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CHL - A Finite Element Scheme for Shock Capturing_2

Tham khảo tài liệu 'chl - a finite element scheme for shock capturing_2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Simpo PDF Merge and Split Unregistered Version - http www.simpopdf.com V I VQh0 -q 4 where q is the relative discharge. Equation 3 may be written in a fixed grid as vwlh I 5 where the symbol implies the jump in the quantities across the discontinuity e.g. h - ÌĨQ. 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 s - 1 z 2 po ị sho pl 4 l2 q - 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 y.2 1 Tz rr2 TZ __ p VqAqơq Pl l - PqVq 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 Simpo PDF Merge and Split Unregistered Version - http www.simpopdf.com dE dt p 2. q vị - Vg 2 1 1 - poyo 11 p 1 q yỵ - vo Vi - vo 2 PiVi - PqVq Now substituting Equation 8 yields dE dt p 1 Po - Pi Vi vo 2 PiVi - povo 12 or finally dE _ pqg 0 - 1 dt 4 iQỈỉi 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 dE dt 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 Vị 0 implies h0 hj. The jump progresses downstream through our fluid element Figure 2 . b. Case 2 Vq Vỵ 0 implies ÌỈQ hj. The fluid passes downstream through the jump Figure 3 . Figure 2. Example of Case 1 jump passes downstream through the fluid element Here we have arbitrarily chosen the flow to be from left to right if we had chosen the opposite direction one would simply have horizontal mirror images of Figures 2 and 3. A fluid particle that is about to be swept into or caught by the jump is considered in front of the jump. A fluid element that has passed through the jump is now behind it. Therefore we may conclude .