Rarefaction effects on separation of hypersonic laminarflows

Gennady N. Markelov, Alexei N. Kudryavtsev, and Mikhail S. Ivanov

Computational Aerodynamics LaboratoryInstitute of Theoretical and Applied Mechanics

Siberian Division of Russian Academy of Sciences, Novosibirsk, Russia

Abstract. A hypersonic monoatomic gas (argon) flow around a hollow cylinder flare was numericallysimulated for low-to-moderate Reynolds numbers. A shock wave induced by the flare causes the formation ofa separation region, and flow rarefaction near the leading edge affects the separation extent. The continuum(Navier-Stokes equations) and kinetic (the DSMC method) approaches were used to study the influence ofthe Mach number and wall-to-freestream temperature ratio on the flow structure and separation. To takeinto account rarefaction effects, slip boundary conditions are used for the Navier-Stokes solver.

INTRODUCTION

The problem of the shock wave/laminar boundary layer interaction has been actively studied lately bothby numerical and experimental methods. This interaction can cause the formation of a separation region andlead, for example, to a decrease in the control surface efficiency and to an increase in the heat flux on the wallin the vicinity of the reattachment point.

The simplest example of the flow near an aerodynamic control surface is the flow around a hollow cylinderflare. This axisymmetric configuration allows one to study the shock wave/ boundary layer interaction and toavoid 3D effects inherent in a flow near a compression ramp. Numerical analysis of such flows is traditionallyperformed using Navier-Stokes equations, and the initial effects of rarefaction are taken into account throughslip velocity and temperature jump. The use of slip conditions is caused by the fact that rarefaction effectscan be observed for slender bodies even for rather high Reynolds numbers (Re = 20,000 — 30,000). Moreover,the use of the continuum approach for an analysis of flow in the vicinity of the leading edge of slender bodiesis not justified, even if the slip velocity and the temperature jump are taken into account. The flow physicsis considerably complicated in case when flow separation is formed, for example, by a flare, and simulation ofsuch flows by the continuum approach is questionable due to the strong rarefaction effects in the vicinity ofthe leading edge.

The state-of-the-art of the algorithms of the DSMC method (adaptive grids, variable time step, etc.) [1,2]allows one to simulate flows at such a rather high Reynolds number Re ~ 20,000. This makes realistic thestudy of the area of applicability of the continuum approach for numerical analysis of laminar separated flows.Such studies were started by the authors of [3] and were continued in [4] where the detailed numerical modelingof axisymmetric shock wave/ laminar boundary layer interaction for the ONERA R5Ch wind tunnel conditionsby using the continuum (Navier-Stokes equations) and kinetic (the DSMC method) approaches was performed.The DSMC results revealed the presence of a high slip velocity and temperature jump near the wall. Therefore,it is necessary to use slip conditions for the Navier-Stokes solver. The application of these conditions has asubstantial effect on the entire flow field and allows one to obtain reasonable agreement with the DSMC resultsfor flow fields and distributions of the pressure and heat-transfer coefficients. However, the NS solver with slipconditions predicts a greater length of the separation region than the DSMC method. Probably, this differenceis caused by different descriptions of translational/rotational energy exchange for both methods.

To eliminate the effect of rotational-translational exchange, a continuum and kinetic simulation of a laminarseparated flow of a monoatomic gas (argon) was performed in the paper. The effect of rarefaction, Machnumber, and wall temperature on laminar separation was also studied.

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

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FLOW CONDITIONS AND NUMERICAL APPROACHES

The hollow-cylinder flare configuration was taken from [5]. The hollow cylinder has a sharp leading edge andis aligned parallel to the oncoming flow. The compression flare is inclined at 30° to the cylinder and is followedby a hollow cylindrical section. The total length of the model is 0.17 m. The reference length L = 0.1017 m isthe distance from the leading edge to the flare. The computations were carried out only for the upper part ofthe model, and the flow inside the model was not considered.

