robust and efficient discontinuous galerkin methods for under resolved turbulent incompressible flows abstract we present a robust and accurate discretization approach for incompressible turbulent flows based on high order discontinuous galerkin methods the dg discretization of the incompressible navier stokes equations uses the local lax friedrichs flux for the convective term the symmetric interior penalty method for the viscous term and central fluxes for the velocity pressure coupling terms stability of the discretization approach for under resolved turbulent flow problems is realized by a purely numerical stabilization approach consistent penalty terms that enforce the incompressibility constraint as well as inter element continuity of the velocity field in a weak sense render the numerical method a robust discretization scheme in the under resolved regime the penalty parameters are derived by means of dimensional analysis using penalty factors of order 1 applying these penalty terms in a postprocessing step leads to an efficient solution algorithm for turbulent flows the proposed approach is applicable independently of the solution strategy used to solve the incompressible navier stokes equations i e it can be used for both projection type solution methods as well as monolithic solution approaches since our approach is based on consistent penalty terms it is by definition generic and provides optimal rates of convergence when applied to laminar flow problems robustness and accuracy are verified for the orr sommerfeld stability problem the taylor green vortex problem and turbulent channel flow moreover the accuracy of high order discretizations as compared to low order discretizations is investigated for these flow problems a comparison to state of the art computational approaches for large eddy simulation indicates that the proposed methods are highly attractive components for turbulent flow solvers

constrained lqr problems in elliptic distributed control systems with point observations in this paper we study bound constrained lqr problems in distributed control systems governed by the elliptic equation with point observations which are motivated by problems in corrosion engineering and contemporary smart materials several regularity and characterization theorems have been established in particular three decomposition formulas are obtained to characterize the optimal control and optimal layer density and are used to direct the numerical computations these results cannot be obtained by the traditional galerkin variational method in the process several useful lemmas are established which are of independent interest we point out that the classical lagrangian multiplier method lmm may fail to provide a reliable numerical algorithm based on our characterization results and the boundary element method two algorithms are proposed to carry out numerical computations it has been shown by our numerical experiments that both algorithms are efficient and insensitive to the partition number of the boundary an adaptive local refinement scheme has also been designed to handle the rough behavior of the optimal solution around sensor locations

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

an overlapping schwarz method for spectral element solution of the incompressible navier stokes equations efficient solution of the navier stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems for unsteady incompressible flows the pressure operator is the leading contributor to stiffness as the characteristic propagation speed is infinite in the context of operator splitting formulations it is the pressure solve which is the most computationally challenging despite its elliptic origins we examine several preconditioners for the consistent l2poisson operator arising in the pn pn 2spectral element formulation of the incompressible navier stokes equations we develop a finite element based additive schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior gauss points for large two dimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation

on the error estimates for the rotational pressure correction projection methods in this paper we study the rotational form of the pressure correction method that was proposed by timmermans minev and van de vosse we show that the rotational form of the algorithm provides better accuracy in terms of the h1 norm of the velocity and of the l2 norm of the pressure than the standard form

on the eddy resolving capability of high order discontinuous galerkin approaches to implicit les under resolved dns of euler turbulence we present estimates of spectral resolution power for under resolved turbulent euler flows obtained with high order discontinuous galerkin dg methods the 1 rule based on linear dispersion diffusion analysis introduced by moura et al 2015 10 is here adapted for 3d energy spectra and validated through the inviscid taylor green vortex problem the 1 rule estimates the wavenumber beyond which numerical diffusion induces an artificial dissipation range on measured energy spectra as the original rule relies on standard upwinding different riemann solvers are tested very good agreement is found for solvers which treat the different physical waves in a consistent manner relatively good agreement is still found for simpler solvers the latter however displayed spurious features attributed to the inconsistent treatment of different physical waves it is argued that in the limit of vanishing viscosity such features might have a significant impact on robustness and solution quality the estimates proposed are regarded as useful guidelines for no model dg based simulations of free turbulence at very high reynolds numbers

