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

on the structural stability of the euler voigt and navier stokes voigt models abstract we consider the euler voigt equations and the navier stokes voigt equations which are obtained by an inviscid regularization from the corresponding equations the main result we show is the structural stability of the system in terms of the variations of both viscosity and regularization parameters

on the construction of suitable solutions to the navier stokes equations and questions regarding the definition of large eddy simulation abstract this paper discusses the notion of large eddy simulation les as a mathematical concept and proposes to use the concept of suitable weak solutions as a building block for the development of a mathematical theory of les various techniques for constructing suitable weak solutions to the three dimensional navier stokes equations are reviewed and many features of these mathematical constructions are shown to comply with well accepted heuristic characterizations of les

length scale estimates for the lans α equations in terms of the reynolds number abstract foias holm and titi c foias d d holm e s titi the three dimensional viscous camassa holm equations and their relation to the navier stokes equations and turbulence theory j dynam differential equations 14 2002 1 35 have settled the problem of existence and uniqueness for the 3d lans equations on periodic box 0 l 3 there still remains the problem first introduced by doering and foias c r doering c foias energy dissipation in body forced turbulence j fluid mech 467 2002 289 306 for the navier stokes equations of obtaining estimates in terms of the reynolds number r e whose character depends on the fluid response as opposed to the grashof number whose character depends on the forcing r e is defined as r e u l where u is a bounded spatio temporally averaged navier stokes velocity field and l the characteristic scale of the forcing it is found that the inverse kolmogorov length is estimated by l k 1 c l 1 4 r e 5 8 moreover the estimate of foias holm and titi for the fractal dimension of the global attractor in terms of r e comes out to be d f a c v v l 1 2 l 2 1 9 8 r e 9 4 where v l l 1 2 3 and v l l l 3 it is also shown that there exists a series of time averaged inverse squared length scales whose members n 0 2 are estimated as n 1 l 2 n 0 2 c n v n 1 n r e 11 4 7 4 n ln r e 1 n c 1 r e ln r e the upper bound on the first member of the hierarchy 1 0 2 coincides with the inverse squared taylor micro scale to within log corrections

on the global regularity of generalized leray alpha type models abstract we generalize leray alpha type models studied in cheskidov et al 2005 1 and linshiz and titi 2007 4 via fractional laplacians and employ besov space techniques to obtain global regularity results with the logarithmically supercritical dissipation

on the existence of global attractors of the approximate deconvolution models of turbulence abstract in this report we consider the problem of the existence of global attractors of the weak solution of the approximate deconvolution models of turbulence the l 2 global attractors are known to exist for large eddy simulation models such as leray alpha leray deconvolution or ns omega models in the case of the approximate deconvolution model as the l 2 norm of the weak solution of the adm is not controlled by the l 2 norm of the initial data and the approximate deconvolution model is an approximation of the filtered nse we are led to the question of existence of h 1 global attractors we also prove better regularity of the weak solution of adm

note on helicity balance of the galerkin method for the 3d navier stokes equations we study the helicity balance of the galerkin method for the 3d navier stokes equations and show that although it does not appear to correctly balance helicity in the usual sense it instead admits a slightly altered helicity balance that matches that of the underlying physics up to boundary conditions we consider helicity treatment in the commonly used galerkin discretization for the 3d navier stokes equations nse it is well known that the nse conserves helicity 13 in the absence of viscous or body forces and admits a precise helicity balance when these forces are present this balance is believed critical in flow structure development 12 and is the basis for the cascade of helicity joint with energy through the inertial range 4 therefore it is very desirable that the helicity balance be preserved in a numerical scheme however in contrast to the energy balance the helicity balance does not hold in the usual way for the standard galerkin approximations see 2 7 and thus creative and naturally more expensive discretizations have been developed that do so first in 10 for axisymmetric flow and later in 14 for the full 3d nse we show herein that the standard galerkin method instead balances a slightly altered discrete helicity type quantity

global regularity for a birkhoff rott α approximation of the dynamics of vortex sheets of the 2d euler equations abstract we present an regularization of the birkhoff rott equation induced by the two dimensional euler equations for the vortex sheet dynamics we show that an initially smooth self avoiding vortex sheet remains smooth for all times under the regularized dynamics provided the initial density of vorticity is an integrable function over the curve with respect to the arc length measure

finite element error analysis for a projection based variational multiscale method with nonlinear eddy viscosity the paper presents a finite element error analysis for a projection based variational multiscale vms method for the incompressible navier stokes equations in the vms method the influence of the unresolved scales onto the resolved small scales is modeled by a smagorinsky type turbulent viscosity

edge detection of potential field sources using scale space monogenic signal fundamental principles abstractwe have developed a new phase based filter to enhance the edges of geologic sources from potential field data using the local phase in the poisson scale space monogenic signal the poisson scale space representation of a potential field data is equivalent to performing an upward continuation of the data we created a band pass filter by taking the differences between two poisson scale space representations of the data the local phase was defined as the arctangent of the ratio of the magnitude of the x and y components of the first order riesz transform of the filtered data to these data these components were computed in the wavenumber domain and then transformed back into the space domain by the inverse fourier transform in the wavenumber domain we found that these components are the multiplication of the fourier transform of the filtered data by a fourier domain kernel which in turn is the multiplication of the first order horizontal derivative filter by the first order vertical integral fi

