Analysis of High-Altitude Ionization Gauge Measurements Using the Direct Simulation Monte Carlo Method

Transcription

1 th AIAA Thermohysics Conference June - July, Portland, Oregon AIAA - Analysis of High-Altitude Ionization Gauge Measurements Using the Direct Simulation Monte Carlo Method Quanhua Sun *, Chunei Cai, and Iain D. Boyd University of Michigan, Ann Arbor, MI 9 James H. Clemmons and James H Hecht ** The Aerosace Cororation, EI Segundo, CA 9 The rarefied, three-dimensional flows exerienced during the Turbulent Oxygen Mixing Exeriment (TOMEX) at altitudes between km and km are simulated using the direct simulation Monte Carlo (DSMC) method. The resent study focuses on ionization gauge measurements obtained by TOMEX. The ayload is modeled in detail and several comlex meshes are generated for the simulations. The simulations show that a bow shock wave is generated in front of the ayload at low altitude that becomes diffusive at higher altitudes. When the altitude increases, the ressure in the channels of the ionization gauge and the ressure variation around the ayload are both decreased. The DSMC results agree very well with data redicted by comressible flow theory and free molecular flow theory when alicable. Comarison between the DSMC results and the TOMEX flight data shows generally good agreement. I. Introduction number of chemical and dynamical rocesses that have minor significance throughout much of the atmoshere Alay a major role in determining the structure and energetics at altitudes between km and km that is often referred to as the MALT (Mesoshere and Lower Thermoshere) region. Such rocesses include the roduction and diffusion of atomic oxygen, interactions of waves and tides, and breakdown of atmosheric gravity waves (AGWs). While the relationshi between eddy diffusion, turbulence, instability structures, dynamical activity, and constituent transort, has been often discussed and seculated uon, an imortant question is how the secies resent in the MALT region are actually affected when otentially unstable conditions occur. The Turbulent Oxygen Mixing Exeriment (TOMEX) was designed to measure the actual atmosheric resonse to the existence of unstable layers as determined by wind and temerature measurements from km to km. The TOMEX combined Na lidar measurements, from Starfire Otical Range in Albuquerque, New Mexico, with a launch of a mother-daughter ayload from White Sands Missile Range, located a little over km from Starfire. The ayload included a trimethyl aluminum (TMA) release (on the mother ayload) to measure winds and diffusion, a five-channel ionization gauge (IG) (on the daughter ayload) to measure neutral density fluctuations at high vertical resolution, and a three-channel hotometer exeriment (on the daughter ayload) to measure atomic oxygen related airglow. The ayload was launched at 9 UT on October and successfully obtained data from all the exeriments. The neutral density measurement emloyed a five-channel ionization gauge. Each channel consisted of a miniature sensor and an accommodation chamber. The ionization gauge was sealed before launch and oened by breaking the seal once the ayload attained altitude. Electronics inside the ayload rovided the circuitry to control and measure the emitted electron current as well as to measure the collected ion current. High-resolution, high-seed analog-to-digital converters were used in order to rovide raid (khz) measurements sanning a large dynamic range in ressure. In order to obtain the ressure (density) distribution around the ayload, one channel was laced forward-looking in the front of the ayload, whereas the other four were equally saced around the ayload. * Post-Doctoral Research Fellow, Deartment of Aerosace Engineering, Member AIAA. Graduate Student Research Assistant, Deartment of Aerosace Engineering. Professor, Deartment of Aerosace Engineering, Associate Fellow AIAA. Research Scientist, Sace Science Alications Laboratory, M-, P.O.Box 99, Los Angeles, CA 99 ** Senior Scientist, Sace Science Alications Laboratory, M-9, P.O.Box 99, Los Angeles, CA 99 Coyright by the, Inc. All rights reserved.

