Appendix A————————————————————–

Governing Equations for Flow in GeneralCoordinates

We collect the transport equations for a multi-component, multi-temperaturenon-equilibrium flow.1 All transport equations have been discussed in Chap-ter 4, as well as in Chapters 5, 6, 7 (see also [2]–[4]). We write the equationsin (conservative) flux-vector formulation, which we have used already for theenergy equation in Sub-Section 4.3.2, and for three-dimensional Cartesiancoordinates:

∂Q

∂t+

∂(E + Evisc)

∂x+

∂(F + F visc)

∂y+

∂(G+Gvisc)

∂z= S. (A.1)

Q is the conservation vector, E, F , G are the convective (inviscid) and Evisc,Fvisc, Gvisc the viscous fluxes in x, y, and z direction. S is the source termof mass and of vibration energy.

The conservation vector Q has the form

Q = [ρi, ρu, ρv, ρw, ρet, ρmevibr,m], (A.2)

where ρi are the partial densities of the involved species i, Section 2.2, ρ isthe density, u, v, w are the Cartesian components of the velocity vector V , et= e + 1/2 V 2 is the mass-specific total energy (V = |V |), Sub-Section 4.3.2,and evibr,m the mass-specific vibration energy of the molecular species m.

The convective and the viscous fluxes in the three directions read

E =

⎡

⎢⎢⎢⎢⎢⎢⎣

ρiuρu2 + pρuvρuwρuht

ρmuevibr,m

⎤

⎥⎥⎥⎥⎥⎥⎦, Evisc =

⎡

⎢⎢⎢⎢⎢⎢⎣

jixτxxτxyτxz

qx −∑i ji,xhi + uτxx + vτxy + wτxz

−kvibr,m∂Tvibr,m

∂x + jmxevibr,m

⎤

⎥⎥⎥⎥⎥⎥⎦,

(A.3)

1 We include only the transport of non-equilibrium vibration energy. Equations forother entities, like rotation energy, can be added. For a more general formulationsee, e.g., [1].

414 Appendix A Governing Equations for Flow in General Coordinates

F =

⎡

⎢⎢⎢⎢⎢⎢⎣

ρivρuv

ρv2 + pρvwρvht

ρmvevibr,m

⎤

⎥⎥⎥⎥⎥⎥⎦, F visc =

⎡

⎢⎢⎢⎢⎢⎢⎣

jiyτyxτyyτyz

qy −∑

i ji,yhi + uτyx + vτyy + wτyz−kvibr,m

∂Tvibr,m

∂y + jmyevibr,m

⎤

⎥⎥⎥⎥⎥⎥⎦,

(A.4)

G =

⎡

⎢⎢⎢⎢⎢⎢⎣

ρiwρuwρvw

ρw2 + pρwht

ρmwevibr,m

⎤

⎥⎥⎥⎥⎥⎥⎦, Gvisc =

⎡

⎢⎢⎢⎢⎢⎢⎣

jizτzxτzyτzz

qz −∑

i ji,xhi + uτzx + vτzy + wτzz−kvibr,m

∂Tvibr,m

∂z + jmzevibr,m

⎤

⎥⎥⎥⎥⎥⎥⎦.

(A.5)The convective flux vectors E, F , G represent from top to bottom the

transport of mass, Sub-Section 4.3.3, momentum, Sub-Section 4.3.1, of totalenergy, Sub-Section 4.3.2, and of non-equilibrium vibration energy, Section5.4.2 In the above ht = et + p/ρ is the total enthalpy, eq. (5.7).

In the viscous flux vectors Evisc, F visc, Gvisc, the symbols jix etc. rep-resent the Cartesian components of the diffusion mass-flux vector j

iof the

species i, Section 4.3.3, and τxx, τxy etc. the components of the viscous stresstensor τ , eqs. (7.10) to (7.15) in Section 7.1.3. In the fifth line each we havethe components of the energy-flux vector q

e, eq. (4.58), Sub-Section 4.3.2,

which we have summarized as generalized molecular heat-flux vector in eqs.(4.61) and (4.62), the latter being the equation for the heat flux in the gas atthe wall in the presence of slip flow. In the sixth line, finally, we find the termsof molecular transport of the non-equilibrium vibration energy in analogy tothe terms of eq. (4.59).

The components of the molecular heat flux vector q in the viscous fluxvectors in eqs. (A.3) to (A.4) read, due to the use of a multi-temperaturemodel with m vibration temperatures

⎛

⎝qxqyqz

⎞

⎠ = −ktrans∇Ttrans +∑

m

kvibr,m∇Tvibr,m. (A.6)

The source term S finally contains the mass sources Smi of the speciesi due to dissociation and recombination, eq. (4.84) in Sub-Section 4.3.3, andthe energy sources Qk,m

2 We have not given the equation(s) for the transport of vibrational energy inSection 5.4, but have referred instead to [5].

Appendix A Governing Equations for Flow in General Coordinates 415

S =

⎡

⎢⎢⎢⎢⎢⎢⎣

Smi

0000∑

k

∑mQk,m

⎤

⎥⎥⎥⎥⎥⎥⎦. (A.7)

The terms Qk,m represent the k mechanisms of energy exchange of them vibration temperatures, for instance between translation and vibration,vibration and vibration, etc., Sub-Section 5.4.

To compute the flow past configurations with general geometries, theabove equations are transformed from the physical space x, y, z into thecomputation space ξ, η, ζ:

ξ = ξ(x, y, z),

η = η(x, y, z),

ζ = ζ(x, y, z).

(A.8)

This transformation, which goes back to H. Viviand [6] and to M. Vinokur[7], regards only the geometry, and not the velocity components. This isin contrast to the approach for the general boundary-layer equations, Sub-Section 7.1.3, where both are transformed. ξ usually defines the main-streamdirection, η the lateral direction, and ζ the wall-normal direction, however ingeneral not in the sense of locally monoclinic coordinates, Sub-Section 7.1.3.

The transformation results in

∂Q

∂t+

∂(E + Evisc)

∂ξ+

∂(F + F visc)

∂η+

∂(G+ Gvisc)

∂ζ= S, (A.9)

which is the same as the original formulation, eq. (A.1).The transformed conservation vector, the convective flux vectors, and the

source term are now

U = J−1U,

E = J−1[ξxE + ξyF + ξzG],

F = J−1[ηxE + ηyF + ηzG],

G = J−1[ζxE + ζyF + ζzG],

S = J−1S,

(A.10)

with J−1 being the Jacobi determinant of the transformation. The trans-formed viscous flux vectors have the same form as the transformed convectiveflux vectors.

The fluxes, eqs. (A.3) to (A.5), are transformed analogously, however wedo not give the details, and refer instead to, for instance, [8, 9].

416 Appendix A Governing Equations for Flow in General Coordinates

References

1. Hirschel, E.H., Weiland, C.: Selected Aerothermodynamic Design Problems ofHypersonic Flight Vehicles, AIAA, Reston, Va. Progress in Astronautics andAeronautics, vol. 229. Springer, Heidelberg (2009)

2. Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena, 2nd edn.John Wiley, New York (2002)

3. Vincenti, W.G., Kruger, C.H.: Introduction to Physical Gas Dynamics. JohnWiley & Sons, New York (1965), Reprint edition, Krieger Publishing Comp.,Melbourne, Fl (1967)

4. Hoffmann, K.A., Chiang, S.T.L., Siddiqui, M.S., Papadakis, M.: FundamentalEquations of Fluid Mechanics. Engineering Education System, Wichita (1996)

5. Park, C.: Nonequilibrium Hypersonic Flow. John Wiley & Sons, New York (1990)6. Viviand, H.: Conservative Forms of Gas Dynamic Equations. La Recherche

Aerospatiale 1, 65–68 (1974)7. Vinokur, M.: Conservative Equations of Gas-Dynamics in Curvilinear Coordi-

nates. J. Comp. Physics 14, 105–125 (1974)8. Pulliam, T.H., Steger, J.L.: Implicit Finite-Difference Simulations of Three-

Dimensional Compressible Flows. AIAA J. 18(2), 159–167 (1980)9. Hirsch, C.: Numerical Computation of Internal and External Flows. Fundamen-

tals of Numerical Discretization, vol. 1. J. Wiley & Sons, New York (1988)

Appendix B————————————————————–

Constants, Functions, SI Dimensions andConversions

We provide in Appendix B.1 constants and air properties. In Appendix B.2the dimensions of the most important quantities are given. The dimensionsare in general the SI units (Systeme International d’unites), see [1, 2], wherealso the constants given in Appendix B.1 can be found. The basic units, thederived units, and conversions1 to US customary units (English units) aregiven. Note that we write the dimensions as ‘a/bc’ instead of ‘a/(bc)’ or ‘ab−1 c−1’.

