lie symmetries of a coupled nonlinear burgers heat equation system a symmetry group analysis of a coupled burgers heat equation system of partial differential equations initially introduced in a study of non classical similarity solutions of the heat equation is carried out the point lie algebra of the system g is shown to possess a maximal solvable ideal a with quotient algebra b g a approximately sl2 r where sl2 r is the lie algebra of 2 2 matrices with zero trace analysis of the prolongation structure of the system yields the backlund transformation obtained previously by painleve analysis the backlund transformation can be expressed as a map onto two coupled linear heat equations or alternatively as a map onto burgers equation the role of sl2 r in a seven dimensional lie algebra l1 obtained by truncating the open ended algebraic prolongation structure is emphasised

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

symmetry classification and exact solutions of the kramers equation the complete classification of lie symmetries for the kramers equation is described group invariance under infinitesimal transformations is used to generate a wide class of exact solutions some of the known solutions are obtained as particular cases the general results can also be used to solve some types of moving boundary problems we researched some q conditional symmetry properties of this equation

on the generation and maintenance of waves and turbulence in simulations of free surface turbulence one of the challenges in numerical simulation of wave turbulence interaction is the precise setup and maintenance of wave and turbulence fields in this paper we investigate techniques for the generation and suppression of specific surface wave modes the generation of turbulence in an inhomogeneous physical domain with a wavy boundary fitted grid and the generation and maintenance of waves and turbulence during the complex wave turbulence interaction process we apply surface pressure to generate and suppress waves based on the solution of linearized cauchy poisson problem we derive three pressure expressions which lead to a d function method a time segment method and a gradual method numerical experiments show that these methods generate waves as specified and eliminate spurious waves effectively the nonlinear wave effect is accounted for with a time relaxation method for turbulence generation we extend the linear forcing method to an inhomogeneous physical domain with a curvilinear computational grid effects of force distribution and computational grid distortion are examined for wave turbulence interaction we develop an algorithm to instantaneously identify specific progressive and standing waves to precisely control the wave amplitude in a complex turbulent flow field we further develop an energy controlling method finally a simulation example of wave turbulence interaction is presented results show that turbulence has unique features in the presence of waves velocity fluctuations are found to be strongly dependent on the wave phase variations of these fluctuations are explained by the pressure strain correlation associated with the wave induced strain field

equivalence and symmetries for variable coefficient linear heat type equations i a systematic and unified approach to transformations and symmetries of a class of variable coefficient heat type linear partial differential equations pdes is presented the complete symmetry group classification is re performed in the current context a useful criterion which is necessary and sufficient for being reducible to the standard heat equation by point transformations is established a similar criterion is also valid for the equations to have a four or six dimensional symmetry group nontrivial symmetry groups in this situation the basis elements are listed in terms of coefficients a number of illustrative examples are given in particular some applications from the recent literature are re examined in our new approach multidimensional parabolic pdes of heat and schrodinger type are also considered

direct numerical simulation of flow past elliptic cylinders flow over elliptic cylinders can be considered prototypical of flow over a range of bluff bodies since the geometry allows one to study the effect of both thickness and angle of attack on the flow field therefore a careful study of this flow should provide valuable insight into the phenomenon of unsteady separation and the structure of bluff body wakes with this in mind a spectral collocation technique has been developed to simulate the full three dimensional incompressible flow over elliptic cylinders and unlike spectral element and spectral multidomain techniques here the flow is solved in a single domain the equations are discretized on a body fitted elliptic cylindrical grid and properties of the metric associated with this coordinate system are used to solve the governing equations in an efficient manner key issues including the inflow and outflow boundary conditions and time discretization are discussed in detail with the hope that this will facilitate future simulations of similar flows finally we present results of two and three dimensional simulations for a range of flow and geometric parameters the results are compared with available experimental data and it is found that important quantities like strouhal numbers and drag coefficients match well with established values

