Daniel Webster College, Nashua, NH, March 24, 2012
July 10—15, 2010 ~ Asilomar Conference Grounds ~ Pacific Grove, California, USA
RAREFACTION EFFECTS IN
HYPERSONIC AERODYNAMICS
Vladimir V. Riabov, Ph.D.
Professor of Computer Science & Math
Rivier College
Nashua, New Hampshire
vriabov@rivier.edu
http://www.rivier.edu/faculty/vriabov/
Topics for Discussing
• Experimental and numerical simulation of
hypersonic rarefied-gas flows in air, nitrogen,
carbon dioxide, argon, and helium;
• Study of aerodynamics of simple-shape bodies
(plate, wedge, cone, disk, sphere, side-by-side
plates and cylinders, torus, and rotating cylinder);
• Analysis of the role of various similarity
parameters in low-density aerothermodynamics;
• Evaluation of various rarefaction and kinetic
effects on drag, lift, pitching moment, and heat
transfer.
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Techniques & Tools
• Experiments in hypersonic vacuum chamber;
• Direct Simulation Monte-Carlo technique:
– DS2G (version 3.2) code of Dr. Graeme A. Bird
– Knudsen numbers Kn∞,L from 0.01 to 10
– (Reynolds numbers Re0,L from 200 to 0.2);
• Solutions of the Navier-Stokes 2-D equations;
• Solutions of the Thin-Viscous-Shock-Layer equations;
• Similarity principles applied to hypersonic rarefied-gas
flows.
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Earth and Mars Atmospheric Parameters
Pressures and temperatures in Mars and Earth atmospheres.
From D. Paterna et al. (2002). Experimental and Numerical Investigation of Martian Atmosphere Entry,
Journal of Spacecraft and Rockets, Vol. 39, No. 2, March–April 2002, pp. 227-236
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Entry velocity envelopes for Earth & Mars missions
Earth Entry
Mars Entry
Entry velocity envelopes for Earth & Mars missions with return to Earth.
From “Capsule Aerothermodynamics,” AGARD Report No. 808, May 1997
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Typical trajectories of hypersonic spacecraft
From: J. J. Bertin and R. M. Cummings, “Critical Hypersonic Aerothermodynamic
Phenomena”, Annual Review of Fluid Mechanics, 2006, Vol. 38, pp.129-157
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European Space Agency Projects
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Flow regimes and thermochemical phenomena
Flow regimes and thermochemical phenomena in the stagnation
region of a 30.5 cm radius sphere flying in air.
From R. N. Gupta et al. NASA-RP-1232, 1990.
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Estimating Transport Coefficients
From Journal of Chemical Physics, 2004, Vol. 198, p. 424;
Riabov’s data from Journal of Thermophysics & Heat Transfer, 1998, Vol. 10, N. 2
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Similarity parameters
• Knudsen number, Kn,L = λ /L
• or equivalent Reynolds number, Re0,L = ρU∞L/μ(T0) ~ 1/Kn,L
• Interaction parameter χ for pressure approximation:
• Viscous-interaction parameter V for skin-friction approximation:
•
•
•
•
•
•
Temperature factor, tw = Tw/T0
Specific heat ratio, γ = cp/cv
Viscosity parameter, n: μ Tn
Upstream Mach number, M∞
Hypersonic similarity parameter, K∞ = M∞ × sinθ
Spin rate, W = ΩD/2U∞
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Choosing a Mathematical Model
The validity of the conventional mathematical models as a function of the local
Knudsen number. From J. N. Moss and G. A. Bird, AIAA Paper, No. 1984-0223.
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Estimating the local Knudsen number
VSL
Kn, local
DSMC
Density distribution along the stagnation streamline of the reentering Shuttle Orbiter
at 92.35 km altitude (Kn∞,R = 0.028). From G. Bird, AIAA Paper No. 1985-994.
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Modeling of Hypersonic Flight Regimes in
Wind Tunnels
The catalogs “Wind Tunnels in the Western and Eastern
Hemispheres” (U.S. Congress, 2008) profiles 65 hypersonic wind
tunnels used for aeronautical testing. The countries represented in
the catalogs include those in America (Brazil [1] and USA [12]), Asia
(Australia [7], China [5], and Japan [4]), Europe (Belgium [2], France
[5], Germany [2], Italy [1], the Netherlands [1], and Russia [21]), and
the Middle East and Central and South Asia (India [3] and Israel [1]).
Aerial view of the Thermal Protection
Laboratory at NASA Ames, California
DWC, March 24, 2012
Nozzle and test chamber of the H2K
hypersonic wind tunnel, Germany
Vladimir V. Riabov: Hypersonic
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T-117 hypersonic wind tunnel, TsAGI,
Moscow region, Russia
13
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Research on Hypersonics in TsAGI (Moscow region, Russia)
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Studies of “Buran” Aerothermodynamics (TsAGI, Russia)
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International Cooperation: Space Projects
British HOTOL and Mriya-225 Launcher tested in T-128 TsAGI Wind Tunnel, Moscow region, Russia
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Ground based testing in Germany
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Ground based testing in Germany
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HERMES Project, Germany &
France, 1987-1992
HERMES-Columbus Project,
Germany & Italy, 1987-1992
Testing ranges of some facilities (after Walpot L, Tech. Rept. Memo. 815, Univ Delft, NL,
1997). Bold lines indicate the individual range of each facility. Note that the times given are
maximal run times and not necessarily testing times for constant conditions in all cases.
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The International Space Station Mission
The ISS effort involves more than 100,000 people in space agencies and at 500
contractor facilities in 37 U.S. states and in 16 countries.
http://www.boeing.com/defense-space/space/spacestation/gallery/
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The International Space Station Mission
World ISS Team: USA, Canada, ESA (Belgium, Denmark, France, Germany,
Italy, the Netherlands, Norway, Spain, Sweden, Switzerland, the United Kingdom),
Japan, Russia, Italy, and Brazil
http://science.nationalgeographic.com/science/space/space-exploration/
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Space Shuttle Orbiter
approximate heat-transfer model
A) Representative flow models
B) Heat flux on fuselage lower centerline
From: J. J. Bertin and R. M. Cummings, Critical Hypersonic Aerothermodynamic
Phenomena, Annual Review of Fluid Mechanics, 2006, Vol. 38, pp.129-157
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Applications of Underexpanded Jets in
Hypersonic Aerothermodynamic Research
• A method of underexpanded hypersonic viscous jets has been
developed to acquire experimental aerodynamic data for simpleshape bodies (plates, wedges, cones, spheres and cylinders) in the
transitional regime between free-molecular and continuum regimes.
• The kinetic, viscous, and rotational nonequilibrium quantum
processes in the jets of He, Ar, N2, and CO2 under various
experimental conditions have been analyzed by asymptotic methods
and numerical techniques.
• Fundamental laws for the characteristics and similarity parameters
are revealed.
• In the case of hypersonic stabilization, the Reynolds number Re0 (or
Knudsen number Kn) and temperature factor are the main similarity
parameters.
• The acquired data could be used for research and prediction of
aerodynamic characteristics of hypersonic vehicles during their flights
under atmospheric conditions of Earth, Mars, and other planets.
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Underexpanded Hypersonic Viscous Jet
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Inviscid Gas Jets
• Ashkenas and Sherman [1964], Muntz [1970], and
Gusev and Klimova [1968] analyzed the structure of
inviscid gas jets in detail.
• The flow inside the jet bounded by shock waves
becomes significantly overexpanded relative to the
outside pressure pa.
• If the pressure pj » pa , the overexpansion value is
determined by the location of the front shock wave
("Mach disk") on the jet axis rd [Muntz, 1970]:
rd/rj = 1.34 (ps/pa)½
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(1)
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The asymptotic solution [Riabov, 1995] of the
Euler equations in a hypersonic jet region:
u'=u0+u1×z2(γ-1) +...; z=r*'(φ)/r'
(2)
u0=[(γ+1)/(γ-1)]½
(7)
v'=v1×z2(γ-1)×d/dφ[lnr*'(φ)] +...
