Euler Equations

The Euler equations describe compressible flow in the absence of dissipative effects (like viscosity). In one-dimension, they appear as:

\[\begin{split} \begin{align*} \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} &= 0 \\ \frac{\partial (\rho u)}{\partial t} + \frac{\partial}{\partial x} (\rho u^2 + p) &= 0 \\ \frac{\partial (\rho E)}{\partial t} + \frac{\partial}{\partial x} \left [ (\rho E + p) u \right ] &= 0 \end{align*} \end{split}\]

Here, \(E\) is the specific total energy which is related to the specific internal energy as:

\[ \rho E = \rho e + \frac{1}{2} \rho u^2 \]

and the system is closed by an equation of state:

\[p = p(\rho, e)\]

A common equation of state is a gamma-law EOS:

\[ p = \rho e (\gamma - 1)\]

where \(\gamma\) is a constant. For an ideal gas, \(\gamma\) is the ratio of specific heats, \(c_p / c_v\).

Note

In ppmpy we will assume a gamma-law equation of state

Conservative form

As expressed above, the Euler equations are in conservative form. We can define the conservative state, \({\bf U}\) as:

\[\begin{split}{\bf U} = \left ( \begin{array}{c} \rho \\ \rho u \\ \rho E \end{array} \right )\end{split}\]

and the flux, \({\bf F}({\bf U})\) as:

\[\begin{split}{\bf F} = \left ( \begin{array}{c} \rho u \\ \rho u^2 + p \\ (\rho E + p) u \end{array} \right )\end{split}\]

and then our system in conservative form is:

\[{\bf U}_t + [{\bf F}({\bf U})]_x = 0\]

and we can do the same technique of discretizing the domain into cells and integrating over the volume of a cell to get the finite-volume conservative update for the system:

\[\frac{\partial {\bf U}_i}{\partial t} = - \frac{1}{\Delta x} \left [ {\bf F}({\bf U}_{i+1/2}) - {\bf F}({\bf U}_{i-1/2}) \right ]\]

This means that we will be able to use the same basic solution methodology from advection and Burgers’ equation with the Euler equations, so long as we can find the fluxes. This primarily means that we need to understand the Riemann problem for the Euler equations.

Primitive variable form

We can alternately express the Euler equations in terms of the primitive variables, \({\bf q}\):

\[\begin{split}{\bf q} = \left ( \begin{array}{c} \rho \\ u \\ p \end{array} \right )\end{split}\]

and the evolution equations are:

\[\begin{split} \left ( \begin{array}{c} \rho \\ u \\ p \end{array} \right )_t + \left ( \begin{array}{c} u & \rho & 0 \\ 0 & u & 1/\rho \\ 0 & \Gamma_1 p & u \end{array} \right ) \left ( \begin{array}{c} \rho \\ u \\ p \end{array} \right )_x = 0 \end{split}\]

where \(\Gamma_1 = d \log p/d \log \rho |_s\)

or compactly:

\[{\bf q}_t + {\bf A}({\bf q}) {\bf q}_x = 0\]

Another useful quantity is the speed of sound, defined as:

\[c = \sqrt{\frac{\Gamma_1 p}{\rho}}\]

Characteristic form

We call a system hyperbolic if the eigenvalues are real and finite. For our matrix \({\bf A}({\bf q})\) above, the eigenvalues are:

\[\begin{split} \begin{align} \lambda^{(-)} &= u - c \\ \lambda^{(0)} &= u \\ \lambda^{(+)} &= u + c \\ \end{align} \end{split}\]

These are the speeds at which information propagates in our system.

We can also find the right and left eigenvectors:

\[{\bf A} {\bf r}^{(\nu)} = \lambda^{(\nu)} {\bf r}^{(\nu)}\]

\[{\bf l}^{(\nu)} {\bf A} = \lambda^{(\nu)} {\bf l}^{(\nu)}\]

giving:

\[\begin{split} {\bf r}^{(-)} = \left ( \begin{array}{c} 1 \\ -c/\rho \\ c^2 \end{array} \right ) % \qquad {\bf r}^{(0)} = \left ( \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right ) % \qquad {\bf r}^{(+)} = \left ( \begin{array}{c} 1 \\ c/\rho \\ c^2 \end{array} \right ) \end{split}\]

and

\[\begin{split} \begin{align*} {\bf l}^{(-)} &= \left ( \begin{array}{ccc} 0 & -\frac{\rho}{2c} & \frac{1}{2c^2} \end{array} \right ) \\ {\bf l}^{(0)} &= \left ( \begin{array}{ccc} 1 & 0 & -\frac{1}{c^2} \end{array} \right ) \\ {\bf l}^{(+)} &= \left ( \begin{array}{ccc} 0 & \frac{\rho}{2c} & \frac{1}{2c^2} \end{array} \right ) \\ \end{align*} \end{split}\]

These are normalized such that

\[{\bf l}^{(i)} \cdot {\bf r}^{(j)} = \delta_{ij}\]

The final form of the Euler equations we will need is the characteristic form. We can construct square matrices \({\bf R}\) and \({\bf L}\) by grouping together the eigenvectors:

\[{\bf R} = \left ( {\bf r}^{(-)} | {\bf r}^{(0)} | {\bf r}^{(+)} \right )\]

\[\begin{split}{\bf L} = \left ( \begin{array}{c} {\bf l}^{(-)} \\ {\bf l}^{(0)} \\ {\bf l}^{(+)} \end{array} \right )\end{split}\]

These satisfy:

\[{\bf L}{\bf R} = {\bf R} {\bf L} = {\bf I}\]

We then define the characteristic variables via:

\[d{\bf w} = {\bf L} d{\bf q}\]

Since we are nonlinear, we cannot simply integrate this. Using this definition, we can rewrite our system as:

\[{\bf w}_t + {\bf \Lambda} {\bf w}_x = 0\]

where \({\bf \Lambda}\) is a diagonal matrix with the eigenvalues on the diagonal:

\[\begin{split} {\bf \Lambda} = {\bf LAR} = \left ( \begin{array}{ccc} \lambda^{(-)} & & \\ & \lambda^{(0)} & \\ & & \lambda^{(+)} \end{array} \right ) \end{split}\]

In this form, the 3 equations are decoupled and are just advection equations for each of the characteristic variables. If our system were linear, we’d be done – each characteristic variable would advect at its given wave speed without interacting with one-another.