3D DSMC Simulation of Rarefied Hypersonic Flowover a Sharp Flat Plate

>H __ skHc #H. Yamaguchi , N. Tsuboi , Y. Matsumoto

* Department of Mechanical Engineering, the University of Tokyo,7-3-1, Hongo, Bunkyo, Tokyo 113-8656, Japan

** Research Division for Space Transportation, Institute of Space and Astronautical Science,3-1-1, Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan

Abstract. The flow over a sharp flat plate with a finite leading edge angle located in a rarefied hypersonic flow has beeninvestigated numerically with the direct simulation Monte Carlo (DSMC) method using the Dynamic Molecular Collision(DMC) Model based on Molecular Dynamics (MD) simulation of nitrogen molecules. The results of the two-dimensionalnumerical simulations of the zero leading edge angle plate agree fairly well with the experimental results [Lengrand, J.C.et al. (1992)] in the middle cross section of the span direction. Thus to investigate much more precisely about the detailedphysical phenomena in the rarefied hypersonic flow over the sharp flat plate, the three-dimensional numerical simulationshave been made. At first, to compare with two-dimensional simulation, the effect of the existence of the finite leadingedge angle - "the leading edge effect", and the effect of the three-dimensional flow structure due to the side edge of theplate - "the finite span effect" are investigated. From these numerical simulations, the flow field around the plate isshown three-dimensionally, and the effects are evaluated to determine the three-dimensional flow structure over the plate.

INTRODUCTION

Many researchers have investigated the rarefied hypersonic flows around space vehicles with respect to thenonequilibrium characteristics. The external flow field and the boundary layer growth around a body of given shapehave the mutual interaction called viscous interaction. This viscous interaction affects the flight characteristics of thevehicles. Thus it is important to analyse this phenomenon. In rarefied regime, the shock wave and the boundary layer"merge" near a leading edge, and in that region, it becomes difficult to distinguish the shock wave from theboundary layer. This region is called the merged layer. A strong nonequilibrium between translational and internaldegrees of freedom of molecules could be observed in this region. However the nonequilibrium in the merged layerhas not been discussed in detail.

In numerical simulations, the direct simulation Monte Carlo (DSMC) method [1] is widely used to simulate therarefied gas flows. The gas-gas interaction model has very important role in this method. To investigatenonequilibrium between the translational and the internal energy of diatomic molecules, the relaxation of the internaldegrees of freedom should be simulated in detail. Thus the molecular collisions, which is micro-scale phenomena,and the macro-scale phenomena of the flow field around a body of given shape, should be treated at one time,because micro-scale phenomena affect macro-scale phenomena and vice versa. The analysis should be done overmulti scales, micro to macro. Thus the Dynamic Molecular Collision (DMC) model [2] is chosen as the collisionmodel. The DMC model is derived from a lot of Molecular Dynamics (MD) simulations of diatomic molecule (N2)two-body collisions. It is the statistical model based on micro-scale phenomena for the macro-scale analysingmethod. The schematic diagram of their relations is shown in Fig.l.

To make the analysis easier, a sharp flat plate is chosen as an object. The another reason of deciding an object isthat there exists the experimental result [3]. To investigate the merged layer in detail, the numerical simulation ismade and compared with the experimental result. There are two elements that could affect the three-dimensionalflow structure around the object - the finite leading edge angle and the existence of the side edge of the plate. Thusthree-dimensional and two-dimensional numerical simulations have made and compared to investigate these effects- the leading edge effect and the finite span effect.

CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00

764

Rarefied Gas

Shock wave

Interaction betweenatomic nucleus and electron

Relaxation ofmolecular collisions

V

Nonequilibrium flow

Modelling

Probability

Translation^/Rotational energy

Collision model

FIGURE 1. Schematic diagram of the present multi-scale analysis model.

NUMERICAL METHOD AND CONDITIONS

To make numerical simulations, the DSMC method is applied. In the DSMC method, relaxation of diatomicmolecules in nonequilibrium flow is the state of art. From the view of the multi scale analysis, the DMC model,which is derived from a lot of MD simulations of diatomic molecules (N2) two-body collisions, is chosen as acollision model of diatomic molecules. The collision frequency is calculated with Null-Collision method [4]. For thegas-surface interaction, the diffuse model is applied as the conclusion of Ref.5.

The flow conditions and the plate sizes are same as those of experiment in Ref.3, and listed in table 1. Thecoordinate is taken as Fig.l. Almost all over the plate belongs to the merged layer regime in this condition.

The numerical simulation is made with four ways.a) 2-dimensional with zero leading edge angle (infinitely thin plate approximation)b) 2-dimensional with finite leading edge anglec) 3-dimensional with zero leading edge angle (infinitely thin plate approximation)d) 3-dimensional with finite leading edge angle

Three-dimensional calculation needs too much memory and time. Thus three-dimensional case is simulated withparallel computing.

FIGURE 2. Coordinate system for the numerical simulation.

765

TABLE 1. Flow Conditions andProperties

Moo

TolK]Pol Pa]T»[K]

P*[Pa]T*[K]

Re (based on 50mm)Kn (based on 50mm)

Length of the plate [mm]Width of the plate [mm]Thickness of the plate [mm]Leading edge angle[deg.]

