Hierarchical Expansion Method in the Solution of the Navier-Stokes Equations for Incompressible Fluids in Laminar Two-Dimensional Flow

Among the several methods used to solve the Navier-Stokes equations Hierarchical Expansion Method has demonstrated satisfactory results. This work aimed to apply the expansion of the variables in hierarchical functions for the solution of the Navier-Stokes equations for incompressible fluids in two dimensions in laminar flow. This method is based on the finite element method. The expansion functions in this study were based on Legendre polynomials, adjusted in the rectangular elements in such a way that corner, side and area functions were defined. The order of the expansion functions associated with the sides and with the area of the elements is adjusted to the necessary or desired degree. This method is denominated by Hierarchical Expansion Method. In order to validate the proposed numeric method three well-known problems of the literature in two dimensions were analyzed;however, for this paper only one problem was presented. The results demonstrated that method was able to provide precise results. From the results obtained in this paper it is possible to conclude that the hierarchical expansion method can be effective for the solution of fluid dynamic problems that involve incompressible fluids.

UNSTEADY/STEADY NUMERICAL SIMULATION OF THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON ARTIFICIAL COMPRESSIBILITY

A mixed algorithm of central and upwind difference scheme for the solution of steady/unsteady incompressible Navier-Stokes equations is presented. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The numerical flux of semi-discrete equations is computed by using the Roe approximation. Time accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The algebraic turbulence model of Baldwin-Lomax is ulsed in this work. As examples, the solutions of flow through two dimensional flat, airfoil, prolate spheroid and cerebral aneurysm are computed and the results are compared with experimental data. The results show that the coefficient of pressure and skin friction are agreement with experimental data, the largest discrepancy occur in the separation region where the lagebraic turbulence model of Baldwin-Lomax could not exactly predict the flow.

分数阶不可压缩Navier-Stokes方程解的爆破性准则

采用Fourier分析及其标准技巧,研究分数阶不可压缩Navier-Stokes方程在齐次Sobolev-Gevrey空间H_(a,σ)^(s)(R^(3))(a>0,σ>1,5/2-2α<s<3/2,1≤α≤5/4)中的初值问题.首先证明当初值u_(0)∈H_(a,σ)^(s)(R^(3))方程存在唯一解u∈C(0,T*);H_(a,σ)^(s)(R^(3));其次证明当T*<∞时,解的指数型爆破准则.

二维不可压缩Navier-Stokes-Landau-Lifshitz方程组的全局强解

本文在二维光滑有界区域中研究不可压缩的Navier-Stokes-Landau-Lifshitz方程组的初边值问题.在初始密度包含真空的情况下,证明在具有任意大的初始速度以及初始时刻宏观分子取向力梯度变化适当小的条件下,该问题全局强解的存在唯一性.

BOUNDARY LAYER ASYMPTOTIC BEHAVIOR OF INCOMPRESSIBLE NAVIER-STOKES EQUATION IN A CYLINDER WITH SMALL VISCOSITY

The object of this article is to study the boundary layer appearing at large Reynolds number （small viscosity ε） incompressible Navier Stokes Equation in a cylinder in space dimension three. These are Navier-Stokes equations linearized around a fixed velocity flow： the authors study the convergence as ε →0 to the inviscid type equations, the authors define the correctors needed to resolve the boundary layer and obtain convergence results valid up to the boundary and the authors also study the behavior of the boundary layer when, simultaneously, time and the Reynolds number tend to infinity, in which case the boundary layer tends to pervade the whole domain.

STABILITY OF VISCOUS SHOCK WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

We study the large-time behavior toward viscous shock waves to the Cauchy problem of the one-dimensional compressible isentropic Navier-Stokes equations with density- dependent viscosity. The nonlinear stability of the viscous shock waves is shown for certain class of large initial perturbation with integral zero which can allow the initial density to have large oscillation. Our analysis relies upon the technique developed by Kanel~ and the continuation argument.

Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations

This paper studies a low order mixed finite element method （FEM） for nonstationary incompressible Navier-Stokes equations. The velocity and pressure are approximated by the nonconforming constrained Q1^4ot element and the piecewise constant, respectively. The superconvergent error estimates of the velocity in the broken H^1-norm and the pressure in the L^2-norm are obtained respectively when the exact solutions are reasonably smooth. A numerical experiment is carried out to confirm the theoretical results.