2 The density measurement using the five-channel ionization gauge can infer the flow density, temerature and velocity, in rincile. The comlexity of the flow, however, can make such interretation difficult. For instance, there were momentum and energy transfer on the ayload surface. A bow shock wave was formed in front of the ayload. The mean free ath of the atmoshere in the MALT region was comarable to the ayload size, which indicates that the flow was in the transitional or even free molecular regime. There is no simle theory that can account for the comlicated flow exerienced during the TOMEX. Numerical simulations are therefore necessary to interret the measurement data. The objective of this aer is to hel understand the flow exerienced during the TOMEX using numerical simulations. Additional findings from the TOMEX can be found in several other aers. - In this aer, the ayload geometry is modeled (section II) using some necessary simlifications. Flows in and around the ayload are simulated (section III) using the direct simulation Monte Carlo (DSMC) method, which is an efficient kinetic aroach widely used in simulating rarefied gas flows. Simulated results are then analyzed and comared with the measurement data (section IV) followed by some concluding remarks (section V). II. Payload Modeling During the TOMEX exeriment, the daughter ayload was ejected above km at a searation of a few meters er second. It traveled at about km/s through the km to km region with a sin rate of almost exactly Hz. The ayload has a cylindrical surface with several instruments installed inside and outside as shown in Fig.. On the to of the ayload are three hotometer telescoes and a channel of the ionization gauge. There are four equally laced oenings on the side surface of the ayload where each oening faces one channel of the ionization gauge. Three deck lates are used to mount the internal instruments. Flow is negligible through the uer and lower deck lates, whereas the cutouts in the middle deck late take about half of the deck late area. The regions between the uer and lower deck lates are the only internal art of the ayload having significant flow from the atmoshere. The ionization gauge is an instrument that rovides a measurement of the atmosheric neutral density and its variations. The technique is based on a simle rincile: a volume of gas to be measured is artially ionized by a stream of electrons, then the resulting ions are collected by a biased collector. The gas density can be determined because it is roortional to the ratio of the emitted electron current to the collected ion current. Figure shows a schematic view of a channel of the ionization gauge where the vacuum door is oened during the measurement. Sensor Vacuum Tee Vacuum Door Vacuum Flange Pinch-Off Tube Figure. The Aerosace Cororation -Channel Photometer and Ionization Gauge Daughter Payload. Figure. Schematic View of a Channel of the Ionization Gauge. The ayload has to be simlified and modeled for numerical simulations because it is imossible and unnecessary to include all the details of the ayload. Figure shows the modeled ayload that accounts for the general flow in and around the daughter ayload. The figure shows the inside by removing art of surface. The three hotometer telescoes are modeled as solid cylinders and the inside region over the uer deck late is masked. The middle deck late is also removed for simlicity. The to channel of the ionization gauge is modeled slightly differently from the other four to account for the difference of auxiliary structures. The to channel is centered on the to late, and the other four channels have a -degree deviation from the line that the hotometer telescoes are

3 located on. The diameter of the ayload is slightly less than.m, and its height is close to.m. The height of the modeled ayload, however, is extended to roughly.m, because additional ayload was attached to the exerimental art. Nevertheless, the additional ayload was not exected to affect the oeration of the ionization gauges since the flight of the ayload was suersonic. The ionization gauge is also simlified as shown in Fig.. The vacuum door is removed, and the oening has a diameter slightly larger than.m. The vacuum tee has a diameter of.m and a height of.m. The distance between the oening and the vacuum tee is about.m. Figure. Modeled Daughter Payload. Figure. Modeled Ionization Gauge Channel. III. DSMC Simulation The TOMEX exeriment was erformed in the mesoshere and low thermoshere region. The atmoshere in this region is very rarefied. For instance, the mean free ath of the atmoshere is about.m at km,.m at km, m at km, and m at km, which are all comarable to the diameter of the ayload,.m. Therefore, the direct simulation Monte Carlo (DSMC) method is used to simulate the flow. A. Direct Simulation Monte Carlo Method The direct simulation Monte Carlo method is a widely used article method and is an effective numerical technique to simulate rarefied, nonequilibrium gas flows. It simulates rarefied gas flows based on hysical rinciles instead of mathematical formulation. The DSMC method models gas flows through comutations of motions and collisions of reresentative articles. The rimary aroximation is to uncoule the molecular motions and intermolecular collisions over small time intervals, where article motions are modeled deterministically whereas the collisions are treated statistically. The flow is simulated using the state-of-the-art DSMC research code MONACO. It is a general-urose code for modeling -D, axi-symmetric, or D rarefied gas flows. The code is fully arallelized, and the comutational domain can be automatically decomosed using the artitioning ackage called Metis. The variable hard shere (VHS) molecular model is secified in this case, which assumes that the total collision cross section deends on the relative collision velocity. B. Comutational Mesh The hysics of a roblem determines the required comutational domain and its boundary conditions, which in this case indicates that the size of the necessary comutational domain should vary with altitude for the TOMEX flows. In the DSMC simulation, the comutational domain is divided into cells and filled with simulated articles where the cell size should be less than the mean free ath of the atmoshere in rincile. Enough cells are also required to faithfully reresent the modeled ayload. Besides the hysical requirements, a ractical restraint is to limit the total number of comutational cells due to the limitation of the comuter resources. For the simulated flows, small cells are used around the ayload, esecially for the ionization gauges, whereas relative large cells are emloyed for the rest of the comutational domain. Following the revious rinciles, several tetrahedral meshes are generated using the commercial software Hyermesh. 9 A tyical surface mesh is shown in Fig., where the axis is assigned to the Z direction and the oenings face the X or Y direction. The surface is modeled as a solid wall whose temerature is K and the

