a family of dynamic finite difference schemes for large eddy simulation a family of dynamic low dispersive finite difference schemes for large eddy simulation is developed the dynamic schemes are constructed by combining taylor series expansions on two different grid resolutions the schemes are optimized dynamically during the simulation according to the flow physics and dispersion errors are minimized through the real time adaption of the dynamic coefficient in case of dns resolution the dynamic schemes reduce to the standard taylor based finite difference schemes with formal asymptotic order of accuracy when going to les resolution the schemes seamlessly adapt to dispersion relation preserving schemes the schemes are tested for large eddy simulation of burgers equation and numerical errors are investigated as well as their interaction with the subgrid model very good results are obtained

symmetry preserving discretization of navier stokes equations on collocated unstructured grids a fully conservative discretization is presented in this paper the same principles followed by verstappen and veldman 2003 3 are generalized for unstructured meshes here a collocated mesh scheme is preferred over a staggered one due to its simpler form for such meshes the basic idea behind this approach remains the same mimicking the crucial symmetry properties of the underlying differential operators i e the convective operator is approximated by a skew symmetric matrix and the diffusive operator by a symmetric positive definite matrix a novel approach to eliminate the checkerboard spurious modes without introducing any non physical dissipation is proposed to do so a fully conservative regularization of the convective term is used the supraconvergence of the method is numerically showed and the treatment of boundary conditions is discussed finally the new discretization method is successfully tested for a buoyancy driven turbulent flow in a differentially heated cavity

explicit combinatorial interpretation of kerov character polynomials as numbers of permutation factorizations abstract we find an explicit combinatorial interpretation of the coefficients of kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups s n in terms of free cumulants r 2 r 3 of the corresponding young diagram our interpretation is based on counting certain factorizations of a given permutation

iterative across time solution of linear differential equations krylov subspace versus waveform relaxation the aim of this paper is two fold first we propose an efficient implementation of the continuous time waveform relaxation wr method based on block krylov subspaces second we compare this new wr krylov implementation against krylov subspace methods combined with the shift and invert sai technique some analysis and numerical experiments are presented since the wr krylov and sai krylov methods build up the solution simultaneously for the whole time interval and there is no time stepping involved both methods can be seen as iterative across time methods the key difference between these methods and standard time integration methods is that their accuracy is not directly related to the time step size

adaptive mesh finite volume calculation of 2d lid cavity corner vortices abstract an anisotropic refinement criterion suitable for finite volume methods is presented and navier stokes solutions are reported for the lid driven cavity flow configuration at re 1000 with adaptive anisotropic meshes h refinement the a posteriori error estimation criterion is based on the assessment of the goodness of fit of the least squares regression used to perform the variables profile reconstruction and it is capable of detecting both large scale and small scale flow phenomena the criterion allowed to capture in detail the large scale flow structure and also the sequence of creeping flow small sharp corner s 1 eddies up to the fourth corner vortex in addition to the primary cavity vortex the smallest corner vortex detected is o 10 16 smaller in velocity magnitude compared with the cavity primary recirculation flow

parallel direct poisson solver for discretisations with one fourier diagonalisable direction in the context of time accurate numerical simulation of incompressible flows a poisson equation needs to be solved at least once per time step to project the velocity field onto a divergence free space due to the non local nature of its solution this elliptic system is one of the most time consuming and difficult to parallelise parts of the code in this paper a parallel direct poisson solver restricted to problems with one uniform periodic direction is presented it is a combination of a direct schur complement based decomposition dsd and a fourier diagonalisation the latter decomposes the original system into a set of mutually independent 2d systems which are solved by means of the dsd algorithm since no restrictions are imposed in the non periodic directions the overall algorithm is well suited for solving problems discretised on extruded 2d unstructured meshes the load balancing between parallel processes and the parallelisation strategy are also presented and discussed the scalability of the solver is successfully tested using up to 8192 cpu cores for meshes with up to 10 9 grid points finally the performance of the dsd algorithm as 2d solver is analysed by direct comparison with two preconditioned conjugate gradient methods for this purpose the turbulent flow around a circular cylinder at reynolds numbers 3900 and 10 000 are used as problem models

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

analysis of numerical errors in large eddy simulation using statistical closure theory this paper develops a dynamic error analysis procedure for the numerical errors arising from spatial discretization in large eddy simulation the analysis is based on edqnm closure theory and is applied to the les of decaying isotropic turbulence first the effects of finite differencing truncation error aliasing error and the dynamic smagorinsky model are independently considered the time evolution of kinetic energy and spectra predicted by the analysis are compared to actual les using the navier stokes equations and good agreement is obtained the analysis is then extended to simultaneously consider all sources of error in a second order discretely energy conserving central difference les solver good agreement between the analysis and actual les is obtained the analysis is used to compare the contribution of the subgrid model to that of numerical errors and it is shown that the contribution of the subgrid scale model is much higher than the numerical errors the proposed one dimensional edqnm les model shows potential as a more general tool for the analysis of numerical error and sgs model in simulations of turbulent flow

reduced aliasing formulations of the convective terms within the navier stokes equations for a compressible fluid the effect on aliasing errors of different formulations describing the cubically nonlinear convective terms within the discretized navier stokes equations is examined in the presence of a non trivial density spectrum fourier analysis shows that the existing skew symmetric forms of the convective term result in reduced aliasing errors relative to the conservation form several formulations of the convective term including a new formulation proposed for cubically nonlinear terms are tested in direct numerical simulation dns of decaying compressible isotropic turbulence both in chemically inert small density fluctuations and reactive cases large density fluctuations and for different degrees of resolution in the dns of reactive turbulent flow the new cubic skew symmetric form gives the most accurate results consistent with the spectral error analysis and at the lowest cost in marginally resolved dns and les poorly resolved by definition the new cubic skew symmetric form represents a robust convective formulation which minimizes both aliasing and computational cost while also allowing a reduction in the use of computationally expensive high order dissipative filters

a multi scale formulation for compressible turbulent flows suitable for general variational discretization techniques based on the recently introduced variational multi scale vms approach to large eddy simulation les as introduced in t j r hughes l mazzei k e jansen large eddy simulation and the variational multiscale method comput visual sci 3 2001 47 59 s s collis monitoring unresolved scales in multiscale turbulence modeling phys fluids 13 6 2001 1800 1806 we present a vms formulation which can be used in the simulation of compressible flows special attention is given to obtain a vms formulation which is suitable for complex flow domains and general variational discretization techniques a generalization of the favre averaging procedure is introduced such that the formulation resembles the favre filtered navier stokes equations traditionally used in les of compressible flow and no explicit subgrid terms arise in the continuity equation also we show that with the use of discretization methods other than fourier spectral methods the vms projection no longer commutes with differentiation this results in additional subgrid scale terms which resemble the commutator error as encountered in the traditional filtering approach to les

