Hypersonic and High-Temperature Gas Dynamics Chapter 2 Notes

Chapter 2 is titled "Hypersonic Shock and Expansion-Wave Relations"
==Introduction==
Please read.
==Basic Hypersonic Shock Relations==
Shock waves are extremely thin contact discontinuities in continuum gas (air) across which large changes in fluid and flow properties are present. Shock wave relationships derived for supersonic flow hold in hypersonic flow as well, but due to M∞ becoming very large, some linearization or simplification to these equations can be made.
Anderson starts with an oblique shock equation for pressure ratio:
:\frac{p_2}{p_1} = 1+\frac{2\gamma}{\gamma+1}\left(M_1^2 \sin^2\beta - 1\right)
where β is the wave angle (not the angle of the wedge that created it). See Fig. 2.2, pg. 37, for oblique shock wave geometry.
As M1 gets very large, even for small β,
:\frac{p_2}{p_1} = \frac{2\gamma}{\gamma+1}M_1^2 \sin^2\beta
Anderson shows that the hypersonic limit equations for an oblique shock are:
:\frac{p_2}{p_1} \rightarrow \frac{2\gamma}{\gamma+1}M_1^2 \sin^2\beta
:\frac{\rho_2}{\rho_1} \rightarrow \frac{\gamma+1}{\gamma-1}
:\frac{T_2}{T_1} \rightarrow \frac{2\gamma(\gamma-1)}{(\gamma+1)^2}M_1^2 \sin^2\beta
:\frac{u_2}{V_1} \rightarrow 1 - \frac{2\sin^2\beta}{\gamma+1}
:\frac{v_2}{V_1} \rightarrow \frac{\sin^2(2\beta)}{\gamma+1}
:C_p \rightarrow \frac{4}{\gamma+1}\sin^2\beta
:\beta \rightarrow \frac{\gamma+1}{2}\theta
Note that u2 and v2 are not the usual u and v components parallel and perpendicular to the shock.
Looking more closely at Cp:
:C_p \equiv \frac{p_2 - p_1}{q_1} = \frac{p_1\left(\frac{p_2}{p_1} - 1\right)}{\frac{\gamma p_1 M_1^2}{2}}
:C_p = \frac{2 \left(\frac{p_2}{p_1} - 1 \right)}{\gamma M_1^2}
:C_p = \frac{2 \left(1+\frac{2\gamma}{\gamma+1}\left(M_1^2 \sin^2\beta - 1\right) - 1 \right)}{\gamma M_1^2}
:C_p = \frac{4\gamma \left(M_1^2\sin^2\beta -1\right)}{\gamma M_1^2 (\gamma+1)} \Rightarrow \frac{4\sin^2\beta}{\gamma+1}
Anderson provides the exact shock relations relating the deflection angle θ to the shock angle β for 2-D sharp-nosed surfaces.
:\tan\theta = 2 \cot \beta \left[ \frac{M_1^2\sin^2\beta - 1}{M_1^2 \left(\gamma + \cos(2\beta)\right) + 2} \right]
This relationship is plotted in Fig. 2.3, pg. 41, for γ = 1.4.
For small θ and β associated with hypersonic flow we know that:
:\tan\theta \approx \sin\theta \approx \theta 
:\cos(2\beta) \approx 1
:\sin\beta \approx \beta 
:\cot\beta = \frac{1}{\tan \beta} \approx \frac{1}{\beta}
Therefore:
: \theta \approx \frac{2}{\beta} \left[ \frac{M_1^2\beta^2 - 1}{M_1^2 \left(\gamma + 1)\right) + 2} \right]
In the high Mach number limit, we can simplify this further:
: \theta \approx \frac{2}{\beta} \left[ \frac{M_1^2\beta^2}{M_1^2 (\gamma + 1)} \right] = \frac{2\beta^2}{\beta(\gamma +1)} = \frac{2\beta}{\gamma +1}
or,
: \frac{\beta}{\theta} \approx \frac{\gamma +1}{2} \Rightarrow \text{ for air where } \gamma =1.4 \text{, } \beta = 1.2\theta
===θ-M-β Charts===
For γ = 1.4, NACA 1135 is a report that provides a summary of compressible flow theory for isentropic expansions, normal and oblique shocks. Tables of isentropic flow parameters and property ratios are provided, from M = 0 to M = ∞. Also, charts are provided of β vs. θ and M1, and p2/p1 and M2 versus θ and M1.
