Research ArticleDSMC Simulation and Experimental Validation ofShock Interaction in Hypersonic Low Density Flow

Hong Xiao Yuhe Shang and Di Wu

School of Power and Energy Northwestern Polytechnical University Xirsquoan Shaanxi 710072 China

Correspondence should be addressed to Hong Xiao xhongnwpueducn

Received 6 August 2013 Accepted 11 October 2013 Published 30 January 2014

Academic Editors G Liang and S Longo

Copyright copy 2014 Hong Xiao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Direct simulationMonte Carlo (DSMC) of shock interaction in hypersonic low density flow is developedThree collisionmolecularmodels including hard sphere (HS) variable hard sphere (VHS) and variable soft sphere (VSS) are employed in the DSMC studyThe simulations of double-cone and Edneyrsquos type IV hypersonic shock interactions in low density flow are performed Comparisonsbetween DSMC and experimental data are conducted Investigation of the double-cone hypersonic flow shows that three collisionmolecularmodels can predict the trend of pressure coefficient and the StantonnumberHSmodel shows the best agreement betweenDSMC simulation and experiment among three collision molecular models Also it shows that the agreement between DSMC andexperiment is generally good for HS and VHS models in Edneyrsquos type IV shock interaction However it fails in the VSS modelBoth double-cone and Edneyrsquos type IV shock interaction simulations show that the DSMC errors depend on the Knudsen numberand the models employed for intermolecular interaction With the increase in the Knudsen number the DSMC error is decreasedThe error is the smallest in HS compared with those in the VHS and VSS models When the Knudsen number is in the level of 10minus4the DSMC errors for pressure coefficient the Stanton number and the scale of interaction region are controlled within 10

1 Introduction

Shock interactions play an important role in hypersonicflows In the past decade significant efforts in computa-tional fluid dynamics (CFD) have been exerted to developprediction techniques for simulating these complex flowstructures [1ndash3] Most of these works were simulated usingthe continuum transport equations (Navier-Stokes) whichdescribe the transport of mass momentum and energyThese equations are based on the hypothesis that the meanfree-path length 120582 of gas molecules is very small comparedwith the characteristic dimension 119871 of the flow This char-acteristic dimension can be defined by the flowing gradientlength scale

119871 =120601

(120597120601120597119909) (1)

Here 120601 is the flow property and it can be density or pressureand so on Also the Knudsen number Kn is usually used todescribe low density flow

Kn =120582

119871 (2)

Flows with Kn gt 10 are called ldquofree molecular flowsrdquo Inthis regime intermolecular collisions rarely occur and theflow is completely dominated by the interaction between thegas and the wall Flows in the free molecular regime canbe simulated using molecular dynamics (MD) or ballisticmodels In the intermediate (001 lt Kn lt 10) or rarefiedregime both collisions with solid surfaces and with gasmolecules are important and therefore have to be includedin the simulation to obtain an accurate result The directsimulation Monte Carlo (DSMC) method developed by Bird[4] is the perfect practical engineering method that can beused in the rarefied regime It is also valid in the free-molecular and continuum regimes although computationalexpenses become very large in the latter case Its compu-tational expenses in fact scale with the Knudsen numberand become prohibitively large when Kn becomes lower thansim005 Usually gas flow can be simulated for Kn lt 001

using continuum-based CFD models for Kn gt 005 using

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 732765 10 pageshttpdxdoiorg1011552014732765

2 The Scientific World Journal

the particle-based DSMCmethods and for 005 lt Kn lt 001

using the coupled NSDSMC methodsUnfortunately hypersonic shock is not an entirely

understood domain Flow property 120601 changes significantly inthe inner of shock Also the gradient of flow property 120597120601120597119909is large because characteristic dimension 119871 is small It causesthe Knudsen number becomes big in the inner of shockAnd the Knudsen number is also big in the far field whichresults in the simulation difficulty of low density hypersonicflows

In low-density hypersonic flows both CFD and DSMCare used in the prediction of pressure and heat charac-teristics However conclusions drawn from the simulationand experiment are different Unfortunately many CFD andDSMC simulations are incompatible with precision direction[5 6]

Some double-cone flow simulation studies includingCandlerrsquos ect CFD [7] and Moss DSMC simulation [8] havebeen conducted In these studies comparisons between simu-lation and experiment reveal significant numerical problemsthat can be encountered when predicting strong gradientsExperimental data are used for comparisons as conductedin ONERA R5Ch Mach 10 low-density wind tunnel [8] InMossrsquos DSMC research VHS model was used HS and VSSmodels were not included in Mossrsquos DSMC simulation BothinCandlerrsquos ect CFD [7] and inMossrsquos DSMC [8] research theexperimental conditions of Run numbers 26 32 33 36 and38 are not included as shown in Table 1

For studying the influence of the DSMCmolecular modelon hypersonic shock interaction DSMC simulations areconducted by three molecular models namely HS [9 10]VHS [9 10] and VSS [11] The simulations are focused onexperimental conditions of Run numbers 26 32 33 36 and38These conditions are not included in previous studies Forfurther reliability Edneyrsquos type IV shock interaction is alsostudied

2 DSMC Model

In DSMC gas is represented by the velocity componentsand positions of a large number of simulated moleculesThe position coordinates velocity components and internalstates of the simulated molecule data are stored Theseparameters are updated with time because the moleculesare concurrently followed by representative collisions andboundary interactions in the simulated domain [12] Theprimary approximation of DSMC is that molecular motionsand intermolecular collisions are decoupled over short timeintervalsTheDSMCrsquos flowchart in the current study is shownin Figure 1

The core of the DSMC algorithm consists of four pri-mary processes moving the particles indexing and cross-referencing the particles simulating collisions and samplingthe flow field [9]

21 Moving the Particles In the first process all the sim-ulated molecules are moved through distances which areproportionate to their velocity components and discrete time

Start

Distribute particles into cells with their initial position and velocity

Move individual particles and compute interactions with boundaries

Index particles into cells

Select collision pairs and perform intermolecular collisions

Sample flow properties

Output results

N gt Ninteraction

Stop

Figure 1 DSMC flowchart

step Appropriate action is taken if the molecule crossesboundaries such as solid surfaces and symmetry boundaryCollisions with surfaces can be treated as being fully secularfully diffuse or any combination of the two Secular collisionsinvolve a simple reversal of themolecular velocity componentnormal to the incident surface Diffused collisions cause arandom reorientation of the reflected molecule where thepostcollision velocity is based on surface temperature

22 Indexing and Tracking the Particles The second DSMCprocess involves indexing and tracking the particles Thismolecular referencing scheme is needed as prerequisite forthe next two steps modeling the collisions and sampling theflow field

23 Simulation of Collisions The third step simulating col-lisions is an important process And it is different from theMD (molecular dynamics) method In DSMC simulationelastic collision model is used in which no interchange oftranslational and internal energy is considered And also inthe collision linear momentum and energy are conserved

231 Impact Parameters and Collision Cross Section Inhomogenous gas the probability of two molecules collisionis proportional to their relative speed and total collision crosssection In DSMC cross section 120590 is defined as [13]

120590 =119887

sin120594

100381610038161003816100381610038161003816100381610038161003816

119889119887

119889120594

100381610038161003816100381610038161003816100381610038161003816

(3)

Here 119887 is the distance of the closest approach to theundisturbed trajectories and 120594 is the angle of deflection asshown in Figure 2

The Scientific World Journal 3

Table 1 Test case in [1]

Run 119881infin

(ms) 120588infin

(Kgm3times103) 119875

infin

(Pa) Ma Rem (times105) Nose26 2581 1093 487 1034 1704 Sharp28 2520 0727 300 1050 1106 Sharp35 2575 0608 185 1249 0945 Sharp32 2572 0758 321 1056 1177 R025033 2636 1288 598 1035 2050 R025036 2584 1108 496 1034 1729 R025038 2453 0498 139 1247 0738 R0288

db

Cr d120576

120576120576

120594

A

O

Rd120594

d120594

dΩ

Crlowast

bd120576

Rsin120594d120576

Figure 2 Illustration of the impact parameters [4]

b

A

O

cr

120579A

120579A120594

crlowast

Figure 3 Collision geometry of HS molecules [4]

Then the total collision cross section can be written as

120590119879

= 2120587int

120587

0

120590 sin120594119889120594 = 2120587int 119887119889119887 (4)

Here the distance of the closest approach 119887 is given indifferent molecular model In the current paper three molec-ular models including HS VHS and VSS are used in thesimulation

(1) HS Model In the HS model the intermolecular force iseffective at

119903 =1

2(1198891

+ 1198892

) = 11988912

(5)

Then as Shown in Figure 3 we can get the closest approach 119887

119887 = 11988912

cos(12120594)

100381610038161003816100381610038161003816100381610038161003816

119889119887

119889120594

100381610038161003816100381610038161003816100381610038161003816

=1

211988912

sin(12120594)

120590 =1198892

12

4

(6)

Therefore the total collision cross section is defined as

120590119879

= 1205871198892

12

(7)

In HS model the molecule diameter can be calculated as

119889 = [(516) (119898119896119879ref120587)

12

120583ref]

12

(8)

where 120583ref is the reference viscosity at temperature 119879ref and119898is the molecular mass

119896 is the Boltzmann constant and 119896 = 1380658times10minus23 JK

(2) VHS Model In the VHS model molecular diameter 119889 isdefined as a function of relative velocity 119888

119903

that is

119889VHS = 119889ref(119888119903ref

119888119903

)

120585

= [(158) (119898120587)

12

(119896119879ref)120596

Γ ((92) minus 120596) 120583ref 120598120596minus(12)

119905

]

12

(9)

Here 120583ref is the reference viscosity at temperature119879ref120596 is thepower law which is expressed as 120596 = (12) + 120585 120585 is relativespeed exponent of VHSmodel and it can be adjusted to meetthe power law 120583 prop 119879

120596 120598119905

is the relative mean kinetic energyexpressed as 120598

119905

equiv (12)119898119903

1198882

119903

119888119903

is relative speed between thecollision molecules 119888

119903ref is reference relative speed betweenthe collision molecules

In the VHS model the closest approach 119887 can be definedas

119887 = 119889VHS cos(1

2120594) (10)

The total collision cross section is expressed as

120590119879

= 120590119879ref(

119889VHS119889ref

)

2

(11)

4 The Scientific World Journal

where 120590119879ref is reference collision cross section at the reference

molecule diameter 119889ref

(3) VSS Model In the VSS model molecular diameter 119889 isdefined as

119889VSS = [5 (120572 + 1) (120572 + 2)(119898120587)

12

(119896119879ref)120596

16120572Γ ((92) minus 120596) 120583ref 120598120596minus(12)

119905

]

12

(12)

Here 120572 is the scattering coefficient (In the case where 120572 = 1the VSS model is the same as the VHS model)

In the VSS model the closest approach 119887 is defined as

119887 = 119889VSScos120572

(1

2120594) (13)

The total collision cross section is defined as

120590119879

= 1205871198892

VSS (14)

232 Gas-Surface Interaction Diffuse reflection [14]with fullaccommodation to the surface temperature is implementedin the simulation It means that molecules diffused underthe Maxwell distribution condition at 119879 = 119879

119882

during thesimulation

24 Sampling The sampling of the macroscopic flow proper-ties is the final process in DSMCThe spatial coordinates andvelocity components of the molecules in a particular cell areused to calculate the macroscopic quantities at the geometriccenter of the cell Consider

120588 = 119899119898

119875 = 120588(119888 minus 119888)2

(15)

where 119899 is the number density of the molecules 119888 is thevelocity of the molecule and 119888 is the average velocity of themolecule

25 Parameters Theparameters selected in the present paperare pressure coefficient and the Stanton number

119862119875

=119875 minus 119875infin

(12) 1205881198812infin

119878119905

=119902

(12) 1205881198813infin

(16)

where 119875 is the flow-field pressure 119875infin

is the far-field pressureof the flow field 120588 is the density of flow-field density 119881

infin

isthe far-field velocity of the flow field and 119902 is the thermal fluxof the solid surface

3 Hypersonic Double-Cone Flow Simulation

31 Moss and Boydrsquos DSMC Predictions for Hypersonic 25∘55∘Double-Cone Flow To validate the DSMC method three25∘55∘ hypersonic double-cone configurations are selectedunder five conditions The experimental data with four

76254

55∘

10309

25∘

Figure 4 Number 1 configuration (in) (sharp nose 25∘55∘ doublecone)

4

55∘

10309

25∘

7283

R025

Figure 5 Number 2 configuration (in) (R025 blunt nose 25∘55∘double cone)

configurations as shown in Figures 4 5 6 and 7 andare reported in [1] in which Moss and Boydrsquos DSMC pre-dictions are compared with the experimental data undersome experimental conditions Boydrsquos predictions using theDSMC code for Run number 35 are shown in Figure 8These predictions clearly underestimated the length of theseparated region and although the heating in the forebodyis slightly underpredicted a good agreement existed betweentheory and experiment for the pressures and the heat transferdownstream of the shockshock interaction in the secondcone [12]

Figures 9 and 10 show the results obtained using theDSMCmethod byMoss [8] andBoyd [15] which significantlyunderpredicted the scale of the interaction region and theposition andmagnitude of the properties in and downstreamof the region of shockshock interaction Moss calculationsshown in Figure 9 indicated that the predicted separatedregion is approximately 25 of the length measured inthe experiment Downstream of the interaction regions thepressure and the heating levels were relatively well predictedBoydrsquos solution for Run number 28 also significantly under-predicted the length of the separated region and unlikeMossrsquos calculation the heat transfer rates both ahead and

The Scientific World Journal 5

4

55∘

1030925

∘

7232

R0288

Figure 6 Number 3 configuration (in) (R0288 blunt nose 25∘55∘double cone)

4

55∘

1030925∘

6805

R06

Figure 7 Number 4 configuration (in) (R06 blunt nose 25∘55∘double cone)

Table 2 Properties of nitrogen molecule in DSMC

Degree of freedom 5Molecular mass times 10

27 Kg 465Viscosity coefficient 1656Viscosity index 074120572 (for VSS model) 136Molecular diameter (VHS) times 10

10m 417Molecular diameter (VSS) times 10

10m 411Molecular diameter (HS) times 10

10m 3784

Table 3 Flow conditions for Run Numbers 26 and 36

Ma Rem KnRun 26 1034 1704 times 10

5

905 times 10minus5

Run 36 1034 173 times 105

891 times 10minus5

downstream of the interaction were underpredicted Signifi-cant improvement of these calculations is doubtful even withthe incorporation of vibrational nonequilibrium effects

32 Our DSMC Simulation In [1] Run numbers 26 32 3336 and 38 were not simulated by DSMC and CFDThereforewe selected these cases to validate the DSMC code with theHS VHS and VSS models

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

Boyd CpBoyd St

S t

Figure 8 Boydrsquos DSMC for Run number 35 [1 15]

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

S t

Moss CpMoss St

Figure 9 Mossrsquos DSMC for Run number 28 [8]

The computational grid employed for Run numbers 2632 33 36 and 38 consists of 2048 cells along the body by 512cells normal to the bodyThis grid is the same as that of BoydrsquosDSMC simulation [15] The base time step for the simulationwas 10minus9 s although this was reduced in very high densityregionsThe solutions were obtained using approximately 64million particles and averaged over a sampling interval of05ms At this time the solution had attained steady state Inparticular the size of the separation region did not change

The properties of nitrogen molecule used in the currentstudy are presented in Table 2

321 DSMC Simulation of Run Numbers 26 and 36 In thissection the DSMC simulation result of Run numbers 26 and36 is presented As shown in Table 3 these two cases wereconducted in the sameMa and Re conditions with sharp noseand R025 in blunt nose respectively

6 The Scientific World Journal

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

Boyd CpBoyd St

S tFigure 10 Boydrsquos DSMC for Run number 28 [15]

000 025 050 075 100 125 150 175 200 225

00

05

10

15

20

25

30

Cp

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L =92075 mm) Run number 26

Figure 11 DSMC simulation of Run number 26 (119862119875

)

The pressure coefficient and the Stanton numberobtained with HS VHS and VSS are shown in Figures11ndash14 These results underpredict the scale of interactionregion and the position and also the magnitude of theparameters is underpredicted in and downstream The HScalculation shown in Figure 11 for Run number 26 indicatesthat the predicted interaction region is approximately 76of the measured experimental length Those of VHS andVSS are 34 and 62 respectively The HS results shownin Figure 12 for Run number 36 indicate that the predictedinteraction region is approximately 76 of the measuredexperimental length Those of VHS and VSS are 30 and57 respectively These predictions are much improvedcompared with Mossrsquos DSMC calculation which was only25 of the lengthmeasured in the experiment [8]The resultsof three models overpredicted the ahead pressure coefficient

000 025 050 075 100 125 150 175 200 225

00

05

10

15

20

25

30

Cp

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 36

Figure 12 DSMC simulation of Run number 36 (119862119875

)

000 025 050 075 100 125 150 175 200 225

0200

0175

0150

0125

0100

0075

0050

0025

0000

minus0025

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 26-

S t

St

Figure 13 DSMC simulation of Run number 26 (119878119905

)

peak value In the downstream of the interaction regionsthe pressure levels got by the three models were relativelywell predicted And also the three models overpredictedthe Stanton number in comparison with the experimentaldata The position of the Stanton number peak was wellpredicted by the three models Figure 13 shows that thepeak values of the Stanton number using HS VHS and VSSare approximately 142 1857 and 1485 respectivelyof the data measured in the experiment for Run number26 Figure 14 shows that the peak values of the Stantonnumber using HS VHS and VSS are approximately 1781928 and 178 respectively of the data measured in theexperiment for Run number 36

322 DSMC Simulation of Run Number 33 In this sectionthe DSMC simulation result of Run number 33 is presented

The Scientific World Journal 7

000 025 050 075 100 125 150 175 200 225

016015014013012011010009008007006005004003002001000

minus001

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 36-

S t

St

Figure 14 DSMC simulation of Run number 36 (119878119905

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

10

05

00

minus05

Cp

XL (L = 92075 mm) Run number 33

Figure 15 DSMC simulation of Run number 33 (119862119875

)

Table 4 Flow conditions for Run Numbers 33 and Numbers 36

Ma Rem KnRun 33 1035 205 times 10

5

753 times 10minus5

Run 36 1034 173 times 105

891 times 10minus5

Table 5 Flow conditions for Run Numbers 32 and 38

Ma Rem KnRun 32 1056 1177 times 10

5

134 times 10minus4

Run 38 1247 0738 times 105

252 times 10minus4

As shown in Table 4 Run numbers 33 and 36 were conductedin the same configuration with different Kn condation

The pressure coefficient and the Stanton numberobtained from HS VHS and VSS using the DSMC

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

020

015

010

005

000

XL (L = 92075 mm) Run number 33-St

S t

Figure 16 DSMC simulation of Run number 33 (119878119905

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

10

05

00

Cp

XL (L = 92075 mm) Run number 32

Figure 17 DSMC simulation of Run number 32 (119862119875

)

method are shown in Figures 15 and 16 These results alsounderpredicted the scale of interaction region and positionmagnitude of the properties in and downstream ofthe shock interaction region The HS result shown inFigure 15 indicates that the predicted interaction regionis approximately 533 of the length measured in theexperiment Those of VHS and VSS are 266 and466 respectively These predictions are not improvedin comparison with Run numbers 26 and 36 calculationsThe three modelsrsquo calculation also overpredicted the aheadpressure coefficient peak value In the downstream of theinteraction region the pressure levels from three models arerelatively well predicted Also three models overpredictedthe Stanton number compared with the experimentaldata But the position of the Stanton number peak is wellpredicted in the three modelsrsquo calculation Figure 16 shows

8 The Scientific World Journal

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

05

10

00

Cp

XL (L = 92075 mm) Run number 38

Figure 18 DSMC simulation of Run number 38 (119862119875

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

016

014

012

010

008

006

004

002

000

minus002

XL (L = 92075 mm) Run number 32-

S t

St

Figure 19 DSMC simulation of Run number 32 (119878119905

)

that the peak values of the Stanton number in HS VHS andVSS are approximately 290 309 and 345 respectivelyof the data measured in the experiment for Run number 33

323 DSMC Simulation for Run Numbers 32 and 38 In thissection theDSMCsimulation results forRunnumbers 32 and38 as shown in Table 5 are presented Run numbers 33 38and 36 were conducted in the same configuration with R025in blunt nose

