Editing Euler equations (fluid dynamics) (section)

==Constraints==
It has been shown that Euler equations are not a complete set of equations, but they require some additional constraints to admit a unique solution: these are the [[equation of state]] of the material considered. To be consistent with [[thermodynamics]] these equations of state should satisfy the two laws of thermodynamics. On the other hand, by definition non-equilibrium system are described by laws lying outside these laws. In the following we list some very simple equations of state and the corresponding influence on Euler equations.

===Ideal polytropic gas===
{{See also|ideal gas}}
For an ideal polytropic gas the fundamental [[equation of state]] is:{{sfn|Quartapelle|Auteri|2013|p=A-61|loc=Appendix E}}

<math display="block">e(v, s) = e_0 e^{(\gamma-1)m\left(s-s_0\right)} \left({v_0 \over v}\right)^{\gamma-1},</math>

where <math>e</math> is the specific energy, <math>v</math> is the specific volume, <math>s</math> is the specific entropy, <math>m</math> is the molecular mass, <math>\gamma</math> here is considered a constant ([[polytropic process]]), and can be shown to correspond to the [[heat capacity ratio]]. This equation can be shown to be consistent with the usual equations of state employed by thermodynamics.

{{hidden
|Demonstration of consistency with the thermodynamics of an ideal gas
|By the thermodynamic definition of temperature:

<math display="block">T(e) \equiv {\partial e \over \partial s} = (\gamma - 1) m e</math>

Where the temperature is measured in energy units. At first, note that by combining these two equations one can deduce the '''[[ideal gas law]]''':

<math display="block">p v = m T,</math>

or, in the usual form:

<math display="block">p = n T,</math>

where: <math>n \equiv \frac m v</math> is the number density of the material. On the other hand the ideal gas law is less strict than the original fundamental equation of state considered.

Now consider the molar heat capacity associated to a process ''x'':

<math display="block">c_x = \left(m T {\partial s \over \partial T}\right)_x</math>

according to the first law of thermodynamics:

<math display="block">d e(v,s)=-p dv + T \, ds</math>

it can be simply expressed as:

<math display="block">c_x \equiv m \left({\partial e \over \partial T}\right)_x + m p \left({\partial v \over \partial T}\right)_x</math>

Now inverting the equation for temperature T(e) we deduce that for an ideal polytropic gas the isochoric heat capacity is a constant:

<math display="block">c_v \equiv m \left({\partial e \over \partial T}\right)_v = m {d e \over dT} = \frac {1}{(\gamma -1)}</math>

and similarly for an ideal polytropic gas the isobaric heat capacity results constant:

<math display="block">c_p \equiv m \left({\partial e \over \partial T}\right)_p + m p  \left({\partial v \over \partial T}\right)_p = m {d e \over dT} + p \left({\partial v \over \partial T}\right)_p = \frac {1}{(\gamma -1)} + 1</math>

This brings to two important [[relations between heat capacities]]: the constant gamma actually represents the '''[[heat capacity ratio]]''' in the ideal polytropic gas:

<math display="block">\frac {c_p}{c_v}= \gamma</math>

and one also arrives to the '''Meyer's relation''':

<math display="block">c_p = c_v+1</math>

The specific energy is then, by inverting the relation T(e):

<math display="block">e(T) = \frac {mT} {\gamma - 1} = c_v m T</math>

The specific enthalpy results by substitution of the latter and of the ideal gas law:

<math display="block">h(T) \equiv e(T) + (p v)(T) = c_v m T + m T = c_p m T</math>

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From this equation one can derive the equation for pressure by its thermodynamic definition:

<math display="block">p(v,e) \equiv - {\partial e \over \partial v} = (\gamma - 1) \frac e v.</math>

By inverting it one arrives to the mechanical equation of state:

<math display="block">e(v,p) = \frac {pv}{\gamma - 1}.</math>

Then for an ideal gas the compressible Euler equations can be simply expressed in the ''mechanical'' or ''primitive variables'' specific volume, flow velocity and pressure, by taking the set of the equations for a thermodynamic system and modifying the energy equation into a pressure equation through this mechanical equation of state. At last, in convective form they result:

{{Equation box 1
|indent=:
|title='''Euler equations for an ideal polytropic gas'''<br/>(''convective form''){{sfn|Toro|1999|p= 91|loc=par 3.1.2 Nonconservative formulations}}
|equation=<math>\begin{align} 
           {Dv \over Dt} &= v\nabla \cdot \mathbf{u} \\[1.2ex]
  \frac{D\mathbf{u}}{Dt} &= v\nabla p + \mathbf{g} \\[1.2ex]
           {Dp \over Dt} &= -\gamma p\nabla \cdot \mathbf{u}
\end{align}</math>
|cellpadding
|border
|border colour = #FF00FF
|background colour = #ECFCF4
}}

and in one-dimensional quasilinear form they results:

<math display="block"> \frac{\partial \mathbf y}{\partial t} + \mathbf A \frac{\partial \mathbf y}{\partial x}  = {\mathbf 0}. </math>

where the conservative vector variable is:

