Journal of Applied Mathematics and Physics, 2014, 2, 115-123
Published Online April 2014 in SciRes. http://www.scirp.org/journal/jamp
How to cite this paper: Khlopkov, Yu.I., et al. (2014) Computer Modelling of Aerothermodynamic Characteristics for Hy-
personic Vehicles. Journal of Applied Mathematics and Physics, 2, 115-123. http://dx.doi.org/10.4236/jamp.2014.25015
Computer Modelling of Aerothermodynamic
Characteristics for Hypersonic Vehicles
Yuri Ivanovich Khlopkov, Anton Yurievich Khlopkov, Zay Yar Myo Myint
Department of Aeromechanics and Flight Engineering, Moscow Institute of Physics and Technology, Zhukovsky,
Email: email@example.com, firstname.lastname@example.org
Received Dec emb er 2013
The purpose of this work is to describe the suitable methods for aerodynamic characteristics cal-
culation of hypersonic vehicles in free molecular flow and the transitional regimes. Moving of the
hypersonic vehicles at high altitude, it is necessary to know the behavior of its aerodynamic cha-
racteristics for all flow regimes. Nowadays, various engineering approaches have been developed
for modelling of aerodynamics of aircraft vehicle designs at initial state. The engineering method
that described in this paper provides good results for the aerodynamic characteristics of various
geometry designs of hypersonic vehicles in the transitional regime. In this paper present the cal-
culation results of aerodynamic characteristics of various hypersonic vehicles in all range of re-
gimes by using engineering method.
Aerodynamic Characteristics, Computational Aerodynamics, Hypersonic Technology, Rarefied Gas
Dynamics; Engineering Method, Aerodynamics in Transitional Regime
Theoretical studies of hypersonic flows associated with the creation of “Space Shuttle” to transport people and
cargo into Earth orbit began in the last century. Research had been focused mainly on the department of TsAGI
named after N.Y. Zhukovsky (Central Aerohydrodynamic Institute). Practical work on the creation of aerospace
systems had been instructed engineering centre of the experimental design bureau named after A.I. Mikoyan.
Air Force Research Institute has developed an original concept of space system, which efficiently integrated the
ideas of the aircraft, rocket plane and space object in 1960. The project was called “Spiral” and represented as a
complex system. Powerful hypersonic aircraft (weight 52 tons, length 38 m, wingspan 16.5 m), which was dis-
persed to six times the speed of sound (Mach = 6), then at height of 28-30 km from its back, manned orbital
plane 10 ton (8 m long and 7.4 m span) was supposed to start.
The “Spiral” (Figure 1) was a response to the U.S. space program to create an interceptor reconnaissance
bomber, the X-20 “Dyna Soar” (Figure 2). As shown, the implementation of project “Dyna Soar” is not suc-
cessful as a “Spiral”. In the end both projects have been folded, although at different stages of development .
Since 1980 aerospace vehicle programs are developed in many developed countries as the U.S. and the Soviet
Yu. I. Khlopkov et al.
Figure 1. Russian project “Spiral”.
Figure 2. USA project “Dyna Soar”.
Union. For example, in England “HOTOL” (Figure 3), Germany “Zenger” (Figure 4), France “Hermes” (Fig-
ure 5), Japan “Hope” (Figu re 6), China “Shenlong” (Figur e 7) and India “AVATAR” (Figure 8). Some of
them have been folded.
Russia is developing new generation reusable spacecraft programs “Clipper”, “RUS” to deliver crews and
cargo to low Earth orbit and the space station since 2000. The first flight is expected in 2015. United States is
developing a new spacecraft “Orion” to deliver crew and cargo to low Earth orbit and back. First full-size test
flight is in 2014, the first flight to the moon planned in 2020.
The DARPA (Defense Advanced Research Projects Agency) is developed “FALCON” (Force Application
and Launch from CONtinental United States) since 2003. The Falcon Hypersonic Technology Vehicle (HTV-2)
is a multiyear research and development effort to increase the technical knowledge base and advance critical
technologies to make long-duration hypersonic flight a reality. Falcon HTV-2 is an unmanned, rocket-launched,
maneuverable aircraft that glides through the Earth’s atmosphere at incredibly fast speeds—Mach 20.
