Hypersonic and High-Temperature Gas Dynamics Chapter 4 Notes

==Introduction==
(Please Read!)
To increase our understanding of the physics of hypersonic flow, the basic equations of motion must be considered. Approximate solution methods and their results are presented in this chapter.
The equations of motion comprise "conservation" equations. The conservation equations represent conservation of mass, momentum, and energy.
This chapter presents classical theoretical solutions to inviscid, hypersonic flow equations that have been reduced to special forms by assumptions and approximations. These include:
*Mach number independence
*Hypersonic small disturbance equations
*Hypersonic equivalence principle
*Blast-wave theory for blunt-nosed bodies
*Thin shock applications
==Governing Equations==
Inviscid, compressible Euler equations:
Continuity:
:\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0
X-momentum:
:\rho \frac{\partial u}{\partial t} + \rho u \frac{\partial u}{\partial x} + \rho v \frac{\partial u}{\partial y} + \rho w \frac{\partial u}{\partial z} = - \frac{\partial p}{\partial x}
Y-momentum:
:\rho \frac{\partial v}{\partial t} + \rho u \frac{\partial v}{\partial x} + \rho v \frac{\partial v}{\partial y} + \rho w \frac{\partial v}{\partial z} = - \frac{\partial p}{\partial y}
Z-momentum:
:\rho \frac{\partial w}{\partial t} + \rho u \frac{\partial w}{\partial x} + \rho v \frac{\partial w}{\partial y} + \rho w \frac{\partial w}{\partial z} = - \frac{\partial p}{\partial z}
Energy:
:\frac{\partial s}{\partial t} + u \frac{\partial s}{\partial x} + v \frac{\partial s}{\partial y} + w \frac{\partial s}{\partial z} = 0
or
:\frac{\text{D}s}{\text{D}t} = 0
The energy equation means that the flow is isentropic, i.e. inviscid and adiabatic. For an isentropic flow, pressure, temperature, and density, along a streamline are related by equations of state for a calorically perfect gas:
:\frac{p}{T^{\frac{\gamma}{\gamma - 1}}} = \text{ const and } \frac{p}{\rho ^\gamma} = \text{ const}
Therefore, if entropy is constant along a streamline, the energy equation can be written as:
:\frac{\partial}{\partial t}\left(\frac{p}{\rho^\gamma}\right) + u \frac{\partial}{\partial x}\left(\frac{p}{\rho^\gamma}\right) + v \frac{\partial}{\partial y}\left(\frac{p}{\rho^\gamma}\right) + w \frac{\partial}{\partial z}\left(\frac{p}{\rho^\gamma}\right) = 0
==Mach Number Independence==
Anderson notes that Fig. 3.15, pg. 73, shows that for a slender cone, the surface pressure coefficient becomes independent of M∞ as M∞ reaches the hypersonic range. This is the essence of the Mach Number Independence Principle:
:"At high Mach numbers, certain aerodynamic quantities such as pressure coefficient, lift, and wave-drag coefficients, and flow field structure (shock-wave shapes and Mach-wave patterns) become essentially independent of Mach number."
Newtonian theory is Mach number independent.
Newton-Lees theory is almost Mach number independent at high M∞.
===Mathematical Principle of Mach Number Independence===
Non-dimensionalize the variables in equations of motion:
:\bar x = \frac{x}{l} \qquad \bar y = \frac{y}{l} \qquad \bar z = \frac{z}{l}
:\bar u = \frac{u}{V_\infty} \qquad \bar v = \frac{v}{V_\infty} \qquad \bar w = \frac{w}{V_\infty} 
:\bar p = \frac{p}{\rho V_\infty^2} \qquad \bar \rho = \frac{\rho}{\rho_\infty}
l is a characteristic length; the ∞ subscript refers to the free stream conditions.
If we assume steady flow (i.e. partials with respect to t are zero), substitution into the governing equations yields:
(long list of non-dimensionalized governing equations)
===Boundary Conditions===
For steady, inviscid flow, the boundary condition is usually that of zero normal velocity at the surface.
:v_n = \vec V \cdot \hat n = 0
For any suction or blowing type boundary condition
:v_n = \vec V \cdot \hat n = \pm v_w
If
:\hat n = n_x \hat i + n_y \hat j + n_z \hat k
at the surface, then
:\vec V \cdot \hat n = v_x n_x + v_y n_y + v_z n_z = u n_x + v n_y + w n_z 
Here, nx, ny, and nz are the direction cosines of the normal vector. It follows that the normalized velocity and space coordinates
:\vec \bar V \cdot \hat \bar n = \bar V_x \bar n_x + \bar V_y \bar n_y + \bar V_z \bar n_z = \bar u \bar n_x + \bar v \bar n_y + \bar w \bar n_z
But nx, ny, and nz are already normalized, hence
:\vec \bar V \cdot \hat \bar n = \bar u n_x + \bar v n_y + \bar w n_z
===Boundary Conditions Behind Oblique Shock Waves===
The exact relationships for pressure, temperature, density, and velocity behind a planar oblique shock wave were given in Chapter 2 Notes, but I'll repeat here for convenience.
