Electric drag

Planet. Space Sci. 1970, Vol. 18, pp. 1035 to 1041.

Pergamon Press. Printed in Northern Ireland

ELECTRIC

DRAG

G. FOURNIER OfficeNational d’Etudeset de RecherchesMrospatiales, 92-Chatillon,France (Received

12January 1970)

Abstract-Starting from a hybrid calculusof the collisionlessplasmaflowaround a cylinder,the coulombiandrag on the body is computed. Three methodsare comparedand their resultscheck the validity of the calculus. However, one of these methods is shown to give a poor accuracy. 1. INTRODUCTION Two sophisticated methods are numerically used and compared with a more elementary one. These computations test a hybrid calculus of mesothermal flows of collisionless plasmas around cylindrical bodies (Fournier, 1969) and show that one of the methods is not accurate at all. With a spherical body, Prager (1967) tried to compare the two methods and succeeded despite many approximations. When a rarefied plasma flows around a conducting body at mesothermal speed, an ionless space occurs downstream, producing a deformation of equipotential surfaces (blown sheath). The charged body (at a negative potential of about 1 V with respect to the plasma potential for an ionospheric satellite by night) acts upon the ions so that: (1) the number of ions striking the body surface is different from what it would be without any electric field; (2) the momentum of an ion arriving on the surface is changed from its initial value; (3) there is an exchange of momentum by coulombian interaction between the body and the ions which do not impinge on it. All these contributions are gathered in the coulombian part of electric drag, quantitatively studied with the following assumptions: (1) the body is a conducting cylinder with circular cross-section and infinite length (no end-effects); (2) the plasma is collisionless in the interaction region with the body; (3) it is made of neutral particles (n), electrons (e) and one kind of singled charged ions (i); (4) out of the perturbation region, each species is in Maxwell-Boltzmann equilibrium at a temperature Tno, T,, or T,, respectively and the plasma is neutral: nLo= nio, n being the density; (5) the component of the plasma mean speed perpendicular to the cylinder axis is mesothermal, i.e. greater than the ion thermal speed and lower than the electron one; (6) the magnetic field is neglected, so that the inductive part of electric drag is not taken into account; (7) the electrons have a Maxwell-Boltzmann equilibrium in the electric field: the body potential with respect to the plasma is negative enough so that this is true in the computation; (8) a steady flow can occur; (9) there is no photoelectric emission by the surface, the impinging electrons are absorbed by the body surface and the ions neutralized; their reemission as neutrals is not considered and the following drag is only due to ions; i.e. an ion with mass M and speed w, when striking the body, gives it a momentum Mw. Because of the low mass of an electron compared to that of any ion and because of the 5th assumption (mean speed of electrons much lower than their thermal speed), the momentum brought to the body by impinging electrons is neglected. The neutral drag is not taken into account here. 1035

1036

G. FOLJRNIER

The coordinate system is bound to the cylinder, the axis of which is Ox; the streaming speed is in the xOy plane: MKSA units are used. 2. FIRST METHOD FOR COMPUTING

THE DRAG

Let 5” be a surface surrounding all space where there is an electric field due to interactions between body and plasma: the only particles whose momentum is changed move inside S’. According to a 10th additional assumption-every particle falling into S’ leaves it or is absorbed by the body-the plasma loses, in unit time, a momentum corresponding to the drag on the body, which is equal to the y-component of the resultant of the moments going across the surface S’ during unit time at steady state (8th assumption), This method is especially interesting when the ion temperature is neglected because the ion stream is monokinetic (Jastrow and Pearse, 1957; Davis and Harris, 1961; Maslennikov and Sigov, 1967). It is then possible to distinguish between impact drag and deflection drag : (1) the impact drag is given by the momentum brought to the surface S’ in unit time by the particles about to impinge on the body surface. When the surface is cylindrical or spherical, these particles have an impact parameter lower than the critical impact parameter corresponding to the grazing trajectory; (2) the deflection drag corresponds to the momentum given to the body in unit time by the particles of which the interaction with the body is only coulombian; their impact parameter is greater than the critical one. The difficulty lies in computing the difference of incoming and outgoing momentum of a particle when crossing S’. In the case of electric fields obtained by rough approximations, these considerations are used in simple computations, where deflection drag is often small compared to impact drag. It is important to note here that impact drag is not the drag due to impacts. This is because the particles, when they strike the body, have a momentum different from that which they had on entering the surface S’. The difficulty increases when the ion temperature is taken into account: it is then necessary to know the ion distribution function at every point of S’ and it is no longer possible to argue about a critical impact parameter. The calculus requires the solution of a system of self-consistent equations (Vlasov-Poisson) connecting the electric potential V and the distribution functions. Taylor (1967) gives a numerical result with a first order computation. The details of formulation will be given in Section 4. 3. SECOND METHOD FOR COMPUTING

THE DRAG

Prager (1967) draws up a balance of forces acting upon the surface S of the body. The1 e are two kinds of forces: (1) those due to ion impacts: it is the momentum along 0y absorbed in unit time by S; (2) those due to coulombian interactions, called Maxwell drag, and computed from the values of the field on S (Section 5). This method requires the self-consistent results for electric potential and distribution functions on S. Analytical appro~mations led to a rough estimation for a sphere (Prager, 1967). A hybrid calculus of the flow (Fournier, 1969) gives all data necessary for obtaining the drag by both methods which ate then rather easy to use. Both methods bring into play the determination of momentum crossing a surface in unit time, as considered below.

