connection between the burgers equation with an elastic forcing term and a stochastic process in this paper a complete analytical resolution of the one dimensional burgers equation with the elastic forcing term k2x f t k is an element of r is presented two methods existing for the case k 0 are adapted and generalized using variable and functional transformations valid for all values of space and time the emergence of a fokker planck equation in the method allows the establishment of a connection between the burgers equation and the ornstein uhlenbeck process

a mesh decoupled height function method for computing interface curvature in this paper a mesh decoupled height function method is proposed and tested the method is based on computing height functions within columns that are not aligned with the underlying mesh and have variable dimensions because they are decoupled from the computational mesh the columns can be aligned with the interface normal vector which is found to improve the curvature calculation for under resolved interfaces where the standard height function method often fails a computational geometry toolbox is used to compute the heights in the complex geometry that is formed at the intersection of the computational mesh and the columns the toolbox reduces the complexity of the problem to a series of straightforward geometric operations using simplices the proposed scheme is shown to compute more accurate curvatures than the standard height function method on coarse meshes a combined method that uses the standard height function where it is well defined and the proposed scheme in under resolved regions is tested this approach achieves accurate and robust curvatures for under resolved interface features and second order converging curvatures for well resolved interfaces

on large start up error of bdf2 abstract the second order backward difference bdf2 method requires solutions at two previous time steps and is often started with the first order backward difference bdf1 method the initial time step is typically taken to be small in order to minimize the effect of the first order error however contrary to expectations the solution computed in this way generates a very large error for stiff problems and the error grows as the initial time step is further reduced as shown in this note the problem is originated from the fact that the variable coefficient bdf2 method combined with any first or higher order one step method is asymptotically equivalent to the trapezoidal method and therefore not l stable based on the study presented it is strongly recommended that bdf2 method be started with a self starting second order implicit runge kutta method which guarantees second order accuracy and l stability at no additional cost

positive numerical integration methods for chemical kinetic systems abstract chemical kinetics conserves mass and renders nonnegative solutions a good numerical simulation would ideally produce mass balanced positive concentration vectors many time stepping methods are mass conservative however unconditional positivity restricts the order of a traditional method to one the projection method presented in this paper ensures mass conservation and positivity first a numerical approximation is computed with one step of a mass preserving traditional scheme if there are negative components the nearest vector in the reaction simplex is found by solving a quadratic optimization problem this vector is shown to better approximate the true solution a simpler version involves just one projection step and stabilizes the reaction simplex this technique works best when the underlying time stepping scheme favors positivity projected methods are more accurate than clipping and allow larger time steps for kinetic systems which are unstable outside the positive quadrant

conservative volume of fluid method for free surface simulations on cartesian grids this paper contributes to the state of the art in cartesian grid methods through development of new advection and reconstruction volume of fluid vof algorithms which are applicable to two and three dimensional flows a computationally efficient and second order vof reconstruction method is presented which uses no inversions to determine the interface normal direction next the lack of conservation of fluid volume in previous vof advection methods are shown to be due to improper treatment of one dimensional stretching in the velocity field this paper uses simple explicit time stepping and a cell center estimate of the volume fraction in the dilatation term to achieve a completely conservative advection method the new methods are simple robust and shown to out perform existing approaches for canonical test problems relevant to breaking wave flows

on donating regions lagrangian flux through a fixed curve lagrangian flux through a fixed curve segment naturally gives rise to the notion of a donating region a compact point set in which each particle will pass through the curve and contribute to the flux a precise geometric determination of the donating region has not been available until now the author proposes an explicit constructive and analytical definition of the donating region based on two characteristic curves of the flow field viz streaklines and timelines it is also shown that within a time interval of finite size each particle in the proposed donating region has a net effect of going across the fixed curve once the presented analysis might be potentially useful for flow topology studies lagrangian flux calculation and explicit interface tracking

an alternating digital tree adt algorithm for 3d geometric searching and intersection problems a searching algorithm is presented for determining which members of a set of n points in an n dimensional space lie inside a prescribed space subregion the algorithm is then extended to handle finite size objects as well as points in this form it is capable of solving problems such as that of finding the objects from a given set which intersect with a prescribed object the suitability of the algorithm is demonstrated for the problem of three dimensional unstructured mesh generation using the advancing front method