an eulerian lagrangian approach to the navier stokes equations we present a formulation of the incompressible viscous navier stokes equation based on a generalization of the inviscid weber formula in terms of a diffusive back to labels map and a virtual velocity we derive a generalization of the inviscid cauchy formula and obtain certain bounds for the objects introduced

the p eigenvalue estimates and lq p harmonic forms on submanifolds of hadamard manifolds abstract in this paper we first shows a lower bound estimate for the first eigenvalue of the p laplacian on a complete submanifold in a simply connected manifold of negative sectional curvature in terms of l n norm of the mean curvature in the second part we prove some vanishing theorems of l q p harmonic l forms on complete noncompact submanifolds in hadamard manifolds by assuming the index of certain schrodinger operators are zero or by assuming integral or pointwise pinching conditions on the mean curvature and the traceless second fundamental form

an optimally accurate discrete regularization for second order timestepping methods for navier stokes equations abstract we propose a new optimally accurate numerical regularization stabilization for a family of second order timestepping methods for the navier stokes equations nse the method combines a linear treatment of the advection term together with stabilization terms that are proportional to discrete curvature of the solutions in both velocity and pressure we rigorously prove that the entire new family of methods are unconditionally stable and o t 2 accurate the idea of curvature stabilization is new to cfd and is intended as an improvement over the commonly used speed stabilization which is only first order accurate in time and can have an adverse effect on important flow quantities such as drag coefficients numerical examples verify the predicted convergence rate and show the stabilization term clearly improves the stability and accuracy of the tested flows

irreducibility and exponential mixing of some stochastic hydrodynamical systems driven by pure jump noise in this paper we derive several results related to the long time behavior of a class of stochastic semilinear evolution equations in a separable hilbert space h d mathfrak u t rm a mathfrak u t rm b mathfrak u t mathfrak u t dt dl t quad mathfrak u 0 x in rm h here a is a positive self adjoint operator and b is a bilinear map and the driving noise l is basically a d rm a 1 2 valued levy process satisfying several technical assumptions by using a density transformation theorem type for levy measure we first prove a support theorem and an irreducibility property of the ornstein uhlenbeck processes associated to the nonlinear stochastic problem second by exploiting the previous results we establish the irreducibility of the nonlinear problem provided that for a certain gamma in 0 1 4 b is continuous on d rm a gamma times d rm a gamma with values in d rm a 1 2 using a coupling argument the exponential ergodicity is also proved under the stronger assumption that b is continuous on rm h times rm h while the latter condition is only satisfied by the nonlinearities of goy and sabra shell models the assumption under which the irreducibility property holds is verified by several hydrodynamical systems such as the 2d navier stokes magnetohydrodynamics equations the 3d leray varvec alpha model the goy and sabra shell models

numerical study of a regularization model for incompressible flow with deconvolution based adaptive nonlinear filtering abstract we study a trapezoidal in time finite element in space discretization of a new leray regularization model that locally chooses the filtering radius using a deconvolution based indicator function to identify regions where regularization is needed because this indicator function is mathematically based it allows us to establish a rigorous analysis of the resulting numerical algorithm we prove well posedness unconditional stability and convergence of the proposed algorithm and test the model on several benchmark problems

inertial manifolds for the hyperviscous navier stokes equations abstract we prove the existence of inertial manifolds for the incompressible hyperviscous navier stokes equations on the two or three dimensional torus u t u u u p f t x r t d div u 0 where d 2 or 3 and 3 2 since the spectral gap condition is not necessarily satisfied for the aforementioned problem in three dimensions we employ the spatial averaging method introduced by mallet paret and sell in 26

chebychev optimized approximate deconvolution models of turbulence abstract if the navier stokes equations are averaged with a local spacial convolution type filter g the resulting system is not closed due to the filtered nonlinear term uu an approximate deconvolution operator d is a bounded linear operator which satisfies u d u o where is the filter width and 2 using a deconvolution operator as an approximate filter inverse yields the closure uu d u d u o the residual stress of this model and related models depends directly on the deconvolution error u d u this report derives deconvolution operators yielding an effective turbulence model which minimize the deconvolution error for velocity fields with finite kinetic energy we also give a convergence theory of deconvolution as 0 an ergodic theorem as the deconvolution order n and estimate the increase in accuracy obtained by parameter optimization the report concludes with numerical illustrations

existence of solutions for the mhd leray alpha equations and their relations to the mhd equations in this paper we derive some new equations and we call them mhd leray alpha equations which are similar to the mhd equations we put forward the concept of weak and strong solutions for the new equations whether the 3 dimensional mhd equations have a unique weak solution is unknown however there is a unique weak solution for the 3 dimensional mhd leray alpha equations the global existence of strong solution and the gevrey class regularity for the new equations are also obtained furthermore we prove that the solutions of the mhd leray alpha equations converge to the solution of the mhd equations in the weak sense as the parameter e in the new equations converges to zero