1037

ELECTRIC DRAG 4. CO~UTA~ON

OF THE MOAN CROSSING A CUBICAL SURFACE CELL IN UNIT TIME

Let dS be a circular cylindrical surface cell of unit length at a radius p from body axis, the unit vector of which forms an angle 0 with the Oy axis; let F be the ion distribution function along Oy and Oz at a point of coordinates (p, 0). The stream-wise (Oy) momentum, .%Jrentering the surface cell, dS, in unit time is +m -Mw,(w, cos 8 -l- w, sin 0)F dw, dw,, _C+ D,dS=dS JJ --m where M is the ion mass, w, and w, the components of its velocity w at point (p, 8). Introduction of non-dimensional variables f=:0%$ c=1/2k;&4 ’

and

dI=--&,

where k is BoI~ann

constant and n,, the unperturbed ion density, leads to +m (cv2 cos 8 + cycp sin @fdcydcB. d,=-2 (1) ss The computation of the scale: distribution function, f,has already been discussed (Fournier, 1969) and will be merely sketched here. The problem is governed by the VIasov equation for ions, Lowell-Boltzmann equilibria in a potential, k’, for electrons and the Poisson equation for the potential. An iteration between the Poisson equation, handled by electric analogies, and the Vlasov equation, keeping the value of the distribution function along an ionic trajectory, gives a self-consistent solution for densities and electric potential. The integral (1) is only an option of the trajectories program.

5. COMPUTATION OF THE MAXWELLDRAG To compute the stream-wise (Oy) force upon the surface S due to coulombian interactions with the plasma, it suffices to use the electrostatic pressure, computed with the total field given by the Poisson equation: this is the scalar pressure exerted on a conducting surface cell, dS, of coordinates (p, S) by the plasma charges, and the whole superficial charge of the body except that of dS. If - V V is the electric field on dS, the electrostatic pressure .c~(VV)~/~and the stream-wise (Oy) force acting on dS is: D,=-

%tv02 cos 8 2

’

where E,,is the dielectric constant in vacuum. Integrating D, over S, the interaction forces among the different parts of the body cancel and the value obtained is indeed the resultant of the body-plasma coulombian interactions. The introduction of non-dimensional variables,

-qv

P

&I

d,,, = n&T, where 4 is the electron charge and R the body radius, leads to P=kT,’

‘==?I

and

,

(2) la = ~&~~~~~~~q2 being the Debye length.

1038

G. FOURNIER 6. CALCULATIONS

The calculations carried out concern a flow determined by the four following nondimensional parameters:

T,Q T,=”

R - = 15, h

Al*= -2.15,

U,=

6,

where vB is the scaled body potential, and U, (Mach number) is bound to the unperturbed plasma stream speed by the relation

(3) The first method is used with cylindrical surfaces S’, the cross-section of which is made of a half-square DABC joined to a half-circle, CD, of radius p (Fig. 1). On the contour

Fro. 1.

t!hJRFACE s’.

DABC, with p > 5R, the ion distribution function is that of the ambient plasma, i.e. a drifting Maxwellian. At point E on Oy axis, formula (1) becomes: cu2exp [-cz2 - (c, - U&2]dc,, dc,,

(4)

or d, = 1 + 2U,,2,

(w2U02

for

UO> 1).

The ambient distribution function being symmetric about the plane xOy, the mean momentum crossing DA and BC is null and the unit length surface DABC with OD = p is crossed by a momentum in unit time: %(D~LBC)= (1 + 2%2)2~~&Tro.

(5)

Along the contour CD, the result is obtained only numerically. Some difficulties arise in choosing p; S’ was defined as a surface out of which ‘there is no more field’. The calculation supposes, indeed, that the plasma is unperturbed at p = 100 R and that for p > 10 R the interactions are negligible; it is theoretically possible to check this with two values of

1039

ELECTRIC DRAG

p between 10 and 100 R. Results are given with p = 18.18 R and 10 R, 9zcco, = 2472Rn,,, kTio and L@ICoD, = 1296Rni,,kTi, resp. ; the accuracy is about 2 per cent in both cases. The drag on a unit length cylinder is then: 9 = (180 f 40)&2,&T,,

from

p = 18.18 R

( 9 = (164 5 24)Rni&Ti0,

from

p = 10 R.

