RAREFACTION EFFECTS IN HYPERSONIC AERODYNAMICS

RAREFACTION EFFECTS IN HYPERSONIC AERODYNAMICS

Vladimir V. Riabov, Ph.D.Professor of Computer Science & Math Rivier CollegeNashua, New [email protected]://www.rivier.edu/faculty/vriabov/

July 10—15, 2010 ~ Asilomar Conference Grounds ~ Pacific Grove, California, USA

Daniel Webster College, Nashua, NH, March 24, 2012

DWC, March 24, 2012 Vladimir V. Riabov: Hypersonic Rarefied Aerothermodynamics

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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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• 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.

Techniques & Tools

Earth and Mars Atmospheric Parameters


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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

Entry velocity envelopes for Earth & Mars missions


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Entry velocity envelopes for Earth & Mars missions with return to Earth.

From “Capsule Aerothermodynamics,” AGARD Report No. 808, May 1997

Mars EntryEarth Entry

Typical trajectories of hypersonic spacecraft


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From: J. J. Bertin and R. M. Cummings, “Critical Hypersonic Aerothermodynamic Phenomena”, Annual Review of Fluid Mechanics, 2006, Vol. 38, pp.129-157

European Space Agency Projects


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Flow regimes and thermochemical phenomena


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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.

Estimating Transport Coefficients


9

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 = ρU∞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∞

Choosing a Mathematical Model


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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.

Estimating the local Knudsen number


12

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.

VSL

DSMC

Kn, local

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]).


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T-117 hypersonic wind tunnel, TsAGI, Moscow region, Russia

Nozzle and test chamber of the H2K hypersonic wind tunnel, Germany

Aerial view of the Thermal Protection Laboratory at NASA Ames, California


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15

Research on Hypersonics in TsAGI (Moscow region, Russia)


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Studies of “Buran” Aerothermodynamics (TsAGI, Russia)

International Cooperation: Space Projects


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British HOTOL and Mriya-225 Launcher tested in T-128 TsAGI Wind Tunnel, Moscow region, Russia

Ground based testing in Germany


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Ground based testing in Germany


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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.

HERMES Project, Germany & France, 1987-1992

HERMES-Columbus Project, Germany & Italy, 1987-1992

http://en.wikipedia.org/wiki/File:Hermes_Spaceplane_ESA.jpg

http://en.wikipedia.org/wiki/File:ESA_Columbus_MTFF.jpg

The International Space Station Mission

Vladimir V. Riabov: Hypersonic Rarefied Aerothermodynamics

21http://www.boeing.com/defense-space/space/spacestation/gallery/

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.

DWC, March 24, 2012

The International Space Station Mission

Vladimir V. Riabov: Hypersonic Rarefied Aerothermodynamics

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http://science.nationalgeographic.com/science/space/space-exploration/

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

DWC, March 24, 2012

Space Shuttle Orbiter approximate heat-transfer model


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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

Applications of Underexpanded Jets in Hypersonic Aerothermodynamic Research

• A method of underexpanded hypersonic viscous jets has been developed to acquire experimental aerodynamic data for simple-shape 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

DWC, March 24, 2012Vladimir V. Riabov: Hypersonic Rarefied Aerothermodynamics

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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 (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)

v'=v1×z2(γ-1)×d/dφ[lnr*'(φ)] +... (3)

ρ'=1/u0×z2 +... (4)

T'=θ1×z2(γ-1) +... (5)

p'=θ1/u0×z2γ +... (6)

u0=[(γ+1)/(γ-1)]½ (7)

u1=-θ1/(γ2-1)½ (8)

v1=-2θ1/{u0[1-2(γ-1)]} (9)

θ1=1/(u0)(γ-1) (10)

r*'(φ)= r*(φ)/rj (11)


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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)

1

10

100

1 10 100Distance along the jet axis, r/r j

Mac

h nu

mbe

r, M

gamma=5/3

gamma=1.4

gamma=9/7


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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)

V=v'/ξλ (14)

Θ=T'/ξλ (15)

X=r*'(0)/(r'ξω) (16)

ξ=4/[3Rej r*'(0)], ξ → 0 (17)

W=-Θ/[u0(γ-1)] - u0Θn/(r02X) (18)

Θ={γ(γ+1)(1-n)ω/(r02X)

+θ11-n(r0X)(1-ω)/ω}1/(1-n)

(19)

r02(∂V/∂X-V/X)-(γu0X)-1∂(r0

2Θ)/∂φ+nu0Θn-1(2X2)-1∂Θ/∂φ = 0

(20)

r0 =r*'(φ)/[r*'(0)] (21)

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.

