cutfem topology optimization of 3d laminar incompressible flow problems abstract this paper studies the characteristics and applicability of the cutfem approach of burman et al 2014 as the core of a robust topology optimization framework for 3d laminar incompressible flow and species transport problems at low reynolds numbers r e 200 cutfem is a methodology for discretizing partial differential equations on complex geometries by immersed boundary techniques in this study the geometry of the fluid domain is described by an explicit level set method where the parameters of a discretized level set function are defined as functions of the optimization variables the fluid behavior is modeled by the incompressible navier stokes equations species transport is modeled by an advection diffusion equation the governing equations are discretized in space by a generalized extended finite element method face oriented ghost penalty terms are added for stability reasons and to improve the conditioning of the system the boundary conditions are enforced weakly via nitsche s method the emergence of isolated volumes of fluid surrounded by solid during the optimization process leads to a singular analysis problem an auxiliary indicator field is modeled to identify these volumes and to impose a constraint on the average pressure numerical results for 3d steady state and transient problems demonstrate that the cutfem analyses are sufficiently accurate and the optimized designs agree well with results from prior studies solved in 2d or by density approaches

numerical analysis of a finite element volume penalty method we present here some contributions to the numerical analysis of the penalty method in the finite element context we are especially interested in the ability provided by this approach to use cartesian non boundary fitted meshes to solve elliptic problems in complicated domain in the spirit of fictitious domains the initial problem is replaced by a penalized one posed over a simply shaped domain which covers the original one this method relies on two parameters namely h space discretization parameter and varepsilon penalty parameter we propose here a general strategy to estimate the error in both parameters and we present how it can be applied to various situations we pay special attention to a scalar version of the rigid motion constraint for fluid particle flows

iterative brinkman penalization for remeshed vortex methods we introduce an iterative brinkman penalization method for the enforcement of the no slip boundary condition in remeshed vortex methods in the proposed method the brinkman penalization is applied iteratively only in the neighborhood of the body this allows for using significantly larger time steps than what is customary in the brinkman penalization thus reducing its computational cost while maintaining the capability of the method to handle complex geometries we demonstrate the accuracy of our method by considering challenging benchmark problems such as flow past an impulsively started cylinder and normal to an impulsively started and accelerated flat plate we find that the present method enhances significantly the accuracy of the brinkman penalization technique for the simulations of highly unsteady flows past complex geometries

modeling a no slip flow boundary with an external force field abstract a novel technique related to peskin s immersed boundary approach is used to introduce solid surfaces into a simulated flow field the navier stokes equations permit the presence of an externally imposed body force that may vary in space and time forces are chosen to lie along a desired surface and to have a magnitude and direction opposing the local flow such that the flow is brought to rest on an element of the surface for unsteady viscous flow the direct calculation of the needed force is facilitated by a feedback scheme in which the velocity is used to iteratively determine the desired value in particular we determine the surface body force from the relation f x 5 t t 0 u x 5 t dt u x 5 t for surface points x 5 velocity u time t and negative constants and examples are presented which include 2d flow around cylinders 3d turbulent channel flow where one boundary is simulated with a force field and turbulent channel flow over a riblet covered surface while the new method may be applied to complex geometries on a non cartesian mesh we have chosen to use a simple cartesian grid all simulations are done with a spectral code in a single computational domain without any mapping of the mesh

a high resolution spatially adaptive vortex method for separating flows part i two dimensional domains a grid free high resolution spatially adaptive vortex method for two dimensional incompressible flow in bounded domains is presented the computational algorithm is based on operator splitting in which convection and diffusion are handled separately every time step in the convection step computational elements are convected with velocities obtained by fast approximations of the biot savart superposition with second order runge kutta time integration scheme diffusion is performed using the smooth redistribution method that employs a gaussian basis function for vorticity in the interior near solid walls the core functions are modified to conserve circulation the no slip boundary condition is enforced by creating of a vortex sheet that is redistributed to neighboring elements using the redistribution method the proposed method enables accurate and smooth recovery of the vorticity and does not require explicit use of vortex images or occasional re meshing algorithms for reduction in computational cost by accurately removing elements in overcrowded regions and for spatial adaptivity that allows for variable core sizes and variable element spacing are presented computations of flow around an impulsively started cylinder for reynolds number values of 1000 3000 and 9500 are preformed to investigate various aspects of the proposed method

a general fictitious domain method with immersed jumps and multilevel nested structured meshes this study addresses a new fictitious domain method for elliptic problems in order to handle general and possibly mixed embedded boundary conditions e b c robin neumann and dirichlet conditions on an immersed interface the main interest of this fictitious domain method is to use simple structured meshes possibly uniform cartesian nested grids which do not generally fit the interface but define an approximate one a cell centered finite volume scheme with a non conforming structured mesh is derived to solve the set of equations with additional algebraic transmission conditions linking both flux and solution jumps through the immersed approximate interface hence a local correction is devised to take account of the relative surface ratios in each control volume for the robin or neumann boundary condition then the numerical scheme conserves the first order accuracy with respect to the mesh step this opens the way to combine the e b c method with a multilevel mesh refinement solver to increase the precision in the vicinity of the interface such a fictitious domain method is very efficient the l 2 and l norm errors vary like o h l where h l is the grid step of the finest refinement level around the interface until the residual first order discretization error of the non refined zone is reached the numerical results reported here for convection diffusion problems with dirichlet robin and mixed dirichlet and robin boundary conditions confirm the expected accuracy as well as the performances of the present method

a galerkin implementation of the generalized helmholtz decomposition of vorticity formulations abstract vorticity formulations for the incompressible navier stokes equations have certain advantages over primitive variable formulations including the fact that the number of equations to be solved is reduced however the accurate implementation of the boundary conditions seems to continue to be an impediment to the acceptance and use of numerical methods based on vorticity formulations velocity boundary conditions can be implicitly satisfied by maintaining the kinematic compatibility of the velocity and vorticity fields as described by the generalized helmholtz decomposition ghd this can be accomplished in one of two ways by either solving for boundary vorticity leading to a dirichlet boundary condition for the vorticity equation or solving for boundary vortex sheet strengths leading to a neumann condition in the past vortex sheet strengths have often been determined by solving an over specified set of linear equations the over specification arose because integral constraints were imposed on the vortex sheet strengths these integral constraints are not necessary and typically are included to mitigate errors in determining the vortex sheet strengths themselves further the constraints overspecify the linear system requiring least squares solution techniques to more accurately satisfy both components of the velocity boundary conditions a galerkin formulation is applied to the generalized helmholtz decomposition this formulation implicitly satisfies an integral constraint that is more general than many of the integral constraints that have been explicitly imposed two implementations of the galerkin ghd are considered in the current work one based on determining the boundary vorticity and one based on determining the boundary vortex sheet strengths a finite element method fem is implemented to solve the vorticity equation along with the boundary data generated from the ghd