(go grab all of those equations)
Note that the non-dimensional pressure is
:\bar p = \frac{p}{\rho V_\infty^2}
Therefore,
:p_2 = \bar p_2 \rho_\infty V_\infty^2 
and
:\frac{p_2}{p_\infty} = \frac{\bar p_2 \rho_\infty V_\infty^2}{p_\infty} = \bar p_2 \frac{V_\infty^2}{RT_\infty} = \bar p_2 \frac{\gamma V_\infty^2}{\gamma RT_\infty} = p_2 \frac{\gamma V_\infty^2}{a_\infty^2} = p_2 \gamma M^2
(missed a little bit here right before break 7/2/2013)
Boundary conditions for non-dimensionalized pressure, density, and velocity as free stream Mach number approaches infinity
In the limit as free stream Mach number approaches infinity, the boundary conditions (post-oblique shock) for non-dimensional pressure, density, and velocity are:
: \bar p_2 \rightarrow \frac{2\sin ^2 \beta}{\gamma + 1}
: \bar \rho_2 \rightarrow \frac{\gamma + 1}{\gamma - 1}
: \bar u_2 \rightarrow 1 - \frac{2\sin^2 \beta}{\gamma + 1}
: \bar v_2 \rightarrow \frac{2 M_\infty^2 \sin^2 \beta \cot \beta}{(\gamma +1) M_infty^2} = \frac{2\sin\beta \cos\beta}{\gamma+1}
Now if β is small,
: \bar v_2 \rightarrow \frac{2\beta}{\gamma+1}
Mach Number Independence
Anderson points out the governing equations in non-dimensional variables do not show explicit dependence on M∞; only through the boundary conditions does M∞ have and influence, and this influence goes away as M∞ → ∞.
===Pressure Coefficient===
:C_p = \frac{p_2 - p_\infty}{\frac{1}{2}\rho_\infty V_\infty^2} = \frac{\bar p_2 \rho_\infty V_\infty^2 - p_\infty}{\frac{1}{2}\rho_\infty V_\infty^2} = 2\left( \bar p_2 - \bar p_\infty \right) \quad \text{where} \quad \bar p_\infty = \frac{p_\infty}{\rho_\infty V_\infty^2}
Therefore, CN, CL, and CD show independence of Mach number at high enough values of M∞. Fig. 4.2, pg. 110, shows plots of sphere drag and cone-cylinder drag as functions of M∞. There is a transonic drag rise regime in the range of 0.9 ≤ M∞ ≤ 1.6~1.7, followed by a drag reduction and eventual independence of M∞.
Sphere drag appears to achieve Mach number independence at lower M∞ than the cone-cylinder. Blunt bodies tend to achieve "Mach number independent behavior at lower M∞ than do slender bodies."
"It is the non-dimensional variables that become Mach number independent [...] Some of the dimensional variables, such as p, are not Mach number independent. Indeed p → ∞ as M∞ → ∞"
==Hypersonic Small-Disturbance Equations==
The hypersonic small disturbance equations are an approximate form or subset of the Euler equations that apply to slim or slender bodies at hypersonic speed. These equations express relationships between, not flow variables, by between perturbations of these variables.
===Perturbation Forms===
:u = V_\infty + u'
:v = v'
u and v are small relative to V∞.
Fig. 4.3, pg. 112, shows an illustration of the perturbation velocities. At the tangent point, the surface inclination angle, θ, is
:\theta = \tan^{-1}\frac{v}{u} = \tan^{-1}\frac{\text{d}y}{\text{d}x}
:\frac{\text{d}y}{\text{d}x} = \frac{v}{u} = \frac{v'}{V_\infty + u'} = O[\tau]
Where O[...] is the "order of" symbol and
:\tau = \frac{d}{l} \equiv \quad \text{slenderness ratio of the body}
To the order of τ
:\frac{\text{d}y}{\text{d}x} = \frac{v'}{V_\infty + u'} = \frac{v'}{V_\infty} = O[\tau]
Multiply the last expression by V∞/a∞ to get
:\frac{v'}{a_\infty} = \frac{V_\infty}{a_\infty} O[\tau] = O[M_\infty \tau]
M∞τ is a measure of the radial perturbation to the flow field caused by the body. Now express the flow field variables in terms of the perturbation forms.
:u=V_\infty + u' \qquad v=v' \qquad w=w'
and the governing equations become:
(list out all of the governing equations here with perturbation variables)
These are the perturbation forms of the dimensional Euler equations.
===Small Angle Approximation===
:\frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma + 1} \left( M_1^2 \sin^2 \beta -1 \right)
For the small angle approximation
:\frac{p_2}{p_1} \approx 1 + \frac{2\gamma}{\gamma + 1} M_1^2 \theta^2 = O[M_1^2 \tau^2]
Therefore, a non-dimensionalizing parameter for pressure is the group γ p∞ M∞2 τ.
