Tham khảo tài liệu 'fundamentals of geophysical fluid dynamics part 1', khoa học tự nhiên, vật lý phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Fluid Dynamics 29 Energy Conservation The principle of energy conservation is a basic law of physics but in the context of fluid dynamics it is derived from the governing equations and boundary conditions Secs. rather than independently specified. The derivation is straightforward but lengthy. For definiteness and sufficient for most purposes in GFD we assume that the force potential is entirely gravitational gz or equivalently that any other contributions to V are absorbed into F. Multiplying the momentum equation by pu gives d pdt u Vp gpw pu V pu F after making use of w Dtz from . Multiplying the mass equation by u2 2 gives dp dt -1 u2V pu The sum of these equations is d_ dt pV u gpw V u 12 p 2 pu pu F It expresses how the local kinetic energy density pu2 2 changes as the flow evolves. Energy is the spatial integral of energy density. To obtain a principle for total energy density E two other local conservation laws are derived to accompany . One comes from multiplying the mass equation by gz viz. h gzp gpw -V u gzp dt This says how gzp the local potential energy density changes. Note that the first right-side term is equal and opposite to the first right-side term in gpw is therefore referred to as the local energy conversion rate between kinetic and potential energies. The second accompanying relation comes from and and has the form d 777 pe -pV u -V- u pe pQ dt This expresses the evolution of local internal energy density pe. Its first right-side term is the conversion rate of kinetic energy to internal energy pV u associated with the work done by compression as discused following . 30 Fundamental Dynamics The sum of - yields the local energy conservation relation @E -@t -V- u p E p u F Q where the total energy density is defined as the sum of the kinetic potential and internal components E 1 pu2 gzp pe . All of the conversion terms have canceled each other in