Thepressure coefficient and the Stanton number obtainedby HS VHS and VSS using the DSMC method are shownin Figures 17ndash20 These results well predicted the scale of theinteraction region and the position magnitude of the proper-ties in and downstream of the shock interaction region TheHS result shown in Figure 17 for Run number 32 indicates

000 025 050 075 100 125 150 175 200 225minus002

000

002

004

006

008

010

012

014

016

018

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 38-

S t

St

Figure 20 DSMC simulation of Run number 38 (119878119905

)

35635

Rn = 8

53L2

L3

L1

L

L 950 to 1055

20∘ 10∘10∘

Dimensions in (mm)

y

x

Origin of coordinate system

L1 = L2 = 50771L3 = 100

Figure 21 Configuration of Edneyrsquos Type IV shock interaction

that the predicted interaction region is approximately 100of the length measured in the experiment Those of VHS andVSS are 41 and 69 respectively The HS result shown inFigure 18 for Run number 38 indicates that the predictedinteraction region is approximately 1068 of the lengthmeasured in the experimentThose ofVHS andVSS are 276and 6896 respectively These predictions are improvedin comparison with Run numbers 26 and 36 calculationsAlso the three modelsrsquo calculation overpredicted the aheadpressure coefficient peak value And in the downstreamof interaction region the pressure levels from the threemodels are relatively well predicted The three models alsooverpredicted the Stanton number in comparison with theexperimental data And also the position of the Stanton num-ber peak is well predicted in the three modelsrsquo calculationFigure 19 shows that the peak values of the Stanton numberin HS VHS and VSS are approximately 1052 1578 and

The Scientific World Journal 9

Experiment

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100minus2

0

2

4

6

8

10

12

14

16

Cp

Angle (deg)

DSMC-HSDSMC-VSSDSMC-VHS

Figure 22 DSMC of type IV shock interaction (119862119875

)

1474 respectively of the data measured in the experimentfor Run number 32 Figure 20 shows that the peak value ofthe Stanton number in HS VHS and VSS are approximately1052 1684 and 1421 respectively of the datameasuredin the experiment for Run number 38

Table 6 shows that the DSMC error depends on theKn number and the models employed for intermolecularinteraction With the increase in the Kn number the DSMCerror decreases The error is the smallest in the HS modelcompared with those in the VHS and VSS models Withthe Kn number increasing the Stanton number DSMC errordecreases When the Kn number is in the level of 10minus4 theDSMC error for pressure coefficient and Stanton number iscontrolled within 10

4 Type IV Shock Interaction Simulation

In the present study results of the numerical simulations forMach 10 air flow as shown in Table 7 were presented for arange of Edneyrsquos type IV shock interactionThe experimentaldata for validation were obtained from [16] The configura-tion is shown in Figure 21

The results of the grid sensitivity investigation are pre-sented in [16] The number of cellssubcells used for oursimulation was 97060958490 proven in [16] to be thefinestThe computational time step was 12 nsThe simulationconsisted of 156 times 10

5 time steps to ensure steady stateconditions

The DSMC data of the pressure coefficient and theStanton number are shown in Figures 22 and 23 The agree-ment between the DSMC calculation and the experiment isgenerally good BothHS andVHSmodels gain improvementsin the errors The error between the HS-DSMC data and theexperimental data is controlled within 5

The result from theDSMC calculation for Edneyrsquos type IVshock interaction also confirmed the same conclusion The

ExperimentDSMC-HS

DSMC-VSSDSMC-VHS

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100

15

10

05

00

Angle (deg)

S t

Figure 23 DSMC of type IV shock interaction (119878119905

)

error is the smallest in the HS model compared with those inthe VHS and VSS models In the region of large Kn numberDSMC using the HS model could be used in pressure andthermal prediction of hypersonic complex flow

5 Conclusion

In the present study the double-cone and Edneyrsquos type IVshock interactions were studied Molecular models wereemployed with DSMC including the HS VHS and VSSmodels Comparisons between DSMC and the experimentwere also conducted

Drawing firm conclusions from this study prior to fullcomparisons of these results is somewhat difficult Howeversome issues are immediately clear from our attempt tocompute these flows

(1) The DSMC method appears to predict qualitativelythe structure of flow separation found in previousCFD calculations

(2) Investigation of the double-cone hypersonic flowreveals that the three collisionsmodels can predict thepressure coefficient and the Stanton number trendsThe agreement between the DSMC simulation andthe experiment is the best in the HS collisions model

(3) Investigation of type IV shock interaction shows thatthe agreement between the DSMC simulation andthe experiment is generally good in the HS and VHScollision models However it fails in the VSS model

(4) Both double-cone flow and type IV shock interactionsimulations show that the DSMC error dependson the Kn number and the models employed forintermolecular interaction Increasing the Kn num-ber decreases the DSMC error The error is the small-est for the HS model compared with those for the

10 The Scientific World Journal

Table 6 Comparisons of calculations ((ValueDSMC minus Valueexperiment)(Valueexperiment))

Run KnThe scale of interaction region

(error in compared with measured data )Peak value of the Stanton number

(error in compared with measured data )HS VHS VSS HS VHS VSS

33 753 times 10minus5 467 734 534 190 209 245

36 891 times 10minus5 24 70 43 78 928 78

26 905 times 10minus5 24 66 38 42 857 485

32 134 times 10minus4 0 59 31 52 578 474

38 252 times 10minus4 68 724 3104 52 684 421

Table 7 Flow conditions of Edneyrsquos type IV shock interaction

Ma Rem Kn10 103 times 10

5

145 times 10minus4

VHS and VSS models When the Kn number is in thelevel of 10minus4 the DSMC error for pressure coefficientthe Stanton number and scale of interaction region iscontrolled within 10

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the Chinese AerospaceInnovation and in part by the Basic Research Fund ofNorthwestern Polytechnical University

References

[1] M S Holden T P Wadhams J K Harvey and G V CandlerldquoComparisons between measurements in regions of laminarshockwave boundary layer interaction in hypersonic flowswithnavier-stokes and DSMC solutionsrdquo Tech Rep RTO-TR-AVT-007-V3 NATO Science and Technology Organization 2006

[2] J Olejniczak and G V Candler ldquoComputation of hyper-sonic shock interaction flow fieldsrdquo in Proceedings of the 7thAIAAASME JointThermophysics and Heat Transfer ConferenceAIAA Paper 98-2446 June 1998

[3] G V Candler I Nompelis M-C Druguet I D Boyd W-LWang and S M Holden ldquoCFD validation for hypersonic flighthypersonic double-cone flow simulationsrdquo in Proceedings of the40th Aerospace Sciences Meeting amp Exhibit AIAA 2002-0581Reno Nev USA January 2002

[4] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Oxford Clarendon Press New York NY USA 1994

[5] Harvey K John M S Holden and T P Wadhams ldquoCode vali-dation study of laminar shockboundary layer and shockshockinteractions in hypersonic flow Part B comparison withnavier-stokes and DSMC solutionsrdquo in Proceedings of the 39thAerospace Sciences Meeting amp Exhibit AIAA 2001-1031 RenoNev USA January 2001

[6] M Gallis C Roy J Payne and T Bartel ldquoDSMC and navier-stokes predictions for hypersonic laminar interacting flowsrdquo inProceedings of the 39th Aerospace Sciences Meeting and ExhibitAIAA 2001-1030 Reno Nev USA January 2001

[7] G V Candler I Nompelis and M-C Druguet ldquoNavier-stokespredictions of hypersonic double-cone and cylinder-flare flowfieldsrdquo in Proceedings of the 39th Aerospace Sciences Meeting ampExhibit AIAA Paper no 2001-1024 Reno Nev USA January2001

[8] J Moss ldquoDSMC computations for regions of shockshock andshockboundary layer interactionrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA 2001-1027 RenoNev USA January 2001

[9] G A Bird The DSMC Method Create Space IndependentPublishing Platform 2013

[10] M Woo and I Greber ldquoMolecular dynamics simulation ofpiston-driven shock wave in hard sphere gasrdquo AIAA Journalvol 37 no 2 pp 215ndash221 1999

[11] K Koura and H Matsumoto ldquoVariable soft sphere molecularmodel for air speciesrdquo Physics of Fluids A vol 4 no 5 pp 1083ndash1085 1992

[12] P S Prasanth and J K Kakkassery ldquoDirect simulation MonteCarlo (DSMC) a numerical method for transition-regimeflows a reviewrdquo Journal of the Indian Institute of Science vol86 no 3 pp 169ndash192 2006

[13] T Tokumasu and Y Matsumoto ldquoDynamic molecular collision(DMC) model for rarefied gas flow simulations by the DSMCmethodrdquo Physics of Fluids vol 11 no 1 pp 1907ndash1920 1999

[14] S F Borisov ldquoProgress in gas-surface interaction studyrdquo inProceedings of the 24th International Rarefied Gas DynamicsSymposium M Capitelli Ed pp 933ndash940 The AmericanInstitute of Physics 2005

[15] I D Boyd and W-L Wang ldquoMonte Carlo computationsof hypersonic interacting flowsrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA Paper no 2001-1029 Reno Nev USA January 2001

[16] J N Moss T Pot B Chanetz and M Lefebvre ldquoDSMCsimulation of shockshock interactions emphases on type IVinteractionrdquo in Proceedings of the 22nd International Symposiumon Shock Waes Paper no 3570 Imperial College London UK1999

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2 The Scientific World Journal

the particle-based DSMCmethods and for 005 lt Kn lt 001

using the coupled NSDSMC methodsUnfortunately hypersonic shock is not an entirely

understood domain Flow property 120601 changes significantly inthe inner of shock Also the gradient of flow property 120597120601120597119909is large because characteristic dimension 119871 is small It causesthe Knudsen number becomes big in the inner of shockAnd the Knudsen number is also big in the far field whichresults in the simulation difficulty of low density hypersonicflows

In low-density hypersonic flows both CFD and DSMCare used in the prediction of pressure and heat charac-teristics However conclusions drawn from the simulationand experiment are different Unfortunately many CFD andDSMC simulations are incompatible with precision direction[5 6]

Some double-cone flow simulation studies includingCandlerrsquos ect CFD [7] and Moss DSMC simulation [8] havebeen conducted In these studies comparisons between simu-lation and experiment reveal significant numerical problemsthat can be encountered when predicting strong gradientsExperimental data are used for comparisons as conductedin ONERA R5Ch Mach 10 low-density wind tunnel [8] InMossrsquos DSMC research VHS model was used HS and VSSmodels were not included in Mossrsquos DSMC simulation BothinCandlerrsquos ect CFD [7] and inMossrsquos DSMC [8] research theexperimental conditions of Run numbers 26 32 33 36 and38 are not included as shown in Table 1

For studying the influence of the DSMCmolecular modelon hypersonic shock interaction DSMC simulations areconducted by three molecular models namely HS [9 10]VHS [9 10] and VSS [11] The simulations are focused onexperimental conditions of Run numbers 26 32 33 36 and38These conditions are not included in previous studies Forfurther reliability Edneyrsquos type IV shock interaction is alsostudied

2 DSMC Model

In DSMC gas is represented by the velocity componentsand positions of a large number of simulated moleculesThe position coordinates velocity components and internalstates of the simulated molecule data are stored Theseparameters are updated with time because the moleculesare concurrently followed by representative collisions andboundary interactions in the simulated domain [12] Theprimary approximation of DSMC is that molecular motionsand intermolecular collisions are decoupled over short timeintervalsTheDSMCrsquos flowchart in the current study is shownin Figure 1

The core of the DSMC algorithm consists of four pri-mary processes moving the particles indexing and cross-referencing the particles simulating collisions and samplingthe flow field [9]

21 Moving the Particles In the first process all the sim-ulated molecules are moved through distances which areproportionate to their velocity components and discrete time

Start

Distribute particles into cells with their initial position and velocity

Move individual particles and compute interactions with boundaries

Index particles into cells

Select collision pairs and perform intermolecular collisions

Sample flow properties

Output results

N gt Ninteraction

Stop

Figure 1 DSMC flowchart

step Appropriate action is taken if the molecule crossesboundaries such as solid surfaces and symmetry boundaryCollisions with surfaces can be treated as being fully secularfully diffuse or any combination of the two Secular collisionsinvolve a simple reversal of themolecular velocity componentnormal to the incident surface Diffused collisions cause arandom reorientation of the reflected molecule where thepostcollision velocity is based on surface temperature

22 Indexing and Tracking the Particles The second DSMCprocess involves indexing and tracking the particles Thismolecular referencing scheme is needed as prerequisite forthe next two steps modeling the collisions and sampling theflow field

23 Simulation of Collisions The third step simulating col-lisions is an important process And it is different from theMD (molecular dynamics) method In DSMC simulationelastic collision model is used in which no interchange oftranslational and internal energy is considered And also inthe collision linear momentum and energy are conserved

231 Impact Parameters and Collision Cross Section Inhomogenous gas the probability of two molecules collisionis proportional to their relative speed and total collision crosssection In DSMC cross section 120590 is defined as [13]

120590 =119887

sin120594

100381610038161003816100381610038161003816100381610038161003816

119889119887

119889120594

100381610038161003816100381610038161003816100381610038161003816

(3)

Here 119887 is the distance of the closest approach to theundisturbed trajectories and 120594 is the angle of deflection asshown in Figure 2

The Scientific World Journal 3

Table 1 Test case in [1]

Run 119881infin

(ms) 120588infin

(Kgm3times103) 119875

infin

(Pa) Ma Rem (times105) Nose26 2581 1093 487 1034 1704 Sharp28 2520 0727 300 1050 1106 Sharp35 2575 0608 185 1249 0945 Sharp32 2572 0758 321 1056 1177 R025033 2636 1288 598 1035 2050 R025036 2584 1108 496 1034 1729 R025038 2453 0498 139 1247 0738 R0288

db

Cr d120576

120576120576

120594

A

O

Rd120594

d120594

dΩ

Crlowast

bd120576

Rsin120594d120576

Figure 2 Illustration of the impact parameters [4]

b

A

O

cr

120579A

120579A120594

crlowast

Figure 3 Collision geometry of HS molecules [4]

Then the total collision cross section can be written as

120590119879

= 2120587int

120587

0

120590 sin120594119889120594 = 2120587int 119887119889119887 (4)

Here the distance of the closest approach 119887 is given indifferent molecular model In the current paper three molec-ular models including HS VHS and VSS are used in thesimulation

(1) HS Model In the HS model the intermolecular force iseffective at

119903 =1

2(1198891

+ 1198892

) = 11988912

(5)

Then as Shown in Figure 3 we can get the closest approach 119887

119887 = 11988912

cos(12120594)

100381610038161003816100381610038161003816100381610038161003816

119889119887

119889120594

100381610038161003816100381610038161003816100381610038161003816

=1

211988912

sin(12120594)

120590 =1198892

12

4

(6)

Therefore the total collision cross section is defined as

120590119879

= 1205871198892

12

(7)

In HS model the molecule diameter can be calculated as

119889 = [(516) (119898119896119879ref120587)

12

120583ref]

12

(8)

where 120583ref is the reference viscosity at temperature 119879ref and119898is the molecular mass

119896 is the Boltzmann constant and 119896 = 1380658times10minus23 JK

(2) VHS Model In the VHS model molecular diameter 119889 isdefined as a function of relative velocity 119888

119903

that is

119889VHS = 119889ref(119888119903ref

119888119903

)

120585

= [(158) (119898120587)

12

(119896119879ref)120596

Γ ((92) minus 120596) 120583ref 120598120596minus(12)

119905

]

12

(9)

Here 120583ref is the reference viscosity at temperature119879ref120596 is thepower law which is expressed as 120596 = (12) + 120585 120585 is relativespeed exponent of VHSmodel and it can be adjusted to meetthe power law 120583 prop 119879

120596 120598119905

is the relative mean kinetic energyexpressed as 120598

119905

equiv (12)119898119903

1198882

119903

119888119903

is relative speed between thecollision molecules 119888

119903ref is reference relative speed betweenthe collision molecules

In the VHS model the closest approach 119887 can be definedas

119887 = 119889VHS cos(1

2120594) (10)

The total collision cross section is expressed as

120590119879

= 120590119879ref(

119889VHS119889ref

)

2

(11)

4 The Scientific World Journal

where 120590119879ref is reference collision cross section at the reference

molecule diameter 119889ref

(3) VSS Model In the VSS model molecular diameter 119889 isdefined as

119889VSS = [5 (120572 + 1) (120572 + 2)(119898120587)

12

(119896119879ref)120596

16120572Γ ((92) minus 120596) 120583ref 120598120596minus(12)

119905

]

12

(12)

Here 120572 is the scattering coefficient (In the case where 120572 = 1the VSS model is the same as the VHS model)

In the VSS model the closest approach 119887 is defined as

119887 = 119889VSScos120572

(1

2120594) (13)

The total collision cross section is defined as

120590119879

= 1205871198892

VSS (14)

232 Gas-Surface Interaction Diffuse reflection [14]with fullaccommodation to the surface temperature is implementedin the simulation It means that molecules diffused underthe Maxwell distribution condition at 119879 = 119879

119882

during thesimulation

24 Sampling The sampling of the macroscopic flow proper-ties is the final process in DSMCThe spatial coordinates andvelocity components of the molecules in a particular cell areused to calculate the macroscopic quantities at the geometriccenter of the cell Consider

120588 = 119899119898

119875 = 120588(119888 minus 119888)2

(15)

where 119899 is the number density of the molecules 119888 is thevelocity of the molecule and 119888 is the average velocity of themolecule

25 Parameters Theparameters selected in the present paperare pressure coefficient and the Stanton number

119862119875

=119875 minus 119875infin

(12) 1205881198812infin

119878119905

=119902

(12) 1205881198813infin

(16)

where 119875 is the flow-field pressure 119875infin

is the far-field pressureof the flow field 120588 is the density of flow-field density 119881

infin

isthe far-field velocity of the flow field and 119902 is the thermal fluxof the solid surface

3 Hypersonic Double-Cone Flow Simulation

31 Moss and Boydrsquos DSMC Predictions for Hypersonic 25∘55∘Double-Cone Flow To validate the DSMC method three25∘55∘ hypersonic double-cone configurations are selectedunder five conditions The experimental data with four

76254

55∘

10309

25∘

Figure 4 Number 1 configuration (in) (sharp nose 25∘55∘ doublecone)

4

55∘

10309

25∘

7283

R025

Figure 5 Number 2 configuration (in) (R025 blunt nose 25∘55∘double cone)

configurations as shown in Figures 4 5 6 and 7 andare reported in [1] in which Moss and Boydrsquos DSMC pre-dictions are compared with the experimental data undersome experimental conditions Boydrsquos predictions using theDSMC code for Run number 35 are shown in Figure 8These predictions clearly underestimated the length of theseparated region and although the heating in the forebodyis slightly underpredicted a good agreement existed betweentheory and experiment for the pressures and the heat transferdownstream of the shockshock interaction in the secondcone [12]

Figures 9 and 10 show the results obtained using theDSMCmethod byMoss [8] andBoyd [15] which significantlyunderpredicted the scale of the interaction region and theposition andmagnitude of the properties in and downstreamof the region of shockshock interaction Moss calculationsshown in Figure 9 indicated that the predicted separatedregion is approximately 25 of the length measured inthe experiment Downstream of the interaction regions thepressure and the heating levels were relatively well predictedBoydrsquos solution for Run number 28 also significantly under-predicted the length of the separated region and unlikeMossrsquos calculation the heat transfer rates both ahead and

The Scientific World Journal 5

4

55∘

1030925

∘

7232

R0288

Figure 6 Number 3 configuration (in) (R0288 blunt nose 25∘55∘double cone)

4

55∘

1030925∘

6805

R06

Figure 7 Number 4 configuration (in) (R06 blunt nose 25∘55∘double cone)

Table 2 Properties of nitrogen molecule in DSMC

Degree of freedom 5Molecular mass times 10

27 Kg 465Viscosity coefficient 1656Viscosity index 074120572 (for VSS model) 136Molecular diameter (VHS) times 10

10m 417Molecular diameter (VSS) times 10

10m 411Molecular diameter (HS) times 10

10m 3784

Table 3 Flow conditions for Run Numbers 26 and 36

Ma Rem KnRun 26 1034 1704 times 10

5

905 times 10minus5

Run 36 1034 173 times 105

891 times 10minus5

downstream of the interaction were underpredicted Signifi-cant improvement of these calculations is doubtful even withthe incorporation of vibrational nonequilibrium effects