Plate Sizes.Ref.[3],[6]

20.21100

3.5-105

13.320.06839

290566

0.047100100520

PARALLEL COMPUTING

It takes much time and computing cost when three-dimensional simulation is done. Thus parallel computing isapplied to that case. The three-dimensional DSMC code is modified with MPI (Message Passing Interface) library.When there is a need to transfer data from one process to another process, the data is restructured as a matrix andtotal transfer data length is shorten.

The parallel computers are Hitachi SR2201, IBM RS/6000 SP-2, Visual Technology Alpha-21164A 600MHz.The last computer is not parallel machine, but workstation cluster, which are connected with 100Base-TX Ethernetnetwork. The computing time to calculate speed up is only CPU time. It does no include I/O process. Almost allcases, the linear speed-ups are achieved. From this result, this DSMC parallel computing code shows very goodperformance and does not depend on the system nor architectures of the machine.

This parallel computation is used for the three-dimensional simulation written below.

a.•oCD8.03

10

1

1 I I I I I 111SR2201 ":

SP-2VT-Alphaideal

14 6 8

102 4 6

number of processors

FIGURE 3. Parallel speed-up of the 3D DSMC code to the number of processors used to calculate (log scale).

766

RESULTS AND DISCUSSIONS

To compare the experimental results and the numerical results, it is better that the condition of the numericalsimulation is simple. Thus ordinary simulation is made with some approximations. In this case, the hypersonicrarefied gas flows around the flat plate could be simulated with two-dimensional infinitely thin plate condition. Now,from the view of the multi scale analysis, the validity of numerical simulation is checked.

Four cases are simulated on the same condition as the experiment [3] -a) 2-dimensional with zero leading edge angle (infinitely thin plate approximation)b) 2-dimensional with finite leading edge anglec) 3-dimensional with zero leading edge angle (infinitely thin plate approximation)d) 3-dimensional with finite leading edge angle.

At first, the density distributions of one case of the results of three-dimensional numerical simulation with finiteleading edge angle are shown in Fig.4. From them, the special density distributions become clear and the size of theleading edge effect and the finite span effect might be appears.

To determine the size of three-dimensional effects - e.g. the leading edge effect and the finite span effect, thedensity contours of the flow field are compared between four cases (Fig.5). From the left side four contours, not alarge difference could be seen. But the right side contours of the cross-sections at X/L = \.5 shows differentdistribution around Z/L = 1.0, where the side edge of the plate exists. The difference of the contour map should bethe effect of boundary condition that is applied to the numerical simulation. The comparison is made at the middlecross section of the span Z/L = 0.0, thus it is not affect the conclusion.

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0,5 1.0X/L

0.00 0.25 0,50 0.75 1.00 1.25Z<t

ML

Dens ity distributions

FIGURE 4. Density contours over the flat plate (upper left: cross section at Z/L = 0, upper right: crosssection at X/L = 1.5 , lower left: cross section at Y/L = 0 , lower right: special distribution overthe half width of the plate).

767

1.0

0.8

0.6

0.4

0.2

0.0

•'$>r,1 *0

O> !

^IIE^* """~~'"T~Z^^$;i.4:t5.^ l̂t-.4- -..-....- -^g^.™ ̂ ^5 ̂

Jf ' ^^^^^ .̂feSf^?/>^i

_.....̂ !!....|t

0.0 0.5 1.0 1.5 2.0

X/L

a) 2D infinitely thin plate

1.0

0.8

0.6

0.4

0.2

o.o0.0 0.5 1.0 1.5 2.0

X/L

b) 2D with leading edge angle 20deg.

1.0

0.8

0.6

0.4

0.2

o.oo.o

:o^°-9 ;ro.7^.....i......gr:...............l.0.5 1.0 1.5

X/L

c) 3D infinitely thin plate

2.0

1.0

0.8

0.6

0.4

0.2

0.0

_rr-rr1.4

-:i-3^T1-2q:...... :..':!; 0.9 1..:::::t.̂ Sr:::0^̂ ^0.00 0.25 0.50 0.75 1.00 1.25

Z/L

c) 3D infinitely thin plate

1.50 1.75

1.0

0.8

0.6

0.4

0.2

o.o

1.1-

o.o 0.5 1.0 1.5 2.0

X/L

d) 3D with leading edge angle 20deg.

1.0

0.8

0.6

0.4

0.2

0.00.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75

Z/L

d) 3D with leading edge angle 20deg.

FIGURE 5. Density contours over the flat plate (from top: condition a), b), c), d); left: cross sectionat Z/L = 0, right: cross section at X/L = 1.5 ).

768

1.0

0.8

0.6

0.4

0.2

0.0

«lr- 2DleOO~2Dle20SDIeOO-3Dle20Exp. I

0.8 1.0 1.2 1.4

P/Pinf

16

14

12

10

8

6

0.0 0.5I

1.0

X/L

Jr- 2DleOO2Dle20.

- SDIeOO3DI020-Exp.

