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Calculation of the pressure on the valves of a sluice

This paper is devoted to a nwnerical method for calculating the pressure on the vertical two-dimensional valve basing on Navier Stokes equaions. Numerical solutions at interior points are established by splitting Navie Stokes unsteady two-dimensional equations into two unsteady one-dimensional equations. | Journal of Mechanics, NCNST of Vietnam T. XIX, 1997, No 3 (25 - 34) CALCULATION OF THE PRESSURE ON THE VALVES OF A SLUICE TRAN GIA LICH - LE KIM LUAT- HAN QUOC TRINH Institute of Mathematics 1 Hanoi, Vietnam Abstract. This paper is devoted to a nwnerical method for calculating the pressure on the vertical two-dimensional valve basing on Navier-Stokes equa~ions. Numerical solutions at interior points are established by splitting Navie~-Stokes unsteady two-dimensional equations into two unsteady one-dimensional equations. An implicit scheme is obtained and the solution for these equations is established by the double sweep method. The values at the boundary points are calculated by the method of characteristics. This algorithm is applied to the concrete case presented at the end of this paper 1. Introduction It is very difficult to calculate the pressure fields, especially, in the case of solid boundary. Much attention has been paid to this problem. The aim of this paper is to present a numerical method for calculating the pressure on the vertical two~dimensional valve in hydraulic engineering. It is well known that the Navier-Stock equations for viscous incompressible fluid flows have the dimensional form as following: av at 1 - '\7. + (V · V')V = --V'P+vb.V + F, - v P (1.1} = 0, where V is velocity vector, P - pressure, F - external force, p - density, Let take p = 1. l.l - kinematics viscosity. It is difficult to find directly numerical solutions of equations (1.1). To avoid it, the artificial compression component is added to the continuity equation (see [1, 2]), and we obtain a modification for the Navier-Stokes equations as follows: av Bt + (V · V'}V = -V'P+ vb.V +F, a(P+ vz) ~--::,---'4'-'-. at (1.2} + '\7 . v = 0. We suppose that either the channel has large enough width (in Oy-direction) or the velocity of fluid flow -changes slowly :in Oy~direction, then we can rewrite equations {1.2} in the vertical two-dimensional equations of .