Efficient Prediction of the Temperature History of a Hypersonic Vehicle Throughout the Mission Trajectory with an Aerodynamic Thermal Load Element

  • Gyubin Kim
  • Yeon Cheol Kang
  • Jeongmin Woo
  • Jeong Ho Kim
  • Jin Yeon ChoEmail author
Original Paper


This work introduces an efficient approach for aero-thermo-mechanical analysis to predict the temperature history of a hypersonic vehicle during its full mission trajectory. The approach uses a recently proposed aerodynamic thermal load element, in which the effects of aerodynamic pressure and aerodynamic heating are efficiently considered using local piston theory and the Eckert reference temperature method, respectively. This element is implemented as a user-subroutine in commercial software to handle realistic models for the aero-thermo-mechanical analysis, such as the X-43A. A finite element model of the X-43A is constructed for a benchmark test. Using the model, an aero-thermo-mechanical analysis is carried out while considering the full mission trajectory of the X-43A. The predicted temperature results are compared with recorded flight test data from the X-43A. Additionally, the aero-thermo-mechanical behavior of the hypersonic vehicle is investigated according to various parameters. The investigation confirms that the approach can be efficiently used in the design of hypersonic vehicles with considerable savings in computational cost.


Hypersonic vehicle Aero-thermo-mechanical analysis Piston theory Eckert reference temperature Aerodynamic thermal load element X-43A 

List of Symbols

\( a_{( \bullet )} \)

Speed of sound, \( {\text{m/s}} \)

\( C_{1} \)

Coefficient of Sutherland’s law, \( \text{kg} /\left( {m\,{\text{s}}\sqrt {\text{K}} } \right) \)

\( c_{\text{f}}^{*} \)

Skin friction coefficient at reference temperature

\( c_{p} \)

Specific heat at constant pressure, \( {\text{J/K/kg}} \)

\( {\mathbf{F}}_{I}^{\text{ATL}} \)

Aerodynamic thermal load element for node I

\( h \)

Convective heat transfer coefficient, \( {\text{W/m}}^{ 2} / {\text{K}} \)

\( M_{( \bullet )} \)

Mach number

\( {\mathbf{n}}_{{\mathbf{0}}} \)

The outward normal unit vector of the surface of the initial configuration

\( {\mathbf{n}}_{\text{t}} \)

The outward unit normal vector of the surface of the deformed configuration

\( p_{( \bullet )} \)

Pressure, \( \text{Pa} \)

\( { \Pr } \)

Prandtl number

\( q_{\text{aero}} \)

Aerodynamic heat flux, \( {\text{W/m}}^{ 2} \)

\( R \)

Ideal gas constant, \( {\text{J/K/kg}} \)

\( \text{Re}_{\text{cr}} \)

Critical Reynold’s number

\( \text{Re}^{*} \)

Reynold’s number at reference temperature

\( r \)

Recovery factor

\( S \)

Coefficient of Sutherland’s law, \( {\rm K} \)

\( {\text{St}}^{*} \)

Stanton number at reference temperature

\( T_{\text{aw}} \)

Adiabatic wall temperature, \( {\rm K} \)

\( T_{\text{w}} \)

Wall temperature, \( {\rm K} \)

\( T^{*} \)

Reference temperature, \( {\rm K} \)

\( T_{( \bullet )} \)

Temperature, \( {\rm K} \)

\( {\mathbf{V}}_{\text{L}} \)

Velocity vector of flow, \( {\text{m/s}} \)

\( {\mathbf{V}}_{\text{S}} \)

Velocity vector of structure, \( {\text{m/s}} \)

\( V_{( \bullet )} \)

Velocity, \( {\text{m/s}} \)

\( W \)

Downwash at each point, \( {\text{m/s}} \)

\( x \)

Distance from leading edge to location of interest, \( \text{m} \)

\( \phi_{I} \)

Shape function for node I

\( \gamma \, \)

Specific heat ratio

\( \mu^{*} \)

Coefficient of viscosity at reference temperature, \( {\text{kg/m/s}} \)

\( \rho_{( \bullet )} \)

Density, \( {\text{kg/m}}^{ 3} \)

\( \rho^{*} \)

Density at reference temperature, \( {\text{kg/m}}^{ 3} \)


\( ( \bullet ) \)

Subscripts such as \( {\text{L}} \), \( {\text{LPT}} \), t or \( \infty \)

\( {\text{L}} \)

Steady-state physical quantity at each local position

\( {\text{LPT}} \)

Updated physical quantity through the local piston theory


Total or stagnation property

\( \infty \)

Physical quantity of freestream



This work was conducted at the High-Speed Vehicle Research Center of KAIST with the support of the Defense Acquisition Program Administration and the Agency for Defense Development under Contract UD170018CD.


