Evaluation of the Gas-Discharge Plasma Characteristics in the Absence and Presence of a Magnetic Field in a High-Speed Flow Based on Experimental Data and Predictive Modeling

Abstract

The main objective of this study is to obtain the average parameters of gas-discharge plasma when controlling the steady position of the bow shock wave (BSW) using the combined action of a gas discharge initiated by a current from an external source and a magnetic field near the frontal surface of the model. The studies were carried out using both experimental and numerical methods in xenon and air. A comparison of the numerical and experimental dependences of the relative distance of the steady BSW from the model on the discharge power showed good agreement. Based on the conducted flow modeling, taking into account the dependence of the adiabatic index on the degree of ionization and the degree of nonequilibrium, and using the theory of Burm et al., gas-discharge plasma characteristics were obtained, such as the degree of ionization and the degree of nonequilibrium, the electron density and the electron temperature in the absence and presence of a magnetic field. By this way an integrated experimental–computational system was formed in which the measured characteristics of the discharge and BSW, as well as the numerically obtained averaged plasma parameters in the impact zone, are combined with the theory of Burm et al. to clarify the thermodynamic state of the medium and determine the corresponding characteristics of the gas-discharge plasma. The obtained results can be used for assessing the characteristics of plasma gas dynamic and magnetohydrodynamic phenomena in high-speed flows; for example, in the development of control systems that take into account the influence of plasma parameters and the electric and magnetic fields.

1. Introduction

Low-temperature plasma magnetohydrodynamics has been developing since the late 1950s, initially in close connection with the direct conversion of high-speed conducting-flow energy. The classical foundations of magnetohydrodynamics (MHD) interaction in weakly ionized gases and plasmas are presented in the monographs [1,2], while conducting low-temperature plasma flows in channels and magnetic fields are discussed in detail in [3,4]. The governing physical mechanism is the Lorentz force acting on charged particles. When an ionized medium moves across a magnetic field, charge separation produces a magnetically induced electromotive force, and a closed external circuit with wall electrodes generates an electric current. In the early stage, this mechanism was mainly considered for MHD generators and expansion channels, where the incoming flow was usually treated as continuous and free of pronounced dissipative structures.

The development of rocket technology and supersonic aviation shifted attention from energy-conversion applications toward active control of supersonic flows. Behind a BSW, high temperatures can produce natural ionization, so the closure of an induced ring current in this region provides a way to exert an electromagnetic force on the flow. Early experiments and studies of blunt-body flows showed that a magnetic field can change the bow-shock position and increase the bow-shock stand-off distance relative to the undisturbed case [5,6,7]. A further group of studies examined MHD control of shock wave structures in air intakes and external supersonic flows. In work on hypersonic inlet configurations, different flight altitudes and regimes, magnetic-field geometries, and methods for increasing air conductivity were considered, including electron-beam ionization [8]. These formulations established the basis for computational and experimental analysis of acceleration, deceleration, and restructuring of weakly ionized supersonic flows.

Advances in computational fluid dynamics made it possible to simulate MHD interaction in shock-containing high-speed flows. The early models coupled the compressible Navier–Stokes equations with Ohm’s law and electromagnetic source terms accounting for the Lorentz force and Joule heating. Under the low-magnetic-Reynolds-number approximation, the flow-induced magnetic field was neglected and the applied field was prescribed in dipole form. Poggie and Gaitonde [9] used second-order central differences with blended second- and fourth-order artificial dissipation and the implicit Beam–Warming method. Otsu et al. [10] employed a finite-volume formulation with the AUSM-DV flux scheme, matrix-free Gauss–Seidel iterations, and point-implicit time integration. Later finite-volume simulations incorporated multispecies thermochemical nonequilibrium, separate energy equations for electrons and heavy particles, and Hall effects [11,12]. These calculations showed that electromagnetic deceleration increases the bow-shock stand-off distance and reduces the near-wall temperature gradient and stagnation-point heat flux. The same flow-control mechanism was subsequently considered for reentry vehicles [10,11,12,13] and the MHD-parachute concept [14,15,16,17] to increase aerodynamic drag and mitigate thermal loading.

The experimental basis of MHD aerodynamics includes plasma-flow studies and direct force measurements. Experiments with argon plasma recorded changes in thermal loading under magnetic-field action [18], while measurements in an expansion channel confirmed that an increase in aerodynamic drag can be registered during MHD interaction. These results are important because they connect the computational picture of Lorentz-force action and induced currents with measurable flow characteristics.

