Contains reports on two research projects.WADC Contract AF33(616)-3984Servomechanisms Research Laborator

Lewis, A. T.

Nisbet, I. C. T.

Oddson, J. K.

Woodson, H. H.

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III. PLASMA DYNAMICSProf. S. C. Brown Prof. H. H. Woodson D. T. KyrazisProf. O. K. Mawardi Dr. I. C. T. Nisbet A. T. LewisProf. D. O. Akhurst D. A. East J. K. OddsonProf. D. C. Pridmore-Brown Z. J. J. SteklyA. MAGNETICALLY DRIVEN SHOCK TUBE STUDIESThe steady-flow solution of shock waves, which is valid for pressure-driven shocktubes, is not applicable to magnetically driven tubes because the conditions of constantdensity and pressure behind the shock front and constant driving pressure are not met.A treatment that approximately satisfies all the conditions, however, is G. I. Taylor'sblast-wave solution (1).This report gives a one-dimensional solution that uses Taylor's method. It isassumed that there is an instantaneous release of energy, E, per unit area in an infiniteplane at x = 0, t = 0. The boundary condition at x = 0 is such that the shock propagatesonly in the positive x-direction. A cylindrical section perpendicular to the plane ofenergy release may be taken as the shock tube with all heat conduction, viscosity, andradiation losses neglected. The gas is assumed to be polytropic (ideal with constantheat capacities). Strong shocks, with no ionization and with particle-isentropic flowbehind the shock, are assumed.The flow equations behind the shock are:Equation of motion du + 1 dp= 0 (1)dt p dxEquation of continuity p+ - (pu)= 0 (2)Adiabatic equationS = 0  (3)where u is the flow velocity, p is the pressure, p is the density, S is the entropy, andy is the ratio of specific heats.The solutions of these equations are known as "progressive waves" (2); the velocity,pressure, and density distributions do not change in shape, but only in scale, as theshock wave advances. We may then writep fl(Po X (4)P = = 4 ) (5 )POThis work was supported in part by WADC Contract AF33(616)-3984 with theServomechanisms Research Laboratory, M. I. T.(III. PLASMA DYNAMICS)(6)X1/2where n = x/X is the normalized position, with X, the shock position.the flow equations and separating X, we haveX1/2 dX = Pof 1dt ox1/2 dX =1 +dt+ 1Substituting in(7)(8)Y fl %'TfI - f 1We find that there is a solution if, and only if,X1/2 dX Adtwhere A is a constant. After integrating, we find thatx = ( A) 2/3 t2/3(10)(11)The functions fl(T) and P1 (-q) mayknown that for strong shocksbe made nondimensional. At the shock front, it isp _Zy M 2p y 1p _ +lP y -lu 2Vs y+lswhere V s = dX/dt, the shock velocity; and M = Vs/a o , the shock Mach number, with a o ,the velocity of sound ahead of the shock. Therefore, we can write2f (rl) =  f(r)a 0Equations 7, 8, and 9 can now be solved for the functions f(TI), i(,]), and 4().result is(12)(13)(14)(15)(16)The+1?~x1/2 dX 1 (Yfl fi(III. PLASMA DYNAMICS), 2f' - 4-- 2' (-Y (17)2 (T1I-)' (18)f, _f y + 2(11-) (19)2 f - (0-) 2These equations can be solved numerically with the use of boundary conditions(Eqs. 12-14):f(1) 2y (20)y +1(1) 2 (21)y+ 1+(1) - y 1 (22)y-1The solution is plotted in Fig. III-I for y- = 1. 4, the value for hydrogen gas.The constant A can now be evaluated from the condition that the total energybehind the shock remains constant. Here, the original internal energy of the gas isneglected in the strong-shock approximation.Fig. III-1. Normalized pressure, velocity, and density distributionsfor y = 1. 4 in an impulse-driven shock tube.(III. PLASMA DYNAMICS)1 2E X  VT )E = p vT+ Tu dx=Substituting Eqs. 4, 5, 6, 15 and 16 in Eq. 23, we obtainE = p A2  1 f(r) 2 ) )d (24)or E = CO po A2 , where Co is the integral that depends on y alone. CO can be evaluatedby numerical integration and is found to be 2. 