a deconvolution enhancement of the navier stokes αβ model a deconvolution enhancement of the email protected b model for turbulent flow is introduced the energy and energy dissipation rate for the enhanced model are derived it is also shown that the consistency error relative to the navier stokes equations and the microscale of the enhanced model are less than those of the email protected b model the proposed model is used to simulate the taylor green vortex problem and results show a qualitatively improved representation of the mean square vorticity when compared to the email protected b model numerical studies of the energy spectrum and the alignment between the vorticity and the eigenvectors of the stretching tensor for three dimensional turbulent flows with re 200 are used to explore the utility of the model a benchmark problem of a two dimensional channel flow over a step for re 600 also indicates that this model can be applied to more general flows than those involving periodic boundary conditions

a right inverse for the divergence operator in spaces of piecewise polynomials in the first part of this paper we study in detail the properties of the divergence operator acting on continuous piecewise polynomials on some fixed triangulation more specifically we characterize the range and prove the existence of a maximal right inverse whose norm grows at most algebraically with the degree of the piecewise polynomials in the last part of this paper we apply these results to thep version of the finite element method for a nearly incompressible material with homogeneous dirichlet boundary conditions we show that thep version maintains optimal convergence rates in the limit as the poisson ratio approaches 1 2 this fact eliminates the need for any reduced integration such as customarily used in connection with the more standardh version of the finite element method

a sensitivity study of the navier stokes α model abstract we present a sensitivity study of the navier stokes model with respect to perturbations of the differential filter length the parameter sensitivity is evaluated using the sensitivity equations method once formulated the sensitivity equations are discretized and computed alongside the ns model using the same finite elements in space and crank nicolson in time we provide a complete stability analysis of the scheme along with the results of several benchmark problems in both 2d and 3d we further demonstrate a practical technique to utilize sensitivity calculations to determine the reliability of the ns model in problem specific settings lastly we investigate the sensitivity and reliability of important functionals of the velocity and pressure solutions

impact of the inherent separation of scales in the navier stokes αβ equations we study the effect of the length scales alpha and beta in the navier stokes alphabeta equations on the energy spectrum and the alignment between the vorticity and the eigenvectors of the stretching tensor in three dimensional homogeneous and isotropic turbulent flows in a periodic cubic domain including the limiting cases of the navier stokes alpha and navier stokes equations a significant increase in the accuracy of the energy spectrum at large wave numbers arises for beta alpha the vorticity structures predicted by the navier stokes alphabeta equations also improve as beta decreases away from alpha however optimal choices for alpha and beta depend not only on the problem of interest but also on the grid resolution

on the accuracy of the rotation form in simulations of the navier stokes equations the rotation form of the navier stokes equations nonlinearity is commonly used in high reynolds number flow simulations it was pointed out by a few authors and not widely known apparently that it can also lead to a less accurate approximate solution than the usual email protected u form we give a different explanation of this effect related to i resolution of the bernoulli pressure and ii the scaling of the coupling between velocity and pressure error with respect to the reynolds number we show analytically that i the difference between the two nonlinearities is governed by the difference in the resolution of the bernoulli and kinematic pressures and ii a simple linear grad div stabilization ameliorates much of the bad scaling of the velocity error with respect to re the rotation form does have superior conservation properties to the alternatives and it appears to be amenable to more efficient preconditioners thus the rotational form with grad div stabilization is a promising method we also give experiments that show bad velocity approximation is tied to poor pressure resolution in either form

on a leray α model of turbulence in this paper we introduce and study a new model for three dimensional turbulence the leray model this model is inspired by the lagrangian averaged navier stokes model of turbulence also known navier stokes model or the viscous camassa holm equations as in the case of the lagrangian averaged navier stokes model the leray model compares successfully with empirical data from turbulent channel and pipe flows for a wide range of reynolds numbers we establish here an upper bound for the dimension of the global attractor the number of degrees of freedom of the leray model of the order of l l d 12 7 where l is the size of the domain and l d is the dissipation length scale this upper bound is much smaller than what one would expect for three dimensional models i e l l d 3 this remarkable result suggests that the leray model has a great potential to become a good sub grid scale large eddy simulation model of turbulence we support this observation by studying analytically and computationally the energy spectrum and show that in addition to the usual k 5 3 kolmogorov power law the inertial range has a steeper power law spectrum for wavenumbers larger than 1 finally we propose a prandtl like boundary layer model induced by the leray model and show a very good agreement of this model with empirical data for turbulent boundary layers

