Hypersonic and High-Temperature Gas Dynamics Chapter 3 Notes

Chapter 3 is titled "Local Surface Inclination Methods"
==Introduction==
"Hypersonic flow is essentially non-linear," even including inviscid compressible flow theory. Supersonic compressible flow can be linearized, for example:
:C_p = \frac{2\theta}{\sqrt{M_\infty^2 - 1}}
where θ = θ(x) is the local surface angle with respect to freestream flow. This θ(x) is the local "inclination" method for describing surface pressure at x. (Assumes slender bodies)
Even though hypersonic flow is inherently non-linear, there are local inclination methods that work.
==Newtonian Flow==
Newton hypothesized (guessed) that the aerodynamic force acting on a flat plate varied with the square of the sine of the plate deflection angle.
F \approx \sin^2\theta
This Newtonian theory does not give good results, except at hypersonic speeds.
Momentum conservation gives the following:
:Change in v-velocity on the plate:
:\Delta v = V_\infty \sin \theta
:Mass flux affected
:\dot m_v = \rho_\infty V_\infty \sin\theta
:Force acting on the plate per unit area
:\frac{F}{A} = \dot m_v \Delta v = \rho_\infty V_\infty \sin\theta V_\infty \sin\theta = \rho_\infty V^2_\infty \sin^2\theta
Newton's impact theory corresponded to a stream of individual particles with no (significant) random thermal motion, only directed motion.
But for continuum fluids, the fluid particles do have random thermal motion; λ << Lc. Consequently, the F/A must be due to a pressure difference across the plate.
:\frac{F}{A} = \rho_\infty V^2_\infty \sin^2\theta = p - p_\infty
Since:
:C_p \equiv \frac{p - p_\infty}{\frac{1}{2} \rho_\infty V^2_\infty} \Rightarrow \rho^2_\infty V^2_\infty = \frac{p - p_\infty}{\frac{1}{2} C_p} 
Then by substitution:
:\frac{F}{A} = \frac{p - p_\infty}{\frac{1}{2} C_p} \sin^2\theta = p - p_\infty
or
:C_p = 2\sin^2 \theta
This relation, Cp = 2sin2θ, is call the Newtonian sine-squared pressure coefficient. It works for thin shock layers where it appears as if freestream flow passes through the shock on the body, and then passes downstream parallel to the body.
===Local Inclination Angle===
Newtonian sine-squared theory has been applied to bodies with surface curvature, here θ is the local inclination angle, the angle between the freestream direction and the local tangent to the surface.
Cp thus does not depend on any aspect of the surrounding flowfield (of course, speed must be high enough for the shock to lie close enough to the body surface, i.e. this shock layer).
===Cp-Normal Force Relation for Flat Plate===
:L = N\cos\theta \qquad D = N\sin\theta
:\frac{L}{D} = \frac{1}{\tan\theta} = \cot \theta
The normal force coefficient is:
:C_N = \frac{N}{q_\infty S} = \frac{N}{q_\infty (1) c}
:N = \int_0^c \left( C_{pw} - C_{pl} \right) q_\infty (1) \text{d}x
:N = C_{p_t} q_\infty c
:C_N = \frac{C_{p_t} q_\infty c}{q_\infty (1) (c)} = C_{p_t}
:C_N = 2\sin^2\theta
Similarly, the lift coefficient is,
:C_L = \frac{L}{q_\infty S} = \frac{N\cos\theta}{q_\infty (1)(c)} = \frac{C_{p_t}\cos\theta q_\infty c}{q_\infty c}
:C_L = C_{p_t}\cos\theta = 2\sin^2\theta \cos\theta
The drag coefficient is,
:C_D = \frac{D}{q_\infty S} = \frac{N\sin\theta}{q_\infty (1)(c)} = \frac{C_{p_t}\sin\theta q_\infty c}{q_\infty c}
:C_D = C_{p_t}\sin\theta = 2\sin^3\theta
The lift-to-drag ratio is:
:\frac{L}{D} = \frac{C_L}{C_D} = \frac{\cos\theta}{\sin\theta} = \cot\theta
Anderson plots L/D, CL, and CD vs θ for inviscid hypersonic flow in Fig. 3.7, pg. 60. The plots show that
#L/D → ∞ as θ → 0 (inviscid only, L/D → 0 as θ → 0 for viscous hypersonic flow, since then CD has a non-zero shear force component)
#CL peaks near 55° (maximum lift)CD peaks near 90° (maximum drag)Anderson says that θ ≈ 55° is a "fairly reasonable" angle of attack for "many practical hypersonic vehicles."
#The CL curve is not linear in the small θ range (0° ≤ θ ≤ 10°) which is directly contrary to subsonic and supersonic CL.
