experiments with implicit upwind methods for the euler equations abstract a number of implicit integration schemes for the one dimensional euler equations with conservative upwind spatial differencing are tested on a problem of steady discontinuous flow fastest convergence quadratic for the first order backward euler scheme is obtained with the upwind switching provided by van leer s differentiable split fluxes which easily linearize in time with roe s nondifferentiable split flux differences the iterations may get trapped in a limit cycle this also happens in a second order scheme with split fluxes if the matrix coefficients arising in the implicit time linearization are not properly centered in space the use of second order terms computed from split fluxes degrades the accuracy of the solution especially if these are subjected to a limiter for the sake of monotonicity preservation second order terms computed from the characteristic variables perform best

an entropy stable discontinuous galerkin approximation for the incompressible navier stokes equations with variable density and artificial compressibility abstract we present a provably stable discontinuous galerkin spectral element method for the incompressible navier stokes equations with artificial compressibility and variable density stability proofs which include boundary conditions that follow a continuous entropy analysis are provided we define a mathematical entropy function that combines the traditional kinetic energy and an additional energy term for the artificial compressibility and derive its associated entropy conservation law the latter allows us to construct a provably stable split form nodal discontinuous galerkin dg approximation that satisfies the summation by parts simultaneous approximation term sbp sat property the scheme and the stability proof are presented for general curvilinear three dimensional hexahedral meshes we use the exact riemann solver and the bassi rebay 1 br1 scheme at the inter element boundaries for inviscid and viscous fluxes respectively and an explicit low storage runge kutta rk3 scheme to integrate in time we assess the accuracy and robustness of the method by solving a manufactured solution the kovasznay flow a lid driven cavity the inviscid taylor green vortex and the rayleigh taylor instability

adjoint complement to viscous finite volume pressure correction methods abstract a hybrid adjoint navier stokes method for the pressure based computation of hydrodynamic objective functional derivatives with respect to the shape is systematically derived in three steps the underlying adjoint partial differential equations and boundary conditions for the frozen turbulence reynolds averaged navier stokes equations are considered in the first step in step two the adjoint discretisation is developed from the primal unstructured finite volume discretisation such that adjoint consistent approximations to the adjoint partial differential equations are obtained following a so called hybrid adjoint approach a unified discrete boundary description is outlined that supports high and low reynolds number turbulent wall boundary treatments for both the adjoint boundary condition and the boundary based gradient formula the third component focused in the development of the industrial adjoint cfd method is the adjoint counterpart to the primal pressure correction algorithm the approach is verified against the direct differentiation method and an application to internal flow problems is presented

blending the eddy viscosity and reynolds stress models using uniform high order discretization accuracy robustness and efficiency are the three major goals of turbulence closure generally eddy viscosity models evms with the boussinesq approximation are numerically stable and efficient

on the quantification and visualization of transient periodic instabilities in pulsatile flows turbulent like flows without cycle to cycle variations are more frequently being reported in studies of cardiovascular flows the associated stimuli might be of mechanobiological relevance but how to quantify them objectively is not obvious classical reynolds decomposition where the flow is separated into mean and fluctuating velocity components is not applicable as the phase average is zero we therefore expanded on established techniques and present the idea analogous to reynolds decomposition to decompose a flow with transient instabilities into low versus high frequency components respectively to discriminate flow instabilities from the underlying cardiac pulsatility transient wall shear stress and velocity signals derived from computational fluid dynamic simulations were transferred to the frequency domain a high pass filter was applied to subtract the 99 most energy containing frequencies which gave a cut off frequency of 25hz we introduce here the spectral power index and compute the fluctuating kinetic energy based on the high pass filtered velocity components both being frequency based operators the efficacy was evaluated in an aneurysm model for multiple flow rates demonstrating transition to turbulent like flows the frequency based operators were found to better correlate with the qualitatively observed flow instabilities compared to conventional descriptors like time averaged wall shear stress or oscillatory shear index we demonstrate how the high frequencies beyond the physiological range could be analyzed and or transferred back to the time domain for quantification and visualization purposes we have introduced general frequency based operators easily extendable to other cardiovascular territories based on a posteriori heuristic filtering that allows for separation isolation and quantification of cycle invariant turbulent like flows

error estimation and anisotropic mesh refinement for 3d laminar aerodynamic flow simulations this article considers a posteriori error estimation and anisotropic mesh refinement for three dimensional laminar aerodynamic flow simulations the optimal order symmetric interior penalty discontinuous galerkin discretization which has previously been developed for the compressible navier stokes equations in two dimensions is extended to three dimensions symmetry boundary conditions are given which allow to discretize and compute symmetric flows on the half model resulting in exactly the same flow solutions as if computed on the full model using duality arguments an error estimation is derived for estimating the discretization error with respect to the aerodynamic force coefficients furthermore residual based indicators as well as adjoint based indicators for goal oriented refinement are derived these refinement indicators are combined with anisotropy indicators which are particularly suited to the discontinuous galerkin dg discretization two different approaches based on either a heuristic criterion or an anisotropic extension of the adjoint based error estimation are presented the performance of the proposed discretization error estimation and adaptive mesh refinement algorithms is demonstrated for 3d aerodynamic flows

multitarget error estimation and adaptivity in aerodynamic flow simulations important quantities in aerodynamic flow simulations are the aerodynamic force coefficients including the pressure induced and the viscous stress induced drag lift and moment coefficients in addition to the exact approximation of these quantities it is of increasing importance in particular in the field of uncertainty quantification to estimate the error in the computed quantities in recent years a posteriori error estimation and goal oriented refinement approaches have been developed for the accurate and efficient computation of single target quantities the current approaches are based on computing an adjoint solution related to each of the specific target quantities under consideration in this paper we extend this approach to the accurate and efficient computation of multiple target quantities instead of computing multiple adjoint solutions one for each target functional the new approach is based on the solution to one discrete adjoint problem and one discrete error problem this way only two auxiliary problems are required irrespective of the number of target functionals the practical performance of this approach is demonstrated for a laminar compressible flow in particular the proposed approach is compared to the standard approach of error estimation and goal oriented refinement as well as to residual based refinement the performance of the algorithms is measured in terms of computing resources required for meeting industrial as well as academic accuracy requirements on the computed force coefficients

transonic separated flow predictions with an eddy viscosity reynolds stress closure model on evalue un modele de fermeture developpe precedemment pour une serie d interactions fortes transsoniques visqueuses non visqueuses avec des degres de decollement variables

hybrid turbulence model for unsteady boundary layers a hybrid model is developed and evaluated for a single data set for a broader perspective the results of algebraic one equation and two equation models are also presented the algebraic model was composed of the van driest near wall mixing length with the cebeci keller unsteady modification and the cebeci smith outer layer eddy viscosity the one equation model was based on the norris reynolds prandtl energy model with a prescribed mixing length the two equation model was based on that of jones and launder making use of wall functions for computional efficiency

formulation of a galerkin spectral element fourier method for three dimensional incompressible flows in cylindrical geometries a primitive variable formulation for simulation of time dependent incompressible flows in cylindrical coordinates is developed spectral elements are used to discretise the meridional semi plane coupled with fourier expansions in azimuth unlike previous formulations where special distributions of nodal points have been used in the radial direction the current work adopts standard gauss lobatto legendre nodal based expansions in both the radial and axial directions using a galerkin projection of the symmetrised cylindrical navier stokes equations all geometric singularities are removed as a consequence of either the fourier mode dependence of axial boundary conditions or the shape of the weight function applied in the galerkin projection this observation implies that in a numerical implementation geometrically singular terms can be naively treated by explicitly zeroing their contributions on the axis in integral expressions without recourse to special treatments such as l hopital s rule exponential convergence of the method both in the meridional semi plane and in azimuth is demonstrated through application to a three dimensional analytical solution of the navier stokes equations in which flow crosses the axis