exact solutions for a forced burgers equation with a linear external force abstract we investigate the solutions of the burgers equation t u x t d x 2 u x t x f x t u x t u x t x u x t x t where f x t is an external force and x t represents a forcing term this equation is first analyzed in the absence of the forcing term by taking f x t k 1 t k 2 t x into account for this case the solution obtained extends the usual one present in the ornstein uhlenbeck process and depending on the choice of k 1 t and k 2 t it can present a stationary state or an anomalous spreading afterwards the forcing terms x t 1 t 2 t x and x t 3 x 4 x 3 are incorporated in the previous analysis and exact solutions are obtained for both cases

a high precision calculation method for interface normal and curvature on an unstructured grid in the volume of fluid algorithm the calculations of the interface normal and curvature are crucially important for accurately simulating interfacial flows however few methods have been proposed for the high precision interface calculation on an unstructured grid in this paper the authors develop a height function method that works appropriately on an unstructured grid in the process the definition of the height function is discussed and the high precision calculation method of the interface normal is developed to meet the necessary condition for a second order method this new method has highly reduced computational cost compared with a conventional high precision method because the interface normal calculation is completed by solving relatively simple algebraic equations the curvature calculation method is also discussed and the approximated quadric curve of an interface is employed to calculate the curvature following a basic verification the developed height function method is shown to successfully provide superior calculation accuracy and highly reduced computational cost compared with conventional calculation methods in terms of the interface normal and curvature in addition the height function method succeeds in calculating accurately the slotted disk revolution problem and the oscillating drop on unstructured grids therefore the developed height function method is confirmed to be an efficient technique for the high precision numerical simulation of interfacial flows on an unstructured grid

an improved height function technique for computing interface curvature from volume fractions an improved height function technique for calculating the interface curvature from volume fractions in three dimensions is presented a detailed analysis of the performance of the height function technique shows that depending on the interface orientation relative to the computational mesh appropriate discretization of the partial derivatives of the height function may considerably reduce the computed interface curvature errors and thus the spurious currents induced in flows with surface tension effects the accuracy and efficiency of the improved height function technique for 3d problems is compared with those of a convolution technique

analytic relations for reconstructing piecewise linear interfaces in triangular and tetrahedral grids in volume of fluid methods for interfacial flow simulations one essential process is the so called interface reconstruction in which an approximate interface is reconstructed from a given discrete volume fraction field in j comput phys 164 2000 228 237 scardovelli and zaleski presented analytical relations connecting linear interfaces and volume fractions in rectangular grids here we present analytical relations connecting linear interfaces and volume fractions in triangular and tetrahedral grids for computing the volume of fluid in an arbitrary polygonal or polyhedral fluid element we also cite some of the most efficient formulas for polygon area and polyhedron volume computations simple test cases show that this analytic method of interface reconstruction is about 18 times faster than an iterative method in two dimensions and four to six times faster in three dimensions the results can be in general applied to other fields as well

anomalous scaling in homogeneous isotropic turbulence the anomalous scaling exponents n of the longitudinal structure functions sn for homogeneous isotropic turbulence are derived from the navier stokes equations by using field theoretic methods to develop a low energy approximation in which the kolmogorov theory is shown to act effectively as a mean field theory the corrections to the kolmogorov exponents are expressed in terms of the anomalous dimensions of the composite operators which occur in the definition of sn these are calculated from the anomalous scaling of the appropriate class of nonlinear green function using an uv fixed point of the renormalization group which thereby establishes the connection with the dynamics of the turbulence the main result is an algebraic expression for n which contains no adjustable constants it is valid at orders n below g 1 where g is the fixed point coupling constant this expression is used to calculate n for orders in the range n 2 10 and the results are shown to be in good agreement with experimental data key examples being 2 0 7 3 1 and 6 1 8

