Numerical simulation of plasma flows in curved coaxial ducts with longitudinal magnetic field

The results of numerical researches of plasma flows in coaxial plasma accelerators ducts of a various geometry with an external longitudinal magnetic field are presented. The non-steady-state two-dimensional MHD-model of relaxation of transonic super-Alfven flows with acceleration is considered. The paper focuses on the cumulative influence of a duct geometry and longitudinal magnetic field on flow characteristics.


Plasma accelerators. History and applications
Physical processes in plasma are diverse and have a wide range of effective technological applications. The design and research of plasma accelerators, in which electromagnetic energy transforms into kinetic energy of plasma fluxes emerging from them, is the major direction in the area of plasma processes using. The idea of plasma acceleration in crossed electric and magnetic fields belongs to Alexey Morozov [1]. The successful development of these devices was launched in the 1950s in USSR under Lev Artsimovich [2] and in Gersh Budker's laboratory. The development, research and design of different varieties of plasma accelerators over the following decades were mainly conducted by the same Alexey Morozov [3].
On the one hand, these include powerful high-current quasi-stationary plasma accelerators (QSPA) (figure 1), which demonstrated record velocities and energies of plasma fluxes emerging from them [4]. Its thrust is commensurate with rocket engine thrust, and, in terms of its technological applications, these devices can be used as electric jets for air and space flights [5]. They have a number of advantages over liquid jets working on chemical fuel, but require a powerful and compact source of electric power. On another the hand -low-power small stationary plasma thrusters (SPT) (figure 2) [6] with a long lifetime, which actively apply for orbital corrections, stabilization and maneuvers of artificial satellites.
Plasma accelerators scope of application is not limited to cosmic and transport directions. They also can be used as a magnetoplasma compressor, which is used to generate high-temperature plasma and for high-energy plasma fluxes injection into magnetic traps developed at the issue of thermonuclear fusion. Finally, these devices can be used for effective processing and surface properties modification of various materials.

Longitudinal magnetic field
A flow acceleration mechanism in plasma accelerator ducts is based on an interaction between an electric discharge current passing through plasma with own transverse magnetic field generated by an electric current flowing into the plant. The further technological development of the plasma acceleration scheme is the imposition of a longitudinal magnetic field induced by conductors external to the accelerator. The existence of a longitudinal magnetic field complicated the picture of plasma motion in the accelerator duct. Depending on the value of an external magnetic field, in terms of a classification, there are two regimes of plasma flows significantly varying on their characteristics [7]: super-Alfven and sub-Alfven (in relation to Alfven velocity corresponding to a longitudinal magnetic field) in a sufficiently weak and strong longitudinal magnetic field, respectively. In an accelerator duct at the same time there may be areas of super-Alfven and sub-Alfven flows.

Mathematical modelling and its role in plasma processes research
Mathematical modelling of plasma processes and numerical solution of corresponding mathematical problems play an important role in the successful developments and researches of plasma accelerators. This approach allows to determine basic qualitative regularities in properties of these processes, and quantitative results provide an opportunity to reduce costs of expensive experiments.
Plasma flows in plasma accelerator ducts in own transverse magnetic field have been well understood in the previous studies [8]. In [9] it has been demonstrated that, in terms of acceleration effectiveness, ducts with a curved central electrode are preferable to accelerators with a curved external electrode. Plasma flows with an external longitudinal magnetic field are less studied. The first steps in this field lied in the two-dimensional calculations of flows in the duct with the weakly curved central electrode [10] (see also [11]). The paper [12] is focused on the researches of ionization and radiation transport processes in plasma accelerator ducts. The two-dimensional MHD-model of flows in plasma accelerators as in injectors for magnetic traps is considered in [13].

Objective of the work
The objective of the present work is mathematical modelling and numerical simulation of accelerating plasma flows in QSPA-type plasma accelerator ducts of a various geometry formed by markedly curved electrodes with a longitudinal magnetic field.
So the object of the simulation in the present paper is plasma flow in plasma accelerator ducts in the external longitudinal magnetic field.

Formulation of the problem
In the present paper, plasma is considered as a continuous medium consisting of ions and electrons with common macro-parameters. Plasma flux injected into the accelerator is axisymmetric ( 0).     Two-dimensional problems therefore are formulated in the duct section by the plane const   in a cylindrical coordinate system ( , , ) r z  .
Plasma with a specified density 0  , temperature 0 T and pressure 0 p enters the duct through its input The discharge current in the input section of the duct has strictly a radial direction, therefore azimuthal magnetic field at the duct input is set as

Method of calculation
For the numerical solution of the problem, it is convenient to use a dimensionless form of variables. The units of measurement are the dimensional quantities participating in the formulation of the problem. In order to get rid of curved boundaries of the calculation area, new dimensional coordinates ( , ) z y , which transform the duct area to a square (figure 5), have been introduced: 1 2 , (1 ) ( ) ( ) z z r y r z yr z     (1) Figure 5. Transformation of curved calculation area of a duct to a square The apparatus of modelling is based on the numerical solution of magnetohydrodynamics (MHD) equations. Axisymmetric flows of a continuous electrically conductive medium (of neglecting its viscosity and thermal conductivity and on the assumption of its infinite electrical conductivity) are described by the following system of equations [10] (see also [11]) in coordinates (1):   [9]. This is mostly due to the electrode curviness, and the longitudinal magnetic field additionally enhances this effect. The electric current distribution in this duct geometry is characterized by the marked deviation which is counterclockwise from the radial direction ( figure 7b). This deviation intensifies in the longitudinal field. Plasma acceleration near the external electrode is less intensively due to decreasing of an input value of an azimuthal magnetic field (figure 7c). In so doing, the longitudinal magnetic field, as in the previous case 3.1, in general, reduces acceleration properties of the duct.
Trajectories and thin flow tubes formed by them have a tendency to «uncurling» in the output area of the duct (figure 7d). They deviate down from those symmetrical relative to the minimum section. The longitudinal field seeks to restore the mentioned symmetry.

Conclusions
In the present work the non-steady-state two-dimensional MHD-model of relaxation of transonic super-Alfen flows with acceleration in plasma accelerator ducts of a various geometry with an external longitudinal magnetic field has been considered and implemented in calculations.
It has been determined that a longitudinal magnetic field in both of duct configurations leads to the following effects:  It pushes plasma up to an external electrode. In terms of technical applications, this result can be used to reduce an amount of plasma flux artificially injected through an external electrode to solve «anode crisis» problem in QSPA;  It deflects an electric current towards a duct input and turns its lines counterclockwise from the radial axis neutralizing at a specific polarity an influence of Hall effect negative for acceleration;  It slightly reduces acceleration properties of a duct and causes a flux rotation in its transverse direction.