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In the numerical simulation of capillary discharge for the Electrothermal (ET) or ET-chemical (ETC) launch system, one needs to solve the Saha equation to determine the composition of plasma for the transport coefficients. This would cause a relatively longer simulation runtime compared with conventional computational fluid dynamics problems. In this paper, the relatively difficult multidimensional problem of solving the Saha equation is transformed into a one-dimensional problem in the simulation of capillary discharge by constructing an iteration equation about temperature. A coefficient is introduced to ensure the convergence of this iteration equation. In order to improve the computational efficiency, this coefficient is further optimized and the methods of setting the iteration initial value dynamically are introduced. Several simulation tests are conducted to study the performance of these two methods. The results show that the simulation runtime could be significantly reduced with the methods presented in this paper.

Electrothermal (ET) or ET-chemical (ETC) launch technology can achieve a muzzle velocity of more than 2 km/s [

In the ET or ETC launch technology, the typical range of temperature and pressure of the plasma generated by capillary discharge are 10000–40000 K and 100 MPa, respectively [

For the simulation of capillary discharge used in the ET or ETC launch system, a problem of employing the method of Trayner and Głowacki or Zaghloul is that those conversation equations in the mathematical model of capillary discharge cannot give temperature directly whereas it is required as one of the inputting data pieces for solving the Saha equation. In the series researches of Zoler et al. [

This method enables the work of solving the Saha equation to become rather simple in the simulation of capillary discharge; however, one still needs to solve an iteration equation with some numerical algorithm in each cell. Due to the computational amount, the simulation of capillary discharge usually takes longer time compared with conventional computational fluid dynamics problem. Particularly, in the cases that a long tube might be joint with the exit of the capillary to serve as the barrel or cathode or other purposes [

In the simulation research of capillary discharge for the ET or ETC launch technology, a widely used form of the Saha equation is given by [

The method of solving the Saha equation individually listed below is from the work of Trayner and Głowacki and Zaghloul [

Set

Calculate (

Calculate (

Repeat steps (ii)-(iii) for all element species considered in the simulation case.

Calculate (

Set

As one can find from procedures (i)–(vi),

Set

Solve (

Calculate (

Set

As mentioned by Zaghloul, the above method transforms the relatively difficult multidimensional problem into a one-dimensional problem about

Equations (

Set

Solve (

Calculate (

Calculate (

Set

By the above method, the relatively difficult multidimensional problem of solving the Saha equation in the simulation of capillary discharge is transformed into a one-dimensional problem about

In the actual simulation of capillary discharge, it would be tedious and unnecessary to set

The basic principle of setting the iterative initial value is to set it as close as possible to the true root so that less iteration would be needed to find it. Besides, since

In the simulation of capillary discharge, in order to achieve a high quality solution or satisfy the Courant-Friedrichs-Lewy (CFL) condition to maintain the stability of the simulation, space step or the time step is usually set to be relatively small value, so that plasma properties from two spatial or temporal adjacent cells might be very close to each other. This actually provides two easy approaches to set the iterative initial value dynamically. As shown in Figure

Schematic diagram of the method of setting iterative initial value.

Several numerical tests are conducted to study the performance of methods presented in this paper. For the accuracy of the present methods, two tests are conducted. The first one is for procedures (a)–(d). In this test, the Saha equation or (

By this terminal criteria, 5 significant digits could be achieved and this is found to be sufficiently accurate for the present research. Then, with the solution, the conductivity is calculated and compared with the results of Kim [

Calculated conductivity for the polyethylene plasma as a function of temperature and pressure. Solid line represents the results from present calculations and symbols represent the results from [

The second test is for procedures (A)–(D). In this test, the capillary discharge used in the ET or ETC launch system is numerically simulated with the experimental data from shot 8 of series experiments conducted by Powell and Zielinski [

Inputting current profile and the geometry of the capillary discharge [

Simulated plasma flow properties along the axial direction at 500

Based on the above simulation, the efficiency of present methods is studied. In Figure

Simulation runtime with different

In Figure

Spatial-temporal distribution of the loop count of procedures (i)–(vi) with (a) “Initial-S” and

A similarity in Figures

Spatial-temporal distribution of

As mentioned above, the exact value of

A method based on the work of Trayner and Głowacki and Zaghloul discharge is introduced to solve the Saha equation in the simulation of capillary discharge used in the ET or ETC launch system. By modifying the specific energy equation, the multidimensional problem of solving the Saha equation is reduced into a one-dimensional iteration equation about the temperature. A coefficient

The authors declare that there are no conflicts of interest regarding the publication of this paper.