Numerical Simulation of the Discharge in Supersonic Flow Around a Sphere

Valentin A. Bityurin1,, Aleksey N. Bocharov2 Institute of High Temperatures of Russian Academy of Sciences, Moscow, Russia

and

Nikolay A. Popov3 Moscow State University, Moscow, Russia

Introduction ECENTLY the most common opinion on main physical mechanism responsible for aerodynamics effects resulted from electrical discharges created in airflow is that it is the heat release associated with Joule heating.

The temperature increase changes the gas density, speed of sound, gas chemical composition and sometimes causes secondary shock and/or acoustic waves propagating away from the discharge area. Such a modification of flow parameters can result in significant change in local and integral gasdynamics characteristics of aerodynamics bodies in high-speed airflows. From the other hand, there are experimental observations which are difficult to explain only by effects of heat release1. Among those are: the non-symmetrical effect of DC and AC discharges on the drag

reduction when the polarity is changed; the non-symmetrical appearance of the electrical current passing normal to shock waves; polarity effects observed in erosive plasma jets. Several experimental results obtained recently at IVTAN 2-8 have excited the old question on non-thermal mechanisms involved into the plasma aerodynamics effects. A similar evidence of possible importance of the sheath layer phenomena has found in the dielectric barrier discharge

1 Director of MHD and Low Temperature Plasma Division, IVTAN, Izhorskaya 13/19, Moscow, 125412, AIAA member. 2 Head of Laboratory of Computational Plasmadynamics, IVTAN, Izhorskaya 13/19, Moscow, 125412, AIAA member. 3 Senior Researcher, Moscow State University, GSP-2, Leninskiye gory, Moscow, 119992.

R

Figure 1. Combined discharge (EB+DC) near spherical model E in supersonic airflow. Id=200 mA, negative model potential, M~ 1,6; Pst= 15 Torr Ib~ 5-10 mA, D= 10mm

45th AIAA Aerospace Sciences Meeting and Exhibit8 - 11 January 2007, Reno, Nevada

AIAA 2007-223

Copyright © 2007 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

(DBD) experimental study as a part of surface discharge effects in plasma aerodynamics applications (see, for example, 9).

One of the examples of the experimental results discussed preliminarily in paper9 revealed the drag reduction of a sphere depending on the electrical current polarity. The experiment was performed with spherical models of 10-20mm in diameter. The models were mounted in supersonic airflow (Mach ~ 1.6) under static pressure 10-100 Torr and static temperature ~250K. A combined DC discharge (electron beam + DC) was created between the nozzle and the model made of metal. The DC polarity could be changed. A simplified physical model was proposed in paper9 to treat the phenomena.

In the current paper an attempt is made to numerically simulate a DC discharge in the flow around a sphere under conditions close to those of paper1. The test sphere of diameter of 10 mm is positioned at 15 mm downstream of the nozzle exit. The sphere’s cathode operation mode is considered, while the nozzle and other facility elements are assumed to be grounded and functioning as anode. The e-beam gun is positioned within the nozzle and represents central axial body. Attention is paid to effect of e-beam and plasma discharge on the flow structure. Specifically, the mechanisms of drag reduction are investigated within the model considered.

The computational model developed here follows, in general, those reported in papers10,11. It is based on coupled solution of compressible fluid equations and transport equations for the charged particles, electrons and positive ions. The latter are considered in terms of so called drift-diffusion model. Poisson’ equation for electric potential is solved to close transport equations. The neutral gas is assumed to be the perfect gas, for simplicity. Townsend ionization model is used for electron impact reaction and three-body recombination model is used as a backward reaction. Effective reactions rate constants are used which approximate the ionization/recombination processes in air. In the model under consideration the main flow can be influenced by both e-beam and plasma effects. The influence of e-beam is taken into account as spatially shaped distribution of thermal power and distribution of production rate for charged particles. Two plasma effects are considered in the model: the Joule heating from electronic and ionic currents and the forces due to space charge in near electrode regions.

Computational model We assume that the flow and electric discharge characteristics can be described by the set of equations reflecting

conservation of mass, momentum, total energy for the entire fluid, and transport equations for charged particles, electrons and ions. Fluid equations are enclosed with the equations of state, and charged particles equations are enclosed with the Poisson equation for electric potential. The governing equations are given below.

