simple finite element method in vorticity formulation for incompressible flows a very simple and efficient finite element method is introduced for two and three dimensional viscous incompressible flows using the vorticity formulation this method relies on recasting the traditional finite element method in the spirit of the high order accurate finite difference methods introduced by the authors in another work optimal accuracy of arbitrary order can be achieved using standard finite element or spectral elements the method is convectively stable and is particularly suited for moderate to high reynolds number flows

discontinuous galerkin least squares finite element methods for singularly perturbed reaction diffusion problems with discontinuous coefficients and boundary singularities in this paper a discontinuous galerkin least squares finite element method is developed for singularly perturbed reaction diffusion problems with discontinuous coefficients and boundary singularities by recasting the second order elliptic equations as a system of first order equations in a companion paper lin in siam j numer anal 47 89 108 2008 a similar method has been developed for problems with continuous data and shown to be well posed uniformly convergent and optimal in convergence rate in this paper the method is modified to take care of conditions that arise at interfaces and boundary singularities coercivity and uniform error estimates for the finite element approximation are established in an appropriately scaled norm numerical examples confirm the theoretical results

convergence of a compact scheme for the pure streamfunction formulation of the unsteady navier stokes system this paper is devoted to the analysis of a new compact scheme for the navier stokes equations in pure streamfunction formulation numerical results using that scheme have been reported in m ben artzi e j comput phys 205 2005 pp 640 664 the scheme discussed here combines the stephenson scheme for the biharmonic operator and ideas from box scheme methodology consistency and convergence are proved for the full nonlinear system instead of customary periodic conditions the case of boundary conditions is addressed it is shown that in one dimension the truncation error for the biharmonic operator is o h 4 at interior points and o h at near boundary points in two dimensions the truncation error is o h 2 at interior points due to the cross terms and o h at near boundary points hence the scheme is globally of order four in the one dimensional periodic case and of order two in the two dimensional periodic case but of order 3 2 for one and two dimensional nonperiodic boundary conditions we emphasize in particular that there is no special treatment of the boundary thus allowing robust use of the scheme the finite element analogy of the finite difference schemes is invoked at several stages of the proofs in order to simplify their verifications

an efficient compact difference scheme for solving the streamfunction formulation of the incompressible navier stokes equations recently a new paradigm for solving the steady navier stokes equations using a streamfunction velocity formulation was proposed by gupta and kalita m m gupta j c kalita a new paradigm for solving navier stokes equations streamfunction velocity formulation j comput phys 207 2005 52 68 which avoids difficulties inherent in the conventional streamfunction vorticity and primitive variable formulations it is discovered that this formulation can reached second order accurate and obtained accuracy solutions with little additional cost for a couple of fluid flow problems in this paper an efficient compact finite difference approximation named as five point constant coefficient second order compact 5pcc soc scheme is proposed for the streamfunction formulation of the steady incompressible navier stokes equations in which the grid values of the streamfunction and the values of its first derivatives velocities are carried as the unknown variables the derivation approach is simple and can be easily used to derive compact difference schemes for other similar high order elliptic differential equations numerical examples including the lid driven cavity flow problem and a problem of flow in a rectangular cavity with the hight width ratio of 2 are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme the results obtained are compared with ones by different available numerical methods in the literature the present scheme not only shows second order accurate but also proves more effective than the existing second order compact scheme of the streamfunction formulation in the aspect of computational cost

vorticity vector potential method for 3d viscous incompressible flows in time dependent curvilinear coordinates e and liu j comput phys 138 1997 57 82 put forward a finite difference method for 3d viscous incompressible flows in the vorticity vector potential formulation on non staggered grids in this paper we will extend this method to the case of flows in the presence of a deformable surface by use of two kinds of surface differential operators the implementation of boundary conditions on a plane is generalized to a curved smooth surface with given velocity distribution whether this be an inflow outflow interface or a curved wall to deal with the irregular and varying physical domain time dependent curvilinear coordinates are constructed and the corresponding tensor analysis is adopted in deriving the component form of the governing equations therefore the equations can be discretized and solved in a regular and fixed parametric domain numerical results are presented for a 3d lid driven cavity with a deforming surface and a 3d duct flow with a deforming boundary a new way to validate numerical simulations is proposed based on an expression for the rate of strain tensor on a deformable surface