1

10

100


Mac

h nu

mbe

r, M

gamma=5/3Ar,Rej=750He,Rej=250gamma=1.4N2,Rej=750gamma=9/7CO2,Rej=750


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Translational Relaxation in a Spherical Expanding Gas FlowParallel (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]

0.001

0.01

0.1

1

1 10 100Distance along the radius, r/r*

Tem

pera

ture

ratio

s, T

/T*

TTX,Kn=0.015TTY,Kn=0.015TTX,Kn=0.0015TTY,Kn=0.0015ideal gas


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Rotational Relaxation in a Freely Expanding Gas

1.E-06

1.E-05

1.E-04

1.E-03

10 100 1000

Temperature T, K

Para

met

er p

*tau

, kg/

(m*s

)

classicalquantum, j*=4quantum, j*=5quantum, j*=6experiment

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]


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Rotational temperature TR along the nitrogen-jet axis

Experimental data from Marrone (1967) and Rebrov (1976): K* = ρ*u*r*/(pτR)* = 2730, psrj = 240 torr·mm and Ts = 290 K (nitrogen)

10

100


Rot

atio

nal t

empe

ratu

re T

R ,

K

exper. [Rebrov]quantum, j*=6quantum, j*=5quantum, j*=4exper. [Marrone]Parker's model


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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*.

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25

Distance along the radius, r/r*

TR/T

0 an

d Tt

/T0

TR, P=17.6Tt, P=17.6

TR, P=78.9

Tt, P=78.9

TR, P=175.7

Tt, P=175.7

TR, inviscid

Tt, inviscid


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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.

1

10

100

1000

0.1 1 10

Hypersonic parameter, K

fx=(

Cx-

Cx o)

/sin

3 (a)

M=7.5,exper. M=7.5,DSMCM=9,exper. M=9,DSMCM=10.7,exper. M=10.7,DSMC


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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.

1

10

100

0.1 1 10Hypersonic parameter, K

fy=C

y/si

n2 (

a)

M=7.5,exper. M=7.5,DSMCM=9,exper. M=9,DSMCM=10.7,exper. M=10.7,DSMC


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Drag coefficient cx of the wedge (2θ = 40 deg) in helium at Re0 = 4 and various Mach numbers M∞

1.5

2

2.5

3

3.5

4

0 10 20 30 40

Angle of attack, deg

Dra

g co

effic

ient

, Cx M=9.9,exper. M=11.8,exper.

M>9,free-mol. M=9.9,DSMCM=11.8,DSMC


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Lift coefficient cy of the wedge (2θ = 40 deg) in helium at Re0 = 4 and various Mach numbers M∞.

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40

Angle of attack, deg

Lift

coe

ffic

ient

, Cy

M=9.9,exper. M=11.8,exper.M>9,free-mol. M=9.9,DSMCM=11.8,DSMC


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The influence of the specific heat ratio γDrag coefficient cx of the wedge (2θ = 40 deg) for various gases vs. the Reynolds number Re0.

1

1.25

1.5

1.75

2

2.25

2.5

1 10 100Reynolds number, Reo

Dra

g co

effic

ient

, Cx

exper., Ar

exper., N2

exper., CO2

DSMC, Ar

DSMC, CO2


41

The influence of the specific heat ratio γLift coefficient cy of the wedge (2θ = 40 deg) at Re0 = 3 and various gases.

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40Angle of attack

Lift

coe

ffic

ient

, Cy

N2,exper.Ar,exper.N2,DSMCAr,DSMC


42

The influence of the specific heat ratio γLift-drag ratio of the blunt plate (δ = 0.1) at α = 20 deg in Ar and N2 vs. Reynolds number Re0.

0

0.2

0.4

0.6

0.8

1

0.1 1 10 100Reynolds number, Reo

Lift

-dra

g ra

tio, L

/DN2,exper.N2,DSMCAr,DSMCN2,free-mol.Ar,free-mol.