Hence,
:\bar p = \frac{p}{\gamma p_\infty M_\infty^2 \tau^2}
For density,
:\bar \rho = \frac{\rho}{\rho_\infty}
For the x-component of velocity, u, we have the velocity behind an attached shock roughly given by ("exactly" for oblique shock)
:\frac{u_2}{u_1} = \frac{u_2}{V_\infty} = 1 - \frac{2\sin^2\beta}{\gamma + 1}
:\Delta u = u_2 - u_1 = V_\infty - u_2
:u_1 = \frac{\Delta u}{V_\infty} = 1 - \frac{u_2}{V_\infty} \approx \frac{2\beta}{\gamma + 1} = O[\tau^2]
Therefore, the velocity normalization parameter is
:\bar u' = \frac{u'}{V_\infty \tau^2}
For the y-component of velocity, v, we have
:\frac{v_2}{V_\infty} \rightarrow \frac{2\sin\beta \cos\beta}{\gamma+1} = \frac{\sin 2\beta}{\gamma +1} = \frac{2\beta}{\gamma+1} = O[\tau]
So,
:\bar v' = \frac{v'}{V_\infty \tau}
An order of magnitude comparison indicates that v is significantly larger than u, and also w >> u, since we suppose that τ << 1, or if non-dimensional u and v are of the same order, then u << v since
(missed the last little bit on 7/2/2013)
Continuity Equation
:\frac{\partial}{\partial \bar x} \left[ \bar \rho \left( \frac{1}{\tau^2} + \bar u ' \right) \right] \left[\rho_\infty V_\infty \tau^2 \right] + \frac{\partial (\bar \rho \bar v ')}{\partial \bar y} \left[\frac{\rho_\infty V_\infty \tau}{\tau} \right] + \frac{\partial (\bar \rho \bar w ')}{\partial \bar z} \left[\frac{\rho_\infty V_\infty \tau}{\tau} \right] = 0
If we neglect the τ2 term because τ is small and τ2 is very small, the continuity equation becomes:
:\frac{\partial \bar \rho}{\partial \bar x} + \frac{\partial (\bar \rho \bar v ')}{\partial \bar y} + \frac{\partial (\bar \rho \bar w ')}{\partial \bar z} = 0
X-Momentum
:\bar \rho \left( \frac{1}{\tau^2} + \bar u ' \right) \frac{\partial}{\partial \bar x} \left( \frac{1}{\tau^2} + \bar u ' \right) \left[\rho_\infty V_\infty^2 \tau^4 \right] + \bar \rho \bar v ' \frac{\partial}{\partial \bar y} \left( \frac{1}{\tau^2} + \bar u ' \right) \left[\frac{\rho_\infty V_\infty^2 \tau^3}{\tau} \right] + \bar \rho \bar w ' \frac{\partial}{\partial \bar z} \left( \frac{1}{\tau^2} + \bar u ' \right) \left[\frac{\rho_\infty V_\infty^2 \tau^3}{\tau} \right] = -\frac{\partial \bar p}{\partial \bar x} \left[ \gamma M_\infty^2 \tau^2 p_\infty \right]
Dividing by ρ∞V∞2τ2 and noting that ρ∞V∞2 = γp∞/M∞2, the x-momentum equation reduces to:
:\bar \rho \left( 1 + \bar u ' \tau^2 \right) \frac{\partial \bar u '}{\partial \bar x} + \bar \rho \bar v ' \frac{\partial \bar u '}{\partial \bar y} + \bar \rho \bar w ' \frac{\partial \bar u '}{\partial \bar z}= -\frac{\partial \bar p}{\partial \bar x}
Y- and Z-Momentum
Similarly the y- and z-momentum equations are:
:\bar \rho \frac{\partial \bar v '}{\partial \bar x} + \bar \rho \bar v ' \frac{\partial \bar v '}{\partial \bar y} + \bar \rho \bar w ' \frac{\partial \bar v '}{\partial \bar z}= -\frac{\partial \bar p}{\partial \bar y}
:\bar \rho \frac{\partial \bar w '}{\partial \bar x} + \bar \rho \bar v ' \frac{\partial \bar w '}{\partial \bar y} + \bar \rho \bar w ' \frac{\partial \bar w '}{\partial \bar z}= -\frac{\partial \bar p}{\partial \bar z}
Energy Equation
The energy equation reduces to:
:\frac{\partial}{\partial \bar x} \left( \frac{\bar p}{\bar \rho^\gamma} \right) + \bar v ' \frac{\partial}{\partial \bar y} \left( \frac{\bar p}{\bar \rho^\gamma} \right) + \bar w ' \frac{\partial}{\partial \bar z} \left( \frac{\bar p}{\bar \rho^\gamma} \right) = 0
Equations 4.41-4.45, pg. 117, are the Hypersonic Small-Disturbance Equations. These equations hold for:
#Slender bodies, τ = d/l << 1
#Small angles, β ~ θ ~ dy/dx ~ τ
#High Mach numbers
#\bar u' \tau^2
Note that \bar u' only appears in the x-momentum equation, so that the x-momentum is decoupled from the directions perpendicular to motion. Equations 4.41 and 4.43-4.45 form a separate system of equations in all variables but \bar u'.
These equations are still non-linear, even though they have been screened for the correct order of magnitude and scaled (non-dimensionalized). There is no general analytical solution, so far, according to Anderson. On the other hand, these equations are the starting point for quasi-analytical and numerical solutions for hypersonic small disturbance flow field theories.
==Hypersonic Similarity==
Flow similarity is defined as dynamic similarity with geometrically similar streamline shapes in terms of scaled and/or non-dimensionalized coordinates, such that the flow field variables (also in scaled/non-dimensionalized coordinates) are the same. Similarity parameters are the parameter groups that make the flow fields (and their mathematical solutions) similar.
Let's consider the governing equations 4.41 and 4.43-4.45. The boundary conditions for the hypersonic small disturbance equations are developed the same way as before.
The body surface boundary condition
:\vec V \cdot \hat n = 0
: un_x + vn_y + wn_z = 0
: (V_\infty + u')n_x + v'n_y + w'n_z = 0
We'll introduce the hypersonic disturbance quantities:
: V_\infty(1 + \tau^2 u')n_x + V_\infty\tau v'n_y + V_\infty\tau w'n_z = 0
: (1 + \tau^2 u')n_x + \tau v'n_y + \tau w'n_z = 0
The direction cosines nx, ny, and nz are in unscaled x, y, z coordinates. In the scaled coordinates:
: n_x = \tau \bar n_x \qquad n_y = \bar n_y \qquad n_z = \bar n_z 
Using the transformed direction cosines:
: (1 + \tau^2 u')\tau \bar n_x + \tau v' \bar n_y + \tau w' \bar n_z = 0
Dividing by τ and noting that τ2 << 1
: \bar n_x + v' \bar n_y + w' \bar n_z = 0
Post-shock boundary condition
Anderson starts with the density ratio for an oblique shock.