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

output functional control for nonlinear equations driven by anisotropic mesh adaption the navier stokes equations the contribution of this paper is twofold firstly moving from the very well known dual weighted residual dwr method we set up a theoretical framework for a goal oriented a posteriori analysis of nonlinear partial differential equations accounting for different approximations of the primal and dual problems nonhomogeneous dirichlet boundary conditions even different on passing from the primal to the dual problem the error due to data approximation and the effect of a possible stabilization secondly moving from this framework and employing anisotropic interpolation error estimates a sound anisotropic mesh adaption procedure is devised for the numerical approximation of the navier stokes equations by continuous piecewise linear finite elements the resulting adaptive procedure is thoroughly addressed and validated on some relevant test cases

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

on the eigenvalues of the electrostatic integral operator abstract a fundamental result in scattering and potential theory in r 3 states that the spectrum of the electrostatic integral operator lies in the interval 1 1 in the case of a sphere and a prolate spheroid it is known that the spectrum of the operator lies in the interval 1 0 a fundamental question arises whether the spectrum of the operator always lies in this interval or whether there exists a smooth surface for which the electrostatic integral operator has a positive eigenvalue in this paper this question is answered a surface is produced whereby the underlying integral operator has a positive eigenvalue

a posteriori error estimation for the lax wendroff finite difference scheme in many application domains the preferred approaches to the numerical solution of hyperbolic partial differential equations such as conservation laws are formulated as finite difference schemes while finite difference schemes are amenable to physical interpretation one disadvantage of finite difference formulations is that it is relatively difficult to derive the so called goal oriented a posteriori error estimates a posteriori error estimates provide a computational approach to numerically compute accurate estimates in the error in specified quantities computed from a numerical solution widely used for finite element approximations a posteriori error estimates yield substantial benefits in terms of quantifying reliability of numerical simulations and efficient adaptive error control the chief difficulties in formulating a posteriori error estimates for finite difference schemes is introducing a variational formulation and the associated adjoint problem and a systematic definition of residual errors in this paper we approach this problem by first deriving an equivalency between a finite element method and the lax wendroff finite volume method we then obtain an adjoint based error representation formula for solutions obtained with this method results from linear and nonlinear viscous conservation laws are given formulation of the lax wendroff difference scheme for conservation laws as a finite element method goal oriented a posteriori error estimate for the lax wendroff finite difference scheme investigation of accuracy of the computational error estimate

on positivity preserving high order discontinuous galerkin schemes for compressible navier stokes equations we construct a local lax friedrichs type positivity preserving flux for compressible navier stokes equations which can be easily extended to multiple dimensions for generic forms of equations of state shear stress tensor and heat flux with this positivity preserving flux any finite volume type schemes including discontinuous galerkin dg schemes with strong stability preserving runge kutta time discretizations satisfy a weak positivity property with a simple and efficient positivity preserving limiter high order explicit runge kutta dg schemes are rendered preserving the positivity of density and internal energy without losing local conservation or high order accuracy numerical tests suggest that the positivity preserving flux and the positivity preserving limiter do not induce excessive artificial viscosity and the high order positivity preserving dg schemes without other limiters can produce satisfying non oscillatory solutions when the nonlinear diffusion in compressible navier stokes equations is accurately resolved

dealiasing techniques for high order spectral element methods on regular and irregular grids high order methods are becoming increasingly attractive in both academia and industry especially in the context of computational fluid dynamics however before they can be more widely adopted issues such as lack of robustness in terms of numerical stability need to be addressed particularly when treating industrial type problems where challenging geometries and a wide range of physical scales typically due to high reynolds numbers need to be taken into account one source of instability is aliasing effects which arise from the nonlinearity of the underlying problem in this work we detail two dealiasing strategies based on the concept of consistent integration the first uses a localised approach which is useful when the nonlinearities only arise in parts of the problem the second is based on the more traditional approach of using a higher quadrature the main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods thereby reducing aliasing driven instabilities we demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations where in the latter both volumetric and interface approximations must be considered we show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations

on the magnetostatic integral operator for ellipsoids the magnetostatic integral operator in r3is of considerable interest in both scattering and potential theory as well as for its scalar pendant the electrostatic integral operator its eigenvalues and its eigenfunctions play a fundamental role in the theory of permanent magnetization while the eigenvalues of both operators are closely related explicit expressions for the eigenfunctions of the magnetostatic operator are known only in the case when the underlying surface is a sphere for the cases of ellipsoids and spheroids we present explicit expressions for the eigenvalues in terms of lame functions and associated legendre functions and expressions for eigenfunctions in terms of surface ellipsoidal and surface spheroidal harmonics respectively as an application the permanent magnetization of ellipsoids is discussed

assessment of various low reynolds number turbulence models in shock boundary layer interaction assessment of various low re two equation turbulence models in transonic flow with shock boundary layer interaction and separation is presented the study includes seven different versions of the k model as well as the k and k r models the models are implemented in conjunction with a characteristics based scheme and an implicit unfactored method the implicit unfactored solution of the fluid flow and turbulence transport equations is obtained by a newton type method which includes point by point gauss seidel relaxation for the inversion of the system of equations the accuracy and efficiency of the models is assessed for the transonic flow over an axisymmetric bump geometry the differences in the numerical results between various models are mainly presented in the kinetic energy and turbulent shear stress especially in the separated flow region the study also reveals that the number of iterations required for steady state solution depends strongly on the model used

a p multigrid discontinuous galerkin method for the euler equations on unstructured grids a p multigrid p polynomial degree discontinuous galerkin method is presented for the solution of the compressible euler equations on unstructured grids the method operates on a sequence of solution approximations of different polynomial orders a distinct feature of this p multigrid method is to use different time integration schemes on different approximation levels resulting in an accurate fast and low memory method that can be used to accelerate the convergence of the euler equations to a steady state for discontinuous galerkin methods the developed method is used to compute the compressible flows for a variety of test problems on unstructured grids the numerical results obtained strongly indicate the order independent property of this p multigrid method an overall speed up factor more than one order of magnitude for both second and third order solutions of all test cases in comparison with the explicit method is demonstrated

adjoint based error estimation and adaptive mesh refinement for the rans and k ω turbulence model equations abstract in this article we present the extension of the a posteriori error estimation and goal oriented mesh refinement approach from laminar to turbulent flows which are governed by the reynolds averaged navier stokes and k turbulence model rans k equations in particular we consider a discontinuous galerkin discretization of the rans k equations and use it within an adjoint based error estimation and adaptive mesh refinement algorithm that targets the reduction of the discretization error in single as well as in multiple aerodynamic force coefficients the accuracy of the error estimation and the performance of the goal oriented mesh refinement algorithm is demonstrated for various test cases including a two dimensional turbulent flow around a three element high lift configuration and a three dimensional turbulent flow around a wing body configuration

the numerical viscosity of entropy stable schemes for systems of conservation laws i discrete approximations to hyperbolic systems of conservation laws are studied the amount of numerical viscosity present in such schemes is quantified and related to their entropy stability by means of comparison to this end conservative schemes which are also entropy conservative are constructed these entropy conservative schemes enjoy second order accuracy moreover they can be interpreted as piecewise linear finite element methods and hence can be formulated on various mesh configurations it is then shown that conservative schemes are entropy stable if and for three point schemes only they contain more viscosity than that present in the above mentioned entropy conservative ones