Two test cases were considered with Mach numbers M^ = 10 and 4, respectively. In these cases, thesame Knudsen number as for ONERA wind tunnel conditions Kn^ = X^/L = 9.35 • 10~3 was used (Aoois the mean free path in the freestream flow). The Reynolds number Re^ = 19000 and 7600 was based onthe freestream parameters and reference length L. The computations were conducted for a constant surfacetemperature and an adiabatic wall. For ONERA wind tunnel conditions, the wall-to-freestream temperatureratio was TW/T^ = 5.74. The same value of the temperature ratio was used for case 1. Since the Machnumber is much smaller for case 2 (M^ = 4), the simulation results obtained for TW/T00 = 5.74 are close tothe results calculated for an adiabatic wall. Therefore, additional calculations were conducted for this casewith T^/Too = 1.0 to reveal the effect of the wall-to-freestream temperature ratio.

Continuum approach

The Navier-Stokes code [4] used solves the full laminar compressible Navier-Stokes equations with bothno-slip and slip boundary conditions. The code is based on the finite- volume TVD scheme for the convectiveterms and the second-order central difference approximation of the diffusive terms. The inviscid fluxes arecomputed by the HLLEM Riemann solver [6]. The computations were conducted using the perfect gas model.The power-law dependence of the viscosity on the temperature p, oc T" was taken for more straightforwardcomparison with the results of DSMC simulations. The relations for the slip velocity and temperature jumpdeduced by Kogan [7] were used to model slip boundary conditions. Taking into account slip conditionssomewhat complicates the computations for an adiabatic wall since the heat flux is determined with accountof the non- vanishing gas velocity on the wall

where U denotes the velocity component tangential to the solid wall and k is the heat conductivity. The firstterm in this expression is always positive. Therefore, the requirement of zero heat flux can be satisfied only bya negative temperature jump. In this case, after calculating slip velocity, the temperature jump is determinedby the relation

Kinetic approach

The SMILE computational tool [2] was used. It based on the DSMC method [8] and the exact majorantfrequency scheme [9] for modeling collision processes. The variable hard sphere model [8] was assumed forintermolecular collisions. Gas/surface interaction was simulated using diffuse reflection with complete accom-modation energy and diffuse-elastic reflection. The latter means that the magnitude of the molecular velocityremains constant during reflection, but an angular distribution of the velocity vector follows the cosine-law ofdiffuse reflection. As a result, the heat flux to the body surface equals zero. The diffuse-elastic reflection wasused to model an adiabatic wall with the DSMC method.

RESULTS AND DISCUSSIONS

The general structure of the flow is shown in Fig. 1 (Numerical Schlieren picture) with clearly visible shockwaves formed in the flow around a hollow-cylinder flare. The separation shock wave intersects and coalesceswith the reattachment shock wave. The new shock, in turn, intersecs and coalesces with the leading-edge shock

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X/L

FIGURE 1. Numerical Schlieren picture and selected stream lines (NS solver, M^ = 4)

FIGURE 2. Density profiles (Moo = 10)

wave. The line separating the boundary layer and the flow behind the shock waves is also clearly visible. Thisline coincides with the leading-edge shock wave in the vicinity of the leading edge, which indicates a strongviscid-inviscid interaction in this flow region. The shock wave/laminar boundary layer interaction induces aseparation region clearly visible in Fig. 1, which also shows selected streamlines.

Constant surface temperature

The results presented in this section were calculated using the continuum and kinetic approaches for aconstant surface temperature T^/T^ = 5.74 and Tw/T^ = 1.0 for M^ = 10 and M^ = 4, respectively. First,we consider the case with M^ = 10. Figure 2 shows a detailed comparison of density profiles in several crosssections X/L = 0.3,0.6, and 0.9. The last cross section passes through the separation region. Note that thedensity increases inside the boundary layer when approaching the body surface. This is typical for a hypersonicflow around a cold body. Like in the case with a diatomic gas [4], the NS solver with no-slip conditions predictsa slightly higher position of the leading-edge shock wave than the DSMC method. The use of slip conditionsfor the NS solver reduces the slope of the leading-edge shock wave and yields good agreement with the DSMCresults for density profiles at X/L = 0.3 and 0.6. However, a significant difference can be noted in the crosssection X/L = 0.9 with two density peaks. The second peak corresponds to the separation shock wave, andits magnitude is determined by the separation region length. The NS solver and the DSMC method predictdifferent lengths of the separation region, and this peak is almost 30% higher for NS results.