==Modified Newtonian Flow==
Anderson notes that CalTech Professor Lester Lees proposed a modification of Newtonian pressure theory. Instead of:
:C_p = 2\sin^2\theta
Lees proposed:
:C_p = C_{p_{max}}\sin^2\theta
Cpmax is the maximum value of Cp. For example, on a blunt body, Cpmax would occur at the stagnation point behind a normal shockwave.
Thus, for a stagnation point:
(eqns 3.4, session 5)
Using this formula, Cpmax is given by (Anderson's equation 3.19, pg. 62):
==Centrifugal Force Corrections to Newtonian Theory==
Anderson says that applying Newtonian Theory to curved surfaces, as done so far, neglects a centrifugal force correction that applies to curved surfaces. Basically, streamline curvature implies a pressure field that varies transverse to the streamlines. Fig. 3.10, pg. 64, shows a sketch of an element of fluid flow along a curved streamline.
(insert figure here)
The pressure at the upper surface must balance the pressure plus the centrifugal force due to the curved motion.
:\frac{\partial p}{\partial n} = \frac{\rho V^2}{R}
Assuming that the stream extends from the wall to some location normal to the wall, then by integration:
:\int_{p_1}^{p_2}\text{d}p = \int_0^{\Delta n} \frac{\rho V^2}{R}\text{d}n
The question is how do we relate the free stream mass flux and momentum flux in the deflected stream tube? We know the free stream density and velocity, ρ∞ and V∞, but we do not necessarily know the local density and velocity, ρ and V. So, Anderson provides a sketch, Fig. 3.11, pg. 65.
(missed some stuff here)
Still, we do not know V. So, with some additional hand-waving, and Fig. 3.11 showing the stream tube being deflected parallel to the body surface, the relation between V∞ and V is:
:V = V_\infty \cos\theta_i
(this assumes no loss of kinetic energy in the stream even though momentum has been transferred. This may be a bad assumption.)
Then the pressure-momentum relationship is:
:\int_{p_1}^{p_2}\text{d}p = \int_0^{y = y_i+\Delta n \cos\theta_i} \frac{\rho_\infty V_\infty^2}{R}\cos\theta \text{d}y
Carry out this integration:
:p_2 - p_i = \frac{\rho_\infty V_\infty^2}{R} \int_0^{y = y_i+\Delta n \cos\theta_i} \cos\theta \text{d}y 
The radius of curvature, R, can be related to the local surface curvature or slope via:
:R = \left. -\frac{\text{d}s}{\text{d}\theta} \right|_i = \left. -\frac{\text{d}y\sin\theta}{\text{d}\theta} \right|_i
By substitution in the pressure equation:
:p_2 - p_i = -\rho_\infty V_\infty^2 \left. \frac{\text{d}\theta}{\text{d}y} \right|_i \sin\theta_i \int_0^{y = y_i+\Delta n \cos\theta_i} \cos\theta \text{d}y
Solving for pi:
:p_i = p_2 + \rho_\infty V_\infty^2 \left. \frac{\text{d}\theta}{\text{d}y} \right|_i \sin\theta_i \int_0^{y = y_i+\Delta n \cos\theta_i} \cos\theta \text{d}y
Subtract pinf from both sides and divide the result by qinf=1/2rhoinfVinf^2
(missed Cp equation)
Where the upper limit was taken as yi since yi >> Δn cosθi
Cp2 is assumed to be the local impact pressure coefficient, i.e.:
:C_{p_2} = 2\sin^2\theta_i
Hence, the centrifugally-corrected form of the Newtonian theory is:
:C_{p_i} = 2\sin^2\theta_i + 2 \left. \frac{\text{d}\theta}{\text{d}y} \right|_i \sin\theta_i \int_0^{y_i} \cos\theta \text{d}y
For axisymmetric bodies or bodies of revolution:
:C_{p_i} = 2\sin^2\theta_i + 2 \left. \frac{\text{d}\theta}{\text{d}y} \right|_i \frac{\sin\theta_i}{y_i} \int_0^{y_i} y \cos\theta \text{d}y
This "theory" was derived by Adolf Busemann in 1933.
===Slender Body Forms===
For slender blunt cylinders and bodies of revolution, the Cpi equations reduce to the simple forms:
:C_p = 2(\theta^2 - Ky) \text{ for slender blunt bodies and curved semi-infinite wedged bodies}
:C_p = 2\theta^2 - Ky \text{ for slender axisymmetric blunt bodies}
Where:
:K \equiv \text{ curvature of surface at point i } = \left( \frac{\partial \theta}{\partial s} \right)_i 
So, how does the theory work?
It doesn't. The so-called Newton-Busemann theory is (missed this part).