minimal co volume hyperbolic lattices ii simple torsion in a kleinian group this paper represents the nal step in solving the problem posed by siegel in 1945 of determining the minimal co volume lattices of hyperbolic 3 space h also problem 3 60 f in the kirby problem list from 1993 here we identify the two smallest co volume lattices both these groups are two generator arithmetic lattices generated by two elements of nite orders 2 and 3 their co volumes are 0 0390 and 0 0408 the precise values are given in terms of the dedekind zeta function of a number eld via a formula of borel our earlier work dealt with the cases when there is a nite spherical subgroup or high order torsion in the lattice thus here we are concerned with the study of simple torsion of low order and the geometric structure of klein 4 subgroups of a kleinian group we also identify certain universal geometric constraints imposed by discreteness on kleinian groups which are of independent interest to obtain these results we use a range of geometric and arithmetic criteria to obtain information on the structure of the singular set of the associated orbifold and then co volume bounds by studying equivariant neighbourhoods of xed point sets together with a rigorous computer search of certain parameter spaces for two generator kleinian groups

a penalization method to take into account obstacles in incompressible viscous flows from the navier stokes brinkman model a penalization method has been derived by several authors to compute incompressible navier stokes equations around obstacles in this paper convergence theorems and error estimates are derived for two kinds of penalization the first one corresponds to l 2 penalization inducing a darcy equation in the solid body the second one corresponds to a h 1 penalization and induces a brinkman equation in the body numerical tests are performed to confirm the efficiency and accuracy of the method

transitions between streamline topologies of structurally stable hamiltonian flows in multiply connected domains abstract we consider hamiltonian vector fields with a dipole singularity satisfying the slip boundary condition in two dimensional multiply connected domains one example of such hamiltonian vector fields is an incompressible and inviscid flow in exterior multiply connected domains with a uniform flow whose hamiltonian is called the stream function here we are concerned with topological structures of the level sets of the hamiltonian which we call streamlines by analogy from incompressible fluid flows classification of structurally stable streamline patterns has been considered in yokoyama and sakajo 2013 where a procedure to assign a unique sequence of words called the maximal word to these patterns is proposed thanks to this procedure we can identify every streamline pattern with its representing sequence of words up to topological equivalence in the present paper based on the theory of word representations we propose a combinatorial method to provide a list of possible transient structurally unstable streamline patterns between two different structurally stable patterns by simply comparing their maximal word representations without specifying any hamiltonian although this method cannot deal with topological streamline changes induced by bifurcations it reveals the existence of many non trivial global transitions in a generic sense we also demonstrate how the present theory is applied to fluid flow problems with vortex structures

computation of three dimensional multiphase flow dynamics by fully coupled immersed flow fcif solver abstract this work presents a fully coupled immersed flow fcif solver for the three dimensional simulation of fluid fluid interaction by coupling two distinct flow solvers using an immersed boundary ib method the fcif solver captures dynamic interactions between two fluids with disparate flow properties while retaining the desirable simplicity of non boundary conforming grids for illustration we couple an ib based unsteady reynolds averaged navier stokes urans simulator with a depth integrated long wave solver for the application of slug development with turbulent gas and laminar liquid we perform a series of validations including turbulent laminar flows over prescribed wavy boundaries and freely evolving viscous fluids these confirm the effectiveness and accuracy of both one way and two way coupling in the fcif solver finally we present a simulation example of the evolution from a stratified turbulent laminar flow through the initiation of a slug that nearly bridges the channel the results show both the interfacial wave dynamics excited by the turbulent gas forcing and the influence of the liquid on the gas turbulence these results demonstrate that the fcif solver effectively captures the essential physics of gas liquid interaction and can serve as a useful tool for the mechanistic study of slug generation in two phase gas liquid flows in channels and pipes

strongly coupled dynamics of fluids and rigid body systems with the immersed boundary projection method a strong coupling algorithm is presented for simulating the dynamic interactions between incompressible viscous flows and rigid body systems in both two and three dimensional problems in this work the navier stokes equations for incompressible flow are solved on a uniform cartesian grid by the vorticity based immersed boundary projection method of colonius and taira dynamical equations for arbitrary rigid body systems are also developed the proposed coupling method attempts to unify the treatment of constraints in the fluid and structure the incompressibility of the fluid the linkages in the rigid body system and the conditions at the interface through the use of lagrange multipliers the resulting partitioned system of equations is solved with a simple relaxation scheme based on an identification of virtual inertia from the fluid the scheme achieves convergence in only 2 to 5 iterations per time step for a wide variety of mass ratios the formulation requires that only a subset of the discrete fluid equations be solved in each iteration several two and three dimensional numerical tests are conducted to validate and demonstrate the method including a falling cylinder flapping of flexible wings self excited oscillations of a system of many linked plates in a free stream and passive pivoting of a finite aspect ratio plate under the influence of gravity in a free stream the results from the current method are compared with previous experimental and numerical results and good agreement is achieved

a parallel fast multipole method for elliptic difference equations a new fast multipole formulation for solving elliptic difference equations on unbounded domains and its parallel implementation are presented these difference equations can arise directly in the description of physical systems e g crystal structures or indirectly through the discretization of pdes in the analog to solving continuous inhomogeneous differential equations using green s functions the proposed method uses the fundamental solution of the discrete operator on an infinite grid or lattice green s function fast solutions o n are achieved by using a kernel independent interpolation based fast multipole method unlike other fast multipole algorithms our approach exploits the regularity of the underlying cartesian grid and the efficiency of ffts to reduce the computation time our parallel implementation allows communications and computations to be overlapped and requires minimal global synchronization the accuracy efficiency and parallel performance of the method are demonstrated through numerical experiments on the discrete 3d poisson equation

