Implementation of a Re-entry CFD tolbox in the ...

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. AIAA Atmospheric Flight Mechanics 6-9Augu°stf2001Ce Montre^Canacia

A01

-37276

AIAA-2001-4314

Implementation of a Re-entry CFD Toolbox in the GESARED Flight Simulator

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A.Guidi," Q.P. Chu,f J.A. Mulder* J. Buursink,§ Faculty of Aerospace Engineering, TUDelft, Delft, The Netherlands The primary objective is to implement a basic Computational Fluid Dynamics tool in a Flight Simulator in order to be able to optimize aerodynamically the vehicle's shape during the initial stability analysis tradeoffs. The emphasis will be on basic tools required to conduct and analyze engineering calculations. The Delft Aerospace Re-entry Test (DART) demonstrator is a small re-entry vehicle. Its stability is under investigation with a flight simulator. During the initial design phase, strong interaction between different disciplines is demanded for an effective design. One important factor is the vehicle's aerodynamics, it is in fact needed to simulate the vehicle dynamics. However, another necessity during the preliminary phase is to have a simple way of evaluating the dynamics for different configurations of the design. This is necessary so that a quick and iterative process will lead to a final design that meets all the requirements and objectives.Real time calculation of the Aerodynamic Coefficients was implemented in the flight simulator, through the Re-entry CFD toolboxes. The aerodynamic regimes implemented are: rarefied, transitional, and continuum hypersonic, all with the influence of viscosity. Such software is what some avant-garde Aerodynamicists would call a "Virtual tunnel".

Nomenclature a TT p 6 6 pw

-0.5 -

1

0

Figure 1 REV-olution configuration, RN=0.250 m, RB=0.670m, L=l.360 m 91=15°, 32=30°

0.5

0*

-0.5-

1

0

In this design phase, a strong interaction between different disciplines is demanded for an effective design. The vehicle equations of motion are a unifying framework for studying many important aspects of a design and for evaluating problems that arise during the analysis phase of the design. From these equations a model can be built to simulate all the aspects of the design. The software implementation of the equations of motion is the flight simulator. Aerothermodynamics, flight dynamics, structural analysis, thermal protection system, and GNC (guidance, navigation and control) are all based upon the output of the flight simulator. The stability characteristics of an aircraft have their roots in the aerodynamics of the vehicle5. Furthermore, the "classical theory" of the stability is expressed in the language of the aerodynamicists. The aerodynamic stability of a blunted cone varies across the speed regimes (hypersonic rarefied, hypersonic transitional, hypersonic continuum, supersonic, transonic, subso-nic). DART needs to be sufficiently aerodynamically stable to overcome the spin stiffness in order to minimize any angle of attack variation through out all the regimes and maintain a controlled attitude until parachute deployment. Stability is the tendency of an aircraft in flight to remain in its path, maintain upright flight and to return to this attitude if displaced, without corrective action by any other device. The stability analysis of a vehicle can be carried out analytically studying the aerodynamic properties of the vehicle in the small perturbation approximation. In this approximation the aerodynamic characteristics can be linearized (approach followed by Bryan in one of the first studies in stability in 1911). A complete non-linear analysis of the stability of an aircraft cannot be carried out without the aid of a computer. The role played in modern flight dynamics from the computers is determinant. The flight simulator gives to the flight dynamicist the kinematics and dynamic characteristics related to position and attitude, above all the damping of the vehicle due to external disturbances can be extrapolated and conclusions on the static and dynamic behavior of the vehicle can be drawn. Recently, the Control and Simulation Division of Delft Aerospace developed such a tool on demand of ESTEC for the design of GNC systems for the CRV (Crew Return Vehicle) for the International Space Station. GESARED (General Simulator for Atmospheric Re-Entry Dynamics) was adapted also for the study of an axisymmetric body and was used to perform the stability analysis during the preliminary design of DART. (For further information on how the equations of motion were implemented in GESARED see Costa R.R., Chu P.Q. et al AIAA 2000-4086).