  1. 1.
    Fisher RD Jr (2015) US officials confirm sixth Chinese hypersonic manoeuvring strike vehicle test. Jane’s defence weekly. Accessed 30 May 2019
  2. 2.
    Eggers T, Longo J, Hoerschgen M, Stamminger A (2005) The hypersonic flight experiment SHEFEX. In: AIAA/CIRA 13th international space planes and hypersonics systems and technologies conference. Capua, Italy, p 3294Google Scholar
  3. 3.
    Reddy KPJ (2007) Hypersonic flight and ground testing activities in India. In: 16th Australasian fluid mechanics conference. Gold Coast, Australia, pp 32–37Google Scholar
  4. 4.
    Sputnik (2018) Russia’s Avangard hypersonic glider warhead enters production-source. Accessed 30 May 2019
  5. 5.
    Majumdar D (2018) We now know how Russia’s New Avangard hypersonic boost-glide weapon will launch. The National Interest. Accessed 30 May 2019
  6. 6.
    Thompson M (1992) At the edge of space: the X-15 flight program. Smithsonian Institution Press, WashingtonGoogle Scholar
  7. 7.
    McClinton C (2006) X-43-Scramjet power breaks the hypersonic barrier: Dryden lectureship in research for 2006. In: 44th AIAA aerospace sciences meeting and exhibit. Reno, Nevada, p 1Google Scholar
  8. 8.
    Leonard C, Amundsen R, Bruce W (2005) Hyper-X hot structures design and comparison with flight data. In: AIAA/CIRA 13th international space planes and hypersonics systems and technologies conference. Capua, Italy, p 3438Google Scholar
  9. 9.
    Amundsen RM, Leonard CP, Bruce WE III (2004) Hyper-X hot structures comparison of thermal analysis and flight data. NASA Langley Research Center. Paper 109-A0016Google Scholar
  10. 10.
    Berry S, Daryabeigi K, Wurster K, Bittner R (2010) Boundary-layer transition on X-43A. J Spacecr Rockets 47(6):922–934CrossRefGoogle Scholar
  11. 11.
    Wang G, Qin M, Hong L (2017) Fluid-thermal-structural coupling simulation method under hypersonic heating environment. In: 21st AIAA international space planes and hypersonics technologies conference. Xiamen, China, p 2376Google Scholar
  12. 12.
    AJ Culler (2010) Coupled fluid-thermal-structural modeling and analysis of hypersonic flight vehicle structures. Doctoral dissertation. The Ohio State UniversityGoogle Scholar
  13. 13.
    NJ Falkiewicz (2012) Reduced-order aerothermoelastic analysis of hypersonic vehicle structures. Doctoral dissertation. The University of MichiganGoogle Scholar
  14. 14.
    R Klock, CE Cesnik (2015) Aerothermoelastic reduced-order model of a hypersonic vehicle. In: AIAA atmospheric flight mechanics conference. Dallas, TX, p 2711Google Scholar
  15. 15.
    Kang YC, Kim GB, Kim JH, Cho JY, Kim HJ (2018) Development of aerodynamic thermal load element for structural design of hypersonic vehicle. J Korean Soc Aeronaut Space Sci 46(11):892–901Google Scholar
  16. 16.
    Zhang WW, Ye ZY, Zhang CA, Liu F (2009) Supersonic flutter analysis based on a local piston theory. AIAA J 47(10):2321–2328CrossRefGoogle Scholar
  17. 17.
    Eckert ERG (1955) Engineering relations for friction and heat transfer to surfaces in high velocity flow. J Aeronaut Sci 22(8):585–587zbMATHGoogle Scholar
  18. 18.
    Ko WL, Gong L (2001) Thermostructural analysis of unconventional wing structures of a hyper-x hypersonic flight research vehicle for the mach 7 mission. National Aeronautics and Space Administration, Dryden Flight Research Center. NASA TP 2001–210398Google Scholar
  19. 19.
    Karlgaard CD, Tartabini PV, Blanchard RC, Kirsch M, Toniolo MD (2006) Hyper-X post-flight trajectory reconstruction. J Spacecr Rockets 43(1):105–115CrossRefGoogle Scholar
  20. 20.
    Bahm C, Baumann E, Martin J, Bose D, Beck R, Strovers B (2005) The X-43A Hyper-X Mach 7 flight 2 guidance, navigation, and control overview and flight test results. In: AIAA/CIRA 13th International space planes and hypersonics systems and technologies conference. Capua, Italy, p 3275Google Scholar
  21. 21.
    Anderson JD Jr (2010) Fundamentals of aerodynamics. McGraw-Hill, New YorkGoogle Scholar
  22. 22.
    Lighthill MJ (1953) Oscillating airfoils at high Mach number. J Aeronaut Sci 20(6):402–406MathSciNetCrossRefGoogle Scholar
  23. 23.
    Anderson JD Jr (2006) Hypersonic and high-temperature gas dynamics. American Institute of Aeronautics and Astronautics, Reston, VirginiaCrossRefGoogle Scholar
  24. 24.
    Arthur PD, Guard FL, Schultz HD (1966) Flat plate turbulent heat transfer at hypervelocities. J Spacecr Rockets 3(10):1549–1551CrossRefGoogle Scholar
  25. 25.
    Harsha P, Keel L, Castrogiovanni A (2005) X-43A vehicle design and manufacture. In: AIAA/CIRA 13th international space planes and hypersonics systems and technologies conference. Capua, Italy.
  26. 26.
    MatWeb L.L.C. (2019) Material property data, MatWeb [Online]. Accessed 30 May 2019
  27. 27.
    Morgan Advanced Materials (2019) Material property of Min-K, Morgan thermal ceramics. Accessed 30 May 2019
  28. 28.
    IMS (2019) Material property of Saffil, insulation machining services. Accessed 30 May 2019

Copyright information

© The Korean Society for Aeronautical & Space Sciences 2019

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringInha UniversityIncheonRepublic of Korea

Personalised recommendations