B.1 Constants and Air Properties

The molar universal gas constant R0, the Stephan-Boltzmann constant σ,and standard gravitational acceleration g0 are:

Molar universal gas constant R0 = 8.314472·103 kgm2/s2 kg-molK= 4.97201·104 lbm ft2/s2 lbm-mol ◦R,

Stephan-Boltzmann constant σ = 5.670400·10−8 W/m2 K4

= 1.7123·10−9 Btu/hr ft2 ◦R4.

Standard gravitationalacceleration at sea level g0 = 9.80665 m/s2

= 32.174 ft/s2.

In Table B.1 we list some properties of air and its major constituents.The properties were addressed in Chapters 2 and 4. Listed are the molecularweights, the gas constants, and intermolecular force parameters, all for thetemperature domain of our interest.

Fig. B.1 shows the dimensionless collision integrals Ωμ = Ωk, and ΩDAB

of air as functions of the dimensionless temperatures kT/ε or kT/εAB. TheLennard-Jones parameters ε/k [K] for air and its constituents are given inTable B.1.1 Details can be found, for instance, athttp://physics.nist.gov/cuu/Units/units.html or athttp://www.chemie.fu-berlin.de/chemistry/general/si_en.html.

418 Appendix B Constants, Functions, SI Dimensions and Conversions

Table B.2 lists the characteristic rotational, vibrational, and dissociationtemperatures of the molecular air constituents N2, O2, and NO.

Table B.1. Molecular weights, gas constants, and intermolecular force parametersof air constituents for the low temperature domain [3, 4]. ∗ is the U. S. standardatmosphere value, + the value from [4].

Gas Molecular weight Specific gas constant Collision dia- 2nd Lennard-Jonesmeter parameter

M [kg/kg-mol] R [m2/(s2K)] σ · 1010 [m] ε/k [K]

air 28.9644∗ (28.97+) 287.06 3.617 97.0

N2 28.02 296.73 3.667 99.8

O2 32.00 259.83 3.430 113.0

NO 30.01 277.06 3.470 119.0

N 14.01 593.47 2.940 66.5

O 16.00 519.65 2.330 210.0

Ar 39.948 208.13 3.432 122.4

He 4.003 2077.06 2.576 10.2

Table B.2. Characteristic rotational, vibrational, and dissociation temperaturesof air molecules [5].

Gas: N2 O2 NO

Θrot [K] 2.9 2.1 2.5

Θvibr [K] 3,390 2,270 2,740

Θdiss [K] 113,000 59,500 75,500

B.2 Dimensions and Conversions

SI basic units and SI derived units are listed of the major flow, transport,and thermal quantities. In the left column name and symbol are given andin the right column the unit (dimension), with the symbol in → [ ] usedin Appendix C, and in the line below its conversion to the U.S. customaryunits.

B.2 Dimensions and Conversions 419

0 5 10 15 20 25 300

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

kT/ε

Ωμ, Ω

D,A

BΩ

μΩ

D, AB

Fig. B.1. Dimensionless collision integrals Ωμ = Ωk, and ΩDAB of air as functionsof kT/ε or kT/εAB [4, 6].

SI Basic Units

length, L [m], → [L]1 m = 100 cm = 3.28084 ft(1,000 m = 1 km)

mass, m [kg], → [M]1 kg = 2.20462 lbm

time, t [s] (= [sec]), → [t]

temperature, T [K], → [T]1 K = 1.8 ◦R⇒ TRankine = TFahrenheit + 459.67⇒ TKelvin = (5/9) TRankine

⇒ TKelvin = TCelsius + 273.15

amount of substance, mole [kg-mol], → [mole]1 kg-mol = 2.20462 lbm-mol

420 Appendix B Constants, Functions, SI Dimensions and Conversions

SI Derived Units

area, A [m2], → [L2]1 m2 = 10.76391 ft2

volume, V [m3], → [L3]1.0 m3 = 35.31467 ft3

speed, velocity, v, u [m/s], → [L/t]1.0 m/s = 3.28084 ft/s

force, F [N] = [kg m/s2], → [M L/t2]1.0 N = 0.224809 lbf

pressure, p [Pa] = [N/m2], → [M/L t2]1.0 Pa = 10−5 bar = 9.86923·10−6 atm == 0.020885 lbf/ft

2

density, ρ [kg/m3], → [M/L3]1.0 kg/m3 = 0.062428 lbm/ft3

(dynamic) viscosity, μ [Pa s] = [N s/m2], → [M/L t]1.0 Pa s = 0.020885 lbf s/ft2

kinematic viscosity, ν [m2/s], → [L2/t]1.0 m2/s = 10.76391 ft2/s

shear stress, τ [Pa] = [N/m2], → [M/L t2]1.0 Pa = 0.020885 lbf/ft

2

energy, enthalpy, work, [J] = [N m], → [M L2/t2]quantity of heat 1.0 J = 9.47813·10−4 BTU =

= 23.73036 lbmft2/s2 = 0.737562 lbf/s2

(mass specific) internal energy, [J/kg] = [m2/s2], → [L2/t2]enthalpy, e, h 1.0 m2/s2 = 10.76391 ft2/s2

(mass) specific heat, cv,cp [J/kg K] = [m2/s2 K], → [L2/t2T]specific gas constant, R 1.0 m2/s2K = 5.97995 ft2/s2 ◦R

power, work per unit time [W] = [J/s] = [N m/s], → [M L2/t3]1.0 W = 9.47813·10−4 BTU/s == 23.73036 lbmft2/s3

thermal conductivity, k [W/m K] = [N/s K], → [M L/t3T]1.0 W/m K = 1.60496·10−4 BTU/s ft ◦R =

References 421

= 4.018342 lbm ft/s3 ◦R

heat flux, q [W/m2] = [J/m2s], → [M/t3]1.0 W/m2 = 0.88055·10−4 BTU/s ft2 == 2.204623 lbm/s3

(binary) mass diffusivity, DAB [m2/s], → [L2/t]1.0 m2/s = 10.76391 ft2/s

thermo diffusivity, DTA [kg/m s], → [M/L t]

1.0 kg/m s = 0.67197 lbm/ft s

diffusion mass flux, j [kg/m2s], → [M/L2t]1.0 kg/m2s = 0.20482 lbm/ft2s

References

1. Taylor, B.N. (ed.): The International System of Units (SI). US Dept. of Com-merce, National Institute of Standards and Technology, NIST Special Publica-tion 330, 2001, US Government Printing Office, Washington, D.C. (2001)

2. Taylor, B.N. (ed.): Guide for the Use of the International System of Units (SI).US Dept. of Commerce, National Institute of Standards and Technology, NISTSpecial Publication 811, 1995, US Government Printing Office, Washington, D.C.(1995)

3. Hirschfelder, J.O., Curtiss, C.F., Bird, R.B.: Molecular Theory of Gases andLiquids. John Wiley & Sons, New York (1966)

4. Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena, 2nd edn.John Wiley & Sons, New York (2002)

5. Vincenti, W.G., Kruger, C.H.: Introduction to Physical Gas Dynamics. JohnWiley & Sons, New York (1965), Reprint edition, Krieger Publishing Comp.,Melbourne, Fl (1975)

6. Neufeld, P.D., Jansen, A.R., Aziz, R.A.: J. Chem. Phys. 57, 1100–1102 (1972)

Appendix C————————————————————–

Symbols and Acronyms

Only the important symbols are listed. If a symbol appears only locally orinfrequent, it is not included. In general the page number is indicated, wherea symbol appears first or is defined. Dimensions are given in terms of the SIbasic units: length [L], time [t], mass [M], temperature [T], and amount ofsubstance [mole], Appendix B. The dimensions are written as ‘a/bc’ insteadof ‘a/(bc)’ or ‘a b−1 c−1’. For actual dimensions and their conversions seeAppendix B.2.

