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 \(\frac{p_o}{\rho_o {C_p}_o T_o}\)


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