developments of entropy stable residual distribution methods for conservation laws i this paper presents preliminary developments of entropy stable residual distribution methods for scalar problems controlling entropy generation is achieved by formulating an entropy conserved signals distribution coupled with an entropy stable signals distribution numerical results of the entropy stable residual distribution methods are accurate and comparable with the classic residual distribution methods for steady state problems high order accurate extensions for the new method on steady state problems are also demonstrated moreover the new method preserves second order accuracy on unsteady problems using an explicit time integration scheme the idea of the multi dimensional entropy stable residual distribution method is generic enough to be extended to the system of hyperbolic equations which will be presented in the sequel of this paper

high order schemes entropy inequalities and nonclassical shocks we are concerned with the approximation of undercompressive regularization sensitive nonclassical solutions of hyperbolic systems of conservation laws by high order accurate conservative and semidiscrete finite difference schemes nonclassical shock waves can be generated by diffusive and dispersive terms kept in balance particular attention is given here to a class of systems of conservation laws including the scalar equations and the system of nonlinear elasticity and to linear diffusion and dispersion in either the conservative or the entropy variables first we investigate the existence and the properties of entropy conservative schemes a notion due to tadmor math comp 49 1987 pp 91 103 in particular we exhibit a new five point scheme which is third order accurate at least second we study a class of entropy stable and high order accurate schemes satisfying a single cell entropy inequality they are built from any high order entropy conservative scheme by adding to it a mesh independent numerical viscosity which preserves the order of accuracy of the base scheme these schemes can only converge to solutions of the system of conservation laws satisfying the entropy inequality these entropy stable schemes exhibit mild oscillations near shocks and interestingly may converge to classical or nonclassical entropy solutions depending on the sign of their dispersion coefficient then based on a third order entropy conservative scheme we propose a general scheme for the numerical computation of nonclassical shocks importantly our scheme satisfies a cell entropy inequality following hayes and lefloch siam j numer anal 35 1998 pp 2169 2194 we determine numerically the kinetic function which uniquely characterizes the dynamics of nonclassical shocks for each regularization of the conservation laws our results compare favorably with previous analytical and numerical results finally we prove that there exists no fully discrete and entropy conservative scheme and we investigate the entropy stability of a class of fully discrete lax wendroff type schemes

errors in power law estimations of inflow rates for intracranial aneurysm cfd patient specific inflow rates are rarely available for computational fluid dynamics cfd studies of intracranial aneurysms instead inflow rates are often estimated from parent artery diameters via power laws i e q dn reflecting adaptation of conduit arteries to demanded flow the present study aimed to validate the accuracy of these power laws internal carotid artery ica flow rates were measured from 25 ica aneurysm patients via 2d phase contrast mri ica diameters derived from 3d segmentation of rotational angiograms were used to estimate inflow rates via power laws from the aneurysm cfd literature assuming the same inlet wall shear stress wss n 3 velocity n 2 or flow rate n 0 for all cases to illustrate the potential impact of errors in flow rate estimates pulsatile cfd was carried out for four cases having large errors for at least one power law flow rates estimated by n 3 and n 0 power laws had significant p 0 01 mean biases of 22 to 32 respectively but with individual errors ranging from 78 to 120 the n 2 power law had no significant bias but had non negligible individual errors of 58 to 71 cfd showed similarly large errors for time averaged sac wss however these were reduced after normalizing by parent artery wss high frequency wss fluctuations evident in 2 4 aneurysms were also sensitive to inflow rate errors care should therefore be exercised in the interpretation of aneurysm cfd studies that rely on power law estimates of inflow rates especially if absolute vs normalized wss or wss instabilities are of interest

performance of discontinuous galerkin methods for elliptic pdes in this paper we compare the performance of several discontinuous galerkin dg methods for elliptic partial differential equations pdes on a model problem theoretical estimates of the condition number of the stiffness matrix are given for dg methods whose bilinear form is symmetric and which are shown to be numerically sharp then the efficiency of the methods in the computation of both the potential and its gradient is tested on unstructured triangular meshes

legendre pseudospectral viscosity method for nonlinear conservation laws in this paper the legendre spectral viscosity sv method for the approximate solution of initial boundary value problems associated with nonlinear conservation laws is studied the authors prove that by adding a small amount of sv bounded solutions of the legendre sv method converge to the exact scalar entropy solution the convergence proof is based on compensated compactness arguments and therefore applies to certain 2 times 2 systems finally numerical experiments for scalar as well as the one dimensional system of gas dynamics equations are presented which confirm the convergence of the legendre sv method moreover these numerical experiments indicate that by post processing the sv approximation one can recover the entropy solution within spectral accuracy

fully discrete spectral methods for boundary integral equations on slender spheroids we develop fully discrete spectral boundary integral algorithms to solve interior and exterior potential boundary value problems on three dimensional slender spheroids our approach is based on identifying and using analytical solutions of associated generalised eigenvalue problems of boundary integral operators we generalise approximation theory results on the sphere using spheroidally appropriate eigenfunctions and solve an open problem of stability of spectral methods for first kind boundary integral equations on spheroids

discontinuous galerkin discretization of the reynolds averaged navier stokes equations with the shear stress transport model in this article we consider the development of discontinuous galerkin dg methods for the numerical approximation of the reynolds averaged navier stokes rans equations with the shear stress transport sst model by menter this turbulence model is based on a blending of the wilcox email protected model used near the wall and the email protected model used in the rest of the domain where the blending functions depend on the distance to the nearest wall for the computation of the distance of each quadrature point in the domain to the nearest of the curved piecewise polynomial wall boundaries we propose a stabilized continuous finite element fe discretization of the eikonal equation furthermore we propose a new wall boundary condition for the dissipation rate w based on the projection of the analytic near wall behavior of w onto the discrete ansatz space of the dg discretization finally we introduce an artificial viscosity to the discretization of the turbulence kinetic energy k equation to suppress oscillations of k near the underresolved boundary layer edge the wall distance computation based on the continuous fe discretization of the eikonal equation is demonstrated for an internal and three external aerodynamic flow geometries including a three element high lift configuration the dg discretization of the rans equations with the sst model is demonstrated for turbulent flows past a flat plate and the rae2822 airfoil cases 9 and 10 the results are compared to the underlying email protected model and experimental data

stabilisation of spectral hp element methods through spectral vanishing viscosity application to fluid mechanics modelling in this paper we present a formulation of spectral vanishing viscosity svv for the stabilisation of spectral hp element methods applied to the solution of the incompressible navier stokes equations we construct the svv around a filter with respect to an orthogonal expansions and prove that this methodology provides a symmetric semi positive definite svv operator after providing a few simple one and two dimensional examples to demonstrate the utility of the svv we examine how it can be applied to a spectral hp element discretisation of the navier stokes equations using a velocity correction splitting scheme we provide three fluid flow examples to help illustrate the pros and cons of this approach on stability and accuracy 2005 elsevier b v all rights reserved

