Monte Carlo Simulation of Hypersonic Rarefied Gas Flow using Rotationally and Vibrationally Inelastic Cross Section Models

Transcription

1 Monte Carlo Simulation of Hypersonic Rarefied Gas Flow using Rotationally and Vibrationally Inastic Cross Section Mods Hiroaki Matsumoto a a Division of systems research, Yokohama National University 79-5 Tokiwadai Hodogaya Yokohama Japan Abstract. The rotational collision number, vibrational raxation time, and transport coefficients of real gases were used to determine key functions for molecular collision mods. The functions determined were the characteristic function for rotationally inastic collision cross section in the statistical inastic cross section (SICS) mod, parameters for the transient probability density function in the vibrationally inastic collision cross section (VICS) mod, and the probability density function of the deflection angle for these diatomic mods. The validity of the present mods was investigated by applying a Monte Carlo simulation of hypersonic rarefied gas flow around a flat plate and comparing the results with experimental data. The moded profiles of rotational temperature and rotational energy distribution near the plate were in reasonable agreement with those found by experiment. Keywords: Direct Simulation Monte Carlo, Rotationally Inastic Collision, Vibrationally Inastic Collision, Hyper Sonic, Flat Plate. PACS: n INTRODUCTION The direct simulation Monte Carlo (DSMC) method is an efficient tool for the prediction of non-equilibrium phenomena in rarefied gas flow. Realistic and effective molecular collision mods are required in order to apply the DSMC method to rarefied gas flow problems. For astic molecular collisions, accurate and realistic calculation techniques based on scattering theory as wl as simple scattering mods,3,4,5,6 based on kinetic theory have been devoped and applied to various rarefied gas flow problems. For a rotationally and vibrationally inastic molecular collision, a simple mod is more practical for engineering; because inastic collision phenomena are quite complicated; enormous calculation time may be required for an accurate approach. Some attractive mods have been devoped, including the dynamic molecular collision (DMC) mod 7, the phenomenological mod formulated by Borgnakke and Larsen (the BL mod 8 ), the statistical inastic cross section (SICS) mod 9 for rotationally inastic collisions, and the BL mod 0 and the vibrationally inastic cross section (VICS) mod for vibrationally inastic collisions. These diatomic collision mods require the probability density function for deflection angle, the probability density function for inastic collision and energy transition, or the cross sections for inastic collisions and energy transition, in order to express the transport coefficients and the inastic collision number or raxation time of real gases. In the present study, the function for the rotationally inastic collision cross section of the SICS mod, parameters for the vibrationally inastic collision probability of the VICS mod, and the probability density function for deflection angle were defined from transport coefficients, rotational collision number, and vibrational raxation time of nitrogen gas. The validity of the present cross section mod was examined by applying it to the simulation of hypersonic rarefied gas flow around a flat plate and comparing the results with experimental data.