Figure 2 VOLAN configuration, RN=0.392 m, RB=0.6m, 1=1.2 m 9j=ff, 92=15° 2 OF 11 AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2001-4314

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The Test Triangle A way in which several aspects of an aerospace design can be related is summarized as shown in the following "Test Triangle": Wind Tunnel

Flight Simulator

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Purely Computational

Flight Test

Purely Physical

Figure 3 Ttest Triangle

Before any engineering system can be optimized, it goes through the above process several times. It can be called the "testing phase" of the design and can be globally defined as "simulation". This global simulation uses both computational and physical tests to evaluate and refine all aspects of an aerospace system. A flight simulator is a fully computational "simulation" of the system, a flight test is a fully physical "simulation" of the system. In the middle the wind tunnel is a partially physical simulation of the system which is then analyzed through computational simulation. Engineering test and simulation plays an extremely important role in aerospace engineering. In a larger sense DART itself can be seen as a flight test of a larger re-entry study. It is in fact reusable and it will allow Delft Aerospace to iterate the "test triangle" several times and acquire and refine knowledge in re-entry. In this paper the attention will be focused on the Computational Simulation as a possible combination of the two steps of Simulator and Wind tunnel in a situation in which the last one is not available (as in the case of a preliminary design), or when it is not completely reliable (it is not possible to simulate perfectly the several parameters that characterise the hypersonic flight in ground-based facilities). The previous statement explains the need of a flight test like DART in order to acquire further knowledge in the aerothermodynamics of hypersonic flight and the need of research in different fields that then can be validated from data collected during the flight. The opportunity to use real data to improve Computational Simulations both in the field of Aerodynamics (e.g. Computational Fluid Dynamics or CFD) and Flight Dynamics (e.g. a Flight Simulator) is one of the mainsprings of the present work. In order to emphasize the scientific foundations and basic tools required to conduct and analyze engineering tests a consideration of J. Andersen is repeated: "Up to as late as I960, the history of the development of fluid mechanics had involved two dimensions: pure experiment and pure theory. With

the advent of computational fluid dynamics after I960 a new third dimension, namely numerical computation, has been added which complements the previous two. The science of fluid dynamics is now extended and applied by using all three dimensions in concert By it's very nature, any hypersonic Jlowfield analysis before the advent of high speed digital computers had to be in the dimension of pure Theory. This was the only option for the analysis of hypersonic flow during the early development of the discipline"1. The computational simulation is basically the implementation of the "pure theory" in software (e.g. a Flight Simulator). What should be noted is that the growth in computer power radically changes the approach used in many different fields. In reviewing the literature of the growing field of flight tests like DART, it becomes clear that if the natural evolution of the "pure theory" is, within the new technology, the computational simulation, then the natural evolution for the physical simulation as moment of validation of data is the flight test. Together with the new technologies it is getting easier to realize these possibilities, even for not very large institutions. An example is the research in Atmospheric re-entry using high altitude balloons conducted by the National University of Argentina (Pablo De Leon et al.). This becomes more true and more essential the more extreme the environment of flight to be studied becomes. Avant-garde Aerodynamicists think that in a near feature a "Virtual Tunnel" could be built (NASA History of Windtunnels). What they really mean is that aerodynamic theory has improved considerably and electronic computers have more than kept pace so that the Computational Fluid Dynamics (CFD) is much more accurate. Not only can simple aircraft components be studied in depth without recourse to the wind tunnel but, in some situations, complete vehicle configurations. Real wind tunnels, of course, will be called on for research, validation of calculations, and performance assessment where theory and CFD falters. Engineers assert that in absence of large separated-flow and transsonic interactions which occur between shocks and boundary layers, aircraft airloads can be predicted using CFD with nearly the same accuracy as that can be measured in the wind tunnel. Thus the "Test Triangle" will become an iteration between Simulation and Flight test. During this initial part of the stability analysis of DART an assumption of such a philosophy was made. A Re-entry CFD toolbox (R-CFD) was implemented in a Matlab/Simulink environment and was connected to the Flight Simulator in order to compute in real time the aerodynamic coefficients during a re-entry simulation. The Aerodynamic Coefficients of the body were calculated from its geometry and the characteristics of the flow were generated by the flight simulator.

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Re-entry Computational Fluid Dynamics (R-CFD) toolboxes One crucial factor among the several fields of a preliminary design of an aircraft is the aerodynamics of the vehicle body. The estimation and measurement of the Aerodynamic characteristics of the vehicle give its representation in the flight simulator. This simulation gives an essential contribution to a complete understanding of the dynamic behavior of the vehicle. The simulation, in the case of the Aerodynamic analysis, is performed with CFD. Computational Aerodynamics is a relatively young field of study. These kinds of calculations are rather complex and require a large amount of computer power. However, another necessity during the preliminary phase is to have a relatively simple way of evaluation for different solutions of the design. This is necessary so that a quick and iterative process will lead to a final design that meets all the requirements and objectives. The aerodynamic regimes implemented in the RCFD toolbox are: hypersonic rarefied (also known as free molecular flow), hypersonic transitional, and hypersonic continuum, all with the influence of viscosity.