short note on reducing interface curvature computation errors in the height function technique a detailed analysis of the errors involved in computing the interface curvature from volume fraction distributions using a height function technique is presented an improved version of the height function technique is proposed based on introducing a correction of the height function discretization error estimated from the local osculating spheres at interface points by using this error correction and an appropriate discretization of the partial derivatives of the height function a substantial improvement in the accuracy of the interface curvature computation can be efficiently achieved

interface curvature via volume fractions heights and mean values on nonuniform rectangular grids estimating the local curvature of an interface involves the local determination of normals to the interface and the rates that they turn along the interface this is challenging in volume of fluid type methods since the interface between materials is specified by the relative amount it cuts off from the computational cells that it crosses also referred to as volume fraction data rather than by a discrete set of points lying on the interface itself in this work we generalize the height function method to nonuniform rectangular grids we demonstrate analytically and numerically that using three successive adjacent integral mean values or column heights interface curvature can be estimated to second order accuracy the first derivative interface normal to third order accuracy and the curve location to fourth order accuracy each at its own special points we also show that there are special points where the curvature can be estimated to fourth order accuracy when using five successive mean values instead underlying all this is a result about the accuracy of the jth derivative of the kth degree polynomial that interpolates a function f at k 1 stencil points placed irregularly in an interval of width h namely for all smooth enough functions f and for k fixed and h getting small there are k 1 j special points in the interval at which the error in the jth derivative is of order o h k 2 j this is one order higher than the usual o h k 1 j error holding over the whole interval for that derivative the special points in the interval are the k 1 j real zeroes of certain f independent polynomials of degree k 1 j with coefficients depending on the interval s stencil points in our case let f x be an indefinite integral of the unknown interfacial curve f x then f x i at k 1 successive stencil points x i is calculated using cumulative sums of the k successive integral mean values of f weighted by the successive interval sizes the k 1 points x i f x i are now interpolated by a kth degree polynomial pf x it s first derivative pf 1 approximates the unknown curve f while pf 2 and pf 3 respectively approximate f 1 and f 2 thus the results above using three successive mean values correspond to k 3 and j 3 2 1 the curvature result using five successive mean values correspond to k 5 and j 3 for this last case since the curvature k f 2 1 f 1 2 3 2 we use the facts that f 1 f 2 pf 2 o h 4 on the whole stencil interval not just at the three special points having o h 4 accuracy for f 2

interface reconstruction with least squares fit and split advection in three dimensional cartesian geometry in this paper we present and analyze different volume of fluid vof reconstruction and advection algorithms that approximate the interface separating two immiscible fluids in the three dimensional space the paper describes the improvement of the reconstruction when a least square fit algorithm which minimizes a distance functional is applied its performance is tested for several smooth surfaces against other simple reconstruction methods then eulerian lagrangian and mixed split advection schemes are presented and analyzed in particular one advection method is discussed that conserves mass exactly for a divergence free velocity field thus allowing computations to machine precision

instantons in the burgers equation the instanton solution for the forced burgers equation is found this solution describes the exponential tail of the probability distribution function of velocity differences in the region where shock waves are absent that is for large positive velocity differences the results agree with the one found recently by polyakov who used the operator product conjecture if this conjecture is true then our wkb asymptotics of the wyld functional integral should be exact to all orders of perturbation expansion around the instanton solution we also generalized our solution for the arbitrary dimension of the burgers kpz equation as a result we found the asymptotics of the angular dependence of the velocity difference probability distribution function textcopyright 1996 the american physical society

kolmogorov turbulence in a random force driven burgers equation anomalous scaling and probability density functions high resolution numerical experiments described in this work show that velocity fluctuations governed by the one dimensional burgers equation driven by a white in time random noise with the spectrum ensuremath vert f k mathrm ensuremath vert 2 ifmmode bar else textasciimacron fi ensuremath propto mathit k mathrm ensuremath 1 exhibit a biscaling behavior all moments of velocity differences mathit s mathit n mathrm ensuremath le 3 r ensuremath vert u x r u x mathrm ensuremath vert mathit n ifmmode bar else textasciimacron fi ensuremath equiv ensuremath vert ensuremath delta u mathrm ensuremath vert mathit n ifmmode bar else textasciimacron fi ensuremath propto mathit r mathit n 3 while mathit s mathit n g3 r ensuremath propto mathit r mathit n ensuremath xi with ensuremath xi mathit n ensuremath approxeq 1 for real ng0 chekhlov and yakhot phys rev e 51 r2739 1995 the probability density function which is dominated by coherent shocks in the interval ensuremath delta u0 is scrp ensuremath delta u r ensuremath propto ensuremath delta u mathrm ensuremath mathit q with q ensuremath approxeq 4 a phenomenological theory describing the experimental findings is presented