α regularization of the pod galerkin dynamical systems of the kuramoto sivashinsky equation abstract in order to improve the dynamics and stability of the pod galerkin models of strongly stiff systems an like regularization is suggested and assessed in the present article in this method the pod eigenmodes of the non linear terms are replaced by their helmholtz filtered counterparts while the other terms are remained unchanged as an example the method is applied to the pod galerkin models of the one dimensional kuramoto sivashinsky ks equation in a full chaotic regime and the fidelity of the original and regularized models to the direct numerical simulations dns are investigated moreover the effects of regularization on the dynamics of various terms and whole of the systems are analyzed via eigenvalue analysis of each term separately and the total dynamical system as a whole the numerical experiments show definite effectiveness of the method and excellent improvements in the predicted dynamics and stability by minimum number of free parameters

a regularization of the burgers equation using a filtered convective velocity this paper examines the properties of a regularization of the burgers equation in one and multiple dimensions using a filtered convective velocity which we have dubbed as the convectively filtered burgers cfb equation a physical motivation behind the filtering technique is presented an existence and uniqueness theorem for multiple dimensions and a general class of filters is proven multiple invariants of motion are found for the cfb equation which are shown to be shared with the viscous and inviscid burgers equations traveling wave solutions are found for a general class of filters and are shown to converge to weak solutions of the inviscid burgers equation with the correct wave speed numerical simulations are conducted in 1d and 2d cases where the shock behavior shock thickness and kinetic energy decay are examined energy spectra are also examined and are shown to be related to the smoothness of the solutions this approach is presented with the hope of being extended to shock regularization of compressible euler equations

vanishing theorems for l2 harmonic 1 forms on complete submanifolds in a riemannian manifold abstract let m be an n dimensional complete orientable noncompact hypersurface in a complete riemannian manifold of nonnegative sectional curvature for 2 n 6 we prove that if m satisfies the stability inequality 0 1 then there is no nontrivial l 2 harmonic 1 form on m for some constant we also provide sufficient conditions for complete hypersurfaces to satisfy the stability inequality moreover we prove a vanishing theorem for l 2 harmonic 1 forms on m when m is an n dimensional complete noncompact submanifold in a complete simply connected riemannian manifold n with sectional curvature k n satisfying that k 2 k n 0 for some constant k

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

energy and helicity preserving schemes for hydro and magnetohydro dynamics flows with symmetry we propose a class of simple and efficient numerical scheme for incompressible fluid equations with coordinate symmetry by introducing a generalized vorticity stream formulation the divergence free constraints are automatically satisfied in addition with explicit treatment of the nonlinear terms and local vorticity boundary condition the navier stokes mhd respectively equation essentially decouples into 2 4 respectively scalar equation and thus the scheme is very efficient moreover with proper discretization of the nonlinear terms the scheme preserves both energy and helicity identities numerically this is achieved by recasting the nonlinear terms convection vorticity stretching geometric source lorentz force and electro motive force in terms of jacobians this conservative property is valid even in the presence of the pole singularity for axisymmetric flows the exact conservation of energy and helicity has effectively eliminated excessive numerical viscosity numerical examples have demonstrated both accuracy and efficiency of the scheme finally local mesh refinement near the boundary can also be easily incorporated into the scheme without extra cost

analysis of a large eddy simulation model based on anisotropic filtering abstract we consider a new large eddy simulation model derived with the introduction of suitable horizontal anisotropic differential filters one main advantage of this filtering is that for channel flows there is no need for artificial boundary conditions hence we can deal with some realistic problems equipped with dirichlet boundary conditions in special bounded domains at least those bounded only in one direction recent numerical results for a similar model based on a derivation with wave number asymptotics are also recalled after a detailed analysis of the properties of the differential filter we prove that the resulting initial boundary value problem is well posed in suitable anisotropic sobolev spaces giving a strong mathematical support to the model we propose some remarks on higher accuracy approximate deconvolution models are also given in the last section

universal bounds on the attractor of the navier stokes equation in the energy enstrophy plane we continue our examination of the projection of the global attractor of the two dimensional navier stokes equations in a normalized dimensionless energy enstrophy plane in this plane the attractor must lie above the line of slope 1 through the origin an immediate consequence of the poincare inequality we showed in dascaliuc et al j dyn differ equ 17 629 649 2005 that the attractor must also lie below a parabola the square root curve here we optimize this parabola to obtain a sharper bound these bounds are universal they are independent of viscosity domain size and strength of the force the effectiveness of the bounds is demonstrated on galerkin approximations of modest size up to 1372 modes

