Heat Transfer on a Hypersonic Sphere with Diffuse Rarefied-gas Injection

The interaction of a diffusing outgas flow from a sphere nose opposing a hypersonic free stream is studied numerically by the direct simulation MonteCarlo technique under the transitional rarefied-gas-flow regime conditions at Knudsen numbers from 0.016 to 1.5 and blowing factors from 0.15 to 1.5. Strong influences of the blowing factor (the ratio of outgas mass flux to upstream mass flux) and the Knudsen number on the flow structure about a sphere (temperature fields, the configuration of mixing flow zones) and on heat distributions along the spherical surface have been found. At large blowing factors, the injected gas significantly reduces heat flux in wide area near the spherical nose. This effect is more pronounced for light gas (helium) injection in the near-continuum flow. Nomenclature d = orifice diameter, 0.002 m f = mole fraction of helium Gw = mass injection rate ratio (“blowing” factor) Kn∞,R = Knudsen number M = Mach number p = pressure, N/m R = sphere radius, 0.015 m Re0,R = Reynolds number St = Stanton number s = coordinate along a spherical surface T = temperature tw = temperature factor, Tw/T0 x = coordinate along an axis of body symmetry subscripts R = sphere radius as a length-scale parameter w = wall condition 0 = stagnation flow condition ∞ = freestream parameter ___________________________________ *Associate Professor, Department of Computer Science and Mathematics, 420 S. Main Street; [email protected]. Senior Member AIAA. Copyright © 2004 by Vladimir V. Riabov. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. Introduction Numerical and experimental studies of aerothermodynamics of hypersonic vehicles have shown that the temperature in the spacecraft nose region can be extremely high, and the maximum value of the heat flux occurs at small values of the nose radius R and small local Reynolds numbers Re0,R. Mass injection can be considered as an effective way of the reduction of heat transfer to the surface in this area (see Refs. 1-8). The boundary-layer flow with gas blowing was studied by Warren, Libbi and Gresci, and Finley. Only few studies (i.e., Pappas and Lee and Moss) were conducted in the case of small Reynolds numbers. Moss found that mass injection dramatically reduces heat transfer to the surface, and when the mass injection rate equals 0.4 of the free-stream mass flux, the viscous layer is blown completely off the surface, and the heat transfer is zero. The effect of injecting gaseous coolants on heat transfer in hypersonic perfect gas flow near blunt bodies was studied by Gershbein and Kolesnikov and Emelianova and Pavlov on the basis of the complete system of Navier-Stokes equations, and by Moss, Shen et al., Ankundinov, Provotorov and Stepanov, and Botin on the basis of the thin viscous shock layer model developed by Cheng. Provotorov and Stepanov had found universal relations between the relative heat flux and the generalized blowing parameters. Heat transfer in the presence of hydrogen blowing and combustion was studied by Riabov and Botin and Botin et al. 21 These studies have shown that the effectiveness of coolant blowing increases with the increase of the Reynolds number, and becomes significant at Re0,R > 90. Heat transfer experimental data received by the method of two-layer thermal-indicator coating confirm this conclusion. Other applications of the gas blowing include the divert and attitude reaction control systems (see the review of Gimelshein et al. and original papers) and a counterflow drag reduction technique in high-speed systems (see the study of Josyula et al. and others). In the present study, the interaction of a blowing gas flow from a nose of the sphere opposing a hypersonic free stream is studied numerically by the 1 American Institute of Aeronautics and Astronautics 42nd AIAA Aerospace Sciences Meeting and Exhibit AIAA 2004-1176 5-8 January 2004, Reno, Nevada direct simulation Monte-Carlo (DSMC) technique under the transitional rarefied-gas-flow regime conditions at Knudsen numbers Kn∞,R from 0.016 to 1.5 and blowing factors Gw (the ratio of outgas mass flux to upstream mass flux) from 0.15 to 1.5. Strong influences of the blowing factor and the Knudsen number on the flow structure about a sphere (temperature fields, the configuration of mixing flow zones) and on heat distributions along the surface of a sphere have