Abstract:The two-dimensional (2D) lid-driven cavity flow is simulated by applying the ordinary differential quadrature (DQ) method to solve the incompressible Navier-Stokes equations in stream function-vorticity form. There are two boundary conditions, one Dirichlet and one Neumann, for the stream function at each boundary though its governing equation is just second-order. Analysis on this over-specified problem is carried out, based on which a new method is proposed to implement these boundary conditions: the Neumann condition is just considered in the vorticity equation while only the Dirichlet condition is applied in the stream function equation. Availability of this method is verified by comparing its numerical results with benchmark data. Two other methods, the One-layer approach and the Two-layer approach, introduced by Shu and Xue (1998) are also shown as a contrast, and detailed formulations are repeated especially for the One-layer approach to reveal its underlying problem: this method is sensitive to the parity of grid numbers in two directions. Comparison between the present method and the Two-layer approach indicates that the former is more convenient to be used in practice for it avoiding the over-specified problem, and also more accurate, while the latter is relatively more efficient.

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	1. Methods on Applying Stream Function Boundary Conditions in DQ Modeling of 2D Lid-driven Cavity Flow
	WANG Tong,GE Yaojun,CAO Shuyang
	Mechanics 25 September 2011
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	Abstract:The two-dimensional (2D) lid-driven cavity flow is simulated by applying the ordinary differential quadrature (DQ) method to solve the incompressible Navier-Stokes equations in stream function-vorticity form. There are two boundary conditions, one Dirichlet and one Neumann, for the stream function at each boundary though its governing equation is just second-order. Analysis on this over-specified problem is carried out, based on which a new method is proposed to implement these boundary conditions: the Neumann condition is just considered in the vorticity equation while only the Dirichlet condition is applied in the stream function equation. Availability of this method is verified by comparing its numerical results with benchmark data. Two other methods, the One-layer approach and the Two-layer approach, introduced by Shu and Xue (1998) are also shown as a contrast, and detailed formulations are repeated especially for the One-layer approach to reveal its underlying problem: this method is sensitive to the parity of grid numbers in two directions. Comparison between the present method and the Two-layer approach indicates that the former is more convenient to be used in practice for it avoiding the over-specified problem, and also more accurate, while the latter is relatively more efficient.
	TO cite this article:WANG Tong,GE Yaojun,CAO Shuyang. Methods on Applying Stream Function Boundary Conditions in DQ Modeling of 2D Lid-driven Cavity Flow[OL].[25 September 2011] http://en.paper.edu.cn/en_releasepaper/content/4442193


	2. On the Darboux Transformation of the (2+1)-Dimensional Kadomtsev-Petviashvili Equation
	Wang Lei ,Gao YiTian,Gai XiaoLing ,Meng DeXin
	Mechanics 09 January 2009
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	Abstract:The Kadomtsev-Petviashvili (KP) equation describes such situations as the two-dimensional long water waves, surface waves and internal waves in straits or channels. The Darboux transformation (DT) of the KP equation is investigated in this paper. The previously-published four constrain conditions on the DT are hereby proved to be compatible and can be reduced to one. New representations of the solutions are presented, which are more concise than the existing ones.
	TO cite this article:Wang Lei ,Gao YiTian,Gai XiaoLing , et al. On the Darboux Transformation of the (2+1)-Dimensional Kadomtsev-Petviashvili Equation[OL].[ 9 January 2009] http://en.paper.edu.cn/en_releasepaper/content/27526

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	3. Painleve Analysis and N-soliton Solutions for the Hirota-Maccari Equation
	Xin Yu,Yi-Tian Gao,Zhi-Yuan Sun,Xiang-Hua Meng,Ying Liu,Qian Feng,Ming-Zhen Wang
	Mechanics 05 January 2009
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	Abstract:As an extended (2+1)-dimensional Hirota model in fluid dynamics, plasma physics and optical fiber communication, the Hirota-Maccari equation is investigated for its integrability in the Painleve sense. With symbolic computation, such equation is bilinearized by the dependent variable transformations obtained from the truncated Painleve expansion at the constant level term. Furthermore, the corresponding N-soliton solutions are given by the Hirota bilinear method. Those solutions are illustrated and to be shown after the collision having the wave velocity and amplitude of each soliton unchanged except its phase shift in the collision region. Finally, another class of solutions with singular points is presented.
	TO cite this article:Xin Yu,Yi-Tian Gao,Zhi-Yuan Sun, et al. Painleve Analysis and N-soliton Solutions for the Hirota-Maccari Equation[OL].[ 5 January 2009] http://en.paper.edu.cn/en_releasepaper/content/27283


