User:Lothar.brendel
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Lothar Brendel
"Admin" of this Wiki
Scratch Pad
2D flow in polar coordinates \((\rho,\phi)\)
- vorticity: \(\omega=(\vec\nabla\times\vec v)\cdot\vec e_z=\partial_\rho v_\phi+\displaystyle\frac{v_\phi-\partial_\phi v_\rho}{\rho }\)
- its gradient: \(\vec\nabla\omega=\vec e_\rho\partial_\rho\omega+\displaystyle\frac{\vec e_\phi}{\rho}\partial_\phi\omega\)
Energy equation
\begin{align} E' &= \rho e+\frac{\rho}{2}\vec u^2\\ E &= E' + \rho\phi\\ \partial_t E &= \partial_t E' + \phi\partial_t\rho\\ &= -\vec\nabla\cdot\big((E+P)\vec u\big)\\ &= -\vec\nabla\cdot\big((E'+P)\vec u+\phi\rho\vec u\big)\\ &= -\vec\nabla\cdot\big((E'+P)\vec u\big)-\phi\vec\nabla\cdot(\rho\vec u)-\vec u\cdot\rho\vec\nabla\phi\\ \Leftrightarrow\quad \partial_t E' &= -\vec\nabla\cdot\big((E'+P)\vec u\big)+\vec u\cdot\vec f \end{align}