian numerical method for simulating quasi static hypo elastoplastic solids whereby the stress is explicitly updated and then an elliptic problem for the velocity is solved which is used to orthogonally project the stress to maintain the quasi staticity constraint we develop a finite difference implementation of the method and apply it to an elasto viscoplastic model of a bulk metallic glass based on the shear transformation zone theory we show that in a two dimensional plane strain simple shear simulation the method is in quantitative agreement with an explicit method like the fluid projection method it is efficient and numerically robust making it practical for a wide variety of applications we also demonstrate that the method can be extended to simulate objects with evolving boundaries we highlight a number of correspondences between incompressible fluid mechanics and quasi static elastoplasticity creating possibilities for translating other numerical methods between the two classes of physical problems

numerical study of wall effects on buoyant gas bubble rise in a liquid filled finite cylinder the wall effects on the axisymmetric rise and deformation of an initially spherical gas bubble released from rest in a liquid filled finite circular cylinder are numerically investigated the bulk and gas phases are considered incompressible and immiscible the bubble motion and deformation are characterized by the morton number mo eotvos number eo reynolds number re weber number we density ratio viscosity ratio the ratios of the cylinder height and the cylinder radius to the diameter of the initially spherical bubble h h d 0 r r d 0 bubble rise in liquids described by eo and mo combinations ranging from 1 0 01 to 277 5 0 092 as appropriate to various terminal state reynolds numbers re t and shapes have been studied the range of terminal state reynolds numbers includes 0 02 re t 70 bubble shapes at terminal states vary from spherical to intermediate spherical cap skirted the numerical procedure employs a front tracking finite difference method coupled with a level contour reconstruction of the front this procedure ensures a smooth distribution of the front points and conserves the bubble volume for the wide range of eo and mo examined bubble motion in cylinders of height h 8 and r 3 is noted to correspond to the rise in an infinite medium both in terms of reynolds number and shape at terminal state in a thin cylindrical vessel small r the motion of the bubble is retarded due to increased total drag and the bubble achieves terminal conditions within a short distance from release the wake effects on bubble rise are reduced and elongated bubbles may occur at appropriate conditions for a fixed volume of the bubble increasing the cylinder radius may result in the formation of well defined rear recirculatory wakes that are associated with lateral bulging and skirt formation the paper includes figures of bubble shape regimes for various values of r eo mo and re t our predictions agree with existing results reported in the literature

the entropy of fuzzy dynamical systems and generators abstract in this paper the entropy and conditional entropy of stochastical complete repartitions are defined the represented measure satisfies all properties analogous to the properties of entropy of measurable repartitions in the crisp case further the notion of entropy of fuzzy dynamical systems is defined the above notion generalizes the notion of entropy of dynamical systems from the classical probability theory finally the kolmogorov sinaj theorem on generators is proved for the fuzzy case

second order accurate volume tracking based on remapping for triangular meshes this paper presents a second order accurate piecewise linear volume tracking based on remapping for triangular meshes this approach avoids the complexity of extending unsplit second order volume of fluid algorithms advection methods on triangular meshes the method is based on lagrangian eulerian le methods therefore it does not deal with edge fluxes and corner fluxes flux corrections as is typical in advection algorithms the method is constructed of three parts a lagrangian phase a reconstruction phase and a remapping phase in the lagrangian phase the original eulerian grid is projected along trajectories to obtain lagrangian grids in practice this projection is handled through the time integration of velocity field for grid vertices at each time step the reconstruction is based on truncating the volume material polygon for each lagrangian mixed grid since in piecewise linear approximation the interface is represented by a segment line the polygon material truncation is mainly finding the segment interface finding the segment interface is calculating the line normal and line constant at each multi fluid cell details of applying two normal calculation methods differential and geometric least squares gls methods are given while the gls method exhibits second order accurate approximation in reproducing circular interfaces the differential least squares dls method results in a first order accurate representation of the interface the last part of the algorithm which is remapping of the volume materials from the lagrangian grid to the original one is performed by a series of polygon intersection procedures the behavior of the algorithm is investigated for flow fields with constant interface topology and flow fields inducing large interfacial stretching and tearing second order accuracy is obtained if the velocity integration as well as the reconstruction steps are at least second order accurate

