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2 ,1752'8&7,21 Hypersnic waveriders are advanced hypersnic lifting bdies which generate high values f aerdynamic lift-t-drag rati at high Mach numbers. Waverider cnfiguratins, intrduced by Nnweiler (1959), are derived frm a knwn analytical flwfield, such as flw ver a tw-dimensinal wedge r flw arund a slender cne. These cnfiguratins are designed analytically with infinitely sharp leading edges fr shck wave attachment. Because the shck wave is attached t the leading edge f the vehicle, the upper and lwer surfaces f the vehicle can be designed separately. Furthermre, the shck wave acts as a barrier in rder t prevent spillage f higher-pressure airflw frm the lwer side f the vehicle t the upper side, resulting in a high-pressure differential and enhanced lift. Usually, it is extremely difficult t cnstruct a perfectly sharp leading edge. Any manufacturing errr results in a significant deviatin frm the design cntur. Mrever, sharp edges are difficult t maintain because they are easily damaged. Additinally, because heat transfer increases inversely with the leading edge radius, high heating is assciated with sharp edges. Therefre, fr practical hypersnic cnfiguratins, leading edges shuld be blunt fr heat transfer, manufacturing and handling cncerns. Because blunt leading edge prmtes shck wave standff, practical leading edges will have shck detachment, making leading edge blunting a majr cncern in the design and predictin f flwfields ver hypersnic cnfiguratins. The flwfield prperties upstream f the leading edge f a bdy are affected by mlecules reflected frm the edge regin. The degree f the effect is in part cnditined by the edge gemetry. In this cntext, Sants (2002 and 2005) investigated the effect f the flatface thickness f truncated wedges n the flwfield structure and n the aerdynamic surface quantities. The thickness effect was examined fr a range f Knudsen number, based n the thickness f the flat face, cvering frm the transitinal flw regime t the free mlecular flw ne. The emphasis f the wrks was t prvide a critical analysis n maximum allwable gemetric bluntness, dictated by either handling r manufacturing requirements, resulting n reduced departures frm ideal aerdynamic perfrmance f the vehicle. Thus, allwing the blunted leading edge t mre clsely represents the riginal sharp leading edge flwfield. Such analysis is als imprtant when a cmparisn is t be made between experimental results in the immediate vicinity f the leading edge and the theretical results, which generally assume a zer-thickness leading edge. Sants (2003) extended further the analysis presented by Sants (2002) n truncated wedges by perfrming a parametric study n these shapes with emphasis placed n the cmpressibility effects. The primary gal f the wrk was t assess the sensitivity f the stagnatin pint heating, ttal drag and shck wave standff distance t changes n the freestream Mach number. These wrks (Sants, 2002, 2003 and 2005) n hypersnic flw past truncated wedges have been cncentrated primarily n the analysis f the aerthermdynamic surface quantities by cnsidering the diffuse reflectin mdel as being the gas-surface interactin. The diffuse mdel assumes that the mlecules are reflected equally in all directins, quite independently f their incident speed and directin. Hwever, as a space flight vehicle is expsed t a rarefied envirnment ver a cnsiderable time, a departure frm the fully diffuse mdel is bserved, resulting frm the clliding mlecules that clean the surface f the vehicle, which becmes gradually decntaminated. In this cnnectin, Sants (2004) perfrmed a parametric study n truncated wedges with emphasis placed n the gas-surface interactin effects. In this scenari, the primary interest was t assess the sensitivity f the heat transfer and ttal drag cefficients t variatins n the surface accmmdatin cefficients experienced by the leading edges.