a comparative study of penalization and phase field methods for the solution of the diffusion equation in complex geometries we present a comparative study of penalization and phase field methods for the solution of the diffusion equation in complex geometries embedded using simple cartesian meshes the two methods have been widely employed to solve partial differential equations in complex and moving geometries for applications ranging from solid and fluid mechanics to biology and geophysics their popularity is largely due to their discretization on cartesian meshes thus avoiding the need to create body fitted grids at the same time there are questions regarding their accuracy and it appears that the use of each one is confined by disciplinary boundaries here we compare penalization and phase field methods to handle problems with neumann and robin boundary conditions we discuss extensions for dirichlet boundary conditions and in turn compare with methods that have been explicitly designed to handle dirichlet boundary conditions the accuracy of all methods is analyzed using one and two dimensional benchmark problems such as the flow induced by an oscillating wall and by a cylinder performing rotary oscillations this comparative study provides information to decide which methods to consider for a given application and their incorporation in broader computational frameworks we demonstrate that phase field methods are more accurate than penalization methods on problems with neumann boundary conditions and we present an error analysis explaining this result

a stochastic particle mesh scheme for uncertainty propagation in vortical flows abstract a new mesh particle scheme is constructed for uncertainty propagation in vortical flow the scheme is based on the incorporation of polynomial chaos pc expansions into a lagrangian particle approximation of the navier stokes equations the main idea of the method is to use a unique set of particles to transport the stochastic modes of the solution the particles are transported by the mean velocity field while their stochastic strengths are updated to account for diffusive and convective effects induced by the coupling between stochastic modes an integral treatment is used for the evaluation of the coupled stochastic terms following the framework of the particle strength exchange pse methods which yields a conservative algorithm it is also shown that it is possible to apply solution algorithms used in deterministic setting including particle mesh techniques and particle remeshing thus the method combines the advantages of particles discretizations with the efficiency of pc representations validation of the method on uncertain diffusion and convection problems is first performed an example is then presented of natural convection of a hot patch of fluid in infinite domain and the computations are used to illustrate the effectiveness of the approach for both large number of particles and high order pc expansions

universal meshes a method for triangulating planar curved domains immersed in nonconforming meshes summary we introduce a new method to triangulate planar curved domains that transforms a specific collection of triangles in a background mesh to conform to the boundary in the process no new vertices are introduced and connectivities of triangles are left unaltered the method relies on a novel way of parameterizing an immersed boundary over a collection of nearby edges with its closest point projection to guarantee its robustness we require that the domain be c2 regular the background mesh be sufficiently refined near the boundary and that specific angles in triangles near the boundary be strictly acute the method can render both straight edged and curvilinear triangulations for the immersed domain the latter includes curved triangles that conform exactly to the immersed boundary and ones constructed with isoparametric mappings to interpolate the boundary at select points high order finite elements constructed over these curved triangles achieve optimal accuracy which has customarily proven difficult in numerical schemes that adopt nonconforming meshes aside from serving as a quick and simple tool for meshing planar curved domains with complex shapes the method provides significant advantages for simulating problems with moving boundaries and in numerical schemes that require iterating over the geometry of domains with no conformity requirements the same background mesh can be adopted to triangulate a large family of domains immersed in it including ones realized over several updates during the coarse of simulating problems with moving boundaries we term such a background mesh as a universal mesh for the family of domains it can be used to triangulate universal meshes hence facilitate a framework for finite element calculations over evolving domains while using only fixed background meshes furthermore because the evolving geometry can be approximated with any desired order numerical solutions can be computed with high order accuracy we present demonstrative examples using universal meshes to simulate the interaction of rigid bodies with stokesian fluids copyright 2014 john wiley sons ltd

penalization modeling of a limiter in the tokamak edge plasma an original penalization method is applied to model the interaction of magnetically confined plasma with limiter in the frame of a minimal transport model for ionic density and parallel momentum the limiter is considered as a pure particle sink for the plasma and consequently the density and the momentum are enforced to be zero inside comparisons of the numerical results with one dimensional analytical solutions show a very good agreement in particular the penalization scheme followed in this paper tends to ensure an almost sonic plasma condition at the plasma obstacle interface bohm like criterion with relatively weak dependence on the target mach number profile within the obstacle the new system being solved in a periodic obstacle free domain an efficient pseudo spectral algorithm based on a fast fourier transform is also proposed and associated with an exponential filtering of the unphysical oscillations due to gibbs phenomenon finally the efficiency of the method is illustrated by investigating the flow spreading from the plasma core to the scrape off layer at the wall in a two dimensional system with one then two neighboring limiters

a fast lattice green s function method for solving viscous incompressible flows on unbounded domains a computationally efficient method for solving three dimensional viscous incompressible flows on unbounded domains is presented the method formally discretizes the incompressible navier stokes equations on an unbounded staggered cartesian grid operations are limited to a finite computational domain through a lattice green s function technique this technique obtains solutions to inhomogeneous difference equations through the discrete convolution of source terms with the fundamental solutions of the discrete operators the differential algebraic equations describing the temporal evolution of the discrete momentum equation and incompressibility constraint are numerically solved by combining an integrating factor technique for the viscous term and a half explicit runge kutta scheme for the convective term a projection method that exploits the mimetic and commutativity properties of the discrete operators is used to efficiently solve the system of equations that arises in each stage of the time integration scheme linear complexity fast computation rates and parallel scalability are achieved using recently developed fast multipole methods for difference equations the accuracy and physical fidelity of solutions are verified through numerical simulations of vortex rings

asymptotic behavior of positive solutions of a nonlinear integral system abstract in this paper we study the properties of positive solutions of an integral system in r n u x r n v p y u q y x y n y d y v x r n v q y u p y x y n y d y we prove the radial symmetry of positive solutions by means of the method of moving planes in integral forms which is established by chen li ou in addition by using a regularity lifting lemma we obtain an integrability result under the assumption p q 1 0 we prove that those solutions are the ground states finally we estimate the decay rates of such solutions when x

a vortex level set method for the two way coupling of an incompressible fluid with colliding rigid bodies we present a vortex method for the simulation of the interaction of an incompressible flow with rigid bodies the method is based on a penalization technique where the system is considered as a single flow subject to the navier stokes equation with a penalization term that enforces continuity at the solid fluid interface and rigid motion inside the solid level set functions are used to capture interfaces compute rigid motions inside the solid bodies and model collisions between bodies a vortex in cell algorithm is built on this method numerical comparisons with existing 3d methods on problems of sedimentation and collision of spheres are provided to illustrate the capabilities of the method