4 thermal accommodation coefficient is set to. The entire surface is divided into, wall faces for the flow at km and, wall faces for the flow at km. The volume of the entire comutational domain is about.m at km, and is increased to m at km where the flow is subsonic. Figure shows a cut-oen view of the mesh for the 9km case, which dislays the structure of the tetrahedral mesh. The total number of tetrahedral cells varies from,9, at km, to, at km, to, at km. Z X Y Figure. Surface Mesh of the Payload for the 9km Case. Figure. Cut-Oen View of Tetrahedral Mesh for the 9km Case. The cell size for lower altitude simulations is not always less than the mean free ath of the atmoshere. To ensure that collisions only occur between articles within a mean free ath, a near-artner selection technique is used to determine the collision article airs. This technique is slightly different from the subcell technique where a cell is divided into several sub-cells. The near-artner technique allows a article to collide with articles only within a small distance. In the resent investigation, the distance is re-determined and is less than half of the cell edge, which ensures that the distance is less than the local mean free ath throughout the comutational domain. The ayload flew at suersonic seed below km, which allows simulations to emloy a small comutational domain. However, a large comutational domain is referred to imlement effective boundary conditions for subsonic flows exerienced at higher altitude. Simulations show that the relative flow is limited to the inner art of the comutational domain even when a mesh designed for the km case is used to simulate the flow near the aogee (.km). Hence, the meshes are accetable for the resent investigation. C. Particle Weighting In a DSMC simulation, the number of simulated articles is determined by the size of the comutational domain and the number of real molecules that each simulated article reresents. For instance, the comutational domain for the flow at km has a volume of m, and the volume of the minimum cell (tetrahedral) is about - m. If articles are simulated in the minimum cell, a total of. 9 articles have to be simulated at the same time, where the density is assumed to be constant for the estimation. Such a simulation is extremely exensive based on the resent comutational ability. In order to limit the total number of simulated articles, a article weighting strategy is emloyed. In general, a article in larger cells reresents more real molecules such that a similar number of articles are simulated in each cell, which is achieved by assigning a relative article weight to each cell. The comutational cells have different article weights for the simulated articles, which means that a article is to be destroyed or cloned when it moves into another cell. The destroying and cloning is erformed according to the robability based on the relative article weights of the two cells. The ossible effects of this article weighting strategy on the simulations are investigated. First, several sets of article weights are tested. One set is roortional to the cell volume, and another is roortional to the square root of the cell volume, whereas a third set allows the simulation to have roughly articles simulated in each cell. Second, a similar flow having zero velocity is simulated using the article weighting strategy. It is found that the main effect of the weight strategy is that the statistical scatter is increased due to the destroying and cloning, which means that the weighting strategy will not have serious effects on the accuracy of the simulations.