essentially compact schemes for unsteady viscous incompressible flows a new fourth order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced the scheme is based on the vorticity stream function formulation it is essentially compact and has the nice features of a compact scheme with regard to the treatment of boundary conditions it is also very efficient at every time step or runge kutta stage only two poisson like equations have to be solved the poisson like equations are amenable to standard fast poisson solvers usually designed for second order schemes detailed comparison with the second order scheme shows the clear superiority of this new fourth order scheme in resolving both the boundary layers and the gross features of the flow this efficient fourth order scheme also made it possible to compute the driven cavity flow at reynolds number 106on a 10242grid at a reasonable cost fourth order convergence is proved under mild regularity requirements this is the first such result to our knowledge

stable numerical boundary conditions for stokes equations the author works with the vorticity stream function formulation of the stokes equations two boundary conditions are given for the stream function while no boundary conditions are given for the vorticity a derived boundary condition for the vorticity is considered and it is proven that the resulting fully discrete numerical method is stable the results are then extended to the three dimensional stokes equations using a staggered grid

meshless schemes for unsteady navier stokes equations in vorticity formulation using radial basis functions in this paper we provide a new scheme for unsteady incompressible flows in vorticity stream function formulation combined with the radial basis functions method it is an efficient meshless method optimal accuracy can be achieved using this method the efficiency and accuracy are demonstrated by numerical examples

analysis of first order system least squares fosls for elliptic problems with discontinuous coefficients part i first order system least squares fosls is a recently developed methodology for solving partial differential equations among its advantages are that the finite element spaces are not restricted by the inf sup condition imposed for example on mixed methods and that the least squares functional itself serves as an appropriate error measure this paper studies the fosls approach for scalar second order elliptic boundary value problems with discontinuous coefficients irregular boundaries and mixed boundary conditions a least squares functional is defined and ellipticity is established in a natural norm of an appropriately scaled least squares bilinear form for some geometries this ellipticity is independent of the size of the jumps in the coefficients the occurrence of singularities at interface corners cross points reentrant corners and irregular boundary points is discussed and a basis of singular functions with local support around singular points is established a companion paper shows that the singular basis functions can be added at little extra cost and lead to optimal performance of standard finite element discretization and multilevel solver techniques also independent of the size of coefficient jumps for some geometries

high accuracy solutions of incompressible navier stokes equations abstract in recent years we have developed high accuracy finite difference approximations for partial differential equations of elliptic type with particular emphasis on the convection diffusion equation these approximations are of compact type have a local truncation error of fourth order and allow the use of standard iterative schemes to solve the resulting systems of algebraic equations in this paper we extend these high accuracy approximations to the solution of navier stokes equations solutions are obtained for the model problem of driven cavity and are compared with solutions obtained using other approximations and those obtained by other authors it is discovered that the high order approximations do indeed produce high accuracy solutions and have a potential for use in solving important problems of viscous fluid flows

a pure compact scheme for the streamfunction formulation of navier stokes equations a pure streamfunction formulation is introduced for the numerical simulation of the two dimensional incompressible navier stokes equations the idea is to replace the vorticity in the vorticity streamfunction evolution equation by the laplacian of the streamfunction the resulting formulation includes the streamfunction only thus no inter function relations need to be invoked a compact numerical scheme which interpolates streamfunction values as well as its first order derivatives is presented and analyzed a number of numerical experiments are presented including driven and double driven cavities where the reynolds numbers are sufficiently large leading to symmetry breaking of asymptotic solutions

