Shock Capturing Methods - Euler Equation

Euler Equation

The Euler equations are the governing equations for inviscid flows. To implement shock-capturing methods, the conservation form of the Euler equations are used. For a flow without external heat transfer and work transfer (isoenergetic flow), the conservation form of the Euler equation in Cartesian coordinate system can be written as

$\frac{\partial {\bold U}}{\partial t} + \frac{\partial {\bold F}}{\partial x} + \frac{\partial {\bold G}}{\partial y} + \frac{\partial {\bold H}}{\partial z} = 0$

where the vectors U, F, G, and H are given by

${\bold U} = \left[ \begin{array}{c} \rho \\ \rho u \\ \rho v \\ \rho w \\ \rho e_t \\ \end{array} \right] \qquad \quad {\bold F} = \left[ \begin{array}{c} \rho u\\ \rho u^2 + p \\ \rho uv \\ \rho uw \\ (\rho e_t + p)u \\ \end{array} \right] \qquad \quad {\bold G} = \left[ \begin{array}{c} \rho v\\ \rho vu \\ \rho v^2 + p \\ \rho vw \\ (\rho e_t + p)v \\ \end{array} \right] \qquad \quad {\bold H} = \left[ \begin{array}{c} \rho w\\ \rho wu \\ \rho wv \\ \rho w^2 + p \\ (\rho e_t + p)w \\ \end{array} \right] \qquad \qquad$

where is the total energy (internal energy + kinetic energy + potential energy) per unit mass. That is

$e_t = e + \frac{u^2 + v^2 + w^2}{2} + gz$

The Euler equation may be integrated with any of the shock-capturing methods available to obtain the solution.

Read more about this topic: Shock Capturing Methods

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