finite difference preconditioning for compact scheme discretizations of the poisson equation with variable coefficients abstract the finite difference preconditioning for higher order compact scheme discretizations of non separable poisson s equation is investigated an eigenvalue analysis of a one dimensional problem is detailed for compact schemes up to the tenth order the analysis concludes that the spectrum is bounded irrespective of the mesh size and the continuous variable coefficient hence combined to a multigrid method the preconditioned richardson method shows a convergence rate which is independent from the mesh size and the variable coefficient several numerical experiments including the simulation of a flow with large density variations confirm that the spectrum of the preconditioned operator remains bounded

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

accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

effects of the computational time step on numerical solutions of turbulent flow effects of large computational time steps on the computed turbulence were investigated using a fully implicit method in turbulent channel flow computations the largest computational time step in wall units which led to accurate prediction of turbulence statistics was determined turbulence fluctuations could not be sustained if the computational time step was near or larger than the kolmogorov time scale

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

the tangent approximation to one sided brownian exit densities let p t be the density of the first exit time of a brownian motion over a one sided moving boundary and let p1 t be the density at t of the first exit time over the tangent to the boundary at t when is p1 t a good approximation to p t we investigate this question by means of a new integral equation for p t which makes possible explicit estimates for the error of the approximation

boosting the accuracy of finite difference schemes via optimal time step selection and non iterative defect correction abstract in this article we present a unified analysis of the simple technique for boosting the order of accuracy of finite difference schemes for time dependent partial differential equations pdes by optimally selecting the time step used to advance the numerical solution and adding defect correction terms in a non iterative manner the power of the technique which is applicable to time dependent semilinear scalar pdes where the leading order spatial derivative has a constant coefficient is its ability to increase the accuracy of formally low order finite difference schemes without major modification to the basic numerical algorithm through straightforward numerical analysis arguments we explain the origin of the boost in accuracy and estimate the computational cost of the resulting numerical method we demonstrate the utility of optimal time step ots selection combined with non iterative defect correction nidc on several different types of finite difference schemes for a wide array of classical linear and semilinear pdes in one and more space dimensions on both regular and irregular domains

numerical analysis of a finite element projection based vms turbulence model with wall laws abstract this paper deals with the numerical analysis of a finite element projection based vms turbulence model that includes general non linear wall laws only a single mesh and interpolation operators on a virtual coarser mesh are needed to implement the model we include a projection stabilization of pressure to use the same interpolation for velocity and pressure good accuracy is obtained with benchmark turbulent flow problems on coarse grids that justify the interest of this approach also the model solves smooth flows with optimal accuracy

a high order compact difference algorithm for half staggered grids for laminar and turbulent incompressible flows abstract the paper presents a novel efficient and accurate algorithm for laminar and turbulent flow simulations the spatial discretisation is performed with help of the compact difference schemes up to 10th order for collocated and half staggered grid arrangements the time integration is performed by a predictor corrector approach combined with the projection method for pressure velocity coupling at this stage a low order discretisation is introduced which considerably decreases the computational costs it is demonstrated that such approach does not deteriorate the solution accuracy significantly following boersma b j 13 the interpolation formulas developed for staggered uniform meshes are used also in the computations with a non uniform strongly varying nodes distribution in the proposed formulation of the projection method such interpolation is performed twice it is shown that it acts implicitly as a high order low pass filter and therefore the resulting algorithm is very robust its accuracy is first demonstrated based on simple 2d and 3d problems an inviscid vortex advection a decay of taylor green vortices a modified lid driven cavity flow and a dipole wall interaction in periodic flow problems the first two cases the solution accuracy exhibits the 10th order behaviour in the latter cases the 3rd and the 4th order is obtained robustness of the proposed method in the computations of turbulent flows is demonstrated for two classical cases a periodic channel with re 395 and re 590 and a round jet with re 21 000 the solutions are obtained without any turbulence model and also without any explicit techniques aiming to stabilise the solution the results are in a very good agreement with literature dns and les data both the mean and r m s values are predicted correctly

a computational approach to modeling cellular scale blood flow in complex geometry we present a computational methodology for modeling cellular scale blood flow in arbitrary and highly complex geometry our approach is based on immersed boundary methods which allow modeling flows in arbitrary geometry while resolving the large deformation and dynamics of every blood cell with high fidelity the present methodology seamlessly integrates different modeling components dealing with stationary rigid boundaries of complex shape moving rigid bodies and highly deformable interfaces governed by nonlinear elasticity thus it enables us to simulate whole blood suspensions flowing through physiologically realistic microvascular networks that are characterized by multiple bifurcating and merging vessels as well as geometrically complex lab on chip devices the focus of the present work is on the development of a versatile numerical technique that is able to consider deformable cells and rigid bodies flowing in three dimensional arbitrarily complex geometries over a diverse range of scenarios after describing the methodology a series of validation studies are presented against analytical theory experimental data and previous numerical results then the capability of the methodology is demonstrated by simulating flows of deformable blood cells and heterogeneous cell suspensions in both physiologically realistic microvascular networks and geometrically intricate microfluidic devices it is shown that the methodology can predict several complex microhemodynamic phenomena observed in vascular networks and microfluidic devices the present methodology is robust and versatile and has the potential to scale up to very large microvascular networks at organ levels

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

direct numerical simulation of spatial transition to turbulence using fourth order vertical velocity second order vertical vorticity formulation a highly accurate algorithm has been developed to study the process of spatial transition to turbulence the algorithmic details of the direct numerical simulation dns of transition to turbulence in a boundary layer based on a formulation in terms of vertical velocity and vertical vorticity are presented issues concerning the boundary conditions are discussed the linear viscous terms are discretized using an implicit crank nicholson scheme and a low storage runge kutta method is used for the nonlinear terms for the spatial discretization fourth order compact finite differences have been used as these have been found to have better resolution compared to explicit differencing schemes of comparable order the number of grid points that are needed per wavelength is close to the theoretical optimum for any numerical scheme the resulting time discretized fourth order equations are split up into two second order equations resulting in helmholtz and poisson type equations the boundary conditions for the laplacian of the vertical velocity are determined using an influence matrix method a robust multigrid algorithm has been developed to solve the resulting anisotropic elliptical equations for the outflow boundary a buffer domain method which smoothly reduces the disturbances to zero in conjunction with parabolization of the navier stokes equations has been used the validation of the results for the dns solver is made both for linear and weakly nonlinear cases