a fixed mesh method for incompressible flow structure systems with finite solid deformations a fixed mesh algorithm is proposed for simulating flow structure interactions such as those occurring in biological systems in which both the fluid and solid are incompressible and the solid deformations are large several of the well known difficulties in simulating such flow structure interactions are avoided by formulating a single set of equations of motion on a fixed eulerian mesh the solid s deformation is tracked to compute elastic stresses by an overlapping lagrangian mesh in this way the flow structure interaction is formulated as a distributed body force and singular surface force acting on an otherwise purely fluid system these forces which depend on the solid elastic stress distribution are computed on the lagrangian mesh by a standard finite element method and then transferred to the fixed eulerian mesh where the joint momentum and continuity equations are solved by a finite difference method the constitutive model for the solid can be quite general for the force transfer standard immersed boundary and immersed interface methods can be used and are demonstrated we have also developed and demonstrated a new projection method that unifies the transfer of the surface and body forces in a way that exactly conserves momentum the interface is still effectively sharp for this approach the spatial convergence of the method is observed to be between first and second order as in most immersed boundary methods for membrane flows the algorithm is demonstrated by the simulations of an advected elastic disk a flexible leaflet in an oscillating flow and a model of a swimming jellyfish

finite surface discretization for incompressible navier stokes equations and coupled conservation laws abstract we present a new finite surface discretization fsd aiming at the incompressible navier stokes equations nse and other coupled conservation laws this discretization defines the velocities as surfaced averaged values living on the faces of the pressure volumes in which the mass is set to be conserved consequently the calculation of the mass balance on these control volumes is exact which allows more accurate information to be kept in the velocity field and produces a very accurate prediction of the pressure in the next time step the proposed discretization reduces the stencil size of the poisson equation in the projection method compared to the finite volume and finite difference discretizations due to highly accurate mass conservation the compact sixth order approximation of fsd can be used with an explicit fourth order pressure treatment this property greatly reduces the cost and complexity of the implementation we present the discrete evolution equation of the surface averaged velocities together with the enforcement of mass conservation and the solution procedure for the pressure the approximation of the nse under this new discretization uses a combination of finite difference and finite volume methods the proposed method is validated using standard laminar test cases we identify the conditions under which a fourth order pressure treatment can support the sixth order and eighth order approximations of the convection term using fourier analysis the performance of the method is evaluated on turbulent channel flows up to friction reynolds number of 950 the quantitative relationships between the accuracy of the solution and grid size are identified we present two performance indices for comparison with other methods at the error level of 1 per mille the proposed method is 28 times faster than the classic second order scheme

comparison of some lie symmetry based integrators lie symmetry based integrators are constructed in order to preserve the local invariance properties of the equations the geometrical methods leading to discretized equations for numerical computations involve many different concepts therefore they give rise to numerical schemes that vary in the accuracy in the computational cost and in the implementation in this paper a comparison is made between some alternative lie symmetry based methods illustrated on the example of the burgers equation the importance of the symmetry preservation is numerically highlighted

on the effect of numerical errors in large eddy simulations of turbulent flows aliased and dealiased numerical simulations of a turbulent channel flow are performed using spectral and finite difference methods analytical and numerical studies show that aliasing errors are more destructive for spectral and high order finite difference calculations than for low order finite difference simulations numerical errors have different effects for different forms of the nonlinear terms in the navier stokes equations for divergence and convective forms spectral methods are energy conserving only if dealiasing is performed for skew symmetric and rotational forms both spectral and finite difference methods are energy conserving even in the presence of aliasing errors it is shown that discrepancies between the results of dealiased spectral and standard nondialiased finite difference methods are due to both aliasing and truncation errors with the latter being the leading source of differences the relative importance of aliasing and truncation errors as compared to subgrid scale model terms in large eddy simulations is analyzed and discussed for low order finite difference simulations truncation errors can exceed the magnitude of the subgrid scale term

large eddy simulation without filter an large eddy simulation les formalism based on sampling operators instead of filters is developed the major advantage of this approach is that sampling operators commute with the product and their application to nonlinear terms is not at the origin of any closure problem in absence of filters that smooth out the small scale structures in the flow the discretization errors in the les are expected to be important they must be modelled the possible confusion between modelling and discretization errors is however avoided since these two effects are identical in the present formalism a generalized dynamic procedure is proposed for sampling based les which allows for model parameter optimization and does not require a detailed analysis of the discretization error in addition to its interesting mathematical properties for les the velocity obtained by a spatial sampling is much closer to experimental probe data than the filtered velocity field

energy preserving turbulent simulations at a reduced computational cost energy conserving discretizations are widely regarded as a fundamental requirement for high fidelity simulations of turbulent flows the skew symmetric splitting of the nonlinear term is a well known approach to obtain semi discrete conservation of energy in the inviscid limit however its computation is roughly twice as expensive as that of the divergence or advective forms alone a novel time advancement strategy that retains the conservation properties of skew symmetric based schemes at a reduced computational cost has been developed this method is based on properly constructed runge kutta schemes in which a different form advective or divergence for the convective term is adopted at each stage a general framework is presented to derive schemes with prescribed accuracy on both solution and energy conservation simulations of homogeneous isotropic turbulence show that the new procedure is effective and can be considerably faster than skew symmetric based techniques

isotropic finite volume discretization finite volume methods traditionally employ dimension by dimension extension of the one dimensional reconstruction and averaging procedures to achieve spatial discretization of the governing partial differential equations on a structured cartesian mesh in multiple dimensions this simple approach based on tensor product stencils introduces an undesirable grid orientation dependence in the computed solution the resulting anisotropic errors lead to a disparity in the calculations that is most prominent between directions parallel and diagonal to the grid lines in this work we develop isotropic finite volume discretization schemes which minimize such grid orientation effects in multidimensional calculations by eliminating the directional bias in the lowest order term in the truncation error explicit isotropic expressions that relate the cell face averaged line and surface integrals of a function and its derivatives to the given cell area and volume averages are derived in two and three dimensions respectively it is found that a family of isotropic approximations with a free parameter can be derived by combining isotropic schemes based on next nearest and next next nearest neighbors in three dimensions use of these isotropic expressions alone in a standard finite volume framework however is found to be insufficient in enforcing rotational invariance when the flux vector is nonlinear and or spatially non uniform the rotationally invariant terms which lead to a loss of isotropy in such cases are explicitly identified and recast in a differential form various forms of flux correction terms which allow for a full recovery of rotational invariance in the lowest order truncation error terms while preserving the formal order of accuracy and discrete conservation of the original finite volume method are developed numerical tests in two and three dimensions attest the superior directional attributes of the proposed isotropic finite volume method prominent anisotropic errors such as spurious asymmetric distortions on a circular reaction diffusion wave that feature in the conventional finite volume implementation are effectively suppressed through isotropic finite volume discretization furthermore for a given spatial resolution a striking improvement in the prediction of kinetic energy decay rate corresponding to a general two dimensional incompressible flow field is observed with the use of an isotropic finite volume method instead of the conventional discretization c 2014 elsevier inc all rights reserved