:\frac{\rho_2}{\rho_\infty} = \bar \rho_2 = \frac{(\gamma +1)M_\infty^2 \sin^2\beta}{(\gamma -1)M_\infty^2 \sin^2\beta + 2} = \left(\frac{\gamma +1}{\gamma -1} \right) \left[ \frac{M_\infty^2 \sin^2\beta}{M_\infty^2 \sin^2\beta + \frac{2}{\gamma - 1}}\right]
For the slender body at hypersonic conditions:
:\sin \beta \approx \beta \approx \left( \frac{\text{d} y}{\text{d} x} \right)_{surface} = \left( \frac{\text{d} \bar y}{\text{d} \bar x} \right)_{surface} \tau
So,
:\bar \rho_2 = \left(\frac{\gamma +1}{\gamma -1} \right) \left[ \frac{\left( \frac{\text{d} \bar y}{\text{d} \bar x} \right)_{surface}^2}{\left( \frac{\text{d} \bar y}{\text{d} \bar x} \right)_{surface}^2 + \frac{2}{M_\infty^2 \tau^2(\gamma - 1)}}\right]
Define K = M∞τ, then
(missed some stuff on 7/9/2013)
Strictly speaking, K = M∞τ is the slender body hypersonic similarity parameter.
Pressure
\frac{p_2}{p_\infty} = 1 + \frac{2 \gamma}{\gamma +1} (M_\infty^2\sin^2 \beta - 1)
Earlier we selected:
\bar p = \frac{p}{\gamma p_\infty M_\infty^2\tau^2}
Substitute this and others:
\bar p_2 = \frac{1}{\gamma M_\infty^2 \tau^2} + \frac{2}{\gamma + 1} \left[ \left( \frac{\text{d} \bar y}{\text{d} \bar x} \right)_{surface}^2 - \frac{1}{M_\infty^2 \tau^2} \right] = \frac{2}{\gamma +1} \left[\frac{1-\gamma}{2\gamma K^2} + \left( \frac{\text{d} \bar y}{\text{d} \bar x} \right)_{surface}^2 \right]
(he goes through all of the other derivations that are written in the book. 7/9/2013)
The only parameters of these equations are γ and K; therefore, the governing equations will have the same boundary conditions and solutions if γ and K have the same values. This is the principle of hypersonic similarity.
===Modification for Small Angle of Attack, α===
The small angle constraint was that:
:\sin \theta \approx \sin\beta \approx \beta \approx \frac{\text{d}y}{\text{d}x} \approx \tau
Therefore, angle of attack, α, must also be small, α = O[τ]. So, we can modify the equations derived so far to account for small α by defining,
:\bar \alpha = \frac{\alpha}{\tau}
and adding or subtracting αbar to/from dybar/dxbar where it appears in the scaled boundary conditions.
(some plots showing how α affects body shape)
Anderson therefore summarizes hypersonic similarity by noting that for "affinely related bodies," if γ, K, and α/τ are the same, within small perturbation slender body restrictions, their boundary conditions and governing equations, hence flow fields, will be the same.
He derives the pressure coefficient as follows:
(derivation of pressure coefficient and lift coefficient from book 7/9/2013)
===Drag Coefficient===
For the drag coefficient, we have:
:\text{d}F_y = p_u \text{d}s \sin \theta
:\text{d}y(l) = \text{d}s \sin \theta
:\text{d}s = \frac{\text{d}y(1)}{ \sin \theta}
:\text{d}F_y = p_u \frac{\text{d}y(1)}{ \sin \theta} \sin \theta = p_u \text{d}y(1)
Then the drag coefficient per unit span is:
:c_d = \frac{D}{S_D q_\infty} = \frac{\int_0^l (p_L + p_U) \text{d}y}{l(1)\frac{\gamma}{2}p_\infty M_\infty^2} = \frac{1}{l(1)}\int_0^l (C_{p_L} + C_{p_U}) \text{d}y
But, to make this an integral on x, we can write:
:c_d = \frac{1}{l(1)}\int_0^l (C_{p_L} + C_{p_U}) \left( \frac{\text{d}y}{\text{d}x} \right) \text{d}y
===Transformation to Perturbation Coordinates===
Noting that
:\bar x = \frac{x}{l} \qquad \text{d}x = l\text{d} \bar x
:x=0 \rightarrow \bar x = 0
:x=l \rightarrow \bar x = 1
Therefore, we can write the lift coefficient as:
:c_l = \int_0^l (C_{p_L} - C_{p_U}) \text{d}\bar x
Using the hypersonic similarity results,
:\frac{c_l}{\tau^2} = \int_0^l (\frac{C_{p_L}}{\tau^2} - \frac{C_{p_U}}{\tau^2}) \text{d}\bar x = f_2(\gamma,K,\frac{\alpha}{\tau}
(missed a few boards, 7/12/2013)
===Generalization to 3-D Planform===
Anderson notes that the previous definitions for cl and cd referred to the planform projected area, S = l(1) (or per unit span). Extending the methodology to general 3-D bodies, Anderson writes:
:L = \iint_S p(x,y,z) \text{d}x \text{d}y
where the same small angle (slender body) assumptions have been made. In transformed variables,
:L = \gamma p_\infty M_\infty^2 \tau^2 \tau \iint_S \bar p(\bar x,\bar y,\bar z) \text{d}\bar x \text{d}\bar y
Define the 3-D body lift coefficient as,
:C_L = \frac{L}{q_\infty S_B}
where S now refers to the projected base area of the body:
:<math>S_B \propto b_y b_z = \tau^2\bar b_y \bar b_z
where by and bz are the half-heigh and half-width, respectively. (shown in Fig. 4.4, pg. 124)
:C_L \propto \frac{2}{\gamma p_\infty M_\infty^2 \tau^2} \left[ \iint_S \bar p(\bar x,\bar y,\bar z) \text{d}\bar x \text{d}\bar y \right] \gamma p_\infty M_\infty^2 \tau^2 \tau
:\frac{C_L}{\tau} = F_1 \left( \gamma,K,\frac{\alpha}{\tau} \right)
Therefore, when CL is expressed in terms of the base area of a 3-D slender body, then CL/τ is the hypersonic similarity result.