a new stabilization of adaptive step trapezoid rule based on finite difference interrupts the adaptive step trapezoid rule tr is a generally effective numerical integrator but it is prone to ringing instability and solution stall we introduce a new stabilization algorithm based on finite difference interrupts fdi that reduces ringing and prevents stall unlike previously reported stabilization schemes our algorithm achieves stability and second order accuracy without incurring significant computational cost or spurious diffusion tr with fdi is at least as stable and accurate as existing methods when solving the prototypical scalar problem we demonstrate that it has better stability accuracy and cost when solving the tightly coupled system describing free surface flows of viscoelastic liquids though we demonstrate tr with fdi in the context of a finite element method of lines it is applicable to any tr based algorithm

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

on the approximation of the stochastic burgers equation we prove mathematical approximation results for the hyperviscous burgers equation driven by additive gaussian noise in particular we show that solutions of approximating equations driven by a discretized noise converge towards the solution of the original equation when the discretization parameter gets small the convergence takes place in the expected value of arbitrary powers of certain norms i e all moments of the difference of the solutions tend to zero in certain function spaces for the hyperviscous burgers equation these results are applied to justify the approximation of certain correlation functions that play a major role in statistical turbulence theory

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

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

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

an equation for the dissipation rate correlation and its implications for the intermittency exponent μ in turbulence we derive an equation satisfied by the dissipation rate correlation function left langle varepsilon vec x vec r t tau varepsilon vec x t right rangle for the homogeneous isotropic state of fully developed turbulence from the the navier stokes equation in the equal time limit we show that the equation leads directly to two intermittency exponents 1 2 6 and 2 z 4 4 where the s are exponents of velocity structure functions and z 4 is a dynamical exponent characterizing the fourth order structure function we discuss the contributions of the pressure terms to the equation and the consequences of hyperscaling

velocity difference probability density functions for burgers turbulence in this paper the polyakov equation phys rev e bold 52 6183 1995 for the velocity difference probability density functions with the random gaussian external force with the correlation function kappa y approximately 1 minus y sup alpha is analyzed solutions for the cases alpha l brace 2 1 2 1 r brace are found which agree very well with available numerical results it is also argued that the stationary regime of burgers turbulence can depend not only on the distribution of the external force but also on the dissipative regularization copyright ital 1997 ital the american physical society

multicomponent turbulence the spherical limit and non kolmogorov spectra a set of models for homogeneous isotropic turbulence is considered in which the navier stokes equations for incompressible fluid flow are generalized to a set of n coupled equations in n velocity fields it is argued that in order to be useful these models must embody a new group of symmetries and a general formalism is laid out for their construction the work is motivated by similar techniques that have had extraordinary success in improving the theoretical understanding of equilibrium phase transitions in condensed matter systems we consider two classes of models a simpler class model i which does not contain an exact galilean symmetry and a more complicated extended class model ii which does the key result is that these models simplify when n is large the so called spherical limit n ensuremath rightarrow ensuremath infty can be solved exactly yielding closed sets of nonlinear integral equations for the response and correlation functions for model i these equations known as kraichnan s direct interaction approximation equations are solved fully in the scale invariant turbulent regime for model ii these equations are more complicated and their full solution is left for future work implications of these results for real turbulence n 1 are discussed in particular it is argued that previously applied renormalization group techniques based on an expansion in the exponent y that characterizes the driving spectrum are incorrect and that the kolmogorov exponent ensuremath zeta has a nontrivial dependence on n with ensuremath zeta n ensuremath rightarrow ensuremath infty 3 2 for both sets of models this value is close to the experimental result ensuremath zeta ensuremath simeq 5 3 which must therefore result from higher order corrections in powers of 1 n prospects for calculating these corrections are briefly discussed though daunting such calculations might provide a controlled perturbation expansion for the kolmogorov and other exponents our techniques may also be applied to other nonequilibrium dynamical problems such as the kardar parisi zhang equation for interface growth and perhaps to turbulence in nonlinear wave systems