a high order discontinuous galerkin method for 2d incompressible flows in this paper we introduce a high order discontinuous galerkin method for two dimensional incompressible flow in the vorticity stream function formulation the momentum equation is treated explicitly utilizing the efficiency of the discontinuous galerkin method the stream function is obtained by a standard poisson solver using continuous finite elements there is a natural matching between these two finite element spaces since the normal component of the velocity field is continuous across element boundaries this allows for a correct upwinding gluing in the discontinuous galerkin framework while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability the method is efficient for inviscid or high reynolds number flows optimal error estimates are proved and verified by numerical experiments

analysis of finite difference schemes for unsteady navier stokes equations in vorticity formulation in this paper we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible navier stokes equations in vorticity stream function formulation the no slip boundary condition for the velocity is converted into local vorticity boundary conditions thom s formula wilkes formula or other local formulas in the earlier literature can be used in the second order method while high order formulas such as briley s formula can be used in the fourth order compact difference scheme proposed by e and liu the stability analysis of these long stencil formulas cannot be directly derived from straightforward manipulations since more than one interior point is involved in the formula the main idea of the stability analysis is to control local terms by global quantities via discrete elliptic regularity for stream function we choose to analyze the second order scheme with wilkes formula in detail in this case we can avoid the complicated technique necessitated by the strang type high order expansions as a consequence our analysis results in almost optimal regularity assumption for the exact solution the above methodology is very general we also give a detailed analysis for the fourth order scheme using a 1 d stokes model

weakly compressible high order i stable central difference schemes for incompressible viscous flows abstract in this paper we propose a weakly compressible model for numerical simulation of the incompressible viscous flows this model asymptotically approximates the incompressible navier stokes equations when the mach number tends to zero the main advantage of this model is that its numerical discretization avoids any poisson solver thus is very attractive for problems with complicated geometries this model is discretized by high order center differences in space and the so called i stable method for time the linear stability region of an i stable method contains part of the imaginary axis when solving a system of convection diffusion equations with a small viscosity the i stable method allows a very large cell reynolds number thus is particularly suitable for the simulation of fluid flows with large reynolds numbers numerical experiments illustrate the efficiency and robustness of this approach

numerical stabilities study of a decomposed compact method for highly oscillatory helmholtz equations abstract this paper concerns numerical stabilities of a decomposed compact finite difference method for solving helmholtz partial differential equation problems with large wave numbers radially symmetric transverse fields and standard polar coordinates are considered a decomposition is implemented to remove the anticipated singularity in the transverse direction compact scheme structures are introduced in the transverse direction to raise the accuracy for laser and subwavelength optical computations it is proven that while the highly accurate compact algorithm shies away from the conventional stability in the von neumann sense it is asymptotically stable with index one additionally a discussion on theoretical and numerical stabilities for this method is presented numerical experiments further demonstrate the high reliability of the scheme when implemented in highly oscillatory optical self focusing wave propagation simulations

a compact difference scheme for the navier stokes equations in vorticity velocity formulation this paper presents a new numerical method for solving the incompressible unsteady navier stokes equations in vorticity velocity formulation the method is applicable to spatial simulations of transitional and turbulent boundary layer flows it is based on a compact difference discretization of the streamwise and wall normal derivatives in cartesian coordinates a fourier collocation approach is used for the spanwise derivatives important new features of the numerical method are the use of nonequidistant differences in the wall normal direction the use of split compact differences in the streamwise direction a new fast iteration for a semi implicit time integration of the wall normal diffusion terms and an improvement of the buffer domain technique to prevent reflections of waves at the outflow boundary results of test calculations are presented to verify the improvements obtained by the use of these new techniques