For drag, Anderson gives the derivation:
:D = \iint_S p(x,y,z) \text{d}y \text{d}z
under the same small angle assumptions.
Then, in perturbation variables,
:D = \gamma p_\infty M_\infty^2 \tau^2 \tau^2 \iint_S \bar p(\bar x,\bar y,\bar z) \text{d}\bar y \text{d}\bar z
or,
:C_D = \frac{D}{q_\infty S_B} \propto \frac{2}{\gamma p_\infty M_\infty^2 \tau^2} \left[ \iint_S \bar p(\bar x,\bar y,\bar z) \text{d}\bar y \text{d}\bar z \right] \gamma p_\infty M_\infty^2 \tau^2 \tau^2
:\frac{C_D}{\tau^2} = F_2\left( \gamma,K,\frac{\alpha}{\tau} \right)
===Summary of 3-D Bodies===
:\frac{C_L}{\tau} = F_1 \left( \gamma,K,\frac{\alpha}{\tau} \right)
:\frac{C_D}{\tau^2} = F_2\left( \gamma,K,\frac{\alpha}{\tau} \right)
3-D bodies have 1 less order of magnitude dependence on the slenderness ratio than planar bodies.
===Experimental Verifications===
Anderson provides several pages (pg. 126-129) of discussion of experimental results that verify the above general theoretical similarity principles.
Fig. 4.5 a and b (pg. 127) for 2-D ogive cylinders
(some discussion of those figures 7/12/2013)
===Extended High-Speed Similarity Parameter===
The term
:\sqrt{M_\infty^2 -1}
is a supersonic compressible flow similarity parameter. So,
:\tau\sqrt{M_\infty^2 -1} \rightarrow \tau M_\infty \qquad \text{as} \qquad M_\infty \rightarrow \infty
and therefore this parameter could serve as a compressible flow similarity parameter for M∞ > 1 up to M∞ → ∞.
"Over what range of K values does hypersonic similarity hold?"
*Can't be precisely answered
*For "very slender bodies" (half-angles of 5° or less), 0.5 ≤ K ≤ ∞ appears to hold
*For "less slender bodies" (half-angles of 20° or less), K > 1.5
"Hypersonics" coined by Dr. H. S. Tsien, of CalTech in 1946.
==Hypersonic Small-Disturbance Theory: Some Results==
This section deals with the development of the solutions to the similar, small-disturbance equations of motion, derived earlier in chapter 4. The solution approach uses classical methods for reducing partial differential equations (PDEs) to ordinary differential equations (ODEs).
===2-D Planar Equations of Motion===
Continuity
:\frac{\partial \bar \rho}{\partial \bar x} + \frac{\partial}{\partial \bar y}(\bar \rho \bar v ') = 0
Y-momentum
:\bar \rho \frac{\partial \bar v '}{\partial \bar x} + \bar \rho \bar v ' \frac{\partial \bar v '}{\partial \bar y} = - \frac{\partial \bar p}{\partial \bar y}
Energy
:\frac{\partial}{\partial \bar x}\left( \frac{\bar p}{\bar \rho^\gamma} \right) + \bar v ' \frac{\partial}{\partial \bar y}\left( \frac{\bar p}{\bar \rho^\gamma} \right) = 0 
Introduce the stream function ψ such that
:\frac{\partial \psi}{\partial \bar y} = \bar \rho \qquad \frac{\partial \psi}{\partial \bar x} = - \bar \rho \bar v '
Note how these are different from the classical boundary layer stream function:
:\frac{\partial \psi}{\partial y} = u \qquad \frac{\partial \psi}{\partial x} = - v
Never-the-less, this stream function satisfies continuity.
:\frac{\partial \bar \rho}{\partial \bar x} + \frac{\partial}{\partial \bar y}(\bar \rho \bar v ') = 0
:\frac{\partial}{\partial \bar x}\left( \frac{\partial \psi}{\partial \bar y} \right) + \frac{\partial}{\partial \bar y}\left( -\frac{\partial \psi}{\partial \bar x} \right) = 0
:\frac{\partial^2 \psi}{\partial \bar x \partial \bar y} - \frac{\partial^2 \psi}{\partial \bar x \partial \bar y} = 0
Hence, ψ satisfies the requirement for an analytical function of two variables.
Returning to the stream function, let us use it in the y-momentum and energy equations, but first let's introduce some shorthand notation.
:\psi_{\bar x} \equiv \frac{\partial \psi}{\partial \bar x} \qquad \psi_{\bar x \bar x} \equiv \frac{\partial^2 \psi}{\partial \bar x^2} \qquad \psi_{\bar y} \equiv \frac{\partial \psi}{\partial \bar y}
The y-momentum equation becomes:
(missed a whole lot of derivations here, 7/12/2013)
This equation is nonlinear. It represents small disturbance isentropic flow over slender axisymmetric bodies.
Equations 4.83 and 4.89 are both nonlinear PDEs for a single dependent variable ψ (in cartesian or axisymmetric coordinates). ψ satisfies the continuity equation exactly. Both equations represent the conservation of momentum in a direction perpendicular to the direction of motion, i.e., either y-momentum or r-momentum.
The hypersonic slender body does not, essentially, disturb or change the axial or streamwise momentum, only the momentum transverse to the axial flow.
(Missed the first 45 mins or so of class on 7/16/2013)
Anderson makes some interesting points:
#We could have studied the exact solution obtained by Taylor and Maccoll, and solved numerically — why bother with a small-perturbation theory for slender cones?
##Because the small disturbance theory was developed in the early 1950s and we want to see how accurate it is, compared to exact theory (which was incorrectly calculated the first time! Taylor and Maccoll theory had to be recalculated and revised solutions issued).
#Small disturbance theory certain closed-form analytical results, which are quite handy.