5 D. Numerical Error All the simulations have at least articles in each cell, and the time ste is less or much less than the mean collision time of the atmoshere. The total number of the simulated articles is in the order of magnitude of million. A tyical simulation runs for, time stes after the simulation reaches a steady state that generally takes less than, time stes, which requires aroximately one day running on a -rocessor Linux cluster. The simulation results are obtained by samling the simulated articles, which means that the DSMC results exhibit statistical scatter. The statistical scatter can be reduced by increasing the samle size. The magnitude of the scatter, however, can only be estimated because the samles are statistically deendent and article destroying and cloning introduce additional scatter. Based on the simulation for a zero velocity flow and a general flow running much more time stes, the statistical error for the results resented in the next section is estimated to be less than ercent in general, which is accetable given the simlifications made for the geometry modeling. IV. Results and Discussions Simulations are erformed for the TOMEX flows at altitudes ranging from km to.km (aogee). The DSMC simulations aroximate the TOMEX flows as seudo steady state flows using the flow conditions at the same altitude. The neutral temerature and densities are given by the Mass-Sectrometer-Incoherent-Scatter (MSIS) model at 9UT on October. The detailed flow conditions are listed in Table, where H is the altitude, U, V and W are the velocity comonents of the atmoshere relative to the ayload in the x, y, or z direction, T is the temerature of the atmoshere, M is the Mach number of the free stream, while n O, n N, and n O are the number densities of the atmoshere comonent atomic oxygen, molecular nitrogen, and molecular oxygen, resectively. Table: Detailed flow conditions for the simulations H(km) U(m/s) V(m/s) W(m/s) T (K) n O (/m ) n N (/m ) n O (/m ) M A. Flow Patterns The TOMEX flight was exected to generate comlicated flow structures during the exeriment. Figures - illustrate the flow atterns redicted by the DSMC simulation when the altitude is km, where the Mach number of the free stream relative to the ayload is roughly.. A strong bow shock wave is observed (Figs. -9) about cm away from the ayload. The bow shock wave is nearly symmetric on the x= lane although the ayload is unevenly distributed in the front, whereas it is non-symmetric in the y= lane because of the -degree angle of attack. The ressure behind the shock wave is increased due to the shock comression. The increase reaches the maximum on the to of the ayload and decreases downward along the ayload surface. The -degree angle of attack of the flow also means that the side of the ayload facing toward the flow has a larger ressure and the oosite side has a lower ressure. Relatively low ressure is redicted inside the ayload. The ressure in the side channels of the ionization gauge deends on whether the oening faces the flow. The channel in the negative x direction has a large ressure and the channel in the ositive x direction has a smaller ressure, whereas the two channels located in the x= lane agree within numerical error. The ressure contours lotted in Fig. show the overall ressure distribution on the ayload surface. Figures and show the density and temerature contours in the y= lane. The density lot is more similar to the ressure lot as comared with the temerature lot because the temerature on the surface is ket at K. It is also found that there is a large temerature jum around the outer surface of the ayload although the Knudsen number of the flow evaluated as the ratio of the mean free ath of the free stream to the diameter of the ayload is only..

6 P P y (m). Figure. Pressure (Pa) Contours in x = Plane at km Predicted by DSMC Simulation. Figure. Pressure (Pa) Contours in y = Plane at km Predicted by DSMC Simulation.. Z... y (m) P..... Y X... P Figure 9. Pressure (Pa) Contours in z =. Plane at km Predicted by DSMC Simulation. Figure. Surface Pressure (Pa) Contours at km Predicted by DSMC Simulation.. E- E-. E-.. E- E- E- E- ρ 9E- E- E- E- E- E- E- E- E- E- E- E-... E T Figure. Density (kg/m) Contours in y = Plane at km Predicted by DSMC Simulation.. Figure. Temerature (K) Contours in y = Plane at km Predicted by DSMC Simulation.

7 P/P P/P Figure. Pressure Contours in y = Plane at 9km Predicted by DSMC Simulation... P/P P/P Figure. Pressure Contours in y = Plane at km Predicted by DSMC Simulation Figure. Pressure Contours in y = Plane at km Predicted by DSMC Simulation. Figure. Pressure Contours in y = Plane at km Predicted by DSMC Simulation Figure. Pressure Contours in y = Plane at km Predicted by DSMC Simulation. P/P P/P Figure. Pressure Contours in y = Plane at km Predicted by DSMC Simulation.

