tailieunhanh - Wave Propagation 2011 Part 10

Tham khảo tài liệu 'wave propagation 2011 part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Differential Quadrature Method for Linear Long Wave Propagation in Open Channels 257 The celerity used in Eq. 10 depends on the channel cross-section geometry the flow area and the resistance formula Sf . For a trapezoidal channel C is defined as follows C aC 2V1 z y b ZV0 1 Z I 1-----1 ----z---- 2 b 2 11 z2 y0 b 2zy0 14 Here a 1 for Chezy formula a 4 3 for the Manning s formula b is bottom width L and z is the side slope L L . 5. Application of differential quadrature method The equation 10 obtained as a result of an arrangement of of the linearized St. Venant equations can be rewritten for the solution of DQM as seen in Eq. 15 R I N N 1 2Ar sQ r c j-Dh2j Q 0 i 1 s 1 15 r 1 V j 1 j 1 J where N is the number of sampling grid points in the x direction R is the number of sampling grid points in time direction and A B B 2 are weight matrix coefficients. Determining the boundary conditions Eq. 15 is solved for the Q i s values. For example when Q x 0 f1 t and Q 0 t f2 x Eq. 15 yields R I N N 1 2AQ 1 C2Bj i-Dh2B 2 q . -A1Q1 - cBw-DhBli Q1 s r 2 V j 2 j 2 J 16 where i 2 3 .N and s 2 3 .R. Using the properties of Legendre polynomials the weight coefficient matrix can be written as Shu et al. 2004 Li nx - Xi Bi k L--- i k Lk xi - xk Bi i i k k 1 Bk 2 Bi k i k xi - xk N B -2 B i k i i i k 1 17 18a 18b 19a 19b For the numerical discretization several different methods are available. It can be selected with equal intervals or non-equal intervals like Chebyshev-Gauss-Lobatto grid points or 258 Wave Propagation in Materials for Modern Applications with the normalization of the routes of Legendre polynomials Shu 2000 . In the present study several different approximations have been tested and Chebyshev-Gaus-Lobatto grid points have been selected. In the Chebyshev-Gaus-Lobatto approximation by writing _ 1 L i -1 r 2 1 - cosN Ĩ n 20 the locations as the calculation points can be calculated as follows __T - T1 x. rN - r 21 In time domain this yields . r- r1 rR - r 22