43

The influence of the specific heat ratio γ:Drag coefficient cx of the disk at α = 90 deg for argon and nitrogen vs. the Reynolds number Re0

1.61.8

22.22.42.62.8

33.2

0.1 1 10 100

Reynolds number, Reo

Dra

g co

effic

ient

, Cx Ar,exper.

N2,exper.

Ar,DSMC

N2,DSMC

Ar,free-mol.N2,free-mol.

Ar,contin.

N2,contin.


44

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.

1.5

2.5

3.5

0.1 1 10Reynolds number, Reo

Dra

g co

effic

ient

, Cx

exper., Heexper., ArDSMC, Ar


45

The influence of the temperature factor twLift-drag ratio for the blunt plate (δ = 0.1) vs. Reynolds number Re0 in nitrogen at α = 20 deg and various tw

0

0.2

0.4

0.6

0.8

1

0.1 1 10 100Reynolds number, Reo

Lift

-dra

g ra

tio, L

/D

N2,exper.,tw=1N2,DSMC,tw=1N2,DSMC,tw=0.34N2,FM,tw=1N2,FM,tw=0.34N2,exper.,tw=0.34


46

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.


47

Drag of a blunt plate

0

0.1

0.20.3

0.4

0.5

0.6

0.01 0.1 1 10Knudsen Number, Kn

Dra

g C

oeffi

cien

t, C

x

DSMCexperimentfree-molecular

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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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.

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


49

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].

0

0.2

0.4

0.6

0 5 10 15 20 25 30 35 40 45Angle of Attack, deg

Lift-

Dra

g R

atio

, Y/X

M=9.9,exper. M=11.8,exper.M>9,FM regime M=9.9,DSMCM=11.8,DSMC


50

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].

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

0.01 0.1 1 10

Knudsen Number, Kn

Dra

g co

effic

ient

, Cx

Ar,exper.

N2,exper.

Ar,DSMC

N2,DSMC

Ar,FM regime

N2,FM regime

Ar,continuum

N2,continuum


51

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].

0.50.60.70.80.9

11.11.2


Dra

g co

effic

ient

, Cx

exper.,tw=1DSMC,tw=1DSMC,tw=0.34FM,tw=1FM,tw=0.34exper.,tw=0.34

Drag of a blunt plate: Tw effect


52

Lift of a blunt plate: Tw effect

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].

0.1

0.2

0.3

0.4

0.5


Lift

coef

ficie

nt, C

y



53

Pitch moment coefficient Cm0 for a blunt plate (δ = 0.1) vs Reynolds number Re0 in air and argon at α = 20 deg.

-0.35

-0.3

-0.25

-0.2

0.1 1 10 100Reynolds Number

Mom

ent C

oeffi

cien

t

Air,DSMC,tw=1 Air,DSMC,tw=0.34Air,FM,tw=1 Air,FM,tw=0.34Argon,DSMC,tw=1 Argon,FM,tw=1

Pitching moment of a blunt plate: Tw and γ effects


54

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].

0.5

1

1.5

2

2.5

0.1 1 10 100 1000

Reynolds Number

Dra

g C

oeffi

cien

t


Drag of a cone: Tw effect


55

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).


56

Lift and drag of two side-by-side plates

Drag and lift coefficients of the side-by-side plates vs. Knudsen number Kn,L at M = 10 in argon.

0

0.1

0.2

0.30.4

0.5

0.6

0.7


Dra

g C

oeff

icie

nt, C

x

Cx,H=0.25L Cx,H=0.5LCx,H=0.75L Cx,H=1.25LCx,FM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Lift

Coe

ffic

ient

, Cy

Cy,H=0.25LCy,H=0.5LCy,H=0.75Cy,H=1.25L


57

Aerocapture with large inflatable balloon-like decelerators (“ballutes”)

Titan Explore MissionFrom “Ballute Missions” (http://www2.jpl.nasa.gov/adv_tech/ballutes/missions.htm)


58

Mach number contours in nitrogen flow about a torus at Kn∞D = 0.01, M∞ = 10, and various geometrical factors H/R.

(a) H/R = 8

(b) H/R = 6

(c) H/R = 4

(d) H/R = 2

The strong influence of the geometrical factor


59

Mach number contours in flows of argon and carbon dioxide about a torus at Kn,D = 0.01, M = 10, and H/R=8.