On the Incompressible Navier-Stokes Equations with Damping

We consider dynamics system with damping, which are obtained by some transformations from the system of incompressible Navier-Stokes equations. These have similar properties to original Navier-Stokes equations the scaling invariance. Due to the presence of the damping term, conclusions are different with proving the origin of the incompressible Navier-Stokes equations and get some new conclusions. For one form of dynamics system with damping we prove the existence of solution, and get the existence of the attractors. Moreover, we discuss with limit-behavior the deformations of the Navier-Stokes equation.

A class of fully third-order accurate projection methods for solving the incompressible Navier-Stokes equations

In this paper, a fully third-order accurate projection method for solving the incompressible Navier-Stokes equations is proposed. To construct the scheme, a continuous projection procedure is firstly presented. We then derive a sufficient condition for the continuous projection equations to be temporally third-order accurate approximations of the original Navier-Stokes equations by means of the localtruncation-error-analysis technique. The continuous projection equations are discretized temporally and spatially to third-order accuracy on the staggered grids, resulting in a fully third-order discrete projection scheme. The possibility to design higher-order projection methods is thus demonstrated in the present paper. A heuristic stability analysis is performed on this projection method showing the probability of its being stable. The stability of the present scheme is further verified through numerical tests. The third-order accuracy of the present projection method is validated by several numerical test cases.

NONLINEAR STABILITY OF VISCOUS SHOCK WAVES FOR ONE-DIMENSIONAL NONISENTROPIC COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH A CLASS OF LARGE INITIAL PERTURBATION

We study the nonlinear stability of viscous shock waves for the Cauchy problem of one-dimensional nonisentropic compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas. The viscous shock waves are shown to be time asymptotically stable under large initial perturbation with no restriction on the range of the adiabatic exponent provided that the strengths of the viscous shock waves are assumed to be sufficiently small.The proofs are based on the nonlinear energy estimates and the crucial step is to obtain the positive lower and upper bounds of the density and the temperature which are uniformly in time and space.

THE FUNDAMENTAL SOLUTION METHOD FOR INCOMPRESSIBLE NAVIER STOKES EQUATIONS

A complete boundary integral formulation for incompressible Navier Stokes equations with time discretization by operator splitting is developed by using the fundamental solutions of the Helmhotz operator equation with different orders. The numerical results for the lift and the drag hysteresis associated with a NACA0012 aerofoil oscillating in pitch are good in comparison with available experimental data.

Direct spectral domain decomposition method for 2D incompressible Navier-Stokes equations

An efficient direct spectral domain decomposition method is developed coupled with Chebyshev spectral approximation for the solution of 2D, unsteady and in- compressible Navier-Stokes equations in complex geometries. In this numerical approach, the spatial domains of interest are decomposed into several non-overlapping rectangu- lar sub-domains. In each sub-domain, an improved projection scheme with second-order accuracy is used to deal with the coupling of velocity and pressure, and the Chebyshev collocation spectral method （CSM） is adopted to execute the spatial discretization. The influence matrix technique is employed to enforce the continuities of both variables and their normal derivatives between the adjacent sub-domains. The imposing of the Neu- mann boundary conditions to the Poisson equations of pressure and intermediate variable will result in the indeterminate solution. A new strategy of assuming the Dirichlet bound- ary conditions on interface and using the first-order normal derivatives as transmission conditions to keep the continuities of variables is proposed to overcome this trouble. Three test cases are used to verify the accuracy and efficiency, and the detailed comparison be- tween the numerical results and the available solutions is done. The results indicate that the present method is efficiency, stability, and accuracy.

ON A NEW 3D MODEL FOR INCOMPRESSIBLE EULER AND NAVIER-STOKES EQUATIONS

In this paper, we investigate some new properties of the incompressible Euler and Navier-Stokes equations by studying a 3D model for axisymmetric 3D incompressible Euler and Navier-Stokes equations with swirl. The 3D model is derived by reformulating the axisymmetric 3D incompressible Euler and Navier-Stokes equations and then neglecting the convection term of the resulting equations. Some properties of this 3D model are reviewed. Finally, some potential features of the incompressible Euler and Navier-Stokes equations such as the stabilizing effect of the convection are presented.