Current research treats MHD interaction as a tool for controlling flow structure, heat transfer, and aerodynamic braking in real high-temperature gases. Theoretical and numerical studies analyze bow-shock displacement and plasma characteristics during Mars-entry conditions [19,20], and review papers summarize plasma and MHD-control applications in high-speed aerodynamics [21]. Within the MEESST project, simulations using different chemical models reproduced experimental data and confirmed the expected MHD effects [11,12]. Experiments on MHD aerobraking under atmospheric-entry conditions showed an increase in bow shock stand-off distance and a reduction in heat flux, supporting the practical potential of this approach [22].

The presented work continues this line of research for supersonic flows with shock wave structures and localized electric and magnetic forcing. These studies show that a near-surface plasma region can influence the formation and location of the steady BSW in air and xenon, and that the plasma parameters determine both the formation rate and the final shock position [23,24]. Studies of internal and external flows also demonstrate that shock wave positions and aerodynamic characteristics can be controlled by plasma, electric, and magnetohydrodynamic actions [25,26,27]. A distinctive feature of these studies is the use of not only an external magnetic field but also an electric field from an external source for MHD action to generate a gas-discharge current in localized flow regions during the flow, which allows for a wider range of intensities and directions of action.

In the presented work, based on experimental data and conducted modeling, an integrated experimental–computational system was created in which the measured characteristics of the discharge and BSW, as well as the numerically obtained averaged plasma parameters in the impact zone, were combined with the plasma theory of Burm et al. [28] to determine the averaged characteristics of gas-discharge plasma in cases of the absence of a magnetic field and for different directions of the MHD action. Thus, a critical issue of the study carried out here is the assessment of the gas-discharge plasma parameters in the near-surface region, such as the degree of ionization, the degree of nonequilibrium, the adiabatic index, as well as the electron density and electron temperature for active action on supersonic flows with shock wave configurations, in the cases of pure plasma action and plasma-MHD action. In the present study, high-speed flows are characterized by freestream Mach numbers of 4.15 and 6.8 for air and xenon, respectively. The numerical part of this research relies on computational–fluid–dynamics methods, specialized software for calculation of high-speed flows, and complex-conservative difference schemes [29,30,31].

2. Experiment Arrangement and Results

To effectively control supersonic flows containing shock waves using magnetohydrodynamics, it is necessary to create a force action under the neutral gas in a selected, most convenient local region of the flow and in the desired direction. This is achieved by artificially generating the motion of charged particles in external electric and magnetic fields. Specifically, a gas discharge of the desired intensity is created in the selected impact zone, with a current in a given direction and a magnetic field perpendicular to the current. In this case, the charges moving in the magnetic field will be acted upon by the Lorentz force in a direction perpendicular to both the gas-discharge current I and the magnetic induction vector B.

Due to energy transfer during collisions from heavy positive ions to gas particles, a ponderomotive force arises and acts on neutral gas. It is equal in magnitude to the Ampere force F_A acting on a current-carrying conductor, F_A = I(B × L), where B is the magnetic induction vector, and L is the propagation vector, having the direction of the current and the length of the gas-discharge zone. Accordingly, a force of F_L = F_A/(S ∙ L) = J × B, conventionally called the Lorentz force, will act on a unit volume of neutral gas in the affected zone, where S is the cross-sectional area of the gas-discharge zone, and J is the vector with current density magnitude and current direction. If the magnetic field is directed perpendicular to the current, the magnitude of the Lorentz can be calculated as F_L = J∙B.

Thus, the method of active MHD control of supersonic flows involves creating local plasma gas-discharge regions in an external magnetic field using an external current source. The location of the affected zones, discharge parameters, and the magnitude of the magnetic induction are determined based on the specific task. The main requirements for this method include the maximum possible magnetic induction and sufficient conductivity of the medium to increase the gas-discharge current density J and reduce the electric field strength in the discharge zone E in order to increase the Lorentz force and reduce Joule heating of the gas. Depending on the current from the external source, it is possible to create the Lorentz force directed either towards the streamlined model (Case F_L+ or F_L+ force) or away from it (Case F_L− or F_L− force). The specific objective of this study is to implement magnetohydrodynamic control of the BSW position in supersonic flow past a blunt semi-cylindrical model.

The experiment was conducted in a supersonic flat nozzle with a 22° opening angle. Figure 1a shows a diagram of the nozzle with the model positioned on its axis. The diagram denoted the hard material (plexiglass) from which the supersonic nozzle was made (shown in light blue). The metal parts of the chamber are correspondingly highlighted with hatching. During supersonic flow around the model, a BSW is formed in front of it. To impact BSW position, a gas-discharge zone is created near the leading edge of the cylinder by applying an external voltage U_pl to electrodes embedded in the surface of the model and turning on a magnetic field with a magnetic induction vector B directed perpendicular to the gas-discharge current I. The current propagates near the front surface of the model between the BSW and the body along a semicircular trajectory in a direction perpendicular to the magnetic field. In this case, a Lorentz force is exerted on the gas near the front edge of the model in the gas-discharge zone. The direction of this force depends on the direction of the gas-discharge current, and the intensity of this force depends on the magnitude of the gas-discharge current. The intensity of the gas-discharge changes depending on the change in the applied voltage U_pl value. Figure 1b shows a schlieren picture of the supersonic flow around the model, visualizing the position of the stationary bow-shock wave in the absence of a magnetic field.