32 Our DSMC Simulation In [1] Run numbers 26 32 3336 and 38 were not simulated by DSMC and CFDThereforewe selected these cases to validate the DSMC code with theHS VHS and VSS models

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

Boyd CpBoyd St

S t

Figure 8 Boydrsquos DSMC for Run number 35 [1 15]

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

S t

Moss CpMoss St

Figure 9 Mossrsquos DSMC for Run number 28 [8]

The computational grid employed for Run numbers 2632 33 36 and 38 consists of 2048 cells along the body by 512cells normal to the bodyThis grid is the same as that of BoydrsquosDSMC simulation [15] The base time step for the simulationwas 10minus9 s although this was reduced in very high densityregionsThe solutions were obtained using approximately 64million particles and averaged over a sampling interval of05ms At this time the solution had attained steady state Inparticular the size of the separation region did not change

The properties of nitrogen molecule used in the currentstudy are presented in Table 2

321 DSMC Simulation of Run Numbers 26 and 36 In thissection the DSMC simulation result of Run numbers 26 and36 is presented As shown in Table 3 these two cases wereconducted in the sameMa and Re conditions with sharp noseand R025 in blunt nose respectively

6 The Scientific World Journal

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

Boyd CpBoyd St

S tFigure 10 Boydrsquos DSMC for Run number 28 [15]

000 025 050 075 100 125 150 175 200 225

00

05

10

15

20

25

30

Cp

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L =92075 mm) Run number 26

Figure 11 DSMC simulation of Run number 26 (119862119875

)

The pressure coefficient and the Stanton numberobtained with HS VHS and VSS are shown in Figures11ndash14 These results underpredict the scale of interactionregion and the position and also the magnitude of theparameters is underpredicted in and downstream The HScalculation shown in Figure 11 for Run number 26 indicatesthat the predicted interaction region is approximately 76of the measured experimental length Those of VHS andVSS are 34 and 62 respectively The HS results shownin Figure 12 for Run number 36 indicate that the predictedinteraction region is approximately 76 of the measuredexperimental length Those of VHS and VSS are 30 and57 respectively These predictions are much improvedcompared with Mossrsquos DSMC calculation which was only25 of the lengthmeasured in the experiment [8]The resultsof three models overpredicted the ahead pressure coefficient

000 025 050 075 100 125 150 175 200 225

00

05

10

15

20

25

30

Cp

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 36

Figure 12 DSMC simulation of Run number 36 (119862119875

)

000 025 050 075 100 125 150 175 200 225

0200

0175

0150

0125

0100

0075

0050

0025

0000

minus0025

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 26-

S t

St

Figure 13 DSMC simulation of Run number 26 (119878119905

)

peak value In the downstream of the interaction regionsthe pressure levels got by the three models were relativelywell predicted And also the three models overpredictedthe Stanton number in comparison with the experimentaldata The position of the Stanton number peak was wellpredicted by the three models Figure 13 shows that thepeak values of the Stanton number using HS VHS and VSSare approximately 142 1857 and 1485 respectivelyof the data measured in the experiment for Run number26 Figure 14 shows that the peak values of the Stantonnumber using HS VHS and VSS are approximately 1781928 and 178 respectively of the data measured in theexperiment for Run number 36

322 DSMC Simulation of Run Number 33 In this sectionthe DSMC simulation result of Run number 33 is presented

The Scientific World Journal 7

000 025 050 075 100 125 150 175 200 225

016015014013012011010009008007006005004003002001000

minus001

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 36-

S t

St

Figure 14 DSMC simulation of Run number 36 (119878119905

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

10

05

00

minus05

Cp

XL (L = 92075 mm) Run number 33

Figure 15 DSMC simulation of Run number 33 (119862119875

)

Table 4 Flow conditions for Run Numbers 33 and Numbers 36

Ma Rem KnRun 33 1035 205 times 10

5

753 times 10minus5

Run 36 1034 173 times 105

891 times 10minus5

Table 5 Flow conditions for Run Numbers 32 and 38

Ma Rem KnRun 32 1056 1177 times 10

5

134 times 10minus4

Run 38 1247 0738 times 105

252 times 10minus4

As shown in Table 4 Run numbers 33 and 36 were conductedin the same configuration with different Kn condation

The pressure coefficient and the Stanton numberobtained from HS VHS and VSS using the DSMC

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

020

015

010

005

000

XL (L = 92075 mm) Run number 33-St

S t

Figure 16 DSMC simulation of Run number 33 (119878119905

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

10

05

00

Cp

XL (L = 92075 mm) Run number 32

Figure 17 DSMC simulation of Run number 32 (119862119875

)

method are shown in Figures 15 and 16 These results alsounderpredicted the scale of interaction region and positionmagnitude of the properties in and downstream ofthe shock interaction region The HS result shown inFigure 15 indicates that the predicted interaction regionis approximately 533 of the length measured in theexperiment Those of VHS and VSS are 266 and466 respectively These predictions are not improvedin comparison with Run numbers 26 and 36 calculationsThe three modelsrsquo calculation also overpredicted the aheadpressure coefficient peak value In the downstream of theinteraction region the pressure levels from three models arerelatively well predicted Also three models overpredictedthe Stanton number compared with the experimentaldata But the position of the Stanton number peak is wellpredicted in the three modelsrsquo calculation Figure 16 shows

8 The Scientific World Journal

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

05

10

00

Cp

XL (L = 92075 mm) Run number 38

Figure 18 DSMC simulation of Run number 38 (119862119875

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

016

014

012

010

008

006

004

002

000

minus002

XL (L = 92075 mm) Run number 32-

S t

St

Figure 19 DSMC simulation of Run number 32 (119878119905

)

that the peak values of the Stanton number in HS VHS andVSS are approximately 290 309 and 345 respectivelyof the data measured in the experiment for Run number 33

323 DSMC Simulation for Run Numbers 32 and 38 In thissection theDSMCsimulation results forRunnumbers 32 and38 as shown in Table 5 are presented Run numbers 33 38and 36 were conducted in the same configuration with R025in blunt nose

Thepressure coefficient and the Stanton number obtainedby HS VHS and VSS using the DSMC method are shownin Figures 17ndash20 These results well predicted the scale of theinteraction region and the position magnitude of the proper-ties in and downstream of the shock interaction region TheHS result shown in Figure 17 for Run number 32 indicates

000 025 050 075 100 125 150 175 200 225minus002

000

002

004

006

008

010

012

014

016

018

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 38-

S t

St

Figure 20 DSMC simulation of Run number 38 (119878119905

)

35635

Rn = 8

53L2

L3

L1

L

L 950 to 1055

20∘ 10∘10∘

Dimensions in (mm)

y

x

Origin of coordinate system

L1 = L2 = 50771L3 = 100

Figure 21 Configuration of Edneyrsquos Type IV shock interaction

that the predicted interaction region is approximately 100of the length measured in the experiment Those of VHS andVSS are 41 and 69 respectively The HS result shown inFigure 18 for Run number 38 indicates that the predictedinteraction region is approximately 1068 of the lengthmeasured in the experimentThose ofVHS andVSS are 276and 6896 respectively These predictions are improvedin comparison with Run numbers 26 and 36 calculationsAlso the three modelsrsquo calculation overpredicted the aheadpressure coefficient peak value And in the downstreamof interaction region the pressure levels from the threemodels are relatively well predicted The three models alsooverpredicted the Stanton number in comparison with theexperimental data And also the position of the Stanton num-ber peak is well predicted in the three modelsrsquo calculationFigure 19 shows that the peak values of the Stanton numberin HS VHS and VSS are approximately 1052 1578 and

The Scientific World Journal 9

Experiment

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100minus2

0

2

4

6

8

10

12

14

16

Cp

Angle (deg)

DSMC-HSDSMC-VSSDSMC-VHS

Figure 22 DSMC of type IV shock interaction (119862119875

)

1474 respectively of the data measured in the experimentfor Run number 32 Figure 20 shows that the peak value ofthe Stanton number in HS VHS and VSS are approximately1052 1684 and 1421 respectively of the datameasuredin the experiment for Run number 38

Table 6 shows that the DSMC error depends on theKn number and the models employed for intermolecularinteraction With the increase in the Kn number the DSMCerror decreases The error is the smallest in the HS modelcompared with those in the VHS and VSS models Withthe Kn number increasing the Stanton number DSMC errordecreases When the Kn number is in the level of 10minus4 theDSMC error for pressure coefficient and Stanton number iscontrolled within 10

4 Type IV Shock Interaction Simulation

In the present study results of the numerical simulations forMach 10 air flow as shown in Table 7 were presented for arange of Edneyrsquos type IV shock interactionThe experimentaldata for validation were obtained from [16] The configura-tion is shown in Figure 21

The results of the grid sensitivity investigation are pre-sented in [16] The number of cellssubcells used for oursimulation was 97060958490 proven in [16] to be thefinestThe computational time step was 12 nsThe simulationconsisted of 156 times 10

5 time steps to ensure steady stateconditions

The DSMC data of the pressure coefficient and theStanton number are shown in Figures 22 and 23 The agree-ment between the DSMC calculation and the experiment isgenerally good BothHS andVHSmodels gain improvementsin the errors The error between the HS-DSMC data and theexperimental data is controlled within 5

The result from theDSMC calculation for Edneyrsquos type IVshock interaction also confirmed the same conclusion The

ExperimentDSMC-HS

DSMC-VSSDSMC-VHS

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100

15

10

05

00

Angle (deg)

S t

Figure 23 DSMC of type IV shock interaction (119878119905

)

error is the smallest in the HS model compared with those inthe VHS and VSS models In the region of large Kn numberDSMC using the HS model could be used in pressure andthermal prediction of hypersonic complex flow

5 Conclusion

In the present study the double-cone and Edneyrsquos type IVshock interactions were studied Molecular models wereemployed with DSMC including the HS VHS and VSSmodels Comparisons between DSMC and the experimentwere also conducted

Drawing firm conclusions from this study prior to fullcomparisons of these results is somewhat difficult Howeversome issues are immediately clear from our attempt tocompute these flows

(1) The DSMC method appears to predict qualitativelythe structure of flow separation found in previousCFD calculations

(2) Investigation of the double-cone hypersonic flowreveals that the three collisionsmodels can predict thepressure coefficient and the Stanton number trendsThe agreement between the DSMC simulation andthe experiment is the best in the HS collisions model

(3) Investigation of type IV shock interaction shows thatthe agreement between the DSMC simulation andthe experiment is generally good in the HS and VHScollision models However it fails in the VSS model

(4) Both double-cone flow and type IV shock interactionsimulations show that the DSMC error dependson the Kn number and the models employed forintermolecular interaction Increasing the Kn num-ber decreases the DSMC error The error is the small-est for the HS model compared with those for the

10 The Scientific World Journal

Table 6 Comparisons of calculations ((ValueDSMC minus Valueexperiment)(Valueexperiment))

Run KnThe scale of interaction region

(error in compared with measured data )Peak value of the Stanton number

(error in compared with measured data )HS VHS VSS HS VHS VSS

33 753 times 10minus5 467 734 534 190 209 245

36 891 times 10minus5 24 70 43 78 928 78

26 905 times 10minus5 24 66 38 42 857 485

32 134 times 10minus4 0 59 31 52 578 474

38 252 times 10minus4 68 724 3104 52 684 421

Table 7 Flow conditions of Edneyrsquos type IV shock interaction

Ma Rem Kn10 103 times 10

5

145 times 10minus4

VHS and VSS models When the Kn number is in thelevel of 10minus4 the DSMC error for pressure coefficientthe Stanton number and scale of interaction region iscontrolled within 10

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the Chinese AerospaceInnovation and in part by the Basic Research Fund ofNorthwestern Polytechnical University

References

[1] M S Holden T P Wadhams J K Harvey and G V CandlerldquoComparisons between measurements in regions of laminarshockwave boundary layer interaction in hypersonic flowswithnavier-stokes and DSMC solutionsrdquo Tech Rep RTO-TR-AVT-007-V3 NATO Science and Technology Organization 2006

[2] J Olejniczak and G V Candler ldquoComputation of hyper-sonic shock interaction flow fieldsrdquo in Proceedings of the 7thAIAAASME JointThermophysics and Heat Transfer ConferenceAIAA Paper 98-2446 June 1998

[3] G V Candler I Nompelis M-C Druguet I D Boyd W-LWang and S M Holden ldquoCFD validation for hypersonic flighthypersonic double-cone flow simulationsrdquo in Proceedings of the40th Aerospace Sciences Meeting amp Exhibit AIAA 2002-0581Reno Nev USA January 2002

[4] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Oxford Clarendon Press New York NY USA 1994

[5] Harvey K John M S Holden and T P Wadhams ldquoCode vali-dation study of laminar shockboundary layer and shockshockinteractions in hypersonic flow Part B comparison withnavier-stokes and DSMC solutionsrdquo in Proceedings of the 39thAerospace Sciences Meeting amp Exhibit AIAA 2001-1031 RenoNev USA January 2001

[6] M Gallis C Roy J Payne and T Bartel ldquoDSMC and navier-stokes predictions for hypersonic laminar interacting flowsrdquo inProceedings of the 39th Aerospace Sciences Meeting and ExhibitAIAA 2001-1030 Reno Nev USA January 2001

[7] G V Candler I Nompelis and M-C Druguet ldquoNavier-stokespredictions of hypersonic double-cone and cylinder-flare flowfieldsrdquo in Proceedings of the 39th Aerospace Sciences Meeting ampExhibit AIAA Paper no 2001-1024 Reno Nev USA January2001

[8] J Moss ldquoDSMC computations for regions of shockshock andshockboundary layer interactionrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA 2001-1027 RenoNev USA January 2001

[9] G A Bird The DSMC Method Create Space IndependentPublishing Platform 2013

[10] M Woo and I Greber ldquoMolecular dynamics simulation ofpiston-driven shock wave in hard sphere gasrdquo AIAA Journalvol 37 no 2 pp 215ndash221 1999

[11] K Koura and H Matsumoto ldquoVariable soft sphere molecularmodel for air speciesrdquo Physics of Fluids A vol 4 no 5 pp 1083ndash1085 1992

[12] P S Prasanth and J K Kakkassery ldquoDirect simulation MonteCarlo (DSMC) a numerical method for transition-regimeflows a reviewrdquo Journal of the Indian Institute of Science vol86 no 3 pp 169ndash192 2006

[13] T Tokumasu and Y Matsumoto ldquoDynamic molecular collision(DMC) model for rarefied gas flow simulations by the DSMCmethodrdquo Physics of Fluids vol 11 no 1 pp 1907ndash1920 1999

[14] S F Borisov ldquoProgress in gas-surface interaction studyrdquo inProceedings of the 24th International Rarefied Gas DynamicsSymposium M Capitelli Ed pp 933ndash940 The AmericanInstitute of Physics 2005

[15] I D Boyd and W-L Wang ldquoMonte Carlo computationsof hypersonic interacting flowsrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA Paper no 2001-1029 Reno Nev USA January 2001

[16] J N Moss T Pot B Chanetz and M Lefebvre ldquoDSMCsimulation of shockshock interactions emphases on type IVinteractionrdquo in Proceedings of the 22nd International Symposiumon Shock Waes Paper no 3570 Imperial College London UK1999

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The Scientific World Journal 3

Table 1 Test case in [1]

Run 119881infin

(ms) 120588infin

(Kgm3times103) 119875

infin

(Pa) Ma Rem (times105) Nose26 2581 1093 487 1034 1704 Sharp28 2520 0727 300 1050 1106 Sharp35 2575 0608 185 1249 0945 Sharp32 2572 0758 321 1056 1177 R025033 2636 1288 598 1035 2050 R025036 2584 1108 496 1034 1729 R025038 2453 0498 139 1247 0738 R0288

db

Cr d120576

120576120576

120594

A

O

Rd120594

d120594

dΩ

Crlowast

bd120576

Rsin120594d120576

Figure 2 Illustration of the impact parameters [4]

b

A

O

cr

120579A

120579A120594

crlowast

Figure 3 Collision geometry of HS molecules [4]

Then the total collision cross section can be written as

120590119879

= 2120587int

120587

0

120590 sin120594119889120594 = 2120587int 119887119889119887 (4)

Here the distance of the closest approach 119887 is given indifferent molecular model In the current paper three molec-ular models including HS VHS and VSS are used in thesimulation

(1) HS Model In the HS model the intermolecular force iseffective at

119903 =1

2(1198891

+ 1198892

) = 11988912

(5)

Then as Shown in Figure 3 we can get the closest approach 119887

119887 = 11988912

cos(12120594)

100381610038161003816100381610038161003816100381610038161003816

119889119887

119889120594

100381610038161003816100381610038161003816100381610038161003816

=1

211988912

sin(12120594)

120590 =1198892

12

4

(6)

Therefore the total collision cross section is defined as

120590119879

= 1205871198892

12

(7)

In HS model the molecule diameter can be calculated as

119889 = [(516) (119898119896119879ref120587)

12

120583ref]

12

(8)

where 120583ref is the reference viscosity at temperature 119879ref and119898is the molecular mass

119896 is the Boltzmann constant and 119896 = 1380658times10minus23 JK

(2) VHS Model In the VHS model molecular diameter 119889 isdefined as a function of relative velocity 119888

119903

that is

119889VHS = 119889ref(119888119903ref

119888119903

)

120585

= [(158) (119898120587)

12

(119896119879ref)120596

Γ ((92) minus 120596) 120583ref 120598120596minus(12)

119905

]

12

(9)

Here 120583ref is the reference viscosity at temperature119879ref120596 is thepower law which is expressed as 120596 = (12) + 120585 120585 is relativespeed exponent of VHSmodel and it can be adjusted to meetthe power law 120583 prop 119879

120596 120598119905

is the relative mean kinetic energyexpressed as 120598

119905

equiv (12)119898119903

1198882

119903

119888119903

is relative speed between thecollision molecules 119888

119903ref is reference relative speed betweenthe collision molecules

In the VHS model the closest approach 119887 can be definedas

119887 = 119889VHS cos(1

2120594) (10)

The total collision cross section is expressed as

120590119879

= 120590119879ref(

119889VHS119889ref

)

2

(11)

4 The Scientific World Journal

where 120590119879ref is reference collision cross section at the reference

molecule diameter 119889ref

(3) VSS Model In the VSS model molecular diameter 119889 isdefined as

119889VSS = [5 (120572 + 1) (120572 + 2)(119898120587)

12

(119896119879ref)120596

16120572Γ ((92) minus 120596) 120583ref 120598120596minus(12)

119905

]

12

(12)

Here 120572 is the scattering coefficient (In the case where 120572 = 1the VSS model is the same as the VHS model)

In the VSS model the closest approach 119887 is defined as

119887 = 119889VSScos120572

(1

2120594) (13)

The total collision cross section is defined as

120590119879

= 1205871198892

VSS (14)

232 Gas-Surface Interaction Diffuse reflection [14]with fullaccommodation to the surface temperature is implementedin the simulation It means that molecules diffused underthe Maxwell distribution condition at 119879 = 119879

119882

during thesimulation

24 Sampling The sampling of the macroscopic flow proper-ties is the final process in DSMCThe spatial coordinates andvelocity components of the molecules in a particular cell areused to calculate the macroscopic quantities at the geometriccenter of the cell Consider

120588 = 119899119898

119875 = 120588(119888 minus 119888)2

(15)

where 119899 is the number density of the molecules 119888 is thevelocity of the molecule and 119888 is the average velocity of themolecule

25 Parameters Theparameters selected in the present paperare pressure coefficient and the Stanton number

119862119875

=119875 minus 119875infin

(12) 1205881198812infin

119878119905

=119902

(12) 1205881198813infin

(16)

where 119875 is the flow-field pressure 119875infin

is the far-field pressureof the flow field 120588 is the density of flow-field density 119881

infin

isthe far-field velocity of the flow field and 119902 is the thermal fluxof the solid surface

3 Hypersonic Double-Cone Flow Simulation

31 Moss and Boydrsquos DSMC Predictions for Hypersonic 25∘55∘Double-Cone Flow To validate the DSMC method three25∘55∘ hypersonic double-cone configurations are selectedunder five conditions The experimental data with four

