Editing Euler equations (fluid dynamics) (section)

==Quasilinear form and characteristic equations==
Expanding the [[flux]]es can be an important part of constructing [[numerical solution|numerical solvers]], for example by exploiting ([[approximation|approximate]]) solutions to the [[Riemann problem]]. In regions where the state vector '''''y''''' varies smoothly, the equations in conservative form can be put in quasilinear form:
<math display="block"> \frac{\partial \mathbf y}{\partial t} + \mathbf A_i \frac{\partial \mathbf y}{\partial r_i}  = {\mathbf 0}. </math>
where <math>\mathbf A_i</math> are called the flux [[Jacobian matrix and determinant|Jacobian]]s defined as the [[matrix (mathematics)|matrices]]:
<math display="block"> \mathbf A_i (\mathbf y)=\frac{\partial \mathbf f_i (\mathbf y)}{\partial \mathbf y}. </math>

This Jacobian does not exist where the state variables are discontinuous, as at contact discontinuities or shocks.

===Characteristic equations===
The compressible Euler equations can be decoupled into a set of N+2 [[wave]] equations that describes [[sound]] in Eulerian continuum if they are expressed in [[method of characteristics|characteristic variables]] instead of conserved variables.

In fact the tensor '''A''' is always [[Diagonalizable matrix|diagonalizable]]. If the [[eigenvalue]]s (the case of Euler equations) are all real the system is defined ''hyperbolic'', and physically eigenvalues represent the speeds of propagation of information.{{sfn|Toro|1999|p= 44|loc=par 2.1 Quasi-linear Equations}} If they are all distinguished, the system is defined ''strictly hyperbolic'' (it will be proved to be the case of one-dimensional Euler equations). Furthermore, diagonalisation of compressible Euler equation is easier when the energy equation is expressed in the variable entropy (i.e. with equations for thermodynamic fluids) than in other energy variables. This will become clear by considering the 1D case.

If <math>\mathbf p_i</math> is the [[right eigenvector]] of the matrix <math>\mathbf A</math> corresponding to the [[eigenvalue]] <math>\lambda_i</math>, by building the [[projection matrix]]:
<math display="block">\mathbf{P} = \left[\mathbf{p}_1, \mathbf{p}_2, ..., \mathbf{p}_n\right].</math>

One can finally find the ''characteristic variables'' as:
<math display="block">\mathbf{w} = \mathbf{P}^{-1} \mathbf{y}.</math>

Since '''A''' is constant, multiplying the original 1-D equation in flux-Jacobian form with '''P'''<sup>−1</sup> yields the characteristic equations:{{sfn|Toro|1999|p= 52|loc= par 2.3 Linear Hyperbolic System}}
<math display="block">
\frac{\partial w_i}{\partial t} + \lambda_j \frac{\partial w_i}{\partial r_j} = 0_i.
</math>

The original equations have been [[Linear independence|decoupled]] into N+2 characteristic equations each describing a simple wave, with the eigenvalues being the wave speeds.  The variables ''w''<sub>''i''</sub> are called the ''characteristic variables'' and are a subset of the conservative variables. The solution of the initial value problem in terms of characteristic variables is finally very simple. In one spatial dimension it is:
<math display="block">w_i(x, t) = w_i\left(x - \lambda_i t, 0\right).</math>

Then the solution in terms of the original conservative variables is obtained by transforming back:
<math display="block">\mathbf{y} = \mathbf{P} \mathbf{w},</math>
this computation can be explicited as the linear combination of the eigenvectors:
<math display="block">\mathbf{y}(x, t) = \sum_{i=1}^m w_i\left(x - \lambda_i t, 0\right) \mathbf p_i.</math>

Now it becomes apparent that the characteristic variables act as weights in the linear combination of the jacobian eigenvectors. The solution can be seen as superposition of waves, each of which is advected independently without change in shape. Each ''i''-th wave has shape ''w''<sub>''i''</sub>''p''<sub>''i''</sub> and speed of propagation ''λ''<sub>''i''</sub>. In the following we show a very simple example of this solution procedure.

