tailieunhanh - Calculation of the horizontal two - dimensional unsteady flows by the method of characteristics

This paper will be concerned with the characteristic form of the twodimensional Saint-Venant equation system, the supplementary equations at the boundaries, the methods of charact eristics for solving the equation system and some numerical experiments. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 25, 2003, No 1 (49 - 64) CALCULATION OF THE HORIZONTAL TWO-DIMENSIONAL UNSTEADY FLOWS BY THE METHOD OF CHARACTERISTICS TRAN GIA LICH, NGUYEN MINH SON AND LE VIET CUONG Institute of Mathematics, Hanoi, Vietnam ABSTRACT. This paper will be concerned with the characteristic form of the twodimensional Saint-Venant equation system , the supplementary equations at the boundaries, the methods of charact eristics for solving the equation system and some numerical experiments. 1. Two-dimensional Saint-Venant equation system The Saint-Venant equation system describing horizontal two-dimensional unsteady flow without turbulent diffusion components can be written in the following form (see [1 , 2]): ( ) where: 1'1 = = ¢1) (~2 (u~ ~ ~g) 0 , A = , By some transformation of the unknown vector U, the equation () can be written in the symmetric form : () where: v~(;J i3 = (~0 ~c v~) , c=Viif . The equation systems () , () , () are quasi-linear hyperbolic. In fact, for any real values s1 , s2 satisfying si + s~ # 0, the matrix A* = s1 A + s2 B has three different real eigen values Aj as follows: A1 + S2V , w + V(si + s~)gH, =w= > 2 = S1 u >. 3 =w-J(si+s~)gH. 50 It is well known (see [1, 2, 5, 6]) that the boundary problem of linear symmetric hyperbolic equation system: Dav Aav Bav at + ax + ay =-;;. ~' () where D, A, B are symmetric matrixs and D > 0, has a solution if beside of the initial conditions, the number of given boundary conditions is equal to the one of negative eigen values Aj of the matrix An = nxA + nyB . In addition, the equation system () has an unique solution, continuously depending on the initial conditions and its right hand side, if the boundary conditions are dissipative. The boundary condition is said to be dissipative if any vector V satisfying this condition also satisfies the following inequality: JJ ([ntD + mxA + myB]V, V)ds 2 0, () s where