a high order discontinuous galerkin fourier incompressible 3d navier stokes solver with rotating sliding meshes we present the development of a sliding mesh capability for an unsteady high order order 3 h p discontinuous galerkin solver for the three dimensional incompressible navier stokes equations a high order sliding mesh method is developed and implemented for flow simulation with relative rotational motion of an inner mesh with respect to an outer static mesh through the use of curved boundary elements and mixed triangular quadrilateral meshes a second order stiffly stable method is used to discretise in time the arbitrary lagrangian eulerian form of the incompressible navier stokes equations spatial discretisation is provided by the symmetric interior penalty galerkin formulation with modal basis functions in the x y plane allowing hanging nodes and sliding meshes without the requirement to use mortar type techniques spatial discretisation in the z direction is provided by a purely spectral method that uses fourier series and allows computation of spanwise periodic three dimensional flows the developed solver is shown to provide high order solutions second order in time convergence rates and spectral convergence when solving the incompressible navier stokes equations on meshes where fixed and rotating elements coexist in addition an exact implementation of the no slip boundary condition is included for curved edges circular arcs and naca 4 digit airfoils where analytic expressions for the geometry are used to compute the required metrics the solver capabilities are tested for a number of two dimensional problems governed by the incompressible navier stokes equations on static and rotating meshes the taylor vortex problem a static and rotating symmetric naca0015 airfoil and flows through three bladed cross flow turbines in addition three dimensional flow solutions are demonstrated for a three bladed cross flow turbine and a circular cylinder shadowed by a pitching naca0012 airfoil

de aliasing on non uniform grids algorithms and applications we present de aliasing rules to be used when evaluating non linear terms with polynomial spectral methods on non uniform grids analogous to the de aliasing rules used in fourier spectral methods they are based upon the idea of super collocation followed by a galerkin projection of the non linear terms we demonstrate through numerical simulation that both accuracy and stability can be greatly enhanced through the use of this approach we begin by deriving from the numerical quadrature rules used by galerkin type projection methods the number of quadrature points and weights needed for quadratic and cubic non linearities we then present a systematic study of the effects of super collocation when using both a continuous galerkin and a discontinuous galerkin method to solve the one dimensional viscous burgers equation we conclude by examining three direct numerical simulation flow examples incompressible turbulent flow in a triangular duct incompressible turbulent flow in a channel at re 395 and compressible flow past a pitching airfoil at re 45 000

adjoint consistency analysis of discontinuous galerkin discretizations this paper is concerned with the adjoint consistency of discontinuous galerkin dg discretizations adjoint consistency in addition to consistency is the key requirement for dg discretizations to be of optimal order in l 2 as well as measured in terms of target functionals we provide a general framework for analyzing the adjoint consistency of dg discretizations which is also useful for the derivation of adjoint consistent methods this analysis will be performed for the dg discretizations of the linear advection equation the interior penalty dg method for elliptic problems and the dg discretization of the compressible euler equations this framework is then used to derive an adjoint consistent dg discretization of the compressible navier stokes equations numerical experiments demonstrate the link of adjoint consistency to the accuracy of numerical flow solutions and the smoothness of discrete adjoint solutions

entropy stable wall boundary conditions for the three dimensional compressible navier stokes equations non linear entropy stability and a summation by parts framework are used to derive entropy stable wall boundary conditions for the three dimensional compressible navier stokes equations a semi discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator the data at the boundary are weakly imposed using a penalty flux approach and a simultaneous approximation term penalty technique although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy the new boundary conditions are compatible with any diagonal norm summation by parts spatial operator including finite element finite difference finite volume discontinuous galerkin and flux reconstruction correction procedure via reconstruction schemes the proposed boundary treatment is tested for three dimensional subsonic and supersonic flows the numerical computations corroborate the non linear stability entropy stability and accuracy of the boundary conditions

a spectral vanishing viscosity method for large eddy simulations a new simulation approach for high reynolds number turbulent flows is developed combining concepts of monotonicity in nonlinear conservation laws with concepts of large eddy simulation the spectral vanishing viscosity svv first introduced by e tadmor siam j numer anal 26 30 1989 is incorporated into the navier stokes equations for controlling high wavenumber oscillations unlike hyperviscosity kernels the svv approach involves a second order operator which can be readily implemented in standard finite element codes in the work presented here discretization is performed using hierarchical spectral hp methods accommodating effectively an ab initio intrinsic scale separation the key result is that monotonicity is enforced via svv leading to stable discretizations without sacrificing the formal accuracy i e exponential convergence in the proposed discretization several examples are presented to demonstrate the effectiveness of the new approach including a comparison with eddy viscosity spectral les of turbulent channel flow in its current implementation the svv approach for controlling the small scales is decoupled from the large scales but a procedure is proposed that will provide coupling similar to the classical les formulation

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

stable robin solid wall boundary conditions for the navier stokes equations in this paper we prove stability of robin solid wall boundary conditions for the compressible navier stokes equations applications include the no slip boundary conditions with prescribed temperature or temperature gradient and the first order slip flow boundary conditions the formulation is uniform and the transitions between different boundary conditions are done by a change of parameters we give different sharp energy estimates depending on the choice of parameters the discretization is done using finite differences on summation by parts form with weak boundary conditions using the simultaneous approximation term we verify convergence by the method of manufactured solutions and show computations of flows ranging from no slip to almost full slip

split form nodal discontinuous galerkin schemes with summation by parts property for the compressible euler equations fisher and carpenter 2013 12 found a remarkable equivalence of general diagonal norm high order summation by parts operators to a subcell based high order finite volume formulation this equivalence enables the construction of provably entropy stable schemes by a specific choice of the subcell finite volume flux we show that besides the construction of entropy stable high order schemes a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms thus by changing the subcell finite volume flux to a specific choice we are able to generate in a systematic way all common split forms of the compressible euler advection terms such as the ducros splitting and the kennedy and gruber splitting although these split forms are not entropy stable we present a systematic way to prove which of those split forms are at least kinetic energy preserving with this we construct a unified high order split form dg framework we investigate with three dimensional numerical simulations of the inviscid taylorgreen vortex and show that the new split forms enhance the robustness of high order simulations in comparison to the standard scheme when solving turbulent vortex dominated flows in fact we show that for certain test cases the novel split form discontinuous galerkin schemes are more robust than the discontinuous galerkin scheme with over integration

staggered upwind method for solving navier stokes and k omega turbulence model equations a staggered finite volume upwind algorithm for solving the compressible navier stokes equations and the k w turbulence model equations has been developed for computing cascade flows roe s upwind scheme is used to discretize the convective terms of the navier stokes equations and a third order upwind scheme is used for the convective terms of the k equations all of the diffusion terms are discretized with the central difference method by the use of a combination of cell centered and cell vertex schemes the method maintains a small stencil for all of the diffusion terms and makes the navier stokes equations and k equations strongly coupled the algorithm was first tested for a flat plate flow the results compare well with empirical correlations further investigations are conducted on a supersonic wedge cascade flow and a low pressure turbine cascade flow at design and off design conditions the calculation results demonstrate the ability of the algorithm on shock capturing and separation prediction

computation of oscillating airfoil flows with one and two equation turbulence models the ability of one and two quation turbulence models to predict unsteady separated flows over airfoils is evaluated an implicit factorized upwind biased numerical scheme is used for the integration of the compressible reynolds averaged navier stokes equations the turbulent eddy viscosity is obtained from the computed mean flowfield by integration of the turbulent field equations one and two equation turbulence models are first tested for a separated airfoil flow at fixed angle of incidence the same models are then applied to compute the unsteady flowfields about airfoils undergoing oscillatory motion at low subsonic mach numbers experimental cases where the flow has been tripped at the leading edge and where natural transition was allowed to occur naturally are considered the more recently developed turbulence models capture the physics of unsteady separated flow significantly better than the standard k e and k models however certain differences in the hysteresis effects are observed for an untripped high reynolds number flow it was found necessary to take into account the leading edge transitional flow region to capture the correct physical mechanism that leads to dynamic stall

