Determining finite volume elements for the 2D Navier-Stokes equations

We consider the 2D Navier-Stokes equations on a square with periodic boundary conditions. Dividing the square into N equal subsquares, we show that if the asymptotic behavior of the average of solutions on these subsquares (finite volume elements) is known, then the large time behavior of the solution itself is completely determined, provided N is large enough. We also establish a rigorous upper bound for N needed to determine the solutions to the Navier-Stokes equation in terms of the physical parameters of the problem.     

Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations

Consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times. Discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing. In this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time. This paper studies discrete data assimilation for the Lorenz equations and the incompressible two-dimensional Navier-Stokes equations. In both cases we obtain bounds on the time interval h between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution.     

Trajectory attractors for dissipative 2D Euler and Navier-Stokes equations

A trajectory attractor is constructed for the 2D Euler system containing an additional dissipation term −ru, r > 0, with periodic boundary conditions. The corresponding dissipative 2D Navier-Stokes system with the same term −ru and with viscosity v > 0 also has a trajectory attractor, . Such systems model large-scale geophysical processes in atmosphere and ocean (see [1]). We prove that → as v → 0+ in the corresponding metric space. Moreover, we establish the existence of the minimal limit of the trajectory attractors as v → 0+. We prove that is a connected invariant subset of . The connectedness problem for the trajectory attractor by itself remains open. Dedicated to the memory of Leonid Volevich Partially supported by the Russian Foundation for Basic Research (projects no 08-01-00784 and 07-01-00500). The first author has been partially supported by a research grant from the Caprio Foundation, Landau Network-Cento Volta.     

Global stability of large solutions to the 3D Navier-Stokes equations

We prove the stability of mildly decaying global strong solutions to the Navier-Stokes equations in three space dimensions. Combined with previous results on the global existence of large solutions with various symmetries, this gives the first global existence theorem for large solutions with approximately symmetric initial data. The stability of unforced 2D flow under 3D perturbations is also obtained.     

A sharp lower bound for the Hausdorff dimension of the global attractors of the 2D Navier-Stokes equations

For a special class of the Navier-Stokes equations on the two-dimensional torus, we give a lower bound in the formG ^2/3 (whereG is the Grashof number) for the Hausdorff dimension of its global attractor which is optimal up to a logarithmic term.Dedicated to the memory of Professor Nicholas D. Kazarinoff     

Exponential Mixing of the 2D Stochastic Navier-Stokes Dynamics

 We consider the Navier-Stokes equation on a two dimensional torus with a random force which is white noise in time, and excites only a finite number of modes. The number of excited modes depends on the viscosity ν, and grows like ν ^-3 when ν goes to zero. We prove that this Markov process has a unique invariant measure and is exponentially mixing in time. Received: 14 March 2002 / Accepted: 7 May 2002 Published online: 22 August 2002     

Hausdorff measure and the Navier-Stokes equations

Solutions to the Navier-Stokes equations are continuous except for a closed set whose Hausdorff dimension does not exceed two.     

Relativistic Navier-Stokes equations in the Meixner-Prigogine scheme

Viscous effects are included in the relativistic Meixner-Prigogine scheme (see: A. Sandoval-Villalbazo, L.S. García-Colín, Physica A 234 (1996) 358). A relativistic generalization of the Navier-Stokes equations is obtained within this framework. The system obtained is analyzed and compared with related work.     

Navier-Stokes equations and area of interfaces

We present new a priori estimates for the vorticity of solutions of the three dimensional Navier-Stokes equations. These estimates imply that theL ¹ norm of the vorticity is a priori bounded in time and that the time average of the 4/(3+) power of theL ^4/(3+) spatial norm of the gradient of the vorticity is a priori bounded. Using these bounds we construct global Leray weak solutions of the Navier-Stokes equations which satisfy these inequalities. In particular it follows that vortex sheet, vortex line and even more general vortex structures with arbitrarily large vortex strengths are initial data which give rise to global weak solutions of this type of the Navier-Stokes equations. Next we apply these inequalities in conjunction with geometric measure theoretical arguments to study the two dimensional Hausdorff measure of level sets of the vorticity magnitude. We obtain a priori bounds on an average such measure, >. When expressed in terms of the Reynolds number and the Kolmogorov dissipation length , these bounds are

Some unsteady pseudo-plane motions for the Navier-Stokes equations

A R I - Some unsteady motions of a viscous incompressible fluid are determined in which all streamlines lie in the planes z = constant. These include motions in which the fluid in each plane z =...     

