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Hypersonic High-Enthalpy Flow in a Leading-edge Separation
N. R. Deepak1 S. Gai1 J. N. Moss2 S. O Byrne1
1School of Engineering & ITUniversity of New South Wales
Australian Defence Force AcademyCanberra, Australia
2NASA Langley Research CenterHampton, USA
(University of New South Wales, Australian Defence Force Academy) 1 / 32
Introduction Motivation
Motivation
High Enthalpy Separated Flows
Understanding of aerothermodynamics - critical towards successful performance
Typical flow separation configurations - compression corner, flat-plate with steps,blunt bodies
These exhibit pre-existing boundary layer at separation - increasing complexity of theinteraction
Enthalpy range: 3.1 MJ/kg to 6.9 MJ/kg
Geometric Configuration
Capable of producing separation at leading-edge without pre-existing boundary layer
Originally proposed by Chapman et al. (1958) for high Re and low M flows
Considered here for laminar high enthalpy hypersonic conditions
Approach of the Problem
Time-accurate Navier-Stokes (N-S)
Direct Simulation Monte Carlo (DSMC)
(University of New South Wales, Australian Defence Force Academy) 2 / 32
Introduction Flow Features-Leading edge separation
Leading Edge Separation
Expansionfan
Separation (S)
Reattachmentshock
Flow
AD
Expansion fan
Flow separation
Recirculating region
Reattachment
Re-compression shock wave
Characterised by a strong expansion atthe leading edge
Flow separation very close to the leadingedge forming a recirculation regionbetween A, B and C
Reattachment on the compression surface
(University of New South Wales, Australian Defence Force Academy) 3 / 32
Introduction Scope
Scope of Research
Understanding of aerothermodynamics
For a unique flow configuration without any pre-existing boundary layer underhypersonic conditions
Using state-of-the art numerical techniques
To aid in designing the experiments based on numerical results
Testing of Chapmans isentropic recompression theory to estimate the base pressure
Background
Chapmans work for high Reynolds and low Mach numbers supersonic flows
No earlier reported work on the present configuration at hypersonic conditions
(University of New South Wales, Australian Defence Force Academy) 4 / 32
Computational Approach Navier-Stokes & Direct Simulation Monte Carlo
Numerical codes & Models
Navier-Stokes (N-S) Solver - Eilmer-3 (Jacobs and Gollan, 2010)
In-house solver, time-dependent, viscous, chemically reactive
Finite-volume, cell-centred, 3D/axisymmetric discretisation
Second order spatial accuracy: modified van Albada limiter and MUSCL
Mass, momentum & energy flux across the cells: AUSMDV algorithm
Time Integration: Explicit time integration
Direct Simulation Monte Carlo (DSMC) - DS2V (Bird, 2006)
Uses probabilistic (Monte Carlo) simulation to solve the Boltzmann equation
Models fluid flow using simulated molecules which represent a large number of realmolecules
(University of New South Wales, Australian Defence Force Academy) 5 / 32
Computational Approach Geometric Configurations
Leading edge separation
Geometry
x
y
S-1S-2
Le
s
s/Le=0
s/Le=1
s
s/Le=3.26
A
B
C
-1-2
Configuration Details
Surface Length, mm Angle
A B (S-1; expansion) 19.730 1=30.5
B C (S-2; compression) 44.776 2=23.7
Horizontal x distance from A C = 58 mmTotal wetted surface (s) length A 7 B 7 C = 64.506 mm
(University of New South Wales, Australian Defence Force Academy) 6 / 32
Computational Approach Geometric Configurations
Leading edge Flow conditions
T-ADFA freestream conditions
Flow Parameter Condition E Condition ATest gas Air AirRe [1/m] 1.34 106 2.43 105
M 9.66 7.25u [m/s] 2503 3730T [K ] 165 593p [Pa] 290 377
[kg/m3] 0.006 0.002
1.4 1.4
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Computational Approach Modelling Details
Modelling Details: Navier-Stokes
Perfect Gas
Air as calorically perfect & single species assumption