Hypersonic Rarefied Flow Flow regimes differ depending on different levels of rarefaction of the atmosphere, which can be represented by the mean free path between molecules. Because of this definition connected with a length, in order to give a relative importance to different flow regimes a normalization is needed. The Knudsen number Kn represents this normalization and it is the ratio between free path and a characteristic length of the vehicle. For the free molecular flow the kinematics model of interaction particle-surface postulated by Maxwell is used. An inclination method derived from this theory9 was implemented. The following equations give the values of the pressure and friction coefficients:

Cp=2(2-(7N)sin20 C f = 2(7r cos 0 sin 0 where aN is the normal momentum accommodation and aT the tangential momentum accommodation:

a

T

»-j-ir*° —~ _Pi~P •* /

r

w

_*i-*r

i

From the pressure and friction coefficients once projected on the body frame the axial coefficient CA, the normal coefficient CN and the moment coefficient CM can be calculated.

M Force and moment have then to be integrated on the surface of the body in order to obtain the value of the total force and moment acting on the body. This integration for the Free Molecular flow will be made through a panel method explained in the following paragraph.

Hypersonic Transitional Flow The two different flow regimes must be linked so that they cover any specific flight condition. A bridge function is needed. The dominance of one flow on the other is simply dependent on the logarithmic of the Knudsen Number. Defining Mrar the weight of the rarefied flow, l-Mrar will be the weight of the continuum. According to Regan9:

Mrar - a - erf(b - log(Kn) + c) + a where: a -0.5 b -0.30709257318569 c -0.80628539465167 and are obtained statistically.

Hypersonic Continuum Flow For the continuum flow the modified Newtonian law C = C/7max sin 9 was integrated for the biconical blunted cone shape. The modified Newtonian law is a so called surface inclination method. It is a non-linear extension of the result that was obtained for inviscid flow over body in supersonic and subsonic flow where with the well known formula Cp —

2&

simply from the

inclination of the surface was possible to determine the pressure. Cpmax\s the maximum value of the pressure coefficient. This equals exactly the Cp in the stagnation point behind a normal shock wave. The total pressure behind a normal shock wave, at the free-stream Mach number, can be calculated using the exact shock-wave theory1. The force coefficients for a sphere and for a cone need to be determined. For simplicity, an expression for the forces in a body-fixed coordinate system is developed, for instance the axial and the normal force. The axial force and the normal force are given by the following formulas2

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CN =

where CA is the axial force coefficient and CN is the normal force coefficient. / is the unit vector in the x direction, j is the unit vector in the y direction, n is the vector perpendicular to urface S, Rb is the base radius and U^ is the freestream velocity in the x direction. The integral has to be calculated for the three different parts of the blunt biconical shape. For the continuum flow a direct numerical integration was used. Finally an approximation of the base pressure can be calculated using the Gaubeaud formula:

C

""W, U+U

(MJ

(

r+\

-i

where Cpb is the pressure coefficient induced by the base pressure. The base pressure coefficient can be calculated the Cnb and the Cabwith:

For the CM the moment around the sphere-cone apex was calculated through the value of the axial and normal component of the forces. - C +C C ^ Msphere ~ ^ MAsphere ^ ^ MN sphere

c^Mtcone =^MAlcone r ^+r ^ MNtcone The final value of CM is obtained as the sum of all the contributions for the single part of the multibody.

