DOI: 10.17586/2226-1494-2015-15-4-741-747


P. V. Bulat, K. N. Volkov

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Article in Russian

For citation: Bulat P.V., Volkov K.N. Monotonic derivative correction for calculation of supersonic flows with shock waves. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol.15, no. 4, pp. 741–741.

Subject of Research. Numerical solution methods of gas dynamics problems based on exact and approximate solution of Riemann problem are considered. We have developed an approach to the solution of Euler equations describing flows of inviscid compressible gas based on finite volume method and finite difference schemes of various order of accuracy. Godunov scheme, Kolgan scheme, Roe scheme, Harten scheme and Chakravarthy-Osher scheme are used in calculations (order of accuracy of finite difference schemes varies from 1st to 3rd). Comparison of accuracy and efficiency of various finite difference schemes is demonstrated on the calculation example of inviscid compressible gas flow in Laval nozzle in the case of continuous acceleration of flow in the nozzle and in the case of nozzle shock wave presence. Conclusions about accuracy of various finite difference schemes and time required for calculations are made. Main Results. Comparative analysis of difference schemes for Euler equations integration has been carried out. These schemes are based on accurate and approximate solution for the problem of an arbitrary discontinuity breakdown. Calculation results show that monotonic derivative correction provides numerical solution uniformity in the breakdown neighbourhood. From the one hand, it prevents formation of new points of extremum, providing the monotonicity property, but from the other hand, causes smoothing of existing minimums and maximums and accuracy loss. Practical Relevance. Developed numerical calculation method gives the possibility to perform high accuracy calculations of flows with strong non-stationary shock and detonation waves. At the same time, there are no non-physical solution oscillations on the shock wave front.

Keywords: computational fluid dynamics, finite volume method, Riemann problem, difference scheme, nozzle.

Acknowledgements. The research has been carried out under financial support by the Ministry of Education and Science of the Russian Federation (agreement No. 14.575.21.0057).

1. Toro E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics. Berlin, Springer-Verlag, 2009, 724 p. doi: 10.1007/b79761
2. Godunov S.K. A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Sbornik: Mathematics, 1959, vol. 47, no. 8–9, pp. 357–393.
3. Kulikovskii A.G., Pogorelov N.V., Semenov A.Yu. Matematicheskie Voprosy Chislennogo Resheniya Giperbolicheskikh Sistem Uravnenii [Mathematical Problems in the Numerical Solution of Hyperbolic Systems]. Moscow, Fizmatlit Publ., 2001, 608 p.
4. Roe P.L. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 1981, vol. 43, no. 2, pp. 357–372. doi: 10.1016/0021-9991(81)90128-5
5. Osher S. Riemann solvers, the entropy condition, and difference approximations. SIAM Journal on Numerical Analysis, 1984, vol. 21, no. 2, pp. 217–235.
6. Osher S., Chakravarthy S. High resolution schemes and the entropy condition. SIAM Journal on Numerical Analysis, 1984, vol. 21, no. 5, pp. 955–984.
7. Einfeldt B. On Godunov-type methods for gas dynamics. SIAM Journal on Numerical Analysis, 1988, vol. 25, no. 2, pp. 294–318.
8. Donat R., Marquina A. Capturing shock reflections: an improved flux formula. Journal of Computational Physics, 1996, vol. 125, no. 1, pp. 42–58. doi: 10.1006/jcph.1996.0078
9. Capdeville G. A multi-dimensional HLL-Riemann solver for Euler equations of gas dynamics. Computers and Fluids, 2011, vol. 47, no. 1, pp. 122–143. doi: 10.1016/j.compfluid.2011.03.001
10. Bulat P.V., Volkov K.N., Silnikov M.S., Chernyshev M.V. Analiz raznostnykh skhem, osnovannykh na tochnom i priblizhennom reshenii zadachi Rimana [Analysis of finite-difference schemes based on exact and approximate solution of Riemann problem]. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol. 15, no. 1, pp. 139–148.
11. Bulat P.V., Bulat M.P. Definition of the existence region of the solution of the problem of an arbitrary gasdynamic discontinuity breakdown at interaction of flat Supersonic jets with formation of two outgoing compression shocks. Research Journal of Applied Sciences, Engineering and Technology, 2015, vol. 9, no. 1, pp. 65–70.
12. Yeom G.-S., Chang K.-S. A modified HLLC-type Riemann solver for the compressible six-equation twofluid
model. Computers and Fluids, 2013, vol. 76, no. 10, pp. 86–104. doi: 10.1016/j.compfluid.2013.01.021
13. Su Y.-C. On the compressible Euler dynamics equations in transonic flow. Nonlinear Analysis: Theory, Methods and Applications, 2014, vol. 109, pp. 156–172. doi: 10.1016/
14. Volkov K.N. Raznostnye skhemy rascheta potokov povyshennoi razreshayushchei sposobnosti i ikh primenenie dlya resheniya zadach gazovoi dinamiki [High-resolution difference schemes of flux calculation and their application to solving gas dynamics problems]. Vychislitel'nye Metody i Programmirovanie, 2005, vol. 6, no. 1, pp. 146–167.
15. Volkov K.N. Reshenie nestatsionarnykh zadach mekhaniki zhidkosti i gaza na nestrukturirovannykh setkakh [Solution of time-dependant problems of gas and fluid mechanics on unstructured grids]. Matematicheskoe Modelirovanie, 2006, vol. 18, no. 7, pp. 3–23.
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