3 The current prpsed paper extends further the previus analysis n truncated wedges by investigating the impact f the angle f attack n the shck-wave structure. Fr psitive angle f attack, imprtant changes ccur in the flwfield structure and in the aerdynamic surface quantities n blunt leading edges. This invlves the mdificatin f the flwfield prperties and shck strength and, cnsequently, sme effects n aerdynamic frces acting n, and n heat transfer t the bdy surface. Mrever, the knwledge f these prperties at zer angle f attack is nt sufficient t predict with certainty the flw characteristics ver these shapes with incidence. The incidence increase causes an asymmetry in the flw patterns as the stagnatin pint mves frm the symmetry axis t the windward side fr psitive angle f attack. In an effrt t btain further insight int the nature f the flwfield structure f truncated wedges under hypersnic transitinal flw cnditins, the essential characteristics f the angle-f-attack effect n the shck wave will be examined fr psitive angle f attack with 5, 10, 15 and 20 degrees f incidence. Under hypersnic transitinal flw cnditins, at very high speeds and high altitudes, the flw ver a given aerdynamic cnfiguratin may be sufficiently rarefied that the apprpriate mlecular mean free path becmes t large, cmpared t a characteristic length f the vehicle fr the use f cntinuum assumptins but nt large enugh fr the use f the free mlecular thery. In such an intermediate r transitin rarefied gas regime, where a significant degree f nn-equilibrium is bserved in the flws, the Direct Simulatin Mnte Carl (DSMC) methd (Bird, 1994) has been emplyed in rder t slve the prblems invlving flws f rarefied hypersnic aerthermdynamics. /($',1*('*(*(20(75< The gemetry f the leading edges in this wrk is the same as that presented in previus wrk (Sants 2005). The truncated wedges are mdeled by assuming a sharp leading edge f half angle θ with a circular cylinder f radius 5 inscribed tangent t this sharp leading edge. The truncated wedges are als tangent t the sharp leading edge and the cylinder at the same cmmn pint. It was assumed a leading edge half angle f 10 degrees, a circular cylinder diameter f 10-2 m and frntal-face thickness W/λ f 0.01, 0.1 and 1, where λ is the freestream mean free path. Figure 1(a) illustrates schematically this cnstructin. It was assumed that the truncated wedges are infinitely lng but nly the length / is cnsidered, since the wake regin behind the truncated wedges is nt f interest in this investigatin. (a) Tangency pint Truncated wedge Sharp wedge θ Cylinder α Flw (b) II η I ξ III III Figure 1: Drawing illustrating (a) the leading edge shapes and (b) the cmputatinal dmain.

4 &20387$7,21$/722/ The ptin f the numerical apprach in rder t mdel rarefied nn-equilibrium flws relies n the extent f flw rarefactin. Fr near-cntinuum flws, the bundary cnditins f slip velcity and temperature jump are enugh t take int accunt fr the rarefactin effects. These bundary cnditins are cmmnly emplyed in the Navier-Stkes equatins r in the viscus shck layer equatins. The Navier-Stkes equatins can be derived frm the Bltzmann equatin (Cercignani, 1988) under the assumptin f small deviatin f the distributin functin frm equilibrium. Nevertheless, the Navier-Stkes equatins becme unsuitable fr studying rarefied flws where the distributin functin becmes cnsiderable nn-equilibrium. In rder t study flws with a significant degree f nn-equilibrium, the Direct Simulatin Mnte Carl (DSMC) methd (Bird, 1994), pineered by Bird in the 60 s, has becme the standard technique emplyed. The DSMC methd simulates real gas flws with varius physical prcesses by means f a huge number f mdeling particles, each f which is a typical representative f great number f real gas mlecules. DSMC mdels the flw as being a cllectin f discrete particles, each ne with a psitin, velcity and internal energy. The state f particles is stred and mdified with time as the particles mve, cllide, and underg bundary interactins in simulated physical space. The simulatin is always calculated as unsteady flw. Hwever, a steady flw slutin is btained as the large time state f the simulatin. Therefre, the DSMC methd is basically an explicit time-marching algrithm. Cllisins in the present DSMC cde are mdeled by using the variable hard sphere (VHS) mlecular mdel (Bird, 1981) and the n time cunter (NTC) cllisin sampling technique (Bird, 1989). Repartitin energy amng internal and translatinal mdes is cntrlled by the Brgnakke-Larsen statistical mdel (Brgnakke and Larsen, 1975). Simulatins are perfrmed using a nn-reacting gas mdel fr a cnstant freestream gas cmpsitin cnsisting f 76.3% f N 2 and 23.7% f O 2. Energy exchanges between the translatinal and internal mdes, rtatinal and vibratinal, are cnsidered. Relaxatin cllisin numbers f 5 and 50 were used fr the calculatins f rtatin and vibratin, respectively. &20387$7,21$/)/2:'20$,1$1'*5,' The cmputatinal dmain is discretized by using a structured grid. The dmain is divided int an arbitrary number f regins, which are subdivided int cmputatinal cells. The cells are further subdivided int fur subcells, tw subcells/cell in each crdinate directin. The linear dimensins f the cells shuld be small in cmparisn with the scale length f the macrscpic flw gradients nrmal t the streamwise directins, which means that the cell dimensins shuld be f the rder f r even smaller than the lcal mean free path (Alexander et al., 1998 and 2000). In the current DSMC cde, the cell prvides a cnvenient reference fr the sampling f the macrscpic gas prperties, while the cllisin partners are selected frm the same subcell. As a result, the flw reslutin is much higher than the cell reslutin. Clse t the bdy surface, cell spacing nrmal t the bdy shuld be als f the rder f a third f the lcal mean free path. If the cell size near the bdy surface is t large, then energetic mlecules at the far edge f the cell are able t transmit mmentum and energy t mlecules immediately adjacent t the bdy surface. This leads t verpredictin f bth the surface heat flux and the aerdynamic frces n the bdy that wuld ccur in the real gas (Haas and Fallavllita, 1994).