a fast pressure correction method for incompressible two fluid flows we have developed a new pressure correction method for simulating incompressible two fluid flows with large density and viscosity ratios the method s main advantage is that the variable coefficient poisson equation that arises in solving the incompressible navier stokes equations for two fluid flows is reduced to a constant coefficient equation which can be solved with an fft based fast poisson solver this reduction is achieved by splitting the variable density pressure gradient term in the governing equations the validity of this splitting is demonstrated from our numerical tests and it is explained from a physical viewpoint in this paper the new pressure correction method is coupled with a mass conserving volume of fluid method to capture the motion of the interface between the two fluids but in general it could be coupled with other interface advection methods such as level set phase field or front tracking first we verified the new pressure correction method using the capillary wave test case up to density and viscosity ratios of 10 000 then we validated the method by simulating the motion of a falling water droplet in air and comparing the droplet terminal velocity with an experimental value next the method is shown to be second order accurate in space and time independent of the vof method and it conserves mass momentum and kinetic energy in the inviscid limit also we show that for solving the two fluid navier stokes equations the method is 10 40 times faster than the standard pressure correction method which uses multigrid to solve the variable coefficient poisson equation finally we show that the method is capable of performing fully resolved direct numerical simulation dns of droplet laden isotropic turbulence with thousands of droplets using a computational mesh of 1024 3 points

a high order projection method for tracking fluid interfaces in variable density incompressible flows we present a numerical method for computing solutions of the incompressible euler or navier stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties the method is based on a second order projection method for variable density flows using an approximate projection formulation the boundary between the fluids is tracked with a second order volume of fluid interface tracking algorithm we present results for viscious rayleigh taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm we also present computational results for the rayleigh taylor instability in air helium and for bubbles and drops in an air water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts

ripple a new model for incompressible flows with free surfaces a new free surface flow model ripple is summarised ripple obtains finite difference solutions for incompressible flow problems having significant surface tension forces at free surfaces of arbitrarily complex topology the key innovation is the continuum surface force model which represents surface tension as a localised volume force other features include a higher order momentum advection model a volume of fluid free surface treatment and an efficient two step projection solution method ripple s unique capabilities are illustrated with two example problems low gravity jet induced tank flow and the collision and coalescence of two cylindrical rods

a bridge between projection methods and simple type methods for incompressible navier stokes equations a bridge is built between projection methods and simple type methods semi implicit method for pressure linked equation a general second order accurate projection method is developed for the simulation of incompressible unsteady flows by employing a non linear update of pressure term as n pn 1 i n pn where n is a coefficient matrix which may depend on the grid size time step and even velocity it includes three and four step projection methods the standard simple method is written in a concise formula for steady and unsteady flows it is proven that simple type methods have second order temporal accuracy for unsteady flows the classical second order projection method and simple type methods are united within the framework of the general second order projection formula two iteration algorithms of simple type methods for unsteady flows are described and discussed in addition detailed formulae are provided for general projection methods by using the runge kutta technique to update the convective term and crank nicholson scheme for the diffusion term copyright 2007 john wiley sons ltd

projection methods for two velocity two pressure models for flows of heterogeneous mixtures in this paper a numerical analysis of two velocity two pressure models for flows of solid particles and fluids is presented first a formal exploitation of the weak formulation of such models asserts that they are amenable to integration via projection methods the challenging issues in the algorithm development for these models are then documented and suitable numerical methodologies for their remedy are devised subsequently an algorithm for the integration of the models of interest is proposed this is a two phase projection method on collocated grids that utilizes a fractional step time marching scheme it is further endowed with an interface detection and treatment methodology to properly account for the stiffness induced by the presence of moving and deforming material interfaces the efficiency and robustness of the proposed numerical method are assessed in a series of numerical experiments that are delineated in the last part of this paper

a continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

the development of a free surface capturing approach for multidimensional free surface flows in closed containers a new surface capturing method is developed for numerically simulating viscous free surface flows in partially filled containers the method is based on the idea that the flow of two immiscible fluids within a closed container is governed by the equations of motion for an incompressible viscous nonhomogeneous variable density fluid by computing the flow fields in both the liquid and gas regions in a consistent manner the free surface can be captured as a discontinuity in the density field thereby eliminating the need for special free surface tracking procedures the numerical algorithm is developed using a conservative implicit finite volume discretization of the equations of motion the algorithm incorporates the artificial compressibility method in conjunction with a dual time stepping strategy to maintain a divergence free velocity field a slope limited high order muscl scheme is adopted for approximating the inviscid flux terms while the viscous fluxes are centrally differenced the capabilities of the surface capturing method are demonstrated by calculating solutions to several two and three dimensional problems

on the reinitialization procedure in a narrow band locally refined level set method for interfacial flows a local level set algorithm for simulating interfacial flows described by the two dimensional incompressible navier stokes equations is presented the governing equations are solved using a finite difference discretization on a cartesian grid and a second order approximate projection method the level set transport and reinitialization equations are solved in a narrow band around the interface using an adaptive refined grid which is reconstructed every time step and refined using a simple uniform cell splitting operation within the band instabilities at the border of the narrow band are avoided by smoothing the level set function in the outer part of the band the influence of different pde based reinitialization strategies on the accuracy of the results is investigated the ability of the proposed method to accurately compute interfacial flows is discussed using different tests namely the advection of a circle of fluid in two different time reversed vortex flows the advection of zalesak s rotating disk the propagation of small amplitude gravity and capillary waves at the interface between two superposed viscous fluids in deep water and a classical test of rayleigh taylor instability with and without surface tension effects the interface location error and area loss for some of the results obtained are compared with those of a recent particle level set method copyright 2005 john wiley sons ltd

