A STABILITY STUDY OF NAVIER-STOKES EQUATIONS (Ⅲ)

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Analysis of two-dimensional cavity flow by finite elements

The variational principle in terms of stream function ψ for free surface gravity flow is discussed by the formulation of first-order variation in a variable domain. Because of different transversal conditions adopted, there are four forms of variational principle in terms of ψ.A n air-gilled cavity flow with given discharge and total energy is then analysed by finite element method. At the end of the cavity, the free stream line is tangent to a short fictitious plate of given length, which joins the fixed boundary at on angle to be determined. The condition that the free stream line should be tangent to the fixed boundary at the point of separation makes the solution unique.Finally curves giving the cavity length as a function of the Fraude number, cavity pressure and channel bottom slope are presented.     

COMPARISON OF STABILITY BETWEEN NAVIER-STOKES AND EULER EQUATIONS

The stability about Navier-Stokes equation and Euler equation was brought into comparison. And by taking their typical initial value problem for example, the reason of leading to the difference in stability between Navier-Stokes equation and Euler equation was also analyzed.     

A deterministic vortex method for solving the Navier-Stokes equations

In this paper, a new 2-D vortex method is developed, which treats the vorticity diffusion in a deterministical way. The Laplacian operator, which describes vorticity diffusion, is approximated by a contour integral. The numerical results of two model problems show that this method has a good accuracy. A primary error estimation is given, and the self-adaptive vortex blob and the boundary conditions are discussed. The project supported by the National Natural Science Foundation of China     

New conditions of stability and convergence of Stokes and Newton iterations for Navier-Stokes equations

This paper considers Stokes and Newton iterations to solve stationary Navier- Stokes equations based on the finite element discretization. We obtain new sufficient conditions of stability and convergence for the two iterations. Specifically, when 0 < σ = (N‖f‖-1)/v² ≤ 1/($\sqrt{2}$+1), the Stokes iteration is stable and convergent, where N is defined in the paper. When 0 < σ ≤ 5/11, the Newton iteration is stable and convergent. This work gives a more accurate admissible range of data for stability and convergence of the two schemes, which improves the previous results. A numerical test is given to verify the theory.     

Numerical study of Navier-Stokes equations

The problem of a viscous incompressible fluid flow around a body of revolution at incidence, which is described by Navier-Stokes equations, is considered. For low Reynolds numbers, the solutions of these equations are smooth functions. A numerical algorithm without saturation is constructed, which responds to solution smoothness. The calculations are performed on grids consisting of 900 (10 × 10 × 9) and 700 (10 × 10 × 7) nodes. On the grid consisting of 900 nodes, a system of 3600 nonlinear equations is solved by a standard code. The pressures on the shaded side of the body of revolution are compared. It is found that a numerical study (on this grid) is feasible for problems with Re ≈ 1. For high Reynolds numbers, the number of grid nodes has to be increased. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 43–52, September–October, 2007.     

A new implicit factored scheme for the compressible Navier-Stokes equations

A new scheme for solving the compressible Navier-Stokes equations is developed. For the inviscid portion of the equations the single step scheme used by the authors is factored according to the sign of the eigenvalues of Jacobian matrix. For the viscous portion of the equations a scheme corrected with operator addition is factored too. The scheme obtained has second order accuracy in time and in space and is used to solve two-dimensional problem. The numerical results of 2-D shock wave-boundary layer interaction are compared with experimental data.     

Stability of Navier-Stokes equation (I)

In this paper we give an example of non-uniqueness of local solution for some kinds of boundary value problem of Navier-Stokes equation.     

Dimensions of attractors for discretizations for Navier-Stokes equations

In this paper, we discretize the 2-D incompressible Navier-Stokes equations with the periodic boundary condition by the finite difference method. We prove that with a shift for discretization, the global solutions exist. After proving some discrete Sobolev inequalities in the sense of finite differences, we prove the existence of the global attractors of the discretized system, and we estimate the upper bounds for the Hausdorff and the fractal dimensions of the attractors. These bounds are indepent of the mesh sizes and are considerably close to those of the continuous version.     

On the stability of general Navier-Stokes type equation

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Impact of singularity of Navier-Stokes equation upon atmospheric motion equations

Some conclusions about the smooth function classes stability for the basic system of equations of atmospheric motion and instability for Navier-Stokes equation are summarized. On the basis of this, by taking the basic system of equations of atmospheric motion via Boussinesq approximation as example to explain in detail that the instability about some simplified models of the basic system of equations for atmospheric motion is caused by the instability of Navier-Stokes equation, thereby, a principle to guarantee the stability of simplified equation is drawn in simplifying the basic system of equations.     

