UDC 532.785We use the example of the two-dimensional problem of siphon filling of a mold by molten metal to compare two methods of studying the flow of a viscous fluid: in vortex-function flow variables (~, ~) and in pressurevelocity variables (p, v). The superiority of the (w, r is demonstrated. Two figures. Bibliography: titlesOne of the most important difficulties in solving applied problems of fluid dynamics in application to pouring ingots and castings, especially in the case of bodies of complicated shape, is the choice of the most effective numerical method. In the present paper we use the example of solving the two-dimensional problem of siphon filling of a mold with molten metal to study the comparative effectiveness of the method of separation of the physical variables of pressure and velocity with the two-field method using the vortexfunction flow variables (w, ~).In the method of vortex-function flow variables (m,~) the equation of continuity is automatically satisfied, while the Navier-Stokes equation transforms intoO~rwhere Re is the Reynolds number and f is the external force density [1,2]. In solving the problem by this method impermeability and adhesion conditions were imposed on the rigid boundaries (the bottom and wall of the mold), while on the free surface and in the pouring hole the values of the velocities were taken as v j_ = 0.008 m/sec and v^ = 0.533 m/sec. The equations and the boundary conditions were discretized on an 8 x 8 grid with step 0.05 m over the x-axis and 0.15 m over the y-axis. At each time step the system (1)-(2) was solved. To solve Eq.(2) a recursive procedure was applied with 20 iterations. In the case of solving the problem in the variables (p, v) [3, 4] the main difficulty is taking account of the incompressibility of the fluid. Even the small errors that inevitably arise in using a computer lead to violation of this condition and hence to divergence of the solution. To avoid this we used the method of separation of physical variables [4] and broke the equation of motion into the following system:Here T is the time step, V is the auxiliary velocity, 17 t and iTt+zxt are the values of the velocity at different times, and 7r = PIP is the dynamic pressure. In solving the problem by this method we applied a 10 • 10 chessboard grid to get a finite-difference approximation of the system (3)-(5) (finding the components of velocity and pressure at different nodes). To discretize the boundary conditions we used "fictitious" nodes and wrote the conditions themselves as: Vi = 0, t~ = -V A on the rigid boundaries, V_L =0, VA =--VA on the axis of the mold, Vm = V^ = -V A on the free surface, V~ erp,
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