5 Hence, heat transfer cefficient and pressure cefficient were used as the representative prperties fr the grid sensitivity study. Figure 1(b) depicts the physical extent f the cmputatinal dmain fr the present simulatins. Based n this figure, side I is defined by the bdy surface. Diffusin reflectin is the cnditin applied t this side. Side II is the freestream side thrugh which simulated mlecules enter and exit. Finally, the flw at the dwnstream utflw bundary, side III, is predminantly supersnic and vacuum cnditin is specified (Gu and Liaw, 2001). At this bundary, simulated mlecules can nly exit. Numerical accuracy in DSMC methd depends n the grid reslutin chsen as well as n the number f particles per cmputatinal cell. Bth effects were investigated t determine the number f cells and the number f particles required t achieve grid independence slutins. The grid generatin scheme used in this study fllws that prcedure presented by Bird (1994). Alng the uter bundary (side II) and the bdy surface (side I) (see Fig. 1(b)), pint distributins are generated in such way that the number f pints n each side is the same (ξ-directin in Fig. 1(b)). Then, the cell structure is defined by jining the crrespnding pints n each side by straight lines and then dividing each f these lines int segments which are jined t frm the system f quadrilateral cells (η-directin in Fig. 1(b)). The distributin can be cntrlled by a number f different distributin functins that allw the cncentratin f pints in regins where high flw gradients r small mean free paths are expected. A grid independence study was made with three different structured meshes in each crdinate directin. The effect f altering the cell size in the ξ-directin was investigated with grids f 50(carse), 100(standard) and 150(fine) cells, and 60 cells in the η-directin fr the bluntest leading edge investigated, W/λ = 1 case. In analgus fashin, an examinatin was made in the η-directin with grids f 30(carse), 60(standard) and 90(fine) cells, and 100 cells in the ξ-directin fr the W/λ = 1 case. Frm the ttal number f cells in the ξ-directin, 30% are lcated alng the frntal surface and 70% distributed alng the afterbdy surface. In additin, each grid was made up f nn-unifrm cell spacing in bth directins. The effect (nt shwn) f changing the cell size in bth directins n the heat transfer and pressure cefficients was rather insensitive t the range f cell spacing cnsidered, indicating that the standard grid, 100x60 cells, fr the W/λ = 1 case, is essentially grid independent. A similar prcedure was perfrmed fr the ther cases investigated. &20387$7,21$/&21',7,216 DSMC simulatins have been perfrmed fr an altitude f 70 km based n the flw cnditins given by Sants (2005) and summarized in Table 1, and the gas prperties (Bird, 1994) are shwn in Table 2. Referring t Tables 1 and 2, 7, S, ρ, Q, µ, and λ stand respectively fr temperature, pressure, density, number density, viscsity and mean free path, and ;, P, G and ωaccunt respectively fr mle fractin, mlecular mass, mlecular diameter and viscsity index. The freestream velcity 9 is assumed t be cnstant at 3.56 km/s, which crrespnd t a freestream Mach number 0 f 12. The translatinal and vibratinal temperatures in the freestream are in equilibrium at 220 K, and the wedge surface has a cnstant wall temperature 7 f 880 K fr all cases cnsidered. This temperature is chsen t be representative f the surface temperature near the stagnatin pint and is assumed t be unifrm ver the bdies. The verall Knudsen number.q is defined as the rati f the mlecular mean free path λ in the freestream gas t a characteristic dimensin f the flwfield. In the present accunt, the characteristic dimensin was defined as being the thickness W f the frntal face f the leading edges. Fr the thicknesses investigated, W/λ = 0.01, 0.1 and 1, the verall Knudsen numbers

6 crrespnd t.q = 100, 10, and 1. Finally, the Reynlds number 5H cvers the range frm t 19.3, based n cnditins in the undisturbed stream with leading edge thickness W as the characteristic length. Table 1: Freestream Cnditins $OWLWXGH 7 S ρ x 10 5 Q x µ x 10 5 λ x (km) (K) (N/m 2 ) (kg/m 3 ) (m -3 ) (Ns/m 2 ) (m) (m/s) Table 2: Gas Prperties ; P (kg) G (m) ω O x x N x x In rder t simulate the angle-f-attack effect, the DSMC calculatins were perfrmed independently fr five distinct numerical values f α, i.e., 0, 5, 10, 15 and 20 degrees. It is imprtant t mentin that α equal t 0 represents the case investigated previusly (Sants, 2005). &20387$7,21$/352&('85( The prblem f predicting the shape and lcatin f detached shck waves has