Viscosity and thermal conductivity modelled using Sutherland formulation
Real Gas: Chemical & Thermal nonequilibrium
Air as thermally perfect gas mixture (5 neutral species assumption)
Viscosity & thermal conductivity: Curve fits adopted from NASA CEA-Program(Extends beyond: 20000 K)
Transport property mixing: Gupta-Yos mixing rules
Chemical & thermal nonequilibrium: Guptas & Parks Two-temperature model
Translational-vibrational energy exchange: Landau - Teller equation
Vibrational relaxation time: Millikan & White empirical correlation
Wall Conditions
Wall temperature (Tw = 300 K); No-slip; Non-catalytic
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Computational Approach Modelling Details
Modelling Details: Navier-Stokes
Perfect Gas
Air as calorically perfect & single species assumption
Viscosity and thermal conductivity modelled using Sutherland formulation
Real Gas: Chemical & Thermal nonequilibrium
Air as thermally perfect gas mixture (5 neutral species assumption)
Viscosity & thermal conductivity: Curve fits adopted from NASA CEA-Program(Extends beyond: 20000 K)
Transport property mixing: Gupta-Yos mixing rules
Chemical & thermal nonequilibrium: Guptas & Parks Two-temperature model
Translational-vibrational energy exchange: Landau - Teller equation
Vibrational relaxation time: Millikan & White empirical correlation
Wall Conditions
Wall temperature (Tw = 300 K); No-slip; Non-catalytic
(University of New South Wales, Australian Defence Force Academy) 8 / 32
Computational Approach Modelling Details
Modelling Details: Navier-Stokes
Perfect Gas
Air as calorically perfect & single species assumption
Viscosity and thermal conductivity modelled using Sutherland formulation
Real Gas: Chemical & Thermal nonequilibrium
Air as thermally perfect gas mixture (5 neutral species assumption)
Viscosity & thermal conductivity: Curve fits adopted from NASA CEA-Program(Extends beyond: 20000 K)
Transport property mixing: Gupta-Yos mixing rules
Chemical & thermal nonequilibrium: Guptas & Parks Two-temperature model
Translational-vibrational energy exchange: Landau - Teller equation
Vibrational relaxation time: Millikan & White empirical correlation
Wall Conditions
Wall temperature (Tw = 300 K); No-slip; Non-catalytic
(University of New South Wales, Australian Defence Force Academy) 8 / 32
Computational Approach Modelling Details
Modelling Details: Direct simulation Monte Carlo
Real Gas: Chemical & Thermal nonequilibrium
Reacting air gas mixture (3 and 5 neutral species assumption)
Variable hard sphere (VHS) collision model
23 Chemical reactions are used for modelling chemistry
Energy exchange between translation, rotational, and vibrational internal energymodes
Wall Conditions
Wall temperature (Tw = 300 K)
Non-catalytic
Surface accommodation=1.0
Wall slip
(University of New South Wales, Australian Defence Force Academy) 9 / 32
Computational Approach Modelling Details
Modelling Details: Direct simulation Monte Carlo
Real Gas: Chemical & Thermal nonequilibrium
Reacting air gas mixture (3 and 5 neutral species assumption)
Variable hard sphere (VHS) collision model
23 Chemical reactions are used for modelling chemistry
Energy exchange between translation, rotational, and vibrational internal energymodes
Wall Conditions
Wall temperature (Tw = 300 K)
Non-catalytic
Surface accommodation=1.0
Wall slip
(University of New South Wales, Australian Defence Force Academy) 9 / 32
Computational Approach Modelling Details
Modelling Details: Direct simulation Monte Carlo
Real Gas: Chemical & Thermal nonequilibrium
Reacting air gas mixture (3 and 5 neutral species assumption)
Variable hard sphere (VHS) collision model
23 Chemical reactions are used for modelling chemistry
Energy exchange between translation, rotational, and vibrational internal energymodes
Wall Conditions
Wall temperature (Tw = 300 K)
Non-catalytic