With the development of space and rocket technologies, it is required the reliable methods on the aerodyna-
mic and aerothermodynamic characteristics modelling of hypersonic vehicles in the whole range of flow regi-
mes, i. e., from the continuum flow regime up to the free-molecular regime. During de-orbiting, the spacecraft
passes through the free molecular, then through the transitional regime and the finalized flight is in the contin-
As we have known that for flight in the upper atmosphere, where it is necessary to take into account the mole-
cular structure of a gas, kinematics models are applied, in particular, the Boltzmann equation and corresponding
numerical methods of simulation. In the extreme case of free-molecular flow, the integral of collisions in the
Boltzmann equation becomes zero, and its general solution is a boundary function of distribution, which remains
constant along the paths of particles . While aircraft are moving in a low atmosphere, the problems are re-
duced to the problems that can be solved in the frame of continuum theory or, to be more precise, by application
of the Navier-Stokes equations and Euler equations [2,3]. On the transition interval between the free molecular
and continuum regimes numerical methods of solving the Boltzmann equation and its model equations are being
used with success .
Yu. I. Khlopkov et al.
Figure 3. HOTOL.
Figure 4. Zenger.
Figure 5. Hermes.
Figure 6. Hope.
Figure 7. Shenlong.
Figure 8. AV A T AR .
Yu. I. Khlopkov et al.
To correctly simulate hypersonic flows, the flows must be understood and modeled correctly and this is more
true than in the numerical simulation of hypersonic flows. Hypersonic must be dominated by an increased un-
derstanding of fluid mechanics reality and an appreciation between reality and the modeling of that reality .
The benefits of numerical simulation for flight vehicle design are enormous: much improved aerodynamic shape
definition and optimization, provision of accurate and reliable aerodynamic data and highly accurate determina-
tion of thermal and mechanical load . Multi-parametric calculations can be performed only by using an ap-
proximation engineering approach. Computer modeling allows to quickly analyze the aerodynamic characteris-
tics of hypersonic vehicles by using theoretical and experimental research in aerodynamics of hypersonic flows.
Nowadays, the basic quantitative tool for study of hypersonic rarefied flows is direct simulation Monte Carlo
method (DSMC)  . DSMC method required large amount of computer memory and unreasonable expen-
sive at the initial stage of vehicle design and trajectory analysis. The solution for this problem is the approximate
local engineering methods [8-11]. The Monte Carlo method remains the most reliable approach, together with
the local engineering methods that provide good results for the global aerodynamic coefficients. The early work
of  indicated that local engineering methods could have significant effect on aerodynamic characteristics of
various hypersonic vehicles. It is natural to create engineering methods, justified by cumulative data of experi-
mental, theoretical and numerical results, enabling the prediction of aerodynamics characteristics of complex
bodies in the transitional regime .
The purpose of this work is to calculate aerodynamic characteristics of perspective space vehicle “Clipper”
and hypersonic technology vehicle “Falcon HTV-2” by using engineering method. This engineering method is
suitable to calculate with taking into account the various Reynolds number, and provide good results for various
hypersonic vehicle designs.
2. Calculation Methods
2.1. Method for Hypersonic Aerodynamics in the Free Molecular Flow Regime
The problem of determination of the aerodynamic characteristics in free molecular flow is set in a usual way. In
the case of hypersonic flow, the values of pressure p and tangential τ forces and heat flux q for the element of a
surface look as below
2σ2sin σπ sin
where στ—accommodation coefficient, γ—specific heat ratio, Tw, T∞—surface temperature and flow tempera-
ture respectively . The experiment provides the most reliable results for the formulation of a connection between
the flow as like the oncoming particles characteristics and the reflected particles through the coefficients of ac-
commodation. The results of aerodynamic characteristics of semi-sphere, cone with the various semi-angles by a
vertex, blunted semi-cones, and blunted semi-cones with wings are presented in .
2.2. Method for Hypersonic Aerodynamics in the Transitional Regime
In this work we use the expressions for the elementary pressure forces and friction forces are applied in the form
described in [3,11].
where, coefficients p0, p1, τ0 (coefficients of the flow regime) are dependent on the Reynolds number Re0 =
ρ∞V∞L/µ0, in which the viscosity coefficient µ0 is calculated at stagnation temperature T0. Except Reynolds
number the most important parameter is the temperature factor Tw/T0, where T0, Tw are the stagnation tempera-
sin θ +sinθpp p=
τ = τ sinθ cosθ
Yu. I. Khlopkov et al.
ture and surface temperature respectively.
The dependency of the coefficients of the regime in the hypersonic case must ensure the transition to the
free -molecular values at Re0→0, and to the values corresponding to the Newton theory, methods of thin tangent
wedges and cones, at Re0→∞. On the basis of the analysis of computational and experimental data, the empirical
formulas are proposed [15-17]
whe r e h is a relative lateral dimension of the apparatus, which is equal to the ratio of its height to its length.
