A Semi-empirical Model to Predict the Attached Axisymmetric Shock Shape

Document Type : Original Article


Aerospace Division, Department of Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran


In this work, a simple semi-empirical model is proposed, based on Response Surface Model, RSM, to determine the shape of an attached oblique shock wave emanating from a pointed axisymmetric nose at zero angle of attack. Extensive supersonic visualization images have been compiled from various nose shapes at different Mach numbers, along with some others performed by the author for the present paper. The method is based on the relationship between the body shape and the shock shape. The body shape and the free stream Mach number determine the shape of the oblique shock standing ahead. From the statistical data bank containing the visualization tests and employing the RSM, an analytic relationship has been established between the body and the shock shape. From this relationship, knowing the body shape and the Mach number, one can simply determine the shock shape. The visualization tests performed by the author for some other cases have approved the accuracy of the proposed relationship. However, the approach is restricted to attached shocks emanating from sharp noses at zero angle of attack. Despite the limitations, this relationship can effectively be used in model scale determination for wind tunnel tests to prevent shock reflection from the walls that could lead to erroneous results.


1.     Martínez-Ruiz, D., Huete, C., Sánchez, A.L., and Williams, F.A., “Interaction of Oblique Shocks and Laminar Shear Layers”, AIAA Journal, Vol. 56, (2018), pp. 1023-1030. DOI: 10.2514/1.J056302.
2.     Mason, F. and Kumar, R., “Study of Shock Wave Boundary Layer Interactions on an Axisymmetric Body”, AIAA 2019-0342, AIAA Scitech Forum, Shock Boundary Layer Interaction Session, (2019), CA, USA
3.     Farahani, M. and Jaberi, A., “Experimental Investigation of Shock Waves Formation and Development Process in Transonic Flow”, Scientia Iranica, Transaction B, Vol. 24, No. 5 (2017), 2457-2465.DOI: 10.24200/sci.2017.4309.
4.     Kulkarni, M.D., “Shape Sensitivity for High-speed Flows with Shocks”, AIAA 2020-0888, AIAA Scitech Forum, Aerodynamic Shape Optimization Session, (2020), FL, USA
5.     Whitham, G.B., “The Flow Pattern of a Supersonic Projectile”, Communications on Pure and Applied Mathematics, Vol. 5, No. 3, (1920), 301-348.DOI: 10.1002/cpa.3160050305.
6.     Love, E.S. and Long, R.H., “A Rapid Method for predicting Attached-Shock Shape”, NACA TN-4167, 1957.
7.     Martel, J.D., and Jolly, B., “Analytical Shock Standoff and Shape Prediction with Validation for Blunt Face Cylinder”, AIAA 2015-0523, AIAA Atmospheric Flight Mechanics Conference, (2015).
8.     Sinclair, J. and Cui, X., “A theoretical approximation of the shock standoff distance for supersonic flows around a circular cylinder”, Physics of Fluids, Vol. 29, (2017), 026102. DOI: 10.1063/1.4975983
9.     Hunt, R.L., and Gamba, M., “Shock Train Unsteadiness Characteristics, Oblique-to-Normal Transition, and Three-Dimensional Leading Shock Structure”, AIAA Journal, Vol. 56, (2018), 1569-1587.DOI: 10.2514/1.J056344
10.   Davari, Ali, R., and Soltani, M.R., “On the Relationship between Unsteady Forces and Shock Angles on a Pitching Airplane Model”, Scientia Iranica, Transaction B, Vol. 17, No. 2, (2010) 102-107.DOI: 10.1063/1.4821520
11.   Jin J., Li G., Wei Z., Dong J., Zhang J. “Calibration of the Versatile Platform and the Supersonic Integrated Section” in CAAA. The Proceedings of the 2018 Asia-Pacific International Symposium on Aerospace Technology (2019), Springer, Singapore, Vol. 459. 915-929, DOI: 10.1007/978-981-13-3305-7_72
12.   Chernyshev, S.L., Ivanov, A.I., Streltsov, E.V., And Volkova, A.O., “Numerical and Experimental Research of New Methods For Wall Interference Reduction In Wind Tunnels of Transonic and Low Supersonic Velocities”, Proceeding of the 7th European Conference on Computational Fluid Dynamics, ECFD 7, (2018), Glasgow, UK
13.   Martínez-Ruiz, D., Huete, C., Sánchez, A.L., and Williams, F.A., “Interaction of Oblique Shocks and Laminar Shear Layers,” AIAA Journal, Vol. 56, (2018), 1023-1030. DOI: 10.2514/1.J056302
14.   Mason, F. and Kumar, R., “Study of Shock Wave Boundary Layer Interactions on an Axisymmetric Body,” AIAA 2019-0342, AIAA Scitech 2019 Forum, Shock Boundary Layer Interaction Session, (2019), CA, USA, DOI: 10.2514/6.2019-0342
15.   Fenrich, R.W., and Alonso, J., “A Comparison of Response Surface Methods for Reliability Analysis using Directional Simulation”, AIAA 2018-0437, AIAA Non-Deterministic Approaches Conference, (2018). DOI: 10.1016/S0167-4730(03)00022-5
16.   Vasu, A., and Grandhi, R.V., “A Response Surface Model Using the Sorted k-fold Approach”, AIAA 2014-1485, 10th AIAA Multidisciplinary Design Optimization Conference, (2014).DOI: 10.2514/1.J052913
17.   Kucuk, U.C., “Application of Response Surface Methodology to Optimize Aerodynamic Performance of NACA Inlet,” AIAA 2017-4991, 53rd AIAA/SAE/ASEE Joint Propulsion Conference, (2017).
18.   Lawson, J., Design and Analysis of Experiments with R, First Edition, CRC Press, (2015).
20.   Fenrich, R.W., and Alonso, J.J., “A Comparison of Response Surface Methods for Reliability Analysis using Directional Simulation”,  AIAA 2018-0437, 5th AIAA Non-Deterministic Approaches Conference, (2018), FL, USA.DOI: 10.2514/6.2018-0437
21.   Van Dyke, M., An Album of Fluid Motion, The Parabolic Press., (1982). DOI: 10.1002/aic.690280628
22.   D. K. Weimer, C. H. Fletcher, and W. Bleakney, “Transonic Flow in a Shock Tube”, Journal of Applied Physics, Vol. 20, No. 4, (1949), 418-421.DOI: 10.1063/1.1698393
23.   Freeman, N.C., Cash, R.F. and Bedder, D., “An experimental investigation of asymptotic hypersonic flows”, Journal of Fluid Mechanics, Vol. 18, (1964), 379-384.DOI: 10.1017/S0022112064000271
24.   Otsu, N., “A Threshold Selection Method from Gray-Level Histograms,” IEEE Transactions on Systems, Man, and Cybernetics, Vol. 9, No. 1, (1979), 62-66. DOI: 10.1109/TSMC.1979.4310076
25.   Kenett, R.S., Zacks, S., and Amberti, D., Modern Industrial Statistics, Second Edition, John Wiley & Sons Ltd, (2014).DOI: 10.1002/9781118763667
26.           Ferreyra, R.T., “Supersonic Cones at Zero Incidence,” AIAA 2016-4275, 46th AIAA Fluid Dynamics Conference, (2016). DOI: 10.2514/6.2016-4275