76254

55∘

10309

25∘

Figure 4 Number 1 configuration (in) (sharp nose 25∘55∘ doublecone)

4

55∘

10309

25∘

7283

R025

Figure 5 Number 2 configuration (in) (R025 blunt nose 25∘55∘double cone)

configurations as shown in Figures 4 5 6 and 7 andare reported in [1] in which Moss and Boydrsquos DSMC pre-dictions are compared with the experimental data undersome experimental conditions Boydrsquos predictions using theDSMC code for Run number 35 are shown in Figure 8These predictions clearly underestimated the length of theseparated region and although the heating in the forebodyis slightly underpredicted a good agreement existed betweentheory and experiment for the pressures and the heat transferdownstream of the shockshock interaction in the secondcone [12]

Figures 9 and 10 show the results obtained using theDSMCmethod byMoss [8] andBoyd [15] which significantlyunderpredicted the scale of the interaction region and theposition andmagnitude of the properties in and downstreamof the region of shockshock interaction Moss calculationsshown in Figure 9 indicated that the predicted separatedregion is approximately 25 of the length measured inthe experiment Downstream of the interaction regions thepressure and the heating levels were relatively well predictedBoydrsquos solution for Run number 28 also significantly under-predicted the length of the separated region and unlikeMossrsquos calculation the heat transfer rates both ahead and

The Scientific World Journal 5

4

55∘

1030925

∘

7232

R0288

Figure 6 Number 3 configuration (in) (R0288 blunt nose 25∘55∘double cone)

4

55∘

1030925∘

6805

R06

Figure 7 Number 4 configuration (in) (R06 blunt nose 25∘55∘double cone)

Table 2 Properties of nitrogen molecule in DSMC

Degree of freedom 5Molecular mass times 10

27 Kg 465Viscosity coefficient 1656Viscosity index 074120572 (for VSS model) 136Molecular diameter (VHS) times 10

10m 417Molecular diameter (VSS) times 10

10m 411Molecular diameter (HS) times 10

10m 3784

Table 3 Flow conditions for Run Numbers 26 and 36

Ma Rem KnRun 26 1034 1704 times 10

5

905 times 10minus5

Run 36 1034 173 times 105

891 times 10minus5

downstream of the interaction were underpredicted Signifi-cant improvement of these calculations is doubtful even withthe incorporation of vibrational nonequilibrium effects

32 Our DSMC Simulation In [1] Run numbers 26 32 3336 and 38 were not simulated by DSMC and CFDThereforewe selected these cases to validate the DSMC code with theHS VHS and VSS models

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

Boyd CpBoyd St

S t

Figure 8 Boydrsquos DSMC for Run number 35 [1 15]

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

S t

Moss CpMoss St

Figure 9 Mossrsquos DSMC for Run number 28 [8]

The computational grid employed for Run numbers 2632 33 36 and 38 consists of 2048 cells along the body by 512cells normal to the bodyThis grid is the same as that of BoydrsquosDSMC simulation [15] The base time step for the simulationwas 10minus9 s although this was reduced in very high densityregionsThe solutions were obtained using approximately 64million particles and averaged over a sampling interval of05ms At this time the solution had attained steady state Inparticular the size of the separation region did not change

The properties of nitrogen molecule used in the currentstudy are presented in Table 2

321 DSMC Simulation of Run Numbers 26 and 36 In thissection the DSMC simulation result of Run numbers 26 and36 is presented As shown in Table 3 these two cases wereconducted in the sameMa and Re conditions with sharp noseand R025 in blunt nose respectively

6 The Scientific World Journal

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

Boyd CpBoyd St

S tFigure 10 Boydrsquos DSMC for Run number 28 [15]

000 025 050 075 100 125 150 175 200 225

00

05

10

15

20

25

30

Cp

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L =92075 mm) Run number 26

Figure 11 DSMC simulation of Run number 26 (119862119875

)

The pressure coefficient and the Stanton numberobtained with HS VHS and VSS are shown in Figures11ndash14 These results underpredict the scale of interactionregion and the position and also the magnitude of theparameters is underpredicted in and downstream The HScalculation shown in Figure 11 for Run number 26 indicatesthat the predicted interaction region is approximately 76of the measured experimental length Those of VHS andVSS are 34 and 62 respectively The HS results shownin Figure 12 for Run number 36 indicate that the predictedinteraction region is approximately 76 of the measuredexperimental length Those of VHS and VSS are 30 and57 respectively These predictions are much improvedcompared with Mossrsquos DSMC calculation which was only25 of the lengthmeasured in the experiment [8]The resultsof three models overpredicted the ahead pressure coefficient

000 025 050 075 100 125 150 175 200 225

00

05

10

15

20

25

30

Cp

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 36

Figure 12 DSMC simulation of Run number 36 (119862119875

)

000 025 050 075 100 125 150 175 200 225

0200

0175

0150

0125

0100

0075

0050

0025

0000

minus0025

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 26-

S t

St

Figure 13 DSMC simulation of Run number 26 (119878119905

)

peak value In the downstream of the interaction regionsthe pressure levels got by the three models were relativelywell predicted And also the three models overpredictedthe Stanton number in comparison with the experimentaldata The position of the Stanton number peak was wellpredicted by the three models Figure 13 shows that thepeak values of the Stanton number using HS VHS and VSSare approximately 142 1857 and 1485 respectivelyof the data measured in the experiment for Run number26 Figure 14 shows that the peak values of the Stantonnumber using HS VHS and VSS are approximately 1781928 and 178 respectively of the data measured in theexperiment for Run number 36

322 DSMC Simulation of Run Number 33 In this sectionthe DSMC simulation result of Run number 33 is presented

The Scientific World Journal 7

000 025 050 075 100 125 150 175 200 225

016015014013012011010009008007006005004003002001000

minus001

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 36-

S t

St

Figure 14 DSMC simulation of Run number 36 (119878119905

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

10

05

00

minus05

Cp

XL (L = 92075 mm) Run number 33

Figure 15 DSMC simulation of Run number 33 (119862119875

)

Table 4 Flow conditions for Run Numbers 33 and Numbers 36

Ma Rem KnRun 33 1035 205 times 10

5

753 times 10minus5

Run 36 1034 173 times 105

891 times 10minus5

Table 5 Flow conditions for Run Numbers 32 and 38

Ma Rem KnRun 32 1056 1177 times 10

5

134 times 10minus4

Run 38 1247 0738 times 105

252 times 10minus4

As shown in Table 4 Run numbers 33 and 36 were conductedin the same configuration with different Kn condation

The pressure coefficient and the Stanton numberobtained from HS VHS and VSS using the DSMC

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

020

015

010

005

000

XL (L = 92075 mm) Run number 33-St

S t

Figure 16 DSMC simulation of Run number 33 (119878119905

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

10

05

00

Cp

XL (L = 92075 mm) Run number 32

Figure 17 DSMC simulation of Run number 32 (119862119875

)

method are shown in Figures 15 and 16 These results alsounderpredicted the scale of interaction region and positionmagnitude of the properties in and downstream ofthe shock interaction region The HS result shown inFigure 15 indicates that the predicted interaction regionis approximately 533 of the length measured in theexperiment Those of VHS and VSS are 266 and466 respectively These predictions are not improvedin comparison with Run numbers 26 and 36 calculationsThe three modelsrsquo calculation also overpredicted the aheadpressure coefficient peak value In the downstream of theinteraction region the pressure levels from three models arerelatively well predicted Also three models overpredictedthe Stanton number compared with the experimentaldata But the position of the Stanton number peak is wellpredicted in the three modelsrsquo calculation Figure 16 shows

8 The Scientific World Journal

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

05

10

00

Cp

XL (L = 92075 mm) Run number 38

Figure 18 DSMC simulation of Run number 38 (119862119875

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

016

014

012

010

008

006

004

002

000

minus002

XL (L = 92075 mm) Run number 32-

S t

St

Figure 19 DSMC simulation of Run number 32 (119878119905

)

that the peak values of the Stanton number in HS VHS andVSS are approximately 290 309 and 345 respectivelyof the data measured in the experiment for Run number 33

323 DSMC Simulation for Run Numbers 32 and 38 In thissection theDSMCsimulation results forRunnumbers 32 and38 as shown in Table 5 are presented Run numbers 33 38and 36 were conducted in the same configuration with R025in blunt nose

Thepressure coefficient and the Stanton number obtainedby HS VHS and VSS using the DSMC method are shownin Figures 17ndash20 These results well predicted the scale of theinteraction region and the position magnitude of the proper-ties in and downstream of the shock interaction region TheHS result shown in Figure 17 for Run number 32 indicates

000 025 050 075 100 125 150 175 200 225minus002

000

002

004

006

008

010

012

014

016

018

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 38-

S t

St

Figure 20 DSMC simulation of Run number 38 (119878119905

)

35635

Rn = 8

53L2

L3

L1

L

L 950 to 1055

20∘ 10∘10∘

Dimensions in (mm)

y

x

Origin of coordinate system

L1 = L2 = 50771L3 = 100

Figure 21 Configuration of Edneyrsquos Type IV shock interaction

that the predicted interaction region is approximately 100of the length measured in the experiment Those of VHS andVSS are 41 and 69 respectively The HS result shown inFigure 18 for Run number 38 indicates that the predictedinteraction region is approximately 1068 of the lengthmeasured in the experimentThose ofVHS andVSS are 276and 6896 respectively These predictions are improvedin comparison with Run numbers 26 and 36 calculationsAlso the three modelsrsquo calculation overpredicted the aheadpressure coefficient peak value And in the downstreamof interaction region the pressure levels from the threemodels are relatively well predicted The three models alsooverpredicted the Stanton number in comparison with theexperimental data And also the position of the Stanton num-ber peak is well predicted in the three modelsrsquo calculationFigure 19 shows that the peak values of the Stanton numberin HS VHS and VSS are approximately 1052 1578 and

The Scientific World Journal 9

Experiment

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100minus2

0

2

4

6

8

10

12

14

16

Cp

Angle (deg)

DSMC-HSDSMC-VSSDSMC-VHS

Figure 22 DSMC of type IV shock interaction (119862119875

)

1474 respectively of the data measured in the experimentfor Run number 32 Figure 20 shows that the peak value ofthe Stanton number in HS VHS and VSS are approximately1052 1684 and 1421 respectively of the datameasuredin the experiment for Run number 38

Table 6 shows that the DSMC error depends on theKn number and the models employed for intermolecularinteraction With the increase in the Kn number the DSMCerror decreases The error is the smallest in the HS modelcompared with those in the VHS and VSS models Withthe Kn number increasing the Stanton number DSMC errordecreases When the Kn number is in the level of 10minus4 theDSMC error for pressure coefficient and Stanton number iscontrolled within 10

4 Type IV Shock Interaction Simulation

In the present study results of the numerical simulations forMach 10 air flow as shown in Table 7 were presented for arange of Edneyrsquos type IV shock interactionThe experimentaldata for validation were obtained from [16] The configura-tion is shown in Figure 21

The results of the grid sensitivity investigation are pre-sented in [16] The number of cellssubcells used for oursimulation was 97060958490 proven in [16] to be thefinestThe computational time step was 12 nsThe simulationconsisted of 156 times 10

5 time steps to ensure steady stateconditions

The DSMC data of the pressure coefficient and theStanton number are shown in Figures 22 and 23 The agree-ment between the DSMC calculation and the experiment isgenerally good BothHS andVHSmodels gain improvementsin the errors The error between the HS-DSMC data and theexperimental data is controlled within 5

The result from theDSMC calculation for Edneyrsquos type IVshock interaction also confirmed the same conclusion The

ExperimentDSMC-HS

DSMC-VSSDSMC-VHS

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100

15

10

05

00

Angle (deg)

S t

Figure 23 DSMC of type IV shock interaction (119878119905

)

error is the smallest in the HS model compared with those inthe VHS and VSS models In the region of large Kn numberDSMC using the HS model could be used in pressure andthermal prediction of hypersonic complex flow

5 Conclusion

In the present study the double-cone and Edneyrsquos type IVshock interactions were studied Molecular models wereemployed with DSMC including the HS VHS and VSSmodels Comparisons between DSMC and the experimentwere also conducted

Drawing firm conclusions from this study prior to fullcomparisons of these results is somewhat difficult Howeversome issues are immediately clear from our attempt tocompute these flows

(1) The DSMC method appears to predict qualitativelythe structure of flow separation found in previousCFD calculations

(2) Investigation of the double-cone hypersonic flowreveals that the three collisionsmodels can predict thepressure coefficient and the Stanton number trendsThe agreement between the DSMC simulation andthe experiment is the best in the HS collisions model

(3) Investigation of type IV shock interaction shows thatthe agreement between the DSMC simulation andthe experiment is generally good in the HS and VHScollision models However it fails in the VSS model

(4) Both double-cone flow and type IV shock interactionsimulations show that the DSMC error dependson the Kn number and the models employed forintermolecular interaction Increasing the Kn num-ber decreases the DSMC error The error is the small-est for the HS model compared with those for the

10 The Scientific World Journal

Table 6 Comparisons of calculations ((ValueDSMC minus Valueexperiment)(Valueexperiment))

Run KnThe scale of interaction region

(error in compared with measured data )Peak value of the Stanton number

(error in compared with measured data )HS VHS VSS HS VHS VSS

33 753 times 10minus5 467 734 534 190 209 245

36 891 times 10minus5 24 70 43 78 928 78

26 905 times 10minus5 24 66 38 42 857 485

32 134 times 10minus4 0 59 31 52 578 474

38 252 times 10minus4 68 724 3104 52 684 421

Table 7 Flow conditions of Edneyrsquos type IV shock interaction

Ma Rem Kn10 103 times 10

5

145 times 10minus4

VHS and VSS models When the Kn number is in thelevel of 10minus4 the DSMC error for pressure coefficientthe Stanton number and scale of interaction region iscontrolled within 10

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the Chinese AerospaceInnovation and in part by the Basic Research Fund ofNorthwestern Polytechnical University

References

[1] M S Holden T P Wadhams J K Harvey and G V CandlerldquoComparisons between measurements in regions of laminarshockwave boundary layer interaction in hypersonic flowswithnavier-stokes and DSMC solutionsrdquo Tech Rep RTO-TR-AVT-007-V3 NATO Science and Technology Organization 2006

[2] J Olejniczak and G V Candler ldquoComputation of hyper-sonic shock interaction flow fieldsrdquo in Proceedings of the 7thAIAAASME JointThermophysics and Heat Transfer ConferenceAIAA Paper 98-2446 June 1998

[3] G V Candler I Nompelis M-C Druguet I D Boyd W-LWang and S M Holden ldquoCFD validation for hypersonic flighthypersonic double-cone flow simulationsrdquo in Proceedings of the40th Aerospace Sciences Meeting amp Exhibit AIAA 2002-0581Reno Nev USA January 2002

[4] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Oxford Clarendon Press New York NY USA 1994

[5] Harvey K John M S Holden and T P Wadhams ldquoCode vali-dation study of laminar shockboundary layer and shockshockinteractions in hypersonic flow Part B comparison withnavier-stokes and DSMC solutionsrdquo in Proceedings of the 39thAerospace Sciences Meeting amp Exhibit AIAA 2001-1031 RenoNev USA January 2001

[6] M Gallis C Roy J Payne and T Bartel ldquoDSMC and navier-stokes predictions for hypersonic laminar interacting flowsrdquo inProceedings of the 39th Aerospace Sciences Meeting and ExhibitAIAA 2001-1030 Reno Nev USA January 2001

[7] G V Candler I Nompelis and M-C Druguet ldquoNavier-stokespredictions of hypersonic double-cone and cylinder-flare flowfieldsrdquo in Proceedings of the 39th Aerospace Sciences Meeting ampExhibit AIAA Paper no 2001-1024 Reno Nev USA January2001

[8] J Moss ldquoDSMC computations for regions of shockshock andshockboundary layer interactionrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA 2001-1027 RenoNev USA January 2001

[9] G A Bird The DSMC Method Create Space IndependentPublishing Platform 2013

[10] M Woo and I Greber ldquoMolecular dynamics simulation ofpiston-driven shock wave in hard sphere gasrdquo AIAA Journalvol 37 no 2 pp 215ndash221 1999

[11] K Koura and H Matsumoto ldquoVariable soft sphere molecularmodel for air speciesrdquo Physics of Fluids A vol 4 no 5 pp 1083ndash1085 1992

[12] P S Prasanth and J K Kakkassery ldquoDirect simulation MonteCarlo (DSMC) a numerical method for transition-regimeflows a reviewrdquo Journal of the Indian Institute of Science vol86 no 3 pp 169ndash192 2006

[13] T Tokumasu and Y Matsumoto ldquoDynamic molecular collision(DMC) model for rarefied gas flow simulations by the DSMCmethodrdquo Physics of Fluids vol 11 no 1 pp 1907ndash1920 1999

[14] S F Borisov ldquoProgress in gas-surface interaction studyrdquo inProceedings of the 24th International Rarefied Gas DynamicsSymposium M Capitelli Ed pp 933ndash940 The AmericanInstitute of Physics 2005

[15] I D Boyd and W-L Wang ldquoMonte Carlo computationsof hypersonic interacting flowsrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA Paper no 2001-1029 Reno Nev USA January 2001

[16] J N Moss T Pot B Chanetz and M Lefebvre ldquoDSMCsimulation of shockshock interactions emphases on type IVinteractionrdquo in Proceedings of the 22nd International Symposiumon Shock Waes Paper no 3570 Imperial College London UK1999

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 The Scientific World Journal

where 120590119879ref is reference collision cross section at the reference

molecule diameter 119889ref

(3) VSS Model In the VSS model molecular diameter 119889 isdefined as

119889VSS = [5 (120572 + 1) (120572 + 2)(119898120587)

12

(119896119879ref)120596

16120572Γ ((92) minus 120596) 120583ref 120598120596minus(12)

119905

]

12

(12)

Here 120572 is the scattering coefficient (In the case where 120572 = 1the VSS model is the same as the VHS model)

In the VSS model the closest approach 119887 is defined as

119887 = 119889VSScos120572

(1

2120594) (13)

The total collision cross section is defined as

120590119879

= 1205871198892

VSS (14)

232 Gas-Surface Interaction Diffuse reflection [14]with fullaccommodation to the surface temperature is implementedin the simulation It means that molecules diffused underthe Maxwell distribution condition at 119879 = 119879

119882

during thesimulation

24 Sampling The sampling of the macroscopic flow proper-ties is the final process in DSMCThe spatial coordinates andvelocity components of the molecules in a particular cell areused to calculate the macroscopic quantities at the geometriccenter of the cell Consider

120588 = 119899119898

119875 = 120588(119888 minus 119888)2

(15)

where 119899 is the number density of the molecules 119888 is thevelocity of the molecule and 119888 is the average velocity of themolecule

25 Parameters Theparameters selected in the present paperare pressure coefficient and the Stanton number

119862119875

=119875 minus 119875infin

(12) 1205881198812infin

119878119905

=119902

(12) 1205881198813infin

(16)

where 119875 is the flow-field pressure 119875infin

is the far-field pressureof the flow field 120588 is the density of flow-field density 119881

infin

isthe far-field velocity of the flow field and 119902 is the thermal fluxof the solid surface

3 Hypersonic Double-Cone Flow Simulation

31 Moss and Boydrsquos DSMC Predictions for Hypersonic 25∘55∘Double-Cone Flow To validate the DSMC method three25∘55∘ hypersonic double-cone configurations are selectedunder five conditions The experimental data with four

76254

55∘

10309

25∘

Figure 4 Number 1 configuration (in) (sharp nose 25∘55∘ doublecone)

4

55∘

10309

25∘

7283

R025

Figure 5 Number 2 configuration (in) (R025 blunt nose 25∘55∘double cone)

configurations as shown in Figures 4 5 6 and 7 andare reported in [1] in which Moss and Boydrsquos DSMC pre-dictions are compared with the experimental data undersome experimental conditions Boydrsquos predictions using theDSMC code for Run number 35 are shown in Figure 8These predictions clearly underestimated the length of theseparated region and although the heating in the forebodyis slightly underpredicted a good agreement existed betweentheory and experiment for the pressures and the heat transferdownstream of the shockshock interaction in the secondcone [12]

