Discrete-Time Collocation Scheme for Time-Dependent Equations and Numerical Simulation of the Mold
|Keywords||Navier-Stokes equations Collocation method SOLA-VOF method Free surface Velocity boundary|
Configuration method developed in the past two to three years, to meet the pure interpolation constraints, seek numerical methods for the approximate solution of operator equation, without calculating the numerical integration calculation is simple and convergence advantages, widely used in engineering and many other areas of computational mathematics. Incompressible fluid, the flow process subject to the conservation of mass and momentum conservation, its mathematical form is the continuity equation and Navier-Stokes equations, namely the Navier-Stokes equations: (?) U / (?) T (u · ▽) uv △ u ▽ p = f, x ∈ Ω, t ∈ (0, T). Continuity equation: ▽ · u = 0. u = 0, onΓ × (0, T), u (x, 0) = u 0 (x), inΩ. This type of equation is widely used in many fields of fluid mechanics, foundry engineering, aerodynamics, research for the algorithm has been a hot issue for the people. Casting is an ancient industry, the casting process, molten liquid metal filled cavity (casting) and cooling solidification process generally speaking. Therefore, the numerical simulation technology in the field of casting is first focused on the numerical simulation of the casting process and solidification process, the temperature field and flow field. Therefore, the filling process of numerical simulation in the casting, the pressure field of the velocity field is very important to solve this paper, SOLA method in solving the casting process continuity equation and momentum conservation equations applications, and on this basis, VOF method to determine the free surface and the velocity boundary condition may appear and summarized their velocity boundary condition should satisfy the equation, then this are given, so that in the calculation process greatly reduced due to boundary conditions The same repeated to improve the computing speed and practicality of the flow field. The text is divided into two chapters. The first chapter of the development equation using the collocation method combined with the finite difference format fully discrete collocation method to solve the format piecewise bicubic the Hermit difference and finite-difference space variables and time variables discrete Finally, the error estimates. The second chapter begins with basic equation is given as a discrete form of the continuity equation and the Navier-Stokes equations, subsequent references SOLA method of processing speed and pressure field and the velocity field and the pressure field iterative correction processing in the use of the modified VOF method free surface, the combination of theoretical analysis and the actual situation that may arise summarizes the different situations of the free surface velocity boundary should satisfy the boundary condition equations.