Governing Equations

The low mach number flow formulation is based upon the work of J. Principe and R. Codina reproduced here in dimensionless form (see Non-Dimensional Terms.).

\[\begin{eqnarray} S\frac{\partial \rho}{\partial t} + \nabla \cdot \left(\rho \boldsymbol{u} \right ) = 0 \\ \rho \left(S\frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \cdot \nabla \boldsymbol{u}) \right ) + \nabla p - \frac{1}{R}\nabla \cdot \left(2 \mu \boldsymbol{\epsilon'}{}(\boldsymbol{u})\right ) = -\frac{1}{F^2}\rho\hat{\boldsymbol{z}} \\ \rho c_p \left(S \frac{\partial T}{\partial t} + \boldsymbol{u} \cdot \nabla T \right ) - \Gamma \beta T S \frac{d p^{th}}{dt} - \frac{1}{P} \nabla \cdot \left (k \nabla T \right ) = HSQ \end{eqnarray}\]

where:

\[\begin{eqnarray} \boldsymbol{\epsilon'}{}(\boldsymbol{u}) = \boldsymbol{\epsilon}{}(\boldsymbol{u}) - \frac{1}{3}\left(\nabla \cdot \boldsymbol{u} \right )\boldsymbol{I} \\ \boldsymbol{\epsilon}{}(\boldsymbol{u}) = \frac{1}{2} \left(\nabla \boldsymbol{u} + \nabla \boldsymbol{u}^T \right ) \end{eqnarray}\]

The ideal gas equation of state is assumed where \(\rho\) is a function of \(p^{th}\) and \(T\) only, \(\rho^*=\frac{p^{th*}}{RT^*}\). From the relationship between \(p_o\), \(\rho_o\), and \(T_o\) we know \(\rho_o=\frac{p_o}{R_o}\). Combining this with the the definition of the non-dimensional quantities

\[\rho^*=\frac{p^{th*}}{RT^*} \Rightarrow \rho\rho_o=\frac{p^{th}p_o}{RTT_o} \Rightarrow p^{th}=\rho T\]

References

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