Potential Singularity for a Family of Models of the Axisymmetric Incompressible Flow

We study a family of 3D models for the incompressible axisymmetric Euler and Navier–Stokes equations. The models are derived by changing the strength of the convection terms in the equations written using a set of transformed variables. The models share several regularity results with the Euler and Navier–Stokes equations, including an energy identity, the conservation of a modified circulation quantity, the BKM criterion and the Prodi–Serrin criterion. The inviscid models with weak convection are numerically observed to develop stable self-similar singularity with the singular region traveling along the symmetric axis, and such singularity scenario does not seem to persist for strong convection.     

On the stabilizing effect of convection in three‐dimensional incompressible flows

We investigate the stabilizing effect of convection in three‐dimensional incompressible Euler and Navier‐Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. In this paper, we show that the convection term together with the incompressibility condition actually has a surprising stabilizing effect. We demonstrate this by constructing a new three‐dimensional model that is derived for axisymmetric flows with swirl using a set of new variables. This model preserves almost all the properties of the full three‐dimensional Euler or Navier‐Stokes equations except for the convection term, which is neglected in our model. If we added the convection term back to our model, we would recover the full Navier‐Stokes equations. We will present numerical evidence that seems to support that the three‐dimensional model may develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new three‐dimensional model and how the convection term in the full Euler and Navier‐Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time. © 2008 Wiley Periodicals, Inc.     

On singularity formation of a 3D model for incompressible Navier–Stokes equations

We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei (2009) in [15] for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. This model shares many properties of the 3D incompressible Navier–Stokes equations. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3D inviscid model for a class of initial boundary value problems with smooth initial data of finite energy. We also prove the global regularity of the 3D inviscid model for a class of small smooth initial data.     

三维不可压缩Euler方程的H1(R~3)轴对称解

在适当的初值条件下，对三维不可压缩Ｅｕｌｅｒ方程组，我们证明了Ｈ１（Ｒ３）轴对称解的存在性．     

A study on the global regularity for a model of the 3D axisymmetric Navier-Stokes equations

This paper investigates the global regularity issue concerning a model equation proposed by Hou and Lei (2008) [9] to understand the stabilizing effects of the nonlinear terms in the 3D axisymmetric Navier-Stokes and Euler equations. We establish the global regularity of a generalized version of their model with a fractional Laplacian when the fractional power satisfies an explicit condition. This condition is exactly the same as in the case of the 3D generalized Navier-Stokes equations and is due to the balance between a more regular nonlinearity and a less effective (five-dimensional) Laplacian.     

On the finite-time singularities of the 3D incompressible Euler equations

We prove the finite-time vorticity blowup, in the pointwise sense, for solutions of the 3D incompressible Euler equations assuming some conditions on the initial data and its corresponding solutions near initial time. These conditions are represented by the relation between the deformation tensor and the Hessian of pressure, both coupled with the vorticity directions associated with the initial data and solutions near initial time. We also study the possibility of the enstrophy blowup for the 3D Euler and the 3D Navier-Stokes equations, and prove the finite-time enstrophy blowup for initial data satisfying suitable conditions. Finally, we obtain a new blowup criterion that controls the blowup by a quantity containing the Hessian of the pressure. © 2006 Wiley Periodicals, Inc.     

On the Lagrangian dynamics of the axisymmetric 3D Euler equations

We study the dynamics along the particle trajectories for the 3D axisymmetric Euler equations. In particular, by rewriting the system of equations we find that there exists a complex Riccati type of structure in the system on the whole of R³, which generalizes substantially the previous results in [5] (D. Chae, On the blow-up problem for the axisymmetric 3D Euler equations, Nonlinearity 21 (2008) 2053-2060). Using this structure of equations, we deduce the new blow-up criterion that the radial increment of pressure is not consistent with the global regularity of classical solution. We also derive a much more refined version of the Lagrangian dynamics than that of [6] (D. Chae, On the Lagrangian dynamics for the 3D incompressible Euler equations, Comm. Math. Phys. 269 (2) (2007) 557-569) in the case of axisymmetry.     

Limite asymptotique pour le modèle de BGK

We present, for the BGK equation, asymptotic limits leading to various equations of incompressible and compressible fluid mechanics: the Navier-Stokes equations, the linearized Navier-Stokes equations, the Euler equation, the linearized Euler equation, and the compressible Euler equation. We state a convergence theorem for the nonlinear Navier-Stokes, as well as a result for the linear Navier-Stokes case, and for the compressible Euler equation.     

The Inviscid Limit for the Steady Incompressible Navier-Stokes Equations in the Three Dimension*

In this paper, the authors consider the zero-viscosity limit of the three dimensional incompressible steady Navier-Stokes equations in a half space R⁺×R². The result shows that the solution of three dimensional incompressible steady Navier-Stokes equations converges to the solution of three dimensional incompressible steady Euler equations in Sobolev space as the viscosity coefficient going to zero. The method is based on a new weighted energy estimates and Nash-Moser itera...     

Convergence of a singular Euler-Poisson approximation of the incompressible Navier-Stokes equations

In this note, we rigorously justify a singular approximation of the incompressible Navier-Stokes equations. Our approximation combines two classical approximations of the incompressible Euler equations: a standard relaxation approximation, but with a diffusive scaling, and the Euler-Poisson equations in the quasineutral regime.

