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

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)


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


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


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


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


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


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


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


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



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



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


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)


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


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


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


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


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


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


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


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


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


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


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


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



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




0.00

0.01

0.10

1.00

10.00

1 10

y/

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/

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





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


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 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


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


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.

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