a cartesian grid method for modeling multiple moving objects in 2d incompressible viscous flow we present an efficient method for solving 2d incompressible viscous flows around multiple moving objects our method employs an underlying regular cartesian grid to solve the system using a streamfunction vorticity formulation and with discontinuities representing the embedded objects the no penentration condition for the moving geometry is satisfied by superposing a homogenous solution to the poisson s equation for the streamfunction the no slip condition is satisfied by generating vorticity on the surfaces of the objects both the initial poisson solution and the evaluation of the homogenous solution require embedding irregular discontinuities in a fast poisson solver computation time is dictated by the time required to do a fast poisson solution plus solve an integral form of laplace s equation there is no significant increase in computational cost if the geometry of the embedded objects is variable and moving relative to the underlying grid we test the method against the canonical example of flow past a cylinder and obtained new results on the flow and forces of two cylinders moving relative to each other

a multigrid ghost point level set method for incompressible navier stokes equations on moving domains with curved boundaries abstract in this paper we present a numerical approach to solve the navier stokes equations on moving domains with second order accuracy the space discretization is based on the ghost point method which falls under the category of unfitted boundary methods since the mesh does not adapt to the moving boundary the equations are advanced in time by using crank nicholson the momentum and continuity equations are solved simultaneously for the velocity and the pressure by adopting a proper multigrid approach to avoid the checkerboard instability for the pressure a staggered grid is adopted where velocities are defined at the sides of the cell and the pressure is defined at the centre the lack of uniqueness for the pressure is circumvented by the inclusion of an additional scalar unknown representing the average divergence of the velocity and an additional equation to set the average pressure to zero several tests are performed to simulate the motion of an incompressible fluid around a moving object as well as the lid driven cavity tests around steady objects the object is implicitly defined by a level set approach that allows a natural computation of geometrical properties such as distance from the boundary normal directions and curvature different shapes are tested circle ellipse and flower numerical results show the second order accuracy for the velocity and the divergence that decays to zero with second order and the efficiency of the multigrid that is comparable with the tests available in literature for rectangular domains without objects showing that the presence of a complex shaped object does not degrade the performance

fast immersed interface poisson solver for 3d unbounded problems around arbitrary geometries abstract we present a fast and efficient fourier based solver for the poisson problem around an arbitrary geometry in an unbounded 3d domain this solver merges two rewarding approaches the lattice green s function method and the immersed interface method using the sherman morrison woodbury decomposition formula the method is intended to be second order up to the boundary this is verified on two potential flow benchmarks we also further analyse the iterative process and the convergence behavior of the proposed algorithm the method is applicable to a wide range of problems involving a poisson equation around inner bodies which goes well beyond the present validation on potential flows

finite group actions on 3 manifolds if g is a finite group acting smoothly on a closed surface f it is well known that g leaves invariant some riemannian metric of constant curvature on f thus any action of g on the 2 sphere s 2 is conjugate in diff s 2 to an orthogonal action if g acts on the torus sx s 1 there is a g invariant flat metric on s a s 1 and if g acts on a surface f with negative euler number then f admits a g invariant hyperbolic metric recently thurston th 1 th 2 th 3 has described the eight 3 dimensional geometries which provide geometric structures for closed 3 manifolds in the same way that the 2 sphere s 2 the euclidean plane e 2 and the hyperbolic plane h 2 provide geometric structures for surfaces see also the survey article by scott sc4 thurston also conjectured that if m is a closed 3 manifold with a geometric structure modelled on one of these eight geometries say x then any smooth action of a finite group g on m should leave invariant some metric on m inducing the geometry x we will say that g preserves the geometric structure on m in this case it should be noted that the restriction to smooth actions of g on m is essential for bing bi showed that there are involutions of s 3 whose fixed set is a wild 2 sphere however in dimension two it was proved by eilenberg ei that any action of a finite group on a surface is conjugate to a smooth action in this paper our main result asserts that thurston s conjecture holds for five of the eight geometries the result is the following

a new diffusion procedure for vortex methods a new method is proposed for simulating diffusion in vortex methods for two dimensional incompressible flows the method resolves length scales up to the spacing of the vortices the grid free nature of vortex methods is fully retained and the distribution of the vortices can be irregular it is shown for the stokes equations that in principle the method can have any order of accuracy it also conserves circulation linear and angular momentum the method is based on exchanging a conserved quantity between arbitrary computational points this suggests that extensions to more general flows may be possible for the two dimensional incompressible flows studied circulation is exchanged between vortices to simulate diffusion the amounts of circulation exchanged must satisfy a linear system of equations based on stability considerations the exchanged amounts should further be positive a procedure to find a solution to this problem is formulated using linear programming techniques to test the method the decay processes of a single point vortex and of a counterrotating pair of point vortices are computed current limitations of the method are discussed

an adaptive residual based splitting approach for the penalized navier stokes equations the interest on immersed boundary methods ibm is increasing in computational fluid dynamics as they simplify the mesh generation problem in this work we consider an approach based on the addition of a penalty term to the navier stokes equations to account for the wall boundary conditions to discretize the resulting equations we use a residual distribution approach previously developed by some of the authors to adapt the method to the ibm considered we developed a new formulation of residual distribution based on a strang splitting method in time coupling an implicit asymptotic integration procedure of the penalization ode with a simplified explicit residual distribution for the navier stokes equations the first method provides an operator which is exact up to orders 2 with the penalty parameter assuming values of the order of 10 10 a modification of the solution gradient reconstruction necessary for the evaluation of the viscous fluxes is also introduced in the paper this guarantees that correct physical values of the viscous stresses are recovered in vicinity of the solid we show formally and numerically that the approach proposed is second order accurate for smooth solutions we evaluate its potential for ibm by coupling the resulting method with unstructured mesh adaptation on wall boundaries several steady and time dependent tests are used to show the promising features of the method proposed

modeling and simulation of fish like swimming modeling and simulation of two dimensional flows past deformable bodies are considered the incompressible navier stokes equations are discretized in space onto a fixed cartesian mesh and the displacement of deformable objects through the fluid is taken into account using a penalization method the interface between the solid and the fluid is tracked using a level set description so that it is possible to simulate several bodies freely evolving in the fluid as an illustration of the methods fish like locomotion is analyzed in terms of propulsion efficiency underwater maneuvering and school swimming are also explored

the basic equations for the large eddy simulation of turbulent flows in complex geometry abstract the equations for large eddy simulation les of turbulent flows are derived by applying a filtering operation to the navier stokes equations les of inhomogeneous turbulent flows requires the use of filters with variable filter width the use of such filters invalidates the standard derivation of the basic equations for the filtered fields since the filtering operation in general does not commute with the operation of differentiation in this paper an alternate definition of the filtering operation based on the mapping function of the nonuniform grid is introduced it is shown that with this modified definition the filtering and differentiation operations commute up to an error which is second order in the filter width it is also shown that the commutation error can be expressed in terms of the filtered field and its derivatives as an asymptotic series in the square of the filter width these results are then applied to the navier stokes equations to derive the basic equations satisfied by the filtered fields