a new family of stable mixed finite elements for the 3d stokes equations a natural mixed element approach for the stokes equations in the velocity pressure formulation would approximate the velocity by continuous piecewise polynomials and would approximate the pressure by discontinuous piecewise polynomials of one degree lower however many such elements are unstable in 2d and 3d this paper is devoted to proving that the mixed finite elements of this p k p k 1 type when k 3 satisfy the stability condition the babuska brezzi inequality on macro tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedra this type of mesh simplifies the implementation since it has no restrictions on the initial mesh the new element also suits the multigrid method

the camassa holm equations and turbulence abstract in this paper we will survey our results on the camassa holm equations and their relation to turbulence as discussed in s chen c foias d d holm e olson e s titi s wynne the camassa holm equations as a closure model for turbulent channel and pipe flow phys rev lett 81 1998 5338 s chen c foias d d holm e olson e s titi s wynne a connection between the camassa holm equations and turbulent flows in channels and pipes phys fluids in press in particular we will provide a more detailed mathematical treatment of those equations for pipe flows which yield accurate predictions of turbulent flow profiles for very large reynolds numbers there are many facts connecting the camassa holm equations to turbulent fluid flows the dimension of the attractor agrees with the heuristic argument based on the kolmogorov statistical theory of turbulence the statistical properties of the energy spectrum agree in numerical simulation with the kolmogorov power law furthermore comparison of mean flow profiles for turbulent flow in channels and pipes given by experimental and numerical data show acceptable agreement with the profile of the corresponding solution of the camassa holm equations

the asymptotics of quantum max flow min cut the quantum max flow min cut conjecture relates the rank of a tensor network to the minimum cut in the case that all tensors in the network are identical in calegari et al j am math soc 23 1 107 188 2010 this conjecture was shown to be false in cui et al j math phys 57 062206 2016 by an explicit counter example here we show that the conjecture is almost true in that the ratio of the quantum max flow to the quantum min cut converges to 1 as the dimension n of the degrees of freedom on the edges of the network tends to infinity the proof is based on estimating moments of the singular values of the network we introduce a generalization of rainbow diagrams to tensor networks to estimate the dominant diagrams a direct comparison of second and fourth moments lower bounds the ratio of the quantum max flow to the quantum min cut by a constant to show the tighter bound that the ratio tends to 1 we consider higher moments in addition we show that the limiting moments as n agree with that in a different ensemble where tensors in the network are chosen independently this is used to show that the distributions of singular values in the two different ensembles weakly converge to the same limiting distribution we present also a numerical study of one particular tensor network which shows a surprising dependence of the rank deficit on n mod 4 and suggests further conjecture on the limiting behavior of the rank

an interpretation of the navier stokes alpha model as a frame indifferent leray regularization abstract we show in this note that the leray regularization of the navier stokes equations is almost frame indifferent but for one second order term neglecting this term we readily recover the navier stokes alpha equations introduced by holm et al adv math 137 1998 1 phys fluids 11 8 1999 2343 hence as an alternative to the exposition in adv math 137 1998 1 phys fluids 11 8 1999 2343 we propose to interpret the navier stokes alpha model as a frame indifferent perturbation of the leray regularization

tractions balances and boundary conditions for nonsimple materials with application to liquid flow at small length scales using a nonstandard version of the principle of virtual power we develop general balance equations and boundary conditions for second grade materials our results apply to both solids and fluids as they are independent of constitutive equations as an application of our results we discuss flows of incompressible fluids at small length scales in addition to giving a generalization of the navier stokes equations involving higher order spatial derivatives our theory provides conditions on free and fixed boundaries the free boundary conditions involve the curvature of the free surface among the conditions for a fixed boundary are generalized adherence and slip conditions each of which involves a material length scale we reconsider the classical problem of plane poiseuille flow for generalized adherence and slip conditions

slip with friction and penetration with resistance boundary conditions for the navier stokes equations numerical tests and aspects of the implementation we consider slip with friction and penetration with resistance boundary conditions in the steady state navier stokes equations this paper describes some aspects of the implementation of these boundary conditions for finite element discretizations numerical tests on two and three dimensional channel flows across a step using the slip with friction boundary condition study the influence of the friction parameter on the position of the reattachment point and the reattachment line of the recirculating vortex respectively

a numerical study of the navier stokes αβ model abstract we present a numerical study of the ns model which is a recently proposed multiscale variation of the ns model that attempts to recapture scales lost through over regularization by separately modeling dissipation range scales we develop a similarity theory for the new model which shows that it is better equipped than the ns model to capture smaller scale behavior next we propose and study an unconditionally stable optimally accurate and efficient finite element implementation for the ns model rigorous proofs for stability and convergence are provided finally we present results from two numerical experiments that demonstrate the advantages of the ns model over the ns model