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

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

what does finite volume based implicit filtering really resolve in large eddy simulations the study illustrated in this paper completes the topics initially investigated in ref 42 the aim being here to analyze the role of the integral based finite volume fv discretizations in large eddy simulations that exploit the implicit filtering approach specifically a theoretical study on the effective shape and length of three dimensional filters induced by some fv based flux reconstructions is the object of this paper for any integral based flux reconstruction one gets always an approximation of the top hat filter kernel this is not the case of the filters induced by the differential based finite difference operators such as those reported and analyzed in refs 16 24 considering the sub filter resolution parameter email protected e f f h being d e f f the effective filter width and h the computational grid size allows us discerning the effective measure of the approximate built in top hat filter the induced shape and width is analyzed by means of a modified wavenumber like analysis that is developed in the 3d fourier space several evaluation criteria applied on different schemes are considered and the differences in terms of either velocity or flux interpolations on staggered or non staggered grids are analyzed conclusions are reported that depending on the using of either the integral or the differential form of the filtered equations the induced numerical filter is or is not a congruent approximation of the exact top hat transfer function for some value q the need of a suitable estimation of the sub filter parameter q is assessed from several real les computations that make use of the new integral based version of the eddy viscosity dynamic modeling presented in ref 42 in fact it is shown that the test filtering length has to be carefully chosen as a function of the fv based induced filter

max flow and min cut with bounded length paths complexity algorithms and approximation we consider the flow on paths versions of max flow and min cut when we restrict to paths having at most b arcs and for versions where we allow fractional solutions or require integral solutions we show that the continuous versions are polynomial even if b is part of the input but that the integral versions are polynomial only when b 3 however when b 3 we show how to solve the problems using ordinary max flow min cut we also give tight bounds on the integrality gaps between the integral and continuous objective values for both problems and between the continuous objective values for the bounded length paths version and the version allowing all paths we give a primal dual approximation algorithm for both problems whose approximation ratio attains the integrality gap thereby showing that it is the best possible primal dual approximation algorithm

construction of explicit and implicit dynamic finite difference schemes and application to the large eddy simulation of the taylor green vortex a general class of explicit and implicit dynamic finite difference schemes for large eddy simulation is constructed by combining taylor series expansions on two different grid resolutions after calibration for re the dynamic finite difference schemes allow to minimize the dispersion errors during the calculation through the real time adaption of a dynamic coefficient in case of dns resolution these dynamic schemes reduce to taylor based finite difference schemes with formal asymptotic order of accuracy whereas for les resolution the schemes adapt to dispersion relation preserving schemes both the explicit and implicit dynamic finite difference schemes are tested for the large eddy simulation of the taylor green vortex flow and numerical errors are investigated as well as their interaction with the dynamic smagorinsky model and the multiscale smagorinsky model very good results are obtained

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

an energy stable scheme for incompressible navier stokes equations with periodically updated coefficient matrix abstract we present an energy stable scheme for simulating the incompressible navier stokes equations based on the generalized positive auxiliary variable gpav framework in the gpav reformulated system the original nonlinear term is replaced by a linear term plus a correction term where the correction term is put under control by an auxiliary variable the proposed scheme incorporates a pressure correction type strategy into the gpav procedure and it satisfies a discrete energy stability property the scheme entails the computation of two copies of the velocity and pressure within a time step by solving an individual de coupled linear equation for each of these field variables upon discretization the pressure linear system involves a constant coefficient matrix that can be pre computed while the velocity linear system involves a coefficient matrix that is updated periodically once every k 0 time steps in the current work where k 0 is a user specified integer the auxiliary variable being a scalar valued number is computed by a well defined explicit formula which guarantees the positivity of its computed values it is observed that the current method can produce accurate simulation results at large or fairly large time step sizes for the incompressible navier stokes equations the impact of the periodic coefficient matrix update on the overall cost of the method is observed to be small in typical numerical simulations several flow problems have been simulated to demonstrate the accuracy and performance of the method developed herein

a dynamically optimized finite difference scheme for large eddy simulation a low dispersive dynamic finite difference scheme for large eddy simulation is developed the dynamic scheme is constructed by combining taylor series expansions on two different grid resolutions the scheme is optimized dynamically through the real time adaption of a dynamic coefficient according to the spectral content of the flow such that the global dispersion error is minimal in the case of dns resolution the dynamic scheme reduces to the standard taylor based finite difference scheme with formal asymptotic order of accuracy when going to les resolution the dynamic scheme seamlessly adapts to a dispersion relation preserving scheme the scheme is tested for large eddy simulation of burgers equation very good results are obtained

explicit runge kutta schemes for incompressible flow with improved energy conservation properties the application of pseudo symplectic runge kutta methods to the incompressible navier stokes equations is discussed in this work in contrast to fully energy conserving implicit methods these are explicit schemes of order p that preserve kinetic energy to order q with q p use of explicit methods with improved energy conservation properties is appealing for convection dominated problems especially in case of direct and large eddy simulation of turbulent flows a number of pseudo symplectic methods are constructed for application to the incompressible navier stokes equations and compared in terms of accuracy and efficiency by means of numerical simulations

a study of differentiation errors in large eddy simulations based on the edqnm theory this paper is concerned with the investigation of numerical errors in large eddy simulations by means of two point turbulence modeling based on the eddy damped quasi normal markovian edqnm theory a stochastic model is developed in order to predict the time evolution of the kinetic energy spectrum obtained by a large eddy simulation les including the effects of the numerics using this framework the influence of the accuracy of the approximate space differencing schemes on les quality is studied for decaying homogeneous isotropic incompressible turbulence with reynolds numbers rek based on the transverse taylor scale equal to 780 2500 and 8000 the results show that the discretization of the filtered navier stokes equations leads to differentiation and aliasing errors error spectra are also presented and indicate that the numerical errors are mainly originating from the approximate differentiation in addition increasing the order of accuracy of the differencing schemes or using algorithms optimized in the fourier space is found to widen the range of well resolved scales unfortunately for all the schemes the smaller scales with wavenumbers close to the grid cut off wavenumber are badly calculated and generate differentiation errors over the whole energy spectrum the eventual use of explicit filtering to remove spurious motions with short wavelength is finally shown to significantly improve les accuracy

sequential quadratic programming sqp for optimal control in direct numerical simulation of turbulent flow the current work focuses on the development and application of an efficient algorithm for optimization of three dimensional turbulent flows simulated using direct numerical simulation dns or large eddy simulations and further characterized by large dimensional optimization parameter spaces the optimization algorithm is based on sequential quadratic programming sqp in combination with a damped formulation of the limited memory bfgs method the latter is suitable for solving large scale constrained optimization problems whose hessian matrices cannot be computed and stored at a reasonable cost we combine the algorithm with a line search merit function based on an l 1 norm to enforce the convergence from any remote point it is first shown that the proposed form of the damped l bfgs algorithm is suitable for solving equality constrained rosenbrock type functions then we apply the algorithm to an optimal control test problem that consists of finding the optimal initial perturbations to a turbulent temporal mixing layer such that mixing is improved at the end of a simulation time horizon t the controls are further subject to a non linear equality constraint on the total control energy dnss are used to resolve all turbulent scales of motion and a continuous adjoint formulation is employed to calculate the gradient of the cost functionals we compare the convergence speed of the sqp l bfgs algorithm to a conventional non linear conjugate gradient method i e the current standard in dns based optimal control and find that the sqp algorithm is more than an order of magnitude faster than the conjugate gradient method

the ls stag method a new immersed boundary level set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties this paper concerns the development of a new cartesian grid immersed boundary ib method for the computation of incompressible viscous flows in two dimensional irregular geometries in ib methods the computational grid is not aligned with the irregular boundary and of upmost importance for accuracy and stability is the discretization in cells which are cut by the boundary the so called cut cells in this paper we present a new ib method called the ls stag method which is based on the mac method for staggered cartesian grids and where the irregular boundary is sharply represented by its level set function this implicit representation of the immersed boundary enables us to calculate efficiently the geometry parameters of the cut cells we have achieved a novel discretization of the fluxes in the cut cells by enforcing the strict conservation of total mass momentum and kinetic energy at the discrete level our discretization in the cut cells is consistent with the mac discretization used in cartesian fluid cells and has the ability to preserve the five point cartesian structure of the stencil resulting in a highly computationally efficient method the accuracy and robustness of our method is assessed on canonical flows at low to moderate reynolds number taylor couette flow flows past a circular cylinder including the case where the cylinder has forced oscillatory rotations finally we will extend the ls stag method to the handling of moving immersed boundaries and present some results for the transversely oscillating cylinder flow in a free stream

