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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:
::(\;1)\quad \quad\frac \;\; = -\frac\left(\rho u\right)
::(\;2)\quad \quad\frac(\rho u) \, = -\frac\left(\rho u^+p\right)
::(\;3)\quad \quad\frac(E^t) = -\frac\left(),
where
:\rho = \, fluid mass density, ()
:u = \, fluid velocity, ()
:e = \, specific internal energy of the fluid, ()
:p = \, fluid pressure, ()
:t = \, time, ()
:x = \, distance, (), and
:E^t = \rho e+\rho \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
::(\;4)\quad\quad p=\left(\gamma-1\right)\rho e,
is valid, where \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
::(\;5)\quad \quad \frac = \text.
For an extensive list of compressible flow equations, etc., refer to NACA Report 1135 (1953).
Note: For a calorically ideal gas \gamma \, is a constant and for a thermally ideal gas \gamma \, 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)
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