Figure 1. (a) Schematic diagram of the model’s location in a supersonic nozzle. (b) Schlieren picture of the BSW position in the absence of the magnetic field, I = 673 A.

If the gas-discharge current is directed from the upper electrode to the lower one, the Lorentz force F_L+ will act toward the model, pressing the gas against its surface. This action is shown schematically in Figure 2a. In this case, the stationary position of the BSW changes; its location will be further from the body compared to Figure 1b, as evident from the schlieren pattern in Figure 2b, obtained with the magnetic field turned on. The graph of the distance d between the BSW and the model surface on the nozzle axis versus the magnitude of the Lorentz force in Figure 2c shows that as the force intensifies, the BSW moves further away from the body. Possible errors in the experimental data that arose during the process of obtaining and processing schlieren flow patterns are also shown here.

Figure 2. (a) Scheme of the Lorentz force F_L+ action in the direction to the model; (b) Schlieren picture of the position of the BSW under the MHD action to the model, I = 655 A. (c) Stand-off distance of the BSW under the action of F_L+, B = 1.4 T. Dash line—the case of the absence of any impact.

When the current direction changes from the lower to the upper electrode, the F_L− force repels the gas from the model surface (see Figure 3a), and the position of the established BSW also changes, it approaches the model, as can be seen from a comparison of Figure 1b, Figure 2b and Figure 3b, which show the position of the BSW at the same currents, but with different directions of the Lorentz force. However, the magnetohydrodynamic effect in the case of F_L− will occur against the background of the plasma effect, described in detail in [23,24]; therefore, it will have a narrower range of currents and voltages for reducing the distance between the BSW and the model, since the mechanism and degree of plasma action are determined by the amount of power supplied to the discharge zone and the change in plasma characteristics in the impact zone, which prevents the wave from approaching. The range of the impact force capable of prevailing over the plasma effect and bringing the BSW closer to the body will depend on the specific task. In this experiment, to decrease BSW stand-off distance the F_L− must be in the range of 4–6.5 N/cm³, which can be seen from the graph in Figure 3c.

Figure 3. (a) Scheme of the Lorentz force F_L− action in the direction from the model. (b) Schlieren picture of the position of the BSW under the MHD action from the model, I = 597 A. (c) Stand-off distance of the BSW under the action of F_L−, B = 1.4 T. Dash line— the case of the absence of any impact.

3. Numerical Simulations

3.1. Methodology and Statement of the Problem

In the experiment, an electric current from an external energy source is generated, producing a gas discharge and, at the same time, an external magnetic field is organized acting in the gas-discharge plasma region. The flow from the shock tube, upon reaching the body, initiates the formation of a bow shock wave (BSW), which passes through the gas-discharge plasma zone affected by the plasma parameters and the impact of the MHD interactions. The calculations assume that this impact zone arises instantaneously due to the difference in the time scales of gas-discharge plasma formation, the occurrence of the ponderomotive force (Lorentz force) acting on charged particles, and gas-dynamic phenomena.

Thus, the effect of the discharge and the external magnetic field is modeled by the action of a volumetric gas region with increased energy and a modified adiabatic index and impact of the Lorentz force. Figure 4 shows a schematic of a numerical approach to studying the influence of near-surface plasma energy combined with MHD impact, on a supersonic flow around a “semi-cylindrical plate”-shaped body. The zone of influence of the discharge and magnetic field is marked in red.

Figure 4. Statement of the problem for calculation (schematic) and computational domain.

The simulations are based on the Navier–Stokes equations for perfect viscous heat-conducting gas; xenon and air are considered. The full Navier–Stokes system of equations in the divergence form for the dimensionless variables [29], added by the right parts associated with plasma and MHD actions is solved numerically:

$${\displaystyle \frac{\partial \mathbf{U}}{\partial t}}+{\displaystyle \frac{\partial (\mathbf{F}+{\mathbf{F}}_{\mathit{v}})}{\partial x}}+{\displaystyle \frac{\partial (\mathbf{G}+{\mathbf{G}}_{\mathit{v}})}{\partial y}}=\mathbf{H}$$

