Let's do little bit of review of compressible flow. There are certain relationships that plays an important role when determining the flow properties when we allow the flow to be compressed. But first, let's make some assumptions to simply the concept.
Assumptions
- Flow is Steady
- One-dimensional
- Calorically perfect gasses
- Specific heat $\gamma$ is constant
Total or Stagnation Temperature
If you were to bring a moving flow to a complete stop, adiabatically (without adding/loosing heat), the maximum total temperature of the flow would be given by the equation 1: \begin{equation}
T_t = T\Big[1+\frac{\gamma - 1}{2}M^2\Big] \tag1 \end{equation} where $T_t$ - Total Temperature, $T$ - Static Temperature
Total or Static Pressure
If you were to bring a moving flow to a complete stop, isentropically, the total pressure of the flow would be given by the equation 2: \begin{equation} P_t = P\Big[1+\frac{\gamma - 1}{2}M^2\Big]^\frac{\gamma}{\gamma -1} \tag2 \end{equation} where $P_t$ - Total Pressure, $P$ - Static Pressure
Mass Flow Parameter
\begin{equation}
MFP = \frac{\dot{m} \sqrt{T_t}}{P_t A} = M\sqrt{\frac{\gamma g_c}{R}}\Big[1+\frac{\gamma -1}{2}M^2\Big]^\frac{-(\gamma+1)}{2(1-\gamma)} \tag3
\end{equation} The above equation is based on Total Pressure of the given flow. MFP is often used to determine the flow area required to choke a given flow (i.e. at M=1). It also has the recognizable maximum at M=1, at which the flow is choked or sonic and flow per unit area is the greatest.
Static Pressure Mass Flow Parameter
\begin{equation}
MF_p = \frac{\dot{m} \sqrt{T_t}}{P_t A} = M\sqrt{{\frac{\gamma g_c}{R}}\Big[1+\frac{\gamma -1}{2}M^2\Big]}\tag4
\end{equation} The static pressure mass flow parameter is used by experimentalists because it uses static pressure in the formula rather than the total pressure and static pressure is easier to measure as compared to the total pressure. $MF_p$ corresponds to only one mach number where as $MFP$ corresponds to two mach numbers (one for subsonic flow and one for supersonic flow).
Impulse Function
\begin{equation}
I = PA + \dot{m}V = PA\Big[1+\gamma M^2\Big]\tag5
\end{equation}
Dynamic Pressure
Static pressure represents the atmospheric pressure at a given altitude where as dynamic pressure fluid's pressure represents the kinetic energy of the fluid. Dynamic pressure $q$ can be calculated using equation 6 shown below.
\begin{equation}
q=\frac{1}{2}\rho V^2 = \frac{1}{2}\gamma PM^2\tag6
\end{equation}
The second form of the dynamic pressure is useful because it uses $\gamma$ and Mach instead of $\rho$ and velocity. $Pv = nRT$ and $a=\sqrt{\gamma RT}$ were substituted to derive the form.
Newtonian hypersonic flow model uses only geometry and the free-stream dynamic pressure to estimate the pressure and forces on bodies immersed in the flows.
Ratio of Specific Heat
$\gamma$ changes based on the temperature.
$T < 2555^o R \rightarrow \gamma$ = 1.4
$2500^o R < T < 3000^o R \rightarrow \gamma$ = 1.33
$3000^o R < T < 3500^o R \rightarrow \gamma$ = 1.30
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