Although real gas effects and high temperature effects, or variable cp, cv, γ, and R, dominate very high Mach numbers, a working knowledge of constant property isentropic, normal, and oblique shock flows is necessary and useful for aeronautical engineering.
==Hypersonic Shock Relations in Terms of the Hypersonic Similarity Parameter==
The parameter group K=M1θ has been found to be a "hypersonic similarity parameter." The term similarity means that a group (in this case K) can be found that reduces the independent variables of a problem, that is, the number of independent variables of a problem or mathematical statement is reduced. Anderson introduces the notion of a similarity group with the θ-β-M1 equation.
:\tan\theta = 2 \cot\beta \left[ \frac{M_1^2\sin^2\beta - 1}{M_1^2(\gamma+\cos(2\beta))+2}\right]
Under the previous analysis of small β and θ, this reduced to:
:\theta = \frac{2}{\beta}\left[\frac{M_1^2\beta^2 - 1}{M_1^2(\gamma+1)+2}\right]
For large M1
: (\gamma +1)M_1^2 \gg 2
but can we always assume:
:M_1^2\beta^2 \gg 1 \text{?}
Solve for M12β2 - 1:
:M_1^2\beta^2 - 1 = \beta\theta \left[ \frac{M_1^2(\gamma+1)}{2} + 1\right] \approx \beta\theta \left[ \frac{M_1^2(\gamma+1)}{2}\right]
Dividing by M12θ2 and rearranging:
:\left(\frac{\beta}{\theta}\right)^2 - \left(\frac{\gamma+1}{2}\right)\left(\frac{\beta}{\theta}\right) - \frac{1}{M_1^2\theta^2} = 0
which, being a quadratic in (β/θ) yields two roots:
:\frac{\beta}{\theta} = \left(\frac{\gamma+1}{4}\right) \pm \sqrt{\left(\frac{\gamma+1}{4}\right)^2+\frac{1}{M_1^2\theta^2}}
Only the "+" root represents a physically sensible solution, hence:
:\beta = \beta(\theta) = \left(\frac{\gamma+1}{4}\right)\theta + \sqrt{\theta^2\left(\frac{\gamma+1}{4}\right)^2+\frac{1}{M_1^2}}
for the hypersonic limit of small angles and large M1, recall that our previous linearization gave:
:\beta = \frac{\gamma+1}{2}\theta
Therefore, a legitimate question is, for what θ, β, and M1 do these functions begin to differ from one another, from small angle approximation, and from the exact solution for β(θ,M)?
Return to the Hypersonic Similarity Parameter
Anderson obtained, for β = β(θ)
:\frac{\beta}{\theta} = \left(\frac{\gamma+1}{4}\right) + \sqrt{\left(\frac{\gamma+1}{4}\right)^2+\frac{1}{M_1^2\theta^2}}
The term M12θ2, or rather its square root, is a hypersonic similarity group or parameter.