===Waves in 1D inviscid, nonconductive thermodynamic fluid===
If one considers Euler equations for a thermodynamic fluid with the two further assumptions of one spatial dimension and free (no external field: ''g''&nbsp;=&nbsp;0):
<math display="block">\begin{align} 
  {\partial v \over \partial t} + u{\partial v \over \partial x} - v {\partial u \over \partial x} &= 0,\\[1.2ex]
  {\partial u \over \partial t} + u{\partial u \over \partial x} - e_{vv} v {\partial v \over \partial x} - e_{vs}v {\partial s \over \partial x} &= 0,\\[1.2ex]
  {\partial s \over \partial t} + u{\partial s \over \partial x} &= 0.
\end{align}</math>

If one defines the vector of variables:
<math display="block">\mathbf{y} = \begin{pmatrix}v \\ u \\ s\end{pmatrix},</math>
recalling that <math>v</math> is the specific volume, <math>u</math> the flow speed, <math>s</math> the specific entropy, the corresponding jacobian matrix is:
<math display="block">{\mathbf A}=\begin{pmatrix}u & -v & 0 \\ - e_{vv} v & u & - e_{vs} v \\ 0 & 0 & u \end{pmatrix}.</math>

At first one must find the eigenvalues of this matrix by solving the [[characteristic equation (calculus)|characteristic equation]]:
<math display="block">\det(\mathbf A(\mathbf y) - \lambda(\mathbf y) \mathbf I) = 0,</math>

that is explicitly:
<math display="block">\det\begin{bmatrix}u-\lambda & -v & 0 \\ - e_{vv} v & u-\lambda & - e_{vs} v \\ 0 & 0 & u-\lambda \end{bmatrix}=0.</math>

This [[determinant]] is very simple: the fastest computation starts on the last row, since it has the highest number of  zero elements.
<math display="block">(u-\lambda) \det \begin{bmatrix}u-\lambda & -v \\ - e_{vv} v & u -\lambda \end{bmatrix}=0.</math>

Now by computing the determinant 2×2:
<math display="block">(u - \lambda)\left((u - \lambda)^2 - e_{vv} v^2\right) = 0,</math>
by defining the parameter:
<math display="block">a(v,s) \equiv v \sqrt {e_{vv}},</math>
or equivalently in mechanical variables, as:
<math display="block">a(\rho,p) \equiv \sqrt {\partial p \over \partial \rho}.</math>

This parameter is always real according to the [[second law of thermodynamics]]. In fact the second law of thermodynamics can be expressed by several postulates. The most elementary of them in mathematical terms is the statement of convexity of the fundamental equation of state, i.e. the [[hessian matrix]] of the specific energy expressed as function of specific volume and specific entropy:
<math display="block"> \begin{pmatrix}e_{vv} & e_{vs} \\ e_{vs} & e_{ss} \end{pmatrix},</math>
is defined positive. This statement corresponds to the two conditions:
<math display="block">\left\{\begin{align} 
                   e_{vv} &> 0 \\[1.2ex]
  e_{vv}e_{ss} - e_{vs}^2 &> 0
\end{align}\right.</math>

The first condition is the one ensuring the parameter ''a'' is defined real.

The characteristic equation finally results:
<math display="block">(u - \lambda)\left((u - \lambda)^2 - a^2\right) = 0</math>

That has three real solutions:
<math display="block">\lambda_1(v,u,s) = u-a(v,s), \qquad \lambda_2(u)= u, \qquad \lambda_3(v,u,s) = u+a(v,s).</math>

Then the matrix has three real eigenvalues all distinguished: the 1D Euler equations are a ''strictly'' hyperbolic system.

At this point one should determine the three eigenvectors: each one is obtained by substituting one eigenvalue in the eigenvalue equation and then solving it. By substituting the first eigenvalue &lambda;<sub>1</sub> one obtains:
<math display="block">\begin{pmatrix}a  & -v & 0 \\ - e_{vv} v & a & - e_{vs} v \\ 0 & 0 & a \end{pmatrix} \begin{pmatrix}v_1\\ u_1  \\s_1 \end{pmatrix}=0.</math>

Basing on the third equation that simply has solution ''s''<sub>1</sub>=0, the system reduces to:
<math display="block">\begin{pmatrix}a  & -v \\-a^2 /v& a  \end{pmatrix} \begin{pmatrix}v_1\\ u_1 \end{pmatrix}=0</math>

