Arabian Journal for Science and Engineering

, Volume 44, Issue 2, pp 1185–1203 | Cite as

Stagnation Temperature Effect on the Supersonic Flow Around Pointed Airfoils with Application for Air

  • Rahima TakhnouniEmail author
  • Toufik Yahiaoui
  • Abderrazak Allali
Research Article - Mechanical Engineering


The aim of this work is to develop a new numerical calculation program to determine the effect of the stagnation temperature on the calculation of the supersonic flow around a pointed airfoils using the equations for oblique shock wave and the Prandtl–Meyer expansion, under the model at high temperature, calorically imperfect and thermally perfect gas, lower than the dissociation threshold of the molecules. The specific heat at constant pressure does not remain constant and varies with the temperature. The new model allows making corrections to the perfect gas model designed for low stagnation temperature, low Mach number, low incidence angle and low airfoil thickness. The stagnation temperature is an important parameter in our model. The airfoil should be pointed at the leading edge to allow an attached shock solution to be seen. The airfoil is discretized into several panels on the extrados and the intrados, placed one adjacent to the other. The distribution of the flow on the panel in question gives a compression or an expansion according to the deviation of the flow with respect to the old adjacent panel. The program determines all the aerodynamic characteristics of the flow and in particular the aerodynamic coefficients. The calculation accuracy depends on the number of panels considered on the airfoil. The application is made for high values of stagnation temperature, Mach number and airfoil thickness. A comparison between our high temperature model and the perfect gas model is presented, in order to determine an application limit of the latter. The application is for air.


Supersonic flow Pointed airfoil Oblique shock High temperature Aerodynamic coefficients Prandtl–Meyer function Calorically imperfect gas Thermally perfect gas Specific heat at constant pressure Error of computation 

List of Symbols

\(\theta \)

Deviation angle of an airfoil segment

\(\psi \)

Flow angle deviation


Mach number

\(\beta \)

Shock wave deviation

\(\mu \)

Mach angle

\(\gamma \)

Specific heats ratio


Thermodynamic constant of air


Specific heat at constant pressure

\(\nu \)

Prandtl–Meyer function

\(\alpha \)

Angle of incidence of the airfoil


Drag force


Lift force


Pitching moment


Dynamic pressure at upstream infinity


Reference surface





\(\rho \)



Airfoil chord


Pitching moment coefficient


Drag coefficient


Lift coefficient


Maximum thickness of the airfoil

x, y

Position of the point on the airfoil

\(\varepsilon \)

Error of computation


Equation of the airfoil extrados


Equation of the airfoil Intrados


Nodes number on the extrados


Nodes number on the intrados


Total number of node on the airfoil


Perfect gas


High temperature



\(\Delta S\)

Total variation of entropy





Upstream condition


Upstream state at the panel under consideration


Considered panel






Point for the calculation of the pitching moment


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The authors acknowledges Khaoula, AbdelGhani Amine, Ritadj and Assil Yahiaoui and Mouza Ouahiba for Granting time to prepare this manuscript.


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Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  • Rahima Takhnouni
    • 1
    Email author
  • Toufik Yahiaoui
    • 2
  • Abderrazak Allali
    • 1
  1. 1.Aircraft Laboratory, Department of Mechanical Engineering, Faculty of TechnologyUniversity of Blida 1BlidaAlgeria
  2. 2.Institute of Aeronautics and Space StudiesUniversity of Blida 1BlidaAlgeria

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