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

Аuthors

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

Kazan National Research Technical University named after A.N. Tupolev, 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 P_oc = 20…100 atm; minimum radius r_m = 0.006…0.06 m and geometric expansion f_a = 53.0;

- solid propellants (metalized fuel - C_10.8760H_46.546O_25.806AL_9.665CL_1.517N_6.781; nitrocellulose fuel - C_23.498H_30.259O_34.190N_10.011) for Laval nozzles at pressure P_oc = 20…70 atm; minimum radius r_m = 0.005…0.05 m and geometric expansion f_a = 33.9.

The number of steps of the explicit scheme (K₁) was calculated for the Runge-Kutta method of the 4^th order. For the reactive media of liquid propellants, a huge number of potential steps (К₁ ≈ 10⁹) were obtained at high P_oc and r_m values. However, with a decrease of the P_oc and r_m 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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