Hypersonic and High-Temperature Gas Dynamics Chapter 1 Notes

Chapter 1 is titled "Some Preliminary Thoughts"
==Hypersonic Flight — Some Historic Firsts==
What is hypersonic flight? Anderson says that it occurs at Mach numbers greater than 5.
Examples:
*Vostok - Soviet spacecraft
*X-15 - first aircraft
==Hypersonic Flow — Why is it Important?==
Evolution of design for hypersonic aircraft has been drastic (Fig. 1.4 - 1.6)
===Wing Evolution===
X-15
X-20
Space Shuttle Orbiter
==Hypersonic Flow — What is it?==
If the temperature of the plate equalled TR, the maximum temperature in the boundary layer, then
:\left.\frac{dT}{dy}\right|_{y=0} = 0
This temperature, TR, is called the recovery or adiabatic wall temperature.
Boundary layer theory allows us to define a convective surface heat transfer coefficient, hq, by writing:
:h_q = \frac{\dot q|_{y=0}}{T_R-T_P} = \frac{-k_f \left.\frac{dT}{dy}\right|_{y=0}}{T_R-T_P}
Since it is often difficult to determine the fluid temperature gradient at the wall, we use conservation of energy at the wall interface (assuming there is not mechanism for heat storage at the interface) to write:
:\dot q_{wall} = -k_f \left.\frac{dT}{dy}\right|_{y=0^+} = -k_p \left.\frac{dT}{dy}\right|_{y=0^-}
Hence,
:h_q = \frac{\dot q_{wall}}{T_R-T_P} = \frac{-k_p \left.\frac{dT}{dy}\right|_{y=0^-}}{T_R-T_P}
Note that hq is always defined as a positive number, so we treat \dot q _{wall} as a positive number and rearrange TR - TP, or TP - TR, to get hq positive.
That is, hq is often determined experimentally, so measure the magnitude of qdot(doubleprime)wall and set up
:\Delta T = T_R - T_P \or \Delta T = T_P - T_R
to get a positive hq.
In applications, however, hq is usually determined either numerically from solving the boundary layer problem, or hq is found from Nusselt number heat transfer correlations, in the form:
:Nu = Const. Re_y^m Pr^n f\left(\frac{M_\infty}{M_{wall}},\frac{T_\infty}{T_{wall}}\right)
(example problem)
The fact that the boundary layer at high speed can be quite hot means that the boundary layer fluid density can be quite low; hence, the boundary layer spreads outward from the wall, getting quite thick. Anderson notes that the boundary layer thickness grows as
:\delta_x \propto \frac{M_\infty^2}{\sqrt{Re_x}} ; Re_x = \frac{\rho_\infty V_\infty x}{M_\infty}
As M∞ increases, δx can get very large, the laminar boundary layer can grow quite thick. The thick boundary layer can significantly modify or force changes to the outer inviscid flow, in particular, the inviscid pressure distribution over the body. See Fig. 1.16, p. 16.
Of course, the coupling can be two-way, that the outer pressure distribution can affect the boundary layer. So the problem can be non-linear. This is especially true if the boundary layer merges with the outer shock wave.
==Low Density Flow==
This section deals with Chapter 1 Section 1.3.5.
:Kn = \frac{\lambda}{L} ; \lambda \equiv \text{mean free path between particles}
Kn = 0 in continuum flow, Kn > 1 rarefied and free molecular flow. See Fig. 1.19, p. 21, or alternative graphic at right. As you read this section, please make note of the definition of the transition between low density flow and continuum flow. You will see that there is no boundary layer in molecular flow. As density increases, the flow becomes rarefied and a slip condition at the wall is present. When continuum is reached, there is a no-slip condition at the wall.
At Kn ≈ 0.3, the conservation equations will apply.
For this class we will be focussing almost entirely on continuum flow.
==Hypersonic Flight Paths==
Fig. 1.23, pg. 29, shows altitude plotted as a function of velocity for several atmospheric entry flight paths.
(Review section 1.5 in the book)
==Inlet at High Speed (Example)==
(I'm not sure what section this actually falls into, maybe Chapter 2 notes)
Consider a stream entering an inlet in which a normal shock stands in front of the inlet and the throat is choked at A4.
Question: What is the limit for the contraction ratio between A2 and A4?