Returning to the linearized form of the pressure ratio equation across an oblique shock:
:\frac{p_2}{p_1} = 1+\frac{2\gamma}{\gamma+1}\left(M_1^2 \beta^2 - 1\right)
Since the first level linearization of the θ-β-M1 relation gave:
:M_1^2\beta^2 - 1 = \frac{\gamma+1}{2}M_1^2\beta\theta = \frac{\gamma+1}{2}M_1^2\theta^2\frac{\beta}{\theta} 
Substituting the previous relationship for β/θ found above yields:
:M_1^2\beta^2 - 1 = \frac{\gamma+1}{2}M_1^2\theta^2 \left[ \left(\frac{\gamma+1}{4}\right) + \sqrt{\left(\frac{\gamma+1}{4}\right)^2+\frac{1}{M_1^2\theta^2}}\right] 
Hence, by substitution into the relationship for p2/p1:
:\frac{p_2}{p_1} = 1+\frac{2\gamma}{\gamma+1}\left( \frac{\gamma+1}{2}M_1^2\theta^2 \left[ \left(\frac{\gamma+1}{4}\right) + \sqrt{\left(\frac{\gamma+1}{4}\right)^2+\frac{1}{M_1^2\theta^2}}\right] \right)
:\frac{p_2}{p_1} = 1 + \gamma M_1^2 \theta^2 \left[ \left(\frac{\gamma+1}{4}\right) + \sqrt{\left(\frac{\gamma+1}{4}\right)^2+\frac{1}{M_1^2\theta^2}}\right]
Define similarity parameter as:
: K \equiv M_1 \theta \text{ Hypersonic similarity parameter}
Thus,
:\frac{p_2}{p_1} = 1 + \gamma K^2 \left[ \left(\frac{\gamma+1}{4}\right) + \sqrt{\left(\frac{\gamma+1}{4}\right)^2+\frac{1}{K^2}}\right]
===Hypersonic Pressure Coefficient===
:C_p = \frac{p_2 - p_1}{q_1} = \frac{2}{\gamma M_1^2} \left(\frac{p_2}{p_1} - 1 \right)
substitute for p2/p1:
:C_p = \frac{2}{\gamma M_1^2} \left(\gamma K^2 \left[ \left(\frac{\gamma+1}{4}\right) + \sqrt{\left(\frac{\gamma+1}{4}\right)^2+\frac{1}{K^2}}\right] \right)
or,
:C_p = \frac{2K^2}{M_1^2} \left[ \left(\frac{\gamma+1}{4}\right) + \sqrt{\left(\frac{\gamma+1}{4}\right)^2+\frac{1}{K^2}}\right]
Note that since K = M1θ.
:C_p = 2\theta^2 \left[ \left(\frac{\gamma+1}{4}\right) + \sqrt{\left(\frac{\gamma+1}{4}\right)^2+\frac{1}{K^2}}\right]
Dr. Anderson notes that Cp/θ can be written as,
:\frac{C_p}{\theta^2} = 2 \left[ \left(\frac{\gamma+1}{4}\right) + \sqrt{\left(\frac{\gamma+1}{4}\right)^2+\frac{1}{K^2}}\right] = f(\gamma,K)
Anderson notes that "relations analogous to [the previous equation] abound in the theory of hypersonic flow over slender bodies."
==Hypersonic Expansion-Wave Relations==
===Prandtl-Meyer Expansion===
A Prandtl-Meyer expansion fan exists where a supersonic flow reaches a deflection corner that turns the flow away from itself. The expansion corner itself does not have to be sharp, but inviscid ideal theory applies to sharp corners.
Conservation laws allow us to derive M-θ relation in the form:
:\theta = \nu(M_2)-\nu(M_1)
where ν(M) is the Prandtl-Meyer function:
:\nu(M) = \sqrt{\frac{\gamma + 1}{\gamma - 1}} \left[ \tan^{-1} \left(\sqrt{\frac{\gamma - 1}{\gamma + 1}\left(M^2 - 1\right)} \right) \right] - \tan^{-1} \left( \sqrt{M^2 - 1} \right)
Anderson notes that at high Mach numbers
:\sqrt{M^2 - 1} \approx \sqrt{M^2} = M
Thus, ν(M) for large M is:
:\nu(M) = \sqrt{\frac{\gamma + 1}{\gamma - 1}} \left[ \tan^{-1} \left(\sqrt{\frac{\gamma - 1}{\gamma + 1}}M \right) \right] - \tan^{-1} M
By introducing trigonometric identities:
:\tan^{-1} x = \frac{\pi}{2} - \tan^{-1} \left(\frac{1}{x}\right)
and series expansion gives:
:\tan^{-1} \left(\frac{1}{x}\right) = \frac{1}{x} - \frac{1}{3x^3} + \frac{1}{5x^5} - \frac{1}{7x^7} ...
one gets:
:\tan^{-1} \left(\frac{1}{x}\right) = \frac{\pi}{2} - \frac{1}{x} + \frac{1}{3x^3} - \frac{1}{5x^5} + \frac{1}{7x^7} ...