(1)

$$\mathbf{U}=\left(\begin{array}{c}\mathsf{\rho }\\ \mathsf{\rho }u\\ \mathsf{\rho }v\\ E\end{array}\right),\mathbf{F}=\left(\begin{array}{c}\mathsf{\rho }u\\ p+\mathsf{\rho }{u}^2\\ \mathsf{\rho }uv\\ u(E+p)\end{array}\right),\mathbf{G}=\left(\begin{array}{c}\mathsf{\rho }v\\ \mathsf{\rho }uv\\ p+\mathsf{\rho }{v}^2\\ v(E+p)\end{array}\right), \mathbf{H}=\left(\begin{array}{c}0\\ F_{Lx}\\ F_{Ly}\\ \mathsf{\rho }q\end{array}\right),$$

$${\mathbf{F}}_{\mathit{v}}=-\left(\begin{array}{c}0\\ {\displaystyle \frac{\mathsf{\mu }}{\mathrm{R}\mathrm{e}}}\left(4/3u_x-2/3v_y\right)\\ {\displaystyle \frac{\mathsf{\mu }}{\mathrm{R}\mathrm{e}}}\left(v_x+u_y\right)\\ {\displaystyle \frac{\mathsf{\mu }{\pi }_1}{\mathrm{R}\mathrm{e}}}+{\displaystyle \frac{1}{\mathrm{N}}}k{T}_x\end{array}\right),{\mathbf{G}}_{\mathit{v}}=-\left(\begin{array}{c}0\\ {\displaystyle \frac{\mathsf{\mu }}{\mathrm{R}\mathrm{e}}}(v_x+u_y)\\ {\displaystyle \frac{\mathsf{\mu }}{\mathrm{R}\mathrm{e}}}(4/3v_y-2/3u_x)\\ {\displaystyle \frac{\mathsf{\mu }{\pi }_2}{\mathrm{R}\mathrm{e}}}+{\displaystyle \frac{1}{\mathrm{N}}}k{T}_y\end{array}\right)$$

$${\pi }_1=u(4/3u_x-2/3v_y)+v(v_x+u_y),{\pi }_2=v(4/3v_y-2/3u_x)+u(v_x+u_y)$$

$$E=\mathsf{\rho }(\mathsf{\epsilon }+0.5(u^2+v^2\left)\right),\mathrm{N}=\mathrm{RePr}(\mathsf{\gamma }-1)/\mathsf{\gamma }.$$

The specific power in the area of higher gas energy formed by the discharge is q, and the specific internal energy is ε.

$$\mathsf{\epsilon }=p/(\mathsf{\rho }(\mathsf{\gamma }-1)).$$

For the dependence of dynamic viscosity μ on temperature T for air, Sutherland’s law was used,

$$\mathsf{\mu }={\displaystyle \frac{T^{1.5}\left(1+s_1\right)}{T+s_1}},$$

where s₁ = 0.084. For xenon it was assumed that

$$\mathsf{\mu }=T^{0.5}.$$

The coefficient of heat conductivity k is modeled to depend on temperature in the same way:

$$k=T^{0.5}.$$

When the Lorentz force $F_L$ is directed towards the body (Case $F_L$+) the right parts in the equations for the impulse components are:

$$F_{Lx}=\left|F_L\right|cos\alpha ,$$

$$F_{Ly}=-\left|F_L\right|sin\alpha ,$$

And when the Lorentz force $F_L$ is directed from the body (Case $F_L$−) the right parts in the equations for the impulse components are:

$$F_{Lx}=-\left|F_L\right|cos\alpha ,$$

$$F_{Ly}=\left|F_L\right|sin\alpha ,$$

Here α is the acute angle between Lorentz force $F_L$ direction and x-axis; the absolute value of the Lorentz force $F_L$ produced by the external magnetic field is:

$$\left|F_L\right|=\left|J\right|\left|B\right|sin90°= \left|J\right|\left|B\right|,$$

(2)

where J is the current density.

Additional term in the right side of the energy equation in (1) is connected with the specific power of the discharge-created plasma q₀ and the work of the Lorentz force. So, the specific power q (the power per unit of mass) is expressed as follows:

$${q=q_0 +q}_{mhd}=q_0 +\left|F_L\right|k_{mhd}.$$

(3)

Here the work of the Lorentz force is supposed to be proportional to |F_L|$;$k_mhd is the variable coefficient (dimensional). Details of the construction of the model for taking into account the MHD effect, and in particular the derivation of the expression for the coefficient k_mhd are given in [27].

The problem is solved in dimensionless variables; the following normalizing coefficients were used in the calculations:

$${\rho }_n={\rho }_{\infty },p_n=p_{\infty },l_n=D,T_n=T_{\infty },u_n=(p_{\infty }/{\rho }_{\infty }{)}^{0.5},t_n=l_n/u_n,$$

where the index ∞ defines the freestream parameters. In the following figures, where not specifically stated, the values on the axes are taken in dimensionless form.

