Comparison of explicit and implicit difference schemes of calculations for chemical non-equilibrium processes in nozzles

Fluid, gas and plasma mechanics


Аuthors

Krioukov V. G.*, Abdullin A. L.**, Demin A. V.***, Safiullin I. I.****

Kazan National Research Technical University named after A.N. Tupolev, KNRTU-KAI, 10, Karl Marks str., Kazan, 420111, Russia

*e-mail: vkrioukov@mail.ru
**e-mail: ala2000@mail.ru
***e-mail: alexei_demin@mail.ru
****e-mail: saf-iskander@rambler.ru

Abstract

The calculation of chemically non-equilibrium flows in the nozzles of rocket engines is normally performed using numerical implicit schemes (due to the stiffness of chemical kinetics equations). But the progress in developing of the stable explicit methods creates the possibility to use these simple methods instead of the implicit schemes. In this study, we introduce a calculating method of potential number of integration steps for explicit schemes, and their effectiveness is evaluated. The method includes:

-calculation of a chemically non-equilibrium flow along the nozzle length using the implicit scheme;

-parallel determination of the Jacobi matrix eigenvalues;

-calculation of the number of potential steps for the explicit integration scheme based on the stability bound.

The calculation of flows was carried out within the framework of the inverse nozzle problem, using the implicitly differential scheme of Pirumov U. G. Numerical research was carried out for the combustion products:

- liquid propellants (O2 + kerosene; N2O4 + C2H8N2) for Laval nozzles at: excess oxidant ratio; pressure Poc = 20…100 atm; minimum radius rm = 0.006…0.06 m and geometric expansion fa = 53.0;

- solid propellants (metalized fuel - C10.8760H46.546O25.806AL9.665CL1.517N6.781; nitrocellulose fuel - C23.498H30.259O34.190N10.011) for Laval nozzles at pressure Poc = 20…70 atm; minimum radius rm = 0.005…0.05 m and geometric expansion fa = 33.9.

The number of steps of the explicit scheme (K1) was calculated for the Runge-Kutta method of the 4th order. For the reactive media of liquid propellants, a huge number of potential steps (К1 ≈ 109) were obtained at high Poc and rm values. However, with a decrease of the Poc and rm parameters, K1 also decreased (to about К1 ≈ 107). In the subsonic part of the nozzle, the K1 values were approximately 10 times higher than in the supersonic part. For the reactive media of solid propellants, the results show the same trend, but at a lower level of K1 values, especially in the case of nitrocellulose fuel, when max (K1) ≈ 106.

Keywords:

chemical non-equilibrium flows, engine nozzles, mathematical model, eigenvalues

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