Flow Patterns near a Torus in Various Gases

(a) argon flow (b) carbon dioxide flow


60

Contours of constant Mach numbers near a torus in the flow of nitrogen at Kn,D = 0.00013 and M = 7.11.

Comparing Numerical Results with Experimental Data

(a) DSMC calculation (b) experiment [Loirel, et al., ISSW, 2001]


61

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.

-0.5

0

0.5

1

1.5

2

2.5

0 60 120 180 240 300 360Angle, deg

Pres

sure

coe

ffici

ent,

C p

H/R=2 H/R=4

H/R=6 H/R=8

Pressure and skin-friction coefficients along the torus surface

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 60 120 180 240 300 360

Angle, deg

Ski

n-fri

ctio

n co

effic

ient

,C f H/R=2 H/R=4

H/R=6 H/R=8

(a) Pressure coefficient, Cp (b) Skin-friction coefficient, Cf


62

Mach number contours in nitrogen flow about a torus at Kn∞D = 1, M∞ = 10, and various geometrical factors H/R.

(a) H/R = 8

(b) H/R = 6

(c) H/R = 4

(d) H/R = 2

Rarefaction Effects: Influence of the Knudsen number, Kn∞,D


63

Skin-friction coefficient Cf along the torus surface in nitrogen flow at H/R = 2, M = 10, and various Knudsen numbers Kn,D.

Rarefaction Effects on Skin-friction Coefficient

-1.2-1

-0.8-0.6-0.4-0.2

00.20.40.60.8

1

0 60 120 180 240 300 360

Angle, deg

Skin

-fric

tion

coef

ficie

nt,C f

Kn=0.01 Kn=0.1Kn=1 Kn=4


64

Drag of a torus in different gases

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.

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

0.01 0.1 1 10Knudsen number, Kn

Dra

g co

effic

ient

, C x

Ar,H/R=8N2,H/R=8CO2,H/R=8Ar,H/R=2N2,H/R=2CO2,H/R=2Ar,cyl,FMN2,cyl,FMCO2,cyl,FM


65

The ballute model used in experiments [Loirel, et al., AIAA Paper, No. 2894, 2002].

Aerodynamics of Toroidal Ballute Models


66

The computational grid near the model used in simulations.

Flow Simulation near Toroidal Ballute Models


67

Flowfield near a toroidal ballute model

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.

a) dissociating oxygen flow b) perfect-gas oxygen flow


68

Aerodynamics of a spinning cylinder

• At subsonic flow conditions, the speed ratio S = M∞∙(0.5∙ )½ becomes small, and aerodynamic coefficients become very sensitive to its magnitude;

• Transition flow regime has been studied numerically at M∞ = 0.15, = 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 counter-clockwise 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.


69

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.

-20

-15

-10

-5

0

5

10

15

0.001 0.01 0.1 1 10

Knudsen Number

Lift

Coe

ffici

ent

W=1 W=3W=6 W=1,FMW=3,FM W=6,FM


70

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.

0

5

10

15

20

25

30

35

0.001 0.01 0.1 1 10Knudsen Number

Dra

g C

oeffi

cien

t

W=0W=1W=3W=6Free-Molecular


71

Lift coefficient Cy of a spinning cylinder vs Knudsen number KnD at supersonic Mach number, M∞ = 10 and different spin rates W in argon.

0

0.05

0.1

0.15

0.2

0.01 0.1 1 10Knudsen Number

Lift

Coe

ffici

ent

W=0.03W=0.1W=0.03,FMW=0.1,FM


72

Drag coefficient Cx of a spinning cylinder vs Knudsen number KnD at supersonic Mach number, M∞ = 10 and different spin rates W in argon.

1

1.5

2

2.5

3

0.01 0.1 1 10Knudsen Number

Dra

g C

oeffi

cien

t

W=0W=0.03W=0.1Free-Molecular


73

Heat flux at the stagnation point of a sphere

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].