a non iterative direct forcing immersed boundary method for strongly coupled fluid solid interactions a non iterative direct forcing immersed boundary method is presented for the strongly coupled simulations of fluid solid interactions while it retains many advantages of the immersed boundary framework by yang and stern 2012 30 especially the simplified field extension strategy for moving boundary treatment and the pointwise integration of hydrodynamic force using the momentum forcing term the present approach improves upon the previous method in several aspects including optimized computational cost for a strong coupling scheme reduced algorithm complexity for a straightforward implementation and enhanced numerical stability for low density ratio problems central to these improvements is a simple intermediate step in which the velocity fields around solid bodies are predicted on temporary non inertial reference frames attached to moving solid bodies in a one to one manner this step enables the explicit inverse of the implicit equations for rigid body dynamics thus rendering unnecessary the previous predictor corrector scheme for iteratively adjusting the displacements and velocities of the immersed bodies until reaching a convergence in addition a simple generalized procedure is developed to obtain the interpolation coefficients in a local reconstruction stencil explicitly from the geometric relationship for verification and validation the vortex induced vibration of a circular cylinder and the rotational galloping of a rectangular body are considered first then several particulate flow problems including settling and buoyant particles of low density ratios a settling particle in a small container and the kissing drafting tumbling problem of two settling particles are studied the agreement between the present results and the reference data in the literature is excellent an overall second order accuracy of the algorithm is verified in two systematic grid convergence tests the present idea can be easily applied to similar methods for achieving a strong coupling scheme on top of a weak one with a nominal increase in the computational cost details of the algorithm are provided to facilitate its implementation in other solvers using non boundary conforming grids

a co located incompressible navier stokes solver with exact mass momentum and kinetic energy conservation in the inviscid limit the incompressible navier stokes equations conserve mass momentum and kinetic energy in the inviscid limit therefore numerical methods should attempt to mimic these conservation properties as well there exists many well established methods with exact conservation properties based on staggered grids but existing methods based on co located grids introduce numerical dissipation in order to yield smooth pressure fields in the present work a novel method that utilizes null space vectors to yield a smooth pressure field is presented the method exactly preserves mass momentum and kinetic energy on co located grids

a fully conservative second order finite difference scheme for incompressible flow on nonuniform grids a second order accurate finite difference discretization of the incompressible navier stokes is presented that discretely conserves mass momentum and kinetic energy in the inviscid limit in space and time the method is thus completely free of numerical dissipation and potentially well suited to the direct numerical simulation or large eddy simulation of turbulent flow the method uses a staggered arrangement of velocity and pressure on a structured cartesian grid and retains its discrete conservation properties for both uniform and nonuniform gird spacing the predicted conservation properties are confirmed by inviscid simulations on both uniform and nonuniform grids the capability of the method to resolve turbulent flow is demonstrated by repeating the turbulent channel flow simulations of h choi and p moin 1994 j comput phys 113 1 where the effect of computational time step on the computed turbulence was investigated the present fully conservative scheme achieved turbulent flow solutions over the entire range of computational time steps investigated t tu2 0 4 to 5 0 little variation in statistical turbulence quantities was observed up to t 1 6 the present results differ significantly from those reported by choi and moin who observed significant discrepancies in the turbulence statistics above t 0 4 and the complete laminarization of the flow at and above t 1 6

high order compact schemes for incompressible flows a simple and efficient method with quasi spectral accuracy in this paper a finite difference code for direct and large eddy simulation dns les of incompressible flows is presented this code is an intermediate tool between fully spectral navier stokes solvers limited to academic geometry through fourier or chebyshev representation and more versatile codes based on standard numerical schemes typically only second order accurate the interest of high order schemes is discussed in terms of implementation easiness computational efficiency and accuracy improvement considered through simplified benchmark problems and practical calculations the equivalence rules between operations in physical and spectral spaces are efficiently used to solve the poisson equation introduced by the projection method it is shown that for the pressure treatment an accurate fourier representation can be used for more flexible boundary conditions than periodicity or free slip using the concept of the modified wave number the incompressibility can be enforced up to the machine accuracy the benefit offered by this alternative method is found to be very satisfactory even when a formal second order error is introduced locally by boundary conditions that are neither periodic nor symmetric the usefulness of high order schemes combined with an immersed boundary method ibm is also demonstrated despite the second order accuracy introduced by this wall modelling strategy in particular the interest of a partially staggered mesh is exhibited in this specific context three dimensional calculations of transitional and turbulent channel flows emphasize the ability of present high order schemes to reduce the computational cost for a given accuracy the main conclusion of this paper is that finite difference schemes with quasi spectral accuracy can be very efficient for dns les of incompressible flows while allowing flexibility for the boundary conditions and easiness in the code development therefore this compromise fits particularly well for very high resolution simulations of turbulent flows with relatively complex geometries without requiring heavy numerical developments

enhancement of an industrial finite volume code for large eddy type simulation of incompressible high reynolds number flow using near wall modelling abstract we present a validation strategy for enhancement of an unstructured industrial finite volume solver designed for steady rans problems for large eddy type simulation with near wall modelling of incompressible high reynolds number flow different parts of the projection based discretisation are investigated to ensure les capability of the numerical method turbulence model parameters are calibrated by using a minimisation of least squares functionals for first and second order statistics of the basic benchmark problems decaying homogeneous turbulence and turbulent channel flow then the method is applied to the flow over a backward facing step at reh 37 500 of special interest is the role of the spatial and temporal discretisation error for low order schemes for wall bounded flows present results confirm existing best practice guidelines for mesh design for free shear layers a sensor to quantify the resolution quality of the les based on the resolved turbulent kinetic energy is presented and applied to the flow over a backward facing step at reh 37 500

investigation into the finite element analysis of confined turbulent flows using a κ e model of turbulence abstract the finite element method is applied to turbulent confined flow problems adopting a primitive variables formulation the turbulence state is defined by a k e model of turbulence the argument is presented that the velocity components as well as k and e should be interpolated linearly near solid boundaries a wall functions method is employed a new and effective way of estimating the wall shear stress in the application of this method is proposed the model is verified by investigating several examples the results are compared with experiments and finite difference predictions

a parallel block multi level preconditioner for the 3d incompressible navier stokes equations the development of robust and efficient algorithms for both steady state simulations and fully implicit time integration of the navier stokes equations is an active research topic to be effective the linear subproblems generated by these methods require solution techniques that exhibit robust and rapid convergence in particular they should be insensitive to parameters in the problem such as mesh size time step and reynolds number in this context we explore a parallel preconditioner based on a block factorization of the coefficient matrix generated in an oseen nonlinear iteration for the primitive variable formulation of the system the key to this preconditioner is the approximation of a certain schur complement operator by a technique first proposed by kay loghin and wathen siam j sci comput 2002 and silvester elman kay and wathen j comput appl math 128 2001 261 the resulting operator entails subsidiary computations solutions of pressure poisson and convection diffusion subproblems that are similar to those required for decoupled solution methods however in this case these solutions are applied as preconditioners to the coupled oseen system one important aspect of this approach is that the convection diffusion and poisson subproblems are significantly easier to solve than the entire coupled system and a solver can be built using tools developed for the subproblems in this paper we apply smoothed aggregation algebraic multigrid to both subproblems previous work has focused on demonstrating the optimality of these preconditioners with respect to mesh size on serial two dimensional steady state computations employing geometric multi grid methods we focus on extending these methods to large scale parallel three dimensional transient and steady state simulations employing algebraic multigrid amg methods our results display nearly optimal convergence rates for steady state solutions as well as for transient solutions over a wide range of cfl numbers on the two dimensional and three dimensional lid driven cavity problem