development of a dispersively accurate conservative level set scheme for capturing interface in two phase flows a two step interface capturing scheme implemented within the framework of conservative level set method is developed in this study to simulate the gas water two phase fluid flow in addition to solving the pure advection equation which is used to advect the level set function for tracking interface both nonlinear and stabilized features are taken into account for the level set function so that a sharply varying interface can be stably predicted to preserve the conservative property the mapping given by x x u t is performed between two coordinates x and x for the advection diffusion equation in a flow field with velocity u to capture the interface the flux term capable of compressing the level set contours is also adopted in the construction of linear inviscid burgers equation which is indispensable in the level set method to resolve the physically sharp interface without incurring contact discontinuity oscillations a damping term which is nonlinear in terms of the level set function and can render an adequate artificial diffusion to stabilize the contact surface is added into the reinitialization step in the modified level set method for accurately predicting the level set function the advection scheme for solving the linear inviscid burgers equation in the advection step of the modified level set method is developed to accommodate the true dispersion relation the solution computed from the resulting two dimensional dispersion relation preserving advection scheme can minimize the phase error less artificial damping is needed to damp the oscillations in the vicinity of contact surface and the interface can be less numerically smeared for the sake of programming simplicity the incompressible two phase flow will be discretized in non staggered grids without incurring checkerboard oscillations by the developed explicit compact scheme for the approximation of pressure gradient terms for the verification of the proposed two dimensional dispersion relation preserving scheme and the non staggered incompressible flow solver three benchmark problems have been chosen in this study the proposed conservative level set method for capturing the interface in incompressible fluid flows is also verified by solving the dam break bubble rising in water droplet falling in water and rayleigh taylor instability problems good agreements with the referenced solutions are demonstrated for all the four investigated problems

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

fast and robust solvers for pressure correction in bubbly flow problems we consider the numerical simulation of two phase fluid flow where bubbles or droplets of one phase move against a background of the other phase such flows are governed by the navier stokes equations the solution of which may be approximated using a pressure correction approach within such an approach the computational cost is often dominated by the solution of a linear system corresponding to a discrete poisson equation with discontinuous coefficients in this paper we explore the efficient solution of these linear systems using robust multilevel solvers such as deflated variants of the preconditioned conjugate gradient method or robust multigrid techniques we consider these families of methods in more detail and compare their performance in the simulation of bubbly flows some of these methods turn out to be very effective and reduce the amount of work to solve the pressure correction system substantially resulting in efficient calculations for two phase flows on highly resolved grids

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows we present a coupled level set volume of fluid clsvof method for computing 3d and axisymmetric incompressible two phase flows this method combines some of the advantages of the volume of fluid method with the level set method to obtain a method which is generally superior to either method alone we present direct comparisons between computations made with the clsvof method and computations made with the level set method the volume of fluid method and the boundary integral method we also compare our computations to the exact solution for an oscillating ellipse due to lamb and experimental results obtained for a rising gas bubble in liquid obtained by hnat and buckmaster our computational examples focus on flows in which surface tension forces and changes in topology are dominant features of the flow

conservative and bounded volume of fluid advection on unstructured grids abstract this paper presents a novel eulerian lagrangian piecewise linear interface calculation plic volume of fluid vof advection method which is three dimensional unsplit and discretely conservative and bounded the approach is developed with reference to a collocated node based finite volume two phase flow solver that utilizes the median dual mesh constructed from non convex polyhedra the proposed advection algorithm satisfies conservation and boundedness of the liquid volume fraction irrespective of the underlying flux polyhedron geometry which differs from contemporary unsplit vof schemes that prescribe topologically complicated flux polyhedron geometries in efforts to satisfy conservation instead of prescribing complicated flux polyhedron geometries which are prone to topological failures our vof advection scheme the non intersecting flux polyhedron advection nifpa method builds the flux polyhedron iteratively such that its intersection with neighboring flux polyhedra and any other unavailable volume is empty and its total volume matches the calculated flux volume during each iteration a candidate nominal flux polyhedron is extruded using an iteration dependent scalar the candidate is subsequently intersected with the volume guaranteed available to it at the time of the flux calculation to generate the candidate flux polyhedron the difference in the volume of the candidate flux polyhedron and the actual flux volume is used to calculate extrusion during the next iteration the choice in nominal flux polyhedron impacts the cost and accuracy of the scheme however it does not impact the methods underlying conservation and boundedness as such various robust nominal flux polyhedron are proposed and tested using canonical periodic kinematic test cases zalesak s disk and two and three dimensional deformation the tests are conducted on the median duals of a quadrilateral and triangular primal mesh in two dimensions and on the median duals of a hexahedral wedge and tetrahedral primal mesh in three dimensions comparisons are made with the adaptation of a conventional unsplit vof advection scheme to our collocated node based flow solver depending on the choice in the nominal flux polyhedron the nifpa scheme presented accuracies ranging from zeroth to second order and calculation times that differed by orders of magnitude for the nominal flux polyhedra which demonstrate second order accuracy on all tests and meshes the nifpa method s cost was comparable to the traditional topologically complex second order accurate vof advection scheme