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

boundary layer for the navier stokes alpha model of fluid turbulence we study boundary layer turbulence using the navier stokes alpha model obtaining an extension of the prandtl equations for the averaged flow in a turbulent boundary layer in the case of a zero pressure gradient flow along a flat plate we derive a nonlinear fifth order ordinary differential equation which is an extension of the blasius equation we study it analytically and prove the existence of a two parameter family of solutions satisfying physical boundary conditions matching these parameters with the skin friction coefficient and the reynolds number based on momentum thickness we get an agreement of the solutions with experimental data in the laminar and transitional boundary layers as well as in the turbulent boundary layer for moderately large reynolds numbers

the lagrangian averaged navier stokes equation with rough data in sobolev spaces abstract the lagrangian averaged navier stokes equation is a recently derived approximation to the navier stokes equation in this article we prove the existence of short time solutions to the incompressible isotropic lagrangian averaged navier stokes equation with low regularity initial data in sobolev spaces w s p r n for 1 p for l 2 based sobolev spaces we obtain global existence results more specifically we achieve local existence with initial data in the sobolev space h n 2 p p r n for initial data in h 3 4 2 r 3 we obtain global existence improving on previous global existence results which required data in h 3 2 r 3

on the convergence of the uniform attractor of 2d ns α model to the uniform attractor of 2d ns system first the existence of uniform attractors for nonautonomous 2d ns a model with normal force term on the t 2 torus in l l o c 2 r v is proved then we study the relations between the uniform attractor of 2d ns a model and 2d ns system we prove that the uniform attractor of the 2d ns a model converges to the 2d ns system as a tends to zero

regularity criteria for a lagrangian averaged magnetohydrodynamic α model abstract in this paper we study the n dimensional lagrangian averaged magnetohydrodynamic lamhd model with 5 n 7 throughout the whole space various regularity criteria are established for both the cases with and those without viscosity

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

nav edge edge detection of potential field sources using normalized anisotropy variance abstractmost existing edge detection algorithms are based on the derivatives of potential field data and thus enhance high wavenumber information and are sensitive to noise the normalized anisotropy variance method nav edge was proposed for detecting edges of potential field anomaly sources based on the idea of normalized standard deviation nstd the main improvement over the balanced windowed normalized variance method i e nstd used for similar purposes was the application of an anisotropic gaussian function designed to detect directional edges and reduce sensitivity to noise nav edge did not directly use higher order derivatives and was less sensitive to noise than the traditional methods that use derivatives in their calculation the utility of nav edge was demonstrated using synthetic potential field data and real magnetic data compared with several existing methods i e the curvature of horizontal gradient amplitude tilt angle and its total horizontal derivative theta map and nstd

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

onsager s conjecture with physical boundaries and an application to the vanishing viscosity limit author s bardos claude titi edriss s wiedemann emil abstract we consider the incompressible euler equations in a bounded domain in three space dimensions recently the first two authors proved onsager s conjecture for bounded domains i e that the energy of a solution to these equations is conserved provided the solution is h quot older continuous with exponent greater than 1 3 uniformly up to the boundary in this contribution we relax this assumption requiring only interior h quot older regularity and continuity of the normal component of the energy flux near the boundary the significance of this improvement is given by the fact that our new condition is consistent with the possible formation of a prandtl type boundary layer in the vanishing viscosity limit

global regularity and convergence of a birkhoff rott α approximation of the dynamics of vortex sheets of the two dimensional euler equations we present an regularization of the birkhoff rott equation br equation induced by the two dimensional euler equations for the vortex sheet dynamics we show the convergence of the solutions of euler equations to a weak solution of the euler equations for initial vorticity being a finite radon measure of fixed sign which includes the vortex sheets case we also show that provided the initial density of vorticity is an integrable function over the curve with respect to the arc length measure i an initially lipschitz chord arc vortex sheet curve evolving under the br equation remains lipschitz for all times ii an initially holder c1 0 1 chord arc curve remains in c1 for all times and finally iii an initially holder cn n 1 0 1 closed chord arc curve remains so for all times in all these cases the weak euler and the br descriptions of the vortex sheet motion are equivalent 2009 wiley periodicals inc

stochastic 3d navier stokes equations in a thin domain and its α approximation abstract in the thin domain o e t 2 0 e where t 2 is a two dimensional torus we consider the 3d navier stokes equations perturbed by a white in time random force and the leray approximation for this system we study ergodic properties of these models and their connection with the corresponding 2d models in the limit e 0 in particular under natural conditions concerning the noise we show that in some rigorous sense the 2d stationary measure comprises asymptotical in time statistical properties of solutions for the 3d navier stokes equations in o e when e 1

numerical approximation of the voigt regularization for incompressible navier stokes and magnetohydrodynamic flows we study voigt regularizations for the navier stokes equations nses and magnetohydrodynamic mhd equations in the presence of physical boundary conditions in particular we develop the first finite element numerical algorithms for these systems prove stability and convergence of the algorithms and test them computationally on problems of practical interest it is found that unconditionally stable implementations of the voigt regularization can be made from a simple change to existing nse and mhd codes and moreover optimal convergence of the developed algorithms solutions to physical solutions can be obtained if lower order mixed finite elements are used finally we show that for several benchmark problems the voigt regularization on a coarse mesh produces good approximations to nse and mhd systems that is the voigt regularization provides accurate reduced order models for nse and mhd flows