Figures 9 and 10 show the results obtained using theDSMCmethod byMoss [8] andBoyd [15] which significantlyunderpredicted the scale of the interaction region and theposition andmagnitude of the properties in and downstreamof the region of shockshock interaction Moss calculationsshown in Figure 9 indicated that the predicted separatedregion is approximately 25 of the length measured inthe experiment Downstream of the interaction regions thepressure and the heating levels were relatively well predictedBoydrsquos solution for Run number 28 also significantly under-predicted the length of the separated region and unlikeMossrsquos calculation the heat transfer rates both ahead and

The Scientific World Journal 5

4

55∘

1030925

∘

7232

R0288

Figure 6 Number 3 configuration (in) (R0288 blunt nose 25∘55∘double cone)

4

55∘

1030925∘

6805

R06

Figure 7 Number 4 configuration (in) (R06 blunt nose 25∘55∘double cone)

Table 2 Properties of nitrogen molecule in DSMC

Degree of freedom 5Molecular mass times 10

27 Kg 465Viscosity coefficient 1656Viscosity index 074120572 (for VSS model) 136Molecular diameter (VHS) times 10

10m 417Molecular diameter (VSS) times 10

10m 411Molecular diameter (HS) times 10

10m 3784

Table 3 Flow conditions for Run Numbers 26 and 36

Ma Rem KnRun 26 1034 1704 times 10

5

905 times 10minus5

Run 36 1034 173 times 105

891 times 10minus5

downstream of the interaction were underpredicted Signifi-cant improvement of these calculations is doubtful even withthe incorporation of vibrational nonequilibrium effects

32 Our DSMC Simulation In [1] Run numbers 26 32 3336 and 38 were not simulated by DSMC and CFDThereforewe selected these cases to validate the DSMC code with theHS VHS and VSS models

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

Boyd CpBoyd St

S t

Figure 8 Boydrsquos DSMC for Run number 35 [1 15]

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

S t

Moss CpMoss St

Figure 9 Mossrsquos DSMC for Run number 28 [8]

The computational grid employed for Run numbers 2632 33 36 and 38 consists of 2048 cells along the body by 512cells normal to the bodyThis grid is the same as that of BoydrsquosDSMC simulation [15] The base time step for the simulationwas 10minus9 s although this was reduced in very high densityregionsThe solutions were obtained using approximately 64million particles and averaged over a sampling interval of05ms At this time the solution had attained steady state Inparticular the size of the separation region did not change

The properties of nitrogen molecule used in the currentstudy are presented in Table 2

321 DSMC Simulation of Run Numbers 26 and 36 In thissection the DSMC simulation result of Run numbers 26 and36 is presented As shown in Table 3 these two cases wereconducted in the sameMa and Re conditions with sharp noseand R025 in blunt nose respectively

6 The Scientific World Journal

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

Boyd CpBoyd St

S tFigure 10 Boydrsquos DSMC for Run number 28 [15]

000 025 050 075 100 125 150 175 200 225

00

05

10

15

20

25

30

Cp

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L =92075 mm) Run number 26

Figure 11 DSMC simulation of Run number 26 (119862119875

)

The pressure coefficient and the Stanton numberobtained with HS VHS and VSS are shown in Figures11ndash14 These results underpredict the scale of interactionregion and the position and also the magnitude of theparameters is underpredicted in and downstream The HScalculation shown in Figure 11 for Run number 26 indicatesthat the predicted interaction region is approximately 76of the measured experimental length Those of VHS andVSS are 34 and 62 respectively The HS results shownin Figure 12 for Run number 36 indicate that the predictedinteraction region is approximately 76 of the measuredexperimental length Those of VHS and VSS are 30 and57 respectively These predictions are much improvedcompared with Mossrsquos DSMC calculation which was only25 of the lengthmeasured in the experiment [8]The resultsof three models overpredicted the ahead pressure coefficient

000 025 050 075 100 125 150 175 200 225

00

05

10

15

20

25

30

Cp

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 36

Figure 12 DSMC simulation of Run number 36 (119862119875

)

000 025 050 075 100 125 150 175 200 225

0200

0175

0150

0125

0100

0075

0050

0025

0000

minus0025

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 26-

S t

St

Figure 13 DSMC simulation of Run number 26 (119878119905

)

peak value In the downstream of the interaction regionsthe pressure levels got by the three models were relativelywell predicted And also the three models overpredictedthe Stanton number in comparison with the experimentaldata The position of the Stanton number peak was wellpredicted by the three models Figure 13 shows that thepeak values of the Stanton number using HS VHS and VSSare approximately 142 1857 and 1485 respectivelyof the data measured in the experiment for Run number26 Figure 14 shows that the peak values of the Stantonnumber using HS VHS and VSS are approximately 1781928 and 178 respectively of the data measured in theexperiment for Run number 36

322 DSMC Simulation of Run Number 33 In this sectionthe DSMC simulation result of Run number 33 is presented

The Scientific World Journal 7

000 025 050 075 100 125 150 175 200 225

016015014013012011010009008007006005004003002001000

minus001

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 36-

S t

St

Figure 14 DSMC simulation of Run number 36 (119878119905

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

10

05

00

minus05

Cp

XL (L = 92075 mm) Run number 33

Figure 15 DSMC simulation of Run number 33 (119862119875

)

Table 4 Flow conditions for Run Numbers 33 and Numbers 36

Ma Rem KnRun 33 1035 205 times 10

5

753 times 10minus5

Run 36 1034 173 times 105

891 times 10minus5

Table 5 Flow conditions for Run Numbers 32 and 38

Ma Rem KnRun 32 1056 1177 times 10

5

134 times 10minus4

Run 38 1247 0738 times 105

252 times 10minus4

As shown in Table 4 Run numbers 33 and 36 were conductedin the same configuration with different Kn condation

The pressure coefficient and the Stanton numberobtained from HS VHS and VSS using the DSMC

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

020

015

010

005

000

XL (L = 92075 mm) Run number 33-St

S t

Figure 16 DSMC simulation of Run number 33 (119878119905

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

10

05

00

Cp

XL (L = 92075 mm) Run number 32

Figure 17 DSMC simulation of Run number 32 (119862119875

)

method are shown in Figures 15 and 16 These results alsounderpredicted the scale of interaction region and positionmagnitude of the properties in and downstream ofthe shock interaction region The HS result shown inFigure 15 indicates that the predicted interaction regionis approximately 533 of the length measured in theexperiment Those of VHS and VSS are 266 and466 respectively These predictions are not improvedin comparison with Run numbers 26 and 36 calculationsThe three modelsrsquo calculation also overpredicted the aheadpressure coefficient peak value In the downstream of theinteraction region the pressure levels from three models arerelatively well predicted Also three models overpredictedthe Stanton number compared with the experimentaldata But the position of the Stanton number peak is wellpredicted in the three modelsrsquo calculation Figure 16 shows

8 The Scientific World Journal

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

05

10

00

Cp

XL (L = 92075 mm) Run number 38

Figure 18 DSMC simulation of Run number 38 (119862119875

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

016

014

012

010

008

006

004

002

000

minus002

XL (L = 92075 mm) Run number 32-

S t

St

Figure 19 DSMC simulation of Run number 32 (119878119905

)

that the peak values of the Stanton number in HS VHS andVSS are approximately 290 309 and 345 respectivelyof the data measured in the experiment for Run number 33

323 DSMC Simulation for Run Numbers 32 and 38 In thissection theDSMCsimulation results forRunnumbers 32 and38 as shown in Table 5 are presented Run numbers 33 38and 36 were conducted in the same configuration with R025in blunt nose

Thepressure coefficient and the Stanton number obtainedby HS VHS and VSS using the DSMC method are shownin Figures 17ndash20 These results well predicted the scale of theinteraction region and the position magnitude of the proper-ties in and downstream of the shock interaction region TheHS result shown in Figure 17 for Run number 32 indicates

000 025 050 075 100 125 150 175 200 225minus002

000

002

004

006

008

010

012

014

016

018

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 38-

S t

St

Figure 20 DSMC simulation of Run number 38 (119878119905

)

35635

Rn = 8

53L2

L3

L1

L

L 950 to 1055

20∘ 10∘10∘

Dimensions in (mm)

y

x

Origin of coordinate system

L1 = L2 = 50771L3 = 100

Figure 21 Configuration of Edneyrsquos Type IV shock interaction

that the predicted interaction region is approximately 100of the length measured in the experiment Those of VHS andVSS are 41 and 69 respectively The HS result shown inFigure 18 for Run number 38 indicates that the predictedinteraction region is approximately 1068 of the lengthmeasured in the experimentThose ofVHS andVSS are 276and 6896 respectively These predictions are improvedin comparison with Run numbers 26 and 36 calculationsAlso the three modelsrsquo calculation overpredicted the aheadpressure coefficient peak value And in the downstreamof interaction region the pressure levels from the threemodels are relatively well predicted The three models alsooverpredicted the Stanton number in comparison with theexperimental data And also the position of the Stanton num-ber peak is well predicted in the three modelsrsquo calculationFigure 19 shows that the peak values of the Stanton numberin HS VHS and VSS are approximately 1052 1578 and

The Scientific World Journal 9

Experiment

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100minus2

0

2

4

6

8

10

12

14

16

Cp

Angle (deg)

DSMC-HSDSMC-VSSDSMC-VHS

Figure 22 DSMC of type IV shock interaction (119862119875

)

1474 respectively of the data measured in the experimentfor Run number 32 Figure 20 shows that the peak value ofthe Stanton number in HS VHS and VSS are approximately1052 1684 and 1421 respectively of the datameasuredin the experiment for Run number 38

Table 6 shows that the DSMC error depends on theKn number and the models employed for intermolecularinteraction With the increase in the Kn number the DSMCerror decreases The error is the smallest in the HS modelcompared with those in the VHS and VSS models Withthe Kn number increasing the Stanton number DSMC errordecreases When the Kn number is in the level of 10minus4 theDSMC error for pressure coefficient and Stanton number iscontrolled within 10

4 Type IV Shock Interaction Simulation

In the present study results of the numerical simulations forMach 10 air flow as shown in Table 7 were presented for arange of Edneyrsquos type IV shock interactionThe experimentaldata for validation were obtained from [16] The configura-tion is shown in Figure 21

The results of the grid sensitivity investigation are pre-sented in [16] The number of cellssubcells used for oursimulation was 97060958490 proven in [16] to be thefinestThe computational time step was 12 nsThe simulationconsisted of 156 times 10

5 time steps to ensure steady stateconditions

The DSMC data of the pressure coefficient and theStanton number are shown in Figures 22 and 23 The agree-ment between the DSMC calculation and the experiment isgenerally good BothHS andVHSmodels gain improvementsin the errors The error between the HS-DSMC data and theexperimental data is controlled within 5

The result from theDSMC calculation for Edneyrsquos type IVshock interaction also confirmed the same conclusion The

ExperimentDSMC-HS

DSMC-VSSDSMC-VHS

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100

15

10

05

00

Angle (deg)

S t

Figure 23 DSMC of type IV shock interaction (119878119905

)

error is the smallest in the HS model compared with those inthe VHS and VSS models In the region of large Kn numberDSMC using the HS model could be used in pressure andthermal prediction of hypersonic complex flow

5 Conclusion

In the present study the double-cone and Edneyrsquos type IVshock interactions were studied Molecular models wereemployed with DSMC including the HS VHS and VSSmodels Comparisons between DSMC and the experimentwere also conducted

Drawing firm conclusions from this study prior to fullcomparisons of these results is somewhat difficult Howeversome issues are immediately clear from our attempt tocompute these flows

(1) The DSMC method appears to predict qualitativelythe structure of flow separation found in previousCFD calculations

(2) Investigation of the double-cone hypersonic flowreveals that the three collisionsmodels can predict thepressure coefficient and the Stanton number trendsThe agreement between the DSMC simulation andthe experiment is the best in the HS collisions model

(3) Investigation of type IV shock interaction shows thatthe agreement between the DSMC simulation andthe experiment is generally good in the HS and VHScollision models However it fails in the VSS model

(4) Both double-cone flow and type IV shock interactionsimulations show that the DSMC error dependson the Kn number and the models employed forintermolecular interaction Increasing the Kn num-ber decreases the DSMC error The error is the small-est for the HS model compared with those for the

10 The Scientific World Journal

Table 6 Comparisons of calculations ((ValueDSMC minus Valueexperiment)(Valueexperiment))

Run KnThe scale of interaction region

(error in compared with measured data )Peak value of the Stanton number

(error in compared with measured data )HS VHS VSS HS VHS VSS

33 753 times 10minus5 467 734 534 190 209 245

36 891 times 10minus5 24 70 43 78 928 78

26 905 times 10minus5 24 66 38 42 857 485

32 134 times 10minus4 0 59 31 52 578 474

38 252 times 10minus4 68 724 3104 52 684 421

Table 7 Flow conditions of Edneyrsquos type IV shock interaction

Ma Rem Kn10 103 times 10

5

145 times 10minus4

VHS and VSS models When the Kn number is in thelevel of 10minus4 the DSMC error for pressure coefficientthe Stanton number and scale of interaction region iscontrolled within 10

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the Chinese AerospaceInnovation and in part by the Basic Research Fund ofNorthwestern Polytechnical University

References

[1] M S Holden T P Wadhams J K Harvey and G V CandlerldquoComparisons between measurements in regions of laminarshockwave boundary layer interaction in hypersonic flowswithnavier-stokes and DSMC solutionsrdquo Tech Rep RTO-TR-AVT-007-V3 NATO Science and Technology Organization 2006

[2] J Olejniczak and G V Candler ldquoComputation of hyper-sonic shock interaction flow fieldsrdquo in Proceedings of the 7thAIAAASME JointThermophysics and Heat Transfer ConferenceAIAA Paper 98-2446 June 1998

[3] G V Candler I Nompelis M-C Druguet I D Boyd W-LWang and S M Holden ldquoCFD validation for hypersonic flighthypersonic double-cone flow simulationsrdquo in Proceedings of the40th Aerospace Sciences Meeting amp Exhibit AIAA 2002-0581Reno Nev USA January 2002

[4] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Oxford Clarendon Press New York NY USA 1994

[5] Harvey K John M S Holden and T P Wadhams ldquoCode vali-dation study of laminar shockboundary layer and shockshockinteractions in hypersonic flow Part B comparison withnavier-stokes and DSMC solutionsrdquo in Proceedings of the 39thAerospace Sciences Meeting amp Exhibit AIAA 2001-1031 RenoNev USA January 2001

[6] M Gallis C Roy J Payne and T Bartel ldquoDSMC and navier-stokes predictions for hypersonic laminar interacting flowsrdquo inProceedings of the 39th Aerospace Sciences Meeting and ExhibitAIAA 2001-1030 Reno Nev USA January 2001

[7] G V Candler I Nompelis and M-C Druguet ldquoNavier-stokespredictions of hypersonic double-cone and cylinder-flare flowfieldsrdquo in Proceedings of the 39th Aerospace Sciences Meeting ampExhibit AIAA Paper no 2001-1024 Reno Nev USA January2001

[8] J Moss ldquoDSMC computations for regions of shockshock andshockboundary layer interactionrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA 2001-1027 RenoNev USA January 2001

[9] G A Bird The DSMC Method Create Space IndependentPublishing Platform 2013

[10] M Woo and I Greber ldquoMolecular dynamics simulation ofpiston-driven shock wave in hard sphere gasrdquo AIAA Journalvol 37 no 2 pp 215ndash221 1999

[11] K Koura and H Matsumoto ldquoVariable soft sphere molecularmodel for air speciesrdquo Physics of Fluids A vol 4 no 5 pp 1083ndash1085 1992

[12] P S Prasanth and J K Kakkassery ldquoDirect simulation MonteCarlo (DSMC) a numerical method for transition-regimeflows a reviewrdquo Journal of the Indian Institute of Science vol86 no 3 pp 169ndash192 2006

[13] T Tokumasu and Y Matsumoto ldquoDynamic molecular collision(DMC) model for rarefied gas flow simulations by the DSMCmethodrdquo Physics of Fluids vol 11 no 1 pp 1907ndash1920 1999

[14] S F Borisov ldquoProgress in gas-surface interaction studyrdquo inProceedings of the 24th International Rarefied Gas DynamicsSymposium M Capitelli Ed pp 933ndash940 The AmericanInstitute of Physics 2005

[15] I D Boyd and W-L Wang ldquoMonte Carlo computationsof hypersonic interacting flowsrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA Paper no 2001-1029 Reno Nev USA January 2001

[16] J N Moss T Pot B Chanetz and M Lefebvre ldquoDSMCsimulation of shockshock interactions emphases on type IVinteractionrdquo in Proceedings of the 22nd International Symposiumon Shock Waes Paper no 3570 Imperial College London UK1999

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 5

4

55∘

1030925

∘

7232

R0288

Figure 6 Number 3 configuration (in) (R0288 blunt nose 25∘55∘double cone)

4

55∘

1030925∘

6805

R06

Figure 7 Number 4 configuration (in) (R06 blunt nose 25∘55∘double cone)

Table 2 Properties of nitrogen molecule in DSMC

Degree of freedom 5Molecular mass times 10

27 Kg 465Viscosity coefficient 1656Viscosity index 074120572 (for VSS model) 136Molecular diameter (VHS) times 10

10m 417Molecular diameter (VSS) times 10

10m 411Molecular diameter (HS) times 10

10m 3784

Table 3 Flow conditions for Run Numbers 26 and 36

Ma Rem KnRun 26 1034 1704 times 10

5

905 times 10minus5

Run 36 1034 173 times 105

891 times 10minus5

downstream of the interaction were underpredicted Signifi-cant improvement of these calculations is doubtful even withthe incorporation of vibrational nonequilibrium effects

32 Our DSMC Simulation In [1] Run numbers 26 32 3336 and 38 were not simulated by DSMC and CFDThereforewe selected these cases to validate the DSMC code with theHS VHS and VSS models

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

Boyd CpBoyd St

S t

Figure 8 Boydrsquos DSMC for Run number 35 [1 15]

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

S t

Moss CpMoss St

Figure 9 Mossrsquos DSMC for Run number 28 [8]

The computational grid employed for Run numbers 2632 33 36 and 38 consists of 2048 cells along the body by 512cells normal to the bodyThis grid is the same as that of BoydrsquosDSMC simulation [15] The base time step for the simulationwas 10minus9 s although this was reduced in very high densityregionsThe solutions were obtained using approximately 64million particles and averaged over a sampling interval of05ms At this time the solution had attained steady state Inparticular the size of the separation region did not change

The properties of nitrogen molecule used in the currentstudy are presented in Table 2

321 DSMC Simulation of Run Numbers 26 and 36 In thissection the DSMC simulation result of Run numbers 26 and36 is presented As shown in Table 3 these two cases wereconducted in the sameMa and Re conditions with sharp noseand R025 in blunt nose respectively

6 The Scientific World Journal

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

Boyd CpBoyd St

S tFigure 10 Boydrsquos DSMC for Run number 28 [15]

000 025 050 075 100 125 150 175 200 225

00

05

10

15

20

25

30

Cp

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L =92075 mm) Run number 26

Figure 11 DSMC simulation of Run number 26 (119862119875

)

The pressure coefficient and the Stanton numberobtained with HS VHS and VSS are shown in Figures11ndash14 These results underpredict the scale of interactionregion and the position and also the magnitude of theparameters is underpredicted in and downstream The HScalculation shown in Figure 11 for Run number 26 indicatesthat the predicted interaction region is approximately 76of the measured experimental length Those of VHS andVSS are 34 and 62 respectively The HS results shownin Figure 12 for Run number 36 indicate that the predictedinteraction region is approximately 76 of the measuredexperimental length Those of VHS and VSS are 30 and57 respectively These predictions are much improvedcompared with Mossrsquos DSMC calculation which was only25 of the lengthmeasured in the experiment [8]The resultsof three models overpredicted the ahead pressure coefficient

000 025 050 075 100 125 150 175 200 225

00

05

10

15

20

25

30

Cp

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 36

Figure 12 DSMC simulation of Run number 36 (119862119875

)