a study of the effects of grid non orthogonality on the solution of shallow water equations in boundary fitted coordinate systems in the present study an existing two dimensional boundary fitted model j hydraul eng asce 122 9 1996 512 is used to study the effect of grid non orthogonality on the solution of shallow water equations using boundary fitted grids the linearized two dimensional shallow water equations are expressed in terms of the grid angle and aspect ratio the truncation errors of the finite difference approximations used in the solution of the governing equations are shown to be dependent on the grid angle and the aspect ratio the coefficient of the truncation error was shown to increase with the decrease in the grid angle the rms errors in model predicted surface elevations and velocities for the case of seiching in a rectangular basin are found to increase gradually as the grid resolution decreases from 174 to 80 grid points per wavelength or as the grid angle decreases from 90 to 50 and increases rather sharply for a grid angle of 30 at grid resolutions less than 80 gridpoints per wavelength the model predicted surface elevations for the case of tidal forcing in a rectangular basin are found to be insensitive to the grid angle at grid resolutions higher than 600 gridpoints per wavelength the rms error in the model predicted velocities is found to increase gradually as the grid angle decreases from 90 to 30 or as the grid resolution decreases from 1400 gridpoints per wavelength to 400 gridpoints per wavelength and increases sharply as the grid resolution decreases from 400 to 150 gridpoints per wavelength two dimensional depth averaged hydrodynamic modeling of tidal circulation in narragansett bay using three different boundary fitted grids showed that the model predicted surface elevations are insensitive to the grid angle at grid resolutions as low as 200 gridpoints per wavelength however the model predicted velocities were found to increase as the grid resolution decreases from 600 to 200 gridpoints per wavelength we conclude from this study that grid angle and grid resolution affects the accuracy of the model predicted currents and the numerical dispersion increases with the decrease in grid angle or grid resolution and these are in agreement with that reached by sankaranarayanan and spaulding dispersion and stability analyses of shallow water equations in boundary fitted coordinates department of ocean engineering university of rhode island 2001 p 33 through a fourier analysis of the discretized equations in boundary fitted coordinates

chern simons theory matrix integrals and perturbative three manifold invariants the universal perturbative invariants of rational homology spheres can be extracted from the chern simons partition function by combining perturbative and nonperturbative results we spell out the general procedure to compute these invariants and we work out in detail the case of seifert spaces by extending some previous results of lawrence and rozansky the chern simons partition function with arbitrary simply laced group for these spaces is written in terms of matrix integrals the analysis of the perturbative expansion amounts to the evaluation of averages in a gaussian ensemble of random matrices as a result explicit expressions for the universal perturbative invariants of seifert homology spheres up to order five are presented

remodeling the b model we propose a complete new formalism to compute unambiguously b model open and closed amplitudes in local calabi yau geometries including the mirrors of toric manifolds the formalism is based on the recursive solution of matrix models recently proposed by eynard and orantin the resulting amplitudes are non perturbative in both the closed and the open moduli the formalism can then be used to study stringy phase transitions in the open closed moduli space at large radius this formalism may be seen as a mirror formalism to the topological vertex but it is also valid in other phases in the moduli space we develop the formalism in general and provide an extensive number of checks including a test at the orbifold point of ap fibrations where the amplitudes compute the t hooft expansion of vevs of wilson loops in chern simons theory on lens spaces we also use our formalism to predict the disk amplitude for the orbifold mathbb c 3 mathbb z 3

an efficient method for temporal integration of the navier stokes equations in confined axisymmetric geometries a method for temporal integration of the navier stokes equations written in cylindrical coordinates is described the objective is to avoid the severe time step limitation usually encountered in confined axisymmetric geometries e g pipe flow caused by a fine azimuthal grid spacing around the centerline and the desire to refine the grid in the radial direction near walls avoiding severe time step limitations usually involves treating all terms with derivatives in the radial and azimuthal directions with an implicit time integration scheme however this leads to a set of coupled nonlinear equations which generally require complex and costly solution procedures the scheme described in this paper decomposes the computational domain into two regions within each region only the derivatives in one coordinate direction is treated implicitly conditions at the interface between the regions are determined to maintain the overall temporal accuracy of the basic time integration schemes results from a direct numerical simulation dns of turbulent pipe flow are validated against computational and experimental results from the literature it is demonstrated that this new scheme allows for larger time steps than other schemes leading to significant cpu savings

symmetries of the fokker planck equation with a constant diffusion matrix in 2 1 dimensions we completely classify the symmetries of the fokker planck equation in two spatial dimensions with a constant positive definite diffusion matrix we apply these results to construct group invariant solutions for a physically interesting family of fokker planck equations

