Tham khảo tài liệu 'a finite element scheme for shock capturing part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Figure 1. Definition of terms for a moving hydraulic jump Uq velocity at section 0 ỈĨQ depth at section 0 Uị velocity at section 1 hj depth at section 1 Vq Uq - Vw the velocity at section 0 relative to the jump Vy U - Vw relative velocity at section 1 Mass. Now following an infinitesimal fluid element in our moving coordinate system we know that mass is conserved so we have Ặ 0 2 dt where p the fluid density is a constant here. Across the element we have p V h - w 0 3 5 Chapter 1 Introduction or vlhl voho q 4 where q is the relative discharge. Equation 3 may be written in a fixed grid as Vw h C 1 5 where the symbol implies the jump in the quantities across the discontinuity . h ỉ2 - hQ. Momentum. In the same manner we show that momentum is conserved by p P0 - Pl 6 at p W I - Wo p Po - Pl w where J _ 1 z 2 po ị sho pl 4 l2 q ạjỵ - Uq po - Pl 8 which in a fixed coordinate system would be vw Uh U2 h P 9 Energy. Now consider the case of mechanical energy E as it passes through the discontinuity dE 1 TZ r It rr2 - dt p 2 1 1 1 2 Pl l - Po o Pl l - Po o 10 If we multiply 8 by Vw add this to 10 using the relationship for q we have 6 Chapter 1 Introduction p 14 - VQ2 2 P - povo p 1 - vo Vi - vo 2 PiVi - povo Now substituting Equation 8 yields dE n - - . . . p 4 po - 1 1 o 2 PiVi - povo or finally dE _ pqg 0 - 1 dt 4 11 12 13 so that the shallow-water equations lose energy at the jump and it is proportional to the depth differential cubed. If the depth is continuous no energy will be lost. While mathematically an energy gain dEịdt 0 through the jump is a possibility physically it is not. If we restrict ourselves to energy losses through the jump there are two possible cases. a. Case 1 Vq Vj 0 implies h0 hj. The jump progresses downstream through our fluid element Figure 2 . b. Case 2 Vq Vị 0 implies h0 hj. The fluid passes downstream through the jump Figure 3 . TT Figure 2. Example of Case 1 jump ere. T ye . L passes downstream through the flow to be from left