(3)
u1=-θ1/(γ2-1)½
(8)
ρ'=1/u0×z2 +...
(4)
v1=-2θ1/{u0[1-2(γ-1)]}
(9)
×z2(γ-1) +...
(5)
θ1=1/(u0)(γ-1)
(10)
p'=θ1/u0×z2γ +...
(6)
r*'(φ)= r*(φ)/rj
(11)
T'=θ1
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Mach number M along the axis of axisymmetric inviscid
jets of argon (or helium), nitrogen, and carbonic acid:
M=(u0)0.5(γ+1)×[ r'/ r*'(0)](γ-1)
100
Mach number, M
gamma=5/3
gamma=1.4
gamma=9/7
10
1
1
10
100
Distance along the jet axis, r/r j
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The asymptotic solution of the Navier-Stokes equations in
a hypersonic viscous gas jet region
[from V. V. Riabov, Journal of Aircraft, 1995, Vol. 32, No. 3]:
W=(u' - u0)/ξλ, λ=2ω(γ-1)
(13)
W=-Θ/[u0(γ-1)] - u0Θn/(r02X)
(18)
V=v'/ξλ
(14)
Θ={γ(γ+1)(1-n)ω/(r02X)
+θ11-n(r0X)(1-ω)/ω}1/(1-n)
(19)
Θ=T'/ξλ
(15)
r02(∂V/∂X-V/X)-(γu0X)-1∂(r02Θ)/∂φ
+nu0Θn-1(2X2)-1∂Θ/∂φ = 0
(20)
X=r*'(0)/(r'ξω)
(16)
r0 =r*'(φ)/[r*'(0)]
(21)
ξ=4/[3Rej r*'(0)], ξ → 0
(17)
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r' = O(Rejω), ω = 1/[2γ-1-2(γ-1)n]
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Mach number M along the axis of axisymmetric viscous
jets of argon, helium, nitrogen, and carbonic acid.
100
Mach number, M
gamma=5/3
Ar,Rej=750
He,Rej=250
gamma=1.4
N2,Rej=750
gamma=9/7
CO2,Rej=750
10
1
1
10
100
Distance along the jet axis, r/r j
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Translational Relaxation in a Spherical Expanding Gas Flow
Parallel (TTX) and transverse (TTY) temperature distributions in the
spherical expansion of argon into vacuum at Knudsen numbers Kn* =
0.015 and 0.0015. [From V. V. Riabov, RGD-23 Proceedings, 2002]
Temperature ratios, T/T
*
1
0.1
TTX,Kn=0.015
0.01
TTY,Kn=0.015
TTX,Kn=0.0015
TTY,Kn=0.0015
ideal gas
0.001
1
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Distance along the radius, r/r *
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Rotational Relaxation in a Freely Expanding Gas
Relaxation time τR of molecular nitrogen vs. kinetic temperature Tt: solid line Parker's model [1959]; open symbols - quantum rotational levels j* = 4, 5,
and 6 [Lebed & Riabov, 1979]. Experimental data from Brau, Lordi [1970]
Parameter p*tau, kg/(m*s)
1.E-03
1.E-04
classical
1.E-05
quantum, j*=4
quantum, j*=5
quantum, j*=6
experiment
1.E-06
10
100
1000
Tem perature T, K
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Rotational temperature TR along the nitrogen-jet axis
100
Rotational temperature T
R,
K
Experimental data from Marrone (1967) and Rebrov (1976):
K* = ρ*u*r*/(pτR)* = 2730, psrj = 240 torr·mm and Ts = 290 K (nitrogen)
exper. [Rebrov]
quantum, j*=6
quantum, j*=5
quantum, j*=4
exper. [M arrone]
Parker's model
10
1
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Distance along the jet axis, r/r j
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Rotational Relaxation in Viscous Gas Flows
Rotational TR and translational Tt temperatures in spherical flows at
different pressure ratios P = p*/pa under the conditions: Re* = 161.83;
K* = 28.4; Pr = n = 0.75; Tsa = 1.2T*.
TR/T0 and Tt/T0
1.2
TR, P=17.6
1
Tt, P=17.6
0.8
TR, P=78.9
Tt, P=78.9
0.6
TR, P=175.7
0.4
Tt, P=175.7
TR, inviscid
0.2
Tt, inviscid
0
0
5
10
15
20
25
Distance along the radius, r/r*
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Similarity Criteria for Aerodynamic
Experiments in Underexpanded Jets
The interaction parameter χ for pressure approximation:
χ=M∞2/[0.5(γ-1)Re0)]0.5
The viscous-interaction parameter V for skin-friction approximation:
V=1/[0.5(γ-1)Re0)]0.5
The value of the Reynolds number, Re0, can be easily changed by
relocation of a model along the jet axis at different distances (x) from a
nozzle exit (Re0 ~ x--2).
Other criteria: Mach number M∞, temperature factor tw; specific heat ratio γ;
and parameter n in the viscosity coefficient approximation µ ~ Tn.
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The Results of Testing: Influence of Mach Number M∞
Normalized drag coefficient vs. hypersonic parameter K = M∞sinα
for the blunt plate (δ = 0.06) in He (γ = 5/3) at Re0 = 2.46.
fx=(Cx-Cxo)/sin 3(a)
1000
100
10
M =7.5,exper.
M =7.5,DSM C
M =9,exper.
M =9,DSM C
M =10.7,exper.
M =10.7,DSM C
1
0.1
1
10
Hypersonic param eter, K
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The Results of Testing: Influence of Mach Number M∞
Normalized lift coefficient vs. hypersonic parameter K = M∞sinα
for the blunt plate (δ = 0.06) in He (γ = 5/3) at Re0 = 2.46.
fy=Cy/sin 2(a)
100
10
M =7.5,exper.
M =7.5,DSM C
M =9,exper.
M =9,DSM C
M =10.7,exper.
M =10.7,DSM C
1
0.1
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1
Hypersonic param eter, K
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37
The Results of Testing: Influence of Mach Number M∞
Drag coefficient cx of the wedge (2θ = 40 deg) in helium at Re0 = 4
and various Mach numbers M∞
Drag coefficient, Cx
4
M =9.9,exper.
M >9,free-mol.
M =11.8,DSM C
3.5
M =11.8,exper.
M =9.9,DSM C
3
2.5
2
1.5
0
10
20
30
40
Angle of attack, deg
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The Results of Testing: Influence of Mach Number M∞
Lift coefficient cy of the wedge (2θ = 40 deg) in helium at Re0 = 4
and various Mach numbers M∞.
1.2
Lift coefficient, Cy
1
0.8
0.6
0.4
0.2
M =9.9,exper.
M =11.8,exper.
M >9,free-mol.
M =9.9,DSM C
M =11.8,DSM C
0
0
10
20
30
40
Angle of attack, deg
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The influence of the specific heat ratio γ
Drag coefficient cx of the wedge (2θ = 40 deg) for
various gases vs. the Reynolds number Re0.
Drag coefficient, Cx
2.5
exper., Ar
2.25
exper., N2
2
exper., CO2
DSM C, Ar
1.75
DSM C, CO2
1.5
1.25
1
1
10
100
Reynolds num ber, Reo
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The influence of the specific heat ratio γ
Lift coefficient cy of the wedge (2θ = 40 deg) at Re0 = 3
and various gases.
Lift coefficient, Cy
1.2
1
0.8
0.6
N2,exper.
0.4
Ar,exper.
N2,DSM C
0.2
Ar,DSM C
0
0
10
20
30
40
Angle of attack
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The influence of the specific heat ratio γ
Lift-drag ratio of the blunt plate (δ = 0.1) at α = 20 deg
in Ar and N2 vs. Reynolds number Re0.
1
Lift-drag ratio, L/D
N2,exper.
0.8
N2,DSM C
0.6
N2,free-mol.