analysis of the fully discrete fat boundary method the fat boundary method is a method of the fictitious domain class which was proposed to solve elliptic problems in complex geometries with non conforming meshes it has been designed to recover optimal convergence at any order despite of the non conformity of the mesh and without any change in the discrete laplace operator on the simple shape domain we propose here a detailed proof of this high order convergence and propose some numerical tests to illustrate the actual behaviour of the method

numerical simulation of a cylinder in uniform flow in this study a virtual boundary technique is applied to the numerical simulation of stationary and moving cylinders in uniform flow this approach readily allows the imposition of a no slip boundary within the flow field by a feedback forcing term added to the momentum equations in the present work this technique is used with a high order finite difference method effectively eliminating spurious oscillations caused by the feedback forcing when used with spectrally discretized flow solvers very good agreement is found between the present calculations and previous computational and experimental results for steady and time dependent flow at low reynolds numbers

advances in direct numerical simulations of 3d wall bounded flows by vortex in cell methods this paper is devoted to the design of vortex in cell vic methods for the direct numerical simulations of wall bounded flows a first method using body fitted grid is presented in the particular case of a cylinder wake this method which has been used in phys fluids 14 6 2002 2021 to investigate the effect on the wake topology of cylinder rotations is an extension of the vic method presented in j comput phys 175 2002 702 for periodic geometries features of the method that are specific to wall bounded geometries interpolation operators field calculations and vorticity flux formulas to enforce no slip boundary conditions are described in details the accuracy of the method in the calculation of the body forces is investigated by comparisons with experiments and benchmark calculations a second class of methods is in the spirit of the immersed boundary methods the paper in particular shows that the no slip conditions are very naturally handled by the vorficity flux formulas independently of the relative locations of the particles and the body numerical experiments on the test case of a ring impinging on a cylinder suggest that the method is second order accurate

immersed boundary smooth extension the immersed boundary method is a simple efficient and robust numerical scheme for solving pde in general domains yet it only achieves first order spatial accuracy near embedded boundaries in this paper we introduce a new high order numerical method which we call the immersed boundary smooth extension ibse method the ibse method achieves high order accuracy by smoothly extending the unknown solution of the pde from a given smooth domain to a larger computational domain enabling the use of simple cartesian grid discretizations e g fourier spectral methods the method preserves much of the flexibility and robustness of the original ib method in particular it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta functions to communicate information between the computational grid and the boundary we present a fast algorithm for solving elliptic equations which forms the basis for simple high order implicit time methods for parabolic pde and implicit explicit methods for related nonlinear pde we apply the ibse method to solve the poisson heat burgers and fitzhugh nagumo equations and demonstrate fourth order pointwise convergence for dirichlet problems and third order pointwise convergence for neumann problems

a lagrangian vorticity collocation method for viscous axisymmetric flows with and without swirl this paper presents a new lagrangian vorticity collocation method for viscous axisymmetric fluid flows with or without swirl the velocity calculation is performed using a representation of the vorticity field in terms of gaussian vortex rings for off center points and gaussian vortex blobs along the axis of symmetry a matrix equation for the element amplitudes is obtained via a collocation approach by fitting the vorticity representation to the known vorticity values at the control points prior to each velocity evaluation this matrix equation is solved using an iterative procedure which both speeds the calculation and filters out noise in the element amplitudes viscous diffusion is accomplished with use of a diffusion velocity method in which control points are moved with the sum of the local fluid velocity and an additional diffusion velocity that accounts for the effect of viscosity on the spread of vorticity support derivatives are obtained by locally fitting a polynomial function to control points about a given control point via a least square formulation and then differentiating the polynomial this method is found to maintain high accuracy even for very irregularly spaced control points the method introduced for viscous diffusion in the paper can also be used for three dimensional vortex methods in general

an implicit boundary approach for viscous compressible high reynolds flows using a hybrid remeshed particle hydrodynamics method abstract we present an implicit boundary particle method with background mesh adaptation we use a brinkman penalisation to represent the boundary of the domain and a remeshed particle method to simulate viscous flow with high reynolds numbers a penalty term is added to the navier stokes equations to impose the boundary conditions the boundary conditions are enforced to a specific precision with no need to modify the numerical method or change the grid achieving an implicit approach for flow around complex boundaries geometries simulation the main idea of the brinkman penalisation method is to model the solid obstacle as a porous medium the governing equations for the compressible fluid and penalised navier stokes for the porous medium are solved simultaneously without need for interface conditions the accuracy of the method is tested for a number of benchmark problems starting with simple cases such as a periodic laminar flow in a channel poiseuille flow to more complex problems such as the lid driven cavity with high reynolds numbers the reflection and propagation of a compressible shock wave and finally swirling flow in a model two stroke diesel engine the remeshed particle mesh method coupled with brinkman penalisation provides a good quality simulation and the results are in agreement with analytical or reference solutions

convergence of a variable blob vortex method for the euler and navier stokes equations convergence is proved of the variable blob vortex method proposed t hou d papageorgiou g papanicolaou and j zhu an adaptive vortex method for two dimensional viscous and incompressible flows preprint for the incompressible euler and navier stokes equations without boundaries this version of the variable blob method is designed to approximate directly the navier stokes equations without matching the interior flow to the boundary layer solution by choosing vortex blobs adaptively the numerical errors due to local shear are minimized

numerical method for incompressible vortical flows with two unbounded directions a new efficient and accurate method has been developed for computing unsteady incompressible viscous flows in a domain where two dimensions are unbounded the third dimension is periodic and the vorticity is rapidly decaying in the unbounded directions we use the term unbounded to mean doubly infinite no boundaries of any kind this is an extension of the methods described by others for flows with two periodic and one unbounded direction where the irrotational velocities outside the vortical domain are treated analytically the new method is shown to be both accurate and efficient the method presented here has finite but arbitrarily high order formal accuracy and incurs substantial additional cost for a given mesh however this increased cost is more than offset by the reduction in the number of mesh points required for a given accuracy the result is that for accurate computations the present method can be orders of magnitude more efficient than others currently in use this paper presents the method discusses implementation issues validates its accuracy and presents sample calculations