The Boundary Layer Equations and a Dimensional Split Method for Navier-Stokes Equations in Exterior Domain of a Spheroid and Ellipsoid

In this paper, the boundary layer equations (abbreviation BLE) for exterior flow around an obstacle are established using semi-geodesic coordinate system (S-coordinate) based on the curved two dimensional surface of the obstacle. BLE are nonlinear partial differential equations on unknown normal viscous stress tensor and pressure on the obstacle and the existence of solution of BLE is proved. In addition a dimensional split method for dimensional three Navier-Stokes equations is established by applying several 2D-3C partial differential equations on two dimensional manifolds to approach 3D Navier-Stokes equations. The examples for the exterior flow around spheroid and ellipsoid are presents here.

解不可压缩Navier-Stokes方程的非精确块因子分解预处理子

针对相关于不可压缩Navier-Stokes方程数值求解的一类3×3块结构的线性方程组,基于线性方程组的等价形式,构造了一个非精确的块因子分解预处理子,在新的特征值等价矩阵形式的基础上,得到了预处理矩阵特征值实部和虚部的上下界估计.数值实验表明,与已有的预处理子相比,所构造的预处理子可以使得GMRES迭代方法对网格尺寸,网格形式以及粘度系数的依赖性都比较弱,且在迭代步数和CPU时间上都占优.

The proper orthogonal decomposition method for the Navier-Stokes equations

The proper orthogonal decomposition (POD) method for the instationary Navier-Stokes equations is considered. Several numerical approaches to evaluating the POD eigenfunctions are presented. The POD eigenfunctions are applied as a basis for a Galerkin projection of the instationary Navier-Stokes equations. And a low-dimensional ordinary differential models for fluid flows governed by the instationary Navier-Stokes equations are constructed. The numerical examples show that the method is feasible and efficient for optimal control of fluids.

粘性依赖于密度的一维等熵可压缩Navier-Stokes方程组粘性激波的非线性稳定性

该文主要研究粘性系数依赖于密度的一维等熵可压缩Navier-Stokes方程组Cauchy问题整体解的大时间渐近行为,主要研究目的是改进文献[7]的结果至γ>1,κ≥0.注意到γ=2,κ=1时,一维等熵可压缩Navier-Stokes方程组对应于Saint-Venant浅水波方程组,该方程组描述了地表浅水运动的规律,在物理学和海洋学中有重要的应用^([1,4,6])。注意到文献^([7])中通过利用Kanel的方法^([19])来推导比容的一致上下界估计,在得出比容的上界时,该方法要求κ<1/2.对该文所研究的问题而言,需要首先利用Kanel’的方法^([19])来推导比容的一致上下界估计.为了扩大κ的取值范围,还需要对比容的上下界作更为精细的能量估计.在得出比容的一致上下界估计之后,可通过精心设计的连续性技巧,将Navier-Stokes方程组的局部解一步步延拓为整体解,并得到对应的大时间渐近行为.

THE NAVIER-STOKES EQUATIONS WITH THE KINEMATIC AND VORTICITY BOUNDARY CONDITIONS ON NON-FLAT BOUNDARIES

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.

A Short and Simple Solution of the Millennium Problem about the Navier-Stokes Equations and Similarly for the Euler Equations

This paper presents a very short solution to the 4th Millennium problem about the Navier-Stokes equations. The solution proves that there cannot be a blow up in finite or infinite time, and the local in time smooth solutions can be extended for all times, thus regularity. This happily is proved not only for the Navier-Stokes equations but also for the inviscid case of the Euler equations both for the periodic or non-periodic formulation and without external forcing (homogeneous case). The proof is based on an appropriate modified extension in the viscous case of the well-known Helmholtz-Kelvin-Stokes theorem of invariance of the circulation of velocity in the Euler inviscid flows. This is essentially a 1D line density of (rotatory) momentum conservation. We discover a similar 2D surface density of (rotatory) momentum conservation. These conservations are indispensable, besides to the ordinary momentum conservation, to prove that there cannot be a blow-up in finite time, of the point vorticities, thus regularity.

A FINITE ELEMENT SOLVER FOR NAVIER-STOKES EQUATIONS VIA VORTICITY AND VELOCITY

The incompressible Navier Stokes equations are solved via variables of vorticity and velocity. Firstly, a rigorous variational framework with the equivalence between the velocity pressure and the vorticity velocity formulations is presented in a Lipschitz domain. Next, a class of Galerkin finite element approximations of the corresponding variational form is introduced, and a convergence analysis is given for the Stokes problem. Finally, an iterative finite element solver for the Navier Stokes problem is proposed.