     

Numerical Solution of Newtonian and Non-Newtonian Flows

This paper deals with the numerical solution of Newtonian and non-Newtonian flows. The flows are supposed to be laminar, viscous, incompressible and steady. The model used for non-Newtonian fluids is some variant of power-law. Governing equations in this model are incompressible Navier-Stokes equations. For numerical solution one could use artificial compressibility method with three stage Runge-Kutta finite volume method in cell centered formulation for discretization of space derivatives. Following cases of flows are solwed: flow through a bypass connected to main channel in 2D and 3D and non-Newtonian flow through branching channels in 2D. These results are presented for 2D and 3D case. This problem could have an application in the area of biomedicine. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A term by term stabilization algorithm for finite element solution of incompressible flow problems

This paper introduces a stabilization technique for Finite Element numerical solution of 2D and 3D incompressible flow problems. It may be applied to stabilize the discretization of the pressure gradient, and also of any individual operator term such as the convection, curl or divergence operators, with specific levels of numerical diffusion for each one of them. Its computational complexity is reduced with respect to usual (residual-based) stabilization techniques. We consider piecewise affine Finite Elements, for which we obtain optimal error bounds for steady Navier-Stokes and also for generalized Stokes equations (including convection). We include some numerical experiment in well known 2D test cases, that show its good performances. Received March 15, 1996 / Revised version received January 17, 1997     

An implicit finite-difference algorithm for the numerical simulation of supersonic flow over blunted bodies

In order to compute axisymmetric laminar supersonic flow we use an unsteady implicit finite-difference scheme. This scheme solves numerically either the inviscid Euler equations or the ‘thin-layer’ Navier-Stokes equations. In both cases the scheme leads to large sparse non-linear systems, which can be solved by a genuine iteration process. The convergence of this process is shown and numerical results are given.     

On the deformations of the incompressible Euler equations

We consider systems of deformed system of equations, which are obtained by some transformations from the system of incompressible Euler equations. These have similar properties to the original Euler equations including the scaling invariance. For one form of deformed system we prove that finite time blow-up actually occurs for ‘generic’ initial data, while for the other form of the deformed system we prove the global in time regularity for smooth initial data. Moreover, using the explicit functional relations between the solutions of those deformed systems and that of the original Euler system, we derive the condition of finite time blow-up of the Euler system in terms of solutions of one of its deformed systems. As another application of those relations we deduce a lower estimate of the possible blow-up time of the 3D Euler equations. This research was supported partially by the KOSEF Grant no. R01-2005-000-10077-0     

Finite element method in the solution of the Euler and Navier-Stokes equations for internal flow

Finite element solutions of the Euler and Navier-Stokes equations are presented, using a simple dissipation model. The discretization is based on the weak-Galerkin weighted residual method and equal interpolation functions for all the unknowns are permitted. The nonlinearity is iterated upon using a Newton method and at each iteration the linear algebraic system is solved by a direct solver with all unknowns fully coupled. Results are presented for two-dimensional transonic inviscid flows and two- and three-dimensional incompressible viscous flows. Convergence of the algorithm is shown to be quadratic, reaching machine accuracy in very few iterations. The inviscid results demonstrate the existence of nonunique numerical solutions to the steady Euler equations.     

A finite element time relaxation method

We discuss a finite element time-relaxation method for high Reynolds number flows. The method uses local projections on polynomials defined on macroelements of each pair of two elements sharing a face. We prove that this method shares the optimal stability and convergence properties of the continuous interior penalty (CIP) method. We give the formulation both for the scalar convection–diffusion equation and the time-dependent incompressible Euler equations and the associated convergence results. This note finishes with some numerical illustrations.     

A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations

We approximate the regular solutions of the incompressible Euler equations by the solution of ODEs on finite-dimensional spaces. Our approach combines Arnold’s interpretation of the solution of the Euler equations for incompressible and inviscid fluids as geodesics in the space of measure-preserving diffeomorphisms, and an extrinsic approximation of the equations of geodesics due to Brenier. Using recently developed semi-discrete optimal transport solvers, this approach yields a numerical scheme which is able to handle problems of realistic size in 2D. Our purpose in this article is to establish the convergence of this scheme towards regular solutions of the incompressible Euler equations, and to provide numerical experiments on a few simple test cases in 2D.     

Finite-dimensional controllability for systems of Navier-Stokes type

We consider the problem of partial controllability for an evolution equation with a quadratic nonlinearity, in which one should provide, at a given time, a given projection of the solution onto some finite-dimensional subspace by using the action of external forces that belong to one and the same subspace at each time. On the basis of estimates for the solution of a subdifferential Cauchy problem for a system of Navier-Stokes type, we prove the controllability and the existence of a control with minimum norm. We consider applications to the Navier-Stokes equations of a viscous incompressible fluid and a heat convection model.     

Conditions on the Pressure for Vanishing Velocity in the Incompressible Fluid Flows in ℝ N

In this paper we derive various sufficient conditions on the pressure for vanishing velocity in the incompressible Navier-Stokes and the Euler equations in ? ^N .     

On Some Model Equations of Euler and Navier-Stokes Equations

The author proposes a two-dimensional generalization of Constantin-Lax-Majda model. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line(vorticity formulation), the author presents some further model equations. He possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. Some discussions on the possible connection between tur...