Replacement of w
w was defined as a thermodynamic variable:
:w = \frac{\bar p}{\bar \rho ^\gamma}
Now, from the boundary conditions on \bar p and \bar \rho behind the shock,
(more derivations of what's going on in pages 137-139)
(After the break on 7/16/2013)
===Numerical Solution of the Governing Equations===
Anderson says that a "straight-forward" numerical solution can be performed, since we have all of the boundary conditions on the body and on the shock wave. The solutions are parameterized by K = M∞τ.
Anderson notes that the solution depends on β/τ at the shock wave and that β is unknown a'priori. Therefore, the solution is iterative.
Anderson's scheme is to numerically solve Eqn. 4.106, pg 136, by the following method:
#Assume β or β/τ (suggestion of β/τ = 1.1 for γ = 1.4)
#Start your numerical integration at\bar\theta = \bar\theta_s = \frac{\beta}{\tau}and integrate towards the body; \bar\theta going from 1.1 to 1.0, where \bar\theta = 1 is the body surface. If you get to\bar\theta = 1 = \bar\theta_c = \frac{\bar r_c}{\bar x} = \frac{r}{\tau x}and the physical \frac{r_c}{\tau x} does not agree, being either larger or smaller, adjust β/τ and iterate.
#Assuming that the assumed β/τ = \bar\theta_s initial guess allows you to get (missed a couple of things here, 7/16/2013)
#What has been computed are the values for f and f' between the shock wave and the body. The actual flow field variables must be extracted from these variables:
(derivation of how this is done)
The pressure field can be converted to a Cp field be the definition of Cp
:C_p = \frac{2}{\gamma M_\infty^2} \left( \frac{p}{p_\infty} - 1 \right)
:C_p = \frac{2}{\gamma M_\infty^2} \left[ \frac{p}{p_\infty \gamma M_\infty ^2 \tau ^2}(\gamma M_\infty ^2 \tau ^2) -1 \right]
:\frac{C_p}{\tau ^2} = \frac{2}{\gamma K^2} \left[ \gamma K^2 \bar p(\bar\theta) - 1\right]
where \bar p(\bar\theta) was given just previously.
===Cone Surface Pressure Coefficient===
From the equation for Cp
:\frac{C_p}{\tau ^2} = \frac{2}{\gamma K^2} \left[ \gamma K^2 \bar p - 1\right]
and using,
:\bar p = \omega \left(\frac{f'}{\bar\theta} \right) ^\gamma
Then,
:\frac{C_p}{\tau ^2} = \frac{2}{\gamma K^2} \left[ \gamma K^2 \omega \left(\frac{f'}{\bar\theta} \right) ^\gamma - 1\right]
Insert \bar\theta = 1 on the cone surface and the value of f'(1) into Eqn 4.124, pg. 141, (previous equation) to obtain the surface pressure.
Anderson presents a discussion of how Dr. Milton Van Dyke originally developed the solution procedure, and the different approach Van Dyke took to define the similarity variable
:\bar\theta = \frac{\bar r}{\bar x} = \frac{r}{\tau x} 
Van Dyke chose
:\bar\theta = \frac{\bar r}{\bar x} = \frac{r}{\beta x} 
which changes the values of f and f at the shock wave f(1) and f(1) occur at the shock.
Van Dyke's solution was not iterative. He integrated until f(\bar\theta) reached 0, which defined the body position. Anderson argues that usually we know τ, and so Van Dyke's approach is really an inverse solution method (given the shock angle, find the body) whereas Anderson's method is direct (given body, find shock) but iterative.
===Results of the Van Dyke-Anderson Approach===
Fig. 4.8, pg. 142, shows Cp/τ2 plotted versus K.
Van Dyke's theory of Cp/τ2 is plotted against two cases of exact theory by Kopal for two different τ (cone half angles).
The smaller cone angle of 10° has an exact solution closer to the small disturbance theory of Van Dyke.
===Rasmussen Analytical Result===
Anderson does not give details, but M. L. Rasmussen presented a closed-form solution to Eqn. 4.106, pg. 139, by integration and successive approximation. See AIAA Journal Vol. 5 Number 8, August 1967, pg. 1495-1497, for details.
The results of Rasmussen's analysis are as follows:
Shock wave angle as a function of K
:K_\beta = K\sqrt{\frac{\gamma + 1}{2} + \frac{1}{K^2}}
where,
:K_\beta \equiv M_\infty \beta
which is Van Dyke's similarity parameter, and,
:K \equiv M_\infty \theta_c
Cone surface pressure coefficient:
:\frac{C_p}{\theta _c^2} = 1+ \frac{(\gamma +1) K^2 +2}{(\gamma -1) K^2 +2} \ln \left( \frac{\gamma + 1}{2}+ \frac{1}{K^2} \right)
Rasmussen's theory depends on the validity of his assumptions:
:\tau = \tan \theta_c = \theta_c = \frac{r_c}{L}
hence, K is not M∞τ in his theoretical results. Fig. 4.8, pg. 142, shows Rasmussen's theory plotted as dark, solid circles. Agreement with Van Dyke's theory is good for K = M∞θc > 1. Better agreement is obtained if Cp/sin2θc is used in Eqn. 4.126, pg. 143.
Fig. 4.9, pg. 144, shows Rasmussen's modified theory plotted versus K = M∞θc and also some exact results from Kopal. Anderson notes some further work extending Rasmussen's method to small angles of attack, α, which we shall pass over.
==Comment on Hypersonic Small-Disturbance Theory==
Anderson points out that the theoretical results of the small-disturbance theory are essentially non-linear, and the theory has powerful utility still today.
==Hypersonic Equivalence Principle and Blast-Wave Theory==
The hypersonic equivalence principle states that steady hypersonic flow fields in the disturbance direction, y and z, are physically equivalent to two-dimensional, unsteady flow fields in these directions.