==Newtonian Theory — What it Really Means==
Anderson makes the point that exact oblique shock results for Cp approach Newtonian theory as M∞ → ∞ and γ → 1.0. Exact Cp 2-D oblique shocks:
:C_p = \frac{4}{\gamma + 1} \left( \sin^2\beta - \frac{1}{M_\infty^2} \right)
As M∞ → ∞:
:C_p \rightarrow \frac{4}{\gamma + 1} \sin^2\beta 
As M∞ → ∞ and γ → 1.0:
:C_p \rightarrow 2 \sin^2\beta 
β is the wave angle, not the deflection angle, θ, but β → θ as M∞ → ∞ and γ → 1.0.
The exact density ratio for 2-D oblique shocks is:
:\frac{\rho_2}{\rho_infty} = \frac{(\gamma + 1) M_\infty^2 \sin^2 \beta}{(\gamma - 1) M_\infty^2 \sin^2 \beta + 2}
As M∞ → ∞:
:\frac{\rho_2}{\rho_infty} \rightarrow \frac{\gamma + 1}{\gamma - 1}
As M∞ → ∞ and γ → 1.0:
:\frac{\rho_2}{\rho_infty} \rightarrow \infty 
"The density behind the shock wave gets infinitely large."
As M∞ → ∞:
:\beta \rightarrow \frac{\gamma + 1}{2} \theta 
As M∞ → ∞ and γ → 1.0:
:\beta \rightarrow \theta 
Therefore, in the limit as M∞ → ∞ and γ → 1.0, the 2-D exact oblique solution for Cp goes to:
:C_p = 2\sin^2\theta
This limit is precisely the Newtonian theory. Anderson states that in the limit as M∞ → ∞ and γ → 1.0, Newtonian-Busemann theory gives better results than straight Newtonian theory.
(see Fig. 3.13, pg. 72)
===Worked Example: Comparison of Newtonian with Exact Theory===
A worked example of a comparison of predictions of lift and drag for an infinitesimally thin flat plate is made with Newtonian theory and "exact" oblique shock wave theory.
Assumptions:
#Thin flat plate is at α = 15° in M = 8 flow
#γ = 1.4
Tasks:
#Compute Cp on the upper and lower surfaces, L, D, CL, and CD
#Compare Newtonian theory to the "exact" methods using oblique shocks and Prandtl-Meyer expansions
Sketch:
(see figure 3.16, pg. 75)
Note that free stream pressure is not provided.
===="Exact" Solution====
Oblique shock theory for the "pressure surface"
(see section 3.5.1 in book)t
===Commentary===
===Lift-to-Drag Ratio for Hypersonic Bodies — Comment===
==Tangent-Wedge Tangent-Cone Methods==
Local inclination methods include Newtonian methods and tangent-wedge and tangent-cone methods.
===Tangent-Wedge===
Let's consider a pointed-nose body with attached shock.
(Fig. 3.17, pg. 79)
#Draw a tangent to any point on the body and extend the tangent to the axis.
The tangent defines the surface of a "semi-wedge" of angle equal to the tangent angle, θi. At point i the pressure equals the pressure on a "semi-wedge" of the same angle θi.
In order for this to be used, the angles, θi, along the body surface must be less than the maximum deflection angle for γ and M∞.
The pressure, pi, is the p2 value given by the exact oblique shock theory, if the body is 2-D in geometry.
===Tangent-Cone===
If the body is axisymmetric, then the tangent line extended down to the axis of symmetry represents an equivalent or tangent cone.
The pressure, pi, is now the surface pressure of the equivalent tangent cone.
===Justification of these Methods===
They seem to work for some cases of sharp-nosed or pointed bodies at high speeds. See Fig. 3.20, pg. 82, where a tangent-cone method is applied to an ogive body.
The method of characteristics has been used to predict inviscid surface pressures for different values of K = M∞(d/l), where (d/l) is the slenderness ratio of the body. The agreement is remarkable. Newtonian theory applies to both blunt and pointy bodies or sharp wedges, but tangent-wedge and tangent-cone methods only apply to bodies pointy or slender enough to have attached shock waves. The shock cannot be detached for tangent-wedge or tangent-cone methods to apply.
==Shock-Expansion Method==
The shock-expansion method is another local inclination method that only applies to sharp leading edges or sharp-nosed bodies of revolution.
#The deflection at the nose is θn (see Fig. 3.21, pg. 83)
#Compute the properties behind the oblique shock produced by a wedge of the same angle θn.
#Use Prandtl-Meyer theory to relate conditions at other downstream points on the body surface.
Example
A point farther down the body surface has a local inclination angle, θi. Determine the Mach number at this point, i, using Prandtl-Meyer theory.