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

computational design for long term numerical integration of the equations of fluid motion the integral constraints on quadratic quantities of physical importance such as conservation of mean kinetic energy and mean square vorticity will not be maintained in finite difference analogues of the equation of motion for two dimensional incompressible flow unless the finite difference jacobian expression for the advection term is restricted to a form which properly represents the interaction between grid points as derived in this paper it is shown that the derived form of the finite difference jacobian prevents nonlinear computational instability and thereby permits long term numerical integrations

an examination of the homentropic euler equations with averaged characteristics abstract this paper examines the properties of the homentropic euler equations when the characteristics of the equations have been spatially averaged the new equations are referred to as the characteristically averaged homentropic euler cahe equations an existence and uniqueness proof for the modified equations is given the speed of shocks for the cahe equations are determined the riemann problem is examined and a general form of the solutions is presented finally numerically simulations on the homentropic euler and cahe equations are conducted and the behaviors of the two sets of equations are compared

spectral scaling of the leray α model for two dimensional turbulence we present data from high resolution numerical simulations of the navier stokes and the leray models for two dimensional turbulence it was shown previously lunasin et al 2007 j turbul 8 30 that for wavenumbers k such that k 1 the energy spectrum of the smoothed velocity field for the two dimensional navier stokes ns model scales as k 7 t his result is in agreement with the scaling deduced by dimensional analysis of the flux of the conserved enstrophy using its characteristic time scale we therefore hypothesize that the spectral scaling of any model in the sub spatial scales must depend only on the characteristic time scale and dynamics of the dominant cascading quantity in that regime of scales the data presented here from simulations of the two dimensional leray model confirm our hypothesis we show that for k 1 the energy spectrum for the twodimensional leray scales as k 5 as expected by the characteristic time scale for the flux of the conserved enstrophy of the leray model these results lead to our conclusion that the dominant directly cascading quantity of the model equations must determine the scaling of the energy spectrum

the anisotropic lagrangian averaged euler and navier stokes equations the purpose of this paper is twofold first we give a derivation of the lagrangian averaged euler lae and navier stokes lans equations this theory involves a spatial scale and the equations are designed to accurately capture the dynamics of the euler and navier stokes equations at length scales larger than while averaging the motion at scales smaller than the derivation involves an averaging procedure that combines ideas from both the material lagrangian and spatial eulerian viewpoints this framework allows the use of a variant of g i taylor s frozen turbulence hypothesis as the foundation for the model equations more precisely the derivation is based on the strong physical assumption that fluctutations are frozen into the mean flow in this article we use this hypothesis to derive the averaged lagrangian for the theory and all the terms up to and including order 2 are accounted for

numerical analysis and testing of a stable and convergent finite element scheme for approximate deconvolution turbulence models abstract this report presents a stable and convergent finite element scheme for the approximate deconvolution turbulence models adm the adm is a popular turbulence model intensely studied lately but the computation of its numerical solution raises issues in terms of efficiency and accuracy this report addresses this question the proposed scheme presented herein is based on a new interpretation of the adm model recently introduced by the author following this interpretation the solution of the adm is viewed as the average of a perturbed navier stokes system the scheme uses the crank nicolson time discretization and the finite element spatial discretization and is proved to be stable and convergent provided a moderate choice of the time step is made numerical tests to verify the convergence rates and performance on a benchmark problem are also provided and they prove the correctness of this approach to numerically solve the adm

a note on third order structure functions in turbulence starting from the navier stokes equation we rigorously prove that a modified third order structure function stilde3 r asymptotically equals 4 over 3 r is the dissipation rate in an inertial regime from this result we rigorously confirm the kolmogorov four fifths law without the kolmogorov assumption on isotropy our definition of the structure function involves a solid angle averaging over all possible orientations of the displacement vector y besides space time averaging direct numerical simulation for a highly symmetric flow for a taylor reynolds number of up to 155 shows that the flow remains significantly anisotropic and that without solid angle averaging the resulting structure functions approximately satisfy these scaling relations over some range of r l y l for some orientation of y but not for another

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

existence of smooth attractors for the navier stokes omega model of turbulence abstract we consider the navier stokes model given by u t u u p f u u 0 subject to periodic boundary conditions with zero mean the ns model is an outgrowth of ideas in approximate deconvolution models and in ns alpha models like the ns alpha model it is simple and conserves in the appropriate context kinetic energy and helicity 3d or energy and enstrophy 2d in first tests ns was found to be accurate robust and amenable to efficient numerical simulation in this note we prove existence and regularity of a global attractor for the model

a rate of convergence for the lansα regularization of navier stokes equations abstract a rate of convergence of the solutions of the lans equations with periodic boundary to the solutions of the navier stokes equations as 0 is obtained in a mixed l 1 l 2 time space norm for small initial data in besov type function spaces in which global existence and uniqueness of solutions can also be established

variational principles for lagrangian averaged fluid dynamics the lagrangian average la of the ideal fluid equations preserves their fundamental transport structure this transport structure is responsible for the kelvin circulation theorem of the la flow and hence for its potential vorticity convection and helicity conservation we show that lagrangian averaging also preserves the euler poincar e variational framework that implies the exact ideal fluid equations in the eulerian representation this is expressed in the lagrangian averaged euler poincar e laep theorem proved here we illustrate the laep theorem by applying it to incompressible ideal fluids to derive the lagrangian averaged euler equations and thereby recover the generalized lagrangian mean motion equation finally we discuss recent progress in applications of these equations as the basis for new la closure models of fluid turbulence

random attractors for locally monotone stochastic partial differential equations abstract we prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive levy noise the main result is applicable to various types of spde such as stochastic burgers type equations stochastic 2d navier stokes equations the stochastic 3d leray model stochastic power law fluids the stochastic ladyzhenskaya model stochastic cahn hilliard type equations stochastic kuramoto sivashinsky type equations stochastic generalized porous media equations and stochastic generalized p laplace equations

