Hypersonic high enthalpy flow in a leading-edge separation

Hypersonic High-Enthalpy Flow in a Leading-edge Separation
N. R. Deepak1
S. Gai1
J. N. Moss2
S. O’ Byrne1
1
School of Engineering & IT
University of New South Wales
Australian Defence Force Academy
Canberra, Australia
2
NASA Langley Research Center
Hampton, 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 the
interaction
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)

Introduction Flow Features-Leading edge separation
Leading Edge Separation
Expansion
fan
Separation (S)
Reattachment
shock
Flow
A
D
Expansion fan
Flow separation
Recirculating region
Reattachment
Re-compression shock wave
Characterised by a strong expansion at
the leading edge
Flow separation very close to the leading
edge forming a recirculation region
between A, B and C
Reattachment on the compression surface

Introduction Scope
Scope of Research
Understanding of aerothermodynamics
For a unique flow configuration without any pre-existing boundary layer under
hypersonic conditions
Using state-of-the art numerical techniques
To aid in designing the experiments based on numerical results
Testing of Chapman’s isentropic recompression theory to estimate the base pressure
Background
Chapman’s work for high Reynolds and low Mach numbers supersonic flows
No earlier reported work on the present configuration at hypersonic conditions

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 real
molecules

Computational Approach Geometric Conﬁgurations
Leading edge separation
Geometry
x
y
S-1
S-2
Le
s
s/Le=0
s/Le=1
s
s/Le=3.26
A
B
C
θ-1
θ-2
Conﬁguration Details
Surface Length, mm Angle
A ←→ B (S-1; expansion) 19.730 θ − 1=30.5◦
B ←→ C (S-2; compression) 44.776 θ − 2=23.7◦
Horizontal x distance from A ←→ C = 58 mm
Total wetted surface (s) length A −→ B −→ C = 64.506 mm

Computational Approach Geometric Conﬁgurations
Leading edge Flow conditions
T-ADFA freestream conditions
Flow Parameter Condition E Condition A
Test gas Air Air
Re [1/m] 1.34 × 106
2.43 × 105
M∞ 9.66 7.25
u∞ [m/s] 2503 3730
T∞ [K] 165 593
p∞ [Pa] 290 377
ρ∞ [kg/m3
] 0.006 0.002
γ 1.4 1.4

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: Gupta’s & Park’s 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

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 energy
modes
Wall Conditions
Wall temperature (Tw = 300 K)
Non-catalytic
Surface accommodation=1.0
Wall slip

Computational Approach Grid Independence Study
Grid independence study-Leading edge separation
Grid i × j △w
Grid-1 90 × 20 100 µm
Grid-2 185 × 40 50 µm
Grid-3 315 × 64 25 µm
Grid-4 466 × 90 20 µm
Grid-5 571 × 108 20 µm
Grid-6 703 × 108 20 µm
Grid-7 894 × 108 20 µm

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
qw,W/m2
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 much
variation
Downstream: significant
variation at peak location
Separation, reattachment &
peak heat flux location: grid
sensitive
Grid-5 (G-5)-chosen grid; total nodes: 61668; △w = 20µm
Separated flow establishment time: ≈ 1000 µs

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
qw,W/m2
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 much
variation
Downstream: Variation at peak
location
Separation, reattachment &
peak heat flux location: grid
sensitive
Grid-5 (G-5)-chosen grid; total nodes: 61668; △w = 20µm
Separated flow establishment time: ≈ 650 µs

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-Stokes
DSMC
-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
τ,N/m2
s/Le
-10
-5
0
5
0.6 0.8 1 1.2
Navier-Stokes
DSMC
1.0E+04
1.0E+05
1.0E+06
0 0.5 1 1.5 2 2.5 3
qw,W/m2
s/Le
3E+03
8E+03
1E+04
2E+04
0.6 0.8 1 1.2
Navier-Stokes
DSMC
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)