Initial conditions are the freestream flow parameters: density ${\rho }_{\infty }$, pressure $p_{\infty }$, and velocity $u_{\infty }.$ According to the evaluation of the experimental data [24], we suppose that in xenon the adiabatic index in the oncoming flow is γ = 1.217, and the adiabatic index in the plasma area in front of the body is γ_s= 1.258. Therefore, for xenon, the problem is solved taking into account the initial gas ionization. For air, the initial ionization is small [23], and it is not taken into account; the adiabatic index in the oncoming flow is γ = 1.323. At the boundaries of the body, the boundary conditions of solid adiabatic wall with no-slip conditions are used; non-reflecting boundary conditions in the normal direction to the wall are used at the exit boundaries.

The stationary domain with higher gas energy models the near-surface energy release with the use of the right-hand sides of the impulse and energy equations in (1), where |F_L| is defined by (2), and q is the specific power (3) in this region (this area is noted by red in Figure 4). The width of this area in the x-direction is assumed equal to 0.1D, which is in accordance with the experimental schlieren images. This region is assumed to arise instantaneously and is determined by the initial conditions and the values of q and F_L. The mechanism of action of the impact region is associated with the fact that the BSW passes through the already-existing region of the impact during its formation at the non-steady stage; thus, its characteristics at the steady state are influenced by the parameters in this region [23].

The simulation is performed using a domestic computational code [30] based on complex-conservative difference schemes [31]. The schemes have the second order of accuracy in space and time. The stencil of the well-known Lax scheme is used; therefore, a shifted uniform grid is used throughout the computational domain. To increase the order of approximation, differential consequences of system (1) for partial derivatives with respect to x and y are used. Thus, in addition to the main conservative variables, their first derivatives are used as conservative variables. These derivatives are considered as unknown functions and are calculated on the same stencil. The required second derivatives are calculated using the values of the first derivatives at the grid nodes. The boundaries of the AD body are embedded in the computational domain without violating the conservation laws, including the regions adjacent to the body boundaries. This is achieved through the use of fractional cells in these regions. The position of the front part of the body on the computational grid is shown in Figure 5.

Figure 5. The position of the frontal part of the body on a computational grid (enlarged, every fifth node is shown).

Details of the construction of these schemes within the computational domain and near the boundaries of the AD body are presented in [31]. It should be noted that a number of test cases related to the numerical methods used, comparison with experimental results, as well as test cases for grid convergence analysis for the developed software are also presented in [31].

3.2. Numerical Study of Flow Parameters Under the Influence of Plasma and a Magnetic Field

The parameters used to simulate the MHD effect in the discharge zone corresponding to the first five experimental points in the graphs in Figure 2c and Figure 3c are presented in Table 1. The adiabatic index and the specific power estimated in [24] for the discharge zone, and the corresponding experimental values of the Lorentz force (normalized by the appropriate coefficient from Table 2) are used.

Table 1. Parameters used in the calculations for Case F_L+ and Case F_L−. Working gas—xenon.

γs [24]	1.258	1.256	1.250	1.275	1.253
q0 [24]	0	52.0	82.5	119.5	147.2
FL, Case FL+	0	28.86	46.74	52.16	61.92
|FL|, Case FL−	0	26.28	36.99	46.20	58.53

Table 2. Determining parameters of for calculations for xenon and air.

	Description	Dimensional Value	Dimensionless Value	Normalizing Coefficient
Xenon	Mach number in the freestream flow M∞		6.8
	The Reynolds number Re		4558.9
	Prandtl’s number Pr		0.623
	Adiabatic index in freestream flow γ		1.217
	Adiabatic index in the initial plasma region (q0 = 0) γs		1.258
	Freestream gas pressure p∞	3.1 × 103 Pa	1.0	pn = p∞
	Freestream gas density ρ∞	0.040793 kg/m3	1.0	ρn = ρ∞
	Freestream gas temperature T∞	1200 K	1.0	Tn = T∞
	Specific power in the plasma region q0		See Table 1	qn = pn/(tn ρn) = 0.698299 × 106 kW/kg
	Lorentz force FL		See Table 1	FLn = ρn un/tn = 0.103332 × 106 kg/(m2 s2)
	Body’s diameter D	3 × 10−2 m	1.0	ln = D = 3 × 10−2 m
	Velocity u	2067.96 m/s	7.502	un = (pn/ρn)0.5 = 275.668 m/s
	Time t		1.0	tn = ln/un = 108.827 µs
Air	Mach number in the freestream flow M∞		4.153
	The Reynolds number Re		6763.2
	Prandtl’s number Pr		0.703
	Adiabatic index in freestream flow γ		1.323
	Adiabatic index in the initial plasma region (q0 = 0) γs		1.323
	Freestream gas pressure p∞	6790.546 Pa	1.0	pn = p∞
	Freestream gas density ρ∞	0.017572 kg/m3	1.0	ρn= ρ∞
	Freestream gas temperature T∞	1302.024 K	1.0	Tn = T∞
	Specific power in the plasma region q0		43, 57, 75, 101	qn = pn/(tn ρn)
	The work of Lorentz force |FL|kmhd		−10, 0, 10	FLn = ρn un/tn
	Body’s diameter D	3 × 10−2 m	1.0	ln = D = 3 × 10−2 m
	Velocity u	2969.272 m/s	4.776	un = (pn/ρn)0.5 = 621.64 m/s
	Time t		1.0	tn = ln/un = 48.26 µs