On a potential-velocity formulation of Navier-Stokes equations

Computational methods in continuum mechanics, especially those encompassing fluid dynamics, have emerged as an essential investigative tool in nearly every field of technology. Despite being underpinned by a well-developed mathematical theory and the existence of readily available commercial software codes, computing solutions to the governing equations of fluid motion remains challenging: in essence due to the non-linearity involved. Additionally, in the case of free surface film flows the dynamic boundary condition at the free surface complicates the mathematical treatment notably. Recently, by introduction of an auxiliary potential field, a first integral of the two-dimensional Navier-Stokes equations has been constructed leading to a set of equations, the differential order of which is lower than that of the original Navier-Stokes equations. In this paper a physical interpretation is provided for the new potential, making use of the close relationship between plane Stokes flow and plane linear elasticity. Moreover, it is shown that by application of this alternative approach to free surface flows the dynamic boundary condition is reduced to a standard Dirichlet-Neumann form, which allows for an elegant numerical treatment. A least squares finite element method is applied to the problem of gravity driven film flow over corrugated substrates in order to demonstrate the capabilities of the new approach. Encapsulating non-Newtonian behaviour and extension to three-dimensional problems is discussed briefly.     

Convergence of fourier methods for Navier-Stokes equations

The Fourier-Galerkin method is used to simulate fluid flows in two and three dimensions, on domains with periodic boundary conditions. It is proved that the numerical solution converges towards the solution of Navier-Stokes equations. The rate of convergence depends on the smoothness of the mathematical solution. Finally, it is shown that the Fourier-Galerkin method can be interpreted as a projection method. This observation may lead to more sophisticated convergence proofs.     

A five-dimensional truncation of the plane incompressible Navier-Stokes equations

A five-modes truncation of the Navier-Stokes equations for a two dimensional incompressible fluid on a torus is considered. A computer analysis shows that for a certain range of the Reynolds number the system exhibits a stochastic behaviour, approached through an involved sequence of bifurcations.Partially supported by G.N.F.M., C.N.R.     

A seven-mode truncation of the plane incompressible Navier-Stokes equations

A model obtained by a seven-mode truncation of the Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is studied. This model, extending a previously studied five-mode one, exhibits a very rich and varied phenomenology including some remarkable properties of hysteresis (i.e., coexistence of attractors). A stochastic behavior is found for high values of the Reynolds number, when no stable fixed points, closed orbits, or tori are present.     

Self-similar universal homogeneous statistical solutions of the Navier-Stokes equations

In this note we consider a family of statistical solutions of the Navier-Stokes equations (i.e. time dependent solutions of the Hopf equation) which seem to constitute the rigorous mathematical framework for the theory of homogeneous turbulence [1], [13]. The main feature of these solutions is that they are the transforms under suitable scalings of thestationary statistical solutions of a new system of equations (the Eq. (2) below).     

Estimates on the vorticity of solutions to the Navier-Stokes equations

We estimate the vorticity of the flow of an incompressible, viscous, three dimensional fluid near the boundary of its container. We obtain a bound that is valid outside a small subset of space-time with special properties.This research was supported in part by the National Science Foundation Grant MCS-7903361     

Preconditioned pseudo-compressibility methods for incompressible Navier-Stokes equations

This paper investigates the pseudo-compressibility method for the incompressible Navier-Stokes equations and the preconditioning technique for accelerating the time marching for stiff hyperbolic equations,and derives and presents the eigenvalues and eigenvectors of the Jacobian matrix of the preconditioned pseudo-compressible Navier-Stokes equations in generally cur-vilinear coordinates.Based on the finite difference discretization the cored for efficiently solving incompressible flows numerically is established.The reliability of the procedures is demonstrated by the application to the inviscid flow past a circular cylinder,the laminar flow over a flat plate,and steady low Reynolds number viscous incompressible flows past a circular cylinder.It is found that the solutions to the present algorithm are in good agreement with the exact solutions or experimental data.The effects of the pseudo-compressibility factor and the parameter brought by preconditioning in convergence characteristics of the solution are investigated systematically.The results show that the upwind Roe’s scheme is superior to the second order central scheme,that the convergence rate of the pseudo-compressibility method can be effectively improved by preconditioning and that the self-adaptive pseudo-compressibility factor can modify the numerical convergence rate significantly compared to the constant form,without doing artificial tuning depending on the specific flow conditions.Further validation is also performed by numerical simulations of unsteady low Reynolds number viscous incompressible flows past a circular cylinder.The results are also found in good agreement with the existing numerical results or experimental data.