(6)

By this method, the drag is obtained from the small difference of two large quantities and the accuracy is hence poor; thus the far interactions cannot be shown between 10 and 18.18 R. In order to compare these results with simplified theories without ion thermal speeds contribution, it is interesting to compute the number of ions absorbed in unit time by unit length of body, i.e. the ion current divided by a factor -9. As for the momentum, the balance may be computed on the surface itself of the body, or through a remote surface S’ of cross-section ABCD (Fig. 1) and unit length. The current density Ji on a circular cylinder is related by: Ji=

ss

m q(wy cos 8 + w, sin fl)F dw, dw,, _-oo

or, non-dimensionalizing: ii = -

(c, cos 8 + c, sin 0)f dc, dc, = -vi(U,

cos ~9+ U, sin t3),

(7)

with vi =

z

and

ji

=

SO

-Ji

nioqd2kTd~’

This formula (7) defines clearly U, and U,; n, is the local ion density; vi, U, and U, are given by the trajectories program. The integration Ii ofJ over a unit length surface is Ii = 12.32 f 0.24

from p = R,

Ii=

from p = 18.18 R,

15&4

Ii = 14 f 2

(8)

from p = 10 R.

The last two values, though consistent with the first one, are of low interest because of their poor accuracy. This current agrees with that of an ambient monokinetic ion stream: its density is n,,, the stream speed is also w. and the corresponding critical impact parameter is p1 = (l-027 f 0.020) R. Now the grazing trajectory of the stream in the previously computed field (with a non-vanishing ion temperature) has an impact parameter pn = (1.040 & 0.003) R.

To p1 and pz are related impact drags (in sense of Section 2)

(9)

1040

G. FOURNIER

Total drags (6) are also given in function of (R~&G?w,,~)with the help of relation (3) in Table 1. The second method leads to: g1 = (154.6 & 3*1)Rni,&Ti0, for the drag due to impacts (not the previous impact drag). The Maxwell drag is Bm = -4*1Rn,kTi,, with -7f%ni,,kTi,-, on the upstream face and +3.4Rn,kTio in fact a thrust. So the total drag is

on the downstream face; it is

LB = (150.5 i 3*1)Rn,,,kT+ All these results are shown in Table 1, scaled by R~,Mw,,~ instead of RniokT,. TABLE 1. DRAGS SCALEDBY Rni,,MwB2 Drag due to impacts (2nd method) Maxwell drag (2nd method) Total drag (2nd method) Total drag (1st method with p = 18.18 R) Total drag (1st method with p = 10 R) Impact drag corresponding to ioncurrent Impact drag from grazing trajectory

2.15 * 0.04 -0.06 2.09 f 0.04 2-50 f 056 2.26 f 0.33 2.05 f 0.04 2.08 zt 0.01

7. CONCLUSIONS

All the previous results are consistent one with another. This is a good check of the hybrid calculation used. However, the first method has very poor accuracy; a more sensible choice of the surface S’ would not improve it much. It is perhaps the reason why Taylor (1967) finds L8/2Rn,MwOz = 1.77 instead of 1.04 as in the present note; the biggest value found above from Table 1 is, in effect, l-53. In the calculation carried out here, the Maxwell drag is but a corrective term of the same order of magnitude as the error on the drag due to impacts; this is because the electrostatic pressures upon the upstream and downstream faces are partly balanced. The elementary assumption of a monokinetic ion stream is quite correct in this case: the determination of the grazing trajectory leads to an ion current and a drag (though neglecting the deflection drag) more accurate than those of the first sophisticated method. Thus, a complex method (Prager, 1967) has proved the validity of a simple one (and the first in time) (Jastrow and Pearse, 1957), in agreement with experiment (Pitts and Knechtel, 1964), while the difficult method of balance through a large surface has been shown to give poor results. REFERENCES DAVIS, A. H. and HARRIS, I. (1961). Interaction of a charged satellite with the ionosphere. In Rarefied Gus Dynamics (Ed. L. Talbot), Suppl. 1, p. 691. Academic Press, New York. FOURNIER, G. (1969). Sur l’ecoulement mesothermique d’un plasma sans collisions autour d’un cylindre conducteur. C.r. Acdd, Sci. Paris A269, 605. JASTROW,R. and PEARSE,C. A. (1957). Atomspheric drag on the satellite. J. geophys. Res. 62,413. MASLENNMOV,M. V. and SIOOV, Yu. S. (1967). Discrete model of medium in a problem on rarefied plasma stream interaction with a charged body. In Rarefied Gas Dynamics (Ed. C. L. Brundin), Suppl. 4, Vol. 2, p. 1657. Academic Press, New York.

ELECTRIC

DRAG

1041

PIITS, W. C. and KNECHTEL,E. D. (1964). Experimental investigation of electric drag on satellites. AZAA Paper No. 64-32. PRAGER,D. J. (1967). The Flow of a RarefiedPlasma Past a Sphere. University Microfilms Co., AM Arbor, Mich. TAYLOR,J. C. (1967). Disturbance of a rarefied plasma by a supersonic body on the basis of the PoissonVlasov equations; 1, the heuristic method. Planet. Space Sci. 15, 155.