numerical studies of finite element variational multiscale methods for turbulent flow simulations different realizations of variational multiscale vms methods within the framework of finite element methods are studied in turbulent channel flow simulations one class of vms methods uses bubble functions to model resolved small scales whereas the other class contains the definition of the resolved small scales by an explicit projection in its set of equations all methods are employed with eddy viscosity models of smagorinsky type the simulations are performed on grids for which a direct numerical simulation blows up in finite time

time accurate computation of unsteady free surface flows using an ale segregated equal order fem abstract arbitrary lagrangian eulerian ale finite element formulations based on segregated equal order interpolation are presented with the aim of computing unsteady free surface flows time accurately a standing vortex problem is solved using both fixed and moving grids to design a solution method which is time accurate in the sense that it conserves vortex kinetic energy it turns out that the chorin type simple algorithm serves this purpose satisfactorily when it is used in conjunction with the galerkin spatial discretization and the crank nicolson temporal discretization then a small amplitude sloshing problem is solved to assure that the crank nicolson central difference scheme among others used for discretizing the kinematic condition preserves the oscillating amplitude of the free surface lastly the most time accurate numerical technique thus designed is applied to solve a solitary wave propagation problem which shows that the predicted maximum run up heights for various initial heights are in good agreement with existing experiment

time step restrictions using semi explicit methods for the incompressible navier stokes equations time step restrictions using semi explicit methods for the incompressible navier stokes equations

a collocated method for the incompressible navier stokes equations inspired by the box scheme we present a new finite difference numerical method to solve the incompressible navier stokes equations using a collocated discretization in space on a logically cartesian grid the method shares some common aspects with and it was inspired by the box scheme it uses centered second order accurate finite difference approximations for the spatial derivatives combined with semi implicit time integration the proposed method is constructed to ensure discrete conservation of mass and momentum by discretizing the primitive velocity pressure form of the equations the continuity equation is enforced exactly to machine accuracy at the collocated locations whereas the momentum equations are evaluated in a staggered manner this formulation preempts the appearance of spurious pressure modes in the embedded elliptic problem associated with the pressure the method shows uniform order of accuracy both in space and time for velocity and pressure in addition the skew symmetric form of the non linear advection term of the navier stokes equations improves discrete conservation of kinetic energy in the inviscid limit to within the order of the truncation error of the time integrator the method has been formulated to accommodate different types of boundary conditions fully periodic periodic channel inflow outflow and lid driven cavity always ensuring global mass conservation a novel aspect of this finite difference formulation is the derivation of the discretization near boundaries using the weak form of the equations as in the finite element method the method of manufactured solutions is utilized to perform accuracy analysis and verification of the solver to assess the applicability of the new method presented in this paper four realistic flow problems have been simulated and results are compared with those in the literature these cases include a lid driven cavity backward facing step kovasznay flow and fully developed turbulent channel

a transient higher order compact scheme for incompressible viscous flows on geometries beyond rectangular in this paper we propose an implicit high order compact hoc finite difference scheme for solving the two dimensional 2d unsteady navier stokes n s equations on irregular geometries on orthogonal grids our scheme is second order accurate in time and fourth order accurate in space it is used to solve three pertinent fluid flow problems namely the flow decayed by viscosity the lid driven square cavity and the flow in a constricted channel it is seen to efficiently capture both transient and steady state solutions of the n s equations with dirichlet as well as neumann boundary conditions apart from including the good features of hoc schemes our formulation has the added advantage of capturing transient viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variations detailed comparison data produced by the scheme for all the three test cases are provided and compared with analytical as well as established numerical results excellent comparison is obtained in all the cases

fram nonlinear damping algorithms for the continuity equation abstract a nonlinear filtering methodology has been developed for the numerical solution of the convective terms in the eulerian form of the hydrodynamic equations fram filtering remedy and methodology introduces a strong local nonlinear dissipation into each of the governing equations to dampen non physical noise due to the properties of the solution algorithm selected for the convective terms in the equations the technique is developed for the multidimensional continuity equation and tested on several differencing schemes in both one and two spatial dimensions the technique is general in that there are no undetermined coefficients

algebraic splitting for incompressible navier stokes equations fully discretized incompressible navier stokes equations are solved by splitting the algebraic system with an approximate factorization this splitting affects the temporal convergence order of velocity and pressure the splitting error is proportional to the pressure variable and a simple analysis shows that the original convergence order of the time integration scheme can be retained by solving for incremental pressure the combination of splitting and incremental pressure is shown to be equivalent to an error correcting method using the full pressure in numerical experiments employing a third order time integration scheme and various orders for the pressure increment the splitting error is shown to control the convergence order and the full order of the scheme is recaptured for both velocity and pressure the difference between perturbing the momentum or the continuity equation is also explored

spatial finite difference approximations for wave type equations the simplest finite difference approximations for spatial derivatives are centered explicit and applied to regular equispaced grids well established generalizations include the use of implicit compact approximations and staggered grids we find here that the combination of these two concepts together with high formal order of accuracy is very effective for approximating the first derivatives in space that occur in many wave type pdes

a simple and efficient direct forcing immersed boundary framework for fluid structure interactions a direct forcing immersed boundary framework is presented for the simple and efficient simulation of strongly coupled fluid structure interactions the immersed boundary method developed by yang and balaras j yang e balaras an embedded boundary formulation for large eddy simulation of turbulent flows interacting with moving boundaries j comput phys 215 1 2006 12 40 is greatly simplified by eliminating several complicated geometric procedures without sacrificing the overall accuracy the fluid structure coupling scheme of yang et al j yang s preidikman e balaras a strongly coupled embedded boundary method for fluid structure interactions of elastically mounted rigid bodies j fluids struct 24 2008 167 182 is also significantly expedited by moving the fluid solver out of the predictor corrector iterative loop without altering the strong coupling property central to these improvements are the reformulation of the field extension strategy and the evaluation of fluid force and moment exerted on the immersed bodies by taking advantage of the direct forcing idea in a fractional step method several cases with prescribed motions are examined first to validate the simplified field extension approach then a variety of strongly coupled fluid structure interaction problems including vortex induced vibrations of a circular cylinder transverse and rotational galloping of rectangular bodies and fluttering and tumbling of rectangular plates are computed the excellent agreement between the present results and the reference data from experiments and other simulations demonstrates the accuracy simplicity and efficiency of the new method and its applicability in a wide range of complicated fluid structure interaction problems

a second order time accurate finite volume method for unsteady incompressible flow on hybrid unstructured grids a new second order time accurate fractional step method for solving unsteady incompressible navier stokes equations on hybrid unstructured grids is presented the nonstaggered grid method originally developed by rhie and chow 1983 aiaa j 21 1525 for steady flow and further extended by zang et al 1994 j comput phys 114 18 to unsteady flow on structured grids is employed in the present study to enforce mass conservation on hybrid unstructured grids the pressure and cartesian velocity components are defined at the center of each cell while the face normal velocities are defined at the mid points of the corresponding cell faces a second order fully implicit time advancement scheme is used for time integration and the resulting nonlinear equations are linearized without losing the overall time accuracy both the momentum and poisson equations are integrated with the finite volume method and the flow variables at the cell face are obtained using an interpolation scheme independent of cell shape the present numerical method is applied to four different benchmark problems and proves to be accurate and efficient