a novel solution method for unsteady incompressible euler flow using the vorticity bernoulli pressure formulation abstract a novel numerical formulation and time integration method for the incompressible euler flow equations is presented using the vorticity bernoulli pressure system lagrangian advection of vorticity carrying particles is adopted from classical vortex methods that are known to give accurate results for vortex dominated flows and allow larger time steps than explicit eulerian schemes solving only one scalar poisson equation per time step instead of a vector valued equation in vortex methods makes the new scheme theoretically more efficient on large computational grids the equation of motion in rotation form is integrated in time under certain approximation assumptions and solved explicitly for the flow velocity the velocity is eliminated as an unknown in the continuity equation which is efficiently solved for the bernoulli pressure using the predicted vorticity field obtained by lagrangian advection the bernoulli pressure is efficiently obtained from the poisson equation in finite volume formulation an estimated velocity field is calculated from the equation of motion to effectively counterbalance the discretization errors made by the prediction in a corrector step the kinematic relation between vorticity and estimated velocity is reformulated and solved in a lagrangian framework as well the velocity field is corrected at the end of the time step two dimensional cases on cartesian grids are considered numerical results for three flow cases demonstrate the locally smooth and accurate behavior and the second order spatial accuracy of the solution the new scheme produces significantly less artificial vortex distortion than the classical vortex method the favorable properties of the solutions with respect to accuracy and long time stability of the overall kinetic energy and enstrophy are also emphasized

positive solutions to integral systems with weight and bessel potentials abstract in this paper we consider the integral system with weight and the bessel potentials u x r n g x y u y p v y q y d y v x r n g x y v y p u y q y d y where u v 0 0 0 n p q 2 and g x is the bessel potential of order first we get the integrability by regularity lifting lemma then we also establish the regularity of the positive solutions afterwards by the method of moving planes in integral forms we show that the positive solutions are radially symmetric and monotone decreasing about the origin finally by an extension of the idea of lei 14 and analytical techniques we get the decay rates of solutions when x

decay symmetry and existence of solutions of semilinear elliptic systems our utmost aim was originally to establish the existence of solutions of system however in this effort we got sidetracked to consider other questions in this way we come to interesting results on asymptotic behavior of solutions on symmetry properties of such solutions and the existence of ground states

boundary conditions for viscous vortex methods this paper presents a neumann type vorticity boundary condition for the vorticity formulation of the navier stokes equations the vorticity creation process at the boundary due to the no slip condition is expressed in terms of a vorticity flux the scheme is incorporated then into a lagrangian vortex blob method that uses a particle strength exchange algorithm for viscous diffusion the no slip condition is not enforced by the generation of new vortices at the boundary but instead by modifying the strength of the vortices in the vicinity of the boundary

analysis of direct three dimensional parabolic panel methods adherence boundary conditions for time dependent partial differential equations via chorin algorithm can be reduced to a parabolic problem with robin fourier boundary conditions in the three dimensional context in the spirit of panel methods one establishes an integral formulation whose key point is the estimation of the potential density introducing a kind of panel method for tangential kinematic boundary conditions this paper discusses explicit estimations of this density in the general case of an arbitrarily shaped three dimensional body which leads to a fast numerical scheme an error analysis is also provided involving body smoothness the holder exponent of the density and whether the body presents torsion or not

a method for determining the velocity induced by highly anisotropic vorticity blobs resolution of boundary layer flows at moderate or high reynolds numbers with the vortex blob method requires a great many isotropic elements in this paper an approximate method for determination of the induced velocity from highly anisotropic vorticity blobs is presented and issues related to use of anisotropic elements in calculations with vortex blob algorithms for high reynolds number near wall flows are examined the method presented here can be used to determine the induced velocity from smooth blob functions of arbitrary form provided that the vorticity length scale associated with the blob is much less in one direction than in orthogonal directions the ratio of these length scales is called the blob aspect ratio and is used as a small parameter to construct an asymptotic approximation to the induced velocity field this method is applied in the present paper to derive induced velocity expressions for anisotropic gaussian blob functions in both two and three dimensions it is argued using test calculations for a blasius boundary layer that although direct calculation of the induced velocity requires about an order of magnitude more cpu time for anisotropic gaussian elements than for isotropic elements this difference is more than made up for by a reduction of several orders of magnitude in the number of elements needed to resolve boundary layer flows at moderate to high reynolds numbers it is also found that the standard vortex blob representation leads to errors in the calculation of wall slip velocity and wall shear stress due to smoothing of the discontinuity between the real and image vorticity fields at the wall but that these errors can be avoided by placing doublet type elements along the wall

a general deterministic treatment of derivatives in particle methods a unified approach to approximating spatial derivatives in particle methods using integral operators is presented the approach is an extension of particle strength exchange originally developed for treating the laplacian in advection diffusion problems kernels of high order of accuracy are constructed that can be used to approximate derivatives of any degree a new treatment for computing derivatives near the edge of particle coverage is introduced using one sided integrals that only look for information where it is available the use of these integral approximations in wave propagation applications is considered and their error is analyzed in this context using fourier methods finally simple tests are performed to demonstrate the characteristics of the treatment including an assessment of the effects of particle dispersion and their results are discussed

topology optimization for unsteady flow a computational methodology for optimizing the conceptual layout of unsteady flow problems at low reynolds numbers is presented the geometry of the design is described by the spatial distribution of a fictitious material with continuously varying porosity the flow is predicted by a stabilized finite element formulation of the incompressible navier stokes equations a brinkman penalization is used to enforce zero velocities in solid material the resulting parameter optimization problem is solved by a non linear programming method the paper studies the feasibility of the material interpolation approach for optimizing the topology of unsteady flow problems the derivation of the governing equations and the adjoint sensitivity analysis are presented a design dependent stabilization scheme is introduced to mitigate numerical instabilities in porous material the emergence of non physical artifacts in the optimized material distribution is observed and linked to an insufficient resolution of the flow field and an improper representation of the pressure field within solid material by the brinkman penalization two numerical examples demonstrate that the designs optimized for unsteady flow differ significantly from their steady state counterparts copyright 2011 john wiley sons ltd

vortex methods with spatially varying cores the accuracy of vortex methods employing smooth vortex particles blobs is determined by the blob size which can be viewed as a mollifier of the vorticity field for computational efficiency this core size needs to be spatially variable as particles are used to discretize different parts of the flow field such as the boundary layer and the wake in bluff body flows we derive here a consistent approximation for the viscous navier stokes equations using variable size vortex particles this derivation is based on the implementation of mappings that allow the consistent formulation of the diffusion and convection operators of the navier stokes equations in the context of vortex methods several local mappings can be combined giving the capability of mesh embedding to vortex methods it is shown that the proposed variable method offers a significant improvement on the computational efficiency of constant core size methods while maintaining the adaptive character of the method the method is ideally suited to flows such as wakes and shear layers and the validity of the approach is illustrated by showing results from cylinder flows and wall vortex interactions using this scheme previously unattainable simulations of cylinders undergoing rotary oscillations at high reynolds numbers reveal an interesting mechanism for drastic drag reduction