0)( =∇+∂∂ Uρρ

t (1)

FrUUU &+∂−∂=∇+⋅∇+∂∂ /τ)( P

tρρ

(2)

WUqU &=∇+∇++∇+∂

∂ )())( 00

τρρ Pete

(3)

Figure 2. Drag measurements. Ib~ 8 mA, model E- sphere

(Ø9,6 mm), distance from nozzle Z=15 mm, M=1,6; Pst=16torr

Here, ρ is the density, U = (Ux,Uy) is velocity, P is the thermodynamic pressure, e0 is specific total energy,

2

20 Uee +=

Transport equations (1) – (3) are enclosed by the equations of state for perfect gas

2

20 Uee += , ργ /)1( Pe =− , TRP ⋅= ρ (4)

Here, γ = 1.4 is the ratio of specific heats, and T is the temperature. Viscous stress tensor components and heat fluxes are specified as usually:

j

iijij x

U∂∂

−∇= ηηδτ U32

, r

q∂∂

−=Tλ , (5)

where δij – Kronecker symbol, η is viscosity, λ is heat conductivity. Gradient and divergence operators are defined as follows:

yx yx ∂∂

+∂∂

=∂∂ eer

, ξξ y

yyx ∂∂

+∂∂

=∇1

(6)

Where ex and ey are unit base-vectors, ξ=0 for Cartesian coordinate system, and ξ=1 for axi-symmetric one. Plasma properties, ion and electron number densities and electric field are found from the transport equations for

concentrations and Poisson equation for electric potential, assuming that drift-diffusion model is valid:

Qt

ni

i &=∇+∂∂ Γ (7)

Qt

ne

e &=∇+∂∂ Γ (8)

eρε =∇E , )( eie nnq −=ρ (9)

In Eqs. (7) through (9) ni is ion number density, ne is electron number density, ρe is the electric charge density, E is

electric field, r

E∂∂

−=ϕ

, ϕ is the electric potential, q is electron charge, ε is permittivity of vacuum, Q& is the

source term defined later. Ion and electron fluxes, Γi and Γe, are defined as follows

( ) rUΓ ∂∂−+= /V iidiii nDn

( ) rUΓ ∂∂−+= /V eedeee nDn (10)

Here, drift velocities Vdi and Vde are defined as

Eidi µ=V , Eede µ−=V (11)

where µi and µe are ion and electron mobilities, respectively:

µi = 42.0/P*, µe = 0.228/P*, P* = P(torr)*293/T (12)

Diffusion coefficients, Di and De in (10) are defined as

Di = µikBT/q, De = µekBTe/q (13)

In (13) kB is the Boltzman’s constant. Electron temperature Te is regarded as function of the reduced electric field, Er = E/n [Td], where n = P/kBT is the total number density:

,447.0 16.0re ET = if Er > 50, re ET 0167.0= , otherwise. Te is given in ev.

Electric current density is determined as

)( eiei q ΓΓjjj −=+=

Source-terms in Eqs (2),(3),(7) and (8) are determined as follows

EF eρ=& ,

extiiee WW && ++= Ejj )( ηη

( ) extieer QnnEQ && +−= βα Γ (14)

Here, first of these terms is the body force acting on charged plasma. Second term is the heat from flowing electron and ion currents and possible heat release from electron beam. It has been taken in this paper that, ηe = 0.5 and ηi = 1. The last of terms (14) represents electron impact ionization process, recombination and external ionization due to electron beam, respectively. Source terms due to electron beam are calculated in accordance with E-beam model discussed later. It is seen from governing equations that flow, mostly neutral, is influenced by plasma through the source-terms, F& and W& .

Initial and boundary conditions The set of equations (1)-(3), (7)-(9) is closed with specifying initial and boundary conditions for the flow and

plasma variables. At inlet boundary (left-most boundary in Fig.3) all flow variables are specified: (ρ, U, P) = (0.38kg/m3, 420 m/s, 0.33·105 Pa; Ma ~ 1.2). At the sphere surface no-slip and wall temperature conditions are specified. No-slip and adiabatic wall conditions are set at the nozzle walls. Zero normal-velocity condition along with zero normal-gradients for other variables is set at the symmetry axis, lower-most boundary in Fig.3. At the part of outlet boundary (right-most and upper-most boundary in Fig.3) the static pressure, Pout=2140 Pa is set, and extrapolation from interior is applied on the rest of outlet boundary.