on discretely entropy conservative and entropy stable discontinuous galerkin methods abstract high order methods based on diagonal norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic pdes 1 2 these methods can also be interpreted as nodal discontinuous galerkin methods with diagonal mass matrices 3 4 5 6 in this work we describe how use flux differencing quadrature based projections and sbp like operators to construct discretely entropy conservative schemes for dg methods under more arbitrary choices of volume and surface quadrature rules the resulting methods are semi discretely entropy conservative or entropy stable with respect to the volume quadrature rule used numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible euler equations in one and two dimensions

an adjoint view on flux consistency and strong wall boundary conditions to the navier stokes equations inconsistent discrete expressions in the boundary treatment of navier stokes solvers and in the definition of force objective functionals can lead to discrete adjoint boundary treatments that are not a valid representation of the boundary conditions to the corresponding adjoint partial differential equations the underlying problem is studied for an elementary 1d advection diffusion problem first using a node centred finite volume discretisation the defect of the boundary operators in the inconsistently defined discrete adjoint problem leads to oscillations and becomes evident with the additional insight of the continuous adjoint approach a homogenisation of the discretisations for the primal boundary treatment and the force objective functional yields second order functional accuracy and eliminates the defect in the discrete adjoint boundary treatment subsequently the issue is studied for aerodynamic reynolds averaged navier stokes problems in conjunction with a standard finite volume discretisation on median dual grids and a strong implementation of noslip walls found in many unstructured general purpose flow solvers going out from a base line discretisation of force objective functionals which is independent of the boundary treatment in the flow solver two improved flux consistent schemes are presented based on either body wall defined or farfield defined control volumes they resolve the dual inconsistency the behaviour of the schemes is investigated on a sequence of grids in 2d and 3d

an interior penalty stabilised incompressible discontinuous galerkin fourier solver for implicit large eddy simulations abstract we present an implicit large eddy simulation iles h p high order 2 unstructured discontinuous galerkin fourier solver with sliding meshes the solver extends the laminar version of ferrer and willden 2012 34 to enable the simulation of turbulent flows at moderately high reynolds numbers in the incompressible regime this solver allows accurate flow solutions of the laminar and turbulent 3d incompressible navier stokes equations on moving and static regions coupled through a high order sliding interface the spatial discretisation is provided by the symmetric interior penalty discontinuous galerkin ip dg method in the x y plane coupled with a purely spectral method that uses fourier series and allows efficient computation of spanwise periodic three dimensional flows since high order methods e g discontinuous galerkin and fourier are unable to provide enough numerical dissipation to enable under resolved high reynolds computations i e as necessary in the iles approach we adapt the laminar version of the solver to increase controllably the dissipation and enhance the stability in under resolved simulations the novel stabilisation relies on increasing the penalty parameter included in the dg interior penalty ip formulation the latter penalty term is included when discretising the linear viscous terms in the incompressible navier stokes equations these viscous penalty fluxes substitute the stabilising effect of non linear fluxes which has been the main trend in implicit les discontinuous galerkin approaches the ip dg penalty term provides energy dissipation which is controlled by the numerical jumps at element interfaces e g large in under resolved regions such as to stabilise under resolved high reynolds number flows this dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps thus providing an efficient stabilization mechanism for iles the ip dg stabilisation is complemented with a spectral vanishing viscosity svv method in the z direction to enhance stability in the continuous fourier space the coupling between the numerical viscosity in the dg plane and the svv damping provides an efficient approach to stabilise high order methods at moderately high reynolds numbers we validate the formulation for three turbulent flow cases a circular cylinder at re 3900 a static and pitch oscillating naca 0012 airfoil at re 10000 and finally a rotating vertical axis turbine at re 40000 with reynolds based on the circular diameter airfoil chord and turbine diameter respectively all our results compare favourably with published direct numerical simulations large eddy simulations or experimental data we conclude that the dg fourier high order solver with ip svv stabilisation proves to be a valuable tool to predict turbulent flows and associated statistics for both static and rotating machinery

stabilization mechanisms in discontinuous galerkin finite element methods in this paper we propose a new general framework for dg methods which allows to uncover a basic mechanism that ensures suitable stability properties of the methods we show that such a mechanism is essentially the same for apparently unrelated stabilizations such as jump penalty upwinding and hughes franca type residual based stabilizations

well posed boundary conditions for the navier stokes equations in this article we propose a general procedure that allows us to determine both the number and type of boundary conditions for time dependent partial differential equations with those well posedness can be proven for a general initial boundary value problem the procedure is exemplified on the linearized navier stokes equations in two and three space dimensions on a general domain

an artificial compressibility flux for the discontinuous galerkin solution of the incompressible navier stokes equations discontinuous galerkin dg methods have proved to be well suited for the construction of robust high order numerical schemes on unstructured and possibly nonconforming grids for a variety of problems their application to the incompressible navier stokes ins equations has also been recently considered although the subject is far from being fully explored in this work we propose a new approach for the dg numerical solution of the ins equations written in conservation form the inviscid numerical fluxes both in the continuity and in the momentum equation are computed using the values of velocity and pressure provided by the exact solution of the riemann problem associated with a local artificial compressibility perturbation of the equations unlike in most of the existing methods artificial compressibility is here introduced only at the interface flux level therefore resulting in a consistent discretization of the ins equations irrespectively of the amount of artificial compressibility introduced the discretization of the viscous term follows the well established dg scheme named br2 the performance and the accuracy of the method are demonstrated by computing the kovasznay flow and the two dimensional lid driven cavity flow for a wide range of reynolds numbers and for various degrees of polynomial approximation

a spalart allmaras turbulence model implementation in a discontinuous galerkin solver for incompressible flows in this paper the artificial compressibility flux discontinuous galerkin dg method for the solution of the incompressible navier stokes equations has been extended to deal with the reynolds averaged navier stokes rans equations coupled with the spalart allmaras sa turbulence model dg implementations of the rans and sa equations for compressible flows have already been reported in the literature including the description of limiting or stabilization techniques adopted in order to prevent the turbulent viscosity email protected from becoming negative in this paper we introduce an sa model implementation that deals with negative email protected values by modifying the source and diffusion terms in the sa model equation only when the working variable or one of the model closure functions become negative this results in an efficient high order implementation where either stabilization terms or even additional equations are avoided we remark that the proposed implementation is not dg specific and it is well suited for any numerical discretization of the rans sa governing equations the reliability robustness and accuracy of the proposed implementation have been assessed by computing several high reynolds number turbulent test cases the flow over a flat plate re 10 7 the flow past a backward facing step re 37400 and the flow around a naca 0012 airfoil at different angles of attack a 0 o 10 o 15 o and reynolds numbers re 2 88x10 6 6x10 6

entropy stable reduced order modeling of nonlinear conservation laws abstract reduced order models of nonlinear conservation laws in fluid dynamics do not typically inherit stability properties of the full order model we introduce projection based hyper reduced models of nonlinear conservation laws which are globally conservative and inherit a semi discrete entropy inequality independently of the choice of basis and choice of parameters

short note an explicit expression for the penalty parameter of the interior penalty method we derive an explicit expression for the penalty parameter of the interior penalty method for elliptic problems the expression yields a coercive bilinear form and is valid for general meshes comprising of geometrically nonconforming simplical elements