04. Equation 11 then becomes= E 1/3 t2/3 (25)andV = 2 E 1/3 (26)Vs  3\ptThe behavior of the shock as a function of these variables may be determined experi-mentally. Our treatment also gives the distributions of pressure, density, and temper-ature behind the shock which may be taken as approximately correct for those regionsin which the assumptions are valid; that is, immediately behind the shock.S- kI CMFig. III-2. Smear camera photograph of shock wave from 25-kvdischarge into hydrogen at pressure of 1 mm Hg.The T tube described in Quarterly Progress Report No. 51, page 28, has been usedwith a smear camera in an attempt to verify Eqs. 25 and 26. A typical smear photo-graph of a 25-kv discharge in hydrogen at a pressure of 1 mm Hg is shown in Fig. III-2.The dark markings are 1 cm apart, and the light scale is in microseconds. These curves(23).1 I I C -~-.--1~---~~-~-3 --(III. PLASMA DYNAMICS)are plotted with log X versus log t in Fig. III-3 so that the slope is the exponent n inthe relation X ~ t n . This is found to be n = 0. 72, which indicates that the shock is fasterthan the predicted n = 0. 666. The only explanation for this is a continued driving of the3020 -E 'oo- 7o200.3 0.5 0.7 I 2 3 5 7 10 20Time (,.sec)Fig. 11-3. X plotted against t for several smear photographs.shock for the first half-cycle of the capacitor discharge; that is, energy is not put inimpulsively.An attempt was made to check the dependence of shock velocity on energy and densityto verify thatV5 ~s )1E/2 (27)at a given position along the tube. The errors in measured quantities are too large,however, to provide a check. Also, as the density is changed, the amount of energyactually delivered to the gas is changed, and no way has been found to actually measureit.Equation 25 does give an estimate of the efficiency of energy transfer from thecapacitor bank to the shock wave. With r R2 E representing the total energy deliveredto the shock, and W, the stored energy in the capacitors,(III. PLASMA DYNAMICS)rR 2  2r R2 X3= SR 0 (28)W CV 2t2The value was found to be between 0. 10 and 0.20 for most shocks.A. T. Lewis, J. K. Oddson, H. H. WoodsonReferences1. G. I. Taylor, The formation of a blast wave by a very intense explosion,Proc. Roy. Soc. (London) A201, 159-186 (1950).2. R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves (Inter-science Publishers, Inc., New York, 1948), p. 419.B. CHANNELING AND COMPRESSION OF A PLASMA BEAMWork on this problem has been concentrated on analysis of the modifying effects ofcollisions on the results previously obtained for the rate of growth of the boundary layerbetween the plasma beam and the magnetic field. Under the circumstances of the experi-ment, the most important type of collision is that between the ions in the beam and thestationary neutral particles in the vacuum tube. Collisions involving protons will alwaysbe of negligible importance at the densities involved in these experiments, but the plasma+beam also contains H2 ions which have a much higher collision cross section. The anal-+ysis indicates that collisions between H2 ions and neutral molecules will be the maincause of growth of the boundary layer at base pressures greater than approximately3 X 10-4 mm Hg, but will be negligible at the lowest pressures used in the experiments.By adjusting the base pressure, it will thus be possible to isolate the effects of colli-sions and to make an experimental test of the theory.In addition, the integral equation for the boundary-layer thickness has been solvedfor the case in which the angular spread of the particles in the plasma beam is verysmall. Explicit results have been obtained for the rate of growth of the boundary layerand the maximum compression obtainable under the experimental conditions.I. C. T. Nisbet

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