0.1

1

0.001 0.01 0.1 1Knudsen number, Kn

Stan

ton

num

ber,

St TVSLexperimentDSMCN-St,slip

The slip effect is significant as the Knudsen number (~altitude) increases


74

Temperature jump at the Shuttle stagnation point atdifferent Knudsen numbers (after Moss and Bird [1985])

Velocity jump on the Shuttle at x=1.5m for U∞= 7.5 km/s at different Knudsen numbers (after Moss and Bird [1985])

Shuttle stagnation point heat flux versus Knudsennumber as predicted by NS, VSL, and, DSMC computations (after Gupta [1986])

Comparison of flight data with DSMC predictions


75

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.

Extent of Thermodynamic Nonequilibrium


76

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.

Comparison of temperature profiles using DSMC and Viscous Shock Layer models


77

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.

Predicting the heat flux using DSMC and Viscous Shock Layer models


78

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.

Effect of rarefaction on predicted drag using DSMC and Viscous Shock Layer models


79

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.

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 wind-tunnel 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.


80


81

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.


82

TVSL Equations of the Thin Viscous Shock Layer

∂(g2ρu)/∂x + ∂(g1g2ρv)/∂y = 0 (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)

ρ(ug1-1∂Al/∂x + v∂Al/∂y) = -∂Jl/∂y,

l=1, 2, ..., Nel-1 (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)


83

The generalized Rankine-Hugoniot conditions at the TVSL external boundary

sinχcosφ(cosχcosφ - u) = σμ∂u/∂y

(10)

ρv = - sinχcosφ (11)

sinχcosφ(sinφ - w) = σμ∂w/∂y (12)

p=sin2χcos2φ (13)

sinχcosφ(H∞- H) = -q +2σμ(u∂u/∂y+ w∂w/∂y)

(14)

sinχcosφ(Al∞- Al) = - Jl (15)

sinχcosφ(αi∞- αi) = - ji (16)

Here χ is a swept angle;φ is a shock incident angle.


84

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.


85

0.1

1

1 10 100 1000

Reynolds number, Reof

Stan

ton

num

ber,

St

TVSLexper.DSMCN-St,slipN-St,nonslipB-Layer

Stanton numbers St for a sphere vs. Reynolds numbers Reof for different medium models at various wind-tunnel experimental conditions [Chernikova, 1980; Nomura, 1974].


86

0.1

1

1 10 100 1000


Stan

ton

num

ber,

St

TVSL

exper.

Stanton numbers St for a cylinder vs. Reynolds numbers Reof for different medium models at various wind-tunnel experimental conditions [Chernikova, 1980; Nomura, 1974].


87

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

Distance along the spherical surface, s/R

Stan

ton

num

ber,

St

DSMCexper.TVSL

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]


88

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6

Distance along the normal, n/R

P/P o,

T/T o,

and

ρ/ρ o

p/po, N-StT/To, N-St0.1*DEN/DENo, N-Stp/po, TVSLT/To, TVSL0.1*DEN/DENo, TVSL

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.


89

Conditions & Calculation Results

h, km Reof

110 1.49

100 6.97

90 47.3

80 230

70 1220

60 5130

Table: Values of the Reynolds number for a sphere (R = 1m) along the Space Shuttle trajectory

REFERENCES:

Tong H, Buckingham A and Curry D. Computational procedure for evaluations of Space Shuttle TRS requirements. AIAA Paper 74-518, 1974.

Moss J and Bird G. Direct simulation of transitional flow for hypersonic reentry conditions. AIAA Paper No. 84-0223, 1984.

Reof = ρ∞U∞R/µ(Tof)


90

0.01

0.1

1

1 10 100 1000 10000


Stan

ton

num

ber,

St

equilibriumcatal.wallnon-catal.wallexper.N-St,slipSTS-2/3 data

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).


91

0

0.05

0.1

0.15

0.2

0.25

0 0.05 0.1 0.15 0.2 0.25

Distance along a normal coordinate, n/R

Mas

s co

ncen

trat

ions

O,noncatal. O,catal. N,noncatal.N,catal. NO,noncatal. NO,catal.

Effect of Surface CatalyticityMass concentrations αi of the air components in the TVSL at Reof = 230: dash lines - ideally catalytic surface; solid lines - absolutely noncatalytic surface.


92

0

5

10

15

20

25

30

35

1 10 100 1000 10000


Tof TsTd Toe

The values of temperatures Ts, Td, Tof, and Toe as functions of Reynolds number Reof.