C.1 Latin Letters

A amplitude, p. 287A surface, [L2]a speed of sound, p. 153, [L/t]C Chapman-Rubesin factor, p. 363, [-]CD drag coefficient, p. 197, [-]CL lift coefficient, p. 197, [-]CR resultant (total) force coefficient, p. 198, [-]c molar density, p. 29, [mole/L3]c phase velocity, p. 287, [L/t]ci molar concentration of species i, p. 29, [mole/L3]cp pressure coefficient, p. 155, [-]cp (mass) specific heat at constant pressure, p. 116, [L2/t2T]cv (mass) specific heat at constant volume, p. 116, [L2/t2T]D diameter, [L]D drag, [ML/t2]DAM1 first Damkohler number, p. 120, [-]DAM2 second Damkohler number, p. 120, [-]DAB mass diffusivity coefficient of a binary system, p. 86, [L2/t]DT

A thermo-diffusion coefficient of species A, p. 103, [M/Lt]E Eckert number, p. 99, [-]e internal (mass-specific) energy, p. 96, [L2/t2]f degree of freedom, p. 115, [-]f frequency, p. 287, [1/t]G Gortler parameter, p. 306, [-]

424 Appendix C Symbols and Acronyms

H altitude, p. 2, [L]H shape factor, p. 247, [-]h (mass-specific) enthalpy, p. 116, [L2/t2]ht total enthalpy, p. 38, [L2/t2]h∗ reference enthalpy, p. 234, [L2/t2]ji

diffusion mass-flux vector of species i, p. 103, [M/L2t]

K hypersonic similarity parameter, p. 209, [-]K acceleration parameter, p. 306, [-]

Kc equilibrium constant, p. 123, [(L3/mole)ν′′−ν′

]Kn Knudsen number, p. 31, [-]k thermal conductivity, p. 84, [ML/t3T]k roughness height, surface parameter, p. 308, [L]k turbulent energy, p. 313, [L2/t2]

kfr forward reaction rate, p. 123, [L3/moleν′−1 t]

kbr backward reaction rate, p. 123, [L3/moleν′′−1 t]

kwiacatalytic recombination rate, p. 133, [L/t]

L characteristic length, [L]L lift, [ML/t2]Le Lewis number, p. 99, [-]M Mach number, p. 92, [-]M molecular weight, p. 29, [M/mole]M moment, [ML2/t2]MN Mach number normal (locally) to the shock wave, p. 163, [-]Mi molecular weight of species i, p. 29, [M/mole]M∞ flight Mach number, [-]M∗ critical Mach number, p. 154, [-]Pe Peclet number, p. 98, [-]Pr Prandtl number, p. 98, [-]p pressure, p. 29, [M/Lt2]pe pressure at the boundary-layer edge, p. 199, [M/Lt2]pi partial pressure of species i, p. 29, [M/Lt2]q dynamic pressure, p. 154, [M/Lt2]q∞ free-stream dynamic pressure, [M/Lt2]q heat flux, p. 82, [M/t3]qgw heat flux in the gas at the wall, p. 38, [M/t3]qw heat flux into the wall, p. 38, [M/t3]qrad thermal radiation heat flux, p. 44, [M/t3]q′ disturbance amplitude, p. 287R gas constant, p. 29, [L2/t2T]R radius, [L]R0 universal gas constant, p. 29, [ML2/t2moleT]Re Reynolds number, p. 92, [-]Reu unit Reynolds number, p. 26, [1/L]Reθ contamination Reynolds number, p. 300, [-]r recovery factor, p. 39, [-]

C.2 Greek Letters 425

ri volume fraction of species i, p. 30, [-]S source term, p. 413, [-]Sc Schmidt number, p. 104, [-]Smi species source term, p. 102, 122, [M/L3t]Sr Strouhal number, p. 81, [-]St Stanton number, p. 38, [-]s entropy, p. 153, [L2/t2T]T temperature, p. 29, [T]Tgw temperature of the gas at the wall, p. 14, [T]Tra radiation-adiabatic temperature, p. 40, [T]Tt total temperature, p. 39, [T]Tw wall temperature, p. 14, [T]Tr recovery temperature, p. 38, [T]Tu level of free-stream turbulence, p. 312, [-]T ∗ reference temperature, p. 234, [T]u, v, w Cartesian velocity components, p. 78, [L/t]u′, v′, w′ non-dimensional Cartesian velocity components, p. 205, [L/t]u, v velocity components normal and tangential to an oblique

shock wave, p. 163, [L/t]u∞, v∞ free-stream velocity, flight speed, p. 26, [L/t]uvs viscous sub-layer edge-velocity, p. 243, [L/t]u+ non-dimensional velocity, p. 243, [-]uτ friction velocity, p. 243, [L/t]Vm maximum speed, p. 154, [L/t]V1, V2 resultant velocities ahead and behind an oblique shock

wave, p. 163, [L/t]V velocities vector, p. 78, [L/t]V, V interaction parameters, p. 366, [-]v∗i physical velocity components, p. 218, [L/t]v0 surface suction or blowing velocity, p. 245, [L/t]x, y, z Cartesian coordinates, p. 78, [L]xi surface-oriented locally monoclinic coordinates, p. 219, [-]xi mole fraction of species i, p. 29, [-]y+ non-dimensional wall distance, p. 243, [-]Z real-gas factor, p. 112, [-]

C.2 Greek Letters

α angle of attack, p. 6, [◦]α thermal accommodation coefficient, p. 101, [-]α thermal diffusivity, p. 98, [L2/t]α wave number, p. 287, [1/L]γ ratio of specific heats, p. 116, [-]

426 Appendix C Symbols and Acronyms

γeff effective ratio of specific heats, p. 178, [-]γia recombination coefficient of atomic species, p. 132, [-]Δ characteristic boundary-layer thickness, p. 44, [L]Δ0 shock stand-off distance, p. 176, [L]δ flow (ordinary) boundary-layer thickness, p. 93, [L]δ ramp angle, p. 163, [◦]δ shock stand-off distance (δ ≡ Δ0), p. 177, [L]δflow flow (ordinary) boundary-layer thickness (δflow ≡ δ), p. 93, [L]δM mass-concentration boundary-layer thickness, p. 104, [L]δT thermal boundary-layer thickness, p. 100, [L]δsc turbulent scaling thickness, p. 243, [L]δvs viscous sub-layer thickness, p. 242, [L]δ1 boundary-layer displacement thickness (δ1 ≡ δ∗), p. 245, [L]δ2 boundary-layer momentum thickness (δ2 ≡ θ), p. 245, [L]ε emissivity coefficient, p. 44, [-]ε density ratio, p. 176, [-]εf fictitious emissivity coefficient, p. 54, [-]Θdiss characteristic dissociation temperature, p. 123, [T]Θrot characteristic rotational temperature, p. 119, [T]Θvibr characteristic vibrational temperature, p. 121, [T]θ flow angle, p. 78, [◦]θ shock angle, p. 163, [◦]θ∗ sonic shock angle, p. 166, [◦]κ bulk viscosity, p. 90, [M/Lt]λ mean free path, p. 31, [L]λ wave length, p. 287, [L]μ viscosity, p. 83, [M/Lt]μ Mach angle, p. 169, [◦]ν kinematic viscosity, p. 227, [L2/t]ν Prandtl-Meyer angle, p. 189, [◦], [rad]ν �ir, ν

�

ir stoichiometric coefficients, p. 122, [-]ρ density, p. 29, [M/L3]ρi partial density of species i, p. 29, [M/L3]ρt total density, p. 153, [M/L3]ρ∗i fractional density of species i, p. 30, [M/L3]ρ0 surface suction or blowing density, p. 245, [M/L3]σ reflection coefficient, p. 94, [-]τ relaxation time, p. 121, [t]τ thickness ratio, p. 209, [t]τ viscous stress tensor, p. 90, [M/t2L]τxx, τxy, ... components of the viscous stress tensor, p. 221, [M/t2L]τw skin friction, wall shear stress, p. 256, [M/t2L]Φ angle, p. 355, [◦]Φ transported entity, p. 79Φij term in Wilke’s mixing formula, p. 88

