Editing Euler equations (fluid dynamics) (section)

===Rankine–Hugoniot equations===
{{See also|Rayleigh equation|Hugoniot equation}}
Starting from the simplest case, one consider a steady free conservation equation in conservation form in the space domain:

<math display="block">\nabla \cdot \mathbf F = \mathbf 0,</math>

where in general '''F''' is the flux matrix. By integrating this local equation over a fixed volume V<sub>m</sub>, it becomes:

<math display="block"> \int_{V_m} \nabla \cdot \mathbf F \,dV = \mathbf 0.</math>

Then, basing on the [[divergence theorem]], we can transform this integral in a boundary integral of the flux:

<math display="block"> \oint_{\partial V_m} \mathbf F \,ds = \mathbf 0.</math>

This ''global form'' simply states that there is no net flux of a conserved quantity passing through a region in the case steady and without source. In 1D the volume reduces to an [[interval (mathematics)|interval]], its boundary being its extrema, then the divergence theorem reduces to the [[fundamental theorem of calculus]]:

<math display="block"> \int_{x_m}^{x_{m+1}} \mathbf F(x') \,dx' = \mathbf 0,</math>

that is the simple [[finite difference equation]], known as the ''jump relation'':

<math display="block"> \Delta \mathbf F = \mathbf 0.</math>

That can be made explicit as:

<math display="block"> \mathbf F_{m+1} - \mathbf F_m = \mathbf 0,</math>

where the notation employed is:

<math display="block"> \mathbf F_{m} = \mathbf F(x_m).</math>

Or, if one performs an indefinite integral:

<math display="block"> \mathbf F - \mathbf F_0 = \mathbf 0.</math>

On the other hand, a transient conservation equation:

<math display="block">{\partial y \over \partial t} + \nabla \cdot \mathbf F = \mathbf 0 ,</math>

brings to a jump relation:

<math display="block"> \frac{dx}{dt} \, \Delta u = \Delta \mathbf F.</math>

For one-dimensional Euler equations the conservation variables and the flux are the vectors:

<math display="block">\mathbf y = \begin{pmatrix} \frac{1}{v} \\ j \\ E^t \end{pmatrix}, </math>
<math display="block">\mathbf F = \begin{pmatrix} j \\ v j^2 + p \\ v j \left(E^t + p\right) \end{pmatrix}, </math>

where:
* <math>v</math> is the specific volume,
* <math>j</math> is the mass flux.

In the one dimensional case the correspondent jump relations, called the [[Rankine–Hugoniot equation]]s, are:<{{sfn|Chorin|Marsden|2013|p=122|loc= par. 3.2 Shocks}}

<math display="block">\begin{align} 
  \frac{dx}{dt}\Delta \left(\frac{1}{v}\right) &= \Delta j,\\[1.2ex]
                         \frac{dx}{dt} \Delta j &= \Delta(vj^2 + p),\\[1.2ex]
                       \frac{dx}{dt}\Delta E^t &= \Delta(jv(E^t + p)).
\end{align}</math>

In the steady one dimensional case the become simply:

<math display="block">\begin{align} 
                                                            \Delta j &= 0,\\[1.2ex]
                                        \Delta\left(v j^2 + p\right) &= 0,\\[1.2ex]
  \Delta\left(j\left(\frac{E^t}{\rho} + \frac{p}{\rho}\right)\right) &= 0.
\end{align}</math>

Thanks to the mass difference equation, the energy difference equation can be simplified without any restriction:

<math display="block">\begin{align} 
                     \Delta j &= 0, \\[1.2ex]
  \Delta\left(vj^2 + p\right) &= 0, \\[1.2ex]
                   \Delta h^t &= 0,
\end{align}</math>

where <math>h^t</math> is the specific total enthalpy.

These are the usually expressed in the convective variables:

<math display="block">\begin{align}
                                    \Delta j &= 0, \\[1.2ex]
        \Delta\left(\frac{u^2}{v} + p\right) &= 0, \\[1.2ex]
  \Delta\left(e + \frac{1}{2}u^2 + pv\right) &= 0,
\end{align}</math>

where:
* <math>u</math> is the flow speed
* <math>e</math> is the specific internal energy.

The energy equation is an integral form of the '''Bernoulli equation''' in the compressible case. 
The former mass and momentum equations by substitution lead to the Rayleigh equation:

<math display="block"> \frac{\Delta p}{\Delta v} = - \frac {u_0^2}{v_0}. </math>

Since the second term is a constant, the Rayleigh equation always describes a simple [[line (geometry)|line]] in the [[pressure volume diagram|pressure volume plane]] not dependent of any equation of state, i.e. the [[Rayleigh line]]. By substitution in the Rankine–Hugoniot equations, that can be also made explicit as:

<math display="block">\begin{align} 
                               \rho u &= \rho_0 u_0, \\[1.2ex]
                         \rho u^2 + p &= \rho_0 u_0^2 + p_0, \\[1.2ex]
  e + \frac{1}{2}u^2 + \frac{p}{\rho} &= e_0 + \frac{1}{2}u_0^2 + \frac{p_0}{\rho_0}.
\end{align}</math>

One can also obtain the kinetic equation and to the Hugoniot equation. The analytical passages are not shown here for brevity.

These are respectively:

<math display="block">\begin{align} 
  u^2(v, p) &= u_0^2 + (p - p_0)(v_0 + v), \\[1.2ex]
    e(v, p) &= e_0 + \tfrac{1}{2} (p + p_0)(v_0 - v).
\end{align}</math>

The Hugoniot equation, coupled with the fundamental equation of state of the material:

<math display="block"> e = e(v,p),</math>

describes in general in the pressure volume plane a curve passing by the conditions (v<sub>0</sub>, p<sub>0</sub>), i.e. the [[Hugoniot curve]], whose shape strongly depends on the type of material considered.

It is also customary to define a ''Hugoniot function'':{{sfn|Henderson|2000|p=167|loc= par. 2.96 The Bethe–Weyl theorem}}

<math display="block"> \mathfrak h (v,s) \equiv e(v,s) - e_0 + \tfrac{1}{2} (p(v,s) + p_0)(v - v_0),</math>

allowing to quantify deviations from the Hugoniot equation, similarly to the previous definition of the ''hydraulic head'', useful for the deviations from the Bernoulli equation.