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

characters of symmetric groups sharp bounds and applications we provide new estimates on character values of symmetric groups which hold for all characters and which are in some sense best possible it follows from our general bound that if a permutation s n has at most n o 1 cycles of length m then 1 1 m o 1 for all irreducible characters of s n this is a far reaching generalization of a result of fomin and lulov

conservative high order finite difference schemes for low mach number flows three finite difference algorithms are proposed to solve a low mach number approximation for the navier stokes equations these algorithms exhibit fourth order spatial and second order temporal accuracy they are dissipation free and thus well suited for dns and les of turbulent flows the key ingredient common to each of the methods presented is a poisson equation with variable coefficient that is solved for the hydrodynamic pressure this feature ensures that the velocity field is constrained correctly it is shown that this approach is needed to avoid violation of the conservation of kinetic energy in the inviscid limit which would otherwise arise through the pressure term in the momentum equation an existing set of finite difference formulae for incompressible flow is generalized to handle arbitrary large density fluctuations with no violation of conservation through the non linear convective terms an algorithm which conserves mass momentum and kinetic energy fully is obtained when an approximate equation of state is used instead of the exact one results from a model problem are used to show both spatial and temporal convergence rates and several test cases are presented to illustrate the performance of the algorithms

edge degeneracy algorithmic and structural results abstract we consider a cops and robber game where the cops are blocking edges of a graph while the robber occupies its vertices at each round of the game the cops choose some set of edges to block and right after the robber is obliged to move to another vertex traversing at most s unblocked edges s can be seen as the speed of the robber both parts have complete knowledge of the opponent s moves and the cops win when they occupy all edges incident to the robbers position we introduce the capture cost on g against a robber of speed s this defines a hierarchy of invariants namely e 1 e 2 e where e is an edge analogue of the admissibility graph invariant namely the edge admissibility of a graph we prove that the problem asking whether e s g k is polynomially solvable when s 1 2 3 while otherwise it is np complete our main result is a structural theorem for graphs of bounded edge admissibility we prove that every graph of edge admissibility at most k can be constructed using k edge sums starting from graphs whose all vertices except possibly from one have degree at most k our structural result is approximately tight in the sense that graphs generated by this construction always have edge admissibility at most 2 k 1 our proofs are based on a precise structural characterization of the graphs that do not contain r as an immersion where r is the graph on two vertices and r parallel edges

upper bound on the characters of the symmetric groups let c be a conjugacy class in the symmetric group s n and be a partition of n let f be the degree of the irreducible representation s c the character of s at c and r c the normalized character c f we prove that there exist constants b 0 and 1 q 0 such that for n 4 for every conjugacy class c in s n and every irreducible representation s of s n r c max q 1 n 1 n b supp c where supp c is the number of non fixed digits under the action of a permutation in c 1 is the size of the largest part in and 1 is the number of parts in the proof is obtained by enumeration of rim hook tableaux the hook formula and probabilistic arguments combinatorial algebraic and statistical applications follow this result in particular we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes

positivity results for stanley s character polynomials in stanley r p stanley irreducible symmetric group characters of rectangular shape sem lothar combin 50 b50d 2003 11 pp the author introduces expressions for the normalized characters of the symmetric group and states some positivity conjectures for these expressions here we give an affirmative partial answer to stanley s positivity conjectures about the expressions using results on kerov polynomials in particular we use new positivity results in goulden and rattan i p goulden a rattan an explicit form for kerov s character polynomials trans amer math soc in press math co 0505317 november 2005 we shall see that the generating series c t introduced in i p goulden a rattan an explicit form for kerov s character polynomials trans amer math soc in press math co 0505317 november 2005 is critical to our discussion

statistical behavior of the orthogonal subgrid scale stabilization terms in the finite element large eddy simulation of turbulent flows abstract numerical simulations have proved that variational multiscale methods vmm perform well as pure numerical large eddy simulation les models in this paper we focus on the orthogonal subgrid scale oss finite element method and make an analysis of the statistical behavior of its stabilization terms in the quasi static approximation this is done by resorting to results from classical statistical fluid mechanics concerning two point velocity pressure and combined correlation functions of various orders given a fine enough mesh with characteristic element size h in the inertial subrange of a turbulent flow it is shown that the rate of transfer of subgrid kinetic energy provided by the oss stabilization terms does not depend on h and that it equals the molecular physical dissipation rate up to a dimensionless constant that only depends on the finite element shapes for a proper redesign of the standard parameters of the formulation this is a noteworthy fact taking into account that the subgrid stabilization terms do not arise from physical considerations but from the mathematical necessity to allow equal interpolation for the pressure and velocity fields as well as to control convection therefore the obtained results contribute somehow to the line of reasoning supporting that pure numerical approaches i e without introducing additional physical models could probably suffice in the les simulation of turbulent flows

gaussian fluctuations of characters of symmetric groups and of young diagrams we study asymptotics of reducible representations of the symmetric groups s q for large q we decompose such a representation as a sum of irreducible components or alternatively young diagrams and we ask what is the character of a randomly chosen component or what is the shape of a randomly chosen young diagram our main result is that for a large class of representations the fluctuations of characters and fluctuations of the shape of the young diagrams are asymptotically gaussian in this way we generalize kerov s central limit theorem the considered class consists of representations for which the characters almost factorize and this class includes for example the left regular representation plancherel measure irreducible representations and tensor representations this class is also closed under induction restriction outer product and tensor product of representations our main tool in the proof is the method of genus expansion well known from the random matrix theory

artificial neural network approach to large eddy simulation of compressible isotropic turbulence a subgrid scale sgs model for large eddy simulation les of compressible isotropic turbulence is constructed by using a data driven framework an artificial neural network ann based on local stencil geometry is employed to predict the unclosed sgs terms the input features are based on the first order and second order derivatives of filtered velocity and temperature which appear in the second order taylor approximation of the sgs stress and heat flux it is shown that the proposed ann 7 model performs better than the gradient model in the a priori test the correlation coefficient is larger and the relative error is smaller for ann 7 model as compared to those of the gradient model in the a priori test in an a posteriori analysis the performance of ann 7 model shows advantage over the dynamic smagorinsky model and dynamic mixed model in the prediction of spectra and structure functions of velocity and temperature and instantaneous flow structures artificial neural network is a promising tool for understanding the physical fundamentals of sgs unclosed terms with further improvement