Starting with the unsteady Euler equations,
Continuity:
:\frac{\partial \rho}{\partial t} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0
Y-momentum:
:\rho \frac{\partial v}{\partial t} + \rho v \frac{\partial v}{\partial y} + \rho w \frac{\partial v}{\partial z} = - \frac{\partial p}{\partial y}
Z-momentum:
:\rho \frac{\partial w}{\partial t} + \rho v \frac{\partial w}{\partial y} + \rho w \frac{\partial w}{\partial z} = - \frac{\partial p}{\partial z}
Energy:
:\frac{\partial}{\partial t}\left(\frac{p}{\rho^\gamma}\right) + v \frac{\partial}{\partial y}\left(\frac{p}{\rho^\gamma}\right) + w \frac{\partial}{\partial z}\left(\frac{p}{\rho^\gamma}\right) = 0
Non-dimensional variables:
:\tilde y = \frac{y}{l} \qquad \tilde z = \frac{z}{l}
:\tilde v = \frac{v}{V_\infty} \qquad \tilde w = \frac{w}{V_\infty} 
:\tilde p = \frac{p}{\rho V_\infty^2} \qquad \tilde \rho = \frac{\rho}{\rho_\infty} \qquad \tilde t = \frac{t}{\left( \frac{l}{V_\infty} \right) }
The non-dimensional governing equations are:
:\frac{\partial \tilde\rho}{\partial \tilde t} + \frac{\partial (\tilde\rho \tilde v)}{\partial \tilde y} + \frac{\partial (\tilde \rho \tilde w)}{\partial \tilde z} = 0
:\tilde \rho \frac{\partial \tilde v}{\partial \tilde t} + \tilde \rho \tilde v \frac{\partial \tilde v}{\partial \tilde y} + \tilde \rho \tilde w \frac{\partial \tilde v}{\partial \tilde z} = - \frac{\partial \tilde p}{\partial \tilde y}
:\tilde \rho \frac{\partial \tilde w}{\partial \tilde t} + \tilde \rho \tilde v \frac{\partial \tilde w}{\partial \tilde y} + \tilde \rho \tilde w \frac{\partial \tilde w}{\partial \tilde z} = - \frac{\partial \tilde p}{\partial \tilde z}
:\frac{\partial}{\partial \tilde t}\left(\frac{\tilde p}{\tilde \rho^\gamma}\right) + \tilde v \frac{\partial}{\partial \tilde y}\left(\frac{\tilde p}{\tilde \rho^\gamma}\right) + \tilde w \frac{\partial}{\partial \tilde z}\left(\frac{\tilde p}{\rho^\gamma}\right) = 0
These equations are exactly the same form as the steady small-disturbance equations (4.41-4.45, pg. 117). The only difference is that xbar and ttilde derivatives replaced one another. Therefore, if we can argue that they are equivalent,
:\bar x=\frac{x}{l} = \tilde t = \frac{t V_\infty}{l}
then,
Anderson shows in Fig. 4.11, pg 148, how the two flow fields can be thought of as physically equivalent, as far as the flow field transverse to the body is concerned. At a point in space, say at the origin of the y-z coordinate system, before the body passes the plane, one has undisturbed flow at that point. As the body passes by the plane, the flow transverse to the body seems to experience a disturbance caused by a time-varying solid surface that forms and drives a shockwave outward from the body.
Anderson notes that these unsteady equations had already been formulated and solved for hypersonic slender-body aerodynamics problems. These solutions were considered and immediately investigated for use in producing hypersonic steady flow disturbances. See pg. 149.
===2-D Airfoil and 1-D Piston===
Anderson takes us through the equivalence principle with the comparison of a steady planar body in high speed motion as it passes through a point in space. The body is infinite in the y-direction (span). The body surface acts like an infinite piston that suddenly moves upward, reaches a maximum height, then drops back to zero in the z-direction. See Fig. 4.12, pg. 149.
For the airfoil,
:z(x) = z_{max} \times f\left( \frac{x}{c} \right)
For the piston,
:z(t) = z_{max} \times f\left( \frac{t}{t_0} \right)
Clearly c (the chord length) and t0 (the period of the piston motion) must be related by:
:t_0 = \frac{c}{V_\infty}
Let wb be the z-velocity on the body, so from inviscid tangent flow boundary condition,
:w_b = V_\infty \frac{\text{d} z}{\text{d}x}
The piston surface velocity is:
:w_p = \frac{\text{d} z}{\text{d}t} 
The principle of equivalence then has:
or,
:V_\infty \frac{\text{d} z}{\text{d}x} = \frac{\text{d} z}{\text{d}t}
or,
:V_\infty \frac{\text{d} z}{\text{d}t} \frac{\text{d}t}{\text{d}x} = \frac{\text{d} z}{\text{d}t}
but,
:\frac{\text{d}x}{\text{d}t} = V_\infty \rightarrow V_\infty \frac{\text{d} z}{\text{d}t} \frac{1}{V_\infty} = \frac{\text{d} z}{\text{d}t}
therefore, the principle is satisfied. That is, the "free piston" velocity is the same as the velocity component normal to the surface of the body at the equivalent x-location.
Note that dividing wb by a∞ gives:
:\frac{w_b}{a_\infty} = \frac{V_\infty}{a_\infty}\frac{\text{d} z}{\text{d}x} = M_\infty \tan\theta(x)
For small disturbances, tanθ ≈ θ. Also θmax ≈ zmax/c (θ ≈ τ), so that,
:\left( \frac{w_b}{a_\infty} \right) _{max} = \left( \frac{w_p}{a_\infty} \right) _{max} = M_\infty \theta _{max} = O\left[ M_\infty \frac{z_{max}}{c} \right] 
But, zmax/c = tau, so hence,
:\left( \frac{w_b}{a_\infty} \right) _{max} = \left( \frac{w_p}{a_\infty} \right) _{max} = M_\infty \theta _{max} = O\left[ M_\infty \tau \right] = O[K] 
Therefore, the requirements for hypersonic similarity principle to hold are requirements as those for hypersonic small disturbance theory.