The determining flow parameters and normalizing coefficients adopted in modeling the effect of a near-surface discharge in xenon and in air are given in Table 2. For air, experimental data are available only for the action of the plasma region (in the absence of MHD influence); therefore, the MHD effect was modeled assuming that k_mhd = 1 for the Case F_L+, and k_mhd = −1 for the Case F_L−.

In Figure 6, the numerical fields of density and temperature are shown at the steady-state flow mode, at time t = 2.0 for different q₀, γ_s, for the Cases F_L+ and F_L−. It is evident that in the zone of influence of the discharge and external magnetic field, the gas density decreases and temperature increases in the case F_L+, and BSW moves from the body (upper images); and density increases and temperature decreases in the case F_L−, and BSW moves towards the body (bottom images). This confirms the experimental results and gives the instrument for controlling the BSW position (along with the characteristics of AD body).

Figure 6. Fields of density (a) and temperature (b) in xenon under the action of the Lorentz force for the F_L+ (upper) and F_L− (bottom), q₀ = 119.5, γ_s = 1.275, F_L+ = 52.16, |F_L−| = 46.2.

The corresponding results for air are presented in Figure 7. Here, it was supposed that F_L+ = 10, k_mhd = 1 and |F_L−| = 10, k_mhd = −1.

Figure 7. Fields of density (a) and temperature (b) in air under the action of the Lorentz force, q₀ = 101, γ_s = 1.24: F_L+ = 10 and k_mhd = 1 (upper) and |F_L−| = 10 and k_mhd = −1 (bottom).

Based on the obtained values of density and temperature in the gas-discharge plasma zone, their average values in this area were calculated. The average characteristics were calculated, as follows:

$${\mathit{f}}_{\mathit{a}}={\displaystyle \frac{\sum _{\mathit{i},\mathit{j}}\mathit{f}(\mathit{i},\mathit{j})}{\mathit{N}}},$$

where f_a is the averaged value, f(i,j) is the value of f in the grid node (i,j), and N is the amount of nodes included in the plasma area. Only the values f(i,j) contained in the impact region are taken into account.

Figure 8 presents the data obtained in the present study, as well as in [23,27]. Figure 8a,b present the calculated and experimental values of the relative stand-off distance of the steady BSW from the model in xenon and air, accordingly, for different values and different directions of the Lorentz force F_L; average temperature values in the plasma region are also shown. In air, the calculated and experimental values are compared only for the action of the plasma zone, without the MHD action. The figures include the error bars that arose during the processing of schlieren pictures. Note that the values marked with squares on the curves were used in the simulation.

Figure 8. Comparison of calculated and experimental values of the relative stand-off distance of the steady BSW from the model in xenon for different values and different directions of the Lorentz force F_L (a) and in air for F_L = 0 (b); average temperature values in the plasma region are also given. Red squares—experiment, orange triangles—calculation.

Thus, based on the experimental schlieren patterns and the conducted modeling, the average gas-dynamic parameters in the gas-discharge plasma region were obtained. These data will be used in the following sections to evaluate the gas-discharge plasma parameters in xenon and air which is the purpose of this study.

3.3. Evaluation of the Gas-Discharge Plasma Parameters in Xenon and Air

As noted earlier, the MHD effect at the model surface during the creation of gas-discharge zones from an external source occurs against the background of plasma action [23,24] which significantly influences the flow process even in the absence of a magnetic field. The energy input into locally created gas-discharge zones in the near-surface region of the model leads to changes in the plasma characteristics in these zones, which affects the gas-dynamic processes.

Figure 9 shows the dependencies of the power input into the discharge region (Figure 9a) and the changing in electron concentration (Figure 9b) on the gas-discharge current for two working gases: monatomic xenon and air which is approximated by diatomic nitrogen (N₂). These data were found from the experimentally obtained volt-ampere characteristics of the discharge. It is evident that to achieve plasma conditions in air similar to those in xenon, and sufficient to noticeably influence the position of the BSW, the input energy must be at least four times greater than the energy input into xenon.

Figure 9. Input power (a) and electron density (b) in discharge zone vs current in xenon and air.