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

multi grid solution of three dimensional problems with discontinuous coefficients a three dimensional multi grid algorithm is developed and compared to an efficient three dimensional incomplete cholesky conjugate gradient iccg algorithm on a suite of test problems with discontinouous and or anisotropic coefficients the multi grid algorithm used plane relaxation and a definition of coarse grid operators which is an extension of that used for two dimensional problems with discontinuous coefficients numerical results show that the three dimensional results are not nearly as good as the two dimensional results on similar problems

a family of new high order ns α models arising from helicity correction in leray turbulence models abstract this report gives a reinterpretation of the ns model which leads to a family of high order ns deconvolution models with ns as the zeroth order case first we show that the navier stokes model arises by adding helicity correction to leray model higher order leray models have recently been proposed in w layton r lewandowski a high accuracy leray deconvolution model of turbulence and its limiting behavior anal appl 6 1 2008 w layton c manica m neda l rebholz numerical analysis and computational testing of a high accuracy leray deconvolution model of turbulence numer methods partial differential equations in press the so called leray deconvolution models that employ van cittert approximate deconvolution to decrease consistency error we use an analogous helicity correction idea to develop a family of higher order accurate ns type models the ns deconvolution models we prove several mathematical and physical properties for this new family of models and discuss the design of efficient algorithms for them

numerical analysis and computational comparisons of the ns alpha and ns omega regularizations we study stability accuracy and efficiency of algorithms for a new regularization of the nse the ns model in which the vorticity term u is averaged given by u t u q q v u f u 0 this is similar to the ns model in which the nonlinear term is x v x u but the small difference opens attractive algorithmic possibilities we give tests both confirming the predicted rates of convergence and exhibiting some shared limitations of both models the experiments also show the discrete ns simulation has greater accuracy tables 3 and 4 at less cost table 2 and requires significantly fewer degrees of freedom section 1 1 than a comparable ns simulation the experiments suggest consideration of adding grad div stabilization and higher accuracy ns deconvolution models as a next logical step in fact this combination produced accurate results see figs 10 and 11 on the coarsest mesh mesh 1 upon which all other methods models and variants tested failed

a connection between scott vogelius and grad div stabilized taylor hood fe approximations of the navier stokes equations this article studies two methods for obtaining excellent mass conservation in finite element computations of the navier stokes equations using continuous velocity fields with a particular mesh construction the scott vogelius element pair has recently been shown to be inf sup stable and have optimal approximation properties while also providing pointwise mass conservation we present herein the first numerical tests of this element pair for the time dependent navier stokes equations we also prove that the limit of the grad div stabilized taylor hood solutions to the navier stokes problem converges to the scott vogelius solution as the stabilization parameter tends to infinity that is we provide theoretical justification that choosing the grad div parameter large does not destroy the solution numerical tests are provided which verify the theory and show how both scott vogelius and grad div stabilized taylor hood with large stabilization parameter elements can provide accurate results with excellent mass conservation for navier stokes approximations

error estimates of b spline based finite element methods for the stationary quasi geostrophic equations of the ocean abstract this paper presents theoretical error estimates of b spline based finite element methods for the streamfunction formulation of the stationary quasi geostrophic equations which describe the large scale wind driven ocean circulation we introduce variational formulations of the streamfunction formulation inspired by the interior penalty discontinuous galerkin method dirichlet boundary conditions are weakly enforced in the formulations and stabilizations are achieved via nitsche s method existence and uniqueness of the approximation are proved and optimal error estimates in the energy norm are demonstrated under a small data assumption numerical experiments are performed to verify the theoretical error estimates on rectangular and l shape geometries

erratum turbulent kinetic energy and a possible hierarchy of length scales in a generalization of the navier stokes a theory phys rev e 75 056306 2007

divergence free finite elements on tetrahedral grids for k 6 it was shown two decades ago that the p k p k 1 mixed element on triangular grids approximating the velocity by the continuous p k piecewise polynomials and the pressure by the discontinuous p k 1 piecewise polynomials is stable for all k 4 provided the grids are free of a nearly singular vertex the problem with the method in 3d was posted then and remains open the problem is solved partially in this work it is shown that the p k p k 1 element is stable and of optimal order in approximation on a family of uniform tetrahedral grids for all k 6 the analysis is to be generalized to non uniform grids when we can deal with the complicity of 3d geometry for the divergence free elements the finite element spaces for the pressure can be avoided in computation if a classic iterated penalty method is applied the finite element solutions for the pressure are computed as byproducts from the iterate solutions for the velocity numerical tests are provided

an analysis of thep version of the finite element method for nearly incompressible materials in this paper we analyze the behavior of the so calledp version of the finite element method when applied to the equations of plane strain linear elasticity we establish optimal rate error estimates that are uniformly valid independent of the value of the poisson ratio v in the interval 0 1 2 this shows that thep versiondoes not exhibit the degeneracy phenomenon which has led to the use of various only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material