The two equations are redundant as usual, then the eigenvector is defined with a multiplying constant. We choose as right eigenvector:
<math display="block"> \mathbf p_1=\begin{pmatrix}v\\ a  \\0\end{pmatrix}.</math>

The other two eigenvectors can be found with analogous procedure as:
<math display="block"> \mathbf p_2=\begin{pmatrix} e_{vs} \\ 0\\ - \left(\frac a v \right)^2 \end{pmatrix}, \qquad \mathbf p_3 = \begin{pmatrix}v\\ -a \\0\end{pmatrix}.</math>

Then the projection matrix can be built:
<math display="block"> \mathbf P (v,u,s)=( \mathbf{p}_1, \mathbf{p}_2, \mathbf{p}_3) =\begin{pmatrix} v & e_{vs} & v\\ a & 0 & -a \\ 0 &  - \left(\frac a v \right)^2 & 0 \end{pmatrix}.</math>

Finally it becomes apparent that the real parameter ''a'' previously defined is the speed of propagation of the information characteristic of the hyperbolic system made of Euler equations, i.e. it is the ''[[group velocity|wave speed]]''. It remains to be shown that the sound speed corresponds to the particular case of an [[Isentropic process|isentropic transformation]]:
<math display="block">a_s \equiv \sqrt {\left({\partial p \over \partial \rho} \right)_s}.</math>

===Compressibility and sound speed===
Sound speed is defined as the wavespeed of an isentropic transformation:
<math display="block">a_s(\rho,p) \equiv \sqrt {\left({\partial p \over \partial \rho} \right)_s},</math>
by the definition of the isoentropic compressibility:
<math display="block">K_s (\rho,p) \equiv \rho \left({\partial p \over \partial \rho} \right)_s,</math>
the soundspeed results always the square root of ratio between the isentropic compressibility and the density:
<math display="block">a_s \equiv \sqrt {\frac {K_s} \rho}.</math>

====Ideal gas====
The sound speed in an ideal gas depends only on its temperature:
<math display="block">a_s (T) = \sqrt {\gamma \frac T m}.</math>

{{hidden
|Deduction of the form valid for ideal gases
|In an ideal gas the isoentropic transformation is described by the [[Poisson's law]]:
<math display="block">d\left(p\rho^{-\gamma}\right)_s = 0</math>
where ''γ'' is the [[heat capacity ratio]], a constant for the material. By explicitating the differentials:

<math display="block">\rho^{-\gamma} (d p)_s + \gamma p \rho^{-\gamma-1} (d \rho)_s =0</math>

and by dividing for ''ρ''<sup>−''γ''</sup> d''ρ'':

<math display="block">\left({\partial p \over \partial \rho}\right)_s = \gamma p \rho</math>

Then by substitution in the general definitions for an ideal gas the isentropic compressibility is simply proportional to the pressure:

<math display="block">K_s (p) = \gamma p </math>

and the sound speed results ('''Newton–Laplace law'''):

<math display="block">a_s (\rho,p) = \sqrt {\gamma \frac p \rho} </math>

Notably, for an ideal gas the [[ideal gas law]] holds, that in mathematical form is simply:

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

where ''n'' is the [[number density]], and ''T'' is the [[absolute temperature]], provided it is measured in ''energetic units'' (i.e. in [[joules]]) through multiplication with the [[Boltzmann constant]]. Since the mass density is proportional to the number density through the average [[molecular mass]] ''m'' of the material:

<math display="block"> \rho = m n </math>

The ideal gas law can be recast into the formula:

<math display="block"> \frac p \rho = \frac T m </math>

By substituting this ratio in the Newton–Laplace law, the expression of the sound speed into an ideal gas as function of  temperature is finally achieved.

|style = border: 1px solid lightgray; width: 90%;
|headerstyle = text-align:left;
}}

Since the specific enthalpy in an ideal gas is proportional to its temperature:

<math display="block">h = c_p  T = \frac {\gamma}{\gamma-1} \frac T m, </math>

the sound speed in an ideal gas can also be made dependent only on its specific enthalpy:

<math display="block">a_s (h) = \sqrt {(\gamma -1) h} .</math>