===Blast Wave Theory===
The motion of the free piston that causes the disturbance of the air around the piston can be replaced by a "blast" or detonation of energy at the central point. A large amount of energy drives an outwardly-expanding shock wave, just as the piston would. The piston-blast wave analogy is indicated in Fig. 4.11 and 4.13, pg. 148 and 151, by comparison.
Both blunt-nosed slabs and blunt-nosed cylinders can be analyzed for the disturbance they create using blast wave theory. The theory cannot be applied directly to the nose region itself (other theory works for the nose), but the nose drag place a very important role in determining the energy release equivalence for application of blast wave theory.
The equivalence of energy release to nose drag is very simple,
:\text{d}E = D\text{d}x
For the nose moving a unit length,
: E = D
See Fig. 4.15, pg. 153, for an illustration of the principle for an infinite slab. Here E is the energy per unit span per unit length moved by the body and D is the drag per unit span.
The bottom line is that the nose drag of the body defines the blast wave energy needed in the mathematical formulas developed to apply blast wave theory to predict, well blasts.
===Results of Classical Blast Wave Theory===
====Pressure vs Time====
Cylindrical blast:
:p(t) = k_1 \rho_\infty \left( \frac{E}{\rho_\infty} \right) ^\frac{1}{2} t^{-1} 
where,
:k_1 = \frac{\gamma^{\left[ 2(\gamma - 1)/(2-\gamma) \right] }}{2^{\left[ (4 - \gamma)/(2-\gamma) \right] }}
and E is defined as the energy release per unit length.
Planar blast:
:p(t) = k_2 \rho_\infty \left( \frac{E}{\rho_\infty} \right) ^\frac{2}{3} t^{-\frac{2}{3}} 
where,
:k_2 = \frac{2^\frac{7}{3} (2\gamma -1)^{\left[ \frac{(5\gamma -4)}{3(2-\gamma)}\right]}}{9(\gamma +1)^\left[\frac{2(\gamma + 1)}{3(2 - \gamma)} \right] }
and E is defined as the energy release per span per unit length (per area).
====Disturbance Distance (Shock Position) vs Time====
Cylindrical Blast:
:r_s = \left( \frac{E}{\rho_\infty} \right) ^\frac{1}{2} t^\frac{1}{2}
Planar Blast:
:r_s = \left( \frac{E}{\rho_\infty} \right) ^\frac{1}{3} t^\frac{2}{3}
====Application of Blast Wave Theory to Hypersonic Small Disturbance Aerodynamics====
Cylindrical Body:
:E = D = C_D q_\infty S
::q_\infty = \frac{1}{2}\rho_\infty V_\infty^2 = \frac{\gamma p_\infty M_\infty^2}{2}
::S = \frac{\pi}{4}d^2
Using the equivalence principle, replace t with:
:t = \frac{x}{V_\infty}
Substituting into Eqn. 4.144a gives:
:p(x) = k_1 \rho_\infty \sqrt{\frac{\pi}{8}} V_\infty d \sqrt{C_D} \frac{V_\infty}{x} = \frac{k_1 \rho_\infty V_\infty^2}{x} d \sqrt{\frac{\pi C_D}{8}}
or, rearranging, and using \gamma p_\infty M_\infty^2 for \rho_\infty V_\infty^2
:p(x) = k_1 \gamma p_\infty M_\infty^2 \sqrt{\frac{\pi C_D}{8}} \left( \frac{x}{d} \right) ^{-1}
::\sqrt{\frac{\pi}{8}} \approx 0.62657 \qquad \gamma = 1.4
then,
:eqn 4.150
The constant, k1 ≈ 0.07768, so that, for the blunt cylinder (but slender) body,
:eqn 4.151
Looking at the shock position, following the same steps for the equation for shock position gives,
:eqn 4.152
Blunt Planar Body:
:E = D = C_D q_\infty S
::q_\infty = \frac{1}{2}\rho_\infty V_\infty^2 = \frac{\gamma p_\infty M_\infty^2}{2}
::S = d(1) \quad \rightarrow \quad \text{projected area per unit span}
Pressure distribution is:
: eqn 4.154 
Shock position is described by:
: eqn 4.155
Anderson notes that for a cylindrical body,
:p \propto \frac{1}{x} \qquad r_s \propto x^\frac{1}{2}
For blunt-nosed slab:
:p \propto x^{-\frac{2}{3}} \qquad r_s \propto x^\frac{2}{3}
For both:
:p \propto M_\infty^2
Both p(x) and r(x) depend on the nose drag coefficient:
:Newtonian Sphere: CD = 1
:Newtonian Cylinder: CD = 4/3
====How Accurate is Blast Wave Theory?====
Anderson answers the question of the accuracy of blast wave theory applied to slender body hypersonic aerodynamics by presenting some classical comparisons by J. Lukasiewicz (ref. 36 in the text). In these references, a blast wave theory was also applied. They are called the first approximation and the second approximation. Anderson has given us the first approximation theoretical results.
Both first and second approximation results are shown on pages 158-159 from Lukasiewicz and Sakuri for blunt nosed cylinders. Results are displayed in Figs. 4.17-4.24, pg. 159-163. A preliminary remark: second approximation theory gives better agreement with experiment if x in Eqn. 4.158 for the flat plate is replaced by x + (2/3)d.
Fig. 4.18, pg. 160, displays a dashed curve for p/p1 (missed the last couple of boards before break, 7/23/2013)
==Thin Shock-Layer Theory==
==Summary and Comments==