sharp interface immersed boundary level set method for wave body interactions a sharp interface cartesian grid method for the large eddy simulation of two phase turbulent flows interacting with moving bodies is presented the overall approach uses a sharp interface immersed boundary formulation and a level set ghost fluid method for solid fluid and fluid fluid interface treatments respectively a four step fractional step method is used for velocity pressure coupling and a lagrangian dynamic smagorinsky subgrid scale model is adopted for large eddy simulations a simple contact angle boundary condition treatment that conforms to the immersed boundary formulation is developed a variety of test cases of different scales ranging from bubble dynamics water entry and exit landslide generated waves to ship hydrodynamics are performed for validation extensions for high reynolds number ship flows using wall layer models are also considered

a fourth order accurate method for the incompressible navier stokes equations on overlapping grids a method is described to solve the time dependent incompressible navier stokes equations with finite differences on curvilinear overlapping grids in two or three space dimensions the scheme is fourth order accurate in space and uses the momentum equations for the velocity coupled to a poisson equation for the pressure the boundary condition for the pressure is taken as u 0 extra numerical boundary conditions are chosen to make the scheme accurate and stable the velocity is advanced explicitly in time any standard time stepping scheme such as runge kutta can be used the poisson equation is solved using direct or iterative sparse matrix solvers or by the multigrid algorithm computational results in two and three space dimensions are given

two dimensional unsteady euler equation solver for arbitrarily shaped flow regions a new technique is described for solving supersonic fluid dynamics problems containing multiple regions of continuous flow each bounded by a permeable or impermeable surface region boundaries are in general arbitrarily shaped and time dependent discretization of such a region for solution by conventional finite difference procedures is accomplished using an elliptic solver which alleviates the dependence on a particular base coordinate system multiple regions are coupled together through the boundary conditions the technique has been applied to a variety of problems including a shock diffraction problem and supersonic flow over a pointed ogive

scale separating operators for variational multiscale large eddy simulation of turbulent flows a general class of scale separating operators based on combined multigrid operators is proposed and analyzed in this work the operators of this class are designed for variational multiscale large eddy simulation using a finite volume or finite element method two representatives are compared to discrete smooth filters which are widely used in the traditional large eddy simulation literature the comparison shows that they are not only theoretically different but also yield considerable differences in the respective numerical results dynamic as well as constant coefficient based subgrid scale modeling is used within the multiscale environment all of the scale separating operators are implemented in a second order accurate energy conserving finite volume method and tested for the case of a turbulent channel flow one operator shows particularly remarkable results in the framework of the variational multiscale large eddy simulation that is profiles are obtained for velocity and kinetic energy which are considerably closer to the respective profiles from a direct numerical simulation than are the profiles resulting from the application of the other operators considered in the present study furthermore this particular operator proves to be very efficient with regard to the important aspect of computational cost that is a reduction in computing time ranging from about 25 up to about 150 compared to the other operators the introduction of a substantial amount of subgrid viscosity to the small scales particularly in the buffer layer of the channel appears to be crucial for the good results achieved with this method

analysis of velocity components decoupled projection method for the incompressible navier stokes equations we study the temporal accuracy and stability of the velocity components decoupled projection method vdpm for fully discrete incompressible navier stokes equations in particular we investigate the effect of three formulations of the nonlinear convection term which include the advective skew symmetric and divergence forms on the temporal accuracy and stability second order temporal accuracy of the vdpm for both velocity and pressure is verified by establishing global error estimates in terms of a discrete l 2 norm considering the energy evolution we demonstrate that the vdpm is stable when the time step is less than or equal to a constant stability diagrams which display the distributions of the maximum magnitude of the eigenvalues of the corresponding amplification matrices are obtained using von neumann analysis these diagrams indicate that the advective form is more stable than the other formulations of the nonlinear convection term numerical tests are performed in order to support the mathematical findings involving temporal accuracy and stability and the effects of the formulations of the nonlinear convection term are analyzed overall our results indicate that the vdpm along with an advective discrete convection operator is almost unconditionally stable second order accurate in time and computationally efficient because of the non iterative solution procedure in solving the decoupled momentum equations

control of flow induced noise behind a circular cylinder using splitter plates laminar vortex sheddings behind a circular cylinder with and without splitter plates attached to the cylinder base at the reynolds numbers of 100 and 160 are simulated by solving the unsteady two dimensional incompressible navier stokes equations the strouhal number lift and drag rapidly change with the length of the splitter plate acoustic source functions are obtained from the computed near field velocity and pressure using the curle s solution of the lighthill acoustic analogy in the case of no splitter plate the volume quadrupole noise is small at a low mach number compared with the surface dipole noise from the cylinder when a splitter plate is attached to the cylinder there are significant modifications of the dipole and quadrupole sources changes in the noise sources at re 100 are very different from those at re 160 and their differences are closely related to the secondary vortex generated at the tip of the splitter plate scattering effects at the edge of the splitter plate are also considered

analysis and prediction of thin airfoil stall phenomena with hybrid turbulence methodology analysis and prediction of thin airfoil stall phenomena of a naca 64a006 airfoil are numerically investigated using large eddy simulation les reynolds averaged navier stokes rans hybrid methodology with a high order compact differencing scheme subsonic flow of m 0 17 with the high reynolds number of re 5 8 x 10 6 is considered and the angle of attack is varied from 4 0 to 11 0 deg

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

finite difference method for incompressible navier stokes equations in arbitrary orthogonal curvilinear coordinates a finite difference method for solving three dimensional time dependent incompressible navier stokes equations in arbitrary curvilinear orthogonal coordinates is presented the method is oriented on turbulent flow simulations and consists of a second order central difference approximation in space and a third order semi implicit runge kutta scheme for time advancement spatial discretization retains some important properties of the navier stokes equations including energy conservation by the nonlinear and pressure gradient terms numerical tests cover cartesian cylindrical polar spherical cylindrical elliptic and cylindrical bipolar coordinate systems both laminar and turbulent flows are considered demonstrating reasonable accuracy and stability of the method

hermitian approximation of the spherical divergence on the cubed sphere previous work croisille 2013 showed that the cubed sphere grid offers a suitable discrete framework for extending hermitian compact operators collatz 1960 to the spherical setup in this paper we further investigate the design of high order accurate approximations of spherical differential operators on the cubed sphere with an emphasis on the spherical divergence of a tangent vector field the basic principle of this approximation relies on evaluating pointwise hermitian derivatives along a series of great circles covering the sphere several test cases demonstrate the very good accuracy of the approximate spherical divergence calculated with the new scheme

