Euler Equations (fluid Dynamics) - Conservation and Vector Form

Conservation and Vector Form

In vector and conservation form, the Euler equations become:

\frac{\partial \bold m}{\partial t}+ \frac{\partial \bold f_x}{\partial x}+ \frac{\partial \bold f_y}{\partial y}+ \frac{\partial \bold f_z}{\partial z}={\bold 0},

where

{\bold m}=\begin{pmatrix}\rho \\ \rho u \\ \rho v \\ \rho w \\E\end{pmatrix};
{\bold f_x}=\begin{pmatrix}\rho u\\p+\rho u^2\\ \rho uv \\ \rho uw\\u(E+p)\end{pmatrix};\qquad {\bold f_y}=\begin{pmatrix}\rho v\\ \rho uv \\p+\rho v^2\\ \rho vw \\v(E+p)\end{pmatrix};\qquad {\bold f_z}=\begin{pmatrix}\rho w\\ \rho uw \\ \rho vw \\p+\rho w^2\\w(E+p)\end{pmatrix}.

This form makes it clear that fx, fy and fz are fluxes.

The equations above thus represent conservation of mass, three components of momentum, and energy. There are thus five equations and six unknowns. Closing the system requires an equation of state; the most commonly used is the ideal gas law (i.e. p = ρ (γ−1) e, where ρ is the density, γ is the adiabatic index, and e the internal energy).

Note the odd form for the energy equation; see Rankine–Hugoniot equation. The extra terms involving p may be interpreted as the mechanical work done on a fluid element by its neighbor fluid elements. These terms sum to zero in an incompressible fluid.

The well-known Bernoulli's equation can be derived by integrating Euler's equation along a streamline, under the assumption of constant density and a sufficiently stiff equation of state.

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