2 BASIC EQUATION AND MOLECULAR COLLISION MODELS Generalized Boltzmann Equation The motion of a diatomic molecule with continuous translational and rotational energy and discrete vibrational energy is described by the generalized Boltzmann equation as lmax l max ( nfi) ( nfi) ij, i ' j ' + c n ' ' ', ', ξξ, ξ ' ξ ' ξ' ξ' sinχ χ ξ f i f j fi f j gi d d d dєd dc () t r j i', j' f f c, ξ, E, f f c, ξ, E and () ( ) ( ) ( ξ ) ( ξ ) i i j j f ' i' f c', ', Ei', f j' ' f c ', ', Ej', (3) where t is time, r is the physical space coordinate, n is the number density, χ is the deflection angle, є is the azimuth angle, g is the rative vocity, f is the distribution function for vocity class c, rotational energy class ξ and ij,' i j ' vibrational energy of quantum lev i, l max is the maximum vibrational lev, and I is the differential cross section. Variables with a prime ( ) refer to the post-collision state. In this study, the differential cross section is reduced to the following form by assuming that molecular collision is limited to astic, rotationally inastic, and vibrationally inastic collisions. ij,' i j ' Iξξ [ ], ξ ' ξ ' dξ' dξ' sinχdχdє Isin χdχdє + Iξξ, ξ ' ξ ' dξ' dξ' sinχdχdє + I, ' ' sinχ χ ij i j d dє R v, (4) ( χ χ ) + {( ξ' ξ' ξ' ξ' )( )} {( )( )} χ χ + ' ' χ χ t p p d pєdє pr p d d p d pєdє pv pi j p d pєdє where p χ and pє are the probability density functions for deflection and azimuth angles, respectivy. The terms t p, p R, p v, p ξ ξ, p i j are respectivy the total cross section, the probability density function for astic, rotational and vibrational collisions, and the rotational and vibrational transition probabilities. They are defined as R v t + R + ij, p, pr, pv, (5) t t t I sin χdχdє, (6) sξξ, ξ ' ξ ' R sξξ, ξ ' ξ ' d ξ'd ξ', sξξ, ξ ' ξ ' I ξξ, ξ ' ξ ' sin χdχdє, p ξ ' ξ ', and (7) R lmax sij,' i j ' i, j ij, i' j', ij, i' j' ij, i' j' sin χ χ, i' j' i' i, j' j i, j ξξ ξ ξ, ' ' s s I d dє p, (8) respectivy, where is the astic collision cross section, R the rotational collision cross section, ij the vibrational collision cross section, s ξξ, ξ ξ the rotational transition cross section, and s ij,i j the vibrational transition cross section. The DSMC method requires p χ, p є, and a set of cross sections (, s ξξ, ξ ξ, and s ij,i j ) or t and a set of collision probabilities (p, p R, p v, p ξ ξ, p i j ). Molecular Collision Mods a. Elastic Collision Mod The VSS mod for a Lennard-Jones potential 4 is applied here for astic collisions. The cross section is defined as () () () Q Q + Q () () Q Q, (9) where Q () and Q () are cross sections for diffusion and viscosity, respectivy; the definitions for these terms are found in Ref.3 and numerical values are listed in Refs.4 and 6.

3 b. Rotationally Inastic Collision Mod For rotationally inastic collisions, the SICS mod 9 is used. The rotational collision cross section R and the transient probability p ξ ξ is defined as R Z ( E ), and (0) [ E ξ' ξ' ] ( E ξ' ξ' ) pξ', ξ', () max {[ E ξ' ξ' ] ( E ξ' ξ' ) } E Etr + ξ + ξ E ' tr + ξ' + ξ', () where E tr is the translational energy and Z is a characteristic function for the rotationally inastic collision cross section. In this study, the function Z is assumed to be of the following form. i / E a E E i th Z i 0 ε LJ, (3) 0 E < Eth where E th is the threshold energy taken to be 6kθr with the characteristic rotational temperature θr, ε LJ is the depth of the potential wl of the Lennard-Jones potential are fitting parameters defined from the rotational collision number Z R of a rigid rotor mod. Parameters a , a and a are obtained with ε LJ / k 9.5 K (k is the Boltzmann constant), and θr.863 K for nitrogen from the rotational collision number Z R of the rigid rotor mod 4 as shown in Fig.a. c. Vibrationally Inastic Collision Mod For vibrationally inastic collisions, the VICS mod is introduced. The vibrational transition cross section s ij,i j is defined as ΔEij,' i j ' η v ( Etr ) pij,' i j' ( Etr) Etr ΔEij,' i j ', Δ Eij,' i j ' > 0 Etr s 0 E <ΔE, Δ E > 0, (4) ij,' i j' tr ij,' i j' ij,' i j' ( ) η v Etr piji,' j' ( Etr) ΔEiji,' j' 0 / Etr + E' tr Δ Eij,' i j ' Ei ' + Ej ' Ei Ej, E tr, (5) where E i is the vibrational energy of quantum lev i, η v is the vibrational steric factor, and p ij is the vibrational transition probability defined as I! J I J p,' ' ' ', exp( ) ( ) ij i j pij pi j pij q q! LI q, and (6) J / θ π π k cosech v h q, α 6 α 4 α v, (7) v v Etr πlv( mkθv) ( α where L ) n ( x) is the associated Laguerre polynomial, I min(i,i ), J max(j,j ), L v is the steepness parameter, θ v is the characteristic vibrational temperature, and h is Plank s constant. For one-quantum transitions i i ± and j j ±, the vibrational transition probability p ij is also defined as ( ) / IJ π kθv pij,' i j ' sin g, ωv, (8) ωvlv h

