Verification and validation of aerodynamic codes using the example of the flow around sharp and blunt cones




The article describes the code–to–code verification of two CFD codes Euler3n and hySol and their validation on the example of the flow around sharp and blunt cones in a wide range of Mach numbers and angles of attack.

supersonic flow, computational fluid dynamics, unstructured grid, Godunov method, verification and validation


Volume 22, issue 4, 2021 year


Верификация и валидация аэродинамических расчетных комплексов на примере задачи обтекания острых и затупленных конусов

В статье описывается перекрестная верификация двух CFD программ Euler3n и hySol и их валидация на примере обтекания острых и затупленных конусов в широком диапа-зоне чисел Маха и углов атаки.

сверхзвуковые течения, вычислительная аэродинамика, неструктурированная сетка, метод Годунова, верификация и валидация


Volume 22, issue 4, 2021 year



1. Roache P.J. Need for Control of Numerical Accuracy, J. Spacecraft and Rockets, 1990, vol. 27, N 2, pp. 98–102.
2. Roy C.J., Smith T.M., Ober C.C. Verification of a Compressible CFD Code using the Method of Manufactured Solutions, AIAA Paper 2002–3110.
3. Guide for the Verification and Validation of Computational Fluid Dynamics Simulations, American Institute of Aeronautics and Astronautics, AIAA–G–077–1998, Reston, VA, 1998
4. Zheleznyakova A. Verification and validation technologies for gas dynamic simulations, Physical-Chemical Kinetics in Gas Dynamics, 2018, vol. 19, iss. 2, http://chemphys.edu.ru/issues/2018-19-2/articles/687/
5. Bulakh B.M. Nonlinear conical flow, Delft University Press, 1985, p. 326.
6. Babenko K.I., Voskresenskii G.P., Lyubimov A.N., Rusanov V.V. Three Dimensional Flow of Ideal Gases around Smooth Bodies. NASA TT F-380, 1968.
7. Petrov K.P. Aerodynamics of Bodies of the Simplest Shapes, Factorial, Moscow, 1998, 432.
8. Krasil’shchikov A.P., Guryashkin L.P. Experimental Studies of Bodies of Revolution in Hypersonic Flows, Fizmatlit, Moscow, 2007.
9. Taylor G.I., Maccoll J.N., The air pressure on cones moving at high speeds, Proc. Roy. Soc. London, Ser., 1933, vol. A139(A338), pp. 278–311.
10. Kopal Z. Tables of supersonic flow around jawing cones. Cambridge, Massachusetts: MIT. Techn. Rep., 1947, N 3.
11. Kopal Z. Tables of supersonic flow around cones of large jaw. Cambridge, Massachusetts:MIT. Techn. Rep., 1949, N 5.
12. Owens R.V. Aerodynamic characteristics of spherically blunted cones at Mach number from 0.5 to 5.0. Washington: NASA TN D–3088, 1965.
13. Sims J.L. Tables for supersonic flow around right circular cones at zero angle of attack. NASA–SP–3004, 1964.
14. Kraiko A.N., Tillyaeva N.I. Axisymmetric-Conical and Locally Conical Flows without Swirling, J. Applied Mech. Tech. Phys., 2014, N 2, 108–126.
15. Artonkin V.G, Leutin P.G., Petrov K.P., Stolyarov E.P. Aerodynamic characteristics of sharp and blunt cones at subsonic and supersonic speeds, Trudi TsAGI, 1972, vol. 1413, p. 91.
16. Bashkin V.A., Egorov I.V., Ivanov D.V., Pafnut’ev V.V. Sharp Sharp circular cone in a supersonic flow of a viscous perfect gas, Uch. Zap. Tsentr. Aerogidrodin. Inst., 2003, vol. 34, N 3–4, pp. 3–16.
