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	$\frac{u_o}{\sqrt{g_o l_o}}$
$P$	Péclet	$\frac{\rho_o {C_p}_o u_o l_o}{k_o}$
$H$	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.