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

a least squares approach based on a discrete minus one inner product for first order systems the purpose of this paper is to develop and analyze a least squares approximation to a first order system the first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and or nonsymmetric the approach taken here is novel in that the least squares functional employed involves a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions in addition the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned thus providing an efficient method for solving the algebraic equations the preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems a task which is well understood

driven cavity flows by efficient numerical techniques abstract efficient and reliable numerical techniques of high order accuracy are presented for solving problems of steady viscous incompressible flow in the plane and are used to obtain accurate solutions for the driven cavity a solution is obtained at reynolds number 10 000 on a 180 180 grid the numerical methods combine an efficient linear system solver an adaptive newton like method for nonlinear systems and a continuation procedure for following a branch of solutions over a range of reynolds numbers

a continuing exploration of a decomposed compact method for highly oscillatory wave problems this paper concerns a highly effective and decomposed compact scheme for solving a highly oscillatory paraxial helmholtz problem in radially symmetric fields the decomposition is utilized in the transverse direction to eliminate the singularity of the differential equation in polar coordinates numerical stability of the splitting scheme is investigated it is shown that the numerical method introduced is not only highly accurate and efficient due to its straightforward algorithmic structure but also stable under reasonable constraints for practical applications computational examples are presented to illustrate our conclusions

first order system ll fosll for general scalar elliptic problems in the plane this paper develops new first order system ll fosll formulations for scalar elliptic partial differential equations it extends the work of z cal et al siam j numer anal 39 2001 pp 1418 1445 where the fosll methodology was first introduced one focus of that paper was to develop fl formulations that allow the use of h1 conforming finite element spaces and optimal multigrid solution techniques to construct l2 approximations of the dependent variables in the presence of discontinuous coefficients the problems for which this goal was achieved were limited to those with no reaction term and with dirichlet and neumann boundaries that were individually connected that is each had at most one component here new fosll formulations are developed to achieve the same goals on a wider class of problems including problems with reaction terms dirichlet and neumann boundaries with multiple components reentrant corners and points at which dirichlet and neumann boundaries meet with an inner angle greater than pi 2 the efficiency of the improved fosll formulations is illustrated by a series of numerical examples

a computational stream function method for two dimensional incompressible viscous flows this work concerns the development of a numerical method based on the stream function formulation of the navier stokes equations to simulate two dimensional plane or axisymmetric viscous flows the main features of the proposed method are the use of the high order finite difference compact method for the discretization of the stream function equation the implicit pseudo transient newton krylov multigrid matrix free method for the stationary stream function equation and the fourth order runge kutta method for the integration of non stationary flows copyright 2005 john wiley sons ltd

lattice bgk model for incompressible navier stokes equation abstract most of the existing lattice boltzmann bgk models lbgk can be viewed as compressible schemes to simulate incompressible fluid flows the compressible effect might lead to some undesirable errors in numerical simulations in this paper a lbgk model without compressible effect is designed for simulating incompressible flows the incompressible navier stokes equations are exactly recovered from this incompressible lbgk model numerical simulations of the plane poiseuille flow the unsteady 2 d shear decaying flow the driven cavity flow and the flow around a circular cylinder are performed the results agree well with the analytic solutions and the results of previous studies

boundary conditions for open boundaries for the incompressible navier stokes equation abstract in this paper we investigate new boundary conditions for the incompressible time dependent navier stokes equation especially inflow and outflow conditions are considered the equations are linearized around a constant flow so that we can use laplace fourier technique to investigate the strength of boundary layers at open boundaries such layers are unphysical and new boundary conditions are proposed so that these boundary layers are suppressed we also show that the boundary conditions we propose do not produce divergence furthermore they give solutions that do not grow in time as long as the forcing function in the system does not we also discuss the numerical treatment of boundaries when fourth order accurate finite difference operators are used to approximate spatial derivatives using higher order methods introduces eigensolutions with boundary layer thickness of the same order of magnitude as the grid size these eigensolutions have to be suppressed in order to not destroy the fourth order accuracy of the method numerical results for the non linear navier stokes equation together with the new boundary conditions are presented these calculations confirm that the results for the linearized problem hold for the non linear problem as well

