Compressible Flow in a Nozzle

Mathematical Model

Consider the compressible Euler equations for flow in a converging-diverging nozzle:

\[\begin{split} \begin{align} \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) &= 0 \\ \frac{\partial (\rho \mathbf{v})}{\partial t} + \nabla \cdot (\rho \mathbf{v} \otimes \mathbf{v} + p\mathbf{I}) &= 0 \\ \frac{\partial E}{\partial t} + \nabla \cdot \left[(E + p) \mathbf{v}\right] &= 0 \end{align} \end{split}\]

Boundary Conditions

• Inlet: Specify inflow conditions (\(\rho\), \(\mathbf{v}\), \(p\)).

• Outlet: Specify outflow conditions or use characteristic-based boundary conditions.

• Nozzle walls: \(\mathbf{v} \cdot \mathbf{n} = 0\) (no penetration).

Initial Conditions

Specify initial conditions for density, velocity, and pressure.

Weak Formulation

Find \(\rho, \mathbf{v}, p \in H^1(\Omega)\) such that:

\[\begin{split} \begin{align} &\int_{\Omega} \left(\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v})\right) \phi \, d\Omega \\ &+ \int_{\Omega} \left(\frac{\partial (\rho \mathbf{v})}{\partial t} + \nabla \cdot (\rho \mathbf{v} \otimes \mathbf{v} + p\mathbf{I})\right) \cdot \mathbf{w} \, d\Omega \\ &+ \int_{\Omega} \left(\frac{\partial E}{\partial t} + \nabla \cdot \left[(E + p) \mathbf{v}\right]\right) \psi \, d\Omega \\ &= 0 \end{align} \end{split}\]

for all test functions \(\phi, \mathbf{w}, \psi \in H^1(\Omega)\).