the sensitivity and accuracy of fourth order finite difference schemes on nonuniform grids in one dimension abstract we construct local fourth order finite difference approximations of first and second derivatives on nonuniform grids in one dimension the approximations are required to satisfy symmetry relationships that come from the analogous higher dimensional fundamental operators the divergence the gradient and the laplacian for example we require that the discrete divergence and gradient be negative adjoint of each other div grad and the discrete laplacian is defined as lap divgrad the adjointness requirement on the divergence and gradient guarantees that the laplacian is a symmetric negative operator the discrete approximations we derive are fourth order on smooth grids but the approach can be extended to create approximations of arbitrarily high order we analyze the loss of accuracy in the approximations when the grid is not smooth and include a numerical example demonstrating the effectiveness of the higher order methods on nonuniform grids

symmetrized compact scheme for receptivity study of 2d transitional channel flow in many physical problems one must maintain symmetry of numerical schemes the symmetry arising due to geometry and or physical conditions defining the flow the direct simulation of transitional plane poiseuille flow presents one such case that motivated the present study it has been shown here using the method of analyzing non periodic problems in sengupta et al 2003 that compact and other high accuracy schemes display strong directionality of the algorithm and such methods cannot be used for direct simulation of the physical flow this prevalent but unacceptable situation for dns is rectified in the present work and a new scheme is introduced that prevents this asymmetry from contaminating results the simulated transitional flow past 2d channel here using the new scheme are in agreement with experimental results and other recent views of sub critical instabilities

numerical dissipation vs subgrid scale modelling for large eddy simulation this study presents an alternative way to perform large eddy simulation based on a targeted numerical dissipation introduced by the discretization of the viscous term it is shown that this regularisation technique is equivalent to the use of spectral vanishing viscosity the flexibility of the method ensures high order accuracy while controlling the level and spectral features of this purely numerical viscosity a pao like spectral closure based on physical arguments is used to scale this numerical viscosity a priori it is shown that this way of approaching large eddy simulation is more efficient and accurate than the use of the very popular smagorinsky model in standard as well as in dynamic version the main strength of being able to correctly calibrate numerical dissipation is the possibility to regularise the solution at the mesh scale thanks to this property it is shown that the solution can be seen as numerically converged conversely the two versions of the smagorinsky model are found unable to ensure regularisation while showing a strong sensitivity to numerical errors the originality of the present approach is that it can be viewed as implicit large eddy simulation in the sense that the numerical error is the source of artificial dissipation but also as explicit subgrid scale modelling because of the equivalence with spectral viscosity prescribed on a physical basis

representations of symmetric groups and free probability abstract we consider representations of symmetric groupssqfor largeq we give the asymptotic behaviour of the characters when the corresponding young diagrams rescaled by a factorq 1 2 converge to some prescribed shape this behaviour can be expressed in terms of the free cumulants for a probability measure associated with the limit shape of the diagram we also show that the basic operations of representation theory like taking tensor products restriction or induction have a limiting behavior which can be described using the free probability theory of d voiculescu

non linearly stable reduced order models for incompressible flow with energy conserving finite volume methods abstract a novel reduced order model rom formulation for incompressible flows is presented with the key property that it exhibits non linearly stability independent of the mesh of the full order model the time step the viscosity and the number of modes the two essential elements to non linear stability are 1 first discretise the full order model and then project the discretised equations and 2 use spatial and temporal discretisation schemes for the full order model that are globally energy conserving in the limit of vanishing viscosity for this purpose as full order model a staggered grid finite volume method in conjunction with an implicit runge kutta method is employed in addition a constrained singular value decomposition is employed which enforces global momentum conservation the resulting velocity only rom is thus globally conserving mass momentum and kinetic energy for non homogeneous boundary conditions a one time poisson equation is solved that accounts for the boundary contribution the stability of the proposed rom is demonstrated in several test cases furthermore it is shown that explicit runge kutta methods can be used as a practical alternative to implicit time integration at a slight loss in energy conservation

boundary treatment for fourth order staggered mesh discretizations of the incompressible navier stokes equations harlow and welch phys fluids 8 1965 2182 2189 introduced a discretization method for the incompressible navier stokes equations conserving the secondary quantities kinetic energy and vorticity besides the primary quantities mass and momentum this method was extended to fourth order accuracy by several researchers 25 14 21 in this paper we propose a new consistent boundary treatment for this method which is such that continuous integration by parts identities including boundary contributions are mimicked in a discrete sense in this way kinetic energy is exactly conserved even in case of non zero tangential boundary conditions we show that requiring energy conservation at the boundary conflicts with order of accuracy conditions and that the global accuracy of the fourth order method is limited to second order in the presence of boundaries we indicate how non uniform grids can be employed to obtain full fourth order accuracy

compact fourth order finite volume method for numerical solutions of navier stokes equations on staggered grids the development of a compact fourth order finite volume method for solutions of the navier stokes equations on staggered grids is presented a special attention is given to the conservation laws on momentum control volumes a higher order divergence free interpolation for convective velocities is developed which ensures a perfect conservation of mass and momentum on momentum control volumes three forms of the nonlinear correction for staggered grids are proposed and studied the accuracy of each approximation is assessed comparatively in fourier space the importance of higher order approximations of pressure is discussed and numerically demonstrated fourth order accuracy of the complete scheme is illustrated by the doubly periodic shear layer and the instability of plane channel flow the efficiency of the scheme is demonstrated by a grid dependency study of turbulent channel flows by means of direct numerical simulations the proposed scheme is highly accurate and efficient at the same level of accuracy the fourth order scheme can be ten times faster than the second order counterpart this gain in efficiency can be spent on a higher resolution for more accurate solutions at a lower cost

underlying mechanism of numerical instability in large eddy simulation of turbulent flows this paper extends our recent theoretical work concerning the feasibility of stable and accurate computation of turbulence using a large eddy simulation m ida and n taniguchi phys rev e 68 036705 2003 in our previous paper it was shown based on a simple assumption regarding the instantaneous streamwise velocity that the application of the gaussian filter to the incompressible navier stokes equations can result in the appearance of a numerically unstable term that can be decomposed into positive and negative viscosities that result raises the question as to whether an accurate solution can be achieved by a numerically stable subgrid scale model in the present paper based on assumptions regarding the statistically averaged velocity we present similar theoretical investigations to show that in several situations the shears appearing in the statistically averaged velocity field numerically destabilize the fluctuation components because of the derivation of a numerically unstable term that represents negative diffusion in a fixed direction this finding can explain the problematic numerical instability that has been encountered in large eddy simulations of wall bounded flows the present result suggests that this numerical problem is universal in large eddy simulations and that if there is no failure in modeling the resulting subgrid scale model can still have unstable characteristics that is the known instability problems of several existing subgrid scale models are not something that one may remove simply by an artificial technique but must be taken seriously so as to treat them accurately

the cutoff phenomenon in finite markov chains abstract natural mixing processes modeled by markov chains often show a sharp cutoff in their convergence to long time behavior this paper presents problems where the cutoff can be proved card shuffling the ehrenfests urn it shows that chains with polynomial growth drunkard s walk do not show cutoffs the best general understanding of such cutoffs high multiplicity of second eigenvalues due to symmetry is explored examples are given where the symmetry is broken but the cutoff phenomenon persists