Surface accommodation=1.0
Wall slip
(University of New South Wales, Australian Defence Force Academy) 9 / 32
Computational Approach Grid Independence Study
Grid independence study-Leading edge separation
Grid i j wGrid-1 90 20 100 mGrid-2 185 40 50 mGrid-3 315 64 25 mGrid-4 466 90 20 mGrid-5 571 108 20 mGrid-6 703 108 20 mGrid-7 894 108 20 m
(University of New South Wales, Australian Defence Force Academy) 10 / 32
Computational Approach Grid Independence Study
Grid sensitivity, Condition E, Ho = 3.1 MJ/kg
1.0E+04
1.0E+05
1.0E+06
0 0.5 1 1.5 2 2.5 3 3.5
q w,W
/m
2
s/Le
Grid-1 (90X20)Grid-2 (185X40)Grid-3 (315X64)Grid-4 (440X90)Grid-5 (571X108)Grid-6 (703X108)
Perfect gas analysis
Heat flux, skin friction &pressure criteria
0 s/Le 0.15: not muchvariation
Downstream: significantvariation at peak location
Separation, reattachment &peak heat flux location: gridsensitive
Grid-5 (G-5)-chosen grid; total nodes: 61668; w = 20m
Separated flow establishment time: 1000 s
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Computational Approach Grid Independence Study
Grid sensitivity, Condition A, Ho = 6.9 MJ/kg
1.0E+04
1.0E+05
1.0E+06
0 0.5 1 1.5 2 2.5 3 3.5
q w,W
/m
2
s/Le
Grid-1 (90X20)Grid-2 (185X40)Grid-3 (315X64)Grid-4 (440X90)Grid-5 (571X108)Grid-6 (703X108)Grid-7 (894X108)
Perfect gas analysis
Heat flux, skin friction &pressure criteria
0 s/Le 0.5: not muchvariation
Downstream: Variation at peaklocation
Separation, reattachment &peak heat flux location: gridsensitive
Grid-5 (G-5)-chosen grid; total nodes: 61668; w = 20m
Separated flow establishment time: 650 s
(University of New South Wales, Australian Defence Force Academy) 12 / 32
Results Results: Condition E
Pressure, Skin-friction & heat flux: Condition E Ho=3.1 MJ/kg
0.0
0.1
1.0
10.0
100.0
0 0.5 1 1.5 2 2.5 3 3.5
p/p
s/Le
1.0
1.5
2.0
0.6 0.8 1 1.2
Navier-StokesDSMC
-80.0
0.0
80.0
160.0
240.0
320.0
400.0
480.0
0 0.5 1 1.5 2 2.5 3 3.5
,N/
m2
s/Le
-10
-5
0
5
0.6 0.8 1 1.2
Navier-StokesDSMC
1.0E+04
1.0E+05
1.0E+06
0 0.5 1 1.5 2 2.5 3
q w,W
/m
2
s/Le
3E+03
8E+03
1E+04
2E+04
0.6 0.8 1 1.2
Navier-StokesDSMC
Strong expansion at leading edge;followed by flow separation
Separation: N-S (s/Le = 0.145);DSMC (s/Le = 0.08)
Reattachment: N-S (s/Le = 2.42);DSMC (s/Le = 2.46)
(University of New South Wales, Australian Defence Force Academy) 13 / 32
Results Results: Condition E
Chapmans interpretation
Expansionfan
Separation (S)
Recirculation region
Reattachmentshock
Flow (M>>1)
LsReattachment (R)
Expansionfan
Separation (S)
Recirculation region
Reattachmentshock
Flow (M>>1)
Reattachment (R)
Ls 0
(a) Compression corner (b) Leading edge separation
Leading edge separation is a limiting case of separation at a compression corner
Separation distance (Ls) from the leading edge goes to zero
(University of New South Wales, Australian Defence Force Academy) 14 / 32
Results Results: Condition A
Pressure, Skin-friction & heat flux: Condition A Ho=6.9 MJ/kg
0.00
0.01
0.10
1.00
10.00
0 0.5 1 1.5 2 2.5 3 3.5
p/p
s/Le
0.0
0.4
0.8
0.2 0.4 0.6 0.8 1 1.2
Navier-StokesDSMC -40.0
0.0
40.0
80.0
120.0
160.0
200.0
240.0
280.0
320.0
0 0.5 1 1.5 2 2.5 3 3.5
,N/
m2
s/Le
-20
-10
0
10
20
0.6 0.8 1 1.2
Navier-StokesDSMC
1.0E+04
1.0E+05
1.0E+06
0 0.5 1 1.5 2 2.5 3 3.5
q w,W
/m
2
s/Le
1.0E+04
6.0E+04
1.1E+05
0.6 0.8 1 1.2
Navier-StokesDSMC
Between 0.05 s/Le 0.25 in N-S,rate of pressure reduction decreaseswith near constant pressure:Indicative of boundary layer growth
N-S: Separation at s/Le = 0.56;Reattachment at s/Le = 1.87
DSMC: No indication ofseparation/reattachment
(University of New South Wales, Australian Defence Force Academy) 15 / 32
Results Results: Further remarks
Differences between N-S and DSMC
Significant differences between N-S and DSMC for condition A
DSMC predicts almost no separation (except for an infinitesimally small region at the corner)