accurate stable and efficient navier stokes solvers based on explicit treatment of the pressure term we present numerical schemes for the incompressible navier stokes equations based on a primitive variable formulation in which the incompressibility constraint has been replaced by a pressure poisson equation the pressure is treated explicitly in time completely decoupling the computation of the momentum and kinematic equations the result is a class of extremely efficient navier stokes solvers full time accuracy is achieved for all flow variables the key to the schemes is a neumann boundary condition for the pressure poisson equation which enforces the incompressibility condition for the velocity field irrespective of explicit or implicit time discretization of the viscous term in the momentum equation the explicit time discretization of the pressure term does not affect the time step constraint indeed we prove unconditional stability of the new formulation for the stokes equation with explicit treatment of the pressure term and first or second order implicit treatment of the viscous term systematic numerical experiments for the full navier stokes equations indicate that a second order implicit time discretization of the viscous term with the pressure and convective terms treated explicitly is stable under the standard cfl condition additionally various numerical examples are presented including both implicit and explicit time discretizations using spectral and finite difference spatial discretizations demonstrating the accuracy flexibility and efficiency of this class of schemes in particular a galerkin formulation is presented requiring only c0 elements to implement

finite difference methods for 3d viscous incompressible flows in the vorticity vector potential formulation on nonstaggered grids simple efficient and accurate finite difference methods are introduced for 3d unsteady viscous incompressible flows in the vorticity vector potential formulation on nonstaggered grids two different types of methods are discussed they differ in the implementation of the normal component of the vorticity boundary condition and consequently the enforcement of the divergence free condition for vorticity both second order and fourth order accurate schemes are developed a detailed accuracy test is performed revealing the structure of the error and the effect of how the convective terms are discretized near the boundary the influence of the divergence free condition for vorticity to the overall accuracy is studied results on the cubic driven cavity flow at reynolds number 500 and 3200 are shown and compared with that of the mac scheme

compact moving least squares an optimization framework for generating high order compact meshless discretizations abstract a generalization of the optimization framework typically used in moving least squares is presented that provides high order approximation while maintaining compact stencils and a consistent treatment of boundaries the approach which we refer to as compact moving least squares resembles the capabilities of compact finite differences but requires no structure in the underlying set of nodes an efficient collocation scheme is used to demonstrate the capabilities of the method to solve elliptic boundary value problems in strong form stably without the need for an expensive weak form the flexibility of the approach is demonstrated by using the same framework to both solve a variety of elliptic problems and to generate implicit approximations to derivatives finally an efficient preconditioner is presented for the steady stokes equations and the approach s efficiency and high order of accuracy is demonstrated for domains with curvi linear boundaries

a compact streamfunction velocity scheme on nonuniform grids for the 2d steady incompressible navier stokes equations in this paper an effective compact finite difference approximation which carries streamfunction and its first derivatives velocities as the unknown variables for the streamfunction velocity formulation of the steady two dimensional incompressible navier stokes equation is developed on non uniform orthogonal cartesian grids to solve the resulting system of equations a multigrid iterative strategy on nonuniform grids is introduced by using the interpolation techniques numerical experiments involving two test problems with analytical solutions and the lid driven square cavity flow problem are carried out to display the superiority of the currently developed method on nonuniform grid numerical results show that the present method on nonuniform grids gets as similarly efficient convergence rate as on uniform grids viz second order accuracy and the resolution of the computed solutions for the problems with the sharp changes can be significantly improved when the nonuniform grid strategy is utilized the backward facing step flow is also calculated by the present method to exhibit the capability to simulate the distant field using fewer grid points the solution for the natural convection problem reveals further the wide applications of the present method not only in the flow problems but also in the heat transfer problems all of these numerical results demonstrate the accuracy and efficiency of the currently proposed schemes