been stimulated by the necessity fr blunt nses and leading edges cnfiguratins designed fr hypersnic flight in rder t cpe with the aerdynamic heating. In additin, the ability t predict the shape and lcatin f shck waves is f primary imprtance in analysis f aerdynamic interference. Furthermre, the knwledge f the shck wave displacement is especially imprtant in waveriders (Nnweiler, 1959), since these hypersnic cnfiguratins usually rely n shck wave attachment at the leading edges t achieve their high lift-t-drag rati at high-lift cefficient. In this present accunt, the shck-wave structure, defined by shape, thickness and detachment f the shck wave, is predicted by emplying a prcedure based n the physics f the particles. In this respect, the flw is assumed t cnsist f three distinct classes f mlecules; class I mlecules dente thse mlecules frm freestream that have nt been affected by the presence f the leading edge; class II mlecules designate thse mlecules that, at sme time in their past histry, have struck and been reflected frm the bdy surface; and finally, class III mlecules define thse mlecules that have been indirectly affected by the presence f the bdy. Figure 2(a) illustrates the definitin fr the mlecular classes. It is assumed that the class I mlecule changes t class III mlecule when it cllides with class II r class III mlecule. Class I r class III mlecule is prgressively transfrmed int class II mlecule when it interacts with the bdy surface. Als, a class II mlecule remains class II regardless f subsequent cllisins and interactins. Hence, the transitin frm class I mlecules t class III mlecules may represent the shck wave, and the transitin frm class III t class II may define the bundary layer. A typical distributin f class III mlecules alng the stagnatin streamline fr blunt leading edges is displayed in Fig. 2(b) alng with the definitin used t determine the thickness, displacement and shape f the shck wave. In this figure, ; is the distance [ alng the stagnatin streamline (see Fig. 1(b)), nrmalized by the freestream mean free path λ, and

7 I is the number f mlecules fr class III t the ttal amunt f mlecules inside each cell. In a rarefied flw, the shck wave has a finite regin that depends n the transprt prperties f the gas, and it can n lnger be cnsidered as a discntinuity beying the classical Rankine-Hugnit relatins. In this cntext, the shck standff distance is defined as being the distance between the shck wave center and the nse f the leading edge alng the stagnatin streamline. As shwn in Fig. 2(b), the center f the shck wave is defined by the statin that crrespnds t the maximum value fr I. The shck wave thickness δ is defined by the distance between the statins that crrespnd t the mean value fr I. Finally, the shck wave shape (shck wave lcatin ) is determined by the crdinate pints given by the maximum value in the I distributin alng the lines departing frm the bdy surface, i.e., η-directin as shwn in Fig. 1(b). 1.0 (a) 0.8 (b) Class I mlecules Class II mlecules 0.6 Shck center Shck standff distance Class III mlecules Shck thickness δ Bdy surface X Figure 2: (a) Drawing illustrating the classificatin f mlecules and (b) Schematic f shck wave structure. &20387$7,21$/5(68/76$1'',6&866,21 The purpse f this sectin is t discuss and t cmpare differences in the displacement, thickness and shape f the shck wave due t variatins n the angle f attack as well as n the leading edge thickness. Befre prceeding with the analysis f the shck-wave structure, it is desirable t highlight the majr features f the results related t the mlecular class distributin. 0ROHFXODU&ODVV'LVWULEXWLRQ The distributin f mlecules fr classes I, II and III alng the symmetry line is illustrated in Figs. 3(a) and 3(b) fr thickness Knudsen number.q f 100 and 1 (Wλ f 0.01 and 1), respectively. These figures display the distributin f mlecules fr angle f attack f 0 and 20 degrees. Nevertheless, it shuld be als mentined that increasing the incidence causes the expected asymmetry in the flw patterns as the stagnatin pint mves frm the symmetry axis t the lwer windward side f the leading-edge surface. In this scenari, it prves instructive t illustrate the mlecular class distributin alng the new stagnatin streamline. Therefre, fr cmparisn purpse, the distributin f mlecules fr classes I, II and III alng the stagnatin line are displayed in Figs. 4(a) and 4(b) fr.q f 100 and 1, respectively. The class distributins fr the ther cases investigated in this wrk are intermediate t the cases displayed in this set f figures and, therefre, they will nt be shwn.