Flow over most of the expansion surface is in slip flow regime (Moss et al., 2012)
slip velocity = uw (s) = w
(
u
y
)
w
=w
ww (s)
slip temperature = Tg Tw = (T )w =2
+ 1(w cp)
1w k
(
dT
dy
)
w
Rarefaction parameter(V ) =M
Res
C ;Res =
us
;C =
w w
ee
Criterion for slip flow (Talbot, 1963) for Condition A
Location Rarefaction parameter, V Rarefaction parameter, VN-S DSMC
s/Le = 0.25 0.143 0.332s/Le = 0.5 0.1009 0.223s/Le = 1.0 0.0715 0.166
(University of New South Wales, Australian Defence Force Academy) 16 / 32
Results Knudsen number
Local Knudsen (Kn) number - Condition A
(a) DSMC (b) Navier-Stokes
The local Knudsen number Kn is defined by Bird (see Moss et al. (2012)) in terms oflocal density gradients in the flow:
Kn =
{
(
x
)2+
(
y
)2}1/2
local
(University of New South Wales, Australian Defence Force Academy) 17 / 32
Results Knudsen number
Local Knudsen (Kn) number - Condition E
(c) DSMC (d) Navier-Stokes
The local Knudsen number Kn is defined by Bird (see Moss et al. (2012)) in terms oflocal density gradients in the flow:
Kn =
{
(
x
)2+
(
y
)2}1/2
local
(University of New South Wales, Australian Defence Force Academy) 18 / 32
Results Density comparison
Density comparison: Double cone vs Leading edge
0.1
1.0
10.0
100.0
0 0.5 1
/
s/L
Leading-edgeDouble-cone (N. R. Deepak, 2010)
The effect of expansion on wall density in comparison to the effect of compression
Density difference - a factor of 100 between expansion and compression on theforebody
(University of New South Wales, Australian Defence Force Academy) 19 / 32
Results Comparison with Champans Theory
Chapmans isentropic re-compression theory
Chapman et al. (1958) proposed a separated flow model and developed a theory toestimate the base pressure (Pd)
Experimental evidence at high supersonic Mach numbers suggests that the modelworks remarkably well even for pre-existing boundary layer in estimating basepressure
In hypersonic high temperature flows, the efficacy of Chapmans isentropicrecompression model is not rigorously verified
Here, the same leading edge separation model used by Chapman is considered
(University of New South Wales, Australian Defence Force Academy) 20 / 32
Results Comparison with Champans Theory
Comparison with Chapmans isentropic re-compression theory
From Navier-Stokes Simulations
Average pressure in the recirculation region or dead air region (Pd)
Pressure (P ) and Mach number (M) downstream of reattachment
Mach number at the edge of the mixing layer (Me)
M2 = (1 u2d )M2e and u
d = ud/ue
pdp
=
[
1 + ( 12
)M2
1 + ( 12
)M2/(1 u2d )
]/(1)
Flow pd/p pd/p
pd/p ReLe =
uLe
condition N-S simulations DSMC simulations Theory -
E 0.088 0.09 0.330 32.88 103
A 0.08 - 0.345 8.14 103
Simple isentropic flow assumption does not appear to hold in hypersonic flow
Streamlines in the shear layer do not recompress isentropically at reattachment, ratherextend over a finite region
Steep isentropic recompression assumption in theory seems unrealistic in low Re flows withthick shear layers
(University of New South Wales, Australian Defence Force Academy) 21 / 32
Results Dividing streamline profile
Dividing streamline profile (ud vs S)
S = S/Sw
S =
x
0
Cseueey2c dx
Sw =
s
0
CSw eueey2c ds
S is the reduced streamwise distance measured from separation to reattachmentalong the free shear layer
CS =
eeand CSW =
w wee
Edge conditions (e, ue , e) in evaluating S and Cs are obtained on the streamlinerunning between 2 and 3
Edge conditions (e, ue , e) in evaluating Sw and CSw are obtained on thestreamline running between 1 and 2
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Results Dividing streamline profile
Dividing streamline (ud vs S)
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
103 102 101 100 101 102 103
u d
S
S = 0: separationR: reattachment
R R R
ud = 0.587 (Chapman, 1958)Denison & Baum (1963)N-S (Cond E)N-S (Cond A)DSMC: axisym (Hruschka, 2010)Expt: axisym (Hruschka, 2010)N-S: cylinder (Park, 2012)