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

on degrees of freedom of certain conservative turbulence models for the navier stokes equations abstract we study the degrees of freedom of several conservative computational turbulence models that are derived via a non dissipative regularizations of the navier stokes equations for the navier stokes the leray and the navier stokes equations we prove that the longtime behavior of their respective solutions is completely determined by a finite set of grid values and by a finite set of fourier modes for each turbulence model the number of determining nodes and of determining modes is estimated in terms of flow parameters such as viscosity smoothing length forcing and domain size these estimates are global as they do not depend on an individual solution

existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators abstract in this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on a banach space with locally monotone operators which is a generalization of the classical result for monotone operators in particular we show that local monotonicity implies pseudo monotonicity the main results are applied to pde of various types such as porous medium equations reaction diffusion equations the generalized burgers equation the navier stokes equation the 3d leray model and the p laplace equation with non monotone perturbations

l 2 harmonic 1 forms on minimal submanifolds in hyperbolic space abstract in this paper we prove the nonexistence of l 2 harmonic 1 forms on a complete super stable minimal submanifold m in hyperbolic space under the assumption that the first eigenvalue 1 m for the laplace operator on m is bounded below by 2 n 1 n 1 moreover we provide sufficient conditions for minimal submanifolds in hyperbolic space to be super stable

understanding 3d analytic signal amplitude the concept of the analytic signal goes back at least to ville 1948 the analytic signal a x of function f x is a complex quantity defined as where h f x represents the hilbert transform of f x nabighian 1972 1974 applies the analytic signal concept to potential field data in two dimensions for a potential field x measured along the x axis at a constant observation height z and generated by a 2d source aligned parallel to the y axis the horizontal derivative x and the vertical derivative z are a hilbert transform pair

eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space let m be an n dimensional complete non compact submanifold in a hyperbolic space with the norm of its mean curvature vector bounded by a constant alpha 0 in particular when m is minimal we have lambda 1 left m right geq frac 1 4 left n 1 right 2 and this is sharp because equality holds when m is totally geodesic

on the well posedness and conservation laws of a family of multiscale deconvolution models for the magnetohydrodynamics equations abstract we present a mathematical study of a large eddy simulation les model for the incompressible magnetohydrodynamics equations the classical closure problem arising for les models is solved with the multiscale deconvolution technique developed by dunca in 11 we prove the model admits unique regular weak solutions and provide a mathematical study of the modeling error

logarithmically extended global regularity result of lans alpha mhd system in two dimensional space abstract we study the two dimensional generalized lans alpha magnetohydrodynamics system we show that the solution pairs of velocity and magnetic fields to this system preserve their initial regularity in two cases dissipation logarithmically weaker than a full laplacian and zero diffusion zero dissipation and diffusion logarithmically weaker than a full laplacian

attractors of the velocity vorticity voigt model of the 3d navier stokes equations with damping abstract in this paper we prove the existence of global and exponential attractors of optimal regularity of a regularization for the three dimensional navier stokes equations with damping which is called the three dimensional velocity vorticity voigt vvv system proposed by larios pei and rebholz larios et al 2019 differently there is no voigt term 2 a w t in the second equation of system 1 1 below the proof is based on some energy estimates in sobolev spaces and the semigroup decomposition

global well posedness of the velocity vorticity voigt model of the 3d navier stokes equations abstract the velocity vorticity formulation of the 3d navier stokes equations was recently found to give excellent numerical results for flows with strong rotation in this work we propose a new regularization of the 3d navier stokes equations which we call the 3d velocity vorticity voigt vvv model with a voigt regularization term added to momentum equation in velocity vorticity form but with no regularizing term in the vorticity equation we prove global well posedness and regularity of this model under periodic boundary conditions we prove convergence of the model s velocity and vorticity to their counterparts in the 3d navier stokes equations as the voigt modeling parameter tends to zero we prove that the curl of the model s velocity converges to the model vorticity which is solved for directly as the voigt modeling parameter tends to zero finally we provide a criterion for finite time blow up of the 3d navier stokes equations based on this inviscid regularization

data assimilation for the navier stokes α equations abstract the well posedness of the data assimilation problem for the navier stokes equations on a bounded three dimensional domain is investigated the data assimilation procedures under consideration are the adjoint method of variational data assimilation 4d var and the method of continuous data assimilation concerning the adjoint method the existence of optimal initial conditions with respect to an observation dependent cost functional is proven the optimizers are characterized by a first order necessary condition involving the adjoint linearized navier stokes equations and conditions for the uniqueness of the initial conditions are given well posedness of the continuous data assimilation problem is proven and convergence rates in terms of observational resolution are provided