finite difference schemes for incompressible flow based on local pressure boundary conditions in this paper we discuss the derivation and use of local pressure boundary conditions for finite difference schemes for the unsteady incompressible navier stokes equations in the velocity pressure formulation their use is especially well suited for the computation of moderate to large reynolds number flows we explore the similarities between the implementation and use of local pressure boundary conditions and local vorticity boundary conditions in the design of numerical schemes for incompressible flow in 2d in their respective formulations when these local numerical boundary conditions are coupled with a fully explicit convectively stable time stepping procedure the resulting methods are simple to implement and highly efficient unlike the vorticity formulation the use of the local pressure boundary condition approach is readily applicable to 3d flows the simplicity of the local pressure boundary condition approach and its easy adaptation to more general flow settings make the resulting scheme an attractive alternative to the more popular methods for solving the navier stokes equations in the velocity pressure formulation we present numerical results of a second order finite difference scheme on a nonstaggered grid using local pressure boundary conditions stability and accuracy of the scheme applied to stokes flow is demonstrated using normal mode analysis also described is the extension of the method to variable density flows

a three point combined compact difference scheme a new three point combined compact difference ccd scheme is developed for numerical models the major features of the ccd scheme are three point implicit sixth order accuracy and inclusion of boundary values due to its combination of the first and second derivatives the ccd scheme becomes more compact and more accurate than normal compact difference schemes the efficient twin tridiagonal for calculating derivatives and triple tridiagonal for solving partial difference equation with the ccd scheme methods are also presented besides the ccd scheme has sixth order accuracy at periodic boundaries and fifth order accuracy at nonperiodic boundaries the possibility of extending to a three point eighth order scheme is also included

a central difference scheme for a pure stream function formulation of incompressible viscous flow we present a numerical scheme for incompressible viscous flow formulated as an equation for the stream function the pure stream function formulation obviates the difficulty associated with vorticity boundary conditions the resulting biharmonic equation is discretized with a compact scheme and solved with an algebraic multigrid solver the advection of vorticity is implemented with a high resolution central scheme that remains stable and accurate in the presence of large gradients the accuracy and robustness of the method are demonstrated for high reynolds number flows in a lid driven cavity

very high order compact finite difference schemes on non uniform grids for incompressible navier stokes equations this article presents a family of very high order non uniform grid compact finite difference schemes with spatial orders of accuracy ranging from 4th to 20th for the incompressible navier stokes equations the high order compact schemes on non uniform grids developed in shukla and zhong r k shukla x zhong derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation j comput phys 204 2005 404 for linear model equations are extended to the full navier stokes equations in the vorticity and streamfunction formulation two methods for the solution of helmholtz and poisson equations using high order compact schemes on non uniform grids are developed the schemes are constructed so that they maintain a high order of accuracy not only in the interior but also at the boundary second order semi implicit temporal discretization is achieved through an implicit backward differentiation scheme for the linear viscous terms and an explicit adam bashforth scheme for the non linear convective terms the boundary values of vorticity are determined using an influence matrix technique the resulting discretized system with boundary closures of the same high order as the interior is shown to be stable when applied to the two dimensional incompressible navier stokes equations provided enough grid points are clustered at the boundary the resolution characteristics of the high order compact finite difference schemes are illustrated through their application to the one dimensional linear wave equation and the two dimensional driven cavity flow comparisons with the benchmark solutions for the two dimensional driven cavity flow thermal convection in a square box and flow past an impulsively started cylinder show that the high order compact schemes are stable and produce extremely accurate results on a stretched grid with more points clustered at the boundary

on the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients the successful implementation of adaptive finite element methods based on a posteriori error estimates depends on several ingredients an a posteriori error indicator a refinement coarsening strategy and the choice of various parameters the objective of the paper is to examine the influence of these factors on the performance of adaptive finite element methods for a model problem the linear elliptic equation with strongly discontinuous coefficients we derive a new a posteriori error estimator which depends locally on the oscillations of the coefficients around singular points extensive numerical experiments are reported to support our theoretical results and to show the competitive behaviors of the proposed adaptive algorithm