and in turn these inviscid flow changes feed back as change in the boundary layer. In hypersonic flow there are two kinds of interaction. One is due to the particularly thick boundary layer interaction and is known as "Pressure Interaction". The other interaction is due to the impingement of the strong shock wave on the boundary layer, and is called Shock wave-Boundary Layer interaction. This last phenomenon was not analysed in the present study. Numerical methods for CFD are mostly concerned with the solution of the system of partial differential equations of Navier-Stokes. But as already mentioned this will be a basic piece of work for further improvement of the various regimes, and the main goal is to implement only an as complete as possible toolbox of R-CFD methods in the Flight Simulator GESARED. Furthermore, the Navier Stokes integration requests a lot of computer power and on a state-of-art computer still a lot of run time. One of the goals here is an acceptable run time in a standard desktop computer. Therefore an engineering approach was used. The viscous effect generates two forces acting on the body. One is tangential to the surface of the body and comes from the explained phenomenon of the shear stresses in the boundary layer. The second is normal to the shape and is due to the presence of the boundary layer itself. The undisturbed flow sees a body modified by the presence of the boundary layer. The tangential force was implemented by calculating the friction coefficient with an engineering approximation called "Reference Temperature method". This method is based on the assumption of a dependency of the value of Cf from a reference temperature somewhere inside the boundary layer. Without going deep into the explanation of this method1'11, the formulas implemented for the laminar and turbullent boundary layer are respectively:

0.664

0.0592

Viscous influence in the Hypersonic Continuum Flow The viscosity plays a relevant role in the hypersonic regime. Viscous drag is due to the stress on the aerodynamic surface and in the boundary layer. The decreased momentum in the flowfield results in a corresponding loss of momentum of the aerodynamic system. Some of the physicals aspects involved in the viscous drag loss are the presence of shear layers, turbulent transition and boundary layer separation. An other effect due to the presence of the boundary layer is that the flow sees the body shape modified by the presence of a thick hypersonic boundary layer. In hypersonic theory this phenomenology is often called viscous interaction. This phenomenon consists of the fact that the viscous boundary layer changes the nature of the outer flow,

In both case the T' is a function of Mach and in the undisturbed flow:

Both these formulas are for a flat plate. To apply the above results to cones simply multiply the right hand side of the equation by the Mangier fraction, ^ . In the design of DART for the first shape under investigation ( Figure 7) the friction was determined only on the two conical parts of the vehicle. This choice came from considering the shape of DART where the

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spherical surface of the nose is small enough to be neglected. In the second shape (Figure 2) the above formula for the flat plate was implemented in the toolbox of the panels method as made for the coefficient of the Free molecular flow. As mentioned, in the study of the viscosity an important phenomenon is involved: the transition between laminar and turbulent flow. For the accurate prediction of skin friction the knowledge of the 'transitional Reynolds number' is critical. However no theory exists for accurate prediction of ReT . Any knowledge concerning it comes from experimental

data10. For the transition between the two different regimes, turbulent and laminar, also an engineering approximation is used. In this approximation the transition is related to the value of the Reynolds number 4. The value of the ReT is assumed to be function of the Mach number through:

log(Re r )-6.421exp(1.209xlO- 4 -M t 2641 ) R e 7 ,=£fe

Once the value of Re for transition is known it is easy to determine the distance xt where transition occurs. The main effect of the viscous interaction is the variation of the CP due to the presence of the Boundary Layer. In that case also an engineering method related to experimental data is applied. Within the viscous interaction two other phenomena can be identified, and, as it is common in Aerodynamics, they can be related to a similarity parameter: the Viscous Interaction Parameter

J£ is used to determine the transition between the two different kinds of interaction: "strong" and "weak". A large value of ^corresponds to the strong interaction and vice versa. The value associated with transition is for practical purposes assumed to be 3. Also it is directly related to the value of the pressure through:

respectively in the case of strong and weak interaction. It should be noted that the discussed viscous interaction happen in case of a laminar boundary layer. In case of a turbulent boundary layer the same adjustment would have to be made. However, most 6

viscous interaction theory is based on laminar flow. This is because viscous interaction occurs at large Mach numbers and small Re numbers and this condition promote laminar boundary layer. Hence in this basic study, turbulent interaction will not be considered.