93

0.01

0.1

1

1 10 100 1000 10000


d/R

The width of the catalytically influenced zone d as a function of Reynolds number Reof.


94

0

0.2

0.4

0.6

0.8

1

1 10 100 1000 10000


Mas

s co

ncen

trat

ions

O2 O N2

N NO

Mass concentrations αiw of air components on the noncatalytic vehicle surface.


95

500

750

1000

1250

1500

1750

2000

1 10 100 1000 10000


Wal

l tem

pera

ture

Tw

e, K

noncatal.wall

catal.wall

Equilibrium temperature Twe of the spherical surface (R = 1 m) vs. Reynolds numbers Reof.


96

500750

1000125015001750200022502500

0.1 1 10 100 1000


Wal

l tem

pera

ture

Tw

e, K

noncatal.wall

catal.wall

Equilibrium temperature Twe of the cylindrical surface (R = 0.1 m) vs. Reynolds numbers Reof.


97

0

500

1000

1500

2000

0 20 40 60 80

Swept angle χ , deg

Wal

l tem

pera

ture

Tw

e, K

Reo=6.97,noncat. Reo=6.97,catal.Reo=230,noncat. Reo=230,catal.Reo=5130,noncat. Reo=5130,catal.

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.


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.

0

0.1

0.2

0.3

0.4

0 0.01 0.02 0.03 0.04 0.05Distance along the central line, x/Rs

Tem

pera

ture

Rat

io

Rs=1m Rs=0.005m

98

The binary similitude law (ρ∞R = const)


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.

0

1

2

3

4

5


Elec

tron

Con

cent

ratio

n

R=1m R=0.005m

99


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.

0

0.5

1

1.5

2

2.5


Elec

tron

Con

cent

ratio

n

R=1m R=0.005m

100

Computation Fluid Dynamic Challenges


101

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.

CFD prediction of Hyper-X flow field at M∞ = 7 with engine operating. (Courtesy of NASA Dryden Flight Research Center).


Pressure ratio pw/po at the front stagnation point of a sphere vs correlation parameter Res(ρs/ρ∞)½.

0.95

1

1.05

1.1

1.15

10 100 1000 10000

Correlation Parameter

Pres

sure

Rat

io

N-St,gamma=1.4, tw=1 N-St,gamma=1.4, tw<1 N-St,gamma=5/3, tw=1N-St,gamma=5/3, tw<1 exper., gamma=1.4,tw=1 exper.,gamma=5/3,tw=1

102

Solutions of Navier-Stokes Equations: Pressure at the Stagnation Point


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).

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6Distance, x/Rs

Tem

pera

ture

Rat

ios

TR/To,Navier-Stokes T/To,Navier-StokesTt/To,Navier-Stokes TR/To,experiment

103

Rotational-Translational Relaxation in Viscous Flows of Nitrogen near a Sphere


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).

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25 0.3Distance, x/Rs

Tem

pera

ture

Rat

ios

TR/To,Navier-Stokes T/To,Navier-StokesTt/To,Navier-Stokes TR/To,experiment

104

Heat Transfer on a Hypersonic Blunt Bodies with Gas Injection


105

-0.2

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10Knudsen number, KnR

Stan

ton

num

ber,

St

DSMC,Gw=0DSMC,Air,Gw=0.94Exper.,Air,Gw=0

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])


106

Temperature contours at Kn ,R = 0.0163 (Re0,R = 92.8) and air-to-air mass blowing factor Gw = 0.7.

Influence of the Air Mass Blowing Factor Gw


107

Stanton number along the spherical surface at Kn ,R = 0.0163 (Re0,R = 92.8) and various air-to-air mass blowing factors.

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Coordinate along spherical surface, s/R

Stan

ton

num

ber,

St

Gw=0 Gw =0.15 Gw=0.32 Gw =0.55

Gw=0.70 Gw =0.94 Gw=1.5

Influence of the Air Mass Blowing Factor Gw


108

Stanton number along the spherical surface at Kn ,R = 0.0326, T0 = 1000 K and lower air-to-air mass blowing factors.

Comparison with Experimental Data (A. Botin [1987]): Air-to-Air Mass Blowing

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

Coordinate along spherical surface, s/R

Stan

ton

num

ber,

St

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


109

Width of the injection-influenced zone, (s/R)max vs Knudsen number at Gw = 0.94.

