\begin{align}
St &\equiv\; \text{Strouhal number}
&\equiv\;& \frac{\text{Unsteady acceleration}}{\text{Advective acceleration}}
\propto \frac{\partial u / \partial t}{u \, (\partial u / \partial n)}
\propto \frac{\Omega U}{U^2 / L}
& = \frac{\Omega L}{U} \\
Re &\equiv\; \text{Reynolds number}
&\equiv\;& \frac{\text{Inertia force}}{\text{Viscous force}}
\propto \frac{\rho u \left( \partial u / \partial n \right)}{\mu \left( \partial^2 u / \partial n^2 \right)}
\propto \frac{\rho U^2 / L}{\mu U / L^2}
& = \frac{\rho U L}{\mu} \\
Fr &\equiv\; \text{Froude number}
&\equiv\;& \left[ \frac{\text{Inertia force}}{\text{Gravity force}} \right]^{1/2}
\propto \left[ \frac{\rho u \left( \partial u / \partial n \right)}{\rho g} \right]^{1/2}
\propto \left[ \frac{\rho U^2 / L}{\rho g} \right]^{1/2}
& = \frac{U}{\sqrt{gL}}
\end{align}
Mass conservation Eq.:$$
\frac{1}{\rho}\frac{D\rho}{Dt}
= -\,\vec{\nabla} \cdot \vec{u}
\;\;\underset{dp = c^{2} d\rho}{\overset{\text{assuming}}{\longrightarrow}}\;\;
\frac{1}{\rho c^{2}}\frac{Dp}{Dt}
= -\,\vec{\nabla} \cdot \vec{u}
\qquad \text{i.e. sound speed}
$$
Using
$$
\vec{x}^{★} = \frac{\vec{x}}{L},
\quad t^{★} = \frac{Ut}{L},
\quad \vec{u}^{★} = \frac{\vec{u}}{U},
\quad p^{★} = \frac{p - p_{\infty}}{\rho U^2},
\quad \rho^{★} = \frac{\rho}{\rho_{\infty}}
$$
$$
\left[ \frac{U^2}{c^2} \right]
\frac{1}{\rho^{★}} \frac{Dp^{★}}{Dt^{★}}
= - \vec{\nabla}^{★} \cdot \vec{u}^{★}
$$
\(
M \equiv \text{Mach number}
= \left[ \frac{\text{inertial force}}{\text{compressibility force}} \right]^{1/2}
\propto \left[ \frac{\rho U^2 / L}{\rho c^2 / L} \right]^{1/2}
= \frac{U}{c}
\)
Normally for gas flows $M < 0.3$ means incompressible
-
\(M < 1 \;\;\rightarrow\;\; \text{subsonic}\)
- \(M > 1 \;\;\rightarrow\;\; \text{supersonic}\)
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