8 Referring t Figs. 3(a) and 3(b), I, I and I are the rati f the number f mlecules fr class I, II and III, respectively, t the ttal amunt f mlecules inside each cell alng the symmetry line. Als, the curves with full and empty symbls crrespnd t angle f attack α f 0 and 20 degrees, respectively. 1.0 (a) 1.0 = I α = I α = II = 0 III = 0 I α = 20 II = 20 III = 20 = (b) II = 0 III = 0 I α = 20 II = 20 III = X X Figure 3: Distributins f mlecules fr classes I, II and III alng the symmetry line parameterized by the angle f attack fr thickness Knudsen number.q f (a) 100 and (b) (a) 1.0 = I α = I α = II = 5 III = 5 I α = 20 II = 20 III = 20 = (b) II = 5 III = 5 I α = 20 II = 20 III = X X Figure 4: Distributins f mlecules fr classes I, II and III alng the stagnatin streamline parameterized by the angle f attack fr thickness Knudsen number.q f (a) 100 and (b) 1. Of great significance in Figs. 3(a) and 3(b) is the behavir f the class I mlecules fr sharp and blunt leading edges. It shuld be bserved that mlecules frm freestream, represented by class I mlecules, cllide with the nse f the leading edges even after the establishment f the steady state. This is shwn in Fig. 3(a), which represent the sharp leading edge case. In cntrast, mlecules frm freestream basically d nt reach the nse f the leading edge fr thse cases illustrated in Fig. 3(b), which represent blunt leading edges. This is explained by the fact that density increases much mre at the vicinity f the stagnatin regin fr blunt leading edges (Sants, 2006), and reaches its maximum value in the stagnatin pint fr the case f zer degree f incidence. In this cnnectin, the buildup f particle density near the nse f the leading edge acts as a shield fr the mlecules cming

9 frm the undisturbed stream. It may be recgnized frm Figs. 3(a) and 3(b) that the angle-fattack effect n the shck standff distance and n the shck thickness is mre prnunced fr the leading edge case defined by.q f 100 than that fr.q f 1. Basically n appreciable changes n bth prperties are bserved fr.q f 1 alng the symmetry line. Nevertheless, appreciable changes n the shck standff distance and n the shck thickness are bserved alng the stagnatin streamline fr the angle-f-attack variatin investigated, as illustrated by Figs. 4(a) and 4(b). 6KRFN:DYH6WDQGRII'LVWDQFH The shck wave standff distance can be bserved in Figs. 3 and 4 fr the cases shwn. Based n the shck displacement definitin presented in Fig. 2(b), the calculated shck wave standff distance, nrmalized by the freestream mean free path λ, is tabulated in Table 3 fr the cases investigated. As mentined previusly, due t the angle-f-attack effect, the stagnatin streamline mves frm the symmetry line t a new psitin in the windward side. In this way, Table 3 tabulates the shck standff distance alng the symmetry line and alng the stagnatin line. The stagnatin-line data are shwn inside the parenthesis. Table 3: Dimensinless shck wave standff distance λ alng the symmetry line and stagnatin line fr truncated wedges at incidence. α(degree).q = 100.Q = 10.Q = (0.097) (0.192) (0.584) (0.090) (0.213) (0.503) (0.110) (0.208) (0.453) (0.135) (0.243) (0.450) It is apparent frm the results n Table 3 that there is a discrete shck standff distance fr the cases shwn. As wuld be expected, the shck standff distance increases with increasing the leading-edge thickness; since the leading edge becmes blunt with increasing the frntal thickness W. It is als seen that, the shck