multifractal subgrid scale modeling within a variational multiscale method for large eddy simulation of turbulent flow multifractal subgrid scale modeling within a variational multiscale method is proposed for large eddy simulation of turbulent flow in the multifractal subgrid scale modeling approach the subgrid scale velocity is evaluated from a multifractal description of the subgrid scale vorticity which is based on the multifractal scale similarity of gradient fields in turbulent flow the multifractal subgrid scale modeling approach is integrated into a variational multiscale formulation which constitutes a new application of the variational multiscale concept a focus of this study is on the application of the multifractal subgrid scale modeling approach to wall bounded turbulent flow therefore a near wall limit of the multifractal subgrid scale modeling approach is derived in this work the novel computational approach of multifractal subgrid scale modeling within a variational multiscale formulation is applied to turbulent channel flow at various reynolds numbers turbulent flow over a backward facing step and turbulent flow past a square section cylinder which are three of the most important and widely used benchmark examples for wall bounded turbulent flow all results presented in this study confirm a very good performance of the proposed method compared to a dynamic smagorinsky model and a residual based variational multiscale method improved results are obtained moreover it is demonstrated that the subgrid scale energy transfer incorporated by the proposed method very well approximates the expected energy transfer as obtained from appropriately filtered direct numerical simulation data the computational cost is notably reduced compared to a dynamic smagorinsky model and only marginally increased compared to a residual based variational multiscale method

on the use of kinetic energy preserving dg schemes for large eddy simulation abstract recently element based high order methods such as discontinuous galerkin dg methods and the closely related flux reconstruction fr schemes have become popular for compressible large eddy simulation les element based high order methods with riemann solver based interface numerical flux functions offer an interesting dispersion dissipation behavior for multi scale problems dispersion errors are very low for a broad range of scales while dissipation errors are very low for well resolved scales and are very high for scales close to the nyquist cutoff in some sense the inherent numerical dissipation caused by the interface riemann solver acts as a filter of high frequency solution components this observation motivates the trend that element based high order methods with riemann solvers are used without an explicit les model added only the high frequency type inherent dissipation caused by the riemann solver at the element interfaces is used to account for the missing sub grid scale dissipation due to under resolution of vortical dominated structures typical for les type setups element based high order methods suffer from stability issues caused by aliasing errors of the non linear flux terms a very common strategy to fight these aliasing issues and instabilities is so called polynomial de aliasing where interpolation is exchanged with projection based on an increased number of quadrature points in this paper we start with this common no model or implicit les iles dg approach with polynomial de aliasing and riemann solver dissipation and review its capabilities and limitations we find that the strategy gives excellent results but only when the resolution is such that about 40 of the dissipation is resolved for more realistic coarser resolutions used in classical les e g of industrial applications the iles dg strategy becomes quite inaccurate we show that there is no obvious fix to this strategy as adding for instance a sub grid scale models on top doesn t change much or in worst case decreases the fidelity even more finally the core of this work is a novel les strategy based on split form dg methods that are kinetic energy preserving the scheme offers excellent stability with full control over the amount and shape of the added artificial dissipation this premise is the main idea of the work and we will assess the les capabilities of the novel split form dg approach when applied to shock free moderate mach number turbulence we will demonstrate that the novel dg les strategy offers similar accuracy as the iles methodology for well resolved cases but strongly increases fidelity in case of more realistic coarse resolutions

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

turbulence representations and trace preserving actions we establish criteria for turbulence in certain spaces of c algebra repre sentations and apply this to the problem of nonclassiability by countable structures for group actions on a standard atomless probability space x and on the hypernite ii1 factor r we also prove that the conjugacy action on the space of free actions of a countably innite amenable group on r is turbulent and that the conjugacy action on the space of ergodic measure preserving ows on x is generically turbulent