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

on the convergence rate of the euler α an inviscid second grade complex fluid model to the euler equations we study the convergence rate of the solutions of the incompressible euler an inviscid second grade complex fluid equations to the corresponding solutions of the euler equations as the regularization parameter approaches zero first we show the convergence in hs s n 2 1 in the whole space and that the smooth euler solutions exist at least as long as the corresponding solution of the euler equations next we estimate the convergence rate for two dimensional vortex patch with smooth boundaries

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

intermittency and regularity issues in 3d navier stokes turbulence two related open problems in the theory of 3d navier stokes turbulence are discussed in this paper the first is the phenomenon of intermittency in the dissipation field dissipation range intermittency was first discovered experimentally by batchelor and townsend over fifty years ago it is characterized by spatio temporal binary behaviour in which long quiescent periods in the velocity signal are interrupted by short active events during which there are violent fluctuations away from the average the second and related problem is whether solutions of the 3d navier stokes equations develop finite time singularities during these events this paper shows that leray s weak solutions of the three dimensional incompressible navier stokes equations can have a binary character in time the time axis is split into good and bad intervals on the good intervals solutions are bounded and regular whereas singularities are still possible within the bad intervals an estimate for the width of the latter is very small and decreases with increasing reynolds number it also decreases relative to the lengths of the good intervals as the reynolds number increases within these bad intervals lower bounds on the local energy dissipation rate and other quantities such as u t and u t are very large resulting in strong dynamics at sub kolmogorov scales intersections of bad intervals for n 1 are related to the potentially singular set in time it is also proved that the navier stokes equations are conditionally regular provided in a given bad interval the energy has a lower bound that is decaying exponentially in time

limitations of determining density or magnetic boundaries from the horizontal gradient of gravity or pseudogravity data the horizontal gradient method has been used since 1982 to locate density or magnetic boundaries from gravity data cordell 1979 or pseudogravity data cordell and grauch 1985 the method is based on the principle that a near vertical fault like boundary produces a gravity anomaly whose horizontal gradient is largest directly over the top edge of the boundary magnetic data can be transformed to pseudogravity data using fourier techniques e g hildenbrand 1983 so that they behave like gravity data thus the horizontal gradient of pseudogravity also has maximum magnitude directly over the boundary the method normally is applied to gridded data rather than to profiles the horizontal gradient magnitude is contoured and lines are drawn or calculated blakely and simpson 1986 along the contour ridges these lines presumably mark the top edges of magnetic or density boundaries however horizontal gradient magnitude maxima gradient maxima can be offset from a position directly over the boundary f

a note on helicity conservation in leray models of incompressible flow abstract we prove conservation of a regularized helicity h l u w d x for the leray model and its variants of turbulent flow where w is the solution of a leray regularized vorticity equation the usual definition of helicity is h u u d x which is considered by navier stokes flows but is not a conserved quantity of the leray model however if u is a leray solution then the difference between h and h l is that h l uses a regularized vorticity and h uses the curl of a regularized velocity the results are extended to show that the standard crank nicolson finite element method for leray models conserves both discrete energy and discrete regularized helicity

eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space abstract let m n be a complete non compact submanifold in the hyperbolic space h n m we first give an estimate for the bottom of the spectral of the laplace operator on m n under an integral pinching condition on the mean curvature as a consequence of this estimation we show some vanishing theorems for l 2 harmonic forms in certain degrees if the total mean curvature of m n is less than an explicit constant and its total curvature is less than a suitable related constant in addition we obtain some vanishing results under certain pointwise restrictions on the traceless second fundamental form moreover according to the nonexistence of nontrivial l 2 harmonic 1 forms we can further prove some one end theorems

on a regularization of a scalar conservation law with discontinuous coefficients this paper is devoted to a scalar conservation law with a linear flux function involving discontinuous coefficients it is clear that the delta standing wave should be introduced into the riemann solution in some nonclassical situation in order to study the formation of delta standing wave we consider a regularization of the discontinuous coefficient with the helmholtz mollifier and then obtain a regularized system which depends on a regularization parameter 0 the regularization mechanism is a nonlinear bending of characteristic curves that prevents their finite time intersection it is proved rigorously that the solutions of regularized system converge to the delta standing wave solution in the 0 limit compared with the classical method of vanishing viscosity here it is clear to see how the delta standing wave forms naturally along the characteristics