simulation of viscous flows with undulatory boundaries part i basic solver a numerical method for the simulation of viscous flows with undulatory walls and free surfaces is presented the simulation domain is discretized by a boundary fitted and time dependent grid the navier stokes equations subject to fully nonlinear kinematic and dynamic boundary conditions at the free surface and no slip boundary condition at the wall are simulated by a hybrid pseudo spectral and finite difference method in space and a semi implicit fractional step method in time the performance of the method is demonstrated by a series of test cases including flows over wavy boundaries various surface waves and interaction between vortices and free surfaces validation by convergence test and extensive comparisons with previous theoretical experimental and numerical studies indicate the accuracy and efficiency of the method finally a simulation example of turbulence and free surface interaction is presented results show that the rich features of the free surface such as surface waves splats anti splats dimples and scars are captured accurately characteristic vortical structures and variation of turbulence statistics in the near surface region are also elucidated

lie symmetries of some equations of the fokker planck type the structure of the local lie groups of symmetries of some partial differential equations of the fokker planck type in one space dimension is investigated a connection between these groups and the group sl2 r is established in the sense that they are all shown to be locally isomorphic to sl2 r a where a is the radical it is conjectured that the groups of lie symmetries of all fokker planck equations in one space dimension have this structure the notion of partial invariance due to ovsiannikov is applied to the equations studied it appears plausible that the class of partially invariant solutions of these equations is larger than the class of invariant solutions although no explicit demonstration of this claim is available at present

an embedded boundary formulation for large eddy simulation of turbulent flows interacting with moving boundaries a non boundary conforming formulation for simulating complex turbulent flows with dynamically moving boundaries on fixed cartesian grids is proposed the underlying finite difference solver for the filtered incompressible navier stokes equations is based on a second order fractional step method on a staggered grid to satisfy the boundary conditions on an arbitrary immersed interface the velocity field at the grid points near the interface is reconstructed using momentum forcing without smearing the sharp interface the concept of field extension is also introduced to treat the points emerging from a moving solid body to the fluid laminar flow cases and large eddy simulations les are presented to demonstrate the formal accuracy and range of applicability of the method in particular simulations of laminar flow induced by the harmonic in line oscillation of a circular cylinder in quiescent fluid and from a transversely oscillating cylinder in a free stream are presented and compared to reference simulations and experiments les of turbulent flow over a traveling wavy wall and transitional flow through a bileaflet prosthetic heart valve are also shown all results are in very good agreement with reference results in the literature

a non staggered grid fractional step method for time dependent incompressible navier stokes equations in curvilinear coordinates a numerical method for solving three dimensional time dependent incompressible navier stokes equations in curvilinear coordinates is presented the non staggered grid method originally developed by c m rhie and w l chow aiaaj 21 1525 1983 for steady state problems is extended to compute unsteady flows in the computational space the cartesian velocity components and the pressure are defined at the center of a control volume while the volume fluxes are defined at the mid point on their corresponding cell faces the momentum equations are integrated semi implicitly by the approximate factorization technique the intermediate velocities are interpolated onto the faces of the control volume to form the source terms of the pressure poisson equation which is solved iteratively with a multigrid method the compatibility condition of the pressure poisson equation is satisfied in the same manner as in a staggered grid method mass conservation can be satisfied to machine accuracy the pressure boundary condition is derived from the momentum equations solutions of both steady and unsteady problems including the large eddy simulation of a rotating and stratified upwelling flow in an irregular container established the favorable accuracy and efficiency of the present method

a ghost cell immersed boundary method for flow in complex geometry an efficient ghost cell immersed boundary method gcibm for simulating turbulent flows in complex geometries is presented a boundary condition is enforced through a ghost cell method the reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second order accuracy of the base solver both dirichlet and neumann boundary conditions can be treated the current ghost cell treatment is both suitable for staggered and non staggered cartesian grids the accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface numerical results are compared with experimental data and boundary fitted grid results the method is further extended to an existing ocean model mitgcm to simulate geophysical flow over a three dimensional bump the method is easily implemented as evidenced by our use of several existing codes

ovsiannikov s method and the construction of partially invariant solutions a plausibility argument presented by the first two authors in an earlier paper j math phys 26 3042 1985 concerning the existence of partially invariant solutions for some equations of the fokker planck type is made precise by the explicit construction of one such solution in the process a substantial simplification of ovsiannikov s method for finding partially invariant solutions is achieved in addition the class of partially invariant solutions obtained by ovsiannikov for the equations of transonic flow of a gas is enlarged