numerical approximation of incompressible navier stokes equations based on an auxiliary energy variable abstract we present a numerical scheme for approximating the incompressible navier stokes equations based on an auxiliary variable associated with the total system energy by introducing a dynamic equation for the auxiliary variable and reformulating the navier stokes equations into an equivalent system the scheme satisfies a discrete energy stability property in terms of a modified energy and it allows for an efficient solution algorithm and implementation within each time step the algorithm involves the computations of two pressure fields and two velocity fields by solving several de coupled individual linear algebraic systems with constant coefficient matrices together with the solution of a nonlinear algebraic equation about a scalar number involving a negligible cost a number of numerical experiments are presented to demonstrate the accuracy and the performance of the presented algorithm

explicit filtering and exact reconstruction of the sub filter stresses in large eddy simulation explicit filtering has the effect of reducing numerical or aliasing errors near the grid scale in large eddy simulation les we use a differential filter namely the inverse helmholtz operator which is readily applied to unstructured meshes the filter is invertible which allows the sub filter scale sfs stresses to be exactly reconstructed in terms of the filtered solution unlike eddy viscosity models the method of filtering and reconstruction avoids making any physical assumptions and is therefore valid in any flow regime the sub grid scale sgs stresses are not recoverable by reconstruction but the second order finite element method used here is an adequate source of numerical dissipation in lieu of an sgs model results for incompressible turbulent channel flow at re 180 are presented which show that explicit filtering and exact sfs reconstruction is a significant improvement over the standard les approach of implicit filtering and eddy viscosity sgs modelling

high order finite volume approximations of differential operators on nonuniform grids abstract when computing numerical solutions to partial differential equations difference operators that mimic the crucial properties of the differential operators are usually more accurate than those that do not properties such as symmetry conservation stability and the duality relationships and identities between the gradient curl and divergence operators are all important using the finite volume method we have derived local accurate reliable and efficient difference methods t divergence gradient and curl operators are defined using a discrete versions of the divergence theorem and stokes theorem these methods are especially powerful on coarse nonuniform grids and in calculations where the mesh moves to track interfaces or shocks numerical examples comparing local second and fourth order finite volume approximations to conservation laws on very rough grids are used to demonstrate the advantages of the higher order methods

symmetry preserving discretization of turbulent flow we propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators i e the convective operator is represented by a skew symmetric coefficient matrix and the diffusive operator is approximated by a symmetric positive definite matrix mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner we give it a concrete form by noting that a symmetry preserving discretization of the navier stokes equations is stable on any grid and conserves the total mass momentum and kinetic energy for the latter the physical dissipation is to be turned off of coarse being stable on any grid the choice of the grid may be based on the required accuracy solely and the main question becomes how accurate is a symmetry preserving discretization its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies the comparison is carried out for a reynolds number of 5600 which is based on the channel width and the mean bulk velocity based on the channel half width and wall shear velocity the reynolds number becomes 180 the comparison shows that with a fourth order symmetry preserving method a 64 64 32 grid suffices to perform an accurate numerical simulation

the dynamic procedure for accuracy improvement of numerical discretizations in fluid mechanics in cfd computations discretization or truncation errors should be small providing an acceptable level of accuracy in this paper an extension is made of the recently proposed les formalism based on sampling operators it is shown that the sampling based dynamic procedure in combination with an appropriate truncation error model can be used as a technique to increase the numerical accuracy of a discretization the technique is resemblant to the well known richardson extrapolation the procedure is tested on a 1d convection diffusion equation and a 2d lid driven cavity at re 400 using a finite difference method promising results are found

zonal polynomials via stanleyʼs coordinates and free cumulants abstract we study zonal characters which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power sum symmetric functions we show that the zonal characters just like the characters of the symmetric groups admit a nice combinatorial description in terms of stanley s multirectangular coordinates of young diagrams we also study the analogue of kerov polynomials namely we express the zonal characters as polynomials in free cumulants and we give an explicit combinatorial interpretation of their coefficients in this way we prove two recent conjectures of lassalle for jack polynomials in the special case of zonal polynomials

natural discretizations for the divergence gradient and curl on logically rectangular grids abstract this is the first in series of papers creating a discrete analog of vector analysis on logically rectangular nonorthogonal nonsmooth grids we introduce notations for 2 d logically rectangular grids describe both cell valued and nodal discretizations for scalar functions and construct the natural discretizations of vector fields using the vector components normal and tangential to the cell boundaries we then define natural discrete analogs of the divergence gradient and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates such as length of elements of coordinate lines areas of elements of coordinate surfaces and elementary volumes we introduce the discrete volume integral of scalar functions the discrete surface integral and a discrete analog of the line integral and prove discrete versions of the main theorems relating these objects these theorems include the following the discrete analog of relationship div a 0 if and only if a curl b curl a 0 if and only if a grad if a grad then the line integral does not depend on path and if the line integral of a vector function is equal to zero for any closed path then this vector is the gradient of a scalar function last we define the discrete operators div grad and curl in terms of primitive differencing operators based on forward and backward differences and primitive metric operators related to multiplications of discrete functions by length of edges areas of surfaces and volumes of 3 d cells these formulations elucidate the structure of the discrete operators and are useful when investigating the relationships between operators and their adjoints

interactions of breaking waves with a current over cut cells by design of the external and internal wave current generators the objective of this paper is to extend our efficient navier stokes solver t li p troch j de rouck wave overtopping over a sea dike j comput phys 198 2004 686 726 for modelling of interactions between breaking waves and a current over a cut cell grid based on a dynamic subgrid scale sgs model this solver is constructed by a novel vof finite volume approach coupled with surface tension when studying waves following a positive current our external generator creates the combined inflow motions of waves and a current which is viewed as one type of wavy inflow conditions for cases of waves against strong currents our internal generator describes the opposing current by incorporating the source function to the continuity and momentum equations as a net driving force acting on the fluid elements lying within the finite thickness source region the outgoing waves downstream are dissipated with a breaking type wave absorber placed in the tank extremity five test cases recommended are of distinctly different applications of interest characterized by overtopping of following waves over sloping and vertical structures under the grid refinement effects the results in 2d and 3d are in close agreement with the experimental data available in terms of the surface wave additionally the performance of convergence in computations is also investigated including full discussion for waves on beaches between 2d and 3d by visualization of the motions that describe the physics of turbulence it has been shown that our solver can capture most of the significant features in wave current interactions varying with three different current speeds positive zero negative

asymptotics of characters of symmetric groups related to stanley character formula we prove an upper bound for characters of the symmetric groups in particular we show that there exists a constant a 0 with a property that for every young diagram with n boxes r rows and c columns tr tr e i amax a r n c n j j n aoj j where j j is the minimal number of factors needed to write 2 sn as a product of transpositions we also give uniform estimates for the error term in the vershik kerov s and biane s character formulas and give a new formula for free cumulants of the transition measure

regular article finite difference schemes for u t x α δg δu that inherit energy conservation or dissipation property we propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations such as the korteweg de vries kdv equation and the cahn hilliard equation the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly since the inherited properties are kept even if the time mesh size changes in the time evolution process we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions and sufficiently flexible to treat the inheritance of the energy conservation and dissipation properties are verified numerically for the kdv equation and the cahn hilliard equation