been found. At large blowing factors (Gw > 0.5), the injected gas significantly reduces heat flux in wide area near the spherical nose. This effect is more pronounced for light gas (helium) injection in the nearcontinuum flow of air. The computer code was developed by Graeme Bird. DSMC Method The DSMC method has been used in this study as a numerical simulation technique for low-density hypersonic gas flows. The DSMC/DS2G code (version 3.2) is used for numerical calculations. Molecular collisions in air and helium are modeled using the variable hard sphere (VHS) molecular model. The gas-surface interactions are assumed to be fully diffusive with full moment and energy accommodation. The code validation was tested by the author in comparing numerical results with experimental data related to the simple-shape bodies. As an example, the comparison of the DSMC recent numerical results with experimental data in air (without blowing) is shown in Fig. 1 for a wide range of Reynolds numbers from 1 to 92.8 (Knudsen numbers from 1.5 to 0.016 respectively) and flow parameters M∞ = 6.5 and tw = 0.31. The error of experimental data (error bars in Fig. 1) was estimated as 8-12% at different flow regimes (see Refs. 19 and 22 for details). The numerical results correlate well with experimental data at 10 < Re0,R < 90. The methodology from Refs. 34 and 35 has been applied in computations. The cases that had been considered by Bird for airflow in near-continuum regimes were reproduced in this study. The mesh size and number of molecules per cell were varied until independence of the flow profiles and aerothermodynamic characteristics from theses parameters was achieved for each considered case. In calculations at mentioned-above conditions, the total number of cells near a sphere (a half-space of the flow segment) is 4200 in four zones, the molecules are distributed non-evenly, and a total number of 129,500 molecules corresponds to an average 31 molecules per cell. Following the recommendations of Refs. 34 and 35, acceptable results are obtained for an average of at least ten molecules per cell in the most critical region of the flow. The error was pronounced when this number falls below five. The cell geometry has been chosen to minimize the changes in the microscopic properties (pressure, density, and temperature) across the individual cell. The variation in cell width has been based on the geometric progression principle and defined by the ratio c = 20 of the width of the cell adjacent to outer boundary to the width of the cell adjacent to inner boundary. In all cases the usual criterion for the time step ∆tm has been realized, 2×10 8 ≤ ∆tm ≤ 1×10 s. Under these conditions, aerothermodynamic coefficients and gasdynamic parameters have become insensitive to the time step. The ratios of the mean separation between collision partners to the local mean free path and the collision time ratio (CTR parameter) 35 of the time step to the local mean collision time have been well under unity over the flowfield. The DS2S program employed time averaging for steady flows. About 95,000 samples have been studied in the considered cases. The computed results have been stored to the TECPLOT files that have been further analyzed to study whether the DSMC numerical criteria are met. The location of the external boundary with the upstream flow conditions varies from 0.75R to 1.5R. Calculations were carried out on a personal computer with a Pentium III 850-MHz processor. The computing time of each variant was estimated to be approximately 12 80 h. Results Influence of the Air Blowing Factor Gw The flow pattern over sphere is significantly sensitive to the blowing parameter Gw, which is the ratio of counterflow outgas mass flux to upstream mass flux. The influence of this parameter on the flow structure has been studied for hypersonic flow of air at M∞ = 6.5 and Kn∞ ,R = 0.0163 (Re0,R = 92.8). It is assumed that the temperature factor is equal to 0.31. The flow conditions are the same as suggested by Botin for experiments with air blowing in a vacuum chamber (stagnation pressure p0 = 120 torr and stagnation temperature T0 = 1000 K). The sphere radius is 0.015 m. The diffuse outgas (air) is blowing from the orifice with the diameter of 0.002 m that is located in the critical front point of the sphere. The blowing factor varies from 0 to 1.5. The local Mach number contours and temperature contours