a fully implicit fully adaptive time and space discretisation method for phase field simulation of binary alloy solidification a fully implicit numerical method based upon adaptively refined meshes for the simulation of binary alloy solidification in 2d is presented in addition we combine a second order fully implicit time discretisation scheme with variable step size control to obtain an adaptive time and space discretisation method the superiority of this method compared to widely used fully explicit methods with respect to cpu time and accuracy is shown due to the high nonlinearity of the governing equations a robust and fast solver for systems of nonlinear algebraic equations is needed to solve the intermediate approximations per time step we use a nonlinear multigrid solver which shows almost h independent convergence behaviour

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

intermittency of height fluctuations in stationary state of the kardar parisi zhang equation with infinitesimal surface tension in 1 1 dimensions the kardar parisi zhang kpz equation with infinitesimal surface tension dynamically develops sharply connected valley structures within which the height derivative is not continuous we discuss the intermittency issue in the problem of stationary state forced kpz equation in 1 1 dimensions it is proved that the moments of height increments c a behave as x 1 x 2 xi a with xi a a for length scales x 1 x 2 sigma the length scale sigma is the characteristic length of the forcing term we have checked the analytical results by direct numerical simulation

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

turbulence without pressure we develop exact field theoretic methods to treat turbulence when the effect of pressure is negligible we find explicit forms of certain probability distributions demonstrate that the breakdown of galilean invariance is responsible for intermittency and establish the operator product expansion we also indicate how the effects of pressure can be turned on perturbatively

a newton krylov solver for remapping based volume of fluid methods a new conservative remapping method is presented here for generating multi dimensional fluxes used for the time integration of hyperbolic equations through single stage unsplit advection a solution to the lagrangian mesh location is generated that converges the volumes of all remapped flux definitions to the volumes of equivalent one dimensional flux definitions at their corresponding cell faces the method is most feasible within localized subdomains of an eulerian solution domain the main utility of the method is the generation of fluxes in the vicinity of discontinuities and sharp gradients that does not introduce physically spuriously extrema while locally conserving mass in generating solutions to lagrangian mesh location by solving a nonlinear equation system the new remap requires the anchoring of the lagrangian mesh removal of degrees of freedom to make the problem definite the method is based on characteristic tracing along trajectories assumed to be linear this paper investigates the convergence and sensitivity of the lagrangian mesh solver to mesh and timestep size in assessing its merit for application in transient cfd codes for single stage unsplit advection volume of fluid methods the new remap suppresses the introduction of undershoots overshoots and wisps in combination with a conservative remapping phase the new method for fixing the location of the lagrangian mesh eliminates the need for heuristic redistribution or clipping procedures for spurious extrema removal

structure functions in the stochastic burgers equation we study analytically and numerically structure functions s q r in the one dimensional burgers equation driven by noise with variance ensuremath propto k ensuremath beta in fourier space a when the noise is cut off at some length l c and b when it is not we present exact relations satisfied by s 3 r the von karman howarth relation and s 4 r that form the basis of our analysis when there is a cutoff length shocks occur and s q r ensuremath propto r for qg 2 for ensuremath delta lrl l c where ensuremath delta is the shock thickness for all ensuremath beta between 1 and 2 we deduce this behavior from the exact relations along with an ansatz that is verified numerically when there is no cutoff length multifractal behavior is known to occur only when ensuremath beta 0 through a study of exact expression s 3 we highlight the difference between multifractality in this case as compared to the case with a cutoff