Results Results: Condition E
Chapman’s interpretation
Expansion
fan
Separation (S)
Recirculation region
Reattachment
shock
Flow (M>>1)
Ls
Reattachment (R)
Expansion
fan
Separation (S)
Recirculation region
Reattachment
shock
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

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-Stokes
DSMC -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
τ,N/m2
s/Le
-20
-10
0
10
20
0.6 0.8 1 1.2
Navier-Stokes
DSMC
1.0E+04
1.0E+05
1.0E+06
0 0.5 1 1.5 2 2.5 3 3.5
qw,W/m2
s/Le
1.0E+04
6.0E+04
1.1E+05
0.6 0.8 1 1.2
Navier-Stokes
DSMC
Between 0.05 ≤ s/Le ≤ 0.25 in N-S,
rate of pressure reduction decreases
with 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 of
separation/reattachment

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) = λw
∂u
∂y w
=
λw
µw
τw (s)
slip temperature = Tg − Tw = (∆T)w =
2γ
γ + 1
(µw cp)−1
λw k
dT
dy w
Rarefaction parameter(V ) =
M
√
Res
√
C ; Res =
ρus
µ
; C =
ρw µw
ρeµe
Criterion for slip flow (Talbot, 1963) for Condition A
Location Rarefaction parameter, V Rarefaction parameter, V
N-S DSMC
s/Le = 0.25 0.143 0.332
s/Le = 0.5 0.1009 0.223
s/Le = 1.0 0.0715 0.166

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 of
local density gradients in the ﬂow:
Kn =



λ ∂ρ
∂x
2
+ ∂ρ
∂y
2
1/2
ρ



local

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 of
local density gradients in the ﬂow:
Kn =



λ ∂ρ
∂x
2
+ ∂ρ
∂y
2
1/2
ρ



local

Results Density comparison
Density comparison: Double cone vs Leading edge
0.1
1.0
10.0
100.0
0 0.5 1
ρ/ρ∞
s/L
Leading-edge
Double-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 the
forebody

Results Comparison with Champan’s Theory
Chapman’s isentropic re-compression theory
Chapman et al. (1958) proposed a separated flow model and developed a theory to
estimate the base pressure (Pd )
Experimental evidence at high supersonic Mach numbers suggests that the model
works remarkably well even for pre-existing boundary layer in estimating base
pressure
In hypersonic high temperature flows, the efficacy of Chapman’s isentropic
recompression model is not “rigorously” verified
Here, the same leading edge separation model used by Chapman is considered

Results Comparison with Champan’s Theory
Comparison with Chapman’s 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)
M′2
= (1 − u∗2
d )M2
e and u∗
d = ud /ue
pd
p′
=
1 + (γ−1
2
)M′2
1 + (γ−1
2
)M′2/(1 − u∗2
d )
γ/(γ−1)
Flow pd /p′
pd /p′
pd /p′
Re′
Le = u′ρ′Le
µ′
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, rather
extend over a finite region
Steep isentropic recompression assumption in theory seems unrealistic in low Re flows with
thick shear layers

Results Dividing streamline proﬁle
Dividing streamline proﬁle (u∗
d vs S∗)
S∗
= S/Sw
S =
x
0
Cs ρeueµey2
c dx
Sw =
s
0
CSw ρeueµe y2
c ds
S is the reduced streamwise distance measured from separation to reattachment
along the free shear layer
CS =
ρµ
ρeµe
and CSW
=
ρw µw
ρeµe
Edge conditions (ρe, ue, µe ) in evaluating S and Cs are obtained on the streamline
running between 2 and 3
Edge conditions (ρe, ue, µe ) in evaluating Sw and CSw are obtained on the
streamline running between 1 and 2

Results Dividing streamline proﬁle
Dividing streamline (u∗
d vs S∗)
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
10−3 10−2 10−1 100 101 102 103
u∗
d
S∗
S∗ = 0: separation
R: reattachment
R R R
u∗
d = 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)
u∗
d proﬁle in agreement with other data
u∗
d for current data does not reach Champan’s value of u∗
d = 0.587