ud profile in agreement with other data
ud for current data does not reach Champans value of u
d = 0.587
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Results Base pressure vs Mach number
pd/p vs M
0.00
0.20
0.40
0.60
0.80
1.00
1 1.5 2 2.5 3 3.5 4 4.5 5
(pd/
p)
M
isentropic (ud = 0.587)N-S-Cond AN-S-Cond EDSMC-Cond Eud = 0.26-Cond Aud = 0.53-Cond E
Correlates well only under isentropic assumption
Numerical data indicate the recompression and pressure rise is strong dependent onviscous effects
(University of New South Wales, Australian Defence Force Academy) 24 / 32
Results Body normal profiles
Body normal profile - Details
The u and v velocities obtained from the data lines have been resolved in parallel(Up) and normal (Un) components
Expansion surface with an angle () of 30.465
Parallel velocity: Up = u cos() v sin()Normal velocity: Un = u sin() + v cos()
Compression surface with an angle ( ) of 23.702
Parallel velocity: Up = u cos() + v sin()Normal velocity: Un = u sin() + v cos()
y is normalised with the boundary layer thickness () at separationCondition E: N-S 2.5 mm; DSMC 1.4 mmCondition A: N-S 8.0 mm; DSMC : since no separation, N-S value is used
(University of New South Wales, Australian Defence Force Academy) 25 / 32
Results Body normal profiles
Body normal profile: Condition E
0.00
0.01
0.10
1.00
10.00
-500 0 500 1000 1500 2000 2500 3000
y/
Up, Parallel velocity (m/s)
Expansion surface-N-SExpansion surface-DSMC
Vertex-N-SVertex-DSMC
Compression surface-N-SCompression surface-DSMC
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
-1500 -1000 -500 0 500 1000 1500
y/
Un, Normal velocity (m/s)
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
0.1 1 10 100
y/
p/p
Expansion surface-N-SExpansion surface-DSMC
Vertex-N-SVertex-DSMC
Compression surface-N-SCompression surface-DSMC
0.00
0.01
0.10
1.00
10.00
1 10
y/
T/T
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Results Body normal profiles
Body normal profile: Condition A
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
-500 0 500 1000 1500 2000 2500 3000 3500 4000
y/
Up, Parallel velocity (m/s)
Expansion surface-N-SExpansion surface-DSMCVertex-N-SVertex-DSMCCompression surface-N-SCompression surface-DSMC
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
-2000 -1500 -1000 -500 0 500 1000 1500 2000
y/
Un, Normal velocity (m/s)
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
0.01 0.1 1 10
y/
p/p
Expansion surface-N-SExpansion surface-DSMCVertex-N-SVertex-DSMCCompression surface-N-SCompression surface-DSMC
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
1 10
y/
T/T
Expansion surface-N-SExpansion surface-DSMC
Vertex-N-SVertex-DSMC
Compression surface-N-SCompression surface-DSMC
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Results Streamlines
Streamlines, separation and reattachment angles
(e) Condition E (f) Condition A
Comparison of measured angle with Oswatitsch (1957) theory
tans = limx,y0(
vu
)
= 3(
dw /dsdpw /ds
)
s
Angle Condition A Condition A Condition E Condition E- Theory Measured Theory Measured
Separation 37 35 47 40
Reattachment 7.5 10 1.4 4
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Computational Visualisation
Computational Visualisation: Resultant velocity
Condition E and Condition A
(a) Navier-Stokes (b) Direct simulation Monte Carlo
(c) Navier-Stokes (d) Direct simulation Monte Carlo
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Conclusions
Conclusions
Numerical simulations of a unique configuration with no pre-existing boundary layerusing N-S and DSMC under hypersonic flow conditions
This has been attempted for the first time under hypersonic flow conditions
Lower enthalpy (higher freestream density) flow condition E : DSMC predicted a largerseparated region by about 15%. Pressure, shear stress and heat flux show similar features.