minimal index torsion free subgroups of kleinian groups a kleinian group is a discrete subgroup of psl 2 c the full group of orientation preserving isometries of 3 dimensional hyperbolic space in the language of t1 q h3 is a hyperbolic 3 orbifold that is a metric 3 orbifold in which all sectional curvatures are 1 and for which is the orbifold fundamental group see t1 for further details a fuchsian group is a discrete subgroup of psl 2 r and as such acts discontinuously on the hyperbolic plane we define to be of finite co volume resp co area if the volume resp area of the quotient orbifold q is finite by selberg s lemma if is a kleinian resp fuchsian group of finite covolume resp co area it contains a torsion free subgroup of finite index by definition any torsion free subgroup cannot contain any finite subgroups of so the index must be a multiple of the lowest common multiple of all orders of finite subgroups of see cfjr for a proof being of finite co volume implies there are only finitely many conjugacy classes of finite subgroups thus we make the following

short note a fourier based elliptic solver for vortical flows with periodic and unbounded directions we present a computationally efficient adaptive solver for the solution of the poisson and helmholtz equation used in flow simulations in domains with combinations of unbounded and periodic directions the method relies on using ffts on an extended domain and it is based on the method proposed by hockney and eastwood for plasma simulations the method is well suited to problems with dynamically growing domains and in particular flow simulations using vortex particle methods the efficiency of the method is demonstrated in simulations of trailing vortices

the π orbifold group of a link on associe a un enlacement l dans s 3 un groupe o l appele le groupe de variete d orbite de l on etudie certaines proprietes du groupe o l reliees a la geometrie de l enlacement l

flusi a novel parallel simulation tool for flapping insect flight using a fourier method with volume penalization we introduce flusi a fully parallel open source software package for pseudospectral simulations of three dimensional flapping flight in viscous flows it is freely available for noncommercial use from github https github com pseudospectators flusi the computational framework runs on high performance computers with distributed memory architectures the discretization of the three dimensional incompressible navier stokes equations is based on a fourier pseudospectral method with adaptive time stepping the complex time varying geometry of insects with rigid flapping wings is handled using the volume penalization method the modules characterizing the insect geometry flight mechanics and wing kinematics are described validation tests for different benchmarks illustrate the efficiency and precision of the approach finally computations for a model insect in the turbulent regime demonstrate the versatility of the software

interaction ruling animal collective behavior depends on topological rather than metric distance evidence from a field study numerical models indicate that collective animal behavior may emerge from simple local rules of interaction among the individuals however very little is known about the nature of such interaction so that models and theories mostly rely on aprioristic assumptions by reconstructing the three dimensional positions of individual birds in airborne flocks of a few thousand members we show that the interaction does not depend on the metric distance as most current models and theories assume but rather on the topological distance in fact we discovered that each bird interacts on average with a fixed number of neighbors six to seven rather than with all neighbors within a fixed metric distance we argue that a topological interaction is indispensable to maintain a flock s cohesion against the large density changes caused by external perturbations typically predation we support this hypothesis by numerical simulations showing that a topological interaction grants significantly higher cohesion of the aggregation compared with a standard metric one

a fictitious domain approach with spread interface for elliptic problems with general boundary conditions the aim of this article is to solve second order elliptic problems in an original physical domain using a fictitious domain method with a spread interface approach the main idea of the fictitious domain approach consists in immersing the original domain of study into a geometrically bigger and simpler one called fictitious domain as the spatial discretization is being performed in the fictitious domain this method allows the use of structured meshes the discretization is not boundary fitted to the original physical domain this paper describes several ways to impose dirichlet robin or neumann boundary conditions on a spread immersed interface without locally modifying the numerical scheme and without using lagrange multipliers the numerical applications focus on difiusion and convection problems in the unit disk with dirichlet or robin boundary conditions for such problems analytical solutions can be determined for a correctly chosen source term the numerical resolution is performed using a q1 finite element scheme the spread interface approach is then combined with a local adaptive mesh refinement algorithm in order to increase the precision in the vinicity of the immersed boundary the l2 norm of the errors is computed in order to evaluate the capability of the method immersed boundaries are found in many industrial applications like two phase flow simulations fluid structure interaction etc this article represents a first step towards the simulation of these kinds of applications

fourier spectral and wavelet solvers for the incompressible navier stokes equations with volume penalization convergence of a dipole wall collision in this study we use volume penalization to mimic the presence of obstacles in a flow or a domain with no slip boundaries this allows in principle the use of fast fourier spectral methods and coherent vortex simulation techniques based on wavelet decomposition of the flow variables to compute turbulent wall bounded flow or flows around solid obstacles by simply adding one term in the equation convergence checks are reported using a recently revived and unexpectedly difficult dipole wall collision as a benchmark computation several quantities like the vorticity isolines truncation error kinetic energy and enstrophy are inspected for a collision of a dipole with a no slip wall and compared with available benchmark data obtained with a standard chebyshev pseudospectral method we quantify the possible deteriorating effects of the gibbs phenomenon present in the fourier based schemes due to continuity restrictions of the penalized navier stokes equations on the wall it is found that gibbs oscillations have a negligible effect on the flow evolution allowing higher order recovery of the accuracy on a fourier basis by means of postprocessing an advantage of coherent vortex simulations on the other hand is that the degrees of freedom of the flow computation can strongly be reduced in this study we quantify the possible reduction of degrees of freedom while keeping the accuracy for an optimal convergence scenario the penalization parameter has to scale with the number of fourier and wavelet modes in addition an implicit treatment of the darcy drag term in the penalized navier stokes equations is beneficial since this allows one to set the time step independent from the penalization parameter without additional computational or memory requirements