000 025 050 075 100 125 150 175 200 225

0200

0175

0150

0125

0100

0075

0050

0025

0000

minus0025

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 26-

S t

St

Figure 13 DSMC simulation of Run number 26 (119878119905

)

peak value In the downstream of the interaction regionsthe pressure levels got by the three models were relativelywell predicted And also the three models overpredictedthe Stanton number in comparison with the experimentaldata The position of the Stanton number peak was wellpredicted by the three models Figure 13 shows that thepeak values of the Stanton number using HS VHS and VSSare approximately 142 1857 and 1485 respectivelyof the data measured in the experiment for Run number26 Figure 14 shows that the peak values of the Stantonnumber using HS VHS and VSS are approximately 1781928 and 178 respectively of the data measured in theexperiment for Run number 36

322 DSMC Simulation of Run Number 33 In this sectionthe DSMC simulation result of Run number 33 is presented

The Scientific World Journal 7

000 025 050 075 100 125 150 175 200 225

016015014013012011010009008007006005004003002001000

minus001

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 36-

S t

St

Figure 14 DSMC simulation of Run number 36 (119878119905

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

10

05

00

minus05

Cp

XL (L = 92075 mm) Run number 33

Figure 15 DSMC simulation of Run number 33 (119862119875

)

Table 4 Flow conditions for Run Numbers 33 and Numbers 36

Ma Rem KnRun 33 1035 205 times 10

5

753 times 10minus5

Run 36 1034 173 times 105

891 times 10minus5

Table 5 Flow conditions for Run Numbers 32 and 38

Ma Rem KnRun 32 1056 1177 times 10

5

134 times 10minus4

Run 38 1247 0738 times 105

252 times 10minus4

As shown in Table 4 Run numbers 33 and 36 were conductedin the same configuration with different Kn condation

The pressure coefficient and the Stanton numberobtained from HS VHS and VSS using the DSMC

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

020

015

010

005

000

XL (L = 92075 mm) Run number 33-St

S t

Figure 16 DSMC simulation of Run number 33 (119878119905

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

10

05

00

Cp

XL (L = 92075 mm) Run number 32

Figure 17 DSMC simulation of Run number 32 (119862119875

)

method are shown in Figures 15 and 16 These results alsounderpredicted the scale of interaction region and positionmagnitude of the properties in and downstream ofthe shock interaction region The HS result shown inFigure 15 indicates that the predicted interaction regionis approximately 533 of the length measured in theexperiment Those of VHS and VSS are 266 and466 respectively These predictions are not improvedin comparison with Run numbers 26 and 36 calculationsThe three modelsrsquo calculation also overpredicted the aheadpressure coefficient peak value In the downstream of theinteraction region the pressure levels from three models arerelatively well predicted Also three models overpredictedthe Stanton number compared with the experimentaldata But the position of the Stanton number peak is wellpredicted in the three modelsrsquo calculation Figure 16 shows

8 The Scientific World Journal

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

05

10

00

Cp

XL (L = 92075 mm) Run number 38

Figure 18 DSMC simulation of Run number 38 (119862119875

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

016

014

012

010

008

006

004

002

000

minus002

XL (L = 92075 mm) Run number 32-

S t

St

Figure 19 DSMC simulation of Run number 32 (119878119905

)

that the peak values of the Stanton number in HS VHS andVSS are approximately 290 309 and 345 respectivelyof the data measured in the experiment for Run number 33

323 DSMC Simulation for Run Numbers 32 and 38 In thissection theDSMCsimulation results forRunnumbers 32 and38 as shown in Table 5 are presented Run numbers 33 38and 36 were conducted in the same configuration with R025in blunt nose

Thepressure coefficient and the Stanton number obtainedby HS VHS and VSS using the DSMC method are shownin Figures 17ndash20 These results well predicted the scale of theinteraction region and the position magnitude of the proper-ties in and downstream of the shock interaction region TheHS result shown in Figure 17 for Run number 32 indicates

000 025 050 075 100 125 150 175 200 225minus002

000

002

004

006

008

010

012

014

016

018

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 38-

S t

St

Figure 20 DSMC simulation of Run number 38 (119878119905

)

35635

Rn = 8

53L2

L3

L1

L

L 950 to 1055

20∘ 10∘10∘

Dimensions in (mm)

y

x

Origin of coordinate system

L1 = L2 = 50771L3 = 100

Figure 21 Configuration of Edneyrsquos Type IV shock interaction

that the predicted interaction region is approximately 100of the length measured in the experiment Those of VHS andVSS are 41 and 69 respectively The HS result shown inFigure 18 for Run number 38 indicates that the predictedinteraction region is approximately 1068 of the lengthmeasured in the experimentThose ofVHS andVSS are 276and 6896 respectively These predictions are improvedin comparison with Run numbers 26 and 36 calculationsAlso the three modelsrsquo calculation overpredicted the aheadpressure coefficient peak value And in the downstreamof interaction region the pressure levels from the threemodels are relatively well predicted The three models alsooverpredicted the Stanton number in comparison with theexperimental data And also the position of the Stanton num-ber peak is well predicted in the three modelsrsquo calculationFigure 19 shows that the peak values of the Stanton numberin HS VHS and VSS are approximately 1052 1578 and

The Scientific World Journal 9

Experiment

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100minus2

0

2

4

6

8

10

12

14

16

Cp

Angle (deg)

DSMC-HSDSMC-VSSDSMC-VHS

Figure 22 DSMC of type IV shock interaction (119862119875

)

1474 respectively of the data measured in the experimentfor Run number 32 Figure 20 shows that the peak value ofthe Stanton number in HS VHS and VSS are approximately1052 1684 and 1421 respectively of the datameasuredin the experiment for Run number 38

Table 6 shows that the DSMC error depends on theKn number and the models employed for intermolecularinteraction With the increase in the Kn number the DSMCerror decreases The error is the smallest in the HS modelcompared with those in the VHS and VSS models Withthe Kn number increasing the Stanton number DSMC errordecreases When the Kn number is in the level of 10minus4 theDSMC error for pressure coefficient and Stanton number iscontrolled within 10

4 Type IV Shock Interaction Simulation

In the present study results of the numerical simulations forMach 10 air flow as shown in Table 7 were presented for arange of Edneyrsquos type IV shock interactionThe experimentaldata for validation were obtained from [16] The configura-tion is shown in Figure 21

The results of the grid sensitivity investigation are pre-sented in [16] The number of cellssubcells used for oursimulation was 97060958490 proven in [16] to be thefinestThe computational time step was 12 nsThe simulationconsisted of 156 times 10

5 time steps to ensure steady stateconditions

The DSMC data of the pressure coefficient and theStanton number are shown in Figures 22 and 23 The agree-ment between the DSMC calculation and the experiment isgenerally good BothHS andVHSmodels gain improvementsin the errors The error between the HS-DSMC data and theexperimental data is controlled within 5

The result from theDSMC calculation for Edneyrsquos type IVshock interaction also confirmed the same conclusion The

ExperimentDSMC-HS

DSMC-VSSDSMC-VHS

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100

15

10

05

00

Angle (deg)

S t

Figure 23 DSMC of type IV shock interaction (119878119905

)

error is the smallest in the HS model compared with those inthe VHS and VSS models In the region of large Kn numberDSMC using the HS model could be used in pressure andthermal prediction of hypersonic complex flow

5 Conclusion

In the present study the double-cone and Edneyrsquos type IVshock interactions were studied Molecular models wereemployed with DSMC including the HS VHS and VSSmodels Comparisons between DSMC and the experimentwere also conducted

Drawing firm conclusions from this study prior to fullcomparisons of these results is somewhat difficult Howeversome issues are immediately clear from our attempt tocompute these flows

(1) The DSMC method appears to predict qualitativelythe structure of flow separation found in previousCFD calculations

(2) Investigation of the double-cone hypersonic flowreveals that the three collisionsmodels can predict thepressure coefficient and the Stanton number trendsThe agreement between the DSMC simulation andthe experiment is the best in the HS collisions model

(3) Investigation of type IV shock interaction shows thatthe agreement between the DSMC simulation andthe experiment is generally good in the HS and VHScollision models However it fails in the VSS model

(4) Both double-cone flow and type IV shock interactionsimulations show that the DSMC error dependson the Kn number and the models employed forintermolecular interaction Increasing the Kn num-ber decreases the DSMC error The error is the small-est for the HS model compared with those for the

10 The Scientific World Journal

Table 6 Comparisons of calculations ((ValueDSMC minus Valueexperiment)(Valueexperiment))

Run KnThe scale of interaction region

(error in compared with measured data )Peak value of the Stanton number

(error in compared with measured data )HS VHS VSS HS VHS VSS

33 753 times 10minus5 467 734 534 190 209 245

36 891 times 10minus5 24 70 43 78 928 78

26 905 times 10minus5 24 66 38 42 857 485

32 134 times 10minus4 0 59 31 52 578 474

38 252 times 10minus4 68 724 3104 52 684 421

Table 7 Flow conditions of Edneyrsquos type IV shock interaction

Ma Rem Kn10 103 times 10

5

145 times 10minus4

VHS and VSS models When the Kn number is in thelevel of 10minus4 the DSMC error for pressure coefficientthe Stanton number and scale of interaction region iscontrolled within 10

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the Chinese AerospaceInnovation and in part by the Basic Research Fund ofNorthwestern Polytechnical University

References

[1] M S Holden T P Wadhams J K Harvey and G V CandlerldquoComparisons between measurements in regions of laminarshockwave boundary layer interaction in hypersonic flowswithnavier-stokes and DSMC solutionsrdquo Tech Rep RTO-TR-AVT-007-V3 NATO Science and Technology Organization 2006

[2] J Olejniczak and G V Candler ldquoComputation of hyper-sonic shock interaction flow fieldsrdquo in Proceedings of the 7thAIAAASME JointThermophysics and Heat Transfer ConferenceAIAA Paper 98-2446 June 1998

[3] G V Candler I Nompelis M-C Druguet I D Boyd W-LWang and S M Holden ldquoCFD validation for hypersonic flighthypersonic double-cone flow simulationsrdquo in Proceedings of the40th Aerospace Sciences Meeting amp Exhibit AIAA 2002-0581Reno Nev USA January 2002

[4] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Oxford Clarendon Press New York NY USA 1994

[5] Harvey K John M S Holden and T P Wadhams ldquoCode vali-dation study of laminar shockboundary layer and shockshockinteractions in hypersonic flow Part B comparison withnavier-stokes and DSMC solutionsrdquo in Proceedings of the 39thAerospace Sciences Meeting amp Exhibit AIAA 2001-1031 RenoNev USA January 2001

[6] M Gallis C Roy J Payne and T Bartel ldquoDSMC and navier-stokes predictions for hypersonic laminar interacting flowsrdquo inProceedings of the 39th Aerospace Sciences Meeting and ExhibitAIAA 2001-1030 Reno Nev USA January 2001

[7] G V Candler I Nompelis and M-C Druguet ldquoNavier-stokespredictions of hypersonic double-cone and cylinder-flare flowfieldsrdquo in Proceedings of the 39th Aerospace Sciences Meeting ampExhibit AIAA Paper no 2001-1024 Reno Nev USA January2001

[8] J Moss ldquoDSMC computations for regions of shockshock andshockboundary layer interactionrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA 2001-1027 RenoNev USA January 2001

[9] G A Bird The DSMC Method Create Space IndependentPublishing Platform 2013

[10] M Woo and I Greber ldquoMolecular dynamics simulation ofpiston-driven shock wave in hard sphere gasrdquo AIAA Journalvol 37 no 2 pp 215ndash221 1999

[11] K Koura and H Matsumoto ldquoVariable soft sphere molecularmodel for air speciesrdquo Physics of Fluids A vol 4 no 5 pp 1083ndash1085 1992

[12] P S Prasanth and J K Kakkassery ldquoDirect simulation MonteCarlo (DSMC) a numerical method for transition-regimeflows a reviewrdquo Journal of the Indian Institute of Science vol86 no 3 pp 169ndash192 2006

[13] T Tokumasu and Y Matsumoto ldquoDynamic molecular collision(DMC) model for rarefied gas flow simulations by the DSMCmethodrdquo Physics of Fluids vol 11 no 1 pp 1907ndash1920 1999

[14] S F Borisov ldquoProgress in gas-surface interaction studyrdquo inProceedings of the 24th International Rarefied Gas DynamicsSymposium M Capitelli Ed pp 933ndash940 The AmericanInstitute of Physics 2005

[15] I D Boyd and W-L Wang ldquoMonte Carlo computationsof hypersonic interacting flowsrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA Paper no 2001-1029 Reno Nev USA January 2001

[16] J N Moss T Pot B Chanetz and M Lefebvre ldquoDSMCsimulation of shockshock interactions emphases on type IVinteractionrdquo in Proceedings of the 22nd International Symposiumon Shock Waes Paper no 3570 Imperial College London UK1999

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

6 The Scientific World Journal

00 02 04 06 08 10 12 14 16 180

1

2

3

4

Pressure

Cp

000

002

004

006

008

010

012

014

Heat transfer

XL

Boyd CpBoyd St

S tFigure 10 Boydrsquos DSMC for Run number 28 [15]

000 025 050 075 100 125 150 175 200 225

00

05

10

15

20

25

30

Cp

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L =92075 mm) Run number 26

Figure 11 DSMC simulation of Run number 26 (119862119875

)

The pressure coefficient and the Stanton numberobtained with HS VHS and VSS are shown in Figures11ndash14 These results underpredict the scale of interactionregion and the position and also the magnitude of theparameters is underpredicted in and downstream The HScalculation shown in Figure 11 for Run number 26 indicatesthat the predicted interaction region is approximately 76of the measured experimental length Those of VHS andVSS are 34 and 62 respectively The HS results shownin Figure 12 for Run number 36 indicate that the predictedinteraction region is approximately 76 of the measuredexperimental length Those of VHS and VSS are 30 and57 respectively These predictions are much improvedcompared with Mossrsquos DSMC calculation which was only25 of the lengthmeasured in the experiment [8]The resultsof three models overpredicted the ahead pressure coefficient

000 025 050 075 100 125 150 175 200 225

00

05

10

15

20

25

30

Cp

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 36

Figure 12 DSMC simulation of Run number 36 (119862119875

)

000 025 050 075 100 125 150 175 200 225

0200

0175

0150

0125

0100

0075

0050

0025

0000

minus0025

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 26-

S t

St

Figure 13 DSMC simulation of Run number 26 (119878119905

)

peak value In the downstream of the interaction regionsthe pressure levels got by the three models were relativelywell predicted And also the three models overpredictedthe Stanton number in comparison with the experimentaldata The position of the Stanton number peak was wellpredicted by the three models Figure 13 shows that thepeak values of the Stanton number using HS VHS and VSSare approximately 142 1857 and 1485 respectivelyof the data measured in the experiment for Run number26 Figure 14 shows that the peak values of the Stantonnumber using HS VHS and VSS are approximately 1781928 and 178 respectively of the data measured in theexperiment for Run number 36

322 DSMC Simulation of Run Number 33 In this sectionthe DSMC simulation result of Run number 33 is presented

The Scientific World Journal 7

000 025 050 075 100 125 150 175 200 225

016015014013012011010009008007006005004003002001000

minus001

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 36-

S t

St

Figure 14 DSMC simulation of Run number 36 (119878119905

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

10

05

00

minus05

Cp

XL (L = 92075 mm) Run number 33

Figure 15 DSMC simulation of Run number 33 (119862119875

)

Table 4 Flow conditions for Run Numbers 33 and Numbers 36

Ma Rem KnRun 33 1035 205 times 10

5

753 times 10minus5

Run 36 1034 173 times 105

891 times 10minus5

Table 5 Flow conditions for Run Numbers 32 and 38

Ma Rem KnRun 32 1056 1177 times 10

5

134 times 10minus4

Run 38 1247 0738 times 105

252 times 10minus4

As shown in Table 4 Run numbers 33 and 36 were conductedin the same configuration with different Kn condation

The pressure coefficient and the Stanton numberobtained from HS VHS and VSS using the DSMC

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

020

015

010

005

000

XL (L = 92075 mm) Run number 33-St

S t

Figure 16 DSMC simulation of Run number 33 (119878119905

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

10

05

00

Cp

XL (L = 92075 mm) Run number 32

Figure 17 DSMC simulation of Run number 32 (119862119875

)

method are shown in Figures 15 and 16 These results alsounderpredicted the scale of interaction region and positionmagnitude of the properties in and downstream ofthe shock interaction region The HS result shown inFigure 15 indicates that the predicted interaction regionis approximately 533 of the length measured in theexperiment Those of VHS and VSS are 266 and466 respectively These predictions are not improvedin comparison with Run numbers 26 and 36 calculationsThe three modelsrsquo calculation also overpredicted the aheadpressure coefficient peak value In the downstream of theinteraction region the pressure levels from three models arerelatively well predicted Also three models overpredictedthe Stanton number compared with the experimentaldata But the position of the Stanton number peak is wellpredicted in the three modelsrsquo calculation Figure 16 shows

8 The Scientific World Journal

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

05

10

00

Cp

XL (L = 92075 mm) Run number 38

Figure 18 DSMC simulation of Run number 38 (119862119875

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

016

014

012

010

008

006

004

002

000

minus002

XL (L = 92075 mm) Run number 32-

S t

St

Figure 19 DSMC simulation of Run number 32 (119878119905

)

that the peak values of the Stanton number in HS VHS andVSS are approximately 290 309 and 345 respectivelyof the data measured in the experiment for Run number 33

323 DSMC Simulation for Run Numbers 32 and 38 In thissection theDSMCsimulation results forRunnumbers 32 and38 as shown in Table 5 are presented Run numbers 33 38and 36 were conducted in the same configuration with R025in blunt nose

Thepressure coefficient and the Stanton number obtainedby HS VHS and VSS using the DSMC method are shownin Figures 17ndash20 These results well predicted the scale of theinteraction region and the position magnitude of the proper-ties in and downstream of the shock interaction region TheHS result shown in Figure 17 for Run number 32 indicates

000 025 050 075 100 125 150 175 200 225minus002

000

002

004

006

008

010

012

014

016

018

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 38-

S t

St

Figure 20 DSMC simulation of Run number 38 (119878119905

)

35635

Rn = 8

53L2

L3

L1

L

L 950 to 1055

20∘ 10∘10∘

Dimensions in (mm)

y

x

Origin of coordinate system

L1 = L2 = 50771L3 = 100

Figure 21 Configuration of Edneyrsquos Type IV shock interaction

that the predicted interaction region is approximately 100of the length measured in the experiment Those of VHS andVSS are 41 and 69 respectively The HS result shown inFigure 18 for Run number 38 indicates that the predictedinteraction region is approximately 1068 of the lengthmeasured in the experimentThose ofVHS andVSS are 276and 6896 respectively These predictions are improvedin comparison with Run numbers 26 and 36 calculationsAlso the three modelsrsquo calculation overpredicted the aheadpressure coefficient peak value And in the downstreamof interaction region the pressure levels from the threemodels are relatively well predicted The three models alsooverpredicted the Stanton number in comparison with theexperimental data And also the position of the Stanton num-ber peak is well predicted in the three modelsrsquo calculationFigure 19 shows that the peak values of the Stanton numberin HS VHS and VSS are approximately 1052 1578 and

The Scientific World Journal 9

Experiment

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100minus2

0

2

4

6

8

10

12

14

16

Cp

Angle (deg)

DSMC-HSDSMC-VSSDSMC-VHS

Figure 22 DSMC of type IV shock interaction (119862119875

)

1474 respectively of the data measured in the experimentfor Run number 32 Figure 20 shows that the peak value ofthe Stanton number in HS VHS and VSS are approximately1052 1684 and 1421 respectively of the datameasuredin the experiment for Run number 38

Table 6 shows that the DSMC error depends on theKn number and the models employed for intermolecularinteraction With the increase in the Kn number the DSMCerror decreases The error is the smallest in the HS modelcompared with those in the VHS and VSS models Withthe Kn number increasing the Stanton number DSMC errordecreases When the Kn number is in the level of 10minus4 theDSMC error for pressure coefficient and Stanton number iscontrolled within 10

4 Type IV Shock Interaction Simulation

In the present study results of the numerical simulations forMach 10 air flow as shown in Table 7 were presented for arange of Edneyrsquos type IV shock interactionThe experimentaldata for validation were obtained from [16] The configura-tion is shown in Figure 21

The results of the grid sensitivity investigation are pre-sented in [16] The number of cellssubcells used for oursimulation was 97060958490 proven in [16] to be thefinestThe computational time step was 12 nsThe simulationconsisted of 156 times 10

