Tham khảo tài liệu 'applied computational fluid dynamics techniques - wiley episode 1 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | SIMPLE EULER NAVIER-STOKES solvers 163 This is the same as central differencing A stability analysis of the RHS ri -- - ui 1 - Ui-1 2h indicates that only every second node is coupled allowing zero-energy or chequerboard modes in the solution. Therefore stabilizing terms have to be added to re-couple neighbouring nodes. The two most common stabilizing terms added are as follows. 1. Terms of second order h2 d2 dx2 which result in an additional RHS term of Ui 1 - 2ui Ui-1 which equals 4 for the -1 1 -1 chequerboard pattern on a uniform mesh shown in Figure . Most TVD schemes use this type of stabilization term. Figure . Chequerboard mode on a uniform 1-D grid 2. Terms of fourth order h4 d4 dx4 which result in an additional RHS term of Ui 2 - 4ui 1 6ui - 4ui-1 Ui-2 which equals 16 for the -1 1 -1 chequerboard pattern on a uniform mesh shown in Figure . Observe that this type of stabilization term has a much more pronounced effect than second-order terms. Therefore one may use much smaller constants when adding them. The fourth-order operator can be obtained in several ways. One obvious choice is to perform two V2-passes over the mesh Jameson et al. 1986 Mavriplis and Jameson 1990 . Another option is to first obtain the gradients of u at points and then to approximate the third derivatives by taking a difference between first derivatives obtained from the gradients and the first derivatives obtained directly from the unknowns Peraire et al. 1992a . The implications of choosing either of these approaches will be discussed in more depth in Chapter 10. 164 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES . EQUIVALENCY WITH FVM Before going on the GFEM using linear elements will be shown to be equivalent to the FVM. Integration by parts in yields ri - N V- Nj F ũ j dQ Ị V Ni Nj F ũ j dQ Ị . As seen from Section for linear triangles this integral reduces to r A VN Ẹ F ũj 3 2 Si i ẸF uj jel jel and in particular for node a in .