a fourth order numerical method for the planetary geostrophic equations with inviscid geostrophic balance the planetary geostrophic equations with inviscid balance equation are reformulated in an alternate form and a fourth order finite difference numerical method of solution is proposed and analyzed in this article in the reformulation there is only one prognostic equation for the temperature field and the velocity field is statically determined by the planetary geostrophic balance combined with the incompressibility condition the key observation is that all the velocity profiles can be explicitly determined by the temperature gradient by utilizing the special form of the coriolis parameter this brings convenience and efficiency in the numerical study in the fourth order scheme the temperature is dynamically updated at the regular numerical grid by long stencil approximation along with a one sided extrapolation near the boundary the velocity variables are recovered by special solvers on the 3 d staggered grid furthermore it is shown that the numerical velocity field is divergence free at the discrete level in a suitable sense fourth order convergence is proven under mild regularity requirements

boundary layer separation and adverse pressure gradient for 2 d viscous incompressible flow we study the detailed process of bifurcation in the flow s topological structure for a two dimensional 2 d incompressible flow subject to no slip boundary conditions and its connection with boundary layer separation the boundary layer separation theory of m ghil t ma and s wang based on the structural bifurcation concept is translated into vorticity form the vorticity formulation of the theory shows that structural bifurcation occurs whenever a degenerate singular point for the vorticity appears on the boundary this singular point is characterized by nonzero tangential second order derivative and nonzero time derivative of the vorticity furthermore we prove the presence of an adverse pressure gradient at the critical point due to reversal in the direction of the pressure force with respect to the basic shear flow at this point a numerical example of 2 d driven cavity flow governed by the navier stokes equations is presented boundary layer separation occurs the bifurcation criterion is satisfied and an adverse pressure gradient is shown to be present

an implicit method for coupled flow body dynamics we propose an efficient method for computing coupled flow body dynamics the time stepping is implicit and uses an iterative method preconditioned gmres to solve the flow body equations the preconditioner solves a decoupled version of the equations which involves only the inversion of banded matrices and requires a small number of iterations per time step we use the method to probe the instability to horizontal motions of an elliptical body with simple vertical motions flapping and rising in both cases a linear instability to horizontal motion sets in above a critical reynolds number leading to a stable oscillatory state the pressure forces play a destabilizing role against the stabilizing viscous forces with oscillatory time scales set by either external flapping or the intrinsic flow body coupling the latter lowers the instability threshold in reynolds number

flow past an impulsively started circular cylinder using a higher order semicompact scheme in the present investigation an attempt has been made to solve the two dimensional incompressible viscous flow past an impulsively started circular cylinder for reynolds numbers ranging from 20 to 5000 using higher order semicompact scheme hosc unlike conventional higher order compact schemes the hosc scheme has been developed to handle the circular geometry of the chosen problem and the intensive algebraic manipulations have been reduced considerably by relaxing the compactness of the computational stencil for few terms but retained for most of the terms of the discretized equations for the flow past an impulsively started circular cylinder the results obtained at low and moderate reynolds numbers have been validated with the experimental and numerical observations available in the literature for high reynolds number flows the present scheme rightly captures the alpha phenomenon at re 1000 and both beta and alpha phenomena one after the other at re 5000

derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation in this paper simple polynomial interpolation is used to derive arbitrarily high order compact schemes for the first derivative and tridiagonal compact schemes for the second derivative consisting of three second derivative nodes in the interior and two on the boundary on non uniform grids boundary and near boundary schemes of the same order as the interior are also developed using polynomial interpolation and for a general compact scheme on a non uniform grid it is shown that polynomial interpolation is more efficient than the conventional method of undetermined coefficients for finding coefficients of the scheme the high order non uniform schemes along with boundary closure of up to 14th order thus obtained are shown to be stable on a non uniform grid with appropriate stretching so that more grid points are clustered near the boundary the stability and resolution properties of the high order non uniform grid schemes are studied and the results of three numerical tests on stability and accuracy properties are also presented

