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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク 1-D Saint Venant equation ： ウィキペディア英語版 One-dimensional Saint-Venant equation In fluid dynamics, the one-dimensional (1-D) Saint-Venant equation was derived by Adhémar Jean Claude Barré de Saint-Venant, and is commonly used to model transient open-channel flow and surface runoff. It is a simplification of the two-dimensional (2-D) shallow water equations, which are also known as the two-dimensional Saint-Venant equations. The 1-D simplification is used exclusively in models including HEC-RAS, SWMM5, (InfoWorks ), ISIS, Flood Modeller, MIKE 11, and MIKE SHE because it is significantly easier to solve than the full shallow water equations. Common applications of the 1-D Saint-Venant Equation include dam break analyses, storm pulses in an open channel, as well as storm runoff in overland flow. ==Derivation from Navier–Stokes equations== The 1-D Saint-Venant equation can be derived from the Navier–Stokes equations that describe fluid motion. The ''x''-component of the Navier–Stokes equations – when expressed in Cartesian coordinates in the ''x''-direction – can be written as: :$\backslash frac\; +\; u\; \backslash frac\; +\; v\; \backslash frac+\; w\; \backslash frac=\; -\backslash frac\; \backslash frac\; +\; \backslash nu\; \backslash left(\backslash frac\; +\; \backslash frac\; +\; \backslash frac\backslash right)+\; f\_x,$ where ''u'' is the velocity in the ''x''-direction, ''v'' is the velocity in the ''y''-direction, ''w'' is the velocity in the ''z''-direction, ''t'' is time, ''p'' is the pressure, ρ is the density of water, ν is the kinematic viscosity, and ''f''x is the body force in the ''x''-direction. + \frac + \frac\right)= 0. |- | style="vertical-align:top;" | 2. || Assuming one-dimensional flow in the ''x''-direction it follows that: :$v\backslash frac+\; w\; \backslash frac\; =\; 0$ |- | style="vertical-align:top;" | 3. || Assuming also that the pressure distribution is approximately hydrostatic it follows that: :$p\; =\; \backslash rho\; g\; h$ or in differential form: :$\backslash partial\; p\; =\; \backslash rho\; g\; \backslash left(\backslash partial\; h\; \backslash right).$ And when these assumptions are applied to the ''x''-component of the Navier–Stokes equations: :$-\backslash frac\; \backslash frac\; =\; -\backslash frac\backslash frac\; =\; -g\; \backslash frac.$ |- | style="vertical-align:top;" | 4. || There are 2 body forces acting on the channel fluid, gravity, and friction: :$f\_x\; =f\_\; +\; f\_$ where ''f''x,g is the body force due to gravity and ''f''x,f is the body force due to friction. |- | style="vertical-align:top;" | 5. || ''f''x,g can be calculated using basic physics and trigonometry: :$$ F_ = (\sin\theta) gM where ''F''g is the force of gravity in the ''x''-direction, θ is the angle, and ''M'' is the mass. The expression for sin θ can be simplified using trigonometry as: :$$ \sin\theta = \frac. For small θ (reasonable for almost all streams) it can be assumed that: :$$ \sin\theta = \tan\theta = \frac = S and given that ''f''x represents a force per unit mass, the expression becomes: :$$ f_ = gS. |- | style="vertical-align:top;" | 6. || Assuming the energy grade line is not the same as the channel slope, and for a reach of consistent slope there is a consistent friction loss, it follows that: :$$ f_ = S_f g. |- | style="vertical-align:top;" | 7. || All of these assumptions combined arrives at the 1-dimensional Saint-Venant equation in the ''x''-direction: :$\backslash frac\; +\; u\; \backslash frac\; +\; g\; \backslash frac\; +\; g\; (S\; -\; S\_f)\; =\; 0$ :$(a)\backslash quad\; \backslash \; \backslash \; (b)\backslash quad\; \backslash \; \backslash \; \backslash \; (c)\; \backslash qquad\; (d)\; \backslash \; \backslash \; \backslash \; (e)$ where (a) is the local acceleration term, (b) is the convective acceleration term, (c) is the pressure gradient term, (d) is the gravity term, and (e) is the friction term.〔Saint-Venant, A. (1871), ''Theorie du mouvement non permanent des eaux, avec application aux crues des rivieres et a l’introduction de marees dans leurs lits''. Comptes rendus des seances de l’Academie des Sciences.〕 |} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「One-dimensional Saint-Venant equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.