interpolating matrix method a finite difference method for arbitrary arrangeme t of mesh points abstract the interpolating matrix method imm is proposed as a finite difference method applicable to the arbitrary arrangement of mesh points in the imm differential coefficients at a mesh point are expressed with linear combinations of a certain number of neighboring mesh points a difference equation is easily constructed by substituting the linear combinations individually into the differential coefficients of the differential equation a computer code for two dimensional incompressible turbulent flows has been developed and some calculation examples are presented to demonstrate the utility of the imm

a 6th order staggered compact finite difference method for the incompressible navier stokes and scalar transport equations in a previous paper we have developed a staggered compact finite difference method for the compressible navier stokes equations in this paper we will extend this method to the case of incompressible navier stokes equations in an incompressible flow conservation of mass is ensured by the well known pressure correction method 7 21 the advection and diffusion terms are discretized with 6th order spatial accuracy the discrete poisson equation which has to be solved in the pressure correction step has the same spatial accuracy as the advection and diffusion operators the equations are integrated in time with a third order adams bashforth method results are presented for a 1d advection diffusion equation a 2d lid driven cavity at a reynolds number of 1000 and 10 000 and finally a 3d fully developed turbulent duct flow at a bulk reynolds number of 5400 in all cases the methods show excellent agreement with analytical and other numerical and experimental work

analysis of an exact fractional step method an exact fractional step or projection method for solving the incompressible navier stokes equations is analyzed the method is applied to both structured and unstructured staggered mesh schemes there are no splitting errors associated with the method it satisfies the incompressibility condition to machine precision and reduces the number of unknowns the exact projection technique is demonstrated on a two dimensional cavity flow and a multiply connected moving domain with a free surface its performance is compared directly to classic fractional step methods and shown to be roughly twice as efficient boundary conditions and the relationship of the method to streamfunction vorticity methods are discussed

a coarse grid projection method for accelerating incompressible flow computations we present a coarse grid projection cgp method for accelerating incompressible flow computations which is applicable to methods involving poisson equations as incompressibility constraints the cgp methodology is a modular approach that facilitates data transfer with simple interpolations and uses black box solvers for the poisson and advection diffusion equations in the flow solver after solving the poisson equation on a coarsened grid an interpolation scheme is used to obtain the fine data for subsequent time stepping on the full grid a particular version of the method is applied here to the vorticity stream function primitive variable and vorticity velocity formulations of incompressible navier stokes equations we compute several benchmark flow problems on two dimensional cartesian and non cartesian grids as well as a three dimensional flow problem the method is found to accelerate these computations while retaining a level of accuracy close to that of the fine resolution field which is significantly better than the accuracy obtained for a similar computation performed solely using a coarse grid a linear acceleration rate is obtained for all the cases we consider due to the linear cost elliptic poisson solver used with reduction factors in computational time between 2 and 42 the computational savings are larger when a suboptimal poisson solver is used we also find that the computational savings increase with increasing distortion ratio on non cartesian grids making the cgp method a useful tool for accelerating generalized curvilinear incompressible flow solvers

assessment of variational multiscale models for the large eddy simulation of turbulent incompressible flows in this work we study the performance of some variational multiscale models vms in the large eddy simulation les of turbulent flows we consider vms models obtained by different subgrid scale approximations which include either static or dynamic subscales linear or nonlinear multiscale splitting and different choices of the subscale space after a brief review of these models we discuss some implementation aspects particularly relevant to the simulation of turbulent flows namely the use of a skew symmetric form of the convective term and the computation of projections when orthogonal subscales are used we analyze the energy conservation and numerical dissipation of the alternative vms formulations which is numerically evaluated in the numerical study we have considered three well known problems the decay of homogeneous isotropic turbulence the taylor green vortex problem and the turbulent flow in a channel we compare the results obtained using different vms models paying special attention to the effect of using orthogonal subscale spaces the vms results are also compared against classical les scheme based on filtering and the dynamic smagorinsky closure altogether our results show the tremendous potential of vms for the numerical simulation of turbulence further we study the sensitivity of vms to the algorithmic constants and analyze the behavior in the small time step limit we have also carried out a computational cost comparison of the different formulations out of these experiments we can state that the numerical results obtained with the different vms formulations as far as they converge are quite similar however some choices are prone to instabilities and the results obtained in terms of computational cost are certainly different the dynamic orthogonal subscales model turns out to be best in terms of efficiency and robustness

the accuracy of the fractional step method we analyze the accuracy of the fractional step method of kim and moin j comput phys 59 1985 pp 308 323 for the incompressible navier stokes equations we show that the boundary conditions cannot be exactly satisfied in the projection step and that this limits the accuracy of the method we also show that the pressure in any projection method can be at best first order accurate our analysis is simpler and more direct than the previous analyses of this method we also show that there is no numerical boundary layer for velocity or pressure but there is one for the auxiliary pressure variable

energy conserving runge kutta methods for the incompressible navier stokes equations energy conserving methods have recently gained popularity for the spatial discretization of the incompressible navier stokes equations in this paper implicit runge kutta methods are investigated which keep this property when integrating in time firstly a number of energy conserving runge kutta methods based on gauss radau and lobatto quadrature are constructed these methods are suitable for convection dominated problems such as turbulent flows because they do not introduce artificial diffusion and are stable for any time step secondly to obtain robust time integration methods that work also for stiff problems the energy conserving methods are extended to a new class of additive runge kutta methods which combine energy conservation with l stability in this class the radau iia b method has the best properties results for a number of test cases on two stage methods indicate that for pure convection problems the additive radau iia b method is competitive with the gauss methods however for stiff problems such as convection dominated flows with thin boundary layers both the higher order gauss and radau iia b method suffer from order reduction overall the gauss methods are the preferred method for energy conserving time integration of the incompressible navier stokes equations

robust multigrid algorithms for the navier strokes equations abstract anisotropies occur naturally in computational fluid dynamics where the simulation of small scale physical phenomena such as boundary layers at high reynolds numbers causes the grid to be highly stretched leading to a slowdown in convergence of multigrid methods several approaches aimed at making multigrid a robust solver have been proposed and analyzed in the literature using the scalar diffusion equation however they have rarely been applied to solving more complicated models such as the incompressible navier stokes equations this paper contains the first published numerical results of the behavior of two popular robust multigrid approaches alternating plane smoothers combined with standard coarsening and plane implicit smoothers combined with semi coarsening for solving the 3 d incompressible navier stokes equations in the simulation of the driven cavity and a boundary layer over a flat plate on a stretched grid grid size grid stretching and reynolds number are the factors considered in evaluating the robustness of the multigrid methods both approaches yield large increases in convergence rates over cell implicit smoothers on stretched grids the combination of plane implicit smoothers and semi coarsening was found to be fully robust in the flat plate simulation up to reynolds numbers 10 6 and the best alternative in the driven cavity simulation for reynolds numbers above 10 3 the alternating plane approach exhibits a better behavior for lower reynolds numbers below 10 3 in the driven cavity simulation

