Editing Euler equations (fluid dynamics) (section)

==Discontinuities==
{{See also|Shock waves|Burgers equation}}
The Euler equations are [[Differential equation|quasilinear]] [[hyperbolic partial differential equation|hyperbolic]] equations and their general solutions are [[wave]]s. Under certain assumptions they can be simplified leading to [[Burgers equation]]. Much like the familiar oceanic [[waves and shallow water|waves]], waves described by the Euler Equations [[breaking wave|'break']] and so-called [[shock waves]] are formed; this is a nonlinear effect and represents the solution becoming [[multi-valued function|multi-valued]]. Physically this represents a breakdown of the assumptions that led to the formulation of the differential equations, and to extract further information from the equations we must go back to the more fundamental integral form. Then, [[weak solution]]s are formulated by working in 'jumps' (discontinuities) into the flow quantities – density, velocity, pressure, entropy – using the [[Rankine–Hugoniot equations]]. Physical quantities are rarely discontinuous; in real flows, these discontinuities are smoothed out by [[viscosity]] and by [[heat transfer]]. (See [[Navier–Stokes equations]])

Shock propagation is studied – among many other fields – in [[aerodynamics]] and [[rocket|rocket propulsion]], where sufficiently fast flows occur.

To properly compute the continuum quantities in discontinuous zones (for example shock waves or boundary layers) from the ''local'' forms{{efn|Sometimes the local and the global forms are also called respectively ''differential'' and ''non-differential'', but this is not appropriate in all cases. For example, this is appropriate for Euler equations, while it is not for Navier-Stokes equations since in their global form there are some residual spatial first-order derivative operators in all the characteristic transport terms that in the local form contains second-order spatial derivatives.}} (all the above forms are local forms, since the variables being described are typical of one point in the space considered, i.e. they are ''local variables'') of Euler equations through [[finite difference method]]s generally too many space points and time steps would be necessary for the memory of computers now and in the near future. In these cases it is mandatory to avoid the local forms of the conservation equations, passing some [[weak formulation|weak forms]], like the [[finite volume method|finite volume one]].

===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.

===Finite volume form===
{{see also|Finite volume method}}
On the other hand, by integrating a generic conservation equation:

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

on a fixed volume ''V<sub>m</sub>'', and then basing on the [[divergence theorem]], it becomes:

<math display="block"> \frac {d}{dt} \int_{V_m} \mathbf y dV + \oint_{\partial V_m} \mathbf F \cdot \hat n ds = \mathbf S .</math>

By integrating this equation also over a time interval:

<math display="block"> \int_{V_m} \mathbf y(\mathbf r, t_{n+1}) \, dV - \int_{V_m} \mathbf y(\mathbf r, t_n) \, dV+ \int_{t_n}^{t_{n+1}} \oint_{\partial V_m} \mathbf F \cdot \hat n \, ds \, dt = \mathbf 0 .</math>

Now by defining the node conserved quantity:

<math display="block">\mathbf y_{m,n} \equiv \frac 1 {V_m} \int_{V_m} \mathbf y(\mathbf r, t_n) \, dV ,</math>

we deduce the finite volume form:

<math display="block">\mathbf{y}_{m,n+1}=\mathbf{y}_{m,n} - \frac{1}{V_m} \int_{t_n}^{t_{n+1}} \oint_{\partial V_m} \mathbf{F} \cdot \hat{n}\, ds \, dt .</math>

In particular, for Euler equations, once the conserved quantities have been determined, the convective variables are deduced by back substitution:

<math display="block">\begin{align} 
\displaystyle  \mathbf u_{m,n} &= \frac{\mathbf j_{m,n}}{\rho_{m,n}}, \\[1.2ex]
\displaystyle          e_{m,n} &= \frac{E^t_{m,n}}{\rho_{m,n}} - \frac{1}{2} u^2_{m,n}.
\end{align}</math>

Then the explicit finite volume expressions of the original convective variables are:{{sfn|Quartapelle|Auteri|2013|p=161|loc=par. 11.10: Forma differenziale: metodo dei volumi finiti}}

{{Equation box 1
|indent=:
|title='''Euler equations'''<br/>(''Finite volume form'')
|equation=<math>\begin{align} 
       \rho_{m,n+1} &= \rho_{m,n} - \frac{1}{V_m}\int_{t_n}^{t_{n+1}}\oint_{\partial V_m}\rho\mathbf{u} \cdot \hat{n}\, ds\, dt \\[1.2ex]
  \mathbf u_{m,n+1} &= \mathbf u_{m,n} - \frac{1}{\rho_{m,n} V_m}\int_{t_n}^{t_{n+1}}\oint_{\partial V_m} (\rho\mathbf{u} \otimes \mathbf{u} - p\mathbf{I}) \cdot \hat{n}\, ds\, dt \\[1.2ex]
  \mathbf e_{m,n+1} &= \mathbf e_{m,n} - \frac{1}{2}\left(u^2_{m,n+1} - u^2_{m,n}\right) - \frac{1}{\rho_{m,n} V_m}\int_{t_n}^{t_{n+1}}\oint_{\partial V_m} \left(\rho e + \frac{1}{2}\rho u^2 + p\right)\mathbf{u} \cdot \hat{n}\, ds\, dt \\[1.2ex]
\end{align}</math>
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