<math display="block">{\mathbf y}=\begin{pmatrix}v\\ u  \\p \end{pmatrix},</math>

and the corresponding jacobian matrix is:{{sfn|Zingale|2013|p=}}{{sfn|Toro|1999|p= 92}}

<math display="block">{\mathbf A}=\begin{pmatrix}u & -v & 0 \\ 0 & u & v \\ 0 & \gamma p & u \end{pmatrix}.</math>

=== Steady flow in material coordinates {{anchor|Steady flow in streamline coordinates}} ===
In the case of steady flow, it is convenient to choose the [[Frenet–Serret frame]] along a [[Streamlines, streaklines, and pathlines|streamline]] as the [[coordinate system]] for describing the steady [[momentum]] Euler equation:{{sfn|Fay|1994|pp=150-152}}
<math display="block">
\boldsymbol{u}\cdot\nabla \boldsymbol{u} = - \frac{1}{\rho} \nabla p,
</math>

where  <math>\mathbf u</math>, <math>p</math> and <math>\rho</math> denote the [[flow velocity]], the [[pressure]] and the [[density]], respectively.

Let <math>\left\{ \mathbf e_s, \mathbf e_n, \mathbf e_b \right\}</math> be a Frenet–Serret [[orthonormal basis]] which consists of a tangential unit vector, a normal unit vector, and a binormal unit vector to the streamline, respectively. Since a streamline is a curve that is tangent to the velocity vector of the flow, the left-hand side of the above equation, the [[convective derivative]] of velocity, can be described as follows:
<math display="block">
  \boldsymbol{u}\cdot\nabla \boldsymbol{u}= u\frac{\partial}{\partial s}(u\boldsymbol{e}_s) = u\frac{\partial u}{\partial s}\boldsymbol{e}_s + \frac{u^2}{R}\boldsymbol{e}_n,</math>
where 
<math display="block">\begin{align}
   \boldsymbol{u} &= u \boldsymbol{e}_s,\\
   \frac{\partial}{\partial s} &\equiv \boldsymbol{e}_s \cdot \nabla,\\
   \frac{\partial\boldsymbol{e}_s}{\partial s} &= \frac{1}{R}\boldsymbol{e}_n,
\end{align}</math>
and <math>R</math> is the [[radius of curvature (mathematics)|radius of curvature]] of the streamline.

Therefore, the momentum part of the Euler equations for a steady flow is found to have a simple form:
<math display="block">\begin{align}
  \displaystyle u\frac{\partial u}{\partial s} &= -\frac{1}{\rho}\frac{\partial p}{\partial s},\\
  \displaystyle                  {u^2 \over R} &= -\frac{1}{\rho}\frac{\partial p}{\partial n} &({\partial / \partial n}\equiv\boldsymbol{e}_n\cdot\nabla),\\
  \displaystyle                              0 &= -\frac{1}{\rho}\frac{\partial p}{\partial b} &({\partial / \partial b}\equiv\boldsymbol{e}_b\cdot\nabla).
\end{align}</math>

For [[barotropic]] flow <math>(\rho = \rho(p))</math>, [[Bernoulli's equation]] is derived from the first equation:
<math display="block">\frac{\partial}{\partial s}\left(\frac{u^2}{2} + \int\frac{\mathrm{d}p}{\rho}\right) = 0.</math>

The second equation expresses that, in the case the streamline is curved, there should exist a [[pressure gradient]] normal to the streamline because the [[centripetal acceleration]] of the [[fluid parcel]] is only generated by the normal pressure gradient.

The third equation expresses that pressure is constant along the binormal axis.

==== Streamline curvature theorem ====
[[File:Streamlines around a NACA 0012.svg|frame|right|
The "Streamline curvature theorem" states that the pressure at the upper surface of an airfoil is lower than the pressure far away and that the pressure at the lower surface is higher than the pressure far away; hence the pressure difference between the upper and lower surfaces of an airfoil generates a lift force.
]]

Let <math>r</math> be the distance from the center of curvature of the streamline, then the second equation is written as follows:
<math display="block">
\frac{\partial p}{\partial r} = \rho \frac{u^2}{r}~(>0),
</math>

where <math>{\partial / \partial r} = -{\partial /\partial n}.</math>

This equation states:<blockquote>
''In a steady flow of an [[inviscid]] [[fluid]] without external forces, the [[center of curvature]] of the streamline lies in the direction of decreasing radial pressure.''
</blockquote>

Although this relationship between the pressure field and flow curvature is very useful, it doesn't have a name in the English-language scientific literature.{{sfn|Babinsky|2003}} Japanese fluid-dynamicists call the relationship the "Streamline curvature theorem".{{sfn|Imai|1973|p=}}

This "theorem" explains clearly why there are such low pressures in the centre of [[vortex|vortices]],{{sfn|Babinsky|2003}} which consist of concentric circles of streamlines.
This also is a way to intuitively explain why airfoils generate [[lift (force)|lift forces]].{{sfn|Babinsky|2003}}