high order accurate solution of the incompressible navier stokes equations high order methods are of great interest in the study of turbulent flows in complex geometries by means of direct simulation with this goal in mind the incompressible navier stokes equations are discretized in space by a compact fourth order finite difference method on a staggered grid the equations are integrated in time by a second order semi implicit method stable boundary conditions are implemented and the grid is allowed to be curvilinear in two space dimensions the method is extended to three dimensions by a fourier expansion in every time step a system of linear equations is solved for the velocity and the pressure by an outer and an inner iteration with preconditioning the convergence properties of the iterative method are analyzed the order of accuracy of the method is demonstrated in numerical experiments the method is used to compute the flow in a channel the driven cavity and a constricted channel

a shift transformation for fully conservative methods turbulence simulation on complex unstructured grids abstract operator transformations are presented that allow matrix operators for collocated variables to be transformed into matrix operators for staggered variables while preserving symmetries these shift transformations permit conservative skew symmetric convective operators and symmetric positive definite diffusive operators to be obtained for staggered variables using collocated operators shift transformations are not limited to uniform or structured meshes and this formulation leads to a generalization of the works of perot j comput phys 159 2000 58 and verstappen and veldman j comput phys 187 2003 343 a set of shift operators have been developed for and applied to a time adaptive cartesian mesh method with a fractional step algorithm the resulting numerical scheme conserves mass to machine error and conserves momentum and energy to second order in time a mass conserving interpolation is used for the variables during mesh adaptation the interpolation conserves momentum and energy to second order in space turbulent channel flow simulations were conducted at re 180 using direct numerical simulation dns the dns results from the adaptive method compare favourably with spectral dns results despite the use of a formally second order accurate scheme

boundary conditions for a divergence free velocity pressure formulation of the navier stokes equations new sets of boundary conditions for the velocity pressure formulation of the incompressible navier stokes equations are derived the boundary conditions have the same form on both inflow and outflow boundaries and lead to a divergence free solution moreover the specific form of the boundary condition makes it possible derive a symmetric positive definite equation system for the internal pressure numerical experiments support the theoretical conclusions

stability of pressure boundary conditions for stokes and navier stokes equations abstract the stability of a finite difference discretization of the time dependent incompressible navier stokes equations in velocity pressure formulation is studied in paticular we compare the stability for different pressure boundary conditions in a semiimplicit time integration scheme where only the viscous term is treated implicitly the stability is studied in three different ways by a normal mode analysis by numerical computation of the amplification factors and by direct numerical simulation of the governing equations all three approaches identify the same pressure boundary condition as the best alternative this condition implicitly enforces the normal derivative of the divergence to be zero on the boundary by coupling the normal derivative of the pressure to the normal component of the curl of the vorticity using this boundary condition we demonstrate that the time step is determined only by the convective term

approximate factorization as a high order splitting for the implicit incompressible flow equations abstract we apply the method of approximate factorization to derive a second order accurate splitting of the incompressible flow equations stokes or navier stokes equations this is novel because the method of approximate factorization was believed inapplicable to this type of equation system we demonstrate the resulting splitting on a second order crank nicolson discretization and point out its intimate relationship to some existing second order accurate splitting projection methods further we use the approximate factorization method to derive entirely new splittings we first develop a new generalized second order accurate splitting which may be specialized to a variety of applications we indicate its application to finite elements checker board free cell centered discretizations and ocean modeling we then generalized the original splitting to an arbitrarily high order scheme

fully conservative higher order finite difference schemes for incompressible flow conservation properties of the mass momentum and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out in particular it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass momentum and kinetic energy this deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems finite difference schemes in a collocated grid system are also analyzed and a violation of kinetic energy conservation is revealed the predicted conservation properties are demonstrated numerically in simulations of inviscid white noise performed in a two dimensional periodic domain the proposed fourth order schemes in a staggered grid system are generalized for the case of a non uniform mesh and the resulting scheme is used to perform large eddy simulations of turbulent channel flow

a first passage problem for a bivariate diffusion process numerical solution with an application to neuroscience when the process is gauss markov we consider a bivariate gauss markov process and we study the first passage time of one component through a constant boundary we prove that its probability density function is the unique solution of a new integral equation and we propose a numerical algorithm for its solution convergence properties of this algorithm are discussed and the method is applied to the study of the integrated brownian motion and to the integrated ornstein uhlenbeck process finally a model of neuroscience interest is discussed

segregated runge kutta time integration of convection stabilized mixed finite element schemes for wall unresolved les of incompressible flows in this work we develop a high performance numerical framework for the large eddy simulation les of incompressible flows the spatial discretization of the nonlinear system is carried out using mixed finite element fe schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint these additional terms introduced at the discrete level have been proved to act as implicit les models in order to perform meaningful wall unresolved simulations we consider a weak imposition of the boundary conditions using a nitsche s type scheme where the tangential component penalty term is designed to act as a wall law next segregated runge kutta srk schemes recently proposed by the authors for laminar flow problems are applied to the les simulation of turbulent flows by the introduction of a penalty term on the trace of the acceleration these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms srk schemes are excellent for large scale simulations since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level leading to two uncoupled systems the pressure system is a darcy type problem that can easily be preconditioned using a traditional block preconditioning scheme that only requires a poisson solver at the end only coercive systems have to be solved which can be effectively preconditioned by multilevel domain decomposition schemes which are both optimal and scalable the framework is applied to the taylor green and turbulent channel flow benchmarks in order to prove the accuracy of the convection stabilized mixed fes as les models and srk time integrators the scalability of the preconditioning techniques in space only has also been proven for one step of the srk scheme for the taylor green flow using uniform meshes moreover a turbulent flow around a naca profile is solved to show the applicability of the proposed algorithms for a realistic problem

high order accurate solution of the incompressible navier stokes equations on massively parallel computers the emergence of petascale supercomputers requires us to develop today s simulation codes for incompressible flows by codes which are using numerical schemes and methods that are better able to exploit the offered computational power in that spirit we present a massively parallel high order navier stokes solver for large incompressible flow problems in three dimensions the governing equations are discretized with finite differences in space and a semi implicit time integration scheme this discretization leads to a large linear system of equations which is solved with a cascade of iterative solvers the iterative solver for the pressure uses a highly efficient commutation based preconditioner which is robust with respect to grid stretching the efficiency of the implementation is further enhanced by carefully setting the adaptive termination criteria for the different iterative solvers the computational work is distributed to different processing units by a geometric data decomposition in all three dimensions this decomposition scheme ensures a low communication overhead and excellent scaling capabilities the discretization is thoroughly validated first we verify the convergence orders of the spatial and temporal discretizations for a forced channel flow second we analyze the iterative solution technique by investigating the absolute accuracy of the implementation with respect to the different termination criteria third orr sommerfeld and squire eigenmodes for plane poiseuille flow are simulated and compared to analytical results fourth the practical applicability of the implementation is tested for transitional and turbulent channel flow the results are compared to solutions from a pseudospectral solver subsequently the performance of the commutation based preconditioner for the pressure iteration is demonstrated finally the excellent parallel scalability of the proposed method is demonstrated with a weak and a strong scaling test on up to o 10 4 processing units and o 10 1 1 grid points