1.5 2.0

34x10"

32

30

28

26

24

22

20

--Jr- 2DleOO-A- 2Dle20

SDIeOO3Dle20Exp.

0.0 0.5 1.0

X/L

1.5 2.0

FIGURE 6. Density profiles over the plate at X/L = 1.5 (upper), Pressure profiles on the plate at Z/L=Q(lower left), Heat transfer rate profiles on the plate at Z/L = 0 (lower right).(Exp.: [Lengrand, J.C. et al. (1992) [3]]).

Now quantitative comparisons between four cases and experimental results [3] are made to determine the size ofthe effects. Density profiles over the plate at X/L = 1.5 , pressure profiles on the plate at Z/L = 0, heat transfer rateprofiles on the plate at Z/L = 0 are compared in Fig.6. The pressure means the normal stress Fn deduced frommolecular momentum exchange, not thermodynamic pressure p. It is same treatment as Ref.3. From them, severalthings could be apparent. The density profiles show very good agreement between experimental results andnumerical results. The properties on the plate agree well near the leading edge but large discrepancy could be seen inmost part of the downstream region. The profile of the experimental pressure shows a trend of discontinuity. And itis very rarefied condition on and near the plate. Thus the results of experiment and numerical simulation agree wellin reasonable way for this condition.

Comparisons should be done between numerical results in the next step. From the figures, it is clear that thediscrepancy between four lines is not so large as that between experimental result and numerical result. But to seemuch more closely, it appears that four lines are separated to two sets, continuous and dotted line, not triangle anddiamond-shaped markers. It means that two-dimensional and three-dimensional results do not show any discrepancyand zero degrees leading edge angle and finite leading edge angle simulation result different in the cross section of

769

0.50

0.25

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75

0.75

0.50

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75

FIGURE 7. Pressure contour map on the plate (left), Heat transfer rate contour map on the plate (right).

the centre of the plate in span direction. This means that the leading edge effect could be affect the flow field. But inthis condition, the external flow is rarefied to Kn=0.047, hypersonic to M-20.2 and the leading edge angle is smallas 20 deg. and the aspect ratio is 7. All of them make these effects small. Thus it could be concluded that the leadingedge effect and the finite span effect is negligible in this condition. The leading edge effect is larger than the finitespan effect. But the leading edge effect could be negligible around the angle 20 deg. as Ref.7. In Ref.7, the leadingedge angle is set as 75, 30, 45, 60 and 80 deg. It is concluded there that only the case 75 deg., the properties on theplate is not affected. Thus in the middle cross section of the span direction, the flow could be treated with theapproximation of the two-dimensional flow.

But it is clear that without the middle cross section of the span direction, the flow field is different from the resultof two-dimensional simulation. To see these three-dimensional effects, the contour maps of the pressure and heattransfer rate on the plate are shown in fig.7. It is interesting that the region of high value in maps is narrow towardthe leeward of the plate. It is because the external flow passed the side edge affect the flow over the plate. The Heattransfer rate contour map shows that the heat transfer is much larger around the side edge. In Fig.5 - right side ofcase d) 3D with leading edge angle 20deg. of cross section at X/L = 1.5 , the density around the side edge is higher.It shows another proof that the flow from the side edge affects the flow field over the plate.

The three-dimensional flow structure becomes apparent with the investigation of the three-dimensional numericalsimulation.

CONCLUDING REMARKS

The rarefied hypersonic flows over a sharp flat plate have been investigated with the 3D DSMC numericalsimulations. From them, the flow field around the plate has been shown three-dimensionally. From the comparisonbetween numerical and experimental result, the validity of the numerical simulation is confirmed, though there isdiscrepancy between them. To compare the four cases of the numerical simulations, the leading edge effect and thefinite span effect are investigated. These effects could be negligible in the condition here, because of rarefaction,hypersonic flow, small leading edge angle, and large aspect ratio. The leading edge effect is much lager than thefinite span effect. The approximation of the two-dimensional flow could be applied to this condition. From theprofiles on the plate from the three-dimensional numerical simulation, the three-dimensional flow structure over theplate is affected. The three-dimensional flow structure over the plate is analysed.

770

ACKNOWLEDGMENTS

The authors would thank Professor J.C. Lengrand (C.N.R.S - France) for his kindly offer of information about hisexperiments.

REFERENCES

1. Bird, G.A., Molecular Gas Dynamics, Clarendon Press, Oxford, 1976.2. Tokumasu, T. and Matsumoto, Y., Phys. Fluids 11 (7), 1907-1920 (1999).3. Lengrand, J.C., Allegre, J., Chpoun, A., and Raffm, M., 18th Int. Symp. on Rarefied Gas Dynamics, 1992, pp.276-284.4. Koura, K., Phys. Fluids 29 (11), 3509-3511 (1986).5. Tsuboi, N. and Matsumoto, Y., 22^ Int. Symp. on Rarefied Gas Dynamics, to appear.6. Lengrand, J.C., private communications.7. Heffner, K.S., Gottesdiener, L., Chpoun, A. and Lengrand, J.C., AIAA 91-1749, (1991).

771