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

on the stability of projection methods for the incompressible navier stokes equations based on high order discontinuous galerkin discretizations abstract the present paper deals with the numerical solution of the incompressible navier stokes equations using high order discontinuous galerkin dg methods for discretization in space for dg methods applied to the dual splitting projection method instabilities have recently been reported that occur for small time step sizes since the critical time step size depends on the viscosity and the spatial resolution these instabilities limit the robustness of the navier stokes solver in case of complex engineering applications characterized by coarse spatial resolutions and small viscosities by means of numerical investigation we give evidence that these instabilities are related to the discontinuous galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high order accuracy with respect to the temporal discretization this new formulation is stable in the limit of small time steps for both equal order and mixed order polynomial approximations although the dual splitting scheme itself includes inf sup stabilizing contributions we demonstrate that spurious pressure oscillations appear for equal order polynomials and small time steps highlighting the necessity to consider inf sup stability explicitly

a comparison of the dispersion and dissipation errors of gauss and gauss lobatto discontinuous galerkin spectral element methods we examine the dispersion and dissipation properties of the gauss and gauss lobatto discontinuous galerkin spectral element methods dgsems for linear wave propagation problems we show that the inherent underintegration in the gauss lobatto variant can be interpreted as a modal filtering of the highest polynomial mode this in turn has a drastic impact on the dispersion and dissipation relations of the gauss lobatto dgsem compared to the gauss variant we show that the gauss dgsem is typically more accurate than the gauss lobatto variant needing fewer points per wavelength for a given accuracy while on the other hand being more restricted in the explicit time step choice we show that the spectra of the dgsem operators depend on the boundary conditions applied and that the ratio of the time step restrictions of the two schemes depends on the choice of boundary conditions

computational error analysis of a discontinuous galerkin discretization applied to large eddy simulation of homogeneous turbulence a computational error assessment of large eddy simulation les in combination with a discontinuous galerkin finite element method is presented for homogeneous isotropic decaying turbulence the error landscape database approach is used to quantify the total simulation error that arises from the use of the smagorinsky eddy viscosity model in combination with the galerkin discretization we adopt a modified hllc flux allowing an explicit control over the dissipative component of the numerical flux the optimal dependence of the smagorinsky parameter on the spatial resolution is determined for second and third order accurate galerkin methods in particular the role of the numerical dissipation relative to the contribution from the smagorinsky dissipation is investigated we observed an exchange of dissipation principle in the sense that an increased numerical dissipation implied a reduction in the optimal smagorinsky parameter the predictions based on galerkin discretization with fully stabilized hllc flux were found to be less accurate than when a central discretization with mainly smagorinsky dissipation was used this was observed for both the second and third order galerkin discretization suggesting to emphasize central discretization of the convective nonlinearity and stabilization that mimics eddy viscosity as sub filter dissipation

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

grad div stabilization and subgrid pressure models for the incompressible navier stokes equations in this paper the grad div stabilization for the incompressible navier stokes finite element approximations is considered from two different viewpoints i as a variational multiscale approach for the pressure subgrid modeling and ii as a stabilization procedure of least square type some new error estimates for the linearized problem with the grad div stabilization are proved with the help of norms induced by the pressure schur complement operator we discuss the stabilization parameter choice arising in the frameworks of least square and multiscale methods and consider assumptions which allow to relate both approaches

singular maximal abelian subalgebras in continuous von neumann algebras let m be a von neumann algebra a maximal abelian subalgebra a in m is singular if the only unitaries in m that normalize a are those in a this notion was first considered by dixmier who gave an example of a singular maximal abelian subalgebra in the separable hypertinite ii factor see 5 later pukanszky lo and tauer 141 gave other examples also in the hyperfinite ii factor in 111 pukanszky showed that some of his type iii factors have singular maximal abelian subalgebras in 151 takesaki introduced a new type of maximal abelian subalgebras that he called simple and showed that they are singular he gave examples of such algebras in the hyperlinite ii factor in 161 nielsen showed that every continuous iptfi factor has simple and thus singular maximal abelian subalgebras note that if m is a separable discrete factor i e m 9 s for some separable hilbert space r and a c s r is maximal abelian then a is the direct sum of an atomic algebra and a diffuse algebra the atomic part is clearly not singular and by a theorem of von neumann the diffuse part is also nonsingular actually in both cases the normalizer generates all 9 z thus discrete factors do not have singular maximal abelian subalgebras in this article we prove that all separable continuous semifinite von neumann algebras and all separable factors of type iii 0 q 1 1 have singular maximal abelian subalgebras in the semifmite case our result is in fact more general if 5 c aut m is a group of automorphisms of m containing the inner automorphisms s 2 int m and with b int m countable then there exists a maximal abelian subalgebra a in m such that the only automorphisms in g that normalize a are the inner automorphisms 151 0022 1236183 3 00