analytical relations connecting linear interfaces and volume fractions in rectangular grids abstract the computational geometric problems arising when a linear interface cuts a cube are considered they are of interest in particular for the calculation of volume fractions or interface positions in three dimensional interface calculations in the volume of fluid vof methods typically the normal vector is known one then wants to compute the volume fraction knowing the interface position or conversely the interface position knowing the volume fraction explicit expressions of general use are given and the algorithms used to search for solutions are described in detail explicit formulas for cubic roots are found to be less than two thirds as time consuming as newton raphson iterations

accurate interface normal and curvature estimates on three dimensional unstructured non convex polyhedral meshes this paper presents a framework for extending the height function technique for the calculation of interface normals and curvatures to unstructured non convex polyhedral meshes with application to the piecewise linear interface calculation volume of fluid method the methodology is developed with reference to a collocated node based finite volume two phase flow solver that utilizes the median dual mesh requiring a set of data structures and algorithms for non convex polyhedral operations truncation of a polyhedron by a plane intersection of two polyhedra joining of two convex polyhedra volume enforcement of a polyhedron by a plane and volume fraction initialization by a signed distance function by leveraging these geometric tools a geometric interpolation strategy for embedding structured height function stencils in unstructured meshes is developed the embedded height function technique is tested on surfaces with known interface normals and curvatures namely cylinder sphere and ellipsoid tests are performed on the median duals of a uniform cartesian mesh a wedge mesh and a tetrahedral mesh and comparisons are made with conventional methods across the tests the embedded height function technique outperforms contemporary methods and its accuracy approaches the accuracy that the traditional height function technique exemplifies on uniform cartesian meshes

a 3d unsplit forward backward volume of fluid approach and coupling to the level set method this paper presents a novel methodology for interface capturing in two phase flows by combining a lagrangian eulerian volume of fluid approach with a level set method while the volume of fluid transport relies on a robust and accurate polyhedral library any high order level set transport may be used the method is shown to be less restrictive in terms of cfl conditions than split volume of fluid methods various geometric integration schemes are proposed and tested for linear velocity fields mass error is shown to vanish and to be third order otherwise the method is validated on 2d and 3d test cases conservation properties are shown to be excellent while geometrical accuracy remains satisfactory even for complex flows such as the primary breakup of a liquid jet

on the properties and limitations of the height function method in two dimensional cartesian geometry in this study we define the continuous height function to investigate the approximation of an interface line and its geometrical properties with the height function method we show that in each mixed cell the piecewise linear interface reconstruction and the approximation of the derivatives and curvature based on three consecutive height function values are second order accurate we also discuss the quadratic reconstruction and fourth order accurate expressions of the normal and curvature we present a hierarchical algorithm to compute the normal vector and curvature of an interface line with the height function method that switches automatically between second and fourth order approximations and that can be applied also when the local radius of curvature is of the order of the grid spacing

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

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

intermittency exponents and energy spectrum of the burgers and kpz equations with correlated noise we numerically calculate the energy spectrum intermittency exponents and probability density p u 0 of the one dimensional burgers and kpz equations with correlated noise we have used pseudo spectral method for our analysis when of the noise variance of the burgers equation variance k 2 exceeds 3 2 large shocks appear in the velocity prole leading to hju k j 2 i k 2 and structure function hju x r t u x t j q i r suggesting that the burgers equation is intermittent for this range of for 1660 the prole is dominated by noise and the spectrum hjh k j 2 i of the corresponding kpz equation is in close agreement with the medina et al renormalization group predictions in the intermediate range 0 1 2 we have also calculated the energy cascade rates for all and found a constant ux for all 1 2 c 2000 published

from scaling to multiscaling in the stochastic burgers equation we investigate the scaling behavior of the structure functions s q r u r ensuremath u 0 q ensuremath propto r ensuremath zeta q in the stochastic burgers equation as a function of the exponent ensuremath beta that characterizes the scale of noise correlations for 0l ensuremath beta l ensuremath 1 we analyze the exact equations satisfied by s q r q 3 4 5 based on certain ans atze for small negative ensuremath beta kolmogorov like scaling with ensuremath zeta q ensuremath q ensuremath beta 3 is obtained as ensuremath beta ensuremath rightarrow ensuremath 1 an increasing multifractal structure occurs with bifractality for ensuremath beta l ensuremath 1 we determine ensuremath zeta 4 and ensuremath zeta 5 which are piecewise continuous and the associated multifractal scaling exponents

stochastic burgers equation with fractional derivative driven by multiplicative noise abstract this article is devoted to the study of the existence and uniqueness of mild solution to time and space fractional stochastic burgers equation perturbed by multiplicative white noise the required results are obtained by stochastic analysis techniques fractional calculus and semigroup theory moreover we prove the regularity properties of mild solution to this generalized burgers equation in the framework of bochner spaces