5 time steps to ensure steady stateconditions

The DSMC data of the pressure coefficient and theStanton number are shown in Figures 22 and 23 The agree-ment between the DSMC calculation and the experiment isgenerally good BothHS andVHSmodels gain improvementsin the errors The error between the HS-DSMC data and theexperimental data is controlled within 5

The result from theDSMC calculation for Edneyrsquos type IVshock interaction also confirmed the same conclusion The

ExperimentDSMC-HS

DSMC-VSSDSMC-VHS

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100

15

10

05

00

Angle (deg)

S t

Figure 23 DSMC of type IV shock interaction (119878119905

)

error is the smallest in the HS model compared with those inthe VHS and VSS models In the region of large Kn numberDSMC using the HS model could be used in pressure andthermal prediction of hypersonic complex flow

5 Conclusion

In the present study the double-cone and Edneyrsquos type IVshock interactions were studied Molecular models wereemployed with DSMC including the HS VHS and VSSmodels Comparisons between DSMC and the experimentwere also conducted

Drawing firm conclusions from this study prior to fullcomparisons of these results is somewhat difficult Howeversome issues are immediately clear from our attempt tocompute these flows

(1) The DSMC method appears to predict qualitativelythe structure of flow separation found in previousCFD calculations

(2) Investigation of the double-cone hypersonic flowreveals that the three collisionsmodels can predict thepressure coefficient and the Stanton number trendsThe agreement between the DSMC simulation andthe experiment is the best in the HS collisions model

(3) Investigation of type IV shock interaction shows thatthe agreement between the DSMC simulation andthe experiment is generally good in the HS and VHScollision models However it fails in the VSS model

(4) Both double-cone flow and type IV shock interactionsimulations show that the DSMC error dependson the Kn number and the models employed forintermolecular interaction Increasing the Kn num-ber decreases the DSMC error The error is the small-est for the HS model compared with those for the

10 The Scientific World Journal

Table 6 Comparisons of calculations ((ValueDSMC minus Valueexperiment)(Valueexperiment))

Run KnThe scale of interaction region

(error in compared with measured data )Peak value of the Stanton number

(error in compared with measured data )HS VHS VSS HS VHS VSS

33 753 times 10minus5 467 734 534 190 209 245

36 891 times 10minus5 24 70 43 78 928 78

26 905 times 10minus5 24 66 38 42 857 485

32 134 times 10minus4 0 59 31 52 578 474

38 252 times 10minus4 68 724 3104 52 684 421

Table 7 Flow conditions of Edneyrsquos type IV shock interaction

Ma Rem Kn10 103 times 10

5

145 times 10minus4

VHS and VSS models When the Kn number is in thelevel of 10minus4 the DSMC error for pressure coefficientthe Stanton number and scale of interaction region iscontrolled within 10

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the Chinese AerospaceInnovation and in part by the Basic Research Fund ofNorthwestern Polytechnical University

References

[1] M S Holden T P Wadhams J K Harvey and G V CandlerldquoComparisons between measurements in regions of laminarshockwave boundary layer interaction in hypersonic flowswithnavier-stokes and DSMC solutionsrdquo Tech Rep RTO-TR-AVT-007-V3 NATO Science and Technology Organization 2006

[2] J Olejniczak and G V Candler ldquoComputation of hyper-sonic shock interaction flow fieldsrdquo in Proceedings of the 7thAIAAASME JointThermophysics and Heat Transfer ConferenceAIAA Paper 98-2446 June 1998

[3] G V Candler I Nompelis M-C Druguet I D Boyd W-LWang and S M Holden ldquoCFD validation for hypersonic flighthypersonic double-cone flow simulationsrdquo in Proceedings of the40th Aerospace Sciences Meeting amp Exhibit AIAA 2002-0581Reno Nev USA January 2002

[4] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Oxford Clarendon Press New York NY USA 1994

[5] Harvey K John M S Holden and T P Wadhams ldquoCode vali-dation study of laminar shockboundary layer and shockshockinteractions in hypersonic flow Part B comparison withnavier-stokes and DSMC solutionsrdquo in Proceedings of the 39thAerospace Sciences Meeting amp Exhibit AIAA 2001-1031 RenoNev USA January 2001

[6] M Gallis C Roy J Payne and T Bartel ldquoDSMC and navier-stokes predictions for hypersonic laminar interacting flowsrdquo inProceedings of the 39th Aerospace Sciences Meeting and ExhibitAIAA 2001-1030 Reno Nev USA January 2001

[7] G V Candler I Nompelis and M-C Druguet ldquoNavier-stokespredictions of hypersonic double-cone and cylinder-flare flowfieldsrdquo in Proceedings of the 39th Aerospace Sciences Meeting ampExhibit AIAA Paper no 2001-1024 Reno Nev USA January2001

[8] J Moss ldquoDSMC computations for regions of shockshock andshockboundary layer interactionrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA 2001-1027 RenoNev USA January 2001

[9] G A Bird The DSMC Method Create Space IndependentPublishing Platform 2013

[10] M Woo and I Greber ldquoMolecular dynamics simulation ofpiston-driven shock wave in hard sphere gasrdquo AIAA Journalvol 37 no 2 pp 215ndash221 1999

[11] K Koura and H Matsumoto ldquoVariable soft sphere molecularmodel for air speciesrdquo Physics of Fluids A vol 4 no 5 pp 1083ndash1085 1992

[12] P S Prasanth and J K Kakkassery ldquoDirect simulation MonteCarlo (DSMC) a numerical method for transition-regimeflows a reviewrdquo Journal of the Indian Institute of Science vol86 no 3 pp 169ndash192 2006

[13] T Tokumasu and Y Matsumoto ldquoDynamic molecular collision(DMC) model for rarefied gas flow simulations by the DSMCmethodrdquo Physics of Fluids vol 11 no 1 pp 1907ndash1920 1999

[14] S F Borisov ldquoProgress in gas-surface interaction studyrdquo inProceedings of the 24th International Rarefied Gas DynamicsSymposium M Capitelli Ed pp 933ndash940 The AmericanInstitute of Physics 2005

[15] I D Boyd and W-L Wang ldquoMonte Carlo computationsof hypersonic interacting flowsrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA Paper no 2001-1029 Reno Nev USA January 2001

[16] J N Moss T Pot B Chanetz and M Lefebvre ldquoDSMCsimulation of shockshock interactions emphases on type IVinteractionrdquo in Proceedings of the 22nd International Symposiumon Shock Waes Paper no 3570 Imperial College London UK1999

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 7

000 025 050 075 100 125 150 175 200 225

016015014013012011010009008007006005004003002001000

minus001

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 36-

S t

St

Figure 14 DSMC simulation of Run number 36 (119878119905

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

10

05

00

minus05

Cp

XL (L = 92075 mm) Run number 33

Figure 15 DSMC simulation of Run number 33 (119862119875

)

Table 4 Flow conditions for Run Numbers 33 and Numbers 36

Ma Rem KnRun 33 1035 205 times 10

5

753 times 10minus5

Run 36 1034 173 times 105

891 times 10minus5

Table 5 Flow conditions for Run Numbers 32 and 38

Ma Rem KnRun 32 1056 1177 times 10

5

134 times 10minus4

Run 38 1247 0738 times 105

252 times 10minus4

As shown in Table 4 Run numbers 33 and 36 were conductedin the same configuration with different Kn condation

The pressure coefficient and the Stanton numberobtained from HS VHS and VSS using the DSMC

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

020

015

010

005

000

XL (L = 92075 mm) Run number 33-St

S t

Figure 16 DSMC simulation of Run number 33 (119878119905

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

10

05

00

Cp

XL (L = 92075 mm) Run number 32

Figure 17 DSMC simulation of Run number 32 (119862119875

)

method are shown in Figures 15 and 16 These results alsounderpredicted the scale of interaction region and positionmagnitude of the properties in and downstream ofthe shock interaction region The HS result shown inFigure 15 indicates that the predicted interaction regionis approximately 533 of the length measured in theexperiment Those of VHS and VSS are 266 and466 respectively These predictions are not improvedin comparison with Run numbers 26 and 36 calculationsThe three modelsrsquo calculation also overpredicted the aheadpressure coefficient peak value In the downstream of theinteraction region the pressure levels from three models arerelatively well predicted Also three models overpredictedthe Stanton number compared with the experimentaldata But the position of the Stanton number peak is wellpredicted in the three modelsrsquo calculation Figure 16 shows

8 The Scientific World Journal

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

05

10

00

Cp

XL (L = 92075 mm) Run number 38

Figure 18 DSMC simulation of Run number 38 (119862119875

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

016

014

012

010

008

006

004

002

000

minus002

XL (L = 92075 mm) Run number 32-

S t

St

Figure 19 DSMC simulation of Run number 32 (119878119905

)

that the peak values of the Stanton number in HS VHS andVSS are approximately 290 309 and 345 respectivelyof the data measured in the experiment for Run number 33

323 DSMC Simulation for Run Numbers 32 and 38 In thissection theDSMCsimulation results forRunnumbers 32 and38 as shown in Table 5 are presented Run numbers 33 38and 36 were conducted in the same configuration with R025in blunt nose

Thepressure coefficient and the Stanton number obtainedby HS VHS and VSS using the DSMC method are shownin Figures 17ndash20 These results well predicted the scale of theinteraction region and the position magnitude of the proper-ties in and downstream of the shock interaction region TheHS result shown in Figure 17 for Run number 32 indicates

000 025 050 075 100 125 150 175 200 225minus002

000

002

004

006

008

010

012

014

016

018

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 38-

S t

St

Figure 20 DSMC simulation of Run number 38 (119878119905

)

35635

Rn = 8

53L2

L3

L1

L

L 950 to 1055

20∘ 10∘10∘

Dimensions in (mm)

y

x

Origin of coordinate system

L1 = L2 = 50771L3 = 100

Figure 21 Configuration of Edneyrsquos Type IV shock interaction

that the predicted interaction region is approximately 100of the length measured in the experiment Those of VHS andVSS are 41 and 69 respectively The HS result shown inFigure 18 for Run number 38 indicates that the predictedinteraction region is approximately 1068 of the lengthmeasured in the experimentThose ofVHS andVSS are 276and 6896 respectively These predictions are improvedin comparison with Run numbers 26 and 36 calculationsAlso the three modelsrsquo calculation overpredicted the aheadpressure coefficient peak value And in the downstreamof interaction region the pressure levels from the threemodels are relatively well predicted The three models alsooverpredicted the Stanton number in comparison with theexperimental data And also the position of the Stanton num-ber peak is well predicted in the three modelsrsquo calculationFigure 19 shows that the peak values of the Stanton numberin HS VHS and VSS are approximately 1052 1578 and

The Scientific World Journal 9

Experiment

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100minus2

0

2

4

6

8

10

12

14

16

Cp

Angle (deg)

DSMC-HSDSMC-VSSDSMC-VHS

Figure 22 DSMC of type IV shock interaction (119862119875

)

1474 respectively of the data measured in the experimentfor Run number 32 Figure 20 shows that the peak value ofthe Stanton number in HS VHS and VSS are approximately1052 1684 and 1421 respectively of the datameasuredin the experiment for Run number 38

Table 6 shows that the DSMC error depends on theKn number and the models employed for intermolecularinteraction With the increase in the Kn number the DSMCerror decreases The error is the smallest in the HS modelcompared with those in the VHS and VSS models Withthe Kn number increasing the Stanton number DSMC errordecreases When the Kn number is in the level of 10minus4 theDSMC error for pressure coefficient and Stanton number iscontrolled within 10

4 Type IV Shock Interaction Simulation

In the present study results of the numerical simulations forMach 10 air flow as shown in Table 7 were presented for arange of Edneyrsquos type IV shock interactionThe experimentaldata for validation were obtained from [16] The configura-tion is shown in Figure 21

The results of the grid sensitivity investigation are pre-sented in [16] The number of cellssubcells used for oursimulation was 97060958490 proven in [16] to be thefinestThe computational time step was 12 nsThe simulationconsisted of 156 times 10

5 time steps to ensure steady stateconditions

The DSMC data of the pressure coefficient and theStanton number are shown in Figures 22 and 23 The agree-ment between the DSMC calculation and the experiment isgenerally good BothHS andVHSmodels gain improvementsin the errors The error between the HS-DSMC data and theexperimental data is controlled within 5

The result from theDSMC calculation for Edneyrsquos type IVshock interaction also confirmed the same conclusion The

ExperimentDSMC-HS

DSMC-VSSDSMC-VHS

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100

15

10

05

00

Angle (deg)

S t

Figure 23 DSMC of type IV shock interaction (119878119905

)

error is the smallest in the HS model compared with those inthe VHS and VSS models In the region of large Kn numberDSMC using the HS model could be used in pressure andthermal prediction of hypersonic complex flow

5 Conclusion

In the present study the double-cone and Edneyrsquos type IVshock interactions were studied Molecular models wereemployed with DSMC including the HS VHS and VSSmodels Comparisons between DSMC and the experimentwere also conducted

Drawing firm conclusions from this study prior to fullcomparisons of these results is somewhat difficult Howeversome issues are immediately clear from our attempt tocompute these flows

(1) The DSMC method appears to predict qualitativelythe structure of flow separation found in previousCFD calculations

(2) Investigation of the double-cone hypersonic flowreveals that the three collisionsmodels can predict thepressure coefficient and the Stanton number trendsThe agreement between the DSMC simulation andthe experiment is the best in the HS collisions model

(3) Investigation of type IV shock interaction shows thatthe agreement between the DSMC simulation andthe experiment is generally good in the HS and VHScollision models However it fails in the VSS model

(4) Both double-cone flow and type IV shock interactionsimulations show that the DSMC error dependson the Kn number and the models employed forintermolecular interaction Increasing the Kn num-ber decreases the DSMC error The error is the small-est for the HS model compared with those for the

10 The Scientific World Journal

Table 6 Comparisons of calculations ((ValueDSMC minus Valueexperiment)(Valueexperiment))

Run KnThe scale of interaction region

(error in compared with measured data )Peak value of the Stanton number

(error in compared with measured data )HS VHS VSS HS VHS VSS

33 753 times 10minus5 467 734 534 190 209 245

36 891 times 10minus5 24 70 43 78 928 78

26 905 times 10minus5 24 66 38 42 857 485

32 134 times 10minus4 0 59 31 52 578 474

38 252 times 10minus4 68 724 3104 52 684 421

Table 7 Flow conditions of Edneyrsquos type IV shock interaction

Ma Rem Kn10 103 times 10

5

145 times 10minus4

VHS and VSS models When the Kn number is in thelevel of 10minus4 the DSMC error for pressure coefficientthe Stanton number and scale of interaction region iscontrolled within 10

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the Chinese AerospaceInnovation and in part by the Basic Research Fund ofNorthwestern Polytechnical University

References

[1] M S Holden T P Wadhams J K Harvey and G V CandlerldquoComparisons between measurements in regions of laminarshockwave boundary layer interaction in hypersonic flowswithnavier-stokes and DSMC solutionsrdquo Tech Rep RTO-TR-AVT-007-V3 NATO Science and Technology Organization 2006

[2] J Olejniczak and G V Candler ldquoComputation of hyper-sonic shock interaction flow fieldsrdquo in Proceedings of the 7thAIAAASME JointThermophysics and Heat Transfer ConferenceAIAA Paper 98-2446 June 1998

[3] G V Candler I Nompelis M-C Druguet I D Boyd W-LWang and S M Holden ldquoCFD validation for hypersonic flighthypersonic double-cone flow simulationsrdquo in Proceedings of the40th Aerospace Sciences Meeting amp Exhibit AIAA 2002-0581Reno Nev USA January 2002

[4] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Oxford Clarendon Press New York NY USA 1994

[5] Harvey K John M S Holden and T P Wadhams ldquoCode vali-dation study of laminar shockboundary layer and shockshockinteractions in hypersonic flow Part B comparison withnavier-stokes and DSMC solutionsrdquo in Proceedings of the 39thAerospace Sciences Meeting amp Exhibit AIAA 2001-1031 RenoNev USA January 2001

[6] M Gallis C Roy J Payne and T Bartel ldquoDSMC and navier-stokes predictions for hypersonic laminar interacting flowsrdquo inProceedings of the 39th Aerospace Sciences Meeting and ExhibitAIAA 2001-1030 Reno Nev USA January 2001

[7] G V Candler I Nompelis and M-C Druguet ldquoNavier-stokespredictions of hypersonic double-cone and cylinder-flare flowfieldsrdquo in Proceedings of the 39th Aerospace Sciences Meeting ampExhibit AIAA Paper no 2001-1024 Reno Nev USA January2001

[8] J Moss ldquoDSMC computations for regions of shockshock andshockboundary layer interactionrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA 2001-1027 RenoNev USA January 2001

[9] G A Bird The DSMC Method Create Space IndependentPublishing Platform 2013

[10] M Woo and I Greber ldquoMolecular dynamics simulation ofpiston-driven shock wave in hard sphere gasrdquo AIAA Journalvol 37 no 2 pp 215ndash221 1999

[11] K Koura and H Matsumoto ldquoVariable soft sphere molecularmodel for air speciesrdquo Physics of Fluids A vol 4 no 5 pp 1083ndash1085 1992

[12] P S Prasanth and J K Kakkassery ldquoDirect simulation MonteCarlo (DSMC) a numerical method for transition-regimeflows a reviewrdquo Journal of the Indian Institute of Science vol86 no 3 pp 169ndash192 2006

[13] T Tokumasu and Y Matsumoto ldquoDynamic molecular collision(DMC) model for rarefied gas flow simulations by the DSMCmethodrdquo Physics of Fluids vol 11 no 1 pp 1907ndash1920 1999

[14] S F Borisov ldquoProgress in gas-surface interaction studyrdquo inProceedings of the 24th International Rarefied Gas DynamicsSymposium M Capitelli Ed pp 933ndash940 The AmericanInstitute of Physics 2005

[15] I D Boyd and W-L Wang ldquoMonte Carlo computationsof hypersonic interacting flowsrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA Paper no 2001-1029 Reno Nev USA January 2001

[16] J N Moss T Pot B Chanetz and M Lefebvre ldquoDSMCsimulation of shockshock interactions emphases on type IVinteractionrdquo in Proceedings of the 22nd International Symposiumon Shock Waes Paper no 3570 Imperial College London UK1999

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 The Scientific World Journal

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

30

25

20

15

05

10

00

Cp

XL (L = 92075 mm) Run number 38

Figure 18 DSMC simulation of Run number 38 (119862119875

)

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

000 025 050 075 100 125 150 175 200 225

016

014

012

010

008

006

004

002

000

minus002

XL (L = 92075 mm) Run number 32-

S t

St

Figure 19 DSMC simulation of Run number 32 (119878119905

)

that the peak values of the Stanton number in HS VHS andVSS are approximately 290 309 and 345 respectivelyof the data measured in the experiment for Run number 33

323 DSMC Simulation for Run Numbers 32 and 38 In thissection theDSMCsimulation results forRunnumbers 32 and38 as shown in Table 5 are presented Run numbers 33 38and 36 were conducted in the same configuration with R025in blunt nose

Thepressure coefficient and the Stanton number obtainedby HS VHS and VSS using the DSMC method are shownin Figures 17ndash20 These results well predicted the scale of theinteraction region and the position magnitude of the proper-ties in and downstream of the shock interaction region TheHS result shown in Figure 17 for Run number 32 indicates

000 025 050 075 100 125 150 175 200 225minus002

000

002

004

006

008

010

012

014

016

018

ExperimentDSMC VHS model

DSMC VSS modelDSMC HS model

XL (L = 92075 mm) Run number 38-

S t

St

Figure 20 DSMC simulation of Run number 38 (119878119905

)

35635

Rn = 8

53L2

L3

L1

L

L 950 to 1055

20∘ 10∘10∘

Dimensions in (mm)

y

x

Origin of coordinate system

L1 = L2 = 50771L3 = 100

Figure 21 Configuration of Edneyrsquos Type IV shock interaction

that the predicted interaction region is approximately 100of the length measured in the experiment Those of VHS andVSS are 41 and 69 respectively The HS result shown inFigure 18 for Run number 38 indicates that the predictedinteraction region is approximately 1068 of the lengthmeasured in the experimentThose ofVHS andVSS are 276and 6896 respectively These predictions are improvedin comparison with Run numbers 26 and 36 calculationsAlso the three modelsrsquo calculation overpredicted the aheadpressure coefficient peak value And in the downstreamof interaction region the pressure levels from the threemodels are relatively well predicted The three models alsooverpredicted the Stanton number in comparison with theexperimental data And also the position of the Stanton num-ber peak is well predicted in the three modelsrsquo calculationFigure 19 shows that the peak values of the Stanton numberin HS VHS and VSS are approximately 1052 1578 and