Since the adiabatic index of the medium, which determines gas-dynamic processes in the plasma zone, depends on the degree of ionization and the degree of nonequilibrium [28], it will vary significantly in the affected zone, as can be seen from Figure 10. Figure 10a shows the dependencies of the adiabatic index on the degree of nonequilibrium, calculated according to the theory of Burm et al. [28] using the experimental values of the electron concentration and the calculated values of the averaged gas density and temperature in the impact zone for xenon. The figure demonstrates the range of possible values of the adiabatic index under the experimental conditions. The same calculations were made for air (Figure 10b).

Figure 10. Adiabatic index: (a) according to the theory of Burm et al. [28] for xenon; (b) according to the theory [28] for air; (c) calculated from the theory [28] and numerical modeling vs current for xenon and air.

The dependencies of the values obtained in the numerical simulation on the current for the two gases are shown in Figure 10c. A comparison of these data showed that in xenon the adiabatic index calculated according to the theory of Burm et al. [28] practically coincides with the value in the model experiment under the condition of thermal equilibrium, when θ = 1 (see, also, [24]). In air, the model values coincide at θ = 0.3, while at θ = 1, the values are significantly lower. This can be explained by the presence of dissociation of diatomic molecules under the heating conditions in the experiment, which was not taken into account in the model used for air, as well as by the possible nonequilibrium state of the air in the impact zone. Accounting for these factors is the task of future numerical studies. Figure 10c shows the adiabatic indices, both from the model experiment and calculated using the theory [28] at θ = 1.

Table 3 and Table 4 summarize the main gas parameters in the impact zone at the leading edge of the model for different gas-discharge intensities. It should be noted that the plasma parameters in the gas-discharge zone, given in Table 4, were refined compared to [23] by averaging the experimental current-voltage characteristics of the discharge using the least-squares method. The corresponding data for each current were taken from the average curve.

Table 3. The main gas parameters in the impact zone for xenon.

I, A	Upl, V	P, kW	q0, W/kg	ne × 10−22, m−3	ρa, kg/m3	α	Ta, K	Θ	γs	(d − d0)/d0
0	0	0	0	1.65	0.19180	0.0187	7530	1 1	1.260	0
373	94	35.1	52	2.03	0.17759	0.0249	8507	1	1.262	0.033
604	90	54.4	82.5	2.70	0.17275	0.0340	9004	1	1.245	0.049
673	101	68.0	119.5	3.0	0.14608	0.0446	10,905	1	1.269	0.115
800	109	87.2	147.2	3.57	0.15475	0.0502	10,647	1	1.252	0.082

Table 4. The main gas parameters in the impact zone for air.

I, A	Upl, V	P, kW	q0, W/kg	ne × 10−22, m−3	ρa, kg/m3	α	Ta, K	Θ	γs	(d − d0)/d0
0	0	0	0	0.50	0.0672	0.0035	4369	0.28	1.323	0
607	345	209.4	43	1.80	0.0550	0.0151	6145	0.32	1.300	0.147
640	354	226.6	55	1.90	0.0515	0.0172	6857	0.30	1.300	0.19
645	357	230.3	57	1.95	0.0511	0.0177	6960	0.30	1.300	0.206
760	391	297.2	75	2.30	0.0483	0.0221	7857	0.32	1.280	0.265

3.4. Evaluation of the Gas-Discharge Parameters Under the MHD Action of Different Directions in Xenon

To diagnose the state of the impact zone in xenon at different values of external fields, the volt-ampere characteristics of the discharge were measured in the experiment, namely, the magnitude of the gas-discharge current and the corresponding value of the voltage on the plasma gap were measured. Based on the obtained data, the power supplied to the circuit and the electron concentration in the gas-discharge zone were estimated. The dependences of these parameters on the current for two directions of MHD impact are shown in Figure 11. The range of the input energy is 0–100 kW, while the resulting electron concentrations of the gas-discharge plasma increase from 1.5 × 10²² to 3.5 × 10²² m⁻³. Using the calculated values of the averaged gas density and temperature, the degree of ionization was determined and the values of the adiabatic index were found, the value of which, according to the theory [28], is determined by the plasma characteristics of the medium, namely the degree of ionization α and the degree of nonequilibrium θ = T_a/T_e.

Figure 11. Input power (a) and electron density (b) in discharge zone vs current.

Figure 12 shows the dependences of the adiabatic index on the degree of nonequilibrium for the obtained degrees of ionization and gas temperature for the F_L+ case. The main parameters of the affected zone and the refined values of the adiabatic index are presented in Table 5. Since the medium in the affected zone can be considered to be in equilibrium during abrupt flow deceleration in front of the body, the refined data were taken at θ = 1. Similar data for the F_L− case are shown in Figure 13 and Table 6.