Ar,DSM C
Ar,free-mol.
0.4
0.2
0
0.1
1
10
100
Reynolds num ber, Reo
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The influence of the specific heat ratio γ:
Drag coefficient, Cx
Drag coefficient cx of the disk at α = 90 deg for argon
and nitrogen vs. the Reynolds number Re0
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
Ar,exper.
N2,exper.
Ar,DSM C
N2,DSM C
Ar,free-mol.
N2,free-mol.
Ar,contin.
N2,contin.
0.1
1
10
100
Reynolds number, Reo
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The influence of the viscosity parameter n: µ ~ Tn
Drag coefficient cx of the plate at α = 90 deg in helium
and argon vs. the Reynolds number Re0.
Drag coefficient, Cx
3.5
2.5
exper., He
exper., Ar
DSM C, Ar
1.5
0.1
1
10
Reynolds num ber, Reo
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The influence of the temperature factor tw
Lift-drag ratio for the blunt plate (δ = 0.1) vs. Reynolds number
Re0 in nitrogen at α = 20 deg and various tw
Lift-drag ratio, L/D
1
N2,exper.,tw=1
N2,DSM C,tw=1
N2,DSM C,tw=0.34
0.8
N2,FM ,tw=1
N2,FM ,tw=0.34
0.6
N2,exper.,tw=0.34
0.4
0.2
0
0.1
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1
10
Reynolds num ber, Reo
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Direct Simulation Monte-Carlo (DSMC) method
• The DSMC method [G. Bird, 1994] and DS2G code (version 3.2) [G.
Bird, 1999] are used in this study;
• Variable Hard Sphere (VHS) molecular collision model in air, nitrogen,
carbon dioxide, helium, and argon;
• Gas-surface interactions are assumed to be fully diffusive with full
moment and energy accommodation;
• Code validation was established [Riabov, 1998] by comparing
numerical results with experimental data [Gusev et al., 1977; Riabov,
1995] related to the simple-shape bodies;
• TEST: a single plate in air flow at 0.02 < Kn,L < 3.2, M = 10, tw = 1;
• Independence of flow profiles and aerodynamic characteristics from
mesh size and number of molecules has been evaluated;
• EXAMPLE: 12,700 cells in eight zones, 139,720 molecules;
• The G. Bird’s criterion for the time step is used: 1×10-8  tm  1×10-6 s;
• Ratio of the mean separation between collision partners to the local
mean free path and the CTR ratio of the time step to the local mean
collision time have been well under unity over flowfield;
• Computing time of each variant on Intel IV PC is variable: 4 – 60 hours.
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Drag Coefficient, Cx
Drag of a blunt plate
0.6
0.5
0.4
0.3
DSMC
experiment
free-molecular
0.2
0.1
0
0.01
0.1
1
Knudsen Number, Kn
10
Fig. 1 Total drag coefficient of the plate vs. Knudsen number Kn∞,L in air at M∞
= 10, tw = 1, and α = 0 deg. Experimental data is from [V. Gusev, et al., 1977].
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Table: Drag coefficient of a single plate in airflow at
Kn ,L = 0.13, M = 10, γ = 1.4, tw = 1, and different
numerical parameters
Number of
cells
Number of
molecules
per cell
Drag
coefficient
Time of
calculation
12,700
11
0.4524
12 h. 28 min.
12,700
22
0.4523
21 h. 03 min.
49,400
11
0.4525
62 h. 06 min.
203,200
11
0.4526
187 h. 11 min.
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Lift-Drag Ratio, Y/X
0.6
0.4
0.2
M =9.9,exper.
M >9,FM regime
M =11.8,exper.
M =9.9,DSM C
M =11.8,DSM C
0
0
5
10
15
20
25
30
35
40
45
Angle of Attack, deg
Lift-drag ratio Y/X for a wedge (θ = 20 deg) in helium flow at Kn∞,L = 0.3
(Re0 = 4) and M∞ = 9.9 and 11.8. Experimental data from [Gusev, et al., 1977].
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3.2
Ar,exper.
Drag coefficient, Cx
3
2.8
2.6
2.4
N2,exper.
Ar,DSMC
N2,DSMC
Ar,FM regime
N2,FM regime
Ar,continuum
2.2
N2,continuum
2
1.8
1.6
0.01
0.1
1
10
Knudsen Number, Kn
Drag coefficient Cx for a disk (α = 90 deg) vs Knudsen number Kn∞,D in argon
(triangles) and nitrogen (squares). Experimental data from [Gusev, et. al., 1977].
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Drag of a blunt plate: Tw effect
Drag coefficient, Cx
1.2
1.1
1
exper.,tw=1
0.9
DSM C,tw=1
0.8
DSM C,tw=0.34
0.7
FM ,tw=1
0.6
FM ,tw=0.34
exper.,tw=0.34
0.5
0.01
0.1
1
10
Knudsen Number, Kn
Drag coefficient Cx for a blunt plate (δ = 0.1) vs Knudsen number Kn∞,L in air at
α = 20 deg. Experimental data from [Gusev, et. al., 1977].
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
51
Lift of a blunt plate: Tw effect
Lift coefficient, Cy
0.5
0.4
0.3
exper.,t w=1
DSMC,t w=1
DSMC,t w=0.34
0.2
FM,t w=1
FM,t w=0.34
exper.,t w=0.34
0.1
0.01
0.1
1
10
Knudsen Number, Kn
Lift coefficient Cy for a blunt plate (δ = 0.1) vs Knudsen number Kn∞,L in air at α =
20 deg. Experimental data from [Gusev, et. al., 1977].
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
52
Pitching moment of a blunt plate: Tw and γ effects
Moment Coefficient
-0.2
-0.25
-0.3
-0.35
0.1
1
10
100
Reynolds Number
Air,DSM C,t w=1
Air,FM ,t w=1
Argon,DSM C,t w=1
Air,DSM C,t w=0.34
Air,FM ,t w=0.34
Argon,FM ,t w=1
Pitch moment coefficient Cm0 for a blunt plate (δ = 0.1) vs Reynolds number
Re0 in air and argon at α = 20 deg.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
53
Drag of a cone: Tw effect
Drag Coefficient
2.5
2
1.5
1
exper.,t w=1
DSM C,t w=1
DSM C,t w=0.34
FM ,t w=1
FM ,t w=0.34
exper.,t w=0.34
0.5
0.1
1
10
100
1000
Reynolds Number
Drag coefficient Cx for a sharp cone (θc = 10 deg) in air at α = 0 deg and tw = 1
and tw = 0.34. Experimental data from [Gusev, et. al., 1977].
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
54
Body Interference in Hypersonic Flows:
Flowfield near two side-by-side plates
Mach number contours in argon flow about a side-by-side plate at Kn ,L =
0.024, H/L = 0.5 (left), and H/L = 1 (right).