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

lessons from lesfoil project on large eddy simulation of flow around an airfoil a synopsis of important results obtained by seven partners in the brite euram project lesfoil sponsored by the european union is presented this project was devoted to assessing the feasibility of large eddy simulations les forthecomputationofthee owaroundanairfoil asatestcase theaerospatialea airfoilat re 2 10 6 and 13 3 deg angle of attack was chosen all partners performed simulations of this e ow with various methods most of which employed e nite volume schemes and various subgrid scale models of eddy viscosity type the key results of theindividual partners publicationsarepresented ina comparativeway itis demonstratedthatthiscone guration with realistic conditions for aeronautical e ows is extremely demanding for les one reason is the need to achieve a threshold resolution for the adequate discretization of the attached and mildly separating boundary layer the second and even more demanding requirement is the need for an adequate treatment of transition which in the present computations was achieved by increased resolution

tvd interpolating matrix method prediction of flow with heat transfer the use of the interpolating matrix method to investigate the dynamical behaviour of natural convection flow will form the main focus of the work we design an original and efficient numerical algorithm that mimic tvd constraints to rule out spurious overshoots and oscillations that appear near steep gradients to solve fluid flow and heat transfer equations and compare the results to some benchmark numerical solutions

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

time dependent subgrid scales in residual based large eddy simulation of turbulent channel flow the present study investigates the effect of taking into account the time dependency of the fine subgrid scales in a residual based variational multiscale approach for large eddy simulation the residual based variational multiscale method with time dependent dynamic subgrid scales is presented and the impact of the time dependency is studied for the well known test case of turbulent channel flow results are presented from computations for various values of the reynolds number namely ret 180 ret 395 and ret 590 and several time step sizes a generalized a time integration scheme is employed results from our numerical experiments with dynamic subgrid scales are compared to results obtained with an approximation not explicitly taking the time dependency of the subgrid scales into account for all re t values and time step sizes under consideration results for resolved quantities computed by both approaches are very similar this statement applies to both mean streamwise velocity and root mean square velocity fluctuations however it provides a model for the subgrid scales not depend ing on the time step size and enabling a more robust representation of unresolved scales thus we conclude that the time dependent subgrid scale approximation is not capable of producing more accurate results for this type of flow if the time step size is chosen within an optimal range however we expect it to be more advantageous for more complex problems since our results indicate that it provides a more reliable representation of unresolved scales

an improved bounded semi lagrangian scheme for the turbulent transport of passive scalars an improved bounded semi lagrangian scalar transport scheme based on cubic hermite polynomial reconstruction is proposed in this paper boundedness of the scalar being transported is ensured by applying derivative limiting techniques single sub cell extrema are allowed to exist as they are often physical and help minimize numerical dissipation this treatment is distinct from enforcing strict monotonicity as done by d l williamson and p j rasch 5 and allows better preservation of small scale structures in turbulent simulations the proposed bounding algorithm although a seemingly subtle difference from strict monotonicity enforcement is shown to result in significant performance gain in laminar cases and in three dimensional turbulent mixing layers the scheme satisfies several important properties including boundedness low numerical diffusion and high accuracy performance gain in the turbulent case is assessed by comparing scalar energy and dissipation spectra produced by several bounded and unbounded schemes the results indicate that the proposed scheme is capable of furnishing extremely accurate results with less severe resolution requirements than all the other bounded schemes tested additional simulations in homogeneous isotropic turbulence with scalar timestep size unconstrained by the cfl number show good agreement with spectral scheme results available in the literature detailed analytical examination of gain and phase error characteristics of the original cubic hermite polynomial is also included and points to dissipation and dispersion characteristics comparable to or better than those of a fifth order upwind eulerian scheme

a current density conservative scheme for incompressible mhd flows at a low magnetic reynolds number part i on a rectangular collocated grid system a consistent conservative and accurate scheme has been designed to calculate the current density and the lorentz force by solving the electrical potential equation for magnetohydrodynamics mhd at low magnetic reynolds numbers and high hartmann numbers on a finite volume structured collocated grid in this collocated grid velocity u pressure p and electrical potential f are located in the grid center while current fluxes are located on the cell faces the calculation of current fluxes on the cell faces is conducted using a conservative scheme which is consistent with the discretization scheme for the solution of electrical potential poisson equation a conservative interpolation is used to get the current density at the cell center which is used to conduct the calculation of lorentz force at the cell center for momentum equations we will show that both conservative and consistent are important properties of the scheme to get an accurate result for high hartmann number mhd flows with a strongly non uniform mesh employed to resolve the hartmann layers and side layers of hunt s conductive walls and shercliff s insulated walls a general second order projection method has been developed for the incompressible navier stokes equations with the lorentz force included this projection method can accurately balance the pressure term and the lorentz force for a fully developed core flow this method can also simplify the pressure boundary conditions for mhd flows

an analysis of numerical errors in large eddy simulations of turbulence the reliability of numerical simulations of turbulence depend on our ability to quantify and control discretization errors in the classical literature on error analysis typically simple linear equations are studied estimates of errors derived from such analyses depend on the assumption that each dependent variable can be characterized by a unique amplitude and scale of spatial variation that can be normalized to unity this assumption is not valid for strongly nonlinear problems such as turbulence where nonlinear interactions rapidly redistribute energy resulting in the appearance of a broad continuous spectrum of amplitudes in such situations the numerical error as well as the subgrid model can change with grid spacing in a complicated manner that cannot be inferred from the results of classical error analysis in this paper a formalism for analyzing errors in such nonlinear problems is developed in the context of finite difference approximations for the navier stokes equations when the flow is fully turbulent analytical expressions for the power spectra of these errors are derived by exploiting the joint normal approximation for turbulent velocity fields these results are applied to large eddy simulation of turbulence to obtain quantitative bounds on the magnitude of numerical errors an assessment of the significance of these errors in made by comparing their magnitudes with that of the nonlinear and subgrid terms one method of controlling the errors is suggested and its effectiveness evaluated through quantitative measures although explicit evaluations are presented only for large eddy simulation the expressions derived for the power spectra of errors are applicable to direct numerical simulation as well

