The low mach number flow formulation is based upon the work of J. Principe and R. Codina reproduced here in dimensionless form. For simplicity:

- the non-dimensional form is written without any additional super/subscript
- the dimensional (tradition with units) values uses a * (e.g. \(T^*\))
- and the characteristic scales are denoted with a o (e.g. \(T_o\))

Dimensionless Form | Name | Definition |
---|---|---|

\(\rho\) | Density | \(\rho = \frac{\rho^*}{\rho_o}\) |

\(p\) | Mechanical Pressure | \(p = \frac{p^*}{p_o}\) |

\(p^{th}\) | Thermodynamic Pressure | \(p^{th} = \frac{p^{th*}}{p_o}\) |

\(\boldsymbol{u}\) | Velocity | \(\boldsymbol{u} = \frac{\boldsymbol{u}^*}{u_o}\) |

\(\mu\) | Viscosity | \(\mu = \frac{\mu^*}{\mu_o}\) |

\(k\) | Thermal Conductivity | \(k = \frac{k^*}{k_o}\) |

\(QTEST\) | External Heating | \(QTEST = \frac{QTEST^*}{Q_o}\) |

\(T\) | Absolute Temperature | \(T = \frac{T^*}{T_o}\) |

\(C_p\) | Specific Heat Capacity | \(C_p = \frac{C_p^*}{ {C_p}_o}\) |

\(\beta\) | Thermal expansion coefficient | \(\beta = \beta^* T_o\) |

\(\boldsymbol{x}\) | Spacial Location | \(\boldsymbol{x} = \frac{\boldsymbol{x}^*}{l_o}\) |

\(p_o\), \(\rho_o\), and \(T_o\) are assumed to be related by the equation of state.

The non-dimensional quantities are defined as:

Symbol | Name | Definition |
---|---|---|

\(S\) | Strouhal | \(\frac{l_o}{u_o t_o}\) |

\(R\) | Reynolds | \(\frac{\rho_o u_o l_o}{\mu_o}\) |

\(F\) | Froude | \(\frac{u_o}{\sqrt{g_o l_o}}\) |

\(P\) | Pรฉclet | \(\frac{\rho_o {C_p}_o u_o l_o}{k_o}\) |

\(H\) | heat release | \(\frac{t_o Q_o}{\rho_o {C_p}_o T_o}\) |

\(\Gamma\) | depends upon state equation | \(\frac{p_o}{\rho_o {C_p}_o T_o}\) |

## References

- Principe, J., & Codina, R. (2009). Mathematical models for thermally coupled low speed flows. Advances in Theoretical and Applied Mechanics, 2(2), 93-112.