C.3 Indices 427

ϕ characteristic manifold, p. 227ϕ sweep angle of leading edge or cylinder, p. 202, [◦]χ cross-flow Reynolds number, p. 301, [-]χ, χ viscous interaction parameter, p. 364, [-]Ψ ′ disturbance stream function, p. 288, [1/t]ψ angle, p. 201, [◦]Ω vorticity content vector, p. 218, [L/t]Ωk dimensionless thermal conductivity collision integral, p. 85, [-]Ωμ dimensionless viscosity collision integral, p. 83, [-]ω circular frequency, p. 287, [1/t]ωi mass fraction of species i, p. 29, [-]ωk exponent in the power-law equation of thermal

conductivity (ωk, ωk1, ωk2), p. 85, [-]ωμ exponent in the power-law equation of

viscosity (ωμ, ωμ1, ωμ2), p. 83, [-]

C.3 Indices

C.3.1 Upper Indices

i i = 1, 2, 3, general coordinates and contravariant velocitycomponents

∗i physical velocity componentT thermo-diffusionu unit+, − ionized+ dimensionless sub-layer entity∗ reference-temperature/enthalpy value

C.3.2 Lower Indices

A, B species of binary gas

br backward reaction

c compressible

corr corrected

cr critical

D drag

e boundary-layer edge, external (flow)

elec electronic

eff effective

equil equilibrium

exp experiment

428 Appendix C Symbols and Acronyms

fp flat surface portion

fr forward reaction

gw gas at the wall

Han Hansen

i imaginary part

i, j species

ic incompressible

k thermal conductivity

L lift

l leeward side

lam laminar

M mass concentration

molec molecule

ne non-equilibrium

R resultant

r real part

r reaction

r recovery

ra radiation adiabatic

rad radiation

ref reference

res residence

rot rotational

SL sea level

Suth Sutherland

s shock

s stagnation point

sc turbulent scaling

scy swept cylinder

sp sphere

T thermal

t total

tr,l transition, lower location

tr,u transition, upper location

trans translational

turb turbulent

vibr vibrational

vs viscous sub-layer

w wall

w windward side

μ viscosity

τ friction velocity

0 reference

1 ahead of the shock wave

2 behind the shock wave

C.5 Acronyms 429

∞ infinity

∗ critical

C.4 Other Symbols

O( ) order of magnitude′ non-dimensional and stretched′ fluctuation entityv vectort tensor

C.5 Acronyms

Indicated is the page where the acronym is used for the first time.

AOTV aeroassisted orbital transfer vehicle, p. 3ARV ascent and re-entry vehicle, p. 3BDW blunt delta wing, p. 63CAV cruise and acceleration vehicle, p. 3DNS direct numerical solution, p. 296FESTIP Future European Space Transportation Investigations

Programme, p. 3GETHRA general thermal radiation, p. 54HALIS high alpha inviscid solution (configuration), p. 135LES large eddy simulation, p. 313OMS orbital maneuvering system, p. 52RANS Reynolds-averaged Navier-Stokes, p. 55RHPM Rankine-Hugoniot-Prandtl-Meyer, p. 19RV re-entry vehicle, p. 3SSTO single stage to orbit, p. 3TSTO two stage to orbit, p. 3TPS thermal protection system, p. 11

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Figures reprinted with kind permission:

Fig. 2.5, by W.G. Vincenti and C.H. KrugerFig. 3.3, by S. WuthrichFigs. 3.7, 3.8, by ASMEFig. 6.7, by the American Institute of Aeronautics and Astronautics, Inc.Figs. 6.11, 9.7, 9.8, by ElsevierFig. 6.18, by H.W. Liepmann and A. RoshkoFig. 6.23, by WIT PressFig. 6.40, by J. Zierep

Second edition, in addition:

Figs. 9.3 and 9.4, by Springer Science+Business MediaFigs. 10.6, 10.7, and 10.8, by Springer Science+Business Media

The permissions to reprint the figures provided directly from theauthors—see the Acknowledgements at the beginning of the book—in allcases were given by the authors.

Name Index

Aihara, Y. 326

Alba, C.R. 329

Anderson Jr., J.D. 21, 108, 211, 258,278, 375

Anseaume, Y. 213

Arnal, D. 276, 324, 326, 330

Arrington, J.P. 74, 142

Aupoix, B. 276, 277, 331

Auweter-Kurz, M. 397

Aymer de la Chevalerie, D. 327

Aziz, R.A. 421

Bakker, P.G. 142

Ballmann, J. 35, 109, 142, 211, 212,277, 323, 325, 376, 377

Barton, N.G. 375

Baumann, R. 328

Becker, M. 377, 378

Beckett, C.W. 141

Beckwith, I.E. 278, 328

Beek, van, J.P.A.J. 331

Behr, R. 76, 278, 331

Benedict, S. 141

Benocci, C. 331

Bergemann, F. 142

Berger, K. 325

Berkowitz, A.M. 330

Berry, S.A. 328

Bertin, J.J. 21, 35, 75, 109, 142,211–213, 277, 323, 325, 376–378

Bertolotti, F.P. 108, 299, 314, 315,325, 326, 329–331

Best, J.T. 142

Biolsi, D. 108, 142

Biolsi, L. 108, 142

Bippes, H. 326

Bird, G.A. 375

Bird, R.B. 35, 43, 75, 107, 142, 213,277, 416, 421

Bissinger, N.C. 211Bleilebens, M. 397Blottner, F.G. 331Borrelli, S. 75Boudreau, A.H. 142Bradshaw, P. 331Brauckmann, G.J. 21Brazier, J.Ph. 276, 277Brenner, G. 376, 377Brown, L. 331Bruck, S. 141, 142, 376, 377Brun, R. 327Busemann, A. 213Busen, R. 328Bushnell, D.M. 20

Cabanne, H. 212Candler, G.V. 325, 326, 329Cardone, G. 327Cartigny, D. 375Casalis, G. 324Cattolica, R. 378Cayley, G. 1, 20Cebeci, T. 277, 324Celic, A. 328Chahine, M.T. 212Chambre, P.L. 108, 378Champion, M. 20Chanetz, B. 375, 376Chang, C.-L. 314, 315, 325, 329, 330Chang, T.S. 277Chapman, D.R. 277, 377Chen, K.K. 330Cheng, H.K. 368, 371, 378Chiang, S.T.L. 416Chikhaoui, A.A. 325, 327

434 Name Index

Chokani, M. 326Choudhari, M. 325Chpoun, A. 375Coakley, T.J. 323Cohen, C.B. 213Coleman, G.N. 331Coleman, G.T. 376Coratekin, T. 376Courant, R. 277Cousteix, J. 21, 75, 107, 142, 211, 276,

277, 324, 374, 397Crafton, J.W. 327Crouch, J.D. 327Cumpsty, N.A. 326Curry, D.M. 75Curtiss, C.F. 421Cuttica, S. 376

D’Ambrosio, D. 377Dallmann, U. 76, 314, 325, 329, 375Damkohler, G. 141Davis, R.T. 378De Luca, L. 327Deissler, R.G. 378Delery, J. 376Derry, S.M. 21, 323Desideri, J.-A. 76, 212, 325Detra, R.W. 42, 75Dietz, G. 297, 325Dohr, A. 108, 143Dolling, D.S. 331, 376Dolvin, D.J. 20Donelli, R. 326Drake, R.M. 43, 75, 107, 277Druguet, M.-C. 141Dujarric, Ch. 20Durand, A. 375Durbin, P.A. 331Dussauge, J.-P. 325, 331Dyke, M. van 277

Eberle, A. 211, 212, 375Eckert, E.R.G. 43, 75, 107, 277Edney, B. 342, 353, 376Eggers, Th. 75Egorov, I.V. 397Ehrenstein, U. 314, 329Erlebacher, G. 329Esser, B. 109

Estorf, M. 328Etkin, B. 21

Fano, L. 141Fasel, H.F. 325, 327, 328Fay, J.A. 42, 74, 142, 266, 278Federov, A. 328Fedorov, A.V. 325, 397Fernholz, H.H. 331Ferri, A. 212Fertig, M. 108, 142, 143, 397Fezer, A. 296, 325Finley, P.J. 331Fischer, J. 108, 143, 377Fish, R.W. 324Fonov, S.D. 327Fonteneau, A. 327Fornasier, L. 53, 75Foysi, H. 331Frederick, D. 277Frey, M. 397Friedrich, R. 325, 331Fruhauf, H.-H. 108, 141–143Fu, S. 331

Gagnepain, J.-J. 20Gamberoni, N. 317, 330Gebing, H. 375Georg, H.-U. 213Gerhold, T. 377Germain, P.D. 326Gersten, K. 21, 35, 74, 108, 212, 277,