8 The flow atterns vary with altitude due to changing flow conditions. First, the mean free ath of the atmoshere is increased by three orders of magnitude from km to km, which changes the flow from the near-continuum regime to the free molecular regime. The thickness of the bow shock wave if it exists will also increase because the thickness of a normal shock wave is roughly several mean free aths, which means that a shock wave in a rarefied condition can extend over a large distance. Also, the Mach number of the free stream is decreased from suersonic to subsonic. For subsonic flows, the effects at any location are global. Figures to illustrate the flows at several altitudes. Each figure shows the ressure contours normalized by the free stream ressure and the stream lines in the y= lane where the velocity in the y direction is very small. As shown, a shock-like structure is still observed for the flow at 9km, whereas a shock wave is hardly recognized for the flow at km. The ressure lot for the flow at km resembles that for the subsonic flow at km although the free stream Mach number is roughly.. It is also found that the ressure variation decreases as the altitude increases, and the variation is limited to a relatively small region at higher altitude. B. Pressure in the Ionization Gauges The ionization gauge measures the neutral density of the gas in the vacuum tee, whereas the ressure is usually used to comare with data obtained using other techniques. Therefore, the ressure in the vacuum tee is analyzed before the simulation results are comared with the measurement data. The inviscid comressible flow theory can redict the total ressure using the Mach number of the free stream: γ = M γ γ () where is the total or stagnant ressure, is the free stream ressure, M is the free stream Mach number, can be regarded as the ressure in the vacuum tee of the and γ is the ratio of secific heats. For subsonic flows, to channel because there is no flow in the vacuum tee. For suersonic flows, however, the total ressure will dro across the shock wave. The total ressure behind the shock wave or the ressure in the vacuum tee,, is aroximated following the normal shock wave relations as follows: = γ γ M ( ) ( ) γ γ γ M γ γ ( γ ) M () The ressures in the side channels are generally less than the total ressure because of the viscous interactions between the flow and the surface of the ayload. The ressure in the vacuum tee can also be aroximately redicted by the free molecular flow theory because TOMEX flows at high altitude are already in the free molecular flow region. The exression based on the free molecular theory is as follows: s = g( s ) e ( )[ ] s g s erf s { [ ] ( ) } f π () g ( s ) = s π T T r, s = u m kt () where f is the ressure in the vacuum tee, T is the free stream temerature, T r is the reflected temerature (it is assumed as the wall temerature), m is the average molecular mass, k is the Boltzmann constant, u is the velocity comonent normal to the door of the channel, and erf ( ) is the error function.

9 The ressure in the to channel of the ionization gauge is lotted in Fig. 9. It shows that the ressure redicted by the DSMC simulation is very close to that obtained using the comressible flow theory when the altitude is low, and it is a little smaller than that redicted by the free molecular theory when the altitude is high. Figure shows the ressure in the side channels of the ionization gauge. The DSMC ressure in the channel facing the flow, P x, is about % lower than that redicted by the free molecular theory, which may be due to the effects of the existence of the ayload surface. For the ressure in the channel oosing the flow, P x, the agreement between the DSMC results and those redicted using the free molecular theory is good when the altitude is high, and the large difference at low altitude indicates that the free molecular theory is invalid here although the ressures at the other channels seem close to the DSMC results. ressure in the to channel (Pa) DSMC simulation comressible theory free molecular theory ressure in the side channels (Pa) P -x (DSMC simulation) P -x (free molecular theory) P x (DSMC simulation) P x (free molecular thory) P x= (free molecular theory) P y (DSMC simulation) P -y (DSMC simulation) altitude (km) Figure 9. Pressure in the To Channel - 9 altitude (km) Figure. Pressure in the Side Channels C. Pressure Comarison between Simulation Results and Measurement Data The comuted ressures in the channels of the ionization gauge are comared with the measurement data. During the TOMEX exeriment, the ayload had a sin rate of almost exactly Hz, which means that each of the side channels recorded the ressure that had a seudo-frequency of Hz. Figure shows the ressure in the channels of the ionization gauge obtained from the DSMC simulations and the measurement. All the measurement data has been calibrated and adjusted to account for effects such as gas venting in the ionization gauge. The simulation offers good agreement with the to channel calibrated data exect at the lowest altitude. The calibrated side data shown in the figure is the measurement ressure in one of the side channels of the ionization gauge. The measurement ressures in all side channels are very similar, and their average is also lotted in the figure. It is found that the DSMC simulation redicts a maximum ressure in the channel facing the flow and a minimum ressure in the channel oosing the flow, and these ressures bound the measurement data. The slight difference between the simulation results and the measurement data is not surrising accounting for ossible differences of the flow conditions and ayload configuration. First, the hysical configuration of the ayload is much more comlicated than is modeled. There are many un-modeled auxiliary arts and all kinds of cables. For instance, the middle deck late and the vacuum doors of the channels of the ionization gauge are removed in the simulation. Second, the flight is actually unsteady, which may introduce differences when the flow is treated as a seudo-steady state at each altitude. Third, the atmoshere conditions used in the simulation are based on the MSIS model. The difference between the model data and the real situation may account for some difference. The data obtained from the lidar 9 ressure (Pa) calibrated to DSMC to calibrated side side average DSMC at x DSMC at -x DSMC at x= altitude (km) Figure. Pressure Comarison between DSMC Simulation Results and Calibrated Measurement Data.