17. Kovalenko V.V., Kravtsov A.N., Melnichuk T.Yu. Resistance of conical nose parts in supersonic flow, TsAGI Sci. J., 2011, vol. 42, N 1, pp. 31–36.
18. Surzhikov S.T. Validation of computational code UST3D by the example of experimental aerodynamic data // J. Phys.: Conf. Ser., 2017, 815, 012023.
19. Roy C.J., McWherter–Payne M.A., Oberkampf W.L. Verification and Validation for Laminar Hypersonic Flowfields, Part 1: Verification, AIAA J., 2003, vol. 41, pp. 1934–1943.
20. Godunov S.K., Zabrodin A.V., Ivanov M.Ia., Kraiko A.N., Prokopov G.P. Numerical solution of multidimensional problems of gas dynamics, Moscow, Nauka, 1976, 400 p.
21. Barth T.J., Jespersen D.C. The design and application of upwind schemes on unstructured meshes, AIAA Paper No. 1989–0366, Jun 1989.
22. Kryukov I.A., Ivanov I., Larina E. Software package hySol for the Numerical Simulation of High-speed Flows, Physical-Chemical Kinetics in Gas Dynamics, 2021, vol. 22, iss. 1, http://chemphys.edu.ru/issues/2021-22-1/articles/902/
23. Kryukov I.A., Ivanov I.E., Larina E.V. Verification of the hySol software package for calculating high-speed flows, Materials of the XX Anniversary International Conference on Computational Mechanics and Modern Applied Software Systems (CMMAPS’2017), 24–31 May 2017, Alushta, Moscow MAI, vol. 1, pp. 485–487.
24. Borovikov S.N., Ivanov I.E., Kryukov I.A., Modeling of test ideal gas dynamics problems using tetrahedral meshes, Matematicheskoe modelirovanie, 2006, vol. 18, N 8, pp. 37–48.
25. Liou M.–S., Steffen Jr C.J., A new flux splitting scheme, J. Comp. Phys., 1993, vol. 107, pp. 23–39.
26. Venkatakrishnan V. Convergence to Steady State Solutions of the Euler Equations on Unstructured Grids with Limiters, J. Comp. Phys., 1995, vol. 118, pp. 120–130.
27. Michalak C., Ollivier–Gooch C. Accuracy preserving limiter for the high–order accurate solution of the Euler equations, J. Comp. Phys., 2012, vol. 228, pp. 8693–8711.
28. Weiss J.M., Maruszewski J.P., Smith W.A. Implicit solution of preconditioned Navier–Stokes equations using algebraic multigrid, AIAA J., 1999, vol. 37, pp. 29–36.
29. Shu C.–W., Osher S., Efficient Implementation of Essentially Non–Oscillatory Shock–Capturing Schemes II, J. Comp. Phys., 1989, vol. 83, pp. 32–78.
30. Loitsyanskii L.G. Fluid and Gas Mechanics, Nauka, Moscow, 1973, 736.
31. Neal L., Jr. Aerodynamic Characteristics at a Mach Number of 6.77 of a 9° cone configuration, with and without spherical afterbodies, at angles of attack up to 180° with various degrees of nose blunting, NASA TN D–3312, 1966.
32. A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities // http://gmsh.info
33. Ermakov M., Kryuchkova A. Generation of unstructured tetrahedral meshes for flow past flight vehicles based on open packages, Physical-Chemical Kinetics in Gas Dynamics, 2020, vol. 21, iss. 2, http://chemphys.edu.ru/issues/2020-21-2/articles/897/
34. Ermakov M.K., Kryukov I.A., Supercomputer modeling of flow past hypersonic flight vehicles // J. Phys.: Conference Series, 2017, 815, 012016, 1–5.
35. Makeich G.S., Kharchenko N., Kryukov I. Aerodynamics and flight dynamics simulation of EXPERT re-entry vehicle // Physical-Chemical Kinetics in Gas Dynamics. 2017. V.18, iss. 2. http://chemphys.edu.ru/issues/2017-18-2/articles/738/