With:
: x = \sqrt{\frac{\gamma - 1}{\gamma + 1}} \text{ and } x = M
by substitution:
:\nu(M) = \sqrt{\frac{\gamma + 1}{\gamma - 1}}\left[\frac{\pi}{2} - \frac{\gamma + 1}{\gamma - 1}\frac{1}{M} + ... \right] - \left[\frac{\pi}{2} - \frac{1}{M} + ... \right]
We will throw away higher order terms:
:\nu(M) = \frac{\pi}{2} \left[ \sqrt{\frac{\gamma +1}{\gamma -1}} - 1 \right] - \left(\frac{\gamma +1}{\gamma -1} -1 \right) \frac{1}{M} = 
:\nu(M) = \frac{\pi}{2} \left[ \sqrt{\frac{\gamma +1}{\gamma -1}} - 1 \right] - \frac{2}{\gamma -1} \frac{1}{M} = 
Therefore for large M,
:\theta = \nu(M_2) - \nu(M_1) = \frac{2}{\gamma -1}\left( \frac{1}{M_1} - \frac{1}{M_2} \right)
Since M2 > M1 in a supersonic expansion,θ > 0.
===Pressure Ratio Across an Expansion at High Mach Number===
The supersonic corner expansion is isentropic, so pt2 = pt1 and,
:\frac{p_2}{p_1} = \frac{p_{t2}}{p_{t1}} \left( \frac{1 + \frac{\gamma - 1}{2} M_1^2}{1 + \frac{\gamma - 1}{2} M_2^2} \right) ^{\frac{\gamma}{\gamma - 1}} = \left( \frac{1 + \frac{\gamma - 1}{2} M_1^2}{1 + \frac{\gamma - 1}{2} M_2^2} \right) ^{\frac{\gamma}{\gamma - 1}}
For large M,
:\frac{p_2}{p_1} \rightarrow \left(\frac{M_1}{M_2} \right)^{\frac{2\gamma}{\gamma -1}}
But, for the corner expansion:
:\frac{1}{\theta} = \frac{1}{\frac{2}{\gamma - 1}\left(\frac{1}{M_1} - \frac{1}{M_2}\right)} = \frac{M_1 M_2}{\frac{2}{\gamma - 1} \left( M_2 - M_1 \right)}
We want M1/M2 to substitute into the above expression for p2/p1.
:M_2 M_1 \theta = \frac{2}{\gamma - 1} \left(M_2 - M_1\right)
:M_2 \left(M_1\theta - \frac{2}{\gamma -1}\right) = \frac{-2}{\gamma -1}M_1 
:M_2 = \frac{\frac{-2}{\gamma -1}M_1}{M_1\theta - \frac{2}{\gamma -1}}
:\frac{M_2}{M_1} = \frac{\frac{-2}{\gamma -1}}{\frac{2}{\gamma -1} - M_1\theta} = \frac{1}{1 - \frac{\gamma - 1}{2} M_1\theta}
or
:\frac{M_1}{M_2} = 1 - \frac{\gamma - 1}{2} M_1\theta
Therefore, the pressure ratio for a high Mach number inviscid corner expansion is given by:
:\frac{p_2}{p_1} = \left(1 - \frac{\gamma - 1}{2} M_1\theta\right)^{\frac{2\gamma}{\gamma - 1}}
Using the definition of the hypersonic similarity parameter, K=M1θ
:\frac{p_2}{p_1} = \left(1 - \frac{\gamma - 1}{2}K\right)^{\frac{2\gamma}{\gamma - 1}}
Equation 2.44 is not restricted to small angles (small θ).
===Pressure Coefficient for Inviscid Corner Expansion at High M===
:C_p = \frac{p_2 - p_1}{q_1} = \frac{p_2 - p_1}{\frac{\gamma}{2}p_1 M_1^2} = \frac{2}{\gamma M_1^2} \left(\frac{p_2}{p_1} -1 \right)
By substitution,
:C_p = \frac{2\theta ^2}{\gamma K} \left[ \left( 1 - \frac{\gamma - 1}{2} K\right) ^{\frac{2\gamma}{\gamma -1}} -1 \right] 
:\frac{C_p}{\theta^2} = f(K,\gamma)
for an expansion fan.
==Summary and Comments==
===Shock Waves===
===Expansion Waves===