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

taylor s frozen flow hypothesis in burgers turbulence by detailed analytical treatment of the shock dynamics in the burgers turbulence with large scale forcing we calculate the velocity structure functions between pairs of points displaced both in time and space our analytical treatment verifies the so called taylor s frozen flow hypothesis without relying on any closure and under very general assumptions we discuss the limitation of the hypothesis and show that it is valid up to time scales smaller than the correlation time scale of temporal velocity correlation function we support the analytical calculation by performing numerical simulation of the periodically kicked burgers equation

a balanced force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework a new balanced force algorithm is presented for modeling interfacial flow with surface tension the algorithm is characterized by a pressure correction method with the interfaces represented by volume fractions within this flow algorithm we devise a continuous e g continuum surface tension model and a sharp e g a ghost fluid method interface representation of the surface tension induced interfacial pressure jump condition the sharp interface representation is achieved by temporarily reconstructing distance functions from volume fractions we demonstrate that a flow algorithm designed to legislate force balance retains an exact balance between surface tension forces and the resulting pressure gradients this balance holds for both continuous and sharp representations of interfacial surface tension the algorithm design eliminates one of the elusive impediments to more accurate models of surface tension driven flow the remaining of which is accurate curvature estimation to validate our formulation we present results for an equilibrium static drop in two and three dimensions having an arbitrary density jump across the interface we find that the sharp surface tension method yields an abrupt pressure jump across the interface whereas the continuous surface tension method results in a smoother transition both methods however yield spurious velocities of the same order the origin of which is due solely to errors in curvature dynamic results are also presented to illustrate the versatility of the method

a computational framework for conservative three dimensional unsplit geometric transport with application to the volume of fluid vof method abstract in this work a novel computational framework for calculating convection fluxes is developed and employed in the context of the piecewise linear interface calculation plic volume of fluid vof method the scheme is three dimensional unsplit discretely conservative and bounded the scheme leverages the idea of semi lagrangian transport to estimate the amount of liquid that is fluxed through each face during a time step the present work can be seen as an extension of the two dimensional emfpa method of lopez et al 2004 16 to three dimensions and an improvement of the three dimensional fmfpa 3d method of hernandez et al 2008 17 with the addition of discrete conservation in fmfpa 3d fluxes of liquid volume fraction are calculated by transporting a cell face back in time with a semi lagrangian method that uses cell face edge velocities to produce a flux hexahedron with flat faces the flux hexahedron may overlap with neighboring fluxes hindering the conservation properties of the method the proposed method computes the fluxes by transporting the cell face back in time using a semi lagrangian step based on the cell face corner velocities which results in a three dimensional generalized flux hexahedron that does not typically have flat faces however the flux volumes do not overlap and discrete conservation can be achieved the complex flux volume is partitioned into a collection of simplices and a simple sign convention allows the calculation of the flux to be reduced to a straightforward and systematic algorithm the proposed vof scheme is tested on multiple benchmark cases including zalesak s disk two and three dimensional deformation tests and the evolution of a droplet in homogeneous isotropic turbulence

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 plic vof method suited for adaptive moving grids this paper presents a new variant of the volume of fluid vof color function c advection algorithm based on the piecewise linear interface construction plic method suitable for use on general moving grids from several existing methods for reconstructing the linear interface we adopted the least squares volume of fluid interface reconstruction algorithm lvira which can be easily implemented on general grids the distinguishing step in the advection algorithm that takes into account the grid movement is the construction of the donating region containing the fluid passing through corresponding cell faces in a single time step the donating regions are constructed utilizing fluid velocity in cell corners relative to grid corner velocities the method is conservative as it complies with the space conservation law scl and requires a proper definition of the grid velocities and fluxes due to the grid movement the accuracy of the presented advection algorithm is assessed with standard test cases it is comparable with other plic based algorithms on fixed grids while the applicability on adaptive moving grids enables a considerable reduction in the number of grid cells