4 The vibrational steric factor η v and the steepness parameter L v are determined from the Millikan-White formula. Using a Landau-Tler plot of vibrational raxation obtained by the Millikan-White formula, as shown in Fig.b, and setting θ v 3393 K for nitrogen, we obtain L v 0.5A and η v.. d. Probability density function of deflection angle and azimuth angle The probability density functions for azimuth angle p є and deflection angle p χ for astic, rotationally inastic, and vibrationally inastic collisions are defined as p, (9) p χ π α χ χ α cos sin δ( χ 0) є b b r b>b,and (0) r () () Q Q () () Q Q, () α + respectivy, where δ is the Dirac dta function, b r is the reduced impact parameter 5 which is introduced so as to describe the transport coefficients of real gases, and b is the impact parameter for astic collisions. The reduced and astic impact parameters are defined as br brmax Rnd, b / π, () respectivy, where R nd is an uniform random number in the range [0,], and b rmax is the maximum value of the reduced impact parameter defined as b + + π. (3) ( ) rmax R i, j / CALCULATION OF HYPERSONIC FLOW AROUND A FLAT PLATE AND DISCUSSION A set of inastic cross section mods ( mod) presented in this study was applied to the simulation of hypersonic flow around a flat plate and compared with the experimental results. The computational domain was taken as -0.5 x/l.5, -0.8 y/l 0.7 and divided into collision and data cls Δx/L Δy/L 0.005, where L is the length of the flat plate. The flat plate had a thickness d/l3/6 and the leading edge angle 30 and was set to 0 x/l and -3/6 y/l 0.0. The upstream boundary conditions at x/l-0.5 and y/l-0.8,0.7 were set to the equilibrium uniform flow, and the down stream boundary condition at x/l.5 was set to be ( nfi)/ x 0. The surface of the plate was set to the diffuse reflection with the temperature T w. Initially, the computational domain was set to be an upstream equilibrium condition. Influx molecules across the upstream boundary were assigned an equilibrium Maxwlian distribution by upstream boundary conditions. The time evolution of the position and vocity of each molecule was simulated using the same algorithm used in reference, except for the treatment of estimating molecular collisions. In this paper, collision sampling was conducted using the null-collision technique 6. After the steady state was established, flow properties were calculated and averaged until statistical fluctuations became sufficiently small. Figures (a) and (b) show comparisons of the rotational temperature profiles obtained with the mod and experimental data near the flat plate surface (ymm), and perpendicular to the plate surface. The experimental conditions were as follows: upstream Mach number M4.89, upstream temperature T 9K, surface temperature of the flat plate T w 300K, and upstream Knudsen number Knl 0 /L0.0, where l 0 is the mean free path in the upstream equilibrium state. As shown in Fig. (a), the rotational temperature profile near the flat plate surface obtained by the mods agrees reasonably with experimentally measured values. On the other hand, some discrepancies are observed in the rotational temperature profiles perpendicular to the plate, and these discrepancies increase with increasing x and y as shown in Fig.(b). The rative rotational energy distributions of the mod are in reasonable agreement with those of experiment for the position x5 mm, however discrepancies increase with increasing x and y as shown in Figs.3(a) and (b) which is similar to the rotational temperature distribution in Fig.(b).