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

limit at zero of the brownian first passage density let b t t 0 be a standard brownian motion started at zero let g be an upper function for b satisfying g 0 0 and let be the first passage time of b over g assume that g is c 1 on increasing locally at zero and concave locally at zero then the following identities hold for the density function f of in the sense that if the second and third limit exist so does the first one and the equalities are valid here is the standard normal density these limits can take any value in 0 the method of proof relies upon the strong markov property of b and makes use of real analysis

a high order solver for unsteady incompressible navier stokes equations using the flux reconstruction method on unstructured grids with implicit dual time stepping we report development of a high order compact flux reconstruction method for solving unsteady incompressible flow on unstructured grids with implicit dual time stepping the method falls under the class of methods now referred to as flux reconstruction correction procedure via reconstruction the governing equations employ chorin s classic artificial compressibility formulation with dual time stepping to solve unsteady flow problems an implicit non linear lower upper symmetric gauss seidel scheme with backward euler discretization is used to efficiently march the solution in pseudo time while a second order backward euler discretization is used to march in physical time we verify and validate implementation of the high order method coupled with our implicit time stepping scheme using both steady and unsteady incompressible flow problems the current implicit time stepping scheme is proven effective in satisfying the divergence free constraint on the velocity field in the artificial compressibility formulation within the context of the high order flux reconstruction method this compact high order method is very suitable for parallel computing and can easily be extended to moving and deforming grids

importance of curvature evaluation scale for predictive simulations of dynamic gas liquid interfaces abstract the effect of the scale used to compute the interfacial curvature on the prediction of dynamic gas liquid interfaces is investigated a new interface curvature calculation methodology referred to herein as the adjustable curvature evaluation scale aces is proposed aces leverages a weighted least squares regression to fit a polynomial through points computed on the volume of fluid representation of the gas liquid interface the interface curvature is evaluated from this polynomial varying the least squares weight with distance from the location where the curvature is being computed adjusts the scale the curvature is evaluated on aces is verified using canonical static test cases and compared against second and fourth order height function methods simulations of dynamic interfaces including a standing wave and oscillating droplet are performed to assess the impact of the curvature evaluation scale for predicting interface motions aces and the height function methods are combined with two different unsplit geometric volume of fluid vof schemes that define the interface on meshes with different levels of refinement we find that the results depend significantly on curvature evaluation scale particularly the aces scheme with a properly chosen weight function is accurate but fails when the scale is too small or large surprisingly the second order height function method is more accurate than the fourth order variant for the dynamic tests even though the fourth order method performs better for static interfaces comparing the curvature evaluation scale of the second and fourth order height function methods we find the second order method is closer to the optimum scale identified with aces this result suggests that the curvature scale is driving the accuracy of the dynamics this work highlights the importance of studying numerical methods with realistic dynamic test cases and that the interactions of the various discretizations is as important as the accuracy of one part of the discretization

multigrid defect correction and fourth order compact scheme for poissons equation this paper presents an analysis of a multigrid defect correction to solve a fourth order compact scheme discretization of the poissons equation we focus on the formulation which arises in the velocity pressure decoupling methods encountered in computational fluid dynamics especially the poissons equation results of the divergence gradient formulation and neumann boundary conditions are prescribed the convergence rate of a multigrid defect correction is investigated by means of an eigenvalues analysis of the iteration matrix the stability and the mesh independency are demonstrated an improvement of the convergence rate is suggested by introducing the damped jacobi and incomplete lower upper smoothers based on an eigenvalues analysis the optimal damping parameter is proposed for each smoother numerical experiments confirm the findings of this analysis for periodic domain and uniform meshes which are the working assumptions further numerical investigations allow us to extend the results of the eigenvalues analysis to neumann boundary conditions and non uniform meshes the hodgehelmholtz decomposition of a vector field is carried out to illustrate the computational efficiency especially by making comparisons with a second order discretization of the poissons equation solved with a state of art of algebraic multigrid method

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

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

a variational multiscale method for turbulent flow simulation with adaptive large scale space in turbulent flows it is only feasible to simulate large flow structures variational multiscale vms methods define these flow structures by projections onto appropriate function spaces this paper presents a finite element vms method which chooses the large scale projection space adaptively the adaption controls the influence of an eddy viscosity model and it is based on the size of the so called resolved small scales the adaptive procedure is described in detail numerical studies at a turbulent channel flow and a turbulent flow around a cylinder are presented it is shown that the method selects the large scale space in a reasonable way and that appropriately chosen parameters improve the results compared to the basic method which uses the same local large scale space in the whole domain and for all times

an analysis of the fractional step method the fractional step method for solving the incompressible navier stokes equations in primitive variables is analyzed as a block lu decomposition in this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved in addition it is shown that poor temporal accuracy first order is not due to boundary conditions but due to the method itself a generalized block lu decomposition that overcomes this difficulty is presented allowing arbitrarily high temporal order of accuracy the generalized decomposition is shown to be useful for a wide range of problems including steady problems technical issues such as stability and the appropriate pressure update scheme are also addressed numerical simulations of the steady incompressible navier stokes equations in a square domain confirm the theoretical results 18 refs 4 figs

an efficient immersed boundary treatment for complex moving object an efficient immersed boundary treatment for simulation of flexible moving body immersed in fluid is presented the level set signed distance function is used to indicate body surface a simple mapping strategy is proposed to avoid costly signed distance re initialization computations the strategy is efficient for both rigid and deformable structures and can be extended to 3 d case easily with the use of signed distance function reconstruction of flow variables on a body surface can be easily implemented through linear bilinear or quadratic interpolation in order to suppress the pressure oscillations caused by the role conversion of forcing points and fluid points a modified interpolation scheme is presented by introducing a dynamic weight term to the fluid point together with local grid refinement and using larger time step the pressure oscillations can be effectively diminished by the modified interpolation scheme the proposed method is validated by 2 d numerical simulations on a uniform flow past a fixed cylinder in line and transversely oscillating cylinders a flapping wing a 3 d simulation on a uniform flow past a rectangle plane with an initial attack angle and a moving anguilliform swimmer present numerical results are compared with other cfd solutions as well as experimental results

performance and analysis of saddle point preconditioners for the discrete steady state navier stokes equations we examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of fundamental parameters on convergence we demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers the structure of these distributions is independent of the discretization mesh size but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller these characteristics are directly correlated with the convergence properties of iterative solvers

a consistent and conservative immersed boundary method for mhd flows and moving boundary problems abstract a consistent and conservative immersed boundary method has been developed to accurately and efficiently solve two topics magnetohydrodynamics flows with a complex boundary and moving boundary problems based on a least square interpolation reconstruction method a unified form of boundary condition is described to handle a single domain situation while a mixed boundary condition is used for coupled conditions of multi domain in order to achieve the consistency with a desired wall boundary condition a so called approximation correction procedure is done and an imposing neumann boundary condition is discussed with two kinds of different schemes besides a consistent and conservative scheme is implemented to satisfy both the charge and mass conservation laws on cells around the immersed surface then a conservative interpolation is reconstructed for velocity in moving boundary problems at last the applied numerical method is validated by stationary and moving boundary cases and an excellent agreement is obtained with good accuracy efficiency and conservation especially the consistent and conservative immersed boundary method can obtain almost the same accurate results as those from the cut cell technique seo and mittal 2011 14 for a moving boundary problem by reducing the spurious pressure

a high order finite difference method with richardsonextrapolation for 3d convection diffusion equation in this paper we extend the sun and zhang s 24 work on high order finite difference method which is based on the richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case firstly we employ a fourth order compact difference scheme to get the fourth order accurate solution on the fine and the coarse grids then we use the richardson extrapolation technique by combining the two approximate solutions to get a sixth order accurate solution on coarse grid finally we apply an operator interpolation scheme to achieve the sixth order accurate solution on the fine grid during this process we use alternating direction implicit adi method to solve the resulting linear systems numerical experiments are conducted to verify the accuracy and effectiveness of the present method