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0 2 4 6 8 1 0 1 2T/T <«

FIGURE 3. Temperature profiles (Moo = 10)

TABLE 1. Separation region length.

constant surface temperature adiabatic wallmethod Xsep/L Xreat/L

Moo = 10

Moo =4

NS, no-slipNS, slipDSMC

NS, no-slipNS, slipDSMC

0.7400.7550.790.6400.6540.69

1.301.301.2851.281.281.28

0.8960.9300.850.6920.7450.71

1.151.151.221.2651.251.265

The results for density and velocity (not shown) profiles are rather close inside the boundary layer. At thesame time, a comparison of the temperature profiles in Fig. 3 shows that the difference between the continuumand kinetic results reaches 40%. The use of slip conditions allows one to obtain identical values of temperaturenear the body surface and decrease the difference inside the boundary layer. However, the temperatures insidethe boundary layer for DSMC and NS with slip conditions still differ by 20%. Obviously, the differences observedin the boundary-layer structure affect its stability to the action of the pressure gradient. The coordinates of thepoints of boundary-layer separation Xsep and reattachment Xreat are listed in Table 1, and it is seen that theNS solver predicts a greater length of the separation region than the DSMC method. The use of slip conditionsleads to some displacement of the coordinate Xsep downstream and does not affect Xreat. Nevertheless, theseparation region length is still greater than the DSMC results by 10%.

Figure 4 shows the distributed aerothermodynamic characteristics obtained using the continuum and kineticapproaches. The differences in the pressure Cp and skin friction Cf coefficients, and Stanton number are mainlydetermined by two factors. The first one is the different length of the separation region. An earlier separationof the flow in NS results leads to a lower pressure peak and to a smaller heat flux on the flare surface. Thesecond factor is the strong nonequilibrium of the flow near the leading edge. Its effect is noticeable up toX/L = 0.1 and can be related to the fact that the continuum approach is inapplicable even with account of slipconditions in the vicinity of the leading edge. The measure of this nonequilibrium is the ratio of the streamwisetemperature to the total translational temperature. A 30% deviation of TX/T is observed up to X/L = 0.1(Fig. 5). The magnitude of TX/T decreases downstream, but is still noticeable at a distance X/L = 0.4, whichcorresponds to 400AQQ.

Strong flow nonequilibrium is the reason that the NS solver with no-slip and slip conditions predicts asignificantly greater heat flux (up to a factor of four) than the DSMC method in the vicinity of the leadingedge (Fig. 4). This also leads to a significant decrease in flow temperature downstream and an earlier separationof the boundary layer.

As the freestream Mach number decreases, the level of flow nonequilibrium decreases too (see Fig. 5). Thiscan be expected to lead to a better agreement between the NS and DSMC results. The flow around a hollow-cylinder flare for M^ = 4 was calculated, and fairly good agreement was actually obtained in terms of density,velocity and even temperature profiles (Fig. 6). A small difference is observed only in the value of the heatflux to the surface for 0.6 < X/L > 0.9 (similar to Fig. 4) and in the separation region length (~ 6%). TheNS solver with no-slip and slip conditions predicts a greater length of the separation region than the DSMCmethod (see Table 1).