High Temperature Effect Finally in order to complete an overview of all the problems connected with the hypersonic flow regime the high temperature effect is briefly introduced. In classical compressible aerodynamics the flow is considered to be a continuum, monophase, chemically inert, at constant composition and without electric charge. Basically the flow is considered thermodynamically as a system of 3 degrees of freedom. If M^is high enough, viscous dissipation within the boundary layer causes high temperatures, which in turns causes chemical reactions within the boundary layer, and the above statement is no longer applicable. In such a case the classical system of equations for compressible aerodynamics (Eulerian and NavierStokes) are not totally applicable; diffusion of chemical species and energy changes due to chemical reactions must be included. Nevertheless, the application of the classical equations to relatively moderate hypersonic conditions yield useful results. Considering that DART is a small re-entry vehicle, and that its design constraints limit the maximum skin temperature at 1200°C, its flight can be considered a moderate hypersonic flight. Furthermore because this study starts basically from a stability analysis, an other effect of the high temperature has to be underlined. Flight experience with the space shuttle has indicated a much higher pitching moment at hypersonic speeds then predicted. Maus7 argue that this discrepancy is due to the effects of a chemically reacting shock layer. At first glance there is little difference between the calculation for a non-reacting shock layer with y = 1.4, and the reacting shock layer assuming local chemical equilibrium. Hence pressure distributions are always somewhat insensitive to chemical reaction effects. However for a long moment arm this difference in pressure distribution can result in non-negligible variation in the pitching moment. This is not the case for our design considering that the maximum length of DART is around 1.5 meter. Indeed this is an other proof that the high temperature reacting gas effects will not have high influence in the stability analysis of DART . In order to have an idea of the error magnitude it should be noted that the modified Newtonian law itself gives an error estimated to be about 5% of the real value. Consequently the correction of the viscosity effect already generates an output with an error value of less then 5%. It would be useless in this OF 11

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basic study to introduce an effect in which the uncertainty due to the approximation of the method would possibly be bigger than the improvement itself.

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Integration of the coefficients with the panel method All the methods discussed above are local methods which need to be integrated over the shape of the body. Numerical integration is feasible for a relative simple shape, but can be difficult in the study of a complex structure. In order to maintain the toolbox as general as possible, for the Free Molecular integration the panels method is used. Basically the shape was discretized in the way of a classic Finite Element Method, and then the pressure coefficients are applied to all the panels, summed and divided into the different components. The method is structured in the following way. The tool calculates all the necessary shape parameters from a minimum number of 5 input parameters. Using this data the structure is divided in a number of nodes that can be varied according to the accuracy wanted. The structure is divided in "n" layers in the longitudinal direction and then each resulting circle in "m" parts. Also, the x coordinate in the conical part has a fixed step increment that depends on the number n that is chosen. For the spherical part the x coordinate is calculated keeping a constant increment in the angle from the symmetry axis along the sphere in

order to have the same size on the lateral sides of each panel. X(i)

= RN-(l-cos((i-l)-d&))

where d$ is a constant angular step . The y and z coordinate are found through the calculation of the Radius of each circle at the different x coordinates and its projection on the two axes. From geometrical considerations the radii are:

r(i) = Ratt+(x(i)-xatt)-tan& for the spherical part, and

for the two conical parts. Plotting all the panels for the two shapes under study for DART at the moment results in Figure 7 and Figure 2.

R-CFD implemented in the Flight Simulator GESARED The tools can be used in two different ways. In the classic way an aerodynamic database can be built. A matrix of aerodynamic coefficients at different values of Mach and Alpha is built and then interpolated in the flight simulator. An other more challenging way is to calculate the exact value of the coefficient in real time connecting directly the RCFD tools to the Flight Simulator as shown in Figure 4.

Force & Moment Computatuin

M aerodyn

Figure 4 R-CFD implemented in GESARED

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Gravity and Planet shape Force and Moment computation with RCFD on-line

Figure 5 GESARED interface

GESARED is implemented in SIMULINK that is particularly suitable for such an operation. A short description of the software is given. The GESARED block in Figure 5 is the implementation of the equations of motion, in it once gave as input the dynamics of the vehicle the kinematics is derived. The "Force and Moment computation" block calculates the dynamic forces acting on the vehicle using the aerodynamic model generated by the RCFD toolbox. This sublevel of the block is the one shown in Figure 4. In it can be seen the block of the R-CFD toolbox and the block of the forces and moments computation. The third Block in GESARED is the one that simulates the planet of the Re-entry in this case of course the Earth was modelled.This real time option is highly time consuming, about 10 times (or more) longer, therefore can be executed after a preliminary analysis in order to obtain more accurate data, especially in the critical part of the bridge of different regimes data table. In this case of implementation more parameters can be used in the calculation of the coefficient, where it would be complex to implement and interpolate a table of more then two dimensions. An other aspect of the importance of this multidisciplinary approach in the study of flight dynamics is underlined by the follow sentence of M.V. Cook5: "Modern flight dynamics is concerned not only with dynamics, stability and control of the basic airframe but also with the sometimes complex interaction between aeroplane and flight control

system. Since the flight control system comprises motion sensors, a central computer, central hardware, a study of this subject becomes a multidisciplinary activity". What came out is that future on-board system would be able to use real time CFD in order to achieve better performances.