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
55
Lift and drag of two side-by-side plates
1
0.9
Cy,H=0.25L
Cy,H=0.5L
0.7
Cy,H=0.75
0.7
Drag Coefficient, Cx
Lift Coefficient, Cy
0.8
Cy,H=1.25L
0.6
0.5
0.4
0.3
0.2
0.1
0
0.01
0.1
1
10
0.6
0.5
0.4
0.3
0.2
Cx,H=0.25L
Cx,H=0.75L
0.1
Cx,FM
0
0.01
Knudsen Num ber, Kn
0.1
Cx,H=0.5L
Cx,H=1.25L
1
10
Knudsen Number, Kn
Drag and lift coefficients of the side-by-side plates vs. Knudsen number
Kn,L at M = 10 in argon.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
56
Aerocapture with large inflatable
balloon-like decelerators (“ballutes”)
Titan Explore Mission
From “Ballute Missions” (http://www2.jpl.nasa.gov/adv_tech/ballutes/missions.htm)
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
57
The strong influence of the geometrical factor
(a) H/R = 8
(c) H/R = 4
(b) H/R = 6
(d) H/R = 2
Mach number contours in nitrogen flow about a torus at
Kn∞D = 0.01, M∞ = 10, and various geometrical factors H/R.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
58
Flow Patterns near a Torus in Various Gases
(a) argon flow
(b) carbon dioxide flow
Mach number contours in flows of argon and carbon dioxide about a
torus at Kn,D = 0.01, M = 10, and H/R=8.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
59
Comparing Numerical Results with Experimental Data
(b) experiment [Loirel, et al., ISSW, 2001]
(a) DSMC calculation
Contours of constant Mach numbers near a torus in the flow of nitrogen
at Kn,D = 0.00013 and M = 7.11.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
60
Pressure and skin-friction coefficients along the torus surface
2.5
Pressure coefficient, Cp
2
H/R=2
H/R=4
H/R=6
H/R=8
Skin-friction coefficient,C
f
0.25
1.5
1
0.5
0.2
0.15
H/R=2
H/R=4
0.1
H/R=6
H/R=8
0.05
0
-0.05
0
60
120
180
240
300
Angle, deg
-0.1
-0.15
0
0
60
120
180
240
300
Angle, deg
360
-0.2
-0.25
-0.5
(a) Pressure coefficient, Cp
(b) Skin-friction coefficient, Cf
Pressure and skin-friction coefficients along the torus surface in nitrogen
flow at Kn,D = 0.01, M = 10, and various geometric factors H/R.
From V. V. Riabov, Journal of Spacecraft & Rockets, 1999, Vol. 36, No. 2.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
61
360
Rarefaction Effects: Influence of the Knudsen number, Kn∞,D
(a) H/R = 8
(c) H/R = 4
(b) H/R = 6
(d) H/R = 2
Mach number contours in nitrogen flow about a torus at
Kn∞D = 1, M∞ = 10, and various geometrical factors H/R.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
62
Rarefaction Effects on Skin-friction Coefficient
Skin-friction coefficient,Cf
1
0.8
0.6
0.4
0.2
0
-0.2 0
-0.4
-0.6
Kn=0.01
Kn=0.1
Kn=1
Kn=4
60
120
180
240
300
360
Angle, deg
-0.8
-1
-1.2
Skin-friction coefficient Cf along the torus surface in nitrogen flow
at H/R = 2, M = 10, and various Knudsen numbers Kn,D.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
63
Drag of a torus in different gases
3
2.75
Drag coefficient, Cx
2.5
2.25
A r,H/R=8
N2,H/R=8
CO2,H/R=8
A r,H/R=2
N2,H/R=2
CO2,H/R=2
A r,cyl,FM
N2,cyl,FM
CO2,cyl,FM
2
1.75
1.5
1.25
1
0.01
0.1
1
10
Knudsen number, Kn
Drag coefficient Cx of a torus vs. Knudsen number Kn∞,D at M∞ = 10 and
different geometrical factors H/R in argon, nitrogen, and carbon dioxide.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
64
Aerodynamics of Toroidal Ballute Models
The ballute model used in experiments [Loirel, et al., AIAA Paper, No. 2894,
2002].
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
65
Flow Simulation near Toroidal Ballute Models
The computational grid near the model used in simulations.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
66
Flowfield near a toroidal ballute model
a) dissociating oxygen flow
b) perfect-gas oxygen flow
Mach number contours in dissociating oxygen flow (left) and perfect-gas flow
(right) about a toroidal ballute model at Kn,D = 0.005, D = 0.006 m, A = 0.042
m, U∞ = 5693 m/s, p∞ = 1.28 kPa, and T∞ = 1415 K.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
67
Aerodynamics of a spinning cylinder
• At subsonic flow conditions, the speed ratio S = M∞∙(0.5∙g )½
becomes small, and aerodynamic coefficients become very
sensitive to its magnitude;
• Transition flow regime has been studied numerically at M∞ =
0.15, g = 5/3 (argon gas), and spin ratio W = 1, 3, and 6;
• The incident molecules dominate when KnD < 0.1, and the
reflected molecules dominate when KnD > 0.1;
• Lift coefficient changes sign for the cylinder spinning in counterclockwise direction, and the drag coefficient becomes a function
of the spin rate;
• At supersonic flow conditions, the speed ratio S becomes large,
and the aerodynamic coefficients become less sensitive to its
magnitude.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
68
15
Lift Coefficient
10
W=1
W=6
W=3,FM
W=3
W=1,FM
W=6,FM
5
0
0.001
-5
0.01
0.1
1
10
Knudsen Number
-10
-15
-20
Lift coefficient Cy of a spinning cylinder vs Knudsen number KnD at
subsonic Mach number, M∞ = 0.15 and different spin rates W in argon.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
69
35
Drag Coefficient
30
25
20
W=0
15
W=1
W=3
10
W=6
5
Free-M olecular
0
0.001
0.01
0.1
Knudsen Number
1
10
Drag coefficient Cx of a spinning cylinder vs Knudsen number KnD at
subsonic Mach number, M∞ = 0.15 and different spin rates W in argon.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
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70
Lift Coefficient
0.2
0.15
W=0.03
W=0.1
W=0.03,FM
W=0.1,FM
0.1
0.05
0
0.01
0.1
1
10
Knudsen Number
Lift coefficient Cy of a spinning cylinder vs Knudsen number KnD at
supersonic Mach number, M∞ = 10 and different spin rates W in argon.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
71
Drag Coefficient
3
2.5
2
W=0
W=0.03
1.5
W=0.1
Free-M olecular
1
0.01
0.1
1
Knudsen Number
10
Drag coefficient Cx of a spinning cylinder vs Knudsen number KnD at
supersonic Mach number, M∞ = 10 and different spin rates W in argon.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
72
Heat flux at the stagnation point of a sphere
Stanton number, St
1
TVSL
experiment
DSM C
N-St,slip
0.1
0.001
0.01
0.1
1
Knudsen number, Kn
Stanton numbers St for a sphere vs. Knudsen numbers Kn∞,R for different medium
models at various wind-tunnel experimental conditions [A. Botin, et al., 1990].
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
73
The slip effect is significant as the Knudsen
number (~altitude) increases
Temperature jump at the Shuttle stagnation point at
different Knudsen numbers (after Moss and Bird [1985])
Shuttle stagnation point heat flux versus Knudsen
number as predicted by NS, VSL, and, DSMC
computations (after Gupta [1986])
Velocity jump on the Shuttle at x=1.5m for U∞= 7.5 km/s at
different Knudsen numbers (after Moss and Bird [1985])
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
74
Comparison of flight data with DSMC predictions
Heating rate at the stagnation point of the reentering Shuttle Orbiter for various
altitudes. From G. A. Bird and J. N. Moss, AIAA Paper No. 84-223, 1984.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
75
Extent of Thermodynamic Nonequilibrium
Flowfield structure along the stagnation streamline of the reentering Shuttle Orbiter
at 92.35 km altitude (Kn∞,R = 0.028). From G. A. Bird and J. N. Moss, AIAA Paper
No. 84-223, 1984.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
76
Comparison of temperature profiles using
DSMC and Viscous Shock Layer models
Flowfield structure along the stagnation streamline of the reentering Shuttle Orbiter
at 92.35 km altitude (Kn∞,R = 0.028). From G. A. Bird and J. N. Moss, AIAA Paper
No. 84-223, 1984.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
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77
Predicting the heat flux using DSMC and
Viscous Shock Layer models
Comparison of predicted and measures heating along the surface of the reentering
Shuttle Orbiter at 92.35 km altitude (Kn∞,R = 0.028). From G. A. Bird and J. N. Moss,
AIAA Paper No. 84-223, 1984.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
78
Effect of rarefaction on predicted drag using
DSMC and Viscous Shock Layer models
Comparison of predicted drag coefficients for the Shuttle Orbiter along its reentering
trajectory. From G. A. Bird and J. N. Moss, AIAA Paper No. 84-223, 1984.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
79
Nonequilibrium and Rarefaction Effects in the
Hypersonic Multicomponent Viscous Shock Layers
• The effects of rarefaction and nonequilibrium processes on
hypersonic rarefied-gas flows over blunt bodies have been studied
by the Direct Simulation Monte-Carlo technique (DSMC) and by
solving the full Navier-Stokes equations and the equations of a
thin viscous shock layer (TVSL) under the conditions of windtunnel experiments and hypersonic-vehicle flights at altitudes from
60 to 110 km.
