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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 Rankine-Hugoniot equation ： ウィキペディア英語版 Rankine–Hugoniot conditions The Rankine–Hugoniot conditions, also referred to as Rankine–Hugoniot jump conditions or Rankine–Hugoniot relations, describe the relationship between the states on both sides of a shock wave in a one-dimensional flow in fluids or a one-dimensional deformation in solids. They are named in recognition of the work carried out by Scottish engineer and physicist William John Macquorn Rankine and French engineer Pierre Henri Hugoniot.〔 See also: Hugoniot, H. (1889) ("Mémoire sur la propagation des mouvements dans les corps et spécialement dans les gaz parfaits (deuxième partie)" ) (on the propagation of movements in bodies, especially perfect gases (second part) ), ''Journal de l'École Polytechnique'', vol. 58, pages 1-125.〕 See also Salas (2006) for some historical background. In a coordinate system that is moving with the shock, the Rankine–Hugoniot conditions can be expressed as: :$$ \begin & \rho_1\,u_s = \rho_2(u_s - u_2) &\qquad \text\\ & p_2 - p_1 = \rho_2\,u_2\,(u_s - u_2) = \rho_1\,u_s\,u_2 &\qquad \text\\ & p_2\,u_2 = \rho_1\,u_s\,\left(\tfrac\,u_2^2 + e_2 - e_1\right) &\qquad \text \end where ''u''s is the shock wave speed, ''ρ''1 and ''ρ''2 are the mass density of the fluid behind and inside the shock, ''u''2 is the particle velocity of the fluid inside the shock, ''p''1 and ''p''2 are the pressures in the two regions, and ''e''1 and ''e''2 are the ''specific'' (with the sense of ''per unit mass'') internal energies in the two regions. These equations can be derived in a straightforward manner from equations (12), (13) and (14) below. Using the Rankine-Hugoniot equations for the conservation of mass and momentum to eliminate ''u''s and ''u''2, the equation for the conservation of energy can be expressed as the Hugoniot equation: :$$ e_2 - e_1 = \tfrac\,(p_2 + p_1)\,(v_1-v_2) where ''v''1 and ''v''2 are the uncompressed and compressed specific volumes per unit mass, respectively. ==Basics: Euler equations in one dimension== Consider gas in a one-dimensional container (e.g., a long thin tube). Assume that the fluid is inviscid (i.e., it shows no viscosity effects as for example friction with the tube walls). Furthermore, assume that there is no heat transfer by conduction or radiation and that gravitational acceleration can be neglected. Such a system can be described by the following system of conservation laws, known as the 1D Euler equations, that in conservation form is: ::$(\backslash ;1)\backslash quad\; \backslash quad\backslash frac\; \backslash ;\backslash ;\; =\; -\backslash frac\backslash left(\backslash rho\; u\backslash right)$ ::$(\backslash ;2)\backslash quad\; \backslash quad\backslash frac(\backslash rho\; u)\; \backslash ,\; =\; -\backslash frac\backslash left(\backslash rho\; u^+p\backslash right)$ ::$(\backslash ;3)\backslash quad\; \backslash quad\backslash frac(E^t)\; =\; -\backslash frac\backslash left(),$ where :$\backslash rho\; =\; \backslash ,$ fluid mass density, () :$u\; =\; \backslash ,$ fluid velocity, () :$e\; =\; \backslash ,$ specific internal energy of the fluid, () :$p\; =\; \backslash ,$ fluid pressure, () :$t\; =\; \backslash ,$ time, () :$x\; =\; \backslash ,$ distance, (), and :$E^t\; =\; \backslash rho\; e+\backslash rho\; \backslash fracu^,$ is the total energy density of the fluid, (), while ''e'' is its specific internal energy () Assume further that the gas is calorically ideal and that therefore a polytropic equation-of-state of the simple form ::$(\backslash ;4)\backslash quad\backslash quad\; p=\backslash left(\backslash gamma-1\backslash right)\backslash rho\; e,$ is valid, where $\backslash gamma$ is the constant ratio of specific heats $c\_p/c\_v$. This quantity also appears as the ''polytropic exponent'' of the polytropic process described by ::$(\backslash ;5)\backslash quad\; \backslash quad\; \backslash frac\; =\; \backslash text.$ For an extensive list of compressible flow equations, etc., refer to NACA Report 1135 (1953). Note: For a calorically ideal gas $\backslash gamma\; \backslash ,$ is a constant and for a thermally ideal gas $\backslash gamma\; \backslash ,$ is a function of temperature. In the latter case, the dependence of pressure on mass density and internal energy might differ from that given by equation (4). 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「Rankine–Hugoniot conditions」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.