skew selfadjoint form for systems of conservation laws abstract hyperbolic systems of conservation laws augumented with an entropy inequality are studied it is shown that such systems can be written in a quasilinear skew selfadjoint form centered differencing of such a form under the smooth regime ends up with a systematic recipe for constructing quasiconservative schemes where the global entropy conservation is recovered employing the above formulation in bounded regions under the nonsmooth regime as well a local entropy decay estimate is also concluded examples of the shallow water and the full gasdynamics equations are explicitly treated

grid adaptation for functional outputs application to two dimensional inviscid flows this paper presents an error estimation and grid adaptive strategy for estimating and reducing simulation errors in integral outputs functionals of partial differential equations adaptive criteria are derived using an adjoint based error correction technique that relates the local residual errors of both the primal and adjoint solutions to the global error in the prescribed functional this relationship allows local error contributions to be used as indicators in a grid adaptive strategy designed to produce specially tuned grids for accurately estimating the chosen functional in this paper attention is limited to two dimensional inviscid flows using a standard finite volume discretization although the procedure may be readily applied to other types of multidimensional problems and discretizations numerical examples demonstrate that the proposed adaptive procedure compares favorably in terms of accuracy efficiency and robustness relative to a commonly used gradient based adaptive method

staggered finite volume scheme for solving cascade flow with a k omega turbulence model a k turbulence model by wilcox is applied to cascade flow calculations a staggered finite volume scheme is proposed to solve the compressible reynolds averaged navier stokes equations and the k equations are solved on staggered control volumes with k and defined at the cell vertices of the original grid by use of this combination of the cell centered and cell vertex schemes the method maintains a small stencil for all diffusion terms and closely couples the navier stokes and k equations

linear dispersion diffusion analysis and its application to under resolved turbulence simulations using discontinuous galerkin spectral hp methods we investigate the potential of linear dispersion diffusion analysis in providing direct guidelines for turbulence simulations through the under resolved dns sometimes called implicit les approach via spectral hp methods the discontinuous galerkin dg formulation is assessed in particular as a representative of these methods we revisit the eigensolutions technique as applied to linear advection and suggest a new perspective to the role of multiple numerical modes peculiar to spectral hp methods from this new perspective secondary eigenmodes are seen to replicate the propagation behaviour of a primary mode so that dg s propagation characteristics can be obtained directly from the dispersion diffusion curves of the primary mode numerical dissipation is then appraised from these primary eigencurves and its effect over poorly resolved scales is quantified within this scenario a simple criterion is proposed to estimate dg s effective resolution in terms of the largest wavenumber it can accurately resolve in a given hp approximation space also allowing us to present points per wavelength estimates typically used in spectral and finite difference methods although strictly valid for linear advection the devised criterion is tested against 1d burgers turbulence and found to predict with good accuracy the beginning of the dissipation range on the energy spectra of under resolved simulations the analysis of these test cases through the proposed methodology clarifies why and how the dg formulation can be used for under resolved turbulence simulations without explicit subgrid scale modelling in particular when dealing with communication limited hardware which forces one to consider the performance for a fixed number of degrees of freedom the use of higher polynomial orders along with moderately coarser meshes is shown to be the best way to translate available degrees of freedom into resolution power

wall normal free reynolds stress model for rotating flows applied to turbomachinery a recently developed near wall wall normal free reynolds stress model is extended and applied to the computation of transonic three dimensional e ows in turbomachinery rotors with tip clearance the reynolds stress model used is completely independent of the distance from the wall and of the normal to the wall direction this is achieved by using a pseudonormal direction dee ned by the gradients of turbulence length scale and anisotropy tensor invariants in the pressure strain echo terms the effects of rotation are included through theexact coriolis termsin the reynolds stress transport equations and through the isotropization of absolute e ow production model used for the rapid pressure strain terms the model is initially validated by comparison with measurements in rotating fully developed plane channel e ows computational results obtained with the proposed reynolds stress model are compared with measurements and with k computations for various operating points of the nasa 37 rotor using a 3 10 6 multiblock computational grid with 41 radial stations within the tip clearance gap the proposed model substantially improves the agreement with measurements compared to existing algebraic and two equation models

eigensolution analysis of spectral hp continuous galerkin approximations to advection diffusion problems this study addresses linear dispersion diffusion analysis for the spectral hp continuous galerkin cg formulation in one dimension first numerical dispersion and diffusion curves are obtained for the advection diffusion problem and the role of multiple eigencurves peculiar to spectral hp methods is discussed from the eigencurves behaviour we observe that cg might feature potentially undesirable non smooth dispersion diffusion characteristics for under resolved simulations of problems strongly dominated by either convection or diffusion subsequently the linear advection equation augmented with spectral vanishing viscosity svv is analysed dispersion and diffusion characteristics of cg with svv based stabilization are verified to display similar non smooth features in flow regions where convection is much stronger than dissipation or vice versa owing to a dependency of the standard svv operator on a local peclet number first a modification is proposed to the traditional svv scaling that enforces a globally constant peclet number so as to avoid the previous issues in addition a new svv kernel function is suggested and shown to provide a more regular behaviour for the eigencurves along with a consistent increase in resolution power for higher order discretizations as measured by the extent of the wavenumber range where numerical errors are negligible the dissipation characteristics of cg with the svv modifications suggested are then verified to be broadly equivalent to those obtained through upwinding in the discontinuous galerkin dg scheme nevertheless for the kernel function proposed the full upwind dg scheme is found to have a slightly higher resolution power for the same dissipation levels these results show that improved cg svv characteristics can be pursued via different kernel functions with the aid of optimization algorithms

positivity preservation and adaptive solution for the k model of turbulence a simple change of dependent variables that guarantees positivity of turbulence variables in numerical simulation codes is presented the approach consists of solving for the natural logarithm of the turbulence variables which are known to be strictly positive the approach is valid for any numerical scheme be it finite difference a finite volume or a finite element method the work focuses on the advantages of the proposed change of dependent variables within the framework of an adaptive finite element method the turbulence equations in logarithmic variables are presented for the standard e model error estimation and mesh adaptation procedures are described the formulation is validated on a shear layer case for which an analytical solution is available this provides a framework for rigorous comparison of the proposed approach with the standard solution technique which makes use of k and e as dependent variables the approach is then applied to solve turbulent flow over a naca0012 airfoil for which experimental measurements are available the proposed procedure results in a robust adaptive algorithm improved predictions of turbulence variables are obtained using the proposed formulation

a spectral property of the electrostatic integral operator for an ellipsoid the eigenfunctions of the electrostatic integral operator and its adjoint are governed by the lame polynomials it is shown that the arithmetic mean of the eigenvalues corresponding to the maximum number of linearly independent eigenfunctions of equal order is the same for the ellipsoid as for the sphere