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

field theoretic thermodynamic uncertainty relation general formulation exemplified with the kardar parisi zhang equation we propose a field theoretic thermodynamic uncertainty relation as an extension of the one derived so far for a markovian dynamics on a discrete set of states and for overdamped langevin equations we first formulate a framework which describes quantities like current entropy production and diffusivity in the case of a generic field theory we will then apply this general setting to the one dimensional kardar parisi zhang equation a paradigmatic example of a non linear field theoretic langevin equation in particular we will treat the dimensionless kardar parisi zhang equation with an effective coupling parameter measuring the strength of the non linearity it will be shown that a field theoretic thermodynamic uncertainty relation holds up to second order in a perturbation expansion with respect to a small effective coupling constant the calculations show that the field theoretic variant of the thermodynamic uncertainty relation is not saturated for the case of the kardar parisi zhang equation due to an excess term stemming from its non linearity

a moving mesh interface tracking method for 3d incompressible two phase flows an interface tracking method using an unstructured moving mesh has been developed for simulating three dimensional incompressible and immiscible two phase flows the interface mesh is moved in a lagrangian fashion a local mesh adaptation method is used to capture the changing interface curvature maintain good mesh quality and deal with large deformation the interface is of zero thickness so the jump and continuity conditions across the interface are implemented directly without any smoothing of the properties of the two fluids this is theoretically beneficial compared to other methods that allow the fluids properties to continuously vary in an interface region the curvature for interfacial tension calculation is geometrically computed by a least squares parabola fitting method a mesh separation scheme for interfacial flows is employed to handle topological transition the numerical technique is tested and validated by several cases which include a two layer couette flow an oscillating droplet surrounded by gas and a droplet in shear flow the results obtained from numerical simulations are found to be in excellent agreement with analytical solutions and experimental results a collapsing ligament in a quiescent gas flow and a modulated jet pinching are simulated to show the robustness of the method

three dimensional forced burgers turbulence supplemented with a continuity equation we investigate turbulent limit of the forced burgers equation supplemented with a continuity equation in three dimensions the scaling exponent of the conditional two point correlation function of density i e r x1 r x2 udu ux12x2u 2a 3 is calculated self consistently in the nonuniversal region from which we obtain a 353 also we derive an equation governing the evolution of the probability density function pdf of longitudinal velocity increments in length scale from which a possible mechanism for the dependence of the inertial pdf to one point urms is developed

numerical solution of the kardar parisi zhang equation in one two and three dimensions abstract on a course grained level a family of microscopic growth processes may be described by a stochastic differential equation which is solved numerically for surface dimensions d 1 2 and 3 dimensional analysis shows that the spatial discretization parameter has the meaning of an effective coupling constant the numerical stability of the euler integration scheme is discussed for the strong coupling exponents defined by surface width time the following effective values were obtained d 1 0 330 0 004 and d 2 0 24 0 005 considering the width and its ensemble fluctuations at constant dimensionless time the transition between strong and weak coupling phases is located in d 3 for the largest coupling for which reliable data are available we obtain an effective exponent close to the best estimates on discrete models d 3 0 17

an accurate adaptive solver for surface tension driven interfacial flows a method combining an adaptive quad octree spatial discretisation geometrical volume of fluid interface representation balanced force continuum surface force surface tension formulation and height function curvature estimation is presented the extension of these methods to the quad octree discretisation allows adaptive variable resolution along the interface and is described in detail the method is shown to recover exact equilibrium to machine accuracy between surface tension and pressure gradient in the case of a stationary droplet irrespective of viscosity and spatial resolution accurate solutions are obtained for the classical test case of capillary wave oscillations an application to the capillary breakup of a jet of water in air further illustrates the accuracy and efficiency of the method the source code of the implementation is freely available as part of the gerris flow solver