Equations of state

Equations of state#

The equations of state are required when thermodynamic processes become relevant in the flow field. The equations of state relate a set of state variables to each other.

References:

Nomenclature:

  • \(p\): pressure

  • \(p^{(0)}\): slow pressure component in low-Mach expansion

  • \(T\): temperature

  • \(\rho\): density

  • \(R\): specific gas constant

Currently, two versions of the equation of state are implemented in FELiCS.

1. Ideal gas equation#

Example use case:

Assumptions:

  • Valid for a hypothetical ideal gas.

The ideal gas equation is

\[ p = \rho R T \]

Linearized ideal gas equation#

The linearized form of the ideal gas equation is

\[ p' = R (\rho' \overline{T} +\overline{\rho} T') \]

2. Low-Mach equation#

Assumptions:

  • Ideal gas with constant mean pressure, i.e. mean pressure in the entire flow field is constant due to low Mach numbers and the pressure field is set to the slow component of the pressure \(p^{(0)}\) fixed at some point of the domain, e.g. at the outlet.

  • Density only changes with temperature, but not with pressure.

Example use case:

The low-Mach equation is

\[ \rho = \frac{p^{(0)}}{RT} \]

Linearized low-Mach equation#

The linearized form of the low-Mach equation is

\[ \rho' = -\frac{\overline{\rho}T'}{\overline{T}} \]

Contents