increases to 0.1, so the main beam acquires small drift as the

tail suddenly emerges. The displacement of the main beam is then % increased as 0 rises to 0.3,

At 0=0,005, a few plots to growing M show no emerging reflected ion flux at all.

Fig. 8.1 is a ratio of two more directly interpretable quantities.

Thus the downstream thermal energy of the ions is T^^ = T^ + T^ as # above, while part of this heating is given by the adiabatic (no

Ti-l

increase in energy) compression T\^^ = T^^ * B^re Y^=2 as required by the laminar field dynamics, and as qualitatively

suggested by the experiments of Kornherr and Schumacher. Each of these : t

137.

components having direct physical meaning should give new information, The adiabatic levels are not an unknown of the shock problem, and may be calculated from the Rankine-Hugoniot relations, and the initial ratio T ^ / T . Useful further, for comparison purposes, is the ratio

Downstream temperature is shown in Fig, 8.8. Together with totals, given by the hard curves, are shown the usual transmitted ion contr­

ibutions (dotted curves) and the adiabatic level (dashed curves). Non- adiabatic heating is now the difference between the hard and dashed curves.

FIG 8.8

Such heating is clearly falling more rapidly than the adiabatic. The familiar asymptotic behaviour at large M is shown by the latter. The argument following Fig. 8.1, is easily applied to the explanation of these curves. Adiabatic (and other) levels are seen to rise from

H

M=l, in a region where density is increasing faster than initial temperature is falling, with M at fixed J3. The enormous increase in ion energy in going from 0=0.1 to 0=0.3 is demonstrated. This matches the rapid rise in ^ (of Pig. 8.3), while initial ion

temperatures rise. Then as ^ goes asymptotic to (|)* (approximately) , low ion temperature causes a fall in T^g.

The temperature jump in going from the up- to the downstream is shown in Pig. 8.9

71%

w

FIG 8.9

The Culham shôck is at Jyi^3.5, 0=0.1, In the perpendicular

direction, the ions have been heated about 16 times. The low 0=0.005 shocks, show the ions heated about 5 times (most of this is adiabatic). The transmitted components are quite interesting. While total and transmitted energies increase when 0 increases, the energy jump of the transmitted has decreased. The measure of transmitted ion heating

is made downstream by integration over this beam only, and is quite accurate. In the upstream it is necessary to decide whether the temperature of the ions, to be transmitted, is equal to that of the whole initial ions, or to that of the initial ions with the reflected

Cto be) ions removed. Here the initial ion temperature is used so that an underestimate of transmitted ion heating might occur. The "cut" made by the electric potential in f^, into the two ion types is not simple (Figll.13, below), making estimates of separate beam densities obtainable only by numerical integration of the upstream once the cut has been determined. However, as long as the ions are not too hot, with a few reflections occuring, the error is small This error will increase with P.

The adiabatic heating is calculated using the measured density ^2M* ^^Gn due to increasing loss of density to reflections, with Increasing j3, the temperature jump of the main beam may get smaller. This can be illustrated, too, in the easily and accurately generated Lg=0 case {Fig. 8.2),

The variation of heating with P is quite slow, as may be seen from Fig. 8,10.

FIG 8.10

Taking sections of the heating surface T/T (over M,|3-space) , at constant M, gives variation with (3 at M=l,5, 2.4, 6,0, At small a behaviour similar to that at small M occurs. Thermal spread is so low that no reflections occur and only small heating is

expected. At 0=0.005, the L ^ O and L^=l/6 cases are more or less the same, and even at large Mach number, low-0 heating is small. It has been noted above that at 0^0,04, reflecting ions occur when

. Thus at M=l,5, the curve remains slowly increasing while there is quite rapid increase for higher M=2,4>M* and M=6,0>M*. A s ,0

increases now, or as thermal spread increases, so there is significant continuation of the trend. But now the trend may continue as 0 goes

141.

C

At low 0 there is a fall in When M>M*, ^ then remains roughly constant with 0. For M<M*, the curve falls all the way to 0=0,3, There is again only slow dependence on so that ion thermal spread must be invoked to explain reflection contributions to Fig, 8,10, The lower curve is of interest. No reflecting ions occur at 0=0,005, and

greater than 0,3, so long as the model holds. So, Fig, 8,10 shows 41 "f increased non-adiabatic heating with 0, The behaviour of the M=l,5

curve, contrasted''with that of the transmitted ions at higher M,

again shows the decreasing effect of the main beam as its density • | diminishes at the expense of reflections. |

The variation of initial temperature with 0 may easily be seen from Fig, 7.3, The other important factor in ion heating is the

electric potential. This is shown in Fig, 8,11, *

(p

the model is clearly finding it easier to slow the ions as P increases. Thi.s is of course due to distortion heating of the ion Maxwellian even at M3M*. (The value (})*, required to slow the cold ion beam increases very slowly with ). Now at higher Mach number, much greater ion heating is observed as shown in Fig. 8.10, so that similar and exag­ gerated behaviour of ^ at high M>M* is expected. This is patently not - the case. Growing reflections are forcing an increase in (j) to effect their proper slowing consistent with , downstream. It has been seen above that the model will sometimes fail to adequately slow the ions

(Garching parameters), whatever the value of (}>, This increasing potential at high P heralds the collapse of the laminar model in an extrapolation to higher J3 .

It is expected that as (}> increases, to try and slow the fast ions, so the flux of ions should increase. These are shown in Fig, 8.12.

û'i

The curves are at first sight slightly odd, for (nv)(M=2.4)> {nvj =6) at j8=0,3, where (f) is increasing. This curve should be inspected simultaneously with Fig, 8.4 for flux variation with M- there this fact is explained. Here the variation at constant M is considered. Then the intriguing event is the large slope of these curves at M=2,4, 0=0.3, where reflections are a maximum. The

increase in c}> should be most noticeable in this region, and indeed, the M=2,4 potential shows traces of a more rapid- climb, than does the M=5 curve. At M=10, low flux is apparent^ and previous para­ graphs have suggested that in an extrapolation to high M, reflected, ions disappear entirely. It would then be expected that at M=10, ^ has the same behaviour as the M=1.5 case. Indeed it is found that (j) decreases monotonically on this range of 0 .

Then the evidence above, as concerns the ion heating may be summarized as follows:

(i) First, in variation with M, there is a low M<M* region (for all 0) where few reflections occur, with some heating due to distortion of f^ by the potential—but the shock is weak and the effects likely to be small; when M^M*, significant reflections rapidly appear with a

strong .contribution to downstream ion energy - in this domain a balance is struck between reflections off the electric potential and

the ability of that same potential to properly slow the ions; when M>>M*, the ions become so cool (at fixed 0), that no significant ion behaviour can be expected, heating stays low, and the electric

potential approaches a level where R^^l.

Strong visual support for these points is given by a rapid progression of displays of the downstream ion distribution f^, over its two-dimensional velocity space. Then at 0=0,1 and in an order M=l,5, 2,4, 4, 6, 10, Figs 8.12, 13, 14, 15, 16, are presented.

'— 6 I* li z

Ill

0

6

11

0