Figure 3. Fragment of computational domain and Mach number field, no discharge. Sizes are given in mm.

Plasma calculations start with the flow field obtained without electric discharge, and initial level of concentrations is specified everywhere in the domain: ne/n = ni/n = 10-9. Boundary conditions for plasma equations are as follows. At the sphere surface “cathode conditions” are set, namely:

∂ni/∂rn = 0, (Γe rn) = γe·(Γi rn), φ = 0. (15)

Here, rn is unit normal-vector, γe is the secondary emission coefficient. γe = 0.01 was taken. At the nozzle walls including central insertion, the “anode conditions” were applied:

∂ne/∂rn = 0, ∂ni/∂rn = 0, φ = V. (16)

Second of conditions (16) seems to be not completely correct, ni = 0 looks better. So, anode layer, usually thin, is neglected in the current model. The anode potential value, V, is specified from the external circuit conditions:

∫==+cathode

dIIRV Sj,00 Ε (17)

In (17) R0 is the load resistance, R0 = 10kOhm, I is the total current into cathode (sphere), and E0 is EMF value. At the symmetry axis normal gradients are set to zero for all plasma variables. The same conditions are applied at the outlet boundaries. Calculations were performed as follows. Given EMF value E0 coupled flow/plasma calculations are done until steady state establishes. Then E0 increases and calculations continue. The previous flow/plasma field obtained for lower value of EMF serves as initial guess for the consequent higher value of EMF.

Electron beam computational model The mutual influence of electron beam and airflow is the most impressive result of experimental work2. As

follows from Fig.2 serious drag reduction for the test sphere in the flow was detected with E-beam only, even without the discharge. Within the physical model tried in the paper and described above the only way for E-beam to reduce the drag is heating of gas flow. Numerical estimations of effect of heating by E-beam show that significant reduction of drag can be achieved with total heat power of order 300 W, which corresponds to 30kEV 10mA E-beam.

However, as follows from E-beam model presented, for example, in paper12 and from our own estimations, E-beam relaxation length is of order of 0.5m for the considered conditions. This means that no essential E-beam power loss should take place on the length of 3cm, distance between the E-beam gun and the test sphere. One may assume that the beam energy (except the energy to bulk ionization) goes into the sphere body and heats it up. However, test calculations of flow around a hot sphere didn’t predict a noticeable drag reduction for this situation. We conclude that at the moment we cannot explain the dramatic effect of E-beam on flow and only ionization due to E-beam is considered in the paper.

E-beam ionization model adopted in the paper reduces to specifying a space-shaped source term Q& in equations (7), (8). The total number of ion-electron pairs generated by E-beam is estimated from total E-beam current 3-10mA, E-beam energy losses 200 – 400 EV on the length of 3 cm, and ionization cost (from 12) 34EV. The shape of beam is difficult to predict because electrons of rather high energy can reflect from the sphere surface and can ionize the gas again. Therefore, shape of the beam close to paraboloidal one is specified in the current model. Inlet cross-section of this ‘paraboloid’ is taken to be 3mm (the diameter of the E-beam gun exit) and is located at the gun exit (central body edge in Fig.3). The diameter of the beam at sphere’s centre position was taken 16mm, larger than sphere diameter. Such a choice of the beam shape is thought to take into account both direct and reflected E-beam electrons near the test sphere. Also it is assumed that E-beam electrons can’t propagate to the back side of sphere. Within this shape of E-beam the exponential decay of the ionization rate is assumed on the distance between the gun exit and sphere with large, 0.5m, characteristic decay length. That is, the ionization rate decreases almost linearly and slowly along the central line. In radial direction polynomial shape is assumed such that ionization rate goes to zero at the external (paraboloidal) boundary of the beam. Finally, the ionization rate at each spatial point is calculated in accordance with taken shape functions, the total number of ionized pairs discussed above normalized to the volume of the beam.