===Kantrowitz Limit===
A normal shock pressure recovery area ratio contraction (A4/A2) limit for supersonic inlets described in NACA Wartime Report ARC No. L5D20, May 1945. The derivation of this limit is as follows:
:\dot m_0 = \dot m_2^+ = \dot m_2^- = \dot m_4
:\rho_0 V_0 A_0 = \rho_2^+ V_2^+ A_2^+ = \rho_2^- V_2^- A_2^- = \rho_4 V_4 A_4
:\rho_0 V_0 A_0 = \sqrt{\frac{\gamma}{R}} M_0\left(1 + \frac{\gamma -1}{2} M_0^2\right) ^{-\frac{\gamma+1}{2(\gamma - 1)}} \frac{p_{t0}A_0}{\sqrt{T_{t0}}}
:\rho_4 V_4 A_4 = \sqrt{\frac{\gamma}{R}} M_4\left(1 + \frac{\gamma -1}{2} M_4^2\right) ^{-\frac{\gamma+1}{2(\gamma - 1)}} \frac{p_{t4}A_4}{\sqrt{T_{t4}}}
Letting these mass flow functions equal, and recognizing that the total temperature, ratio of specific heats, and gas constant are constant,
:M_0\left(1 + \frac{\gamma -1}{2} M_0^2\right) ^{-\frac{\gamma+1}{2(\gamma - 1)}} p_{t0}A_0 = M_4\left(1 + \frac{\gamma -1}{2} M_4^2\right) ^{-\frac{\gamma+1}{2(\gamma - 1)}} p_{t4}A_4
Solving for A4/A2,
:\frac{A_4}{A_0} = \frac{M_0}{M_4} \frac{p_{t0}}{p_{t4}} \left(\frac{1 + \frac{\gamma - 1}{2} M_0^2}{1 + \frac{\gamma - 1}{2} M_4^2}\right) ^{-\frac{\gamma+1}{2(\gamma -1)}}
Three assumptions are now made:
#The flow from behind the normal shock in the inlet is isentropic, or pt4 = pt2-
#pt2- is related to pt0 through the normal shock relations
#The flow at station 4 is choked (or sonic), M4 = 1.
:\frac{p_{t0}}{p_{t4}} = \frac{p_{t0}}{p_{t2}^-} = \left[ \frac{(\gamma + 1)M_0^2}{(\gamma - 1)M_0^2 +2}\right] ^{\frac{-\gamma}{\gamma-1}} \left[\frac{\gamma+1}{2\gamma M_0^2 - (\gamma - 1)} \right] ^{\frac{-1}{\gamma-1}}
and, since M4 = 1,
:M_4\left(1 + \frac{\gamma -1}{2} M_4^2\right) ^{-\frac{\gamma+1}{2(\gamma - 1)}} = \left( \frac{\gamma + 1}{2} \right)^{-\frac{\gamma + 1}{2(\gamma - 1)}}
Hence,
:\frac{A_4}{A_0} = M_0 \left(\frac{1 + \frac{\gamma - 1}{2} M_0^2}{\frac{\gamma + 1}{2}}\right)^{-\frac{\gamma + 1}{2(\gamma - 1)}}\left[ \frac{(\gamma + 1)M_0^2}{(\gamma - 1)M_0^2 +2}\right] ^{\frac{-\gamma}{\gamma-1}} \left[\frac{\gamma+1}{2\gamma M_0^2 - (\gamma - 1)} \right] ^{\frac{-1}{\gamma-1}}
This can also be written as,
:\frac{A_4}{A_0} = M_0\left(\frac{\gamma + 1}{2 + (\gamma - 1) M_0^2}\right)^{\frac{\gamma + 1}{2(\gamma - 1)}}\left[ \frac{(\gamma + 1)M_0^2}{(\gamma - 1)M_0^2 +2}\right] ^{\frac{-\gamma}{\gamma-1}}\left[\frac{\gamma+1}{2\gamma M_0^2 - (\gamma - 1)} \right] ^{\frac{-1}{\gamma-1}}
The hypersonic limit of the Kantrowitz limit (A4/A2), in air (γ = 1.4), is approximately 0.63.
 This it taken from Inlet Starting and Contraction Ratio, AIAA 1996-2914.