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\

0.9X/L

0.9X/L

Pressure coefficient Skin friction coefficient Stanton numberFIGURE 4. Distributed aerothermodynamic characteristics (Moo = 10)

FIGURE 5. Thermal nonequilibrium (constant surface temperature, left - MOO = 10, right - MOO = 4)

0.15Yf

0. 9000. 925

1. 0251. 050

1. ISO1. 175

FIGURE 6. Temperature profiles (Moo = 4)

Adiabatic wall

As is shown above, the slip conditions allow one to obtain fair agreement for NS and DSMC results in terms ofdensity and velocity profiles for MQQ = 10, and the difference is observed only in the value of temperature insidethe boundary layer. The heat flux near the leading edge predicted by the continuum approach is severalfoldhigher than that predicted by the kinetic approach. It can be assumed that exactly this is the reason for otherdifferences, for example, the different length of the separation region. Obviously, in the case of continuum andkinetic simulation of the flow around a body with an adiabatic wall, the heat flux is identical (namely, zero).Therefore, a complete coincidence of continuum and kinetic results can be expected.

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6+0007.000

FIGURE 7. Temperature flow field (NS solver, MOO = 10, left - constant surface temperature, right - adiabatic wall)

Ojfk

FIGURE 8. Temperature profiles (Moo = 10)

FIGURE 9. Velocity profiles (M^ = 10)

To verify this assumption, computations for adiabatic wall conditions were conducted. The use of an adiabaticwall for MOO — 10 leads to a significant increase in flow temperature near the body surface. For example, fora constant surface temperature, the temperature in the boundary layer varied from lOToo near the forebodyto ISToo near the flare surface, whereas for an adiabatic wall, the temperature in the boundary layer variesin a narrow range from 27^ to 291 ,̂ which is slightly lower than the total temperature of the flow 34.3^(Fig. 7). The temperature profiles for the continuum and kinetic approaches are compared in Fig. 8. It isof interest to note that the NS solver with no-slip conditions and the DSMC method predict an identicaltemperature near the whole surface of the body. An increase in temperature in the near-wall region leads tosignificant changes in the flow structure, in particular, to an increase in the boundary-layer thickness, which isclearly seen in the temperature and velocity profiles (Fig. 9), and to a greater slope of the leading-edge shockwave.

For an adiabatic wall, the difference in the position of the leading-edge shock wave obtained by the NS solverwith no-slip conditions and the DSMC method is more significant than for the above case with a constantsurface temperature (cf. the density profiles in Fig. 2 and 10). The use of slip conditions strongly decreasesthe slope of the leading-edge shock wave so that it is located even lower than that predicted by the DSMCmethod. Note that the density near the body surface for an adiabatic wall is identical for the continuum andkinetic approaches.

712

FIGURE 10. Density profiles (M^ = 10)

— DSMC— -NS, no-slip—• NS, slip

0.9X/L

0.9X/L

FIGURE 11. Thermal nonequilibrium (left) and pressure (center) and skin friction (right) coefficients (Moo = 10,adiabatic wall)

The use of slip conditions leads to a significant temperature jump and slip velocity on the front part of thecylindrical surface (cross sections X/L = 0.3 and 0.6). However, the flow parameters inside the boundary layerin the cross section X/L = 0.9 are in reasonable agreement for both approaches. In this cross section, thereis no flow separation for NS results, and the DSMC method predicts the beginning of separation. Despitesimilar structures of the boundary layer ahead of the separation point, the DSMC method predicts a greaterlength of the separation region than the NS solver (see Table 1). Taking into account slip conditions leadsto a further decrease in the separation region length. Thus, the computations with an adiabatic wall did notyield an identical length of the separation region for the continuum and kinetic approaches, and even led tothe opposite situation. This is, apparently, caused by two factors. First, an increase in the local mean freepath in the near wall region due to higher wall temperature did not change flow noneqilubrium (cf. Figs. 5and 11) and did not increase the area of the applicability of the NS equations. Second, an increase in theboundary-layer thickness led to a position of the shock-wave interaction region located much farther from theflare surface than in the flow around a hollow-cylinder flare with a constant surface temperature. In this case,the expansion at the flare shoulder stronger affects the shock wave interaction region and decreases the adversepressure gradient and, hence consequently, the separation region length. This is reflected in the distribution ofCp whose magnitude on the flare became of the same order with its value near the leading edge (see Fig. 11).A greater slope of the leading-edge shock wave for the NS solver with no-slip conditions causes an increase inpressure and friction on the forebody.

