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THE NAVIER-STOKES EQUATIONS WITH THE kinematic AND VORTICITY boundary conditionS ON NON-FLAT BOUNDARIES

Acta Mathematica Scientia 2009年第4期29卷 919-948页

作者： Dan Osborne Mathematical Institute University of Oxford 24-29 St Giles

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.

关键词： Navier-Stokes equations incompressible vorticity boundary condition kinematic boundary condition absolute boundary condition non-flat boundary general domain Stokes operator Neumann problem Poisson equation vorticity strong solutions inviscid limit slip boundary condition

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Vertical 2D Modeling of Free Surface Flow with Hydrodynamic Pressure Using SIMPLE Arithmetic in σ Coordinates

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海洋工程：英文版 2004年第1期18卷 79-92页

作者：吴修广沈永明郑永红 State Key Laboratory of Coastal and Offshore Engineering Guangzhou Institute of Energy Conversion

A numerical model for shallow water flow has been developed based on the unsteady Reynolds-averaged Navier-Stokes equations with the hydrodynamic pressure instead of hydrostatic pressure assumption. The equations are transformed into the σ-coordinate system and the eddy viscosity is calculated with the standard k-ε turbulence model. The control volume method is used to discrete the equations, and the boundary conditions at the bed for shallow water models only include vertical diffusion terms expressed with wall functions. And the semi-implicit method for pressure linked equation arithmetic is adopted to solve the equations. The model is applied to the 2D vertical plane flow of a current over two steep-sided trenches for which experiment data are available for comparison and good agreement is obtained. And the model is used to predicting the flow in a channel with a steep-sided submerged breakwater at the bottom, and the streamline is drawn.

关键词： free surface flow hydrodynamic pressure kinematic boundary condition σ-coordinate SIMPLE arithmetic

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