The Scientific World Journal 9

Experiment

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100minus2

0

2

4

6

8

10

12

14

16

Cp

Angle (deg)

DSMC-HSDSMC-VSSDSMC-VHS

Figure 22 DSMC of type IV shock interaction (119862119875

)

1474 respectively of the data measured in the experimentfor Run number 32 Figure 20 shows that the peak value ofthe Stanton number in HS VHS and VSS are approximately1052 1684 and 1421 respectively of the datameasuredin the experiment for Run number 38

Table 6 shows that the DSMC error depends on theKn number and the models employed for intermolecularinteraction With the increase in the Kn number the DSMCerror decreases The error is the smallest in the HS modelcompared with those in the VHS and VSS models Withthe Kn number increasing the Stanton number DSMC errordecreases When the Kn number is in the level of 10minus4 theDSMC error for pressure coefficient and Stanton number iscontrolled within 10

4 Type IV Shock Interaction Simulation

In the present study results of the numerical simulations forMach 10 air flow as shown in Table 7 were presented for arange of Edneyrsquos type IV shock interactionThe experimentaldata for validation were obtained from [16] The configura-tion is shown in Figure 21

The results of the grid sensitivity investigation are pre-sented in [16] The number of cellssubcells used for oursimulation was 97060958490 proven in [16] to be thefinestThe computational time step was 12 nsThe simulationconsisted of 156 times 10

5 time steps to ensure steady stateconditions

The DSMC data of the pressure coefficient and theStanton number are shown in Figures 22 and 23 The agree-ment between the DSMC calculation and the experiment isgenerally good BothHS andVHSmodels gain improvementsin the errors The error between the HS-DSMC data and theexperimental data is controlled within 5

The result from theDSMC calculation for Edneyrsquos type IVshock interaction also confirmed the same conclusion The

ExperimentDSMC-HS

DSMC-VSSDSMC-VHS

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100

15

10

05

00

Angle (deg)

S t

Figure 23 DSMC of type IV shock interaction (119878119905

)

error is the smallest in the HS model compared with those inthe VHS and VSS models In the region of large Kn numberDSMC using the HS model could be used in pressure andthermal prediction of hypersonic complex flow

5 Conclusion

In the present study the double-cone and Edneyrsquos type IVshock interactions were studied Molecular models wereemployed with DSMC including the HS VHS and VSSmodels Comparisons between DSMC and the experimentwere also conducted

Drawing firm conclusions from this study prior to fullcomparisons of these results is somewhat difficult Howeversome issues are immediately clear from our attempt tocompute these flows

(1) The DSMC method appears to predict qualitativelythe structure of flow separation found in previousCFD calculations

(2) Investigation of the double-cone hypersonic flowreveals that the three collisionsmodels can predict thepressure coefficient and the Stanton number trendsThe agreement between the DSMC simulation andthe experiment is the best in the HS collisions model

(3) Investigation of type IV shock interaction shows thatthe agreement between the DSMC simulation andthe experiment is generally good in the HS and VHScollision models However it fails in the VSS model

(4) Both double-cone flow and type IV shock interactionsimulations show that the DSMC error dependson the Kn number and the models employed forintermolecular interaction Increasing the Kn num-ber decreases the DSMC error The error is the small-est for the HS model compared with those for the

10 The Scientific World Journal

Table 6 Comparisons of calculations ((ValueDSMC minus Valueexperiment)(Valueexperiment))

Run KnThe scale of interaction region

(error in compared with measured data )Peak value of the Stanton number

(error in compared with measured data )HS VHS VSS HS VHS VSS

33 753 times 10minus5 467 734 534 190 209 245

36 891 times 10minus5 24 70 43 78 928 78

26 905 times 10minus5 24 66 38 42 857 485

32 134 times 10minus4 0 59 31 52 578 474

38 252 times 10minus4 68 724 3104 52 684 421

Table 7 Flow conditions of Edneyrsquos type IV shock interaction

Ma Rem Kn10 103 times 10

5

145 times 10minus4

VHS and VSS models When the Kn number is in thelevel of 10minus4 the DSMC error for pressure coefficientthe Stanton number and scale of interaction region iscontrolled within 10

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the Chinese AerospaceInnovation and in part by the Basic Research Fund ofNorthwestern Polytechnical University

References

[1] M S Holden T P Wadhams J K Harvey and G V CandlerldquoComparisons between measurements in regions of laminarshockwave boundary layer interaction in hypersonic flowswithnavier-stokes and DSMC solutionsrdquo Tech Rep RTO-TR-AVT-007-V3 NATO Science and Technology Organization 2006

[2] J Olejniczak and G V Candler ldquoComputation of hyper-sonic shock interaction flow fieldsrdquo in Proceedings of the 7thAIAAASME JointThermophysics and Heat Transfer ConferenceAIAA Paper 98-2446 June 1998

[3] G V Candler I Nompelis M-C Druguet I D Boyd W-LWang and S M Holden ldquoCFD validation for hypersonic flighthypersonic double-cone flow simulationsrdquo in Proceedings of the40th Aerospace Sciences Meeting amp Exhibit AIAA 2002-0581Reno Nev USA January 2002

[4] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Oxford Clarendon Press New York NY USA 1994

[5] Harvey K John M S Holden and T P Wadhams ldquoCode vali-dation study of laminar shockboundary layer and shockshockinteractions in hypersonic flow Part B comparison withnavier-stokes and DSMC solutionsrdquo in Proceedings of the 39thAerospace Sciences Meeting amp Exhibit AIAA 2001-1031 RenoNev USA January 2001

[6] M Gallis C Roy J Payne and T Bartel ldquoDSMC and navier-stokes predictions for hypersonic laminar interacting flowsrdquo inProceedings of the 39th Aerospace Sciences Meeting and ExhibitAIAA 2001-1030 Reno Nev USA January 2001

[7] G V Candler I Nompelis and M-C Druguet ldquoNavier-stokespredictions of hypersonic double-cone and cylinder-flare flowfieldsrdquo in Proceedings of the 39th Aerospace Sciences Meeting ampExhibit AIAA Paper no 2001-1024 Reno Nev USA January2001

[8] J Moss ldquoDSMC computations for regions of shockshock andshockboundary layer interactionrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA 2001-1027 RenoNev USA January 2001

[9] G A Bird The DSMC Method Create Space IndependentPublishing Platform 2013

[10] M Woo and I Greber ldquoMolecular dynamics simulation ofpiston-driven shock wave in hard sphere gasrdquo AIAA Journalvol 37 no 2 pp 215ndash221 1999

[11] K Koura and H Matsumoto ldquoVariable soft sphere molecularmodel for air speciesrdquo Physics of Fluids A vol 4 no 5 pp 1083ndash1085 1992

[12] P S Prasanth and J K Kakkassery ldquoDirect simulation MonteCarlo (DSMC) a numerical method for transition-regimeflows a reviewrdquo Journal of the Indian Institute of Science vol86 no 3 pp 169ndash192 2006

[13] T Tokumasu and Y Matsumoto ldquoDynamic molecular collision(DMC) model for rarefied gas flow simulations by the DSMCmethodrdquo Physics of Fluids vol 11 no 1 pp 1907ndash1920 1999

[14] S F Borisov ldquoProgress in gas-surface interaction studyrdquo inProceedings of the 24th International Rarefied Gas DynamicsSymposium M Capitelli Ed pp 933ndash940 The AmericanInstitute of Physics 2005

[15] I D Boyd and W-L Wang ldquoMonte Carlo computationsof hypersonic interacting flowsrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA Paper no 2001-1029 Reno Nev USA January 2001

[16] J N Moss T Pot B Chanetz and M Lefebvre ldquoDSMCsimulation of shockshock interactions emphases on type IVinteractionrdquo in Proceedings of the 22nd International Symposiumon Shock Waes Paper no 3570 Imperial College London UK1999

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 9

Experiment

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100minus2

0

2

4

6

8

10

12

14

16

Cp

Angle (deg)

DSMC-HSDSMC-VSSDSMC-VHS

Figure 22 DSMC of type IV shock interaction (119862119875

)

1474 respectively of the data measured in the experimentfor Run number 32 Figure 20 shows that the peak value ofthe Stanton number in HS VHS and VSS are approximately1052 1684 and 1421 respectively of the datameasuredin the experiment for Run number 38

Table 6 shows that the DSMC error depends on theKn number and the models employed for intermolecularinteraction With the increase in the Kn number the DSMCerror decreases The error is the smallest in the HS modelcompared with those in the VHS and VSS models Withthe Kn number increasing the Stanton number DSMC errordecreases When the Kn number is in the level of 10minus4 theDSMC error for pressure coefficient and Stanton number iscontrolled within 10

4 Type IV Shock Interaction Simulation

In the present study results of the numerical simulations forMach 10 air flow as shown in Table 7 were presented for arange of Edneyrsquos type IV shock interactionThe experimentaldata for validation were obtained from [16] The configura-tion is shown in Figure 21

The results of the grid sensitivity investigation are pre-sented in [16] The number of cellssubcells used for oursimulation was 97060958490 proven in [16] to be thefinestThe computational time step was 12 nsThe simulationconsisted of 156 times 10

5 time steps to ensure steady stateconditions

The DSMC data of the pressure coefficient and theStanton number are shown in Figures 22 and 23 The agree-ment between the DSMC calculation and the experiment isgenerally good BothHS andVHSmodels gain improvementsin the errors The error between the HS-DSMC data and theexperimental data is controlled within 5

The result from theDSMC calculation for Edneyrsquos type IVshock interaction also confirmed the same conclusion The

ExperimentDSMC-HS

DSMC-VSSDSMC-VHS

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100

15

10

05

00

Angle (deg)

S t

Figure 23 DSMC of type IV shock interaction (119878119905

)

error is the smallest in the HS model compared with those inthe VHS and VSS models In the region of large Kn numberDSMC using the HS model could be used in pressure andthermal prediction of hypersonic complex flow

5 Conclusion

In the present study the double-cone and Edneyrsquos type IVshock interactions were studied Molecular models wereemployed with DSMC including the HS VHS and VSSmodels Comparisons between DSMC and the experimentwere also conducted

Drawing firm conclusions from this study prior to fullcomparisons of these results is somewhat difficult Howeversome issues are immediately clear from our attempt tocompute these flows

(1) The DSMC method appears to predict qualitativelythe structure of flow separation found in previousCFD calculations

(2) Investigation of the double-cone hypersonic flowreveals that the three collisionsmodels can predict thepressure coefficient and the Stanton number trendsThe agreement between the DSMC simulation andthe experiment is the best in the HS collisions model

(3) Investigation of type IV shock interaction shows thatthe agreement between the DSMC simulation andthe experiment is generally good in the HS and VHScollision models However it fails in the VSS model

(4) Both double-cone flow and type IV shock interactionsimulations show that the DSMC error dependson the Kn number and the models employed forintermolecular interaction Increasing the Kn num-ber decreases the DSMC error The error is the small-est for the HS model compared with those for the

10 The Scientific World Journal

Table 6 Comparisons of calculations ((ValueDSMC minus Valueexperiment)(Valueexperiment))

Run KnThe scale of interaction region

(error in compared with measured data )Peak value of the Stanton number

(error in compared with measured data )HS VHS VSS HS VHS VSS

33 753 times 10minus5 467 734 534 190 209 245

36 891 times 10minus5 24 70 43 78 928 78

26 905 times 10minus5 24 66 38 42 857 485

32 134 times 10minus4 0 59 31 52 578 474

38 252 times 10minus4 68 724 3104 52 684 421

Table 7 Flow conditions of Edneyrsquos type IV shock interaction

Ma Rem Kn10 103 times 10

5

145 times 10minus4

VHS and VSS models When the Kn number is in thelevel of 10minus4 the DSMC error for pressure coefficientthe Stanton number and scale of interaction region iscontrolled within 10

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the Chinese AerospaceInnovation and in part by the Basic Research Fund ofNorthwestern Polytechnical University

References

[1] M S Holden T P Wadhams J K Harvey and G V CandlerldquoComparisons between measurements in regions of laminarshockwave boundary layer interaction in hypersonic flowswithnavier-stokes and DSMC solutionsrdquo Tech Rep RTO-TR-AVT-007-V3 NATO Science and Technology Organization 2006

[2] J Olejniczak and G V Candler ldquoComputation of hyper-sonic shock interaction flow fieldsrdquo in Proceedings of the 7thAIAAASME JointThermophysics and Heat Transfer ConferenceAIAA Paper 98-2446 June 1998

[3] G V Candler I Nompelis M-C Druguet I D Boyd W-LWang and S M Holden ldquoCFD validation for hypersonic flighthypersonic double-cone flow simulationsrdquo in Proceedings of the40th Aerospace Sciences Meeting amp Exhibit AIAA 2002-0581Reno Nev USA January 2002

[4] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Oxford Clarendon Press New York NY USA 1994

[5] Harvey K John M S Holden and T P Wadhams ldquoCode vali-dation study of laminar shockboundary layer and shockshockinteractions in hypersonic flow Part B comparison withnavier-stokes and DSMC solutionsrdquo in Proceedings of the 39thAerospace Sciences Meeting amp Exhibit AIAA 2001-1031 RenoNev USA January 2001

[6] M Gallis C Roy J Payne and T Bartel ldquoDSMC and navier-stokes predictions for hypersonic laminar interacting flowsrdquo inProceedings of the 39th Aerospace Sciences Meeting and ExhibitAIAA 2001-1030 Reno Nev USA January 2001

[7] G V Candler I Nompelis and M-C Druguet ldquoNavier-stokespredictions of hypersonic double-cone and cylinder-flare flowfieldsrdquo in Proceedings of the 39th Aerospace Sciences Meeting ampExhibit AIAA Paper no 2001-1024 Reno Nev USA January2001

[8] J Moss ldquoDSMC computations for regions of shockshock andshockboundary layer interactionrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA 2001-1027 RenoNev USA January 2001

[9] G A Bird The DSMC Method Create Space IndependentPublishing Platform 2013

[10] M Woo and I Greber ldquoMolecular dynamics simulation ofpiston-driven shock wave in hard sphere gasrdquo AIAA Journalvol 37 no 2 pp 215ndash221 1999

[11] K Koura and H Matsumoto ldquoVariable soft sphere molecularmodel for air speciesrdquo Physics of Fluids A vol 4 no 5 pp 1083ndash1085 1992

[12] P S Prasanth and J K Kakkassery ldquoDirect simulation MonteCarlo (DSMC) a numerical method for transition-regimeflows a reviewrdquo Journal of the Indian Institute of Science vol86 no 3 pp 169ndash192 2006

[13] T Tokumasu and Y Matsumoto ldquoDynamic molecular collision(DMC) model for rarefied gas flow simulations by the DSMCmethodrdquo Physics of Fluids vol 11 no 1 pp 1907ndash1920 1999

[14] S F Borisov ldquoProgress in gas-surface interaction studyrdquo inProceedings of the 24th International Rarefied Gas DynamicsSymposium M Capitelli Ed pp 933ndash940 The AmericanInstitute of Physics 2005

[15] I D Boyd and W-L Wang ldquoMonte Carlo computationsof hypersonic interacting flowsrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA Paper no 2001-1029 Reno Nev USA January 2001

[16] J N Moss T Pot B Chanetz and M Lefebvre ldquoDSMCsimulation of shockshock interactions emphases on type IVinteractionrdquo in Proceedings of the 22nd International Symposiumon Shock Waes Paper no 3570 Imperial College London UK1999

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 The Scientific World Journal

Table 6 Comparisons of calculations ((ValueDSMC minus Valueexperiment)(Valueexperiment))

Run KnThe scale of interaction region

(error in compared with measured data )Peak value of the Stanton number

(error in compared with measured data )HS VHS VSS HS VHS VSS

33 753 times 10minus5 467 734 534 190 209 245

36 891 times 10minus5 24 70 43 78 928 78

26 905 times 10minus5 24 66 38 42 857 485

32 134 times 10minus4 0 59 31 52 578 474

38 252 times 10minus4 68 724 3104 52 684 421

Table 7 Flow conditions of Edneyrsquos type IV shock interaction

Ma Rem Kn10 103 times 10

5

145 times 10minus4

VHS and VSS models When the Kn number is in thelevel of 10minus4 the DSMC error for pressure coefficientthe Stanton number and scale of interaction region iscontrolled within 10

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the Chinese AerospaceInnovation and in part by the Basic Research Fund ofNorthwestern Polytechnical University

References

[1] M S Holden T P Wadhams J K Harvey and G V CandlerldquoComparisons between measurements in regions of laminarshockwave boundary layer interaction in hypersonic flowswithnavier-stokes and DSMC solutionsrdquo Tech Rep RTO-TR-AVT-007-V3 NATO Science and Technology Organization 2006

[2] J Olejniczak and G V Candler ldquoComputation of hyper-sonic shock interaction flow fieldsrdquo in Proceedings of the 7thAIAAASME JointThermophysics and Heat Transfer ConferenceAIAA Paper 98-2446 June 1998

[3] G V Candler I Nompelis M-C Druguet I D Boyd W-LWang and S M Holden ldquoCFD validation for hypersonic flighthypersonic double-cone flow simulationsrdquo in Proceedings of the40th Aerospace Sciences Meeting amp Exhibit AIAA 2002-0581Reno Nev USA January 2002

[4] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Oxford Clarendon Press New York NY USA 1994

[5] Harvey K John M S Holden and T P Wadhams ldquoCode vali-dation study of laminar shockboundary layer and shockshockinteractions in hypersonic flow Part B comparison withnavier-stokes and DSMC solutionsrdquo in Proceedings of the 39thAerospace Sciences Meeting amp Exhibit AIAA 2001-1031 RenoNev USA January 2001

[6] M Gallis C Roy J Payne and T Bartel ldquoDSMC and navier-stokes predictions for hypersonic laminar interacting flowsrdquo inProceedings of the 39th Aerospace Sciences Meeting and ExhibitAIAA 2001-1030 Reno Nev USA January 2001

[7] G V Candler I Nompelis and M-C Druguet ldquoNavier-stokespredictions of hypersonic double-cone and cylinder-flare flowfieldsrdquo in Proceedings of the 39th Aerospace Sciences Meeting ampExhibit AIAA Paper no 2001-1024 Reno Nev USA January2001

[8] J Moss ldquoDSMC computations for regions of shockshock andshockboundary layer interactionrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA 2001-1027 RenoNev USA January 2001

[9] G A Bird The DSMC Method Create Space IndependentPublishing Platform 2013

[10] M Woo and I Greber ldquoMolecular dynamics simulation ofpiston-driven shock wave in hard sphere gasrdquo AIAA Journalvol 37 no 2 pp 215ndash221 1999

[11] K Koura and H Matsumoto ldquoVariable soft sphere molecularmodel for air speciesrdquo Physics of Fluids A vol 4 no 5 pp 1083ndash1085 1992

[12] P S Prasanth and J K Kakkassery ldquoDirect simulation MonteCarlo (DSMC) a numerical method for transition-regimeflows a reviewrdquo Journal of the Indian Institute of Science vol86 no 3 pp 169ndash192 2006

[13] T Tokumasu and Y Matsumoto ldquoDynamic molecular collision(DMC) model for rarefied gas flow simulations by the DSMCmethodrdquo Physics of Fluids vol 11 no 1 pp 1907ndash1920 1999

[14] S F Borisov ldquoProgress in gas-surface interaction studyrdquo inProceedings of the 24th International Rarefied Gas DynamicsSymposium M Capitelli Ed pp 933ndash940 The AmericanInstitute of Physics 2005

[15] I D Boyd and W-L Wang ldquoMonte Carlo computationsof hypersonic interacting flowsrdquo in Proceedings of the 39thAerospace Sciences Meeting and Exhibit AIAA Paper no 2001-1029 Reno Nev USA January 2001

[16] J N Moss T Pot B Chanetz and M Lefebvre ldquoDSMCsimulation of shockshock interactions emphases on type IVinteractionrdquo in Proceedings of the 22nd International Symposiumon Shock Waes Paper no 3570 Imperial College London UK1999

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of