boundary integral solutions of the heat equation the boundary integral method bim has recently become quite popular because of its ability to provide cheap numerical solutions to the laplace equation this paper describes an attempt to apply a similar approach to the time dependent heat equation in two space variables 1 introduction in its simplest form the bim uses the prescribed initial and boundary data together with the known fundamental solution of a given differen tial equation defined on some domain q to construct a second integral equation which is itself to be solved on the boundary of ui in our case the integral equation solution is a layered thermal potential i e it represents a continuous distribution of sources and sinks along the boundary of the two dimensional domain u one then obtains the solution to the given problem by integrating the product of the layered thermal potential and a known kernel over the boundary hence the terminology boundary integral the advantage of such an approach is that the heart of the computation viz the solution of the integral equation is performed on the boundary thus reducing the problem from two space dimensions to one furthermore the resulting domain in one space and time will be rectangular a computational convenience even though the given domain i2 may have been quite irregular see example 5 in the appendix of course as a result of these simplifications we may expect substantial savings in computer time and storage the work outlined below is based on the use of single layered thermal potentials and requires that the domain have a smooth boundary and with some restrictions either a neumann or mixed boundary condition the equation itself must be homogeneous but we do allow inhomogeneous initial data though not shown here it seems to be well within the capabilities of this approach to handle boundaries involving an arbitrary mixture of piecewise c2 curves with dirichlet neumann and or mixed boundary data of course the tradeoff is that to evaluate the solution we must do a double integral for each point at which we want to know the solution if needed at a large number of points the cost of generating the solution dominates and the method becomes impractical in many applications however in particular in semiconductor

the immersed boundary method a projection approach a new formulation of the immersed boundary method with a structure algebraically identical to the traditional fractional step method is presented for incompressible flow over bodies with prescribed surface motion like previous methods a boundary force is applied at the immersed surface to satisfy the no slip constraint this extra constraint can be added to the incompressible navier stokes equations by introducing regularization and interpolation operators the current method gives prominence to the role of the boundary force acting as a lagrange multiplier to satisfy the no slip condition this role is analogous to the effect of pressure on the momentum equation to satisfy the divergence free constraint the current immersed boundary method removes slip and non divergence free components of the velocity field through a projection the boundary force is determined implicitly without any constitutive relations allowing the present formulation to use larger cfl numbers compared to some past methods symmetry and positive definiteness of the system are preserved such that the conjugate gradient method can be used to solve for the flow field examples show that the current formulation achieves second order temporal accuracy and better than first order spatial accuracy in l 2 norms for one and two dimensional test problems results from two dimensional simulations of flows over stationary and moving cylinders are in good agreement with those from previous experimental and numerical studies

reinforcement learning and wavelet adapted vortex methods for simulations of self propelled swimmers we present a numerical method for the simulation of collective hydrodynamics in self propelled swimmers swimmers in a viscous incompressible flow are simulated with a remeshed vortex method coupled with brinkman penalization and projection approach the remeshed vortex methods are enhanced via wavelet based adaptivity in space and time the method is validated on benchmark swimming problems furthermore the flow solver is integrated with a reinforcement learning algorithm such that swimmers can learn to adapt their motion so as to optimally achieve a specified goal such as fish schooling the computational efficiency of the wavelet adapted remeshed vortex method is a key aspect for the effective coupling with learning algorithms the suitability of this approach for the identification of swimming behaviors is assessed on a set of learning tasks

uniform convergence of a linearly transformed particle method for the vlasov poisson system a particle method with linear transformation of the particle shape functions is studied for the 1d 1v vlasov poisson equation and a priori error estimates are proven which show that the approximated densities converge in the uniform norm when compared to standard fixed shape particle methods the present approach can be seen as a way to gain one order in the convergence rate of the particle trajectories at the cost of linearly transforming each particle shape it also allows to compute strongly convergent densities with particles that overlap in a bounded way

natural vorticity boundary conditions on solid walls abstract we derive a new kind of boundary conditions for the vorticity equation with solid wall boundaries for fluid flow problems the formulation uses a dirichlet condition for the normal component of vorticity and neumann type conditions for the tangential components in a galerkin integral formulation the tangential condition is natural i e it is enforced by a right hand side functional and does not impose a boundary constraint on trial and test spaces the functional involves the pressure variable and we discuss several velocity vorticity formulations where the proposed condition is appropriate several numerical experiments are given that illustrate the validity of the new approach

convergence analysis of the q1 finite element method for elliptic problems with non boundary fitted meshes summary the aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain s boundary this problematic of the last century e g the finite difference methodology returns to topical studies with the huge development of domain embedding fictitious domain or cartesiangrid methods these methods use regular structured meshes most often cartesian for non aligned domains although non boundary fitted approaches become more and more applied very few studies are devoted to theoretical error estimates in this paper the convergence of a q1 nonconforming finite element method is analyzed for secondorder elliptic problems with dirichlet robin or neumann boundary conditions the finite element method uses standard q1 rectangular finite elements as the finite element approximate space is not contained in the original solution space this method is referred to as nonconforming a stairstep boundary defined from the cartesian mesh approximates the original domain s boundary the convergence analysis of the finite element method for such a kind of non boundary fitted stairstepped approximation is no treated in the literature the study of dirichlet problems is based on similar techniques as those classically used with boundary fitted linear triangular finite elements the estimates obtained for robin problems are novel and use some more technical arguments the rate of convergence is proved to be in o h 1 2 for the h 1 norm for all general boundary conditions and classical duality arguments allow to obtain an o h error estimate in the l 2 norm for dirichlet problems numerical results obtained with fictitious domain techniques that impose original boundary conditions on a non boundary fitted approximate immersed interface are presented these results confirm the theoretical rates of convergence copyright c 2007 john wiley sons ltd

final states of decaying 2d turbulence in bounded domains influence of the geometry abstract direct numerical simulations of two dimensional decaying turbulence in wall bounded domains are presented the navier stokes equations are solved using a fourier pseudo spectral method with volume penalization starting from random initial conditions we study the influence of the geometry of the domain on the flow dynamics in particular on the long time behaviour circular square triangular and annular domains are considered and we show how the geometry plays a crucial role regarding the decay scenario towards final states three stages can be distinguished formation of coherent vortices from random initial conditions vortex wall interactions and finally relaxation towards a quasi steady structure the eigenvalues estimated from the decay rate of both energy and enstrophy depend on the geometry and agree well with the theoretical eigenvalues based on the stokes mode of the corresponding domain for the final states we find a linear functional relation between vorticity and streamfunction

a discontinuous galerkin based immersed boundary method a numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced the proposed method is conceptually simple and free of user defined parameters starting with a conforming finite element mesh the key ingredient is to switch those elements intersected by the dirichlet boundary to a discontinuous galerkin approximation and impose the dirichlet boundary conditions strongly by virtue of relaxing the continuity constraint at those elements boundary locking is avoided and optimal order convergence is achieved this is shown through numerical experiments in reaction diffusion problems copyright 2008 john wiley sons ltd