Figure 12. Calculated adiabatic index according to the theory [28] for xenon for Case F_L+.

Table 5. The main gas parameters in the impact zone at F_L+ action.

I, A	Upl, V	P, kW	q0, W/kg	ne × 10−22, m−3	ρa, kg/m3	α	Ta, K	Θ	Te, K	γs	(d − d0)/d0
0	0	0	0	1.65	0.19181	0.0187	7530	1	7530	1.260	0
331	96	31.8	52.0	1.99	0.17633	0.0245	8806	1	8806	1.273	0.050
543	101	54.8	82.5	2.50	0.14987	0.0363	11,183	1	11,183	1.297	0.164
655	110	72.1	119.5	2.93	0.11147	0.0571	15,826	1	15,826	1.347	0.279
753	116	87.3	147.2	3.36	0.11928	0.0612	14,989	1	14,989	1.326	0.2

Figure 13. Calculated adiabatic index according to the theory of Burm et al. for xenon for Case F_L−.

Table 6. The main gas parameters in the impact zone at F_L− action.

I, A	Upl, V	P, kW	q0, W/kg	ne × 10−22, m−3	ρa, kg/m3	α	Ta, K	Θ	Te, K	γs	(d − d0)/d0
0	0	0	0	1.65	0.19181	0.0187	7530	1	7530	1.260	0
340	98	33.3	52	2.00	0.15450	0.0281	10,439	1	10,439	1.306	0.033
478	113	54	82.5	2.34	0.16097	0.0316	9800	1	9800	1.276	0
597	130	77.6	119.5	2.70	0.18444	0.0319	7176	1	7176	1.196	−0.148
756	124	93.7	147.2	3.36	0.18789	0.0389	7169	1	7169	1.180	−0.066

The dependences of the degree of ionization, electron temperature, and adiabatic index on the magnitude of the gas-discharge current for the two cases of the Lorentz force direction, shown in Figure 14, demonstrate that different directions of MHD action change the parameters of the affected zone differently. With an increase in the gas-discharge intensity at low currents, the MHD effect is weak, and the parameters are similar for different directions. A turning point occurs at a current of 450 A, and the dependences change dramatically. At F_L+, all parameters increase sharply, while at F_L−, the temperature and adiabatic index decrease, with γ falling below the initial value obtained in the absence of external fields. At higher currents, the parameters tend to converge. It should be noted that the MHD effect in the case of F_L−, for decreasing the distance between the BSW and the model, operates in the current range of 450–800 A. This current range corresponds to a sharp difference in the plasma zone parameters.

Figure 14. Dependence of ionization degree (a), electron temperature (b) and adiabatic index (c) on the gas-discharge current for Cases F_L− and F_L+.

It should be noted that although quantitative results were obtained in this study, these results are estimates, since the model contains limitations associated with a constant average representation of the plasma region, the use of averaged constant values of the adiabatic index, and specific discharge power in this region.

4. Conclusions

In this study, experimental and numerical methods were used to investigate the influence of the MHD impact combined with an artificially generated gas-discharge current from an external source near the front surface of the body for controlling supersonic flow, including the position and shape of the bow shock wave. In the experiment, with two current closure directions and MHD action, and for purely plasma action (without MHD impact) plasma characteristics in the discharge zone were measured, such as the energy deposited in the discharge, the electron concentration, and the degree of ionization. The range of currents within which active control of the bow shock wave position is possible was determined as 450–800 A.

Based on the Burm et al. theory [28] and calculated gas parameters, the electron temperature and the adiabatic index of the medium in the impact zone were determined for two directions of the Lorentz force action and for the case of absence of MHD action. It was shown that the direction of the force impact influences the change in these characteristics with current within the active control range. It was obtained that when the Lorentz force is directed towards the model, these parameters increase; when the Lorentz force is directed from the body, they decrease.

By this way, an integrated experimental–computational system was formed in which the measured characteristics of the discharge and bow shock wave, as well as the numerically obtained averaged plasma parameters in the impact zone, are combined with the Burm et al. plasma theory to clarify the thermodynamic state of the medium and determine the corresponding characteristics of the gas-discharge plasma in cases of the absence of a magnetic field and for different directions of its action.

Thus, based on the possibility of controlling the position of a stationary bow-shock wave both to increase and to decrease its distance from the aerodynamic model using locally organized external electric and magnetic fields, the conducted study assessed and demonstrated the possibility of influencing the plasma parameters in the gas-discharge zone such as the degree of ionization, the degree of nonequilibrium, the adiabatic index, as well as the electron density and electron temperature. The results presented may be of interest for assessing the characteristics of plasmagasdynamic and magnetohydrodynamic phenomena in high-speed flows, for example in the development of control systems based on plasma generation and the action of a magnetic field.