Higher enthalpy (lower freestream density) flow condition A : N-S results predicted a clearlyseparated region whereas the DSMC gave no indication of existence of a separated region
Although DSMC indicated shear stress values very close to zero, over whole of theexpansion surface, they were still distinctly positive
No indication of separation with the DSMC for condition A is attributed to the fact that theDSMC calculations take slip effects into account
Rarefaction effects resulting from the leading edge expansion are strong and could delayseparation further down the expansion surface
The assumption of no-slip in N-S may be inadequate for this configuration with condition A
Isentropic recompression theory of Chapman may not be adequate in hypersonic highenthalpy low Reynolds number flows
(University of New South Wales, Australian Defence Force Academy) 30 / 32
Thanks and Acknowledgements
Thank you
Acknowledgements
Dr. Peter Jacobs (University of Queensland)
UNSW Silver Star Research Grant
For more information
Prof. Sudhir [email protected]
Dr. Deepak [email protected]
School of Engineering & ITUniversity of New South WalesAustralian Defence Force AcademyCanberra, Australia
(University of New South Wales, Australian Defence Force Academy) 31 / 32
References
References
Bird, G. A. (2006), DS2V: Visual DSMC Program for Two-Dimensional and AxiallySymmetric Flows.URL: http://gab.com.au/index.html
Chapman, D. R., Kuehn, D. M. and Larson, H. K. (1958), Investigation of SeparatedFlows in Supersonic and Subsonic Streams with Emphasis on the Effect of Transition,Technical Report 1356, NACA.
Jacobs, P. A. and Gollan, R. J. (2010), The Eilmer3 Code, Technical Report Report2008/07, Department of Mechanical Engineering, University of Queensland.
Moss, J. N., O Byrne., S., Deepak, N. R. and Gai, S. L. (2012), Simulation ofHypersonic, High-Enthalpy Separated Flow over a Tick Configuration, 28thInternational Symposium on Rarefied Gas Dynamics, Zaragoza, July 9-13th, 2012.
Oswatitsch, K. (1957), Die Ablosungsbedingung von Grenzschichten, in Boundary LayerResearch: International Union of Theoretical and Applied Mechanics Symposium,Freiburg, SpringerVerlag, Berlin, pp. 357367.
Talbot, L. (1963), Criterion for Slip near the Leading Edge of a Flat Plate in HypersonicFlow, AIAA Journal 1(5), 11691171.
(University of New South Wales, Australian Defence Force Academy) 32 / 32
IntroductionMotivationFlow Features-Leading edge separationScopeComputational ApproachNavier-Stokes & Direct Simulation Monte CarloGeometric ConfigurationsGeometric ConfigurationsModelling DetailsModelling DetailsGrid Independence StudyGrid Independence StudyGrid Independence StudyResultsResults: Condition EResults: Condition EResults: Condition AResults: Further remarksKnudsen numberKnudsen numberDensity comparisonComparison with Champan's TheoryComparison with Champan's TheoryDividing streamline profileDividing streamline profileBase pressure vs Mach numberBody normal profilesBody normal profilesBody normal profilesStreamlinesComputational VisualisationConclusionsThanks and AcknowledgementsReferences