• The nonequilibrium, equilibrium and “frozen” flow regimes have
been examined for various physical and chemical processes in air.
• The influence of similarity parameters (Reynolds number,
temperature factor, catalyticity parameters, and geometrical
factors) on the flow structure near the blunt body and on its
aerodynamic coefficients in hypersonic streams of dissociating air
is studied.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
80
Approximation of a Thin Viscous
Shock Layer (TVSL)
• The TVSL equations are found from asymptotic
analysis of the Navier-Stokes equations [6, 11] at
ε→0, Reof→∞, and σ = (εReof) = const, where Reof =
ρ∞U∞R/µ(Tof) is Reynolds number, ε =(γ-1)/(2γ), γ is a
specific heat ratio.
– Cheng H. “The blunt-body problem in hypersonic flow at low
Reynolds number,” Paper No. 63-92, IAS, New York, 1963.
– Provotorov V. and Riabov V. “Study of nonequilibrium
hypersonic viscous shock layer,” Trudy TsAGI, Issue 2111,
pp. 142-156, 1981 (in Russian).
– Riabov, V. (2004). "Nonequilibrium and Rarefaction Effects in
the Hypersonic Multicomponent Viscous Shock Layers,"
Proceedings of the 24th ICAS Congress, Paper 34, Japan.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
81
TVSL Equations of the Thin Viscous
Shock Layer
∂(g2ρu)/∂x + ∂(g1g2ρv)/∂y = 0
(1) ρ(ug1-1∂Al/∂x + v∂Al/∂y) = -∂Jl/∂y,
l=1, 2, ..., Nel-1
ρ(ug1-1∂u/∂x + v∂u/∂y)=-εg1-1dpw/dx
+σ∂(μ∂u/∂y)/∂y
(2)
∂p/∂y = Kρu2
(3)
ρ(ug1-1∂w/∂x + v∂w/∂y)
=σ∂(μ∂w/∂y)/∂y
(4)
ρ(ug1-1∂H/∂x + v∂H/∂y) = -∂q/∂y
+2σ∂(μu∂u/∂y)/∂y+2σ∂(μw∂w/∂y)/∂y
(5)
DWC, March 24, 2012
(6)
ρ(ug1-1∂αi/∂x + v∂αi/∂y) = ∂ji/∂y+ωi, i= Nel, ..., N-1
(7)
Σαk = 1, k = 1, 2, ..., N
(8)
p= ρRgT
(9)
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
82
The generalized Rankine-Hugoniot
conditions at the TVSL external boundary
sinχcosφ(cosχcosφ - u) =
σμ∂u/∂y
(10)
sinχcosφ(H∞- H) = -q
+2σμ(u∂u/∂y+ w∂w/∂y)
(14)
ρv = - sinχcosφ
(11)
sinχcosφ(Al∞- Al) = - Jl
(15)
sinχcosφ(sinφ - w) = σμ∂w/∂y
(12)
sinχcosφ(αi∞- αi) = - ji
(16)
p=sin2χcos2φ
(13)
DWC, March 24, 2012
Here χ is a swept angle;
φ is a shock incident angle.
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
83
Numerical Method
• The numerical procedure of Provotorov and Riabov (1981
-1994) was used for the solution of nonlinear partially
differential equations (1)-(7).
• The equation terms have been approximated by using the
two-point second-order Keller's scheme (1974).
• The iteration process converges with the second order
towards the solution.
• The results have been obtained in the whole range of
chemical reaction rates up to the values near equilibrium.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
84
Stanton numbers St for a sphere vs. Reynolds numbers
Reof for different medium models at various wind-tunnel
experimental conditions [Chernikova, 1980; Nomura, 1974].
Stanton number, St
1
TVSL
exper.
DSM C
N-St,slip
N-St,nonslip
B-Layer
0.1
1
10
100
1000
Reynolds num ber, Re of
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
85
Stanton numbers St for a cylinder vs. Reynolds numbers
Reof for different medium models at various wind-tunnel
experimental conditions [Chernikova, 1980; Nomura, 1974].
Stanton number, St
1
TVSL
exper.
0.1
1
10
100
1000
Reynolds num ber, Re of
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
86
Stanton numbers St along the spherical surface
coordinate s/R at Reof = 46.38, M∞ = 6.5, tw = 0.31.
Experimental data from [Chernikova, 1980; Botin, 1987]
Stanton number, St
0.3
0.25
0.2
0.15
DSMC
exper.
TVSL
0.1
0.05
0
0
0.2
0.4
0.6
0.8
1
Distance along the spherical surface, s/R
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
87
Pressure, temperature and density along the normal on
the critical line of a sphere at Reof = 7.33, M∞ = 6.5, and
tw = 0.315.
2.5
P/Po, T/T o, and ρ/ρo
p/po, N-St
T/To, N-St
2
0.1*DEN/DENo, N-St
p/po, TVSL
1.5
T/To, TVSL
0.1*DEN/DENo, TVSL
1
0.5
0
0
0.2
0.4
0.6
Distance along the norm al, n/R
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
88
Conditions & Calculation Results
Table: Values of the Reynolds number for a sphere (R = 1m) along the
Space Shuttle trajectory
Reof = ρ∞U∞R/µ(Tof)
h, km
Reof
110
1.49
100
6.97
REFERENCES:
90
47.3
80
230
Tong H, Buckingham A and Curry D. Computational
procedure for evaluations of Space Shuttle TRS
requirements. AIAA Paper 74-518, 1974.
70
1220
60
5130
DWC, March 24, 2012
Moss J and Bird G. Direct simulation of transitional
flow for hypersonic reentry conditions. AIAA Paper
No. 84-0223, 1984.
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
89
Stanton numbers St vs. Reynolds numbers Reof for a sphere of radius
R=1m along the Space Shuttle trajectory and different medium
models. Wind-tunnel experimental data from Chernikova (1980), Botin
(1987); STS-2 and STS-3 data from Moss & Bird (1984).
Stanton number, St
1
0.1
equilibrium
catal.wall
non-catal.wall
exper.
N-St,slip
STS-2/3 data
0.01
1
10
100
1000
10000
Reynolds num ber, Re of
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
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90
Effect of Surface Catalyticity
Mass concentrations
Mass concentrations αi of the air components in the TVSL at Reof = 230: dash
lines - ideally catalytic surface; solid lines - absolutely noncatalytic surface.
0.25
0.2
0.15
0.1
0.05
0
0
0.05
0.1
0.15
0.2
0.25
Distance along a norm al coordinate, n/R
DWC, March 24, 2012
O,noncatal.
O,catal.
N,noncatal.
N,catal.
NO,noncatal.
NO,catal.
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
91
The values of temperatures Ts, Td, Tof, and Toe as
functions of Reynolds number Reof.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
92
The width of the catalytically influenced zone d as a
function of Reynolds number Reof.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
93
Mass concentrations αiw of air components on
the noncatalytic vehicle surface.
Mass concentrations
1
0.8
0.6
0.4
O2
O
N
NO
N2
0.2
0
1
10
100
1000
10000
Reynolds num ber, Re of
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
94
Wall temperature T
we,
K
Equilibrium temperature Twe of the spherical surface
(R = 1 m) vs. Reynolds numbers Reof.
2000
1750
1500
1250
1000
noncatal.wall
750
catal.wall
500
1
10
100
1000
10000
Reynolds num ber, Re of
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
95
we,
2500
2250
Wall temperature T
K
Equilibrium temperature Twe of the cylindrical surface
(R = 0.1 m) vs. Reynolds numbers Reof.