Numerical procedure Equations (1)-(3) and (7)-(9) are discretized within Control-Volume approach. Details on solving the gasdynamic

part can be found elsewhere 9. On every gasdynamic time-step the plasma equations (7)-(9) are discretized with their own, plasma time-step, which is typically less than gasdynamic one. While plasma time-stepping the flow field is considered as fixed. On every plasma time-step the algebraic analogue of (7)-(9) represents a coupled set of non-linear equations, so sub-iterations are used to obtain solution on every plasma time-step. On every sub-iteration the linearized equations for ions, electrons and electric potential represent sparse-matrix 9-diagonal sets of equations. Each set is solved with Additive Correction MultiGrid technique 13 and Modified Incomplete Factorization 14 elliptic solver is used as smoother. 2 MG cycles were done for concentrations and from 2 to 8 cycles were done for Poisson equation. From 2 to 15 sub-iterations were required to obtain solutions on every plasma time-step. Several plasma time-steps were performed within gasdynamic time-step, and plasma-specific source-terms in equations (2) and (3) were treated as plasma-time averaged. Typically, gasdynamic time-step ranged from 2·10-9 to 4·10-8, plasma time-step ranged from 5·10-11 to 10-9, and typical time to achieve a steady-state solution is of order of 0.5 ms. To accelerate convergence to steady-state global gasdynamic iterations were done. During global iteration plasma-specific source-terms are considered as fixed in time and space, and gasdynamic set of equations is solved until convergence. Then coupled flow/plasma computations were done to correlate plasma distributions with the disturbed flow field. 2 – 3 global iterations were usually required to obtain steady-state solution for any EMF value considered in the paper. Sphere’s surface-fitted computational grid has been used with 90 uniformly spaced mesh points along the surface and 120 points in ‘orthogonal’ direction.

Results Computational domain and original flow are presented in Fig.3, where Mach number field is shown.

Characteristic dimensions shown in figure are given in mm. Specific for the considered flow is subsonic flow between central body and the test sphere. The electric discharge occurs between the central body (anode) and the test

E=1000V, U = 977V, I = 2.32 mA

E = 2000V, U = 1303V, I = 69.7 mA

E = 4000V, U = 1690V, I = 231 mA

Figure 4. Electron number density for three values of EMF, no E-beam.

E=1000V, U = 977V, I = 2.32 mA

E = 2000V, U = 1303V, I = 69.7 mA

E = 4000V, U = 1690V, I = 231 mA

Figure 5. Ion number density for three values of EMF, no E-beam.

sphere (cathode). However, it is assumed that central body and nozzle walls are maintained at the same potential value, so the nozzle walls are also treated as anode walls. In all cases considered here, the electric potential of the sphere’s surface is set to zero and anode potential is found in accordance with equation (17). First, the discharge without E-beam is considered, and then E-beam stimulated discharge is discussed.

No E-beam discharge In Figs. 4 and 5 the electron and ion number

densities are presented corresponding to three EMF values, EMF = 1kV, 2kV and 4kV, respectively. Also, the anode – cathode voltage and total current are given. One can see that the discharge develops in the vicinity of rear stagnation point. While current increases, plasma occupies larger and larger zone downstream the sphere. Specific feature of the discharge is that electric current flows mostly between the edge of nozzle and sphere; the central-body electrode makes no contribution to the current. Both this feature and the shape of plasma domain are the consequence of competition of convection in high-speed non-conducting flow and drift in electric field. The fact that discharge develops near the stagnation point is assigned to the fact that ionization rate is determined by the reduced electric field value E/n. The electric field near the surface varies rather slowly along the surface, while the density varies dramatically. The highest values of the gas density are achieved near the front side of sphere, and the lowest are achieved in the rear separation zone. So, this zone is that of enhanced ionization. This is demonstrated in Fig.6 through Fig.8. In Fig.6 the electric potential and current stream lines are shown corresponding to the cases of Figs.4 and 5. In Fig.7 the surface distributions of ion number density are presented, and ion normal current density distributions are shown in Fig.8. Negative values of ion current density (Fig.8) mean that current come

Figure 7 Surface ion number density distribution, no E-beam.

Figure 8 Surface ion current density distribution, no E-beam.

E=1000V, U = 977V, I = 2.32 mA

E = 2000V, U = 1303V, I = 69.7 mA

E = 4000V, U = 1690V, I = 231 mA

Figure 6. Electric potential (colored) and total current stream lines (black) for three values of EMF, no E-

beam.

into the surface. At low voltages the convection prevails and the current could come into the sphere only from the rear. At higher voltages the drift becomes significant in larger domain. As consequence, current-carrying surface expands toward the upper point of sphere. Note that the vortex in the rear side of sphere also helps to such expansion of plasma.