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

an efficient time advancing strategy for energy preserving simulations energy conserving numerical methods are widely employed within the broad area of convection dominated systems semi discrete conservation of energy is usually obtained by adopting the so called skew symmetric splitting of the non linear convective term defined as a suitable average of the divergence and advective forms although generally allowing global conservation of kinetic energy it has the drawback of being roughly twice as expensive as standard divergence or advective forms alone in this paper a general theoretical framework has been developed to derive an efficient time advancement strategy in the context of explicit runge kutta schemes the novel technique retains the conservation properties of skew symmetric based discretizations at a reduced computational cost it is found that optimal energy conservation can be achieved by properly constructed runge kutta methods in which only divergence and advective forms for the convective term are used as a consequence a considerable improvement in computational efficiency over existing practices is achieved the overall procedure has proved to be able to produce new schemes with a specified order of accuracy on both solution and energy the effectiveness of the method as well as the asymptotic behavior of the schemes is demonstrated by numerical simulation of burgers equation

the projection method for the incompressible navier stokes equations the pressure near a no slip wall an explicit staggered projection method for the incompressible navier stokes equations with no slip walls is analyzed and used in simulations to address several issues related to the pressure boundary condition required when the continuity equation is replaced by the standard pressure poisson equation ppe 2 p u u f first it is shown that a ppe system supplemented with a neumann pressure boundary condition derived from the momentum equation can be made consistent with the navier stokes equations if it is extended with the requirement that 2 u 0 is zero near the wall and the solution is sufficiently smooth this implies that it is possible to formulate a boundary condition for the standard ppe without the necessity to resort to green s functions which is interesting for theoretical reasons second the equivalence is shown between the staggered projection method and the staggered discretization of above ppe system the derivation of the equivalence sheds light upon the so called ppe paradox and leads to an approximation of the wall value of p n which is not required but implied by the staggered projection method third the near wall regularity of a solution of the navier stokes equations is numerically analyzed by means of direct numerical simulation of turbulent channel flow performed with the staggered projection method from the numerical inspection of all terms of the momentum equation in the near wall region it is concluded that the three components of the momentum equation are satisfied on the wall for t 0 for short times but also in the turbulent regime in the limit t 0 the pressure gradient is observed to converge to the initial pressure gradient in the l 2 norm which confirms a disputed theoretical result in literature even in the maximum norm the pressure gradient appears to converge to the initial pressure gradient the only discontinuities observed in the simulations are the discontinuities of the tangential viscous terms and the time derivatives of the tangential velocities on the wall at t 0 thus the numerical results indicate that the regularity of the solution for turbulent channel flow is stronger than claimed by existing theory

a volume of fluid based simulation method for wave impact problems in this paper some aspects of water impact and green water loading are considered by numerically investigating a dambreak problem and water entry problems the numerical method is based on the navier stokes equations that describe the flow of an incompressible viscous fluid the equations are discretised on a fixed cartesian grid using the finite volume method even though very small cut cells can appear when moving an object through the fixed grid the method is stable the free surface is displaced using the volume of fluid method together with a local height function resulting in a strictly mass conserving method the choice of boundary conditions at the free surface appears to be crucial for the accuracy and robustness of the method for validation results of a dambreak simulation are shown that can be compared with measurements a box has been placed in the flow as a model for a container on the deck of an offshore floater on which forces are calculated the water entry problem has been investigated by dropping wedges with different dead rise angles a cylinder and a cone into calm water with a prescribed velocity the resulting free surface dynamics with the sideways jets has been compared with photographs of experiments also a comparison of slamming coefficients with theory and experimental results has been made finally a drop test with a free falling wedge has been simulated

high order finite difference schemes on non uniform meshes with good conservation properties numerical simulation of turbulent flows dns or les requires numerical methods that are both stable and free of numerical dissipation one way to achieve this is to enforce additional constraints such as discrete conservation of mass momentum and kinetic energy the objective of this work is to generalize the high order schemes of morinishi et al to non uniform meshes while maintaining conservation properties of the schemes as much as possible this generalization is achieved by preserving symmetries of the uniform mesh case the proposed schemes do not simultaneously conserve mass momentum and kinetic energy however depending on the form of the convective term conservation of either momentum or energy in addition to mass can be achieved it is shown that the conservation properties of the generalized schemes are as good as those of the standard second order finite difference scheme on non uniform meshes while the accuracy of the new schemes is definitely superior the predicted conservation properties are demonstrated numerically in inviscid flow simulations

mass momentum and energy conserving mamec discretizations on general grids for the compressible euler and shallow water equations the paper explains a method by which discretizations of the continuity and momentum equations can be designed such that they can be combined with an equation of state into a discrete energy equation the resulting mamec discretizations conserve mass momentum as well as energy although no explicit conservation law for the total energy is present essential ingredients are i discrete convection that leaves the discrete energy invariant and ii discrete consistency between the thermodynamic terms of particular relevance is the way in which finite volume fluxes are related to nodal values the method is an extension of existing methods based on skew symmetry of discrete operators because it allows arbitrary equations of state and a larger class of grids than earlier methods the method is first illustrated with a one dimensional example on a highly stretched staggered grid in which the mamec method calculates qualitatively correct results and a non skew symmetric finite volume method becomes unstable a further example is a two dimensional shallow water calculation on a rectilinear grid as well as on an unstructured grid the conservation of mass momentum and energy is checked and losses are found negligible up to machine accuracy

an explicit form for kerov s character polynomials kerov considered the normalized characters of irreducible representations of the symmetric group evaluated on a cycle as a polynomial in free cumulants biane has proved that this polynomial has integer coefficients and made various conjectures recently sniady has proved biane s conjectured explicit form for the first family of nontrivial terms in this polynomial in this paper we give an explicit expression for all terms in kerov s character polynomials our method is through lagrange inversion

numerically stable formulations of convective terms for turbulent compressible flows abstract a systematic analysis of the discrete conservation properties of non dissipative central difference approximations of the compressible navier stokes equations is reported a generalized splitting of the nonlinear convective terms is considered and energy preserving formulations are fully characterized by deriving a two parameter family of split forms previously developed formulations reported in literature are shown to be particular members of this family novel splittings are introduced and discussed as well furthermore the conservation properties yielded by different choices for the energy equation i e total and internal energy entropy are analyzed thoroughly it is shown that additional preserved quantities can be obtained through a suitable adaptive selection of the split form within the derived family local conservation of primary invariants which is a fundamental property to build high fidelity shock capturing methods is also discussed in the paper numerical tests performed for the taylor green vortex at zero viscosity fully confirm the theoretical findings and show that a careful choice of both the splitting and the energy formulation can provide remarkably robust and accurate results

an annotated bibliography on guaranteed graph searching graph searching encompasses a wide variety of combinatorial problems related to the problem of capturing a fugitive residing in a graph using the minimum number of searchers in this annotated bibliography we give an elementary classification of problems and results related to graph searching and provide a source of bibliographical references on this field

improving shock free compressible rans solvers for les on unstructured meshes standard numerical methods used to solve the reynolds averaged navier stokes equations are known to be too dissipative to carry out large eddy simulations since the artificial dissipation they introduce to stabilize the discretization of the convection term usually interacts strongly with the subgrid scale model a possible solution is to resort to non dissipative central schemes unfortunately these schemes are in general unstable a way to reach stability is to select a central scheme that conserves the discrete kinetic energy to that purpose a family of kinetic energy conserving schemes is developed to perform simulations of compressible shock free flows on unstructured grids a direct numerical simulation of the flow past a sphere at a reynolds number of 300 and a large eddy simulation at a reynolds number of 10 000 are performed to validate the methodology