324Gerz, T. 328Ghosh, S. 331Gibson, W.E. 109Glowinski, R. 76, 212, 213, 376, 378Gnoffo, P.A. 142Godard, J.P. 326Gobel, F. 74Gortler, H. 304, 326, 375Goodrich, W.D. 21, 319, 323Gordon, S. 143Grasso, F. 376Green, S.I. 375Greene, F.A. 142Griffith, B.F. 142Gronig, H. 109Groh, A. 108

Name Index 435

Gulhan, A. 292, 375, 386, 397Gupta, R.N. 75

Haase, W. 326, 331, 376Haberle, J. 397Hall, P. 314, 329Hanifi, A. 314, 329Hannemann, K. 142, 377Hannemann, V. 142Hansen, C.F. 85, 108Hartmann, G. 212Harvey, J.K. 212, 378Haupt, M. 75Hayes, J.R. 278, 327Hayes, W.D. 35, 212, 364, 377Head, M.R. 326Hein, S. 297, 314, 325, 328, 329Heiser, W.H. 211Heitmann, D. 327, 328Helms, V.T. 326Henckels, A. 375Hendricks, W.L. 378Henningson, D.S. 211, 324, 329Henze, A. 397Herberg, T. 375Herbert, Th. 314, 329Herdrich, G. 397Hernandez, P. 330Hidalgo, H. 42, 75Hilbert, D. 277Hildebrand, R.B. 75Hilgenstock, A. 76Hill, D.C. 330Hilsenrath, J. 141Hirsch, C. 416Hirschel, E.H. 20, 21, 34, 35, 74–76,

107–109, 141, 142, 211–213, 276–278,323, 324, 328–330, 374, 375, 377, 378,397, 411, 416

Hirschfelder, J.O. 421Hold, R.K. 53, 75, 76, 278, 375, 378Hoffmann, K.A. 416Hoge, J. 141Holden, M.S. 342, 352, 376, 377Holt, M. 108Hornung, H.G. 35, 142, 212, 326, 328,

377Horvath, T.J. 328Houdeville, R. 324

Huang, P.G. 331Hudson, M.L. 326Hung, P.G. 324Hussaini, M.Y. 329

Ingen, van, J.L. 317, 330

Jansen, A.R. 421Jentink, T. 325Johnson, H.A. 277Johnson, H.B. 314, 328, 329Jones, J.J. 74, 142

Kanne, S. 141Katzer, E. 397Kemp, J.H. 366, 377Kendall, J.M. 283, 324Keraus, R. 108, 143Kerschen, E.J. 328Keuk, van, J. 376Khan, M.M.S. 325Kimmel, R.L. 325, 327Kipp, H.W. 318, 326, 330Klebanoff, P.S. 278Kleijn, C.R. 76Kliche, D. 213Kloker, M. 296, 324, 325, 328, 330Klopfer, G.H. 377Knauss, H. 327Koc, A. 75Koelle, D.E. 35Koelle, H.H. 75Kolly, J. 377Konopka, P. 328Koppenwallner, G. 35, 108, 212, 378Kordulla, W. 21, 75, 107, 142, 211,

276, 277, 324, 328, 374, 397Kosmatopoulos, E. 330Krause, E. 212Krier, J.V. 326Krogmann, P. 311, 327, 328Kruger, C.H. 35, 107, 141, 416, 421Kuczera, H. 20, 35Kudriavtsev, V.V. 76Kuehl, J.J. 330Kufner, E. 292, 314, 325, 386Kunz, R. 21Kyriss, C.L. 330

436 Name Index

Laburthe, F. 314, 329Lafon, A. 276Lanfranco, M.J. 142Lang, M. 324Langtry, R.B. 324Laurence, D. 376Lees, L. 213, 278, 292, 324, 325, 377,

378Li, F. 325Li, Q. 327Liepmann, H.W. 211, 212, 326, 375Lightfoot, E.N. 35, 43, 75, 107, 142,

213, 277, 416, 421Lighthill, M.J. 46, 75, 141, 278Likki, S.R. 324Lin, C.C. 292, 324Lin, N. 329Liu, C. 327Longo, J.M. 40, 74, 376Lorenz, V. 213Lu, F.K. 20, 327Ludecke, H. 327Lugt, H.J. 211, 375

Mack, L.M. 283, 285, 292, 293, 296,324, 325, 330

Macrossan, M.N. 142Maeder, Th. 331Malik, M.R. 296, 314, 315, 324, 325,

327, 329, 330Malmuth, N. 328Mangler, W. 278Marini, M. 325, 376Marren, D.E. 20Marrone, P.V. 109Martellucci, A. 330Marvin, J.G. 323Marxen, O. 324Masek, R.V. 318, 330Masi, F. 141Maurer, F. 326, 328Maus, J.R. 142McBride, B.J. 143McCauley, W.D. 324McDonald, H. 324Meier, H.U. 331Meinke, M. 375Meiss, J.-H. 375Menart, J. 327

Mendenhall, M.R. 278Menne, S. 278Menter, F.R. 324, 328Mertens, J. 35Messerschmid, E.W. 141Mharchi, M. 213Miller, D.A. 375Moelwyn-Hughes, E.A. 108Monnoyer, F. 43, 75, 76, 213, 277,

278, 375Morkovin, M.V. 282, 283, 306, 311,

321, 324, 331Moss, J.N. 35, 212Mughal, M.S. 314, 329Mukund, R. 327Mundt, Ch. 74–76, 108, 143, 212, 213,

278, 375Murthy, T.K.S. 324Muylaert, J. 142, 376

Napolitano, L.G. 108, 142, 377Narasimha, R. 212, 306, 327Nelson, H.F. 20Netterfield, M. 277Neuenhahn, T. 394, 397Neufeld, P.D. 421Neumann, R.D. 278, 376Newton, I. 196Norstrud, H. 278Novelli, Ph. 75Nowak, R.J. 328Nutall, L. 141

Obrist, D. 325Oertel, H. 35Olivier, H. 109, 212, 213, 397Oskam, B. 108Oswatitsch, K. 92, 108, 205, 208, 211,

213, 375Owen, P.R. 301, 326Oye, I. 278

Pagella, A. 326Pai, S.I. 108Paredes, P. 326Park, C. 141, 416Paulson Jr., J.W. 21Peake, D.J. 276, 375Peng, S.H. 331Perez, E. 330

Name Index 437

Periaux, J. 20, 35, 76, 109, 142,211–213, 277, 323, 325, 329, 375–378

Perrier, P. 325, 377Perruchoud, G. 47, 75Pettersson Reif, B.A.P. 331Pfitzner, M. 75, 212, 278Pierce, A.J. 327Piquet, J. 331Pironneau, O. 20Platzer, M. 375Poll, D.I.A. 278, 300, 301, 326, 327Pope, S.B. 330Pot, T. 375Prandtl, L. 219, 277Pratt, D.T. 211Probstein, R.F. 35, 212, 364, 377Pulliam, T.H. 416

Radespiel, R. 40, 74, 142, 278, 327,328, 376

Rakich, J.V. 142, 277Randall, D.G. 301, 326Ranuzzi, G. 376Rasheed, A. 328Reed, H.L. 326, 328, 330Reisch, U. 213Reshotko, E. 213, 278, 293, 324, 325,

328Riddell, F.R. 42, 74, 142, 266, 278Riedelbauch, S. 63, 75, 76Rieger, H. 278Rist, U. 324, 326, 328, 330Rizzi, A. 212, 375Robben, F. 378Robinson, S.K. 331Rodi, W. 376Rohsenow, W.M. 76Rosenhead, L. 325Roshko, A. 211, 375Rotta, J.C. 331Rotta, N.R. 212Roy, Ch.J. 331Rubesin, M.W. 277, 377Rubin, S.G. 357, 377Rudman, S. 357, 377Rues, D. 377Rufolo, G.C. 75

Sacher, P.W. 20, 21Saile, D. 338, 375

Salinas, H. 315, 330Salvetti, M.-V. 212Saric, W.S. 325–328Sarkar, S. 331Sarma, G.S.R. 108, 141, 142Sawley, M.L. 47, 75Schaaf, S.A. 108, 378Schall, E. 141Schlager, H. 328Schlichting, H. 21, 35, 74, 108, 212,