The influence of the discharge on the flow can be seen from Fig.9, where the temperature field is shown. The primary effect of the discharge is heating in the cathode layer. The temperature field is also strongly influenced by the vortex. Maximal temperatures are achieved in the region just below the separation point, i.e. in the region where the speed flow is small. While heat power rate is high near the surface within the cathode layer, heating near the rear stagnation point is less intensive because of the presence of convective transport in the rear vortex. Role of Joule heating in the rear side of sphere is the pressure rise in this region. As consequence, the surface pressure also rises along the current-carrying part of the surface. Fig.10 demonstrates effect of the discharge on surface pressure distribution. The increase of pressure in the rear side of surface results in reducing the integral of pressure over the surface, i.e. drag. Fig.11 shows that total drag reduction due to discharge is about 10% and this is almost due to heating from the discharge. Expected effect of electric force in

the cathode layer due to the charge separation appeared to be negligible. Fig.12 demonstrates some plasma distributions (case EMF = 4kV) along the line nearly normal to the sphere surface, which starts at 135 deg of the surface (from the leading sphere’ point). It is seen that typical level of the x-component of the force in the charged layer is about 104 N/m2, and thickness of the layer is about 1 mm. So the pressure drop due to the force is about 10 N, which is much less than typical value of pressure in the rear side of sphere, see Fig.11.

It should be noticed that as the discharge current grows the region of flow disturbance due to plasma approaches the upper sphere’s point, and perturbations of flow field in the flow separation zone become larger (see, for example, Fig.10). Original expectations on the discharge-aided flow control were related with this effect: to influence on the flow precisely at the separation point, to move separation point downstream thus reducing the drag. This effect takes place in the calculations, but, first, its amplitude is small for the reached currents and, second, larger currents are probably needed to move the discharge zone closer to the upper point.

Figure 10 Surface pressure distribution, no E-beam.

E=1000V, U = 977V, I = 2.32 mA

E = 2000V, U = 1303V, I = 69.7 mA

E = 4000V, U = 1690V, I = 231 mA

Figure 9. Gas temperature for three values of EMF, no E-beam.

Increasing current encounters significant numerical difficulties because discharge “jumps” in the region near the leading stagnation point, where uncontrolled rise of concentrations and current occurs.

E-beam stimulated discharge The E-beam stimulated discharge repeats many features found in the calculations discussed above. The shape of

the discharge can be seen from Fig.13 where ion number densities are shown for three values of EMF. Electron number density distributions are quite similar to ion’s one except for that cathode layer of the thickness of ~ 1mm. This discharge shape is completely determined by the specified distribution of the E-beam ionization rate. The

Figure 11 Voltage, total heat power and total drag vs. total current, no E-beam.

E = 500 V, U = 173 V, I = 32.7 mA

E = 2000 V, U = 425 V, I = 157 mA

E = 3500V, U = 565 V, I = 293 mA Figure 13. Ion number density for three values of

EMF, E-beam on.

E = 500 V, U = 173 V, I = 32.7 mA

E = 2000 V, U = 425 V, I = 157 mA

E = 3500V, U = 565 V, I = 293 mA

Figure 14. Gas temperature for three values of

EMF, E-beam on.

Figure 12. Number densities and X-component of force along the line normal to sphere at 135deg. E

= 4kV.

cathode layer is well distinguished and it is seen that ion concentration is non-uniform along the sphere surface. This is assigned to the influence of the leading edge vortex. The size of vortex is larger than cathode layer thickness, which can be seen from Fig.14. Highest gas temperature takes place in the region of vortex. Therefore, in quasi-neutral part of the plasma column ions are driven due to convection rather than drift. Even at highest value of total current there is a region within the vortex where ions move outward the sphere surface. The thickness of cathode layer near the stagnation point is slightly more than over the rest of surface. This is the effect of the vortex: in quasi-neutral region of the vortex near the symmetry line ions are decelerated by the recirculation flow and electrons follow them.