standff distance alng the symmetry line increased with increasing the angle f attack fr thickness Knudsen numbers f 100 and 10. In cntrast, n appreciable changes were bserved fr thickness Knudsen number f 1 at incidence. It is imprtant t mentin that shck standff distance becmes imprtant in hypersnic vehicles such as waveriders, which depend n leading edge shck attachment t achieve their high lift-t-drag rati at high lift cefficient. In this cnnectin, leading edge with smaller frntal face seem t be mre apprpriate, since it presents reduced shck wave detachment distances. Althugh it has lng been knwn that smaller shck detachment distance is assciated with a higher heat lad t the nse f the bdy. 6KRFN:DYH7KLFNQHVV Accrding t the definitin fr shck-wave thickness illustrated in Fig. 2(b), the shck wave thickness δ alng the stagnatin streamline can be calculated frm Figs. 3 and 4 fr the leading edges cases displayed. As a result f the calculatin, Table 4 tabulates the shck-wave thickness δ, nrmalized by the freestream mean free path λ, fr all cases investigated. In a similar way, the values presented are fr the shck-wave thickness alng the symmetry line and the stagnatin line.

10 & ' It is evident frm Table 4 that, in general, the shck-wave thickness fllws the same trend presented by the shck-wave standff distance in that it increases with increasing the angle f attack α fr thickness Knudsen numbers f 100 and 10, and stays cnstant fr thickness Knudsen number f 1. Table 4: Dimensinless shck wave thickness δλ alng the symmetry line and stagnatin line fr truncated wedges at incidence. α(degree).q = 100.Q = 10.Q = (0.318) (0.460) (1.313) (0.357) (0.492) (1.238) (0.412) (0.540) (1.131) (0.474) (0.594) (1.212) 6KRFN:DYH6KDSH The shck-wave shape, defined by the shck wave center lcatin, is btained by calculating the psitin that crrespnds t the maximum I fr class III mlecules in the η- directin alng the bdy surface (see Fig. 2(b)). Cmparisn f the shck wave shape at incidence and at zer angle f attack is illustrated in Fig. 5(a) fr thickness Knudsen number f 1. In an effrt t emphasize pints f interest, a magnified view f the shck wave shapes at the vicinity f the leading-edge nse is shwn in Fig. 5(b). In this set f plts, ; and < are the Cartesian crdinates [ and \ nrmalized by λ (a) α = (b) α = α = 0 2 α = Bdy shape 0 Bdy shape "$#% = 1-4 (*),+ = ! Figure 5: Shck-wave shapes n truncated wedges as a functin f the angle f attack fr thickness Knudsen number.q f 1: (a) shck-wave n the ttal leading edge and (b) magnified view at the vicinity f the leading-edge nse.

11 It is seen frm this set f figures that increasing the incidence causes the expected asymmetry in the shck wave patterns as the stagnatin pint mves frm the axis t the lwer windward side. As a result, the net buildup f particle density decreases in the leeward side and it increases in the windward side (Sants, 2006) with increasing the incidence. Cnsequently, the presence f the leading edge, prpagated by randm mtin f the mlecules, is cmmunicated t a larger distance away frm the bdy in the leeward side than that in the windward side. Hence, the shck wave center lcates mre away f the bdy surface n the leeward side and clser t the bdy surface n the windward side. &21&/8',1*5(0$5.6 This study applies the Direct Simulatin Mnte Carl methd t investigate the shck wave structure fr a family f truncated wedges. The calculatins have prvided infrmatin cncerning the nature f