Influence of the Rarefaction Factor in Air Mass Blowing

00.05

0.10.150.2

0.250.3

0.35

0.01 0.1 1 10Knudsen number, KnR

DSMC,Air-to-Air,Gw=0.94


110

Contours of helium mole fraction at Kn ,R = 0.0163 (Re0,R = 92.8) and helium-to-air mass blowing factor Gw = 0.7.

Diffuse Injection of Helium into Air Stream


111

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.


112

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.

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Coordinate along spherical surface, s/RStan

ton

num

ber,

St

Gw=0 N2,Gw=0.32 N2,Gw=0.70

He,Gw=0.32 He,Gw=0.70

Influence of the Helium and Nitrogen Mass Blowing Factor Gw


113

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 10141;• V. V. Riabov and A. V. Botin, Journal of Thermophysics and

Heat Transfer, 1995; Vol. 9, No. 2, pp. 233-239.


114

Hypersonic Hydrogen Combustion in Thin Viscous Shock Layer (TVSL)


115

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 Navier-Stokes equations;

7 – The wedge.

NOTE: This pattern is for the Type III interference.


116

The Electron-Beam Fluorescence Technique for Flow Visualization

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.

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.

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.


117


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 shock-inclination angles, 20 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.


118


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.

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 .

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].


119

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 free-stream 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 finite-difference program is a second-order accurate scheme.


120

Mach number contours calculated by the Navier-Stokes solver at Re0 = 15.5 (KnR = 0.1) for different types of interaction




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


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.

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Loca

l Sta

nton

Num

ber,

St

-100 -50 0 50 100 Meridian Angle, deg

experiment Navier-Stokes No Interference

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Loca

l Sta

nton

Num

ber,

St




122





0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Loca

l Sta

nton

Num

ber,

St



0

0.05

0.1

0.15

0.2

0.25

Loca

l Sta

nton

Num

ber,

St




123



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].

0

0.05

0.1

0.15

0.2

0.25

Loca

l Sta

nton

Num

ber,

St




124

Mach Number Contours (Types II & III) obtained by the Direct Simulation Monte-Carlo Technique

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.


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.


125

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.



126

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.

0

0.2

0.4

0.6

0.8

1

1.2

1.4 Lo

cal S

tant

on N

umbe

r, S

t


No Interfer. Type II Type III Type V Type VI


127

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.

0

1

2

3

4

5

6 P

ress

ure

Coe

ffici

ent,

Cp


No Interfer. Type II Type III Type V Type VI


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Conclusions• The DSMC method is effective in studies of hypersonic rarefied flows of

various gases near probes in the transition regime between free-molecular 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.


129

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).


130

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. 276-284.

• 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.


131

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 High-Speed, High Knudsen Number Microchannel Flows,” Journal of Thermophysics and Heat Transfer, Vol. 11, No. 4, 1997, pp. 497-505.

• 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.


132

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 Simple-Shape 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.


133

References: Ballute Missions • McRonald A. A light-weight inflatable hypersonic drag device for

planetary entry. AIAA Paper, No. 99-0422, 1999.• Hall J and Le A. Aerocapture trajectories for spacecraft with large,

towed ballutes. AAS/AIAA Space Flight Mechanics Meeting, AAS Paper 01-235, February 2001.

• Rasheed A, Kujii K, Hornung H., and Hall J. Experimental investigation of the flow over a toroidal aerocapture ballute. AIAA Paper, No. 2460, 2001.

• Gnoffo P. Computational aerothermodynamics in aeroassist applications. Journal of Spacecraft and Rockets, Vol. 40, No. 3, pp. 305-312, 2003.

• McIntyre T, Lourel I, Eichmann T, Morgan R, Jacobs P and Bishop A. Experimental expansion tube study of the flow over a toroidal ballute. Journal of Spacecraft and Rockets, Vol. 41, No. 5, pp. 716-725, 2004.

• Moss J. Direct Simulation Monte Carlo simulations of ballute aerothermodynamics under hypersonic rarefied conditions. Journal of Spacecraft and Rockets, Vol. 44, No. 2, pp. 289-298, 2007.


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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.

RAREFACTION EFFECTS IN HYPERSONIC AERODYNAMICS

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