moving mesh finite element methods for the incompressible navier stokes equations this work presents the first effort in designing a moving mesh algorithm to solve the incompressible navier stokes equations in the primitive variables formulation the main difficulty in developing this moving mesh scheme is how to keep it divergence free for the velocity field at each time level the proposed numerical scheme extends a recent moving grid method based on harmonic mapping r li t tang and p w zhang j comput phys 170 2001 pp 562 588 which decouples the pde solver and the mesh moving algorithm this approach requires interpolating the solution on the newly generated mesh designing a divergence free preserving interpolation algorithm is the first goal of this work selecting suitable monitor functions is important and is found challenging for the incompressible flow simulations which is the second goal of this study the performance of the moving mesh scheme is tested on the standard periodic double shear layer problem no spurious vorticity patterns appear when even fairly coarse grids are used

analysis of a fourth order finite difference method for the incompressible boussinesq equations the convergence of a fourth order finite difference method for the 2 d unsteady viscous incompressible boussinesq equations based on the vorticity stream function formulation is established in this article a compact fourth order scheme is used to discretize the momentum equation and long stencil fourth order operators are applied to discretize the temperature transport equation a local vorticity boundary condition is used to enforce the no slip boundary condition for the velocity one sided extrapolation is used near the boundary dependent on the type of boundary condition for the temperature to prescribe the temperature at ghost points lying outside of the computational domain theoretical results of the stability and accuracy of the method are also provided in numerical experiments the method has been shown to be capable of producing highly resolved solutions at a reasonable computational cost

an accurate compact treatment of pressure for colocated variables we propose a new scheme for the pressure treatment in computations of incompressible flow using a colocated grid arrangement to avoid oscillations associated with the sparse non compact stencil we introduce a compact fourth order equivalent of this stencil and study its advantages over the classical second order averaging procedure

vorticity boundary condition and related issues for finite difference schemes this paper discusses three basic issues related to the design of finite difference schemes forunsteadyviscous incompressible flows using vorticity formulations the boundary condition for vorticity an efficient time stepping procedure and the relation between these schemes and the ones based on velocity pressure formulation we show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier we also show that if we couple a standard centered difference scheme with third or fourth order explicit runge kutta methods the resulting schemes havenocell reynolds number constraints for high reynolds number flows these schemes are stable under the cfl condition given by the convective terms finally we show that the classical mac scheme is the same as thom s formula coupled with second order centered differences in the interior in the sense that one can define discrete vorticity in a natural way for the mac scheme and get the same values as the ones computed from thom s formula we use this to derive an efficient fourth order runge kutta time discretization for the mac scheme from the one for thom s formula we present numerical results for driven cavity flow at high reynolds number 105

compact h4 finite difference approximations to operators of navier stokes type a method of obtaining compact finite difference approximations of h4 accuracy to operators of navier stokes type is considered the basic procedure is developed for operators in one space dimension and subsequently applied to problems in more space dimensions and in time four illustrative numerical examples are given which indicate clearly in various cases that excellent accuracy may be obtained using the methods comparisons with previous results and with the results of h2 accurate computations are made

renormalization group and anomalous scaling in a simple model of passive scalar advection in compressible flow field theoretical renormalization group methods are applied to a simple model of a passive scalar quantity advected by the gaussian non solenoidal compressible velocity field with the covariance propto delta t t x x epsilon convective range anomalous scaling for the structure functions and various pair correlators is established and the corresponding anomalous exponents are calculated to the order epsilon 2 of the epsilon expansion these exponents are non universal as a result of the degeneracy of the rg fixed point in contrast to the case of a purely solenoidal velocity field obukhov kraichnan model the correlation functions in the case at hand exhibit nontrivial dependence on both the ir and uv characteristic scales and the anomalous scaling appears already at the level of the pair correlator the powers of the scalar field without derivatives whose critical dimensions determine the anomalous exponents exhibit multifractal behaviour the exact solution for the pair correlator is obtained it is in agreement with the result obtained within the epsilon expansion the anomalous exponents for passively advected magnetic fields are also presented in the first order of the epsilon expansion