the shck-wave detachment distance, shck-wave thickness and shck-wave shape resulting frm variatins n the angle f attack and n the thickness f the frntal face fr the idealized situatin f tw-dimensinal hypersnic rarefied flw. The analysis shwed that the shck-wave structure was affected by changes n the angle f attack. It was fund that the shck-wave standff and the shck-wave thickness increased with the incidence rise fr the sharp leading edge cases investigated. In cntrast, n appreciable changes were bserved in bth prperties fr the blunt leading edge case. As expected, it was als verified that the shck wave center lcates mre away f the bdy surface n the leeward side than that n the windward side. 5()(5(1&(6 Alexander, F. J., Garcia, A. L., & Alder, B. J., 1998, Cell size dependence f transprt cefficient in stchastic particle algrithms. 3K\VLFVRI)OXLGV, vl. 10, n. 6, pp Alexander, F. J., Garcia, A. L., & Alder, B. J., 2000, Erratum: Cell size dependence f transprt cefficient is stchastic particle algrithms. 3K\VLFVRI)OXLGV, vl. 12, n. 3, pp Bird, G. A., 1981, Mnte Carl simulatin in an engineering cntext. In Sam S. Fisher, ed., 3URJUHVVLQ$VWURQDXWLFVDQG$HURQDXWLFV5DUHILHGJDV'\QDPLFV, vl. 74, part I, pp , AIAA, New Yrk. Bird, G. A., 1989, Perceptin f numerical methd in rarefied gasdynamics. In E. P. Muntz, and D. P. Weaver and D. H. Capbell, eds., 5DUHILHG JDV '\QDPLFV 7KHRUHWLFDO DQG &RPSXWDWLRQDO 7HFKQLTXHV, vl. 118, pp , Prgress in Astrnautics and Aernautics, AIAA, New Yrk. Bird, G. A., 1994, 0ROHFXODU*DV'\QDPLFVDQGWKH'LUHFW6LPXODWLRQRI*DV)ORZV. Oxfrd University Press, Oxfrd, England, UK. Brgnakke, C. & Larsen, P. S., 1975, Statistical cllisin mdel fr Mnte Carl simulatin f plyatmic gas mixture. -RXUQDORIFRPSXWDWLRQDO3K\VLFV, vl. 18, n. 4, pp Cercignani, C., 1988, 7KH %ROW]PDQQ (TXDWLRQ DQG,WV $SSOLFDWLRQV, Springer-Verlag, New Yrk, NY. Garcia, A. L., & Wagner, W., 2000, Time step truncatin errr in direct simulatin Mnte Carl. 3K\VLFVRI)OXLGV, vl. 12, n. 10, pp Gu, K. & Liaw, G.-S., 2001, A review: bundary cnditins fr the DSMC methd. In 3URFHHGLQJV RI WKH WK $,$$ 7KHUPRSK\VLFV &RQIHUHQFH, AIAA Paper , Anaheim, CA.

12 Haas, B, L., & Fallavllita, M. A., 1994, Flw reslutin and dmain influence in rarefied hypersnic blunt-bdy flws. -RXUQDORI7KHUPRSK\VLFVDQG+HDW7UDQVIHU, vl. 8, n. 4, pp Hadjicnstantinu, N. G., 2000, Analysis f discretizatin in the direct simulatin Mnte Carl. 3K\VLFVRI)OXLGV, vl. 12, n. 10, pp Nnweiler, T. R. F., 1959, Aerdynamic prblems f manned space vehicles. -RXUQDORIWKH 5R\DO$HURQDXWLFDO6RFLHW\, vl. 63, Sept, pp Sants, W. F. N., 2002, Truncated leading edge effects n flwfield structure f a wedge in lw density hypersnic flight speed. In WK %UD]LOLDQ &RQJUHVV RI 7KHUPDO (QJLQHHULQJ DQG6FLHQFHV(1&,7, Caxambu, MG, Brazil. Sants, W. F. N., 2003, Cmpressibility effects n flwfield structure f truncated leading edge in lw density hypersnic flw. In WK,QWHUQDWLRQDO &RQJUHVV RI 0HFKDQLFDO (QJLQHHULQJ, &2%(0, Sã Paul, SP, Brazil. Sants, W. F. N., 2004, The effect f incmplete surface accmmdatin n heat transfer and drag f truncated wedge in rarefied regime. In UG %UD]LOLDQ &RQJUHVV RI 0HFKDQLFDO (QJLQHHULQJ&21(0, Belém, PA, Brazil. Sants, W. F. N., 2005, Flat-faced leading-edge effects in lw-density hypersnic wedge flw, -RXUQDORI6SDFHFUDIWDQG5RFNHWV, vl. 42, n. 1, pp Sants, W. F. N., 2006, Angle-f-attack effect n rarefied hypersnic wedge flw. In 9,, 6LPSRVLXPRI&RPSXWDWLRQDO0HFKDQLFV6,00(&, Araxá, MG, Brazil.