A. Dimensionless parameters

The equations of motion, Equation 15778.50, Equation 15778.51 and Equation 15778.54, are five scalar equations for the five unknowns w, p, and T in three-dimensional flows. The fluid properties ρ, c_p, β, η, and λ that appear in these equations are known functions of temperature and pressure. In general, all 10 unknowns are functions of position r and time t as well as functions of all initial and boundary values of the unknowns.

The large number of independent variables can be reduced considerably by introducing dimensionless variables and by writing the equations of motion in a dimensionless form.

We introduce the dimensionless variables

\[\label{eq1} \begin{split} &\textbf{r}^{\ast}=\frac{\textbf{r}}{l},\;t^{\ast}=\frac{U_\infty t}{l},\;\;\textbf{w}^{\ast}=\frac{\textbf{w}}{U_\infty},\;\;p^{\ast}=\frac{p - p_\infty}{\rho_\infty U_\infty^2},\;T^{\ast}=\frac{T}{T_\infty},\;\rho ^{\ast}=\frac{\rho}{\rho_\infty},\;c_p^{\ast}=\frac{c_p}{c_{p_\infty}},\;\beta ^{\ast}=\frac{\beta}{\beta_\infty},\\ &\\ &\eta ^{\ast}=\frac{\eta}{\eta_\infty},\;\lambda ^{\ast}=\frac{\lambda}{\lambda_\infty} \end{split} \tag{1}\]