A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations

We note in this study that the Navier-Stokes equations, when expressed in streamfunction-vorticity form, can be approximated to fourth-order accuracy with stencils extending only over a 3 x 3 square of points. The key advantage of the new compact fourth-order scheme is that it allows direct iteration for low-to-medium Reynolds numbers. Numerical solutions are obtained for the model problem of the driven cavity and compared with solutions available in the literature. For Re ? 7500 point-SOR iteration is used and the convergence is fast.     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅱ)

The solvability of the Euler equations about incompressible inviscid fluid based on the stratification theory is discussed. And the conditions for the existence of formal solutions and the methods are presented for calculating all kinds of ill-posed initial value problems. Two examples are given as the evidences that the initial problems at the hyper surface does not exist any unique solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

A high-order characteristics/finite element method for the incompressible Navier-Stokes equations

In this paper we consider a discretization of the incompressible Navier-Stokes equations involving a second-order time scheme based on the characteristics method and a spatial discretization of finite element type. Theoretical and numerical analyses are detailed and we obtain stability results abnd optimal eror estimates on the velocity and pressure under a time step restriction less stringent than the standard Courant-Freidrichs-Levy condition. Finally, some numerical results obtained wiht the code N3S are shown which justify the interest of this scheme and its advantages with respect to an analogous first-order time scheme. © 1997 John Wiley & Sons, Ltd.     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅰ)

The ill posed initial value problem of the Euler equations and the formal solvability of ill posed problem based on stratification theory are discussed. For some ill posed initial value problems, the existence conditions of formal solutions and the methods of how to construct a formal solution are given. Finally, an example is given to discuss the ill posedness of the initial value problem on hyper plane {t=0} in R⁴, and explain that the problem has more than one solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

A new approach to inverse problems of wave equations

We introduce a multi-cost-functional method for solving inverse problems of waveequations.This method has its simplicity,efficiency and good physical interpretation.It hasthe advantage of being programmed for two-or three-(space)dimensional problems as wellas for one-dimensional problems.     

TWO-GRID ERROR ESTIMATES FOR THE STREAM FUNCTION FORM OF NAVIER-STOKES EQUATIONS

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On pressure boundary conditions for the incompressible Navier-Stokes equations

The pressure is a somewhat mysterious quantity in incompressible flows. It is not a thermodynamic variable as there is no ‘equation of state’ for an incompressible fluid. It is in one sense a mathematical artefact—a Lagrange multiplier that constrains the velocity field to remain divergence-free; i.e., incompressible—yet its gradient is a relevant physical quantity: a force per unit volume. It propagates at infinite speed in order to keep the flow always and everywhere incompressible; i.e., it is always in equilibrium with a time-varying divergence-free velocity field. It is also often difficult and/or expensive to compute. While the pressure is perfectly well-defined (at least up to an arbitrary additive constant) by the governing equations describing the conservation of mass and momentum, it is (ironically) less so when more directly expressed in terms of a Poisson equation that is both derivable from the original conservation equations and used (or misused) to replace the mass conservation equation. This is because in this latter form it is also necessary to address directly the subject of pressure boundary conditions, whose proper specification is crucial (in many ways) and forms the basis of this work. Herein we show that the same principles of mass and momentum conservation, combined with a continuity argument, lead to the correct boundary conditions for the pressure Poisson equation: viz., a Neumann condition that is derived simply by applying the normal component of the momentum equation at the boundary. It usually follows, but is not so crucial, that the tangential momentum equation is also satisfied at the boundary.     

Several exact solutions of the simplified Navier-Stokes equations (SNSE)

The Simplified Navier-Stokes equations (SNSE) and their exact solutions for the flow near a rotating disk and the flow in the vicinity of a stagnation point for both two- and three-dimensional flows are presented in this paper. The analysis shows that in the aforementioned cases the exact solutions of the inner-outer-layer-matched SNSE^[4] are completely consistent with those of the complete Navier-Stokes equations (CNSE) and that the exact velocity solutions of D-SNSE^[1,3] agree with those of CNSE, however, the exact pressure solutions of D-SNSE do not agree with those of CNSE. The maximum relative pressure errors between the exact solutions of D-SNSE and CNSE can be as high as a hundred per cent.     

Numerical solution of the Navier-Stokes equations using a finite element method

A finite element algorithm for solving the Navier-Stokes equations is presented for the analysis of high-speed viscous flows. The algorithm uses triangular elements. The unsteady equations are integrated to steady state with a Runge-Kutta time-marching scheme. A postprocessing artificial dissipation term is introduced to stabilize the computations and to dampen dissipation errors. Numerical results are compared with the calculation of uniform flow on a rectangular region which encounters an embedded oblique shock. A shock/turbulent boundary layer problem is also solved and results are compared with experimental data. It is shown that the postprocessing smoothing term and boundary conditions similar to the finite difference method work well in the present numerical studies.