277, 324Schmatz, M.A. 76, 211, 212, 278Schmid, P.J. 211, 324Schmidt, W. 35Schmisseur, J.D. 310, 327Schmitt, H. 213Schneider, S.P. 311, 327, 328Schneider, W. 74, 142, 375Schrauf, G. 314, 328, 329Schroder, W. 278, 375, 397Schubauer, G.B. 278, 283, 324Schubert, A. 376Schulein, E. 327Schulte, P. 328Schulte-Werning, B. 76Schumann, U. 328Schwamborn, D. 331Schwane, R. 376Schwarz, G. 328Scott, C.D. 142Seror, S. 141Sesterhenn, J. 325Shang, J. 327Shapiro, A.H. 107Shea, J.F. 323Shepherd, J.E. 377Sherman, F.S. 378Shidlovskiy, V.P. 378Shih, Y. 327Siddiqui, M.S. 416Simen, M. 314, 329Simeonides, G. 75, 141, 142, 236, 243,

277, 278, 325–327, 330, 376Simmonds, A.L. 75Skramstadt, H.K. 283, 324Smith, A.M.O. 317, 330Smith, R.W. 331Smits, A.J. 331Smolinsky, G. 377

438 Name Index

Soudakov, V.G. 397Spall, R.E. 324, 327Spina, E.F. 331Srinivasan, S. 108, 143Starkloff, M. 108Statnikov, V. 338, 375Staudacher, W. 21, 323, 375Steger, J.L. 416Stetson, K.F. 281, 296, 318, 323Stewart, D.A. 142Stewart, W.E. 35, 43, 75, 107, 142,

213, 277, 416, 421Stollery, J.L. 376, 378Stouflet, B. 20Streeter, V.L. 107, 108Stuckert, G.K. 329Sturtevant, B. 377Su, W.-H. 375Sucipto, T. 326Suzen, Y.B. 324Sweet, S. 377Szechenyi, E. 375

Talbot, L. 358, 377Tani, I. 326Tannehill, J.C. 108, 143, 277Taylor, B.N. 421Temann, R. 212Tescione, D. 75Theofilis, V. 325, 326Thomas, P. 20Thorwald, B. 211, 213Thyson, N.A. 330Tobak, M. 276, 375Touloukian, S. 141Tran, Ph. 326, 330Tsien, H.S. 213Tumin, A. 325Tumino, G. 277

Velazquez, A. 327Vermeulen, J.P. 327Vignau, F. 294

Vigneron, Y.C. 239, 277Vincenti, W.G. 35, 107, 141, 416, 421Vinh, H. 330Vinokur, K. 415, 416Viswanath, P.R. 327Viviand, H. 415, 416Volker, S. 324Volkert, H. 328Vollmers, H. 76Vos, J.B. 74

Wagner, S. 324, 326, 328–330Walpot, L.M.G. 142, 277Wang, K.C. 64, 76Wanie, K.M. 278Weber, C. 331Weiland, C. 20, 21, 34, 74, 109, 141,

211, 212, 278, 323, 331, 375, 397,411, 416

Weilmuenster, K.J. 21, 108, 142, 143Weingartner, S. 35Weise, A. 328White, E.B. 326White, F.M. 277, 327Whitham, G.B. 325Wieting, A.R. 352, 377Wilcox, D.C. 276, 323Willems, S. 292, 386, 397Williams, R.M. 35, 323Williams, S.D. 20, 75Wimbauer, J. 323Wood, R.M. 375Woolley, W. 141Wuthrich, S. 47, 75

Yee, H.C. 377

Zakkay, V. 212Zeitoun, D.E. 141, 325Zemsch, S. 327Zhang, H.-Q. 375Zierep, J. 211Zoby, E.V. 75

Subject Index

Ablative cooling 42, 308, 319Accommodation coefficient 101, 133,

358, 372f.Aerothermoelasticity 7, 12Air– composition 28ff., 96f., 111, 127– constituents 28, 139, 418– mean molecular weight 23, 418– thermo-chemical properties 139f.– transport properties 83, 85, 88–90,

139f.Air data 25Amplification rate 285ff., 292f.Atmospheric fluctuations 7, 311Attachment-line– boundary layer 51, 299– contamination 52, 283, 299–301,

310, 319– flow 255, 299– heating 52f., 67f., 70– instability 298Averaging– Favre 218, 321– Reynolds 218, 241, 321

Barometric height formula 25Binary scaling, see Similarity parameter

106Blunt Delta Wing 63ff., 254, 306,

335f.Boltzmann equation 173, 174Boost-glide vehicle 3

Boundary layer 7ff., 26f., 32, 39f., 44,49–51, 58, 67, 69, 73, 79, 93, 100,105, 137, 145, 149f., 173–176, 181,184–188, 193, 209, 215–219, 225–228,230f., 235f., 239f., 252–256, 259,269f., 275, 279–283, 286–289, 291,292, 295–299, 301–304, 309f., 321f.,333, 344, 360, 362, 373f., 380–384,386, 389f., 396

– 1/7-th-power law 241, 242,247–249, 257, 320, 410

– assumption 219, 228, 245– Blasius 32, 44, 232, 240f., 248f.,

257, 290, 381, 409f.– density gradient 10, 298, 321, 382– laminar 39, 69, 235, 240, 241, 244,

247, 248– mass concentration 104f., 216, 240– over-tripping 310– temperature gradient 10, 14, 82,

188, 223, 228, 321– temperature gradient in the gas at

the wall 13f., 37, 72, 263, 269, 292,381, 383, 386

– thermal 44, 47, 98–100, 125, 216,240, 360

– tripping 260, 281, 283, 309, 310– turbulent 39, 45, 234f., 241f.,

247–249, 283, 311– virtual origin 250f.– wall compatibility conditions

229ff., 381

440 Subject Index

Boundary layer instability– first mode 292ff., 386– second (Mack) mode 292ff., 387Boundary-layer equations/method

34, 54, 225, 229, 233– coupled Euler/second-order 179f.,

186f., 225–227– first order 184, 186, 187, 217, 226f.,

230, 245– second order 226Boundary-layer flow profile 217– cross-flow 217f., 253, 301f.– main-flow 217, 231f., 247f., 301f.– stream-wise 218– three-dimensional 218– two-dimensional 218Boundary-layer thickness 16, 263,

380– characteristic 44, 52f., 61, 67, 73,

185, 242, 279, 344, 381, 410– displacement 228, 240f., 245–249,

253, 288, 306, 308f., 319, 333, 353,356, 360, 362f., 366, 380

– flow (ordinary), 32, 49, 72f., 93, 100,104, 184, 196, 242

– mass concentration 104f.– momentum 247, 248, 252, 288, 318– thermal 44, 46, 99, 100, 126, 223,

354– turbulent scaling 44, 46, 47, 244,

252, 272, 381– viscous sub-layer 44, 240, 242–244,

252, 259, 272, 279, 285, 309, 380f.Bow shock 121, 147–150, 169, 173,

181, 185, 207, 302, 349f., 352, 353,373

– stand-off distance 138, 176ff., 350BURAN 3f., 318

Catalytic surface recombination 12,17, 48f., 111, 118, 126, 131f., 134f.,137f., 266, 349

– finite 48, 102, 105, 133f., 136, 349– fully 48, 102, 105f., 126f., 132–134,

136, 350– non-catalytic wall 48, 126, 133f.,

349Cayley’s design paradigm 1

Chapman-Rubesin– constant 363, 368– criterion 224f.Characteristics 168, 173, 205Cold-spot situation 52, 63, 67, 70, 73,

224, 246, 305Collision 31, 112, 118f., 132– diameter 83, 418– excitation 119– integral 83, 85, 417, 419– number 119, 127– partner 122Cone flow 147, 163, 172f., 296Continuum flow 32ff.Coordinate system– external inviscid streamline-oriented

218Coupled Euler/boundary-layer method

48, 58, 95Crocco’s theorem 183, 207Cross-flow– direction 230– instability 299, 301f., 319– pressure gradient 253– shock 65, 71, 338, 341

Damkohler number, see Similarityparameter 120

De-excitation 119, 127Deflection angle 163, 169ff.Degree of freedom 28, 115f., 118–120,