2000
1750
1500
1250
noncatal.wall
1000
750
catal.wall
500
0.1
1
10
100
1000
Reynolds num ber, Re of
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
96
Equilibrium temperature Twe of the cylindrical surface (R = 1 m)
as a function of the swept angle χ for different values of the
Reynolds number Reof: solid lines - noncatalytic surface;
dashed lines - catalytic surface.
Wall temperature T
we,
K
2000
1500
1000
500
Reo=6.97,noncat.
Reo=6.97,catal.
Reo=230,noncat.
Reo=230,catal.
Reo=5130,noncat.
Reo=5130,catal.
0
0
20
40
60
80
Sw ept angle χ , deg
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
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97
The binary similitude law (ρ∞R = const)
Temperature Ratio
0.4
0.3
0.2
Rs=1m
0.1
Rs=0.005m
0
0
0.01
0.02
0.03
0.04
0.05
Distance along the central line, x/Rs
Temperature T/T0 at the stagnation streamline of the sphere at Re0,R =7.33, u∞ = 7.8
km/s, ρ∞Rs = 5.35∙10-7 kg/m2 and different sphere radii: Rs = 1 and Rs = 0.005m.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
98
Electron Concentration
5
4
3
2
R=1m
1
R=0.005m
0
0
0.01
0.02
0.03
0.04
0.05
Distance along the central line, x/Rs
Electron concentration Ne∙Rs∙10-11 m-2 at the stagnation streamline of the sphere
at Re0,R =7.33, u∞ = 7.8 km/s, ρ∞Rs = 5.35∙10-7 kg/m2 and different sphere radii:
Rs = 1 and Rs = 0.005m. Catalytic surface.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
99
Electron Concentration
2.5
2
R=1m
R=0.005m
1.5
1
0.5
0
0
0.01
0.02
0.03
0.04
0.05
Distance along the central line, x/Rs
Electron concentration Ne∙Rs∙10-14 m-2 at the stagnation streamline of the sphere
at Re0,R =7.33, u∞ = 7.8 km/s, ρ∞Rs = 5.35∙10-7 kg/m2 and different sphere radii:
Rs = 1 and Rs = 0.005m. Noncatalytic surface.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
100
Computation Fluid Dynamic Challenges
CFD prediction of Hyper-X flow field at
M∞ = 7 with engine operating. (Courtesy
of NASA Dryden Flight Research Center).
 The thin shock layer coming from the forebody remains very close to the vehicle
surface for a large distance down the body length;
 Various shock-shock and shock-boundary layer interactions occur in the vicinity of
the ramp inlet;
 Flow interactions with the engine exhaust and lower surface body contouring
create a complex flow field under the back half of the vehicle;
 All of this could take place at flight conditions where chemical reactions would be
important.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
101
Solutions of Navier-Stokes Equations:
Pressure at the Stagnation Point
Pressure Ratio
1.15
1.1
1.05
1
0.95
10
100
1000
10000
Corre lation Param e te r
N-St ,gamma=1.4, t w=1
N-St ,gamma=1.4, t w<1
N-St ,gamma=5/ 3, t w=1
N-St ,gamma=5/ 3, t w<1
exper., gamma=1.4,t w=1
exper.,gamma=5/ 3,t w=1
Pressure ratio pw/po at the front stagnation point of a sphere
vs correlation parameter Res(ρs/ρ∞)½.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
102
Temperature Ratios
Rotational-Translational Relaxation in Viscous
Flows of Nitrogen near a Sphere
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Distance, x/Rs
TR/To,Navier-Stokes
Tt/To,Navier-Stokes
T/To,Navier-Stokes
TR/To,experiment
The nonequilibrium rotational TR, translational Tt, and equilibrium overall T
temperatures at the stagnation stream line near a sphere: Kn∞,R = 0.08 (Re0,R =
16.86), M∞ = 9, T0 = 298 K, tw = 0.3. Experimental data from Tirumalesa (1968).
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
103
Temperature Ratios
1
0.8
0.6
0.4
0.2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Distance, x/Rs
TR/To,Navier-Stokes
Tt/To,Navier-Stokes
T/To,Navier-Stokes
TR/To,experiment
The nonequilibrium rotational TR, translational Tt, and equilibrium overall T
temperatures at the stagnation stream line near a sphere: Kn∞,R = 0.017 (Re0,R =57.4),
M∞ = 18.8, T0 =1600 K, tw = 0.19. Experimental data from Ahouse & Bogdonoff (1969).
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
104
Heat Transfer on a Hypersonic Blunt Bodies
with Gas Injection
Stanton number, St
1
0.8
0.6
0.4
DSM C,Gw=0
0.2
DSM C,Air,Gw=0.94
0
-0.2
0.01
Exper.,Air,Gw=0
0.1
1
10
Knudsen num ber, KnR
TEST: Heat transfer on a sphere in air flow (without blowing)
at 0.016 < Kn,R < 1.5 (92.8 > Re0,R > 1), M = 6.5, tw = 0.31;
Comparison with experiments (Ardasheva, et al [1979])
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
105
Influence of the Air Mass Blowing Factor Gw
Temperature contours at Kn ,R = 0.0163 (Re0,R = 92.8) and air-to-air mass
blowing factor Gw = 0.7.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
106
Influence of the Air Mass Blowing Factor Gw
0.25
0.2
Stanton number, St
0.15
0.1
0.05
Gw =0
Gw =0.15
Gw =0.32
Gw =0.70
Gw =0.94
Gw =1.5
Gw =0.55
0
0
0.1
-0.05
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Coordinate along spherical surface, s/R
-0.1
Stanton number along the spherical surface at Kn ,R = 0.0163 (Re0,R = 92.8)
and various air-to-air mass blowing factors.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
107
Comparison with Experimental Data (A. Botin [1987]):
Air-to-Air Mass Blowing
0.3
Stanton number, St
0.25
0.2
0.15
0.1
DSMC,Gw =0
DSMC,Air,Gw =0.15
DSMC,Air,Gw =0.32
Exper.,Gw =0
Exper.,Air,Gw =0.15
Exper.,Air,Gw =0.32
0.05
0
0
0.2
0.4
0.6
0.8
1
Coordinate along spherical surface, s/R
Stanton number along the spherical surface at Kn ,R = 0.0326, T0 = 1000 K and
lower air-to-air mass blowing factors.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
108
Influence of the Rarefaction Factor in Air Mass Blowing
Width of the injection-influenced zone, (s/R)max vs Knudsen number at Gw = 0.94.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
109
Diffuse Injection of Helium into Air Stream
Contours of helium mole fraction at Kn ,R = 0.0163 (Re0,R = 92.8) and helium-toair mass blowing factor Gw = 0.7.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
110
Influence of the Helium Mass Blowing Factor Gw
Temperature contours at Kn ,R = 0.0163 (Re0,R = 92.8) and helium-to-air mass
blowing factor Gw = 0.7.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
111
Influence of the Helium and Nitrogen Mass Blowing Factor Gw
0.25
0.2
Stanton number, St
0.15
0.1
0.05
0
0
0.1
-0.05
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Coordinate along spherical surface, s/R
-0.1
-0.15
Gw =0
N2,Gw =0.32
He,Gw =0.32
He,Gw =0.70
N2,Gw =0.70
-0.2
Stanton number along the spherical surface at Kn ,R = 0.0163 (Re0,R = 92.8) and
various helium-to-air and nitrogen-to-air mass blowing factors.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
112
Hypersonic Hydrogen Combustion in
Thin Viscous Shock Layer (TVSL)
• Combustion of air-hydrogen mixture (11 components, 35
chemical reactions);
• Two-dimensional flow in thin viscous shock layer (TVSL) near
noncatalytic surface of parabolic cylinder (radius R=0.015 m);
• Flow parameters: V = 2933 m/s, T = 230K, Tw = 600K,
Reynolds number Re0R = 628;
• Slot or uniform (G = G0) injection;
• Modified Newton-Raphson numerical method with exponential
box-schemes;
• Y-Adaptive grid 10141;
• V. V. Riabov and A. V. Botin, Journal of Thermophysics and
Heat Transfer, 1995; Vol. 9, No. 2, pp. 233-239.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
113
Hypersonic Hydrogen Combustion in
Thin Viscous Shock Layer (TVSL)
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
114
Interaction of Shock Waves near Blunt Bodies
NOTATION:
1 – The boundary of shock layer;
2 – The inclined shock;
3 – The sonic line;
4 – The mixing layer;
5 – The inner shock;
6 – The calculation domain for the NavierStokes equations;
7 – The wedge.