on the properties of discrete spatial filters for cfd the spatial filtering of variables in the context of computational fluid dynamics cfd is a common practice most of the discrete filters used in cfd simulations are locally accurate models of continuous operators however when filters are adaptative i e the filter width is not constant or meshes are irregular discrete filters sometimes break relevant global properties of the continuous models they are based on for example the principle of maxima and minima reduction or conservation are eventually infringed in this paper we analyze the properties of analytic continuous convolution filters and extract those we consider to define filtering then we impose the accomplishment of these properties on explicit discrete filters by means of constraints three filters satisfying the derived conditions are deduced and compared to common differential discrete cfd filters on synthetic fields tests on the developed discrete filters show the fulfillment of the imposed properties in particular the problem of maxima and minima generation is resolved for physically relevant cases the tests are conducted on the basis of the eigenvectors of graph laplacian matrices of meshes thus insight into the relations between filtering and oscillation growth on general meshes is provided further tests on singularity fields and on isentropic vortices have also been conducted to evaluate the performance of filters on basic cfd fields results confirm that imposing the proposed conditions makes discrete filters properties consistent with those of the continuous ones

high order fluxes for conservative skew symmetric like schemes in structured meshes developing high order non dissipative schemes is an important research task for both steady and unsteady flow computations we take as a starting point the built in de aliasing property of the discretized skew symmetric form for the non linear terms of the navier stokes equations recalled in kravchenko and moin 1 two families of high order locally conservative schemes matching this discretized skew symmetric form are considered and rewritten in terms of telescopic fluxes for both finite difference and finite volume approximations in the context of compressible flows the jameson s scheme 2 is shown to be the second order member of larger families of skew symmetric like centered schemes the fourth order finite volume and finite difference and the sixth order finite difference schemes which belong to this family are provided the proposed schemes are extended to shock capturing schemes either by modifying the jameson s artificial viscosity or by hybriding the centered flux with weno 3 fluxes an adapted interpolation is proposed to extend the use of the proposed schemes to non regular grids several tests are provided showing that the conjectured order is properly recovered even with irregular meshes and that the shock capturing properties allow us to improve the second order results for standard test cases the improvement due to fourth order is then confirmed for the estimation of the growth of two ts waves and three crow instability dimensional unstable modes for both confined and free shear flows the last application concerns the steady computation using the spalart allmaras model of a separated boundary layer it confirms that the use of a high order scheme improves the results even in this type of steady applications

a computational error assessment of central finite volume discretizations in large eddy simulation using a smagorinsky model we present a framework for the computational assessment and comparison of large eddy simulation methods we apply this to large eddy simulation of homogeneous isotropic decaying turbulence using a smagorinsky subgrid model and investigate the combined effect of discretization and model errors at coarse subgrid resolutions we compare four different central finite volume methods these discretization methods arise from the four possible combinations that can be made with a second order and a fourth order central scheme for either the convective and the viscous fluxes by systematically varying the simulation resolution and the smagorinsky coefficient we determine parameter regions for which a desired number of flow properties is simultaneously predicted with approximately minimal error we include both physics based and mathematics based error definitions leading to different error measures designed to emphasize either errors in large or in small scale flow properties it is shown that the evaluation of simulations based on a single physics based error may lead to inaccurate perceptions on quality we demonstrate however that evaluations based on a range of errors yields robust conclusions on accuracy both for physics based and mathematics based errors parameter regions where all considered errors are simultaneously near optimal are referred to as multi objective optimal parameter regions the effects of discretization errors are particularly important at marginal spatial resolution such resolutions reflect local simulation conditions that may also be found in parts of more complex flow simulations under these circumstances the asymptotic error behavior as expressed by the order of the spatial discretization is no longer characteristic for the total dynamic consequences of discretization errors we find that the level of overall simulation errors for a second order central discretization of both the convective and viscous fluxes the 2 2 method and the fully fourth order 4 4 method is equivalent in their respective multi objective optimal regions mixed order methods i e the 2 4 and 4 2 combinations yield errors which are considerably higher

upper bound on the characters of the symmetric groups for balanced young diagrams and a generalized frobenius formula we study asymptotics of an irreducible representation of the symmetric group sn corresponding to a balanced young diagram a young diagram with at most rows and columns for some fixed constant c in the limit as n tends to infinity we show that there exists a constant d which depends only on c with a property that where p denotes the length of a permutation the minimal number of factors necessary to write p as a product of transpositions our main tool is an analogue of the frobenius character formula which holds true not only for cycles but for arbitrary permutations

a simple approach to discretize the viscous term with spatially varying eddy viscosity a simple approach to discretize the viscous dissipation term in the incompressible navier stokes equations with spatially varying viscosity is presented unlike the case where the viscosity remains constant its discretization may be quite cumbersome especially for high order staggered formulations to circumvent this problem we propose an alternative form of the viscous term whose discretization is straightforward notice that this approach is also suitable for eddy viscosity models for large eddy simulation moreover since it is based on already available operators it can be easily implemented on any structured or unstructured code the supra convergence of the method is numerically shown on both a fourth order cartesian staggered and an unstructured collocated formulation

the linkage of a graph the linkage of a graph is defined to be the maximum min degree of any of its subgraphs it is known that the linkage of a graph is equal to its width for an arbitrary linear ordering of the vertices of the graph consider the maximum with respect to any vertex v of the number of vertices connected with v and preceding it in the ordering the width of the graph is the minimum of these maxima over all possible linear orderings width has been used in artificial intelligence in the context of constraint satisfaction problems csps a more general notion is defined by considering not the number of vertices preceding and connected with v but rather the least number of vertices preceding and connected with any cluster of at most j consecutive vertices extending to the right up to v j is a given integer the graph parameter thus defined is called j width no efficient algorithm was known for computing the j width in this paper we introduce a graph parameter depending on j that refers to the subgraphs of the graph and generalizes the notion of linkage we prove the min max theorem that this graph parameter which we call j linkage is equal to j width and we then give a polynomial time algorithm for computing it for constant j we also find tight lower and upper bounds for the j linkage equivalently the j width of graphs with given numbers of vertices and edges it is interesting to note that a lower bound for the width of a graph had been found by erdos as we show however that bound is not tight moreover we prove that our lower bound for width is also a tight lower bound for treewidth pathwidth and bandwidth graph parameters that may be arbitrarily larger than width finally we show that computing the j linkage is a p complete problem whereas we prove that approximating it is a threshold problem it is in nc for approximation factors 1 2

a further study of numerical errors in large eddy simulations numerical errors in large eddy simulations les arise from aliasing and discretization errors and errors in the subfilter scale sfs turbulence model using a direct numerical simulation dns dataset of stably stratified shear flow to perform a priori tests we compare the numerical error from several finite difference schemes to the magnitude of the sfs force this is an extension of ghosal s analysis j comput phys 125 1996 187 to realistic flow fields by evaluating different grid resolutions as well as different filter grid ratios we provide guidelines for les for a second order finite difference scheme a filter grid ratio of at least four is desired for a sixth order pade scheme a filter grid ratio of two is sufficient