an optimal order interior penalty discontinuous galerkin discretization of the compressible navier stokes equations in this article we propose a new symmetric version of the interior penalty discontinuous galerkin finite element method for the numerical approximation of the compressible navier stokes equations here particular emphasis is devoted to the construction of an optimal numerical method for the evaluation of certain target functionals of practical interest such as the lift and drag coefficients of a body immersed in a viscous fluid with this in mind the key ingredients in the construction of the method include i an adjoint consistent imposition of the boundary conditions ii an adjoint consistent reformulation of the underlying target functional of practical interest iii design of appropriate interior penalty stabilization terms numerical experiments presented within this article clearly indicate the optimality of the proposed method when the error is measured in terms of both the l 2 norm as well as for certain target functionals computational comparisons with other discontinuous galerkin schemes proposed in the literature including the second scheme of bassi and rebay cf f bassi s rebay gmres discontinuous galerkin solution of the compressible navier stokes equations in b cockburn g karniadakis c w shu eds discontinuous galerkin methods lecture notes in comput sci engrg vol 11 springer berlin 2000 pp 197 208 f bassi s rebay numerical evaluation of two discontinuous galerkin methods for the compressible navier stokes equations int j numer methods fluids 40 2002 197 207 the standard sipg method outlined in r hartmann p houston symmetric interior penalty dg methods for the compressible navier stokes equations i method formulation int j numer anal model 3 1 2006 1 20 and an nipg variant of the new scheme will be undertaken

simultaneous approximation term based boundary discretization for moment equations of rarefied gas dynamics abstract previous works have developed boundary conditions that lead to the l 2 boundedness of solutions to the linearised moment equations here we present a spatial discretization that preserves the l 2 stability by recovering integration by parts over the discretized domain and by imposing boundary conditions using a simultaneous approximation term sat we develop three different forms of the sat using i characteristic splitting of moment equation s boundary conditions ii decoupling of moments in moment equations and iii characteristic splitting of boltzmann equation s boundary conditions we discuss how the first two forms differ in terms of their usage and implementation we show that the third form is equivalent to using an upwind kinetic numerical flux along the boundary and we argue that even though it provides stability it prescribes the incorrect number of boundary conditions using benchmark problems we compare the accuracy of moment solutions computed using different sats our numerical experiments also provide new insights into the convergence of moment approximations to the boltzmann equation s solution

velocity correction projection methods for incompressible flows we introduce and study a new class of projection methods namely the velocity correction methods in standard form and in rotational form for solving the unsteady incompressible navier stokes equations we show that the rotational form provides improved error estimates in terms of the h1 norm for the velocity and of the l2 norm for the pressure we also show that the class of fractional step methods introduced in s a orsag m israeli and m deville j sci comput 1 1986 pp 75 111 and k e karniadakis m israeli and s a orsag j comput phys 97 1991 pp 414 443 can be interpreted as the rotational form of our velocity correction methods thus to the best of our knowledge our results provide the first rigorous proof of stability and convergence of the methods in those papers we also emphasize that contrary to those of the above groups our formulations are set in the standard l2 setting and consequently they can be easily implemented by means of any variational approximation techniques in particular the finite element methods

unconditional stability and long term behavior of transient algorithms for the incompressible navier stokes and euler equations abstract this paper examines the long term behavior dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible navier stokes equations including both direct schemes and fractional step projection methods the algorithms are termed nonlinear unconditionally stable if in the absence of a forcing term but for arbitrary initial conditions the computed kinetic energy decreases for arbitrary step sizes i e l 2 stability such an a priori stability estimate is a characteristic feature of the navier stokes system similarly the algorithms are said to exhibit asymptotic t independent long term dissipative behavior if as the continuum flow generated by the navier stokes equations the algorithmic flow also exhibits an absorbing set and a maximal attractor both direct and fractional step methods are described which inherit these fundamental properties of the navier stokes system for any time step size moreover a subclass of algorithms which retain these strong notions of nonlinear stability and long term dissipative behavior is identified which in addition has the remarkable property of being linear within the time step the key implication is that no iterations are required to compute the solution within a time step a specific member of this class is shown to possess the same absorbing set as the continuum navier stokes system a second order accurate unconditionally stable method is also identified which retains the property of linearity within a time step numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations

performance of popular turbulence model for attached and separated adverse pressure gradient flows the performance of four popular eddy viscosity turbulence models under adverse pressure gradient conditions is investigated the baldwin lomax the johnson king the baldwin barth and the wilcox omega models have been implemented into the ins code which solves the incompressible reynolds averaged navier stokes equations results are shown for the well known samuel joubert flow and two new flowfields recently reported by d m driver 1991 the two new flowfields pose a stronger test of the models than the samuel joubert flow because of the more severe retardation of the boundary layer including separation in one case a detailed comparison of the numerical results and the experimental data is shown

corner corrected diagonal norm summation by parts operators for the first derivative with increased order of accuracy combined with simultaneous approximation terms summation by parts sbp operators offer a versatile and efficient methodology that leads to consistent conservative and provably stable discretizations however diagonal norm operators with a repeating interior point operator that have thus far been constructed suffer from a loss of accuracy while on the interior these operators are of degree 2p at a number of nodes near the boundaries they are of degree p and therefore of global degree p meaning the highest degree monomial for which the operators are exact at all nodes this implies that for hyperbolic problems and operators of degree greater than unity they lead to solutions with a global order of accuracy lower than the degree of the interior point operator in this paper we develop a procedure to construct diagonal norm first derivative sbp operators that are of degree 2p at all nodes and therefore can lead to solutions of hyperbolic problems of order 2 p 1 this is accomplished by adding nonzero entries in the upper right and lower left corners of sbp operator matrices with a repeating interior point operator this modification necessitates treating these new operators as elements where mesh refinement is accomplished by increasing the number of elements in the mesh rather than increasing the number of nodes the significant improvements in accuracy of this new family for the same repeating interior point operator are demonstrated in the context of the linear convection equation

entropy stable discontinuous interfaces coupling for the three dimensional compressible navier stokes equations non linear entropy stability and a summation by parts sbp framework are used to derive entropy stable interior interface coupling for the semi discretized three dimensional 3d compressible navier stokes equations a complete semidiscrete entropy estimate for the interior domain is achieved combining a discontinuous entropy conservative operator of any order 1 2 with an entropy stable coupling condition for the inviscid terms and a local discontinuous galerkin ldg approach with an interior penalty ip procedure for the viscous terms the viscous penalty contributions scale with the inverse of the reynolds number re so that for re their contributions vanish and only the entropy stable inviscid interface penalty term is recovered this paper extends the interface couplings presented 1 2 and provides a simple and automatic way to compute the magnitude of the viscous ip term the approach presented herein is compatible with any diagonal norm summation by parts sbp spatial operator including finite element finite volume finite difference schemes and the class of high order accurate methods which include the large family of discontinuous galerkin discretizations and flux reconstruction schemes this note relies on the formalism introduced in 1 3 and complements the new class of interior entropy stable sbp operators of any order for the 3d compressible navier stokes equations on unstructured grids that was proposed in 1 2 to keep the notation as simple as possible a uniform cartesian grid is considered in the derivation however the extension to generalized curvilinear coordinates and unstructured grids follows immediately if the transformation from computational to physical space preserves the semi discrete geometric conservation 4 the proposed interface coupling technique has been successfully combined with a high order entropy stable discretization for the simulation of two dimensional 2d and 3d viscous subsonic and supersonic flows presented in 1 5 6