NOTE: This pattern is for the Type III
interference.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
115
The Electron-Beam Fluorescence Technique for Flow
Visualization
The experiments have been carried out in a vacuum wind tunnel at low-density airflow parameters:
M = 6.5, T0 = 1000 K, p0 = 4000 N/m2. The model is a plate with a cylindrical edge of radius R = 0.01
m. The flow is characterized by the Reynolds number Re0 = 15.5 and the Knudsen number KnR = 0.1.
No interference: The plane oblique shock wave
was generated by a wedge with an apex angle 
= 20 deg, and the angle of the shock inclination 
about 27 deg. A string transverse mechanism
allowed moving the wedge at any position along
the vertical Y-axis to simulate interactions.
DWC, March 24, 2012
Type I interference is characterized by the
formation of two shocks after the
intersection of two oblique shocks of
opposite families. The intersection point is
sufficiently downstream of the sonic point
on the bow shock wave.
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
116
The Electron-Beam Fluorescence Technique for
Flow Visualization
Type II interaction reflects the Mach
phenomenon and produces two triple
points separated by a normal shock after
the intersection of the two oblique shock
waves.
DWC, March 24, 2012
In the Type III interference case, the slip line
reattaches on the body surface. If the oblique
shock crosses the strong bow shock, a slip
line is produced separating subsonic area
from a supersonic flow zone. This case would
be only possible in continuum at small shockinclination angles,   20 deg.
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
117
The Electron-Beam Fluorescence Technique for
Flow Visualization
In the Type V interaction case, both weak
oblique shock waves of the same
direction would interact above the upper
sonic line generating a supersonic jet after
the upper multiple point .
In the Type VI interference case, far from the
leading critical point of the cylinder, a shock, a
slip line and an expansion wave would be
generated behind the triple point above the
considered region.
NOTE: The following cases were not found in the recent experiments: If the slip line is unable to
reattach on the wall, a supersonic jet, bounded by subsonic regions develops (Type IV) [Edney,
1968]. The special case of a curved supersonic jet (Type IVa) was studied by Purpura et al. [1998].
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
118
Solving the Navier-Stokes Equations
•
•
•
•
The full system of Navier-Stokes equations (I. Egorov, JCMMP, 31, 1991)
have been solved numerically. The meridian angle  is varied over the
interval -90    +90 deg. The steady-state equations in the arbitrary
curvilinear coordinate system  = (x, y),  = (x, y), where x and y are
Cartesian coordinates, have been written in conservation form.
The equations have been nondimensionalized with respect to the
parameters in freestream flow.
The outer boundary is divided into two parts: one part with uniform freestream conditions at the constant Mach number M , and the other with the
conditions behind an inclined plane shock. The velocity slip and
temperature jump effects are considered at the body surface.
The numerical method has been described in detail in (A. Botin, Fluid Dyn.,
28, 1993). A set of FORTRAN standardized programs has been used to
solve the problem. The grid contains 44 nodes related to the surface
curvilinear coordinate  and 41 nodes along the normal . The finitedifference program is a second-order accurate scheme.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
119
Mach number contours calculated by the Navier-Stokes solver at
Re0 = 15.5 (KnR = 0.1) for different types of interaction
In the Type III interference case, the slip line
reattaches on the body surface. If the oblique
shock crosses the strong bow shock, a slip
line is produced separating subsonic area
from a supersonic flow zone. This case would
be only possible in continuum at small shockinclination angles,   20 deg.
DWC, March 24, 2012
In the Type V interaction case, both weak
oblique shock waves of the same
direction would interact above the upper
sonic line generating a supersonic jet after
the upper multiple point .
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
120
Experimental and numerical results of local Stanton number
distributions around a cylinder at Re0 = 15.5 (KnR = 0.1) for
different types of shock-wave/shock-layer interference
0.2
Local Stanton Number, St
Local Stanton Number, St
0.2
0.18
0.18
0.16
0.16
0.14
0.14
0.12
0.12
0.1
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
-100
-50
0
50
100
0.02
-100
-50
Meridian Angle, deg
experiment
Navier-Stokes No Interference
Type I interference is characterized by the
formation of two shocks after the
intersection of two oblique shocks of
opposite families. The intersection point is
sufficiently downstream of the sonic point
on the bow shock wave.
DWC, March 24, 2012
0
50
100
Meridian Angle, deg
experiment
Navier-Stokes No Interference
Type II interaction reflects the Mach
phenomenon and produces two triple
points separated by a normal shock after
the intersection of the two oblique shock
waves.
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
121
Experimental and numerical results of local Stanton number
distributions around a cylinder at Re0 = 15.5 (KnR = 0.1) for
different types of shock-wave/shock-layer interference
0.25
Local Stanton Number, St
Local Stanton Number, St
0.35
0.3
0.2
0.25
0.15
0.2
0.15
0.1
0.1
0.05
0.05
0
-100
-50
0
50
100
0
-100
-50
Navier-Stokes No Interference
In the Type III interference case, the slip line
reattaches on the body surface. If the oblique
shock crosses the strong bow shock, a slip
line is produced separating subsonic area
from a supersonic flow zone. This case would
be only possible in continuum at small shockinclination angles,   20 deg.
DWC, March 24, 2012
50
100
Meridian Angle, deg
Meridian Angle, deg
experiment
0
experiment
Navier-Stokes No Interference
In the Type V interaction case, both weak
oblique shock waves of the same
direction would interact above the upper
sonic line generating a supersonic jet after
the upper multiple point .
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
122
Experimental and numerical results of local Stanton number
distributions around a cylinder at Re0 = 15.5 (KnR = 0.1) for
different types of shock-wave/shock-layer interference
Local Stanton Number, St
0.25
0.2
0.15
0.1
0.05
0
-100
-50
0
50
100
Meridian Angle, deg
experiment
Navier-Stokes No Interference
In the Type VI interference case, far from the
leading critical point of the cylinder, a shock, a
slip line and an expansion wave would be
generated behind the triple point above the
considered region.
DWC, March 24, 2012
NOTE: The following cases were not
found in the recent experiments: If the
slip line is unable to reattach on the
wall, a supersonic jet, bounded by
subsonic regions develops (Type IV)
[Edney, 1968]. The special case of a
curved supersonic jet (Type IVa) was
studied by Purpura et al. [1998].
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
123
Mach Number Contours (Types II & III) obtained
by the Direct Simulation Monte-Carlo Technique
The calculations have been carried out at low-density airflow parameters: M = 6.5, T0 = 1000 K, p0 =
4000 N/m2, R = 0.01 m, Re0 = 15.5, KnR = 0.1, wedge apex angle  = 20 deg, and the angle of the
shock inclination  about 27 deg.
Type II interaction reflects the Mach
phenomenon and produces two triple
points separated by a normal shock after
the intersection of the two oblique shock
waves.
DWC, March 24, 2012
In the Type III interference case, the slip line
reattaches on the body surface. If the oblique
shock crosses the strong bow shock, a slip
line is produced separating subsonic area
from a supersonic flow zone.