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

a note to the kolmogorov sinaj entropy of fuzzy dynamical systems abstract in this note the kolmogorov sinaj entropy h m of fuzzy dynamical systems is studied it is showed that h m coincides on isomorphic objects finally we improve the formulation of kolmogorov sinaj s theorem on generators for the fuzzy case

a novel roe solver for incompressible two phase flow problems abstract in this paper a novel roe type riemann solver is developed for solving incompressible two phase flow problems the two phase flow is modeled by coupling the artificial compressibility based navier stokes equations and an interface advection equation in a conservative level set framework the proposed roe solver does not assume a locally constant density field while evaluating the flux jacobian as a result the solver is capable of capturing the exact jump conditions across the contact and shear waves which in turn helps in enhancing the accuracy of the interface motion moreover mass conservation characteristics are significantly improved by adopting conservative level set framework together with finite volume discretization of the system of equations a dual time stepping method with explicit pseudo time marching scheme is used for modeling the unsteady terms the efficacy of the proposed solver is demonstrated using various standard incompressible two phase flow problems involving low amplitude sloshing broken dam rayleigh taylor instability and droplet splash

a geometrical predictor corrector advection scheme and its application to the volume fraction function we present a multidimensional eulerian advection method for interfacial and incompressible flows in two dimensional cartesian geometry in the scheme we advect the grid nodes backwards along the streamlines to compute the pre images of the grid lines these pre images are approximated by continuous piecewise linear lines the enforcement of the discrete version of the incompressibility constraint is a very important issue to determine correctly the flux polygons and to reduce considerably the integration discretization and interpolation numerical errors the proposed method compares favorably with other previous multidimensional advection methods as long as the initial interface line is well reconstructed conversely we show that when the interface is very fragmented the total numerical error is completely dominated by the reconstruction error and in these conditions it is very difficult to assess which advection scheme is the most reliable one

an improved plic vof method for tracking thin fluid structures in incompressible two phase flows a drawback common to plic schemes is the difficulty of reconstructing the interface when filaments of a thickness smaller than the cell size are present in this work we present an improvement of the reconstruction method proposed by lopez et al a volume of fluid method based on multidimensional advection and spline interface reconstruction j comput phys 195 2004 718 for two dimensional flows which allows tracking fluid structures thinner than the cell size by allowing the interface to be represented in each cell by two non contiguous linear segments the method is based on using markers that are placed every time step at the mid points of the reconstructed cell interface segments the markers are used to improve the orientation and location of the interface segments although the method essentially remains of a vof type numerical tests show that a substantial improvement in accuracy is achieved over previous volume of fluid methods and that the proposed method compares well even with a recent hybrid markers and volume of fluid method

a hybrid particle level set method for improved interface capturing in this paper we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field our method uses lagrangian marker particles to rebuild the level set in regions which are underresolved this is often the case for flows undergoing stretching and tearing the overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method our method compares favorably with volume of fluid methods in the conservation of mass and purely lagrangian schemes for interface resolution the method is presented in three spatial dimensions

a stabilized finite element method using a discontinuous level set approach for solving two phase incompressible flows a numerical method for the simulation of three dimensional incompressible two phase flows is presented the proposed algorithm combines an implicit pressure stabilized finite element method for the solution of incompressible two phase flow problems with a level set method implemented with a quadrature free discontinuous galerkin dg method e marchandise j f remacle n chevaugeon a quadrature free discontinuous galerkin method for the level set equation journal of computational physics 212 2006 338 357 the use of a fast contouring algorithm n chevaugeon e marchandise c geuzaine j f remacle efficient visualization of high order finite elements international journal for numerical methods in engineering permits us to localize the interface accurately by doing so we can compute the discontinuous integrals without neither introducing an interface thickness nor reinitializing the level set the capability of the resulting algorithm is demonstrated with large scale numerical examples free surface flows dam break sloshing and small scale ones two phase poiseuille rayleigh taylor instability