compressible

Note

We currently only support two-dimensional domains.

solver

Compressible flow solver configuration

conservative

Definitions of conservative methods

eos

Definitions of equation of states for compressible flow.

riemann_solver

Definitions of Riemann solvers for compressible flow.

viscosity

Definitions of viscous constitutive relations for compressible flow

scaling

Definitions of dimensionless compressible flow equations.

config

Definitions of boundary/domain conditions for compressible flow

Compressible flow equations

The fundamental equations describing the motion of an unsteady, viscous and compressible flow in a space-time cylinder \(\Omega \times (0, t_{end}] \in \mathbb{R}^{d+1}\) with non-empty bounded \(d\)-dimensional spatial domain \(\Omega\), with boundary \(\partial \Omega\), and final time \(t_{end}\), are specified by the Navier-Stokes equations. In terms of conservative variables, \(\vec{U} = \begin{pmatrix} \rho, & \rho \vec{u}, & \rho E \end{pmatrix}^\T\), with density \(\rho\), velocity \(\vec{u}\), and total specific energy \(E\), this system can be expressed in dimensionless form as

\[\begin{align*} \frac{\partial \vec{U}}{\partial t} + \div(\vec{F}(\vec{U}) - \vec{G}(\vec{U}, \nabla \vec{U} )) = \vec{0}. \end{align*}\]

In a general form the convective \(\vec{F}\) and the viscous fluxes \(\vec{G}\) are given by

\[\begin{align*} \vec{F}(\vec{U}) & = \begin{pmatrix} \rho \vec{u}^\T \\ \rho \vec{u} \otimes \vec{u} + p \I \\ \rho H \vec{u}^\T \end{pmatrix}, & \vec{G}(\vec{U}, \nabla \vec{U}) = \begin{pmatrix}\vec{0}^\T \\\mat{\tau} \\ (\mat{\tau} \vec{u} - \vec{q})^\T \end{pmatrix}, \end{align*}\]

where \(p\) denotes the pressure, \(H = E + p/\rho\) the specific enthalpy, \(\mat{\tau}\) the deviatoric stress tensor, and \(\vec{q}\) the heat flux vector.

To close the system of equations we need to specify the equation of state (see eos) and the constitutive relations for the deviatoric stress tensor \(\mat{\tau}\) and the heat flux vector \(\vec{q}\) (see viscosity).

Quasi-linear Euler equations

The hyperbolic nature of the Navier-Stokes equations with respect to time lies in the Euler equations [3]

\[\begin{align*} \frac{\partial \vec{U}}{\partial t} + \div(\vec{F}(\vec{U})) = \vec{0}, \end{align*}\]

which are derived by neglecting the viscous contributions. From a characteristic point of view, it is essential to express these equations in quasi-linear form

\[\begin{align*} \frac{\partial \vec{U}}{\partial t} + \sum_{i=1}^d \mat{A}_i \frac{\partial \vec{U}}{\partial x_i} &= \vec{0}, \\ \end{align*}\]

where the \(\mat{A}_i\) are the directional convective Jacobians.

Examples