	4. Galilean invariance in two-dimensional lattice BGK models
	Ran zheng
	Mechanics 03 February 2007
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	Abstract:The Galilean invariance inherent in Navier-Stokes equations is not really recovered in two dimensional lattice Boltzmann equation, especially for shock calculation. This paper presents a systematic analysis on this subject.
	TO cite this article:Ran zheng. Galilean invariance in two-dimensional lattice BGK models [OL].[ 3 February 2007] http://en.paper.edu.cn/en_releasepaper/content/11027


	5. Lattice Boltzmann Thermohydro-dynamics and Galilean Invariance
	Zheng Ran
	Mechanics 26 January 2007
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	Abstract: From the point view of group-invairance, recovering the Galilean invariance for the isothermal LBGKE (Qian, D.d’Humieres, and P.Lallemand, 1992), induces a new natural thermal-dynamical system, which is compatible with the elementary statistical thermodynamics. New strategy for the thermal LBGKE calculation was also presented.
	TO cite this article:Zheng Ran. Lattice Boltzmann Thermohydro-dynamics and Galilean Invariance[OL].[26 January 2007] http://en.paper.edu.cn/en_releasepaper/content/10923


	6. A Note on Harten’s Entropy Enforcement Condition
	Ran Zheng
	Mechanics 09 January 2006
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	Abstract:The theme of this paper is to understand the nonlinear stability properties of Harten’s TVD scheme from the point views of symmetry inherent in the PDE. Especial attentions are paid to Harten’s entropy enforcement condition. In this paper, we show that Harten’s entropy enforcement condition is consistent with the group invariant conditions.
	TO cite this article:Ran Zheng. A Note on Harten’s Entropy Enforcement Condition[OL].[ 9 January 2006] http://en.paper.edu.cn/en_releasepaper/content/4904


	7. THE RE-EXPLORATION OF THE SPATIAL OSCILLATIONS IN FINITE DIFFERENCE SOLUTIONS FOR NAVIER-STOKES SHOCKS
	Ran Zheng
	Mechanics 28 October 2005
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	Abstract:Through the preliminary numerical calculations, we gained confidence on the use of the NNDschemes, the principle of choice of the coefficients of the third order derivatives is still based on the linearized conclusion. Under the unsteady, nonlinear conditions, the validate is an open question. Symmeries could help us to reach a new conlusion, which domenstrated that the linearized conclusion is met with the similarity inavirants, but not for the Galileo invariants in general case. In this paper, the spatial oscillations in finite difference solutions for Navier-Stokes shocks are re-explored from a point view of group-invariance theory. It is shown that the oscillations of numerical solutions in the upstream and downstream regions of shock have link with the symmetry broken. The symmetry inherent necessary conditions must be satisfied for the choice of finite difference schemes.
	TO cite this article:Ran Zheng. THE RE-EXPLORATION OF THE SPATIAL OSCILLATIONS IN FINITE DIFFERENCE SOLUTIONS FOR NAVIER-STOKES SHOCKS[OL].[28 October 2005] http://en.paper.edu.cn/en_releasepaper/content/3401


	8. On Symmetries of 1D Lattice-Boltzmann Equation
	Ran Zheng
	Mechanics 27 October 2005
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	Abstract:The Galilean and similarity invariance inherent in Navier-Stokes equations are not really recovered in one dimensional lattice Boltzmann equation, especially for shock calculation. Qian’s theoretical development recovering Navier-Stokes equations in LBE is questionable.
	TO cite this article:Ran Zheng. On Symmetries of 1D Lattice-Boltzmann Equation[OL].[27 October 2005] http://en.paper.edu.cn/en_releasepaper/content/3394


	9. On Galilean Invariance of 1D Lattice-Boltzmann Equation
	Ran Zheng
	Mechanics 18 October 2005
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	Abstract:The Galilean invariance is not really recovered in one dimensional lattice Boltzmann equation, especially for shock calculation. Qian’s theoretical development recovering Navier-Stokes equations in LBE is questionable.
	TO cite this article:Ran Zheng. On Galilean Invariance of 1D Lattice-Boltzmann Equation[OL].[18 October 2005] http://en.paper.edu.cn/en_releasepaper/content/3284

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