are shown in Figs. 2 and 3, respectively, for the case of the strong blowing factor (Gw = 0.7). The temperature field is disturbed in the vicinity of the orifice in the subsonic area of the flow behind the strong shock wave. The distributions of the Stanton number St along the spherical surface at various blowing factors are shown in Fig. 4. For the considered near-continuum flow regime conditions, when the mass injection rate equals 0.7 of the free-stream mass flux, 2 American Institute of Aeronautics and Astronautics 42nd AIAA Aerospace Sciences Meeting and Exhibit AIAA 2004-1176 5-8 January 2004, Reno, Nevada the viscous layer is blown completely off the surface, and the heat transfer is zero. At stronger blowing (Gw > 0.7), diffuse outgas flow displaces completely the viscous layer off the sphere surface. The displacement effect spreads both in the upstream direction and along the surface. The width of the “displacement” zone can be characterized by the normalized surface coordinate (s/R)max where the heat transfer is maximum (see Fig. 4). Influence of the Rarefaction Factor (Reynolds Number Re0,R) The rarefaction factor, which can be characterized by the Reynolds number Re0,R (or the equivalent Knudsen number, Kn∞ ,R) plays an important role in the flow structure as well as in aerothermodynamics. The Stanton number reduces significantly with increasing the Reynolds number (see Fig. 1). The numerical data (calculated at Gw = 0) correlate well with experimental data at 10 < Re0,R < 100. The outgas counterflow reduces significantly the heat transfer to the surface. This effect is more pronounced at higher values of the Reynolds number, Re0,R > 20 (see Fig. 1). Also the width of the injection-influenced “displacement” zone (at Gw = 0.94) increases by the factor of 3 at increasing the Reynolds number from 1 to 92.8 (see Fig. 5). Diffuse Injection of Helium into Air Stream Helium has been selected as outgas to study the role of diffusive effects of blowing. Under nearcontinuum flow conditions (Kn∞ ,R = 0.0163 or Re0,R = 92.8), the flow structure with helium blowing has the same features as were discussed above, but the size of the “displacement” zone is larger than in the case of airto-air blowing. The contours of helium mole fraction f(He), local Mach number contours and temperature contours are shown in Figs. 6, 7, and 8, respectively, for the case of the strong helium blowing factor (Gw = 0.7). The mole concentration of helium (Fig. 6) is still significant (up to the value of 0.1) at the distance of 0.2R in upstream flow and 3.5d along the sphere surface. The Mach number contours (Fig. 7) are displaced and temperature contours (Fig. 8) are disturbed more pronouncedly than in the case of air-toair blowing (see Figs. 2 and 3 respectively). Even at moderate blowing (0.7 > Gw > 0.32), diffuse outgas flow displaces completely the viscous layer off the sphere surface, and values of the Stanton number become negative (see Fig. 9). The similar effect was discussed in the experimental study by Botin. Conclusion The influence of the blowing parameter (the ratio of outgas mass flux to upstream mass flux) and the rarefaction factor (Knudsen number or Reynolds number) on the flow structure about a sphere has been studied for hypersonic flow of air. It has been found that at near-continuum flow regime conditions (Re0,R = 92.8), when the mass air injection rate equals 0.7 of the free-stream mass flux, the viscous layer is blown completely off the surface, and the heat transfer is zero. The displacement effect of blowing spreads both in the counterflow direction and along the surface. This effect is more pronounced at higher values of the Reynolds number, Re0,R > 20 and the width of the injectioninfluenced “displacement” zone (at Gw = 0.94) increases by the factor of 3 at increasing the Reynolds number from 1 to 92.8. The Mach number contours are displaced and temperature contours are disturbed more significantly for helium than in the case of air-to-air blowing. Even at moderate helium blowing rates (0.7 > Gw > 0.32), diffuse outgas flow displaces completely the viscous layer off the sphere surface, and values of the Stanton number become negative. AcknowledgmentsThe author would like to express gratitude to G. A.Bird for the opportunity of using the DS2G computerprogram, and to A. V. Botin for fruitful discussions. 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