a high order discontinuous galerkin method for the unsteady incompressible navier stokes equations we present a high order discontinuous galerkin discretization of the unsteady incompressible navier stokes equations in convection dominated flows using triangular and tetrahedral meshes the scheme is based on a semi explicit temporal discretization with explicit treatment of the nonlinear term and implicit treatment of the stokes operator the nonlinear term is discretized in divergence form by using the local lax friedrichs fluxes thus local conservativity is inherent spatial discretization of the stokes operator has employed both equal order p k p k and mixed order p k p k 1 velocity and pressure approximations a second order approximate algebraic splitting is used to decouple the velocity and pressure calculations leading to an algebraic helmholtz equation for each component of the velocity and a consistent poisson equation for the pressure the consistent poisson operator is replaced by an equivalent in stability and convergence operator namely that arising from the interior penalty discretization of the standard poisson operator with appropriate boundary conditions this yields a simpler and more efficient method characterized by a compact stencil size we show the temporal and spatial behavior of the method by solving some popular benchmarking tests for an unsteady stokes problem second order temporal convergence is obtained while for the taylor vortex test problem on both semi structured and fully unstructured triangular meshes spectral convergence with respect to the polynomial degree k is obtained by studying the orr sommerfeld stability problem we demonstrate that the p k p k method yields a stable solution while the p k p k 1 formulation leads to unphysical instability the good performance of the method is further shown by simulating vortex shedding in flow past a square cylinder we conclude that the proposed discontinuous galerkin method with the p k p k formulation is an efficient scheme for accurate and stable solution of the unsteady navier stokes equations in convection dominated flows

estimation of penalty parameters for symmetric interior penalty galerkin methods this paper presents computable lower bounds of the penalty parameters for stable and convergent symmetric interior penalty galerkin methods in particular we derive the explicit dependence of the coercivity constants with respect to the polynomial degree and the angles of the mesh elements numerical examples in all dimensions and for different polynomial degrees are presented we show the numerical effects of loss of coercivity

adaptation strategies for high order discontinuous galerkin methods based on tau estimation in this paper three p adaptation strategies based on the minimization of the truncation error are presented for high order discontinuous galerkin methods the truncation error is approximated by means of a estimation procedure and enables the identification of mesh regions that require adaptation three adaptation strategies are developed and termed a posteriori quasi a priori and quasi a priori corrected all strategies require fine solutions which are obtained by enriching the polynomial order but while the former needs time converged solutions the last two rely on non converged solutions which lead to faster computations in addition the high order method permits the spatial decoupling for the estimated errors and enables anisotropic p adaptation these strategies are verified and compared in terms of accuracy and computational cost for the euler and the compressible navier stokes equations it is shown that the two quasi a priori methods achieve a significant reduction in computational cost when compared to a uniform polynomial enrichment namely for a viscous boundary layer flow we obtain a speedup of 6 6 and 7 6 for the quasi a priori and quasi a priori corrected approaches respectively

fully discrete explicit locally entropy stable schemes for the compressible euler and navier stokes equations abstract recently relaxation methods have been developed to guarantee the preservation of a single global functional of the solution of an ordinary differential equation here we generalize this approach to guarantee local entropy inequalities for finitely many convex functionals entropies and apply the resulting methods to the compressible euler and navier stokes equations based on the unstructured h p adaptive ssdc framework of entropy conservative or dissipative semidiscretizations using summation by parts and simultaneous approximation term operators we develop the first discretizations for compressible computational fluid dynamics that are primary conservative locally entropy stable in the fully discrete sense under a usual cfl condition explicit except for the parallelizable solution of a single scalar equation per element and arbitrarily high order accurate in space and time we demonstrate the accuracy and the robustness of the fully discrete explicit locally entropy stable solver for a set of test cases of increasing complexity

a high order semi explicit discontinuous galerkin solver for 3d incompressible flow with application to dns and les of turbulent channel flow abstract we present an efficient discontinuous galerkin scheme for simulation of the incompressible navier stokes equations including laminar and turbulent flow we consider a semi explicit high order velocity correction method for time integration as well as nodal equal order discretizations for velocity and pressure the non linear convective term is treated explicitly while a linear system is solved for the pressure poisson equation and the viscous term the key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under resolved high reynolds number flows as well as small time steps this penalty method is similar to the grad div stabilization widely used in continuous finite elements we further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations the solver is specifically designed for large scale computations through matrix free linear solvers including efficient preconditioning strategies and tensor product elements which have allowed us to scale this code up to 34 4 billion degrees of freedom and 147 456 cpu cores we validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large eddy simulation of turbulent channel flow at r e 180 as well as 590

adaptive discontinuous galerkin finite element methods for the compressible euler equations in this paper a recently developed approach for the design of adaptive discontinuous galerkin finite element methods is applied to physically relevant problems arising in inviscid compressible fluid flows governed by the euler equations of gas dynamics in particular we employ weighted type i a posteriori bounds to drive adaptive finite element algorithms for the estimation of the error measured in terms of general target functionals of the solution these error estimates involve the product of the finite element residuals with local weighting terms involving the solution of a certain dual problem that must be numerically approximated this general approach leads to the design of economical finite element meshes specifically tailored to the computation of the target functional of interest as well as providing reliable and efficient error estimation the superiority of the proposed approach over mesh refinement algorithms which employ standard unweighted type ii error indicators which do not require the solution of an auxiliary problem are illustrated by a series of numerical experiments here we consider transonic flow through a nozzle as well as subsonic and supersonic flows around different airfoil geometries

adjoint based h p adaptive discontinuous galerkin methods for the 2d compressible euler equations in this paper we investigate and present an adaptive discontinuous galerkin algorithm driven by an adjoint based error estimation technique for the inviscid compressible euler equations this approach requires the numerical approximations for the flow i e primal problem and the adjoint i e dual problem which corresponds to a particular simulation objective output of interest the convergence of these two problems is accelerated by an hp multigrid solver which makes use of an element gauss seidel smoother on each level of the multigrid sequence the error estimation of the output functional results in a spatial error distribution which is used to drive an adaptive refinement strategy which may include local mesh subdivision h refinement local modification of discretization orders p enrichment and the combination of both approaches known as hp refinement the selection between h and p refinement in the hp adaptation approach is made based on a smoothness indicator applied to the most recently available flow solution values numerical results for the inviscid compressible flow over an idealized four element airfoil geometry demonstrate that both pure h refinement and pure p enrichment algorithms achieve equivalent error reductions at each adaptation cycle compared to a uniform refinement approach but requiring fewer degrees of freedom the proposed hp adaptive refinement strategy is capable of obtaining exponential error convergence in terms of degrees of freedom and results in significant savings in computational cost a high speed flow test case is used to demonstrate the ability of the hp refinement approach for capturing strong shocks or discontinuities while improving functional accuracy

convergence of spectral methods for nonlinear conservation laws we discuss the convergence of fourier methods for scalar nonlinear conservation laws that exhibit spontaneous shock discontinuities numerical tests indicate that the convergence may and in fact in some cases we prove it must fail with or without post processing of the numerical solution instead we introduce here a new kind of spectrally accurate vanishing viscosity to augment the fourier approximation of such nonlinear conservation laws using compensated compactness arguments augmented by the assumption of l infty stability we show for the inviscid burgers model equation that this spectral viscosity method prevents oscillations and convergence to the unique entropy solution follows

weak solutions and convergent numerical schemes of modified compressible navier stokes equations lately there has been some interest in modifications of the compressible navier stokes equations to include diffusion of mass in this paper we investigate possible ways to add mass diffusion to the 1 d navier stokes equations without violating the basic entropy inequality as a result we recover brenner s modification of the navier stokes equations as a special case we consider brenner s system along with another modification where the viscous terms collapse to a laplacian diffusion for each of the two modifications we derive a priori estimates for the pde sufficiently strong to admit a weak solution we propose a numerical scheme and demonstrate that it satisfies the same a priori estimates for both modifications we then demonstrate that the numerical schemes generate solutions that converge to a weak solution up to a subsequence as the grid is refined