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Hybrid Scheme
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Traditional Eulerian Representation (Review)
Here we review the traditional Eulerian representation of the Euler Equation, as has been discussed in detail earlier.
in terms of velocity:
The so-called "Eulerian form" of the Euler equation can be straightforwardly derived from the standard Lagrangian representation to obtain,
Eulerian Representation
of the Euler Equation,
<math>~\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla) \vec{v}= - \frac{1}{\rho} \nabla P - \nabla \Phi</math>
in terms of momentum density:
Also, we can multiply this expression through by <math>~\rho</math> and combine it with the continuity equation to derive what is commonly referred to as the,
Conservative Form
of the Euler Equation,
<math>~\frac{\partial(\rho\vec{v})}{\partial t} + \nabla\cdot [(\rho\vec{v})\vec{v}]= - \nabla P - \rho \nabla \Phi</math>
The second term on the left-hand-side of this last expression represents the divergence of the "dyadic product" of the vector momentum density (<math>~\rho</math><math>~\vec{v}</math>) and the velocity vector <math>~\vec{v}</math> and is sometimes written as, <math>\nabla\cdot [(\rho \vec{v}) \otimes \vec{v}]</math>.
Component Forms
Let's split the vector Euler equation into its three scalar components; various examples are identified in Table 1.
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Example # |
Grid Basis |
Grid Rotation |
Momentum Basis |
Momentum Frame |
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1 |
Cartesian |
Nonrotating |
Cartesian |
Inertial |
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2 |
Cylindrical |
Nonrotating |
Cylindrical |
Inertial |
=CartNon & CartNon
Consider the familiar case of transport of
Related Discussions
- Euler equation viewed from a rotating frame of reference or Main Page.
- An earlier draft of this "Euler equation" presentation.
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