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
124
Mach Number Contours (Type V) obtained by
the Direct Simulation Monte-Carlo Technique
The calculations have been carried out at low-density airflow parameters: M = 6.5, T0 = 1000 K, p0 =
4000 N/m2, R = 0.01 m, Re0 = 15.5, KnR = 0.1, wedge apex angle  = 20 deg, and the angle of the
shock inclination  about 27 deg.
In the Type V interaction case, both weak oblique shock waves of the same direction would interact
above the upper sonic line generating a supersonic jet after the upper multiple point .
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
125
Local Stanton Numbers obtained by the Direct
Simulation Monte-Carlo Technique
The calculations have been carried out at low-density airflow parameters: M = 6.5, T0 = 1000 K, p0 =
4000 N/m2, R = 0.01 m, Re0 = 15.5, KnR = 0.1, wedge apex angle  = 20 deg, and the angle of the
shock inclination  about 27 deg. Various types of shock-wave/viscous-layer interaction.
Local Stanton Number, St
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-100
-50
0
50
100
Meridian Angle, deg
No Interfer. Type II
DWC, March 24, 2012
Type III
Type V
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
Type VI
126
Local Pressure Coefficients obtained by the
Direct Simulation Monte-Carlo Technique
The calculations have been carried out at low-density airflow parameters: M = 6.5, T0 = 1000 K, p0 =
4000 N/m2, R = 0.01 m, Re0 = 15.5, KnR = 0.1, wedge apex angle  = 20 deg, and the angle of the
shock inclination  about 27 deg. Various types of shock-wave/viscous-layer interaction.
Pressure Coefficient, Cp
6
5
4
3
2
1
0
-100
-50
0
50
100
Meridian Angle, deg
No Interfer. Type II
DWC, March 24, 2012
Type III
Type V
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
Type VI
127
Conclusions
• The DSMC method is effective in studies of hypersonic rarefied flows of
various gases near probes in the transition regime between freemolecular and continuum regimes.
• For conditions approaching the hypersonic limit, the Knudsen number
(Kn∞) and temperature factor (tw) are the primary similarity parameters.
• The role of the specific heat ratio (γ) is also significant in aerodynamics
of a disk and a plate at various angles of attack.
• Important rarefaction effects that are specific for the transition flow
regime have been found:
– non-monotonic lift and drag of plates,
– strong repulsive force between side-by-side plates and cylinders,
– dependence of drag on torus radii ratio,
– at subsonic upstream conditions, the lift on a rotating cylinder has
different signs in continuum and free-molecule flow regimes. The
sign changes in the transition flow regime (at about Kn∞,D = 0.1).
• The acquired information could be effectively used for investigation and
prediction of the aerodynamic characteristics of probes (plates, wedges,
cylinders, spheres, cones, disks, and torus) in complex atmospheric
conditions of the Earth, Mars, Venus, and other planets and moons.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
128
References: Experiments, DSMC Method
•
•
•
•
Koppenwallner, G., and Legge, H., “Drag of Bodies in Rarefied
Hypersonic Flow,” Thermophysical Aspects of Reentry Flows, edited by
J. N. Moss and C. D. Scott, Vol. 103, Progress in Astronautics and
Aeronautics, AIAA, New York, 1994, pp. 44-59.
Bird, G. A., “Rarefied Hypersonic Flow Past a Slender Sharp Cone,”
Proceedings of the 13th International Symposium on Rarefied Gas
Dynamics, edited by O. M. Belotserkovskii, M. N. Kogan, S. S.
Kutateladze, and A. K. Rebrov, Vol. 1, Plenum Press, New York, 1985,
pp. 349-356.
Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas
Flows, 1st ed., Oxford University Press, Oxford, England, UK, 1994.
Gusev, V. N., Erofeev, A. I., Klimova, T. V., Perepukhov, V. A., Riabov,
V. V., and Tolstykh, A. I., “Theoretical and Experimental Investigations
of Flow Over Simple Shape Bodies by a Hypersonic Stream of Rarefied
Gas,” TsAGI Transactions, No. 1855, 1977, pp. 3-43 (in Russian).
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
129
References (continued)
•
•
•
•
Riabov, V. V., “Comparative Similarity Analysis of Hypersonic Rarefied
Gas Flows near Simple-Shape Bodies,” Journal of Spacecraft and
Rockets, Vol. 35, No. 4, 1998, pp. 424-433.
Gorelov, S. L., and Erofeev, A. I., “Qualitative Features of a Rarefied
Gas Flow About Simple Shape Bodies,” Proceedings of the 13th
International Symposium on Rarefied Gas Dynamics, edited by O. M.
Belotserkovskii, M. N. Kogan, S. S. Kutateladze, and A. K. Rebrov, Vol.
1, Plenum Press, New York, 1985, pp. 515-521.
Lengrand, J. C., Allège, J., Chpoun, A., and Raffin, M., “Rarefied
Hypersonic Flow Over a Sharp Flat Plate: Numerical and Experimental
Results,” Rarefied Gas Dynamics: Space Science and Engineering,
edited by B. D. Shizdal and D. P. Weaver, Vol. 160, Progress in
Astronautics and Aeronautics, AIAA, Washington, DC, 1994, pp. 276284.
Riabov, V. V., “Numerical Study of Hypersonic Rarefied-Gas Flows
About a Torus,” Journal of Spacecraft and Rockets, Vol. 36, No. 2,
1999, pp. 293-296.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
130
References (continued)
• Coudeville, H., Trepaud, P., and Brun, E. A., “Drag Measurements
in Slip and Transition Flow,” Proceedings of the 4th International
Symposium on Rarefied Gas Dynamics, edited by J. H. de Leeuw,
Vol. 1, Academic Press, New York, pp. 444-466.
• Mavriplis, C., Ahn, J. C., and Goulard, R., “Heat Transfer and
Flowfields in Short Microchannels Using Direct Simulation Monte
Carla,” Journal of Thermophysics and Heat Transfer, Vol. 11, No.
4, 1997, pp. 489-496.
• Oh, C. K., Oran, E. S., and Sinkovits, R. S., “Computations of HighSpeed, High Knudsen Number Microchannel Flows,” Journal of
Thermophysics and Heat Transfer, Vol. 11, No. 4, 1997, pp. 497505.
• Bird, G. A., The DS2G Program User’s Guide, Version 3.2, G.A.B.
Consulting Pty, Killara, New South Wales, Australia, 1999, pp.1-56.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
131
References (continued)
• Bisch, C., “Drag Reduction of a Sharp Flat Plate in a Rarefied
Hypersonic Flow” in Rarefied Gas Dynamics, edited by J. L.
Potter, 10th International Symposium Proceedings, Vol. 1, AIAA,
Washington, DC, 1976, pp. 361-377.
• Kogan, M. N., Rarefied Gas Dynamics, Plenum Press, New
York, 1969, pp. 345-390.
• Lourel, I., Morgan, R.G. “The Effect of Dissociation on Chocking
of Ducted Flows.” AIAA Paper 2002-2894. Washington, DC:
AIAA; 2002.
• Riabov, V. V., “Numerical Study of Interference between SimpleShape Bodies in Hypersonic Flows”. (Proceedings of the Fifth
M.I.T. Conference on Computational Fluid and Solid Mechanics,
June 17-19, 2009, edited by K. J. Bathe). Computers and
Structures, Vol. 87, 2009, pp. 651-663.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
132
References: Ballute Missions
•
•
•
•
•
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Acknowledgements
The author would like to express gratitude to G. A.
Bird for the opportunity of using the DS2G computer
program; to J. N. Moss for valuable discussions of
the DSMC technique and results; to V. N. Gusev, M.
N. Kogan, V. P. Provotorov, A. V. Botin, L. G.
Chernikova, T. V. Klimova, S. G. Kryukova, A. I.
Erofeev, V. A. Perepukhov, and Yu. V. Nikol’skiy for
contributions at earlier stages of this research, and to
I. Lourel for providing information about toroidal
ballute models and their experimental conditions.
DWC, March 24, 2012
Vladimir V. Riabov: Hypersonic
Rarefied Aerothermodynamics
134