The technique proved to be good for the calculation of hypersonic flow of convex, not very thin, and spatial
bodies. The calculation fully reflects a qualitative behavior of drag force coefficient CD as a function of the me-
dium rarefaction within the whole range of the angles of attack, and provides a quantitative agreement with ex-
periment and calculation through the Boltzmann equation with an accuracy of 5%. On the accuracy of the rela-
tion of the locality method can be said that they are applied with the smallest error in the case of the bodies that
are close to being spherical, and are not applied in the case of very thin bodies, when the condition is M∞ sin θ >>
This locality method for calculation of aerodynamic characteristics of the bodies in the hypersonic flow of ra-
refied gas in the transitional regime gives a good result for CD for a wide range of bodies, and a qualitatively
right result for lift force coefficient CL. In this case, it is necessary to involve more complete models that take
into account the presence of the boundary layer . In early papers [13,15-19] described the results of aerody-
namic characteristics of various hypersonic vehicles by using this method.
The results of the calculation of the coefficients of drag force CD, lift force CL, pitching moments MZ with value
of angle of attack α from 0 to 90 deg for Russian perspective space vehicle “Clipper, TsAGI model” (Figure 9)
 and USA perspective hypersonic technology vehicle “Falcon HTV-2” (Figure 10) are presented.
The calculation has been carried out through the method described on the above section within the range of
angles of attack α from 0 deg up to 90 deg with a step of 5 deg. The parameters of the problem are the following:
ratio of heat capacities
= 1.4; temperature factor Tw/T0 = 0.01; Reynolds number Rе0 = 0, 10, 100, 10000; ve-
locity ratio = 15.
The dependencies of CD(
) and MZ(
) are presented in Figures 11-13. It can be seen from these results
that when the Reynolds number increased, the drag coefficients CD of vehicle diminished which can be ex-
Figure 9. Geometry view of
space vehicle “Clipper”.
[(2 α)] /
τ3.7 2[6.88exp(0.00720.000016)]R RR
Yu. I. Khlopkov et al.
Figure 10. Geometry view of hyper-
sonic vehicle “Falcon HTV-2”.
Figure 11. Drag coefficients CD for space vehicle “Clipper”.
Figure 12. Lift coefficients CL for space vehicle “Clipper”.
Yu. I. Khlopkov et al.
plained by the decrease of normal and tangent stresses. At high Reynolds number Re0 ≥ 106, characteristics al-
most not changed. The dependency CL(
) is increased at high Reynolds number which can be explained by the
decrease of normal and tangent stresses.
The values of MZ are quite sensitive to the variation of Re0. MZ changes its sign less than zero at Re0 ~ 102. At
Re0 ~ 104, the value of Mz = −0.03 at the angle of attack is reached at α ≈ 40 deg. Results by using local engi-
neering method are compared with the results obtained by DSMC and Newtonian methods.
The dependencies of CD(α), CL(α) and MZ(α) of hypersonic vehicle “Falcon HTV-2” are presented in Figures
14-16. In Figure 14, it can be seen with the increasing of Reynolds number, the drag coefficients CD of vehicle
decreased. It can be explained that by the decrease of normal and tangent stresses. Drag coefficients CD of Fal-
con more than Clipper. The dependency CL(α) is increased, and the value is reached to 0.54 at Re0 ~ 104. The
values of MZ are quite sensitive to the variation of Re0, changes its sign at α ~ 5 deg.
Figure 13. Pitching moment coefficients MZ for space
Figure 14. Drag coefficients CD for hypersonic vehicle
Yu. I. Khlopkov et al.
Figure 15. Lift coefficients CL for hypersonic vehicle “Falcon
Figure 16. Pitching moment coefficients MZ for hypersonic
vehicle “Falcon HTV-2”.
Many analytical and numerical methods to estimate the aerothermodynamics of hypersonic vehicles were ap-
peared with the development of space launch technologies. This paper presents different methods to calculate
aerodynamic characteristics of various perspective hypersonic vehicles in rarefied gas flows. With the use of this
engineering method, it had been calculated aerodynamic characteristics for various hypersonic vehicle design in
rarefied gas flow. Results are compared with the DSMC and traditional Newtonian method. Methods which de-
scribed above give good results and suitable to calculate aerodynamics for various hypersonic vehicle designs.
The reported study was partially supported by RFBR (research project No. 11-07-00300-а).
Yu. I. Khlopkov et al.
The support by RFBR (research project No. 11-07-00300-а) is cordially appreciated by the authors.
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