5 Figure 4 shows the vibrational temperature distribution around the flat plate. Some of the molecules were excited to the vibrational energy lev from the ground state i0 to i by interacting with the flat plate, and traved down stream with very little vibrationally inastic collision probability as shown in Fig.4; this indicates that the flow fid was not affected by the vibrationally inastic collisions in this study. The results of this study suggest that the characteristic function or parameters for rotationally inastic collision mod require some appropriate optimization. ZR Rigid Rotor log( pτ v atm s ) MW T K ( T K ) -/3 FIGURE. (a) Comparison of rotational collision number given by the SICS-VISC and rigid rotor mods, (b) Comparison of the Landau-Tler plot of vibrational raxation obtained by the mod and the Millikan-White formula x 5mm x0mm x0mm T rot K 00 y mm x mm T rot K FIGURE. Comparison between calculation results and experimental data. (a) Rotational temperature profile near the flat plate surface (ymm), (b) Rotational temperature profiles perpendicular to the plate. f ( ξ ) / f (0) y mm y3 mm y5 mm y8 mm f ( ξ ) / f (0) y mm y3 mm y5 mm y8 mm y.5 mm exp ξ K ξ K FIGURE 3. Comparison of rative rotational energy distributions given by the mod and experimental data at (a) x5mm, (b) x0mm.

6 FIGURE 4. Profile of non-dimensional vibrational temperature T v /T around the flat plate. CONCLUDING REMARKS The rotational collision number, vibrational raxation time, and transport coefficients of a real gas were used to determine the parameters of the characteristic function for the rotationally inastic collision cross section in the SICS mod, the fitting parameters for vibrationally inastic transient probability in the VICS mod, and the probability density function of deflection angle for diatomic molecular collision mods. The validity of the present mod was tested by applying it to the simulation of hypersonic flow around a flat plate and comparing the results with the experimental data. The rotational temperature profile near the flat plate surface and the rative rotational energy distribution given by the mods agree reasonably with those obtained by experiment. However discrepancies between the simulation and experiment increased with increasing distance from the flat plate. This study shows that the characteristic function or parameters for the rotationally inastic collision mod require some appropriate optimization. REFERENCES. G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flow (Clarendon press, Oxford, 994).. G. A. Bird, "Monte Carlo Simulation in an Engineering Context," Prog. Astronaut. Aeronaut. AIAA 74, 39 (98). 3. G. A. Bird, "Definition of Mean Free Path for Real Gases," Phys. Fluids 6, 3 (983). 4. K. Koura and H. Matsumoto, "Variable Soft Sphere Molecular Mod for Inverse-Power-Law and Lennard-Jones Potentials," Phys. Fluids A3, 459 (99) 5. K. Koura and H. Matsumoto, "Variable Soft Sphere Molecular Mod for Air Species," Phys. Fluids A4, 083 (99). 6. H. Matsumoto, "Variable sphere molecular mod for inverse power law and Lennard-Jones potentials in Monte Carlo simulations," Phys. Fluids 4, 456 (00). 7. Tokumasu, T. and Matsumoto, Y., Dynamic molecular collision (DMC) mod for rarefied gas flow simulations by the DSMC method, Phys. Fluids,,907(999). 8. Borgnakke, C. and Larsen, P. S., Statistical collision mod for Monte Carlo simulation of polyatomic gas mixture, J. Comput. Phys.,8,405(975). 9. Koura, K., Statistical inastic cross-section mod for the Monte Carlo simulation of molecules with continuous internal energy, Phys. Fluids, A5, 778(993). 0. Boyd, I., Raxation of discrete rotational energy distributions using a Monte Carlo method, Phys. Fluids A5 78 (993).. Koura, K., Improved Null-Collision Technique in the Direct Simulation Monte Carlo Method, Computers Math. Applic, 35(998), N.Tsuboi and Y.Matsumoto, Interaction between Shock Wave and Boundary Layer in Nonequilibrium Nypersonic Rarefied Flow, JSME International Jouanal, B49, 77(006). 3. S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge Univ. Press, 976: 3rd edition, 939: st edition). 4. H.Matsumoto., Rotational Collision Number nad Rotationally Collision Cross Section of Inastic Collision Mod for Nitrogen Molecule in the Direct Simulation Monte Carlo Method, Trans,. Japan Society of Mech. Eng., 73, 93 (007). 5. H.Matsumoto., Transport Coefficients of Inastic Collision mod in the Direct Simulation Monte Carlo Method, Trans,. Japan Society of Mech. Eng., 70, 97 (004). 6. Koura, K., "Null-Collision Technique in the Direct Simulation Monte Carlo Method," Phys. of Fluids 9, (986).