10 measurement do indicate that the temerature of the atmoshere below 9km was a little warmer than that given by the MSIS model. For instance, the ressure in the to channel will decrease roughly % if the ressure of the atmoshere is ket the same while the temerature redicted by the lidar measurement is used in the simulation. The decrease of the channel ressure is because the free stream Mach number is decreased due to the warmer atmoshere. Accounting for all these factors, the DSMC simulations are very successful in studying the TOMEX flows, and show flow details that are not revealed by the measurement. V. Conclusion The rarefied, three-dimensional flows exerienced during the Turbulent Oxygen Mixing Exeriment (TOMEX) at altitudes between km and km were simulated using the direct simulation Monte Carlo (DSMC) method. The details of imortant geometric structures of the ayload were modeled, and several comlex tetrahedral meshes were generated for the simulations. The DSMC simulations emloyed a article weighting technique to limit the total number of simulated articles and a near-artner selection strategy to determine the collision article airs. Simulations showed that a bow shock wave was generated in front of the ayload at low altitude that became diffusive when the altitude was increased even when the Mach number of the free stream was larger than.. It was also found that the ressure in the channels of the ionization gauge decreased monotonically with increasing altitude where the ressure variation was also decreased. The ressure at several locations was estimated using comressible flow theory and free molecular flow theory, and indicated that the DSMC simulations were reasonable. Comarison between the DSMC results and the measurement data showed that the DSMC simulations redicted the TOMEX flows very well. The accuracy of the comutations indicates that DSMC should be used to design future flight exeriments. Acknowledgments This work is funded in art by NASA through grants NAG- (TOMEX) and NAG- (TRIO). The authors are also grateful for comutational resources rovided on the Maui High Performance Comuting Center and on the high erformance clusters in the Deartment of Aerosace Engineering and Mechanics at the University of Minnesota. References Hecht, J. H., Liu, A. Z., Bisho, R. L., Clemmons, J. H., Gardner, C. S., Larsen, M. F., Roble, R. G., Swenson, G. R., and Walterscheid, R. L., An Overview of Observations of Unstable Layers During the Turbulent Oxygen Mixing Exeriment (TOMEX), Journal of Geohysical Research, Vol. 9, DS,. Liu, A. Z., Roble, R. G., Hecht, J. H., Larsen, M. F., and Gardner, C. S., Unstable Layers in the Mesoause Region Observed with Na Lidar during the Turbulent Oxygen Mixing Exeriment (TOMEX) Camaign, Journal of Geohysical Research, Vol. 9, DS,. Bisho, R. L., Larsen, M. F., Hecht, J. H., Liu, A. Z., and Gardner, C. S., TOMEX: Mesosheric and Lower Thermosheric Diffusivities and Instability Layers, Journal of Geohysical Research, Vol. 9, DS,. Hecht, J. H., Liu, A. Z., Walterscheid, R. L., Roble, R. G., Larsen, M. F., and Clemmons, J. H., Airglow Emissions and Oxygen Mixing Ratios from the Photometer Exeriment on the Turbulent Oxygen Mixing Exeriment (TOMEX), Journal of Geohysical Research, Vol. 9, DS,. Larsen, M. F., Liu, A. Z., Bisho, R. L., and Hecht, J. H., TOMEX: A Comarison of Lidar and Sounding Rocket Chemical Tracer Wind Measurements, Geohysical Research Letters, Vol., No.,,. -. Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, 99. Dietrich, S., and Boyd, I. D., Scalar and Parallel Otimized Imlementation of the Direct Simulation Monte Carlo Method, Journal of Comutational Physics, Vol., 99,. -. Karyis G., and Kumar, V., METIS, A Software Package for Partitioning Unstructured Grahs, Partitioning Meshes, and Comuting Fill-Reducing Orderings of Sarse Matrices, version., Technical Reort, University of Minnesota, Deartment of Comuter Science and Engineering, Hyermesh, Version., Altair Engineering, Inc., Troy, Michigan. Hedin, A. E., Extension of the MSIS Thermosheric Model into the Middle and Lower Atmoshere, Journal of Geohysical Research, Vol. 9, 9, 99. Anderson, J. D., Modern Comressible Flow: with Historical Persective, McGraw-Hill Press, 99. Gombosi, T. I., Gaskinetic Theory, Cambridge University Press, 99.