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 (u∗
d = 0.587)
N-S-Cond A
N-S-Cond E
DSMC-Cond E
u∗
d = 0.26-Cond A
u∗
d = 0.53-Cond E
Correlates well only under isentropic assumption
Numerical data indicate the recompression and pressure rise is strong dependent on
viscous eﬀects

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 (α) of 30.465◦
Parallel velocity: Up = u · cos(α) − v · sin(α)
Normal velocity: Un = u · sin(α) + v · cos(α)
Compression surface with an angle (β ) of 23.702◦
Parallel velocity: Up = u · cos(β) + v · sin(β)
Normal velocity: Un = −u · sin(β) + v · cos(β)
y is normalised with the boundary layer thickness (δ) at separation
Condition E: N-S ≈ 2.5 mm; DSMC ≈ 1.4 mm
Condition A: N-S ≈ 8.0 mm; DSMC : since no separation, N-S value is used

Body normal proﬁle: Condition E
0.00
0.01
0.10
1.00
10.00
-500 0 500 1000 1500 2000 2500 3000
y/δ
Up, Parallel velocity (m/s)
Expansion surface-N-S
Expansion surface-DSMC
Vertex-N-S
Vertex-DSMC
Compression surface-N-S
Compression 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/δ
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/δ
p/p∞
Vertex-N-S
Vertex-DSMC
0.00
0.01
0.10
1.00
10.00
1 10
y/δ
T/T∞

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/δ
Up, Parallel velocity (m/s)
Vertex-N-S
Vertex-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/δ
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/δ
p/p∞
Vertex-N-S
Vertex-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/δ
T/T∞
Vertex-N-S
Vertex-DSMC

Results Streamlines
Streamlines, separation and reattachment angles
(e) Condition E (f) Condition A
Comparison of measured angle with Oswatitsch (1957) theory
tanθs = limx,y→0
v
u
= −3 dτw /ds
dpw /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◦

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

Conclusions
Conclusions
Numerical simulations of a unique configuration with no pre-existing boundary layer
using 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 larger
separated 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 clearly
separated 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 the
expansion surface, they were still distinctly positive
No indication of separation with the DSMC for condition A is attributed to the fact that the
DSMC calculations take slip effects into account
Rarefaction effects resulting from the leading edge expansion are strong and could delay
separation 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 high
enthalpy low Reynolds number flows

Thanks and Acknowledgements
Thank you
Acknowledgements
Dr. Peter Jacobs (University of Queensland)
UNSW Silver Star Research Grant
For more information
Prof. Sudhir Gai
s.gai@adfa.edu.au
Dr. Deepak Ramanath
d.ramanath@adfa.edu.au
School of Engineering & IT
University of New South Wales
Australian Defence Force Academy
Canberra, Australia

References
References
Bird, G. A. (2006), ‘DS2V: Visual DSMC Program for Two-Dimensional and Axially
Symmetric Flows’.
URL: http://gab.com.au/index.html
Chapman, D. R., Kuehn, D. M. and Larson, H. K. (1958), Investigation of Separated
Flows 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 Report
2008/07, Department of Mechanical Engineering, University of Queensland.
Moss, J. N., O’ Byrne., S., Deepak, N. R. and Gai, S. L. (2012), Simulation of
Hypersonic, High-Enthalpy Separated Flow over a ’Tick’ Configuration, 28th
International Symposium on Rarefied Gas Dynamics, Zaragoza, July 9-13th, 2012.
Oswatitsch, K. (1957), Die Ablosungsbedingung von Grenzschichten, in ‘Boundary Layer
Research: International Union of Theoretical and Applied Mechanics Symposium,
Freiburg’, Springer–Verlag, Berlin, pp. 357–367.
Talbot, L. (1963), ‘Criterion for Slip near the Leading Edge of a Flat Plate in Hypersonic
Flow’, AIAA Journal 1(5), 1169–1171.

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