symmetry classification and exact solutions of the one dimensional fokker planck equation with arbitrary coefficients of drift and diffusion symmetry properties of the one dimensional fokker planck equations with arbitrary coefficients of drift and diffusion are investigated it is proved that the group symmetry of these equations can be one two four or six parametric and corresponding criteria are obtained the changes of the variables reducing fokker planck equations to the heat equation and schrodinger equation with certain potentials are determined using the substructure of the invariance algebra of the rayleigh process equation we obtain the differential invariants and construct some classes of the exact solutions of this equation

a contribution of the trivial connection to the jones polynomial and witten s invariant of 3d manifolds ii we extend the results of our previous paper 1 from knots to links by using a formula for the jones polynomial of a link derived recently by n reshetikhin we establish a relation between the parameters of this formula and the multivariable alexander polynomial this relation is illustrated by an example of a torus link we check that our expression for the alexander polynomial satisfies some of its basic properties finally we derive a link surgery formula for the loop corrections to the trivial connection contribution to witten s invariant of rational homology spheres

a second order boundary fitted projection method for free surface flow computations this paper describes a new approach to the high fidelity simulation of axisymmetric free surface flows a boundary fitted grid is coupled with a new projection method for the solution of the navier stokes equations with second order accuracy in space and time two variants of this new method are developed by adapting two existing algorithms suitable for prescribed velocity boundary conditions to the case of prescribed normal and tangential stresses at the free boundary a normal mode analysis for a fixed boundary problem confirms the second order accuracy of the algorithms the approach is validated by comparison with a rayleigh plesset solution for an oscillating spherical bubble with an analysis of shape oscillations and with existing results for the buoyant rise of a deforming bubble for reynolds numbers up to 200 and weber numbers up to 12 in addition to the simulation of axisymmetric free surface flows of intrinsic interest the present approach is suitable for the validation of genuinely three dimensional calculations

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 front tracking method for the computations of multiphase flow direct numerical simulations of multiphase flows using a front tracking method are presented the method is based on writing one set of governing equations for the whole computational domain and treating the different phases as one fluid with variable material properties interfacial terms are accounted for by adding the appropriate sources as functions at the boundary separating the phases the unsteady navier stokes equations are solved by a conventional finite volume method on a fixed structured grid and the interface or front is tracked explicitly by connected marker points interfacial source terms such as surface tension are computed on the front and transferred to the fixed grid advection of fluid properties such as density is done by following the motion of the front the method has been implemented for fully three dimensional flows as well as for two dimensional and axisymmetric ones first the method is described for the flow of two or more isothermal phases the representation of the moving interface and its dynamic restructuring as well as the transfer of information between the moving front and the fixed grid are discussed applications and extensions of the method to homogeneous bubbly flows atomization flows with variable surface tension solidification and boiling are then presented

opening mirror symmetry on the quintic aided by mirror symmetry we determine the number of holomorphic disks ending on the real lagrangian in the quintic threefold we hypothesize that the tension of the domainwall between the two vacua on the brane which is the generating function for the open gromov witten invariants satisfies a certain extension of the picard fuchs differential equation governing periods of the mirror quintic we verify consistency of the monodromies under analytic continuation of the superpotential over the entire moduli space we further check the conjecture by reproducing the first few instanton numbers by a localization computation directly in the a model and verifying ooguri vafa integrality this is the first exact result on open string mirror symmetry for a compact calabi yau manifold

classification of the extended symmetries of fokker planck equations the authors obtain a complete classification of the possible types of the extended symmetry of any fokker planck equation together with a necessary and sufficient condition for each type they give also the most general form up to a change of variables of the lie generators of these symmetries and a comparison with the case of the heat equation

potential symmetries and invariant solutions for the generalized one dimensional fokker planck equation abstract pucci and saccomandi j phys a 26 1993 681 have obtained invariant solutions for the fokker planck fp equation via potential symmetries herein we analyze the generalized one dimensional fp equation for various cases corresponding to the physically interesting situations through the application of the same technique some interesting outcomes of this study are the deductions of the new exact solutions that do not seem to have been reported in the literature