:\Delta\theta = \theta_n - \theta_i = \nu(M_i) - \nu(M_n)
:\nu(M_i) = \nu(M_n) + \Delta\theta
That is:
(long write out of the Prandtl-Meyer)
By now, you should have the Prandtl-Meyer function programmed into a function in your programming language of choice (mine is Python), so finding values for Mi should not be an issue.
Once Mi is known, use the isentropic pressure relationships to find pi, given that pn is known from the oblique shock theory. For example, for a sharp-pointed (or sharp-wedge) nose, the local inclination angle at the nose defines the initial nose shock angle, treated as an oblique shock.
(Missed all of the remainder of class on 6/28/2013)
==Summary and Comments==
Final remarks on Newtonian theories for hypersonic pressure distributions from "A Survey of Hypersonic Aerodynamics and Aerothermodynamics for Planetary Entry Capsules," C. Ong, R. D. Braun, and S. M. Ruffin.
#The theories apply to continuum flows, where Kn << 1 or Halt < 92 km.
#Flows are in chemical equilibrium
#United States Standard Atmosphere, 1976
#Pressure theories:
##NewtonianC_p(\theta) = 2\sin^2\theta
##Newton-LeesC_p(\theta,M,\gamma) = C_{p_{max}}\sin^2\thetaC_{p_{max}} = \frac{2}{\gamma M_\infty^2} \left( \left[ \frac{(\gamma+1)^2 M_\infty^2}{4\gamma M_\infty^2 - 2(\gamma - 1)}\right] ^\frac{\gamma}{\gamma - 1}\left[\frac{1 - \gamma - 2\gamma M_\infty^2}{\gamma + 1}\right] -1 \right)
##Newton-BusemannC_{p_i} = 2\sin^2\theta_i + 2 \left. \frac{\text{d}\theta}{\text{d}y} \right|_i \sin\theta_i \int_0^{y_i} \cos\theta \text{d}yfor 2-D bodies.
Note that the authors remark that the Newton-Lees theory "requires post shock γ" which is not mentioned by Dr. Anderson in his hypersonic text. The theory in Anderson's book applies to constant γ = 1.4. Note, however, that from the definition of Cpmax:
:C_{p_{max}} = \frac{p_{max} - p_1}{q_1} = \frac{p_{t2} - p_1}{\gamma_1 \frac{p_1 M_1^2}{2}}
Here, γ1 is the free stream value of γ, i.e. γ1 = 1.4. Rearranging:
:C_{p_{max}} = \frac{2}{\gamma_1 M_1^2} \left(\frac{p_{t2}}{p_1} - 1 \right)
Now, pt2 can be computed using real gas, chemical equilibrium codes, which can account for both variable gas chemistry due to dissociation and variable cp, cv, γ, R effects.
So, we can compute Cpmax directly with a chemistry code such as NASA CEA, which can be used to compute pt2 behind a normal shock for a blunt body.
Finally, the Newton-Busemann theory for centrifugal force corrections, also does not appear to improve straight Newtonian theory or the Lees' modification of it.
==Design Example 3.1==
Anderson describes an industry standard program for estimating surface pressure on hypersonic vehicles called Hypersonic Arbitrary Body Program (HABP). The program predicts surface pressures, shear forces, and heat transfer distributions over arbitrary bodies in hypersonic flow. The program also includes lift, drag, and moment coefficients. HAPB offers 17 different options for predicting surface pressures on windward (pressure) and leeward (suction) sides of surfaces.
Anderson cites ref. 225 by Fisher, who applied HAPB to ogive and shuttle-like configurations and compared results to experimental data. Some of this information from Fisher's study is shown in Figs. 3.26-3.28, pg. 90-93. Fig. 3.26, pg. 90, shows that the combination of surface pressure methods denoted 3,2(K=1.81)(cylinder-boattail) gave the best comparison to the data for the "cylinder-boattail" part of the body and 6,4 for the ogive part of the body.
Other methods in use for similar purposes, such as the NASA Ames code DPLR, are also presented.
See table 3.1, pg. 89, for the various surface pressure prediction methods for windward and leeward surfaces. Anderson notes that use of these methods for various body-parts is a "mix-and-match" trial and error process. It helps (a lot) to have experimental data to compare to!
===Comparison to Shuttle-like Configuration===
Fig. 3.27, pg. 92, shows results from Fisher where a shuttle-like configuration was broken down into separate "panels," each of which was treated as an aerodynamic surface element. Straight Newtonian theory was used to model each panel Cp over the entire set of panels. This yielded CL, CD, and CM for the vehicle shape.
The theoretical values are compared to shuttle performance data at one Mach number (M = 13.5) where α values are matched. The results appear in very reasonable agreement with data.
Fig. 3.28 shows further comparisons