123f., 131Diffusion 38, 79, 82, 90, 99, 100, 102,

103, 105, 119, 206, 414– concentration-gradient driven 82,

103– multi-component 88– pressure-gradient driven 82, 103,

138– temperature-gradient driven 82,

103Direct numerical simulation 296, 302,

312, 316Dissociation 28, 83, 85, 88, 102, 115,

118f., 122f., 125, 127, 130f., 179f.,266, 414, 418

Disturbance environment 281, 283,286, 294, 296, 311, 312

Subject Index 441

Drag 3, 11, 14f., 25, 34, 149, 181,197, 204, 206, 260, 280, 292, 301, 319

– induced 95, 148, 336, 337– pressure/form 14, 148, 240, 256,

274, 291, 336, 383, 390– skin-friction 14, 40, 148, 240, 256,

274, 382– total 248, 256, 291– viscous 12, 56, 95, 240, 275, 280,

308, 336– wave 50, 148f., 209, 210

Eckert number, see Similarityparameter 99

Effective ratio of specific heats 117,178f., 209

Emissivity coefficient 7, 44, 48, 54f.,58, 59, 63

– fictitious 54–56Entropy 148f., 153, 160–163, 165,

182–184, 203, 207, 282Entropy layer 182ff.– instability 297f.– slip-flow profile like 8, 184– swallowing 184–187, 225, 297– wake-like 8, 297Equilibrium flow 120, 123, 130, 139,

180, 350Equipartition principle 116Equivalent inviscid source distribution

245, 246Eucken formula 85f.Euler equations/method 33, 43, 92,

95, 152, 174, 200, 203f., 206, 226,230, 236, 246

Excitation– vibrational, half 117– electronic 96, 114– rotational 116, 119, 418– translational 116, 119– vibrational 28, 85, 86, 112, 115,

117, 119, 121f., 124f., 418Expansion fan 173, 188, 342Expansion limit 153f., 191

Fick’s mass diffusion law 82

Flight

– quasi-steady 17

– steady 17

– trajectory 17

Flight vehicle

– AOTV-type 3, 5, 373

– ARV-type 3, 5, 12, 280, 319

– CAV-type 3, 5, 12, 45, 50f., 111,138, 148–150, 205, 210, 225, 251, 254,256, 259, 260, 263f., 269, 274f., 280,282f., 292, 308, 309, 319, 346, 356,366f.

– compressibility-effects dominated4, 56

– RV-type 3, 5, 12, 17, 27, 45, 48,51, 57, 111, 138, 148f., 205, 225, 251,254, 256, 259, 263f., 279f., 283, 300,301, 308, 309, 319, 366f., 373

– viscosity-effects dominated 4, 280

Flow attachment 217, 236, 254f., 341,345, 390f.

– line 50–52, 58f., 63–65, 67–71, 73,200, 202, 240, 245, 246, 253–255, 260,261, 262f., 266–268, 295, 298–300,305, 306, 335f., 340f.

– shock 342, 347

Flow regimes 30ff.

Flow separation 16f., 34, 67, 149,173, 215, 217, 228f., 236, 245, 281,290, 322, 333–336, 338, 341f., 344f.,382, 388, 390f.

– bubble 291, 336

– flow-off 240, 334, 336f.

– global 336f., 341

– line 51, 52, 63f., 67, 69, 70, 73, 245,246, 253f., 260, 295, 335f., 340

– local 279, 336f., 341

– open 64

– shock 342, 347, 392

– shock induced 322

– squeeze-off 334–336, 338

– unsteady 338

442 Subject Index

Flow three-dimensionality 8Flux-vector formulation 95, 413Formation energy 96Fourier’s heat conduction law 82Free-stream surface 10, 295Frozen flow 120f., 180, 303

Galilean invariance 80Gas– calorically perfect 28f., 111f.– perfect 14, 39, 98, 124, 152, 222,

234, 236– thermally perfect 28f., 78, 96,

111f., 115f., 264Gas constant– specific 29, 116, 418– universal 29, 116, 417Gasdynamic equation 206GETHRA module 54, 55Gortler instability 302f., 306Gravitational acceleration 417

HALIS configuration 135f., 348f.Hansen equation 85Heat flux 37ff., 44, 49f., 53, 82, 305,

310, 354, 414– cold wall 47, 51– convective 38– due to mass diffusion 8, 97ff.– due to slip flow 97– in the gas at the wall 14, 16, 38–42,

132, 134f., 185, 187, 203, 216, 239,253, 263f., 266f., 269, 283, 285, 291,309, 341, 342, 350, 354, 355, 380, 414

– shock-layer radiation 42– stagnation point 42f., 266– surface radiation 16, 40, 42, 44, 49,

67f., 71– tangential to the surface 13, 40f.,

44, 225, 322– wall 13, 16f., 38, 40f., 49f., 54, 102,

263HERMES 3, 26, 43, 58–60, 318, 340f.,

350f.Heterosphere 23HIFiRE 3Homosphere 23Hot experimental technique 16, 379

Hot-spot situation 52, 63, 67, 70–73,224, 246, 285, 305, 310, 341

Hyperboloid-flare configuration 350f.Hypersonic shadow effect 145, 197,

203, 208, 338Hypersonic similarity parameter, see

Similarity parameter 209

Interaction– Edney type 150, 342ff., 351ff., 388,

390– hypersonic viscous 16f., 215, 228,

296, 298, 333, 356, 358ff., 365– pressure 356, 364– shock/boundary-layer 16, 173, 193,

228, 241, 302f., 321f., 333f., 337,339f., 383, 388, 390f., 394

– shock/shock 333, 341, 350f., 351,355

– strong 19, 149, 150, 215, 228, 281,320f., 333f., 337, 358f., 366f.

– strong interaction limit 367f.– vorticity 356– weak 245, 333, 364, 367– weak interaction limit 367f.Intermittency region 285Inviscid flow 67, 79, 92, 145, 152,

157, 180, 183f., 185–187, 209, 216f.,226, 236, 240f., 245, 247, 300, 333,337, 342, 356f., 373

Knudsen number, see Similarityparameter 31

Laminar-turbulent transition 3, 8,12, 27, 34, 48, 52, 138, 185, 215,223, 224, 243, 245, 250, 253, 272f.,279–282, 284, 297, 304, 305, 308, 309,312, 319, 338, 341, 346, 386f.

Large eddy simulation 316Leading edge 10, 14, 40, 47, 50–52,

64, 67, 70, 200, 235, 240, 246, 254,255, 295, 297, 298, 299f., 306, 312,334, 336, 338, 349, 356, 394

Lewis number, see Similarity parameter104

Lift 25, 198, 203, 256, 295, 334, 337Lift to drag ratio 40, 338Lighthill gas 117

Subject Index 443

Loads– acoustic 11– dynamic 11, 334– mechanical 11, 149, 150, 216, 283– pressure 351, 356– static 11– thermal/heat 5, 12, 15–17, 27, 37,

39, 43, 51, 56, 122, 131, 138, 148f.,150, 206, 216, 263, 275, 279f., 283,297, 301, 303f., 319, 334, 336, 346,350, 351

Locality principle 228, 337f.Low-density effects 8, 14, 356f., 367,

368, 374

Mach angle 147, 169f., 192, 209Mach number, see Similarity parameter

13Mach number independence 92, 176,

181, 198– principle 205ff.Mach reflection 341Mangler effect 184f., 253, 297– inverse 253Mass fraction 30, 86, 103, 120, 126,

129f., 180f.Mean free path 31, 34, 123, 173f.,

372f.Merged layer 356, 373Mesosphere 23Micro-aerothermodynamics 12Mixing formula 29, 88Mole fraction 29, 88Molecular weight 30, 83, 116, 418Monte Carlo simulation 134f., 369Multi-temperature model 121, 414

Navier-Stokes/RANS equa-tions/method 33, 54, 55,63f., 67, 69, 90, 93, 102, 124, 127,134–136, 174, 186, 219f., 225, 236f.,241, 245, 269, 271, 273, 286, 288,299, 346, 358, 368f., 371f., 373, 374,383, 388

– hybrid RANS/LES 323, 338Newton– flow 34, 145, 197ff.– limit 32, 118

– model 196, 199, 399– modified model 199ff.Newton’s friction law 82Newtonian fluid 82, 90Non-equilibrium flow 97, 100f., 105,

120f., 124, 130, 139f., 180, 321, 413Non-parallel effects 233, 302, 314,

317Nozzle flow 54, 127, 138– frozen 127ff., 138

Orr-Sommerfeld equation 233

Peclet number, see Similarity parameter98

Parabolization 239, 314Parabolized stability equations/method

286, 314, 317Partial density 29Partial pressure 29Perturbation coupling 186, 246, 333Poincare surface 65, 66, 340Point of inflection 231f., 289–292,