highly turbulent solutions of the lagrangian averaged navier stokes alpha model and their large eddy simulation potential we compute solutions of the lagrangian averaged navier stokes alpha lans alpha model for significantly higher reynolds numbers up to re approximately 8300 than have previously been accomplished this allows sufficient separation of scales to observe a navier stokes inertial range followed by a second inertial range specific to the lans alpha model both fully helical and nonhelical flows are examined up to reynolds numbers of approximately 1300 analysis of the third order structure function scaling supports the predicted l3 scaling it corresponds to a k 1 scaling of the energy spectrum for scales smaller than alpha the energy spectrum itself shows a different scaling which goes as k1 this latter spectrum is consistent with the absence of stretching in the subfilter scales due to the taylor frozen in hypothesis employed as a closure in the derivation of the lans alpha model these two scalings are conjectured to coexist in different spatial portions of the flow the l3 e k approximately k 1 scaling is subdominant to k1 in the energy spectrum but the l3 scaling is responsible for the direct energy cascade as no cascade can result from motions with no internal degrees of freedom we demonstrate verification of the prediction for the size of the lans alpha attractor resulting from this scaling from this we give a methodology either for arriving at grid independent solutions for the lans alpha model or for obtaining a formulation of the large eddy simulation optimal in the context of the alpha models the fully converged grid independent lans alpha model may not be the best approximation to a direct numerical simulation of the navier stokes equations since the minimum error is a balance between truncation errors and the approximation error due to using the lans alpha instead of the primitive equations furthermore the small scale behavior of the lans alpha model contributes to a reduction of flux at constant energy leading to a shallower energy spectrum for large alpha these small scale features however do not preclude the lans alpha model from reproducing correctly the intermittency properties of the high reynolds number flow

convergence of the 2d euler α to euler equations in the dirichlet case indifference to boundary layers abstract in this article we consider the euler system as a regularization of the incompressible euler equations in a smooth two dimensional bounded domain for the limiting euler system we consider the usual non penetration boundary condition while for the euler regularization we use velocity vanishing at the boundary we also assume that the initial velocities for the euler system approximate in a suitable sense as the regularization parameter 0 the initial velocity for the limiting euler system for small values of this situation leads to a boundary layer which is the main concern of this work our main result is that under appropriate regularity assumptions and despite the presence of this boundary layer the solutions of the euler system converge as 0 to the corresponding solution of the euler equations in l 2 in space uniformly in time we also present an example involving parallel flows in order to illustrate the indifference to the boundary layer of the 0 limit which underlies our work

grade two fluids on non smooth domain driven by multiplicative noise existence uniqueness and regularity the author presented a lecture on the subject of this paper at the workshop on numerics for stochastic partial differential equations and their applications held at the johann radon institute for computational and applied mathematics ricam linz austria during the period 04 16 december 2016

numerical analysis of leray tikhonov deconvolution models of fluid motion this report develops and studies a new family of navier stokes equation regularizations leray tikhonov regularizations with time relaxation models this new family of turbulence models is based on a modification consistent with the large scales of tikhonov lavrentiev regularization with this approach we obtain an approximation of the unfiltered solution by one filtering step we introduce the modified tikhonov deconvolution operator and study its mathematical properties we also perform rigorous numerical analysis of a computationally attractive algorithm for this family of models and present numerical experiments using it

weighted volume growth and vanishing properties of f minimal hypersurfaces in a weighted manifold abstract in this paper we prove that a complete noncompact submanifold in a weighted manifold with nonpositive sectional curvature has at least linear weighted volume growth moreover we obtain several sufficient conditions for f minimal hypersurfaces to have infinite weighted volume by using an f laplacian comparison result we obtain a lower bound of the first eigenvalue for the f laplace operator on submanifolds in a weighted manifold we also obtain vanishing results for l f 2 harmonic 1 forms on complete noncompact f minimal hypersurfaces in a weighted manifold finally we prove a topological structure theorem for complete noncompact l f stable f minimal hypersurfaces via a liouville type theorem for f harmonic functions with finite f energy

boundary effects on exact solutions of the lagrangian averaged navier stokes α equations in order to clarify the behavior of solutions of the lagrangian averaged navier stokes lans equations in the presence of solid walls we identify a variety of exact solutions of the full equations and their boundary layer approximations the solutions demonstrate that boundary conditions suggested for the lans equations in the literature 1 for a bounded domain do not apply in a semi infinite domain the convergence to solutions of the navier stokes equations as 0 is elucidated for infinite energy solutions in a semi infinite domain and non uniqueness of these solutions is discussed we also study the boundary layer approximation of lans equations denoted the prandtl equations and report solutions for turbulent jets and wakes our version of the prandtl equations includes an extra term necessary to conserve linear momentum and corrects an earlier result of cheskidov 2

edge enhancement of magnetic data using fractional order derivative filters abstractedge enhancement and detection techniques are fundamental operations in magnetic data interpretation many techniques for edge enhancement have been developed some based on profile data and others designed for grid based data sets methods that are traditionally applied to magnetic data such as total horizontal derivative thd and analytic signal as require the computation of integer order horizontal and vertical derivatives of the magnetic data however if the data set contains features with a large variation in amplitude then the features with small amplitudes may be difficult to outline in addition because most edge enhancement and detection filters are derivative based filters they also amplify high frequency noise content in the data as a result the accuracy of derivative based filters is restricted to data of high quality we suggested the modification of the thd and as filters by combining the amplitude spectra of fractional order derivative filters with ad hoc phase spectra p

well posedness and a numerical study of a regularization model with adaptive nonlinear filtering for incompressible fluid flow we give a detailed analytical study of a leray model of incompressible flow that uses nonlinear filtering based on indicator functions the indicator functions allow for local regularization instead of global regularization which can over smooth and dampen out important flow structures the key to the analysis is the identification of the indicator function as a nemyskii operator after proving well posedness we provide a numerical study which includes proving optimal convergence of finite element method for the model as well as several numerical experiments