Validation The validation of the toolbox was executed in several ways. First the output of the value of a single aerodynamic coefficient over a flat plate is calculated and compared with literature data. Most important validation was obtained from wind tunnel data from the ESA and ASTRIUM mission IRDT (Inflatable Re-entry and Descent Technology) a 45° blunted cone. Figure 6 and Figure 7 shows this last comparison. The error in the axial coefficient and normal coefficient is increasing with the angle of attack. Particularly good results are obtained with the Ca in the continuum and rarefied flow where the two curves match up to an angle of attack of around 50 degrees for the continuum and for all values for the rarefied flow regime. In the case of Cn in continuum flow the error increases significantly beyond a value of alpha of about 20 degrees and up to 35 degrees in the rarefied.

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Conclusion This work was preliminary finalized to the stability analysis of DART therefore approximated methods were implemented. However, it can be a preliminary work for further study on application of CFD tools in flight dynamics opening new challenging field of research and new concept of GNC. This tool easily allows the designer to tradeoff and analyze different shapes within the flight simulator GESARED. Currently under investigation at the Control and Simulation Division of the Faculty of Aerospace Engineering of the Technical University of Delft are 4 different kinds of re-entry vehicles. Often during these preliminary studies there is a lack of data. Such a toolbox will be very useful also for future work to complete missing data in the field of aerodynamics. Further validation of the simulation software shall be achieved with comparison to flight test data from the actual DART flight. In light of this, as already mentioned, DART is a flight test itself for Delft Aerospace in order to acquire knowledge on reentry hypersonic aerodynamics and on validation of different codes developed to the faculty (e.g. the present work in R-CFD and the flight simulator GESARED).

Rockets, vol. 18, no. 1, January-February 1981, pp64-

References ^nderson, J.D. Jr , Hypersonic and high temperature gas dynamics, AIAA 2000 2 Bertin, J.J. , Hypersonic aerothermodynamics AIAA, 1994 3 Buursink J., Van Baten T.J. et al. "DART - The Delft Aerospace Re-entry Test demonstrator" IAFOO-V.4.07; IAF, 2000 4 Bowcutt, K. G., Anderson J.D. , and Capriotti D.: "Viscous Optimized Hypersonic Waveriders", AIAA Paper no. 87-1257, 1986 5 Cook, M. V. "Flight Dynamic Principles" Arnold, London, 1997 6 Costa, R.R. , et al , "Atmospheric Re-entry Modeling and Simulation: Application to a Lifting Body Re-Entry Vehicle", AIAA Paper 2000-4086 7 Maus, J. R., et al. , Hypersonic Mach number and Real gas Effect on Space Shuttle Orbiter Aerodynamics , Journal of Spacecraft and Rockets, vol. 21, nO.2, March-April 1984, pp 136-141 8 Pablo De Leon et al. Research in Atmospheric Re-entry Vehicles Using High Attitude Balloons IAFOO-V.2.08 ,2000 9 Regan, Frank J. , Dynamics of atmospheric reentry, AIAA 1994 10 Reshotko, Eli : "Boundary Layer Stability and Transition" in Annual Review of fluid Mechanics, vol. 8, 1976, pp 311-349 H Zoby, E.V., J.N. Moss, and K. Sutton: "Approximate Convective-Heating Equations for Hypersonic Flows" Journal of Spacecraft and 9 OF 11 AMERICAN INSTITUTE OF AERONAUTIC AND ASTRONAUTIC PAPER 2001-4314

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—Ca windtunnel -CaCFD

CO

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O

-100

-50

0

100

50

a 0.6 -,

-CnCFD

i

• Cn windtunnel |

-100

80

-80

100

-0.5

a Figure 6 Comparison between R- CFD toolbox and wind tunnel data of the axial coefficient CA and the normal coefficient CN in continuum flow for the IRDT vehicle

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2.5 -,

-100

-80

-60

-40

-20

1.4

1.2

—•— Cn windtunnelj --^--CnCFD

0.6 -

0.4 -

0.2

10

-0.2

20

30

40

50

60

70

80

90

100

J

a Figure 7 Comparison between R-CFD toolbox and wind tunnel data of the axial coefficient CA and the normal coefficient CN in free molecular flow for the IRDT vehicle

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