Flow characteristics within the quasi-neutral column of plasma can be seen from Fig.15 through Fig.17. In Fig.15 gas temperature and pressure distributions along the central line between sphere and E-beam gun exit are shown. Fig.16

represents distributions of electron current density and heat power along the same line. In Fig.17 electron and ion number densities near the sphere’s surface are plotted. It is seen that flow characteristics are influenced mainly by heating in quasi-neutral part of plasma. While intensive heat release takes place in the cathode layer, Fig.16, it dissipates in the region of the leading edge vortex. So, both temperature and pressure grow in this part of the discharge. There is also a remarkable heat release in the region near the anode. This is due to strong current concentration on the anode, since the size of anode surface is almost twenty times less than sphere’s one. Heating of flow in the most part of the discharge region is a potential for the total drag reduction, while strong heat release just near the surface should lead to the increase of drag due to local pressure increase. Fig.17 demonstrates typical structure of cathode layer. For the considered conditions, the thickness of layer increases as the total current grows while ion number density almost doesn’t change. As in the previous case, the magnitude of the electric force at the highest current is about 2.5·104 N/m2 on the wall, but equivalent pressure drop is about 10N, which is too small compared to surface pressure values.

Surface plasma characteristics are presented in Fig.18 through Fig.21. In Fig.18 normal-to-surface electric field strength is shown, the ion number density is shown in Fig.19, and ion current density is shown in Fig.20. Distribution

Figure 15. Distribution of pressure and temperature between sphere and E-beam gun exit.

1 – EMF = 500 V, 2 – EMF = 2000 V, 1 – EMF = 3500 V

Figure 16. Distribution of electron current density and heat power rate between sphere and E-beam gun exit.

1 – EMF = 500 V, 2 – EMF = 2000 V, 1 – EMF = 3500 V

of pressure mostly responsible for the sphere’s drag is shown in Fig.21. It is seen that as voltage grows, maximum of all presented characteristics moves to the flow separation point, which is close to the point where both pressure and gas density have maximum. While pressure grows near the stagnation point due to Joule heating in cathode layer, peak pressure decreases due to heating in the quasi-neutral plasma column. As in no E-beam case, the reduction of drag is due to rise of pressure in the rear part of sphere, Fig.21. In turn, rise of pressure is completely due to the heat input from the discharge. The influence of the electric force is small for the reason explained earlier. The effect of drag reduction can be seen from Fig.22, where the voltage-current characteristics and power input from the discharge are also shown. The magnitude of drag reduction is almost same (8%) as in no E-beam case, since total heat input to the flow is responsible for the drag reduction.

Conclusions

Coupled Navier-Stokes/drift-diffusion numerical model has been applied to study DC electric discharge in supersonic laminar flow around the sphere. The problem formulation has tried to simulate the experimental setup of paper2. Both no E-beam discharge and E-beam stimulated discharge was considered. In all simulations sphere was considered as a cathode

The main influence of plasma on the flow was found to be Joule heating in quasi-neutral region of plasma and in positively charged cathode layer. The influence of the electric force taking place in the cathode layer of small thickness appeared very small. This situation was typical for both no E-beam and E-beam stimulated discharge.

Comparison with the experimental results shows that there is significant difference in the influence of plasma on the flow found in current computations and reported in the experimental work2. First of all, drag reduction registered in experiments is much larger than those obtained numerically. Possible reasons for this discrepancy seem to be the following. First, laminar flow model has been considered, while a flow in experiment was probably turbulent. Second, lower pressure and density reported in paper2 could also probably influence on the computation results due to the change in one of the principal parameter, E/n. Third, the simplest plasma model has been considered in the paper which doesn’t take into account non-equilibrium transport of energy in low-density flows. In addition, effect of high-energy ions taking place in cathode layer on the momentum transfer wasn’t also considered. Finally, the validity of the drift-diffusion model, at least within the cathode layer, remains open.

Figure 17. Distribution of electron and ion number density near the sphere’ surface.

1 – EMF = 500 V, 2 – EMF = 2000 V, 1 – EMF = 3500 V

Figure 18. Distribution of normal-to-surface electric field strength along the sphere surface.

1 – EMF=500V, 2 – EMF=2000V, 3 – EMF=3500V

Figure 20. Distribution of ion current density along the sphere surface.

1 – EMF=500V, 2 – EMF= 2000V, 3 – EMF=3500V

Figure 19. Distribution of ion number density

along the sphere surface. 1– EMF=500V, 2– EMF=2000V, 3–

EMF=3500V

Figure 21. Distribution of pressure along the sphere surface.

1– EMF=500V, 2– EMF=2000V, 3– EMF=3500V

Figure 22. Voltage, total power, and total drag vs. total current, E-beam on.

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