382–384– criterion 289Prandtl number, see Similarity

parameter 86Prandtl relation 158f., 164Prandtl-Meyer expansion 145,

188–191, 203Pressure coefficient 155, 198, 200,

205– total 160, 166– vacuum 155Pressure gradient 14, 227, 236, 282,

291, 381– adverse 232, 248, 253, 286, 290,

292, 382, 384, 391– favorable 253, 291, 381– normal 226, 356

Radiation cooling 5, 7, 10, 12, 14, 19,37, 38, 40, 41ff., 49, 51, 53f., 72, 73,79, 101, 149, 184, 215, 219, 240, 253,269, 271, 273f., 297, 305, 315, 321f.,338

– non-convex effects 40, 41ff., 53–56,343

444 Subject Index

Radiation heating 42Radiation-adiabatic temperature 10,

40f., 47–49, 50f., 55f., 58f., 63f., 67,69–71, 73, 94, 135f., 145, 193, 263,267f., 270–272, 283, 344, 349f., 355,380

Ramp flow 169, 172, 304, 305, 342f.,383, 388, 390, 394

Rankine-Hugoniot conditions 95,156, 158, 164, 360f., 373

RANS equations/method 305Rarefaction parameter, see Similarity

parameter 366Rate– effect 111, 119, 121–124– equation 121f.– process 121f., 127Real-gas effects– high temperature 7, 10, 27f., 39,

43, 57, 59, 92, 111, 114f., 138, 169,179, 181, 182, 202, 208f., 219, 234,236, 266, 271, 274, 303, 313, 322, 338,346, 350f., 355, 373

– van der Waals 30, 111–113, 131Receptivity 282, 307ff., 311, 312, 314,

316– model 312, 314Recombination 28, 102, 115, 118f.,

122, 131, 414Recovery– factor 39, 234, 270– temperature 13, 38–41, 43f., 46f.,

49, 57f., 59, 69, 223, 234, 244, 261,269–271, 293, 344

Reference temperature/enthalpy 44,215, 233f., 243

– extension 247, 257, 261, 363Reflection coefficient 94Relaminarization 306f.Reynolds analogy 264Reynolds number, see Similarity

parameter 13RHPM-flyer 19, 198f.Roughness, see Surface property 308

SANGER 3, 5f., 26, 33, 269–275, 280,367

Schmidt number, see Similarityparameter 104

Shape factor 245, 247, 249

Shock wave 34, 145f., 150, 174, 182,185, 188, 216, 342, 360f.

– bow shock 95, 146f.

– capturing 95, 174– computational treatment 34– curved 183– embedded 146, 148, 169, 181, 336

– fitting 95, 168, 176– impingement 302, 352, 354– intersection 173– normal 123f., 152, 156, 157,

159–162, 178, 183– normal direction 163, 174– oblique 138, 145, 150, 152, 163ff.,

166, 167, 174, 178, 356, 359, 362

– sonic limit 166, 170f.– strong 146, 157, 166, 170f.– thickness 32, 124, 157, 173, 174,

176, 373

– weak 146, 166, 170f.– weak limit 170f., 172Shock-formation region 356f., 358f.,

368–373

Shock-on-lip situation 150Shock-slip condition 373Similarity parameter– Tw/T∞, 13, 106

– Binary scaling 106– Damkohler number 120f.– Eckert number 99f.– Hypersonic similarity parameter

209, 362ff.– Knudsen number 31ff., 94f., 101,

358, 373

– Lewis number 99, 104, 266– Mach number 10, 13, 26, 91f.,

146ff.– Peclet number 98f.

– Prandtl number 39, 86f., 98ff., 224– – turbulent 321– Rarefaction parameter 366, 368– Reynolds number 13, 92f.

– – unit 26– – unit Reynolds number effect 311– Schmidt number 104f.– – turbulent 322– Strouhal number 80f.

– Viscous interaction parameter364ff.

Subject Index 445

Skin friction 11, 12, 16, 185, 233, 236,273, 275, 307, 320, 322, 335f., 366

– laminar 73, 257– turbulent 73, 248, 258, 274, 275Slip flow 32, 94, 219, 230–232, 236,

356, 358, 372–374Slip line/surface 91, 151, 173f., 342,

350, 353f.Space Shuttle Orbiter 3ff., 26, 33, 40,

43, 48f., 52, 58, 61, 62, 134f., 138,209, 280, 300, 310, 318f., 338, 351

Space-marching scheme 174, 239, 314Speed– maximum 153, 158– of sound critical 154Speed of sound 39, 91, 121, 124, 146,

153, 158f., 162, 206, 207, 238, 291Spillage flow 396ff.Stagnation point 10, 42f., 50, 58, 64,

113, 121, 125, 131, 149, 156, 167,176, 179, 184, 186, 199f., 224, 240,253f., 260, 263–266, 271, 299f., 303,335, 350, 354, 380

Stanton number 15, 38, 47Stanton-number concept 13Stefan-Boltzmann constant 44, 417Stratosphere 23, 311Strouhal number, see Similarity

parameter 80Surface– blowing 91, 97, 228, 245, 381– coating 12, 17, 132, 135, 256– curvature effect 233, 302, 304, 314,

317– erosion 17, 256– suction 91, 97, 228, 245, 291, 381Surface pressure 11, 91, 209, 334,

347f., 356Surface property 7, 12, 281f., 308f.,

315– necessary 12, 16, 309f.– permissible 12, 16, 253, 309f.Sutherland equation 83

Temperature– boundary-layer edge 10, 188, 233,

321– free-stream 13, 106– radiation-adiabatic, see Radiation-

adiabatic temperature 38

– recovery, see Recovery temperature39

– total 39, 46, 49, 69, 125, 131, 158,233

– wall 10, 12–14, 17f., 37f., 40f., 43,47, 49, 53, 72, 73, 101f., 106, 125,131–134, 149, 215, 223f., 233f., 239,244, 247, 252, 260, 263f., 269, 273,275, 283, 292–294, 303, 313, 338,344f., 367, 379f., 382–388, 390, 392,394

Temperature jump 14, 32, 38, 40,101, 356, 360, 369, 372–374

Thermal conductivity 44f., 82f., 84,86–89, 226, 269

Thermal diffusivity 98

Thermal loads, see Loads 16

Thermal protection system 11f., 40,43, 48f., 54, 256, 283, 300, 308, 310,318f.

Thermal reversal 40, 49f., 73

Thermal state of the surface 11f.,13–16, 37, 44, 72, 101, 106, 131, 215,223, 236, 239, 240, 248, 254, 263,265, 268, 279, 281f., 292, 294, 307ff.,319f., 334, 366, 390

Thermal surface effects 11ff., 15, 37,106, 138, 279, 334, 379ff.

– thermo-chemical 11, 379

– viscous 11, 379ff.

Thermosphere 23

Tollmien-Schlichting

– instability 286, 301, 302

– wave 284, 297, 302

Total-pressure loss 150, 156, 173, 336

Trailing edge 151, 334, 336, 390

Transition criterion 272, 285f., 315ff.

Transpiration cooling 42

Troposphere 23, 311

Turbulence 3, 16, 27, 34, 215, 273,275, 279–284, 300, 313, 320

– fluctuation 290

– free-stream 286, 307, 312

– model 215, 285, 321f., 334, 346–348

– Reynolds-stress model 323

Turbulent

– heat conduction 321

– mass diffusion 321

446 Subject Index

Units– SI 418ff.– U.S. customary (English) 418ff.

Vibration-dissociation coupling 122,127

Viscosity 44, 82–86, 88, 89, 187, 192,226, 231, 233, 235, 250, 256ff., 261,264, 267, 363f., 400

– bulk 90, 119, 221, 236– kinematic 98, 227, 300Viscous interaction parameter, see

Similarity parameter 364Viscous shock layer 158, 368, 373– equations/method 34, 225, 373Viscous stress tensor 221, 414

Viscous sub-layer 44, 240, 242, 243Volume fraction 30Vortex 216, 301f., 334–336, 341– breakdown 338– Gortler 304f.– scrubbing 52– shedding 81, 333f.– sheet 173, 216, 334f.– wake 334Vorticity content 218

X-15 15, 342X-20/Dyna-Soar 3X-38 3, 33, 55, 56, 136f., 305, 306, 322X-43A 3X-51A 3