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.

Read more about this topic:  Euler Equations (fluid Dynamics)

Famous quotes containing the words conservation and/or form:

    “The putting into force of laws which shall secure the conservation of our resources, as far as they may be within the jurisdiction of the Federal Government, including the more important work of saving and restoring our forests and the great improvement of waterways, are all proper government functions which must involve large expenditure if properly performed.”
    —William Howard Taft (1857–1930)

    “Although Samuel had a depraved imagination—perhaps even because of this—love, for him, was less a matter of the senses than of the intellect. It was, above all, admiration and appetite for beauty; he considered reproduction a flaw of love, and pregnancy a form of insanity. He wrote on one occasion: “Angels are hermaphrodite and sterile.””
    —Charles Baudelaire (1821–1867)