铸造充型过程数值模拟特征分数步长法

Charecteristics Fractional Step Method for Melt FlowSimulation in Casting Process1Lu Tongchao, Ge LiangSchool of Mathematics and System Sciences, Shandong University Jinan (250100)E-mail：lutc@AbstractA computational model on the melt flow simulation in casting process is partial differential equations which include the continuous equation and the momentum equation . We use the fractional step method to deal with the momentum equation. The compution is divided into two steps. In the first step, we use the characteristics finite difference method to compute the intermediate value. In the second step, we gain the possion equation of the pressure with help of the continuous equation; the equation is solved with the iterative method. The convergence and the stability of the equation are analyzed in this paper. Keywords: the fractional step method; method of characteristics; convergence; iterative1IntroductionThe substance of casting process is to pour the liquid metal into the mold cavity and make it cool and freeze. The filling of liquid metal is the first phase of the form of casting. A lot of defects such as cold cut, impurity or gas entrapment and the mould erosion are formed in mould filling. So it is important for gaining the high grade casting to know the process of mould filling. But owing to the complexity of the mould filling, people only had to gain the experiental rule by a lot of experiments for the mould filling. Now along with the development of computer, the melt flow simulation in casting process makes great progress[4].In simulation of the mould filling the fluid flow phenomena are governed by a set of non-linear equations. We could use the numerical methods such as finite element or finite difference to solve the Navier-Stokes equation, and gain the flowing role of liquid. In earlier studies, Chorin(1968) and Teman(1969)[6 ] had brought forward the projection method independently. In 1972 Patankar and Spalding [13 ]brought forward the simple method. And then the simplec method had also been brought forward, and Raithby[11 ] gave the simplex method, Sheng [12 ]gave the simplet method.In this article A computational model on the melt flow simulation in casting process is partial differential equations which include the continuous equation and the momentum equation . We use the fractional step method to deal with the momentum equation. The compution is divided into two steps. In the first step, we use the characteristics finite difference method to compute the intermediate value. In the second step, we gain the possion equation of the pressure with help of the continuous equation; the equation is solved with the iterative method. The convergence and the stability of the equation are analyzed in this paper.2Mathematical modelsIn simulation of the mould filling the fluid flow phenomena are governed by mass and momentum conservation equations that can be described as follows [10 ](a) The momentum equations for the liquid phase are as follows:1 Support by Research Fund for Doctoral of High Education by China State Educationg Ministry(20030422049)1x u p V u g u tx μρρ∂∂∂∂=−⋅−++Δr " (2.1)1y v p V v g v t y μρρ∂∂=−⋅−++Δ∂∂r " (2.2)where ,()x y ,∈Ω(0]t J T ∈=,V ui vj =+r r ris the velocity vector, is the pressure, p x g and y g are the body accelerationcomponents, ρ is the average density and μ is the viscosity. (b) The continuity equation for the liquid region is()0V tρρ∂+⋅=∂r " (2.3)[()]()l s l lf V f V tρρρ−∂0+⋅+⋅=∂r r "" (2.4)where sf ,lf are solid and liquid fractions respectively, and ss l l ff ρρρ+=,. Inthis article the molten metal could be seen as the incompressible liquid. And also assume that themolten metal would not solidity before the end of the mold filling. Then the equation could be simplified as 1s l f f +=0u vV x y∂∂⋅=+=∂∂r " (2.5)(c)The velocity boundary conditions for the liquid region areThe wall boundary and the free surface boundary are included in the velocity boundary conditions. In the situation of wall boundary, the velocity in the vertical orientation to the wall is equal to zero. The velocity in the free surface boundary is demand to meet the continuity equation.0u v V x y∂∂⋅=+=∂∂r" Series of iris images, includin3 fractional step characteristics finite difference schemeDivide the domain , the steps in x-direction and y-direction are Ωx Δ,, nodey Δi x i x =Δ,. We will consider a time step j y j =Δy T Nt Δ= and approximate the solution at times, where 。