It was shown that a decrease in the freestream Mach number leads to smaller differences between the con-tinuum and kinetic results. Computations with an adiabatic wall for M^ = 4 also revealed smaller differencesbetween the flow parameters. However, a qualitative difference in the slope of the leading-edge shock wave(see the temperature profiles in Fig. 12), in the distributed characteristics, and in the separation region length(Table 1) is retained.

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FIGURE 12. Temperature profiles (Af*, = 4)

CONCLUSIONS

The computations for the Mach 10 flow of a monoatomic gas (argon) show that the continuum approachpredicts a greater length of the separation region than the DSMC method. The use of slip boundary conditionsfor taking into account initial rarefaction effects allowed us to decrease the difference between the continuumand DSMC results. A comparison of the density, temperature, and velocity profiles shows that significantdifferences are observed only for the temperature inside the boundary layer. This is probably caused by strongflow rarefaction near the leading edge and, as a consequence, by a significant difference in heat fluxes nearthis edge predicted by the NS solver and DSMC method. A decrease in the freestream Mach number leads tosmaller flow nonequilibrium in the vicinity of the leading edge and to better agreement between the continuumand kinetic results. In particular, the difference in the separation region length was reduced from 10% forMOO = 10 to 6% for MOO = 4.

The use of adiabatic wall boundary conditions for flow simulation near a hollow cylinder flare induceda significant increase in the gas temperature near the body surface, which led to substantial changes in theboundary-layer structure. The NS solver with no-slip conditions and the DSMC method predict almost identicalvalues of density and temperature near the body surface. In this case, however, the DSMC method predicts agreater length of the separation region than the NS solver. The use of slip conditions leads to more considerablechanges than in the case of a constant surface temperature. For example, the slope of the leading-edge shockwave becomes even smaller than the DSMC prediction. The qualitative effect of slip conditions on the separationregion length is retained, the latter decreases in size, and the difference between the DSMC and NS resultsincreases.

To study in more detail the above-described quilitative difference between the DSMC and NS predictions ofthe separation region length for cold and adiabatic walls experimental data are indispensable. In the term ofRTO Working Group 10 an experimental study of hypersonic laminar flow around a hollow-cylinder flare witha heated wall is planned and it will clarify the situation.

Acknowledgement This work was supported by the INTAS Grant No. 99-00749 and RFBR Grant No.00-01-00824. This support is gratefully acknowledged.

REFERENCES

1. Moss, J.N., Olejniczak, J., AIAA Paper 98-2668, June 1998.2. Ivanov, M.S. , Markelov, G.N., Gimelshein, S.F., AIAA Paper 98-2669, June 1998.3. Markelov, G.N., Kudryavtsev, A.N., Ivanov, M.S., Proc. of 21 Int. Symp. on Rarefied Gas Dynamics, Cepadues

Editions, 1, 647 (1999).4. Markelov, G.N., Kudryavtsev, A.N., Ivanov, M.S., AIAA Paper 99-3527, June-July, 1999.5. Chanetz, B., TR RT 42/4362 AN, ONERA (1995).6. Einfeldt, B., Munz, C.D., Roe, P.L., Sjogren, B.,J. Comput. Phys., 92, 273 (1991).7. Kogan, M.N., Rarefied gas dynamics, New York: Plenum Press, 1969, ch. 5.8. Bird, G.A., Molecular gas dynamics and the Direct Simulation of gas flows, Oxford: Clarendon Press, 1994.9. Ivanov, M.S., Rogasinsky, S.V., Proc. of 18 Int. Symp. on Rarefied Gas Dynamics, VCH, 629 (1991).

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