17. Characteristics

17.1. Unsteady

Euler equations

\[\begin{split}\begin{aligned} 0 & = D_t \rho + \rho \nabla \cdot \mathbf{u} \\ \mathbf{0} & = \rho D_t \mathbf{u} + \nabla p \\ 0 & = \rho D_t e^t + \nabla \cdot \left( p \mathbf{u} \right) \ . \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} 0 & = \partial_t \rho + \mathbf{u} \cdot \nabla \rho + \rho \nabla \cdot \mathbf{u} \\ \mathbf{0} & = \partial_t \mathbf{u} + \mathbf{u} \cdot \nabla \mathbf{u} + \frac{1}{\rho} \nabla p \\ 0 & = \partial_t e + \mathbf{u} \cdot \nabla e + \frac{p}{\rho} \nabla \cdot \mathbf{u} \ . \end{aligned}\end{split}\]

with \(p(\rho, e)\). In Cartesian coordinates

\[\begin{split} \begin{bmatrix} 1 & \cdot & \cdot & \cdot \\ \cdot & 1 & \cdot & \cdot \\ \cdot & \cdot & 1 & \cdot \\ \cdot & \cdot & \cdot & 1 \\ \end{bmatrix} \partial_t \begin{bmatrix} \rho \\ u \\ v \\ e \end{bmatrix} + \begin{bmatrix} u & \rho & \cdot & \cdot \\ \frac{1}{\rho}\left(\frac{\partial p}{\partial \rho}\right)_e & u & \cdot & \frac{1}{\rho}\left(\frac{\partial p}{\partial e}\right)_\rho \\ \cdot & \cdot & u & \cdot \\ \cdot & \frac{p}{\rho} & \cdot & u \\ \end{bmatrix} \partial_x \begin{bmatrix} \rho \\ u \\ v \\ e \end{bmatrix} + \begin{bmatrix} v & \cdot & \rho & \cdot \\ \cdot & v & \cdot & \cdot \\ \frac{1}{\rho}\left(\frac{\partial p}{\partial \rho}\right)_e & \cdot & v & \frac{1}{\rho}\left(\frac{\partial p}{\partial e}\right)_\rho \\ \cdot & \cdot & \frac{p}{\rho} & v \\ \end{bmatrix} \partial_y \begin{bmatrix} \rho \\ u \\ v \\ e \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \ . \end{split}\]

and looking for curves \(\mathbf{R}(t)\), s.t. the equations become ODEs, for the function \(\mathbf{U}(t) = \mathbf{u}(\mathbf{R}(s(t)), t)\). Since

\[d_t \mathbf{U} = \partial_t \mathbf{u} + \frac{d \mathbf{R}}{d s} \frac{d s}{dt} \cdot \nabla \mathbf{u} = \partial_t \mathbf{u} + X'(s) \dot{s} \partial_x \mathbf{u} + Y'(s) \dot{s} \partial_y \mathbf{u} \ ,\]

todo Uncomment. Fix and continue

17.2. Steady

\[\begin{split}\begin{aligned} 0 & = \mathbf{u} \cdot \nabla \rho + \rho \nabla \cdot \mathbf{u} \\ \mathbf{0} & = \mathbf{u} \cdot \nabla \mathbf{u} + \frac{1}{\rho} \nabla p \\ 0 & = \mathbf{u} \cdot \nabla e + \frac{p}{\rho} \nabla \cdot \mathbf{u} \ . \end{aligned}\end{split}\]

with \(p(\rho, e)\). In Cartesian coordinates

\[\begin{split} \begin{bmatrix} u & \rho & \cdot & \cdot \\ \frac{1}{\rho}\left(\frac{\partial p}{\partial \rho}\right)_e & u & \cdot & \frac{1}{\rho}\left(\frac{\partial p}{\partial e}\right)_\rho \\ \cdot & \cdot & u & \cdot \\ \cdot & \frac{p}{\rho} & \cdot & u \\ \end{bmatrix} \partial_x \begin{bmatrix} \rho \\ u \\ v \\ e \end{bmatrix} + \begin{bmatrix} v & \cdot & \rho & \cdot \\ \cdot & v & \cdot & \cdot \\ \frac{1}{\rho}\left(\frac{\partial p}{\partial \rho}\right)_e & \cdot & v & \frac{1}{\rho}\left(\frac{\partial p}{\partial e}\right)_\rho \\ \cdot & \cdot & \frac{p}{\rho} & v \\ \end{bmatrix} \partial_y \begin{bmatrix} \rho \\ u \\ v \\ e \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \ . \end{split}\]