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

on reference solutions and the sensitivity of the 2d kelvin helmholtz instability problem abstract two dimensional kelvin helmholtz instability problems are popular examples for assessing discretizations for incompressible flows at high reynolds number unfortunately the results in the literature differ considerably this paper presents computational studies of a kelvin helmholtz instability problem with high order divergence free finite element methods reference results in several quantities of interest are obtained for three different reynolds numbers up to the beginning of the final vortex pairing a mesh independent prediction of the final pairing is not achieved due to the sensitivity of the considered problem with respect to small perturbations a theoretical explanation of this sensitivity to small perturbations is provided based on the theory of self organization of 2d turbulence possible sources of perturbations that arise in almost any numerical simulation are discussed

a stable and high order accurate discontinuous galerkin based splitting method for the incompressible navier stokes equations abstract in this paper we consider discontinuous galerkin dg methods for the incompressible navier stokes equations in the framework of projection methods in particular we employ symmetric interior penalty dg methods within the second order rotational incremental pressure correction scheme the major focus of the paper is threefold i we propose a modified upwind scheme based on the vijayasundaram numerical flux that has favourable properties in the context of dg ii we present a novel postprocessing technique in the helmholtz projection step based on h div reconstruction of the pressure correction that is computed locally is a projection in the discrete setting and ensures that the projected velocity satisfies the discrete continuity equation exactly as a consequence it also provides local mass conservation of the projected velocity iii numerical results demonstrate the properties of the scheme for different polynomial degrees applied to two dimensional problems with known solution as well as large scale three dimensional problems in particular we address second order convergence in time of the splitting scheme as well as its long time stability

short note a short note on the discontinuous galerkin discretization of the pressure projection operator in incompressible flow this note reports on the issue of spurious compressibility artifacts that can arise when the popular pressure projection pp method is used for unsteady simulations of incompressible flow using the symmetric interior penalty discontinuous galerkin sip dg method through a spectral analysis of the projection operator s sip dg discretization we demonstrate that the eigenfunctions of the operator do not form a basis that allows for the correct enforcement of the incompressibility constraint this short coming can cause numerical instabilities for inviscid advection dominated and density stratified flow simulations especially for long time integrations and or under resolved situations to remedy this problem we propose a local post processing projection that enforces incompressibility exactly to allow for stable and robust long time integrations

comparison of implicit and explicit hybridizable discontinuous galerkin methods for the acoustic wave equation we describe implicit and explicit formulations of the hybridizable discontinuous galerkin method for the acoustic wave equation based on state of the art numerical software and quantify their efficiency for realistic application settings in the explicit scheme the trace of the acoustic pressure is computed from the solution on the two elements adjacent to the face at the old time step tensor product shape functions for quadrilaterals and hexahedra evaluated with sum factorization are used to ensure low operation counts for applying the inverse mass matrix of lagrangian shape functions with full gaussian quadrature a new tensorial technique is proposed as time propagators diagonally implicit and explicit runge kutta methods are used respectively we find that the computing time per time step is 25 to 200 times lower for the explicit scheme with an increasing gap in three spatial dimensions and for higher element degrees our experiments on realistic 3d wave propagation with variable material parameters in a photoacoustic imaging setting show an improvement of two orders of magnitude in terms of time to solution despite stability restrictions on the time step of the explicit scheme operation counts and a performance model to predict performance on other computer systems accompany our results

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

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