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A magnetohydrodynamics study using an electromagnetic shock tube Offenberger, Allan Anthony 1963

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A MAGNETOHTDRODYNAMJ.es STUDY USING AN ELECTROMAGNETIC SHOCK TUBE by ALLAN ANTHONY OFFENBERGER B.A.Sc, University of Bri t i s h Columbia, 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of PHYSIOS accept this thesis as conforming t o the standard required for candidates for the degree of MASTER OF APPLIED SCIENCE THE UNIVERSITY OF BRITISH COLUMBIA February, 1963 In presenting this thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of "t^ k.^  SI l C . -^  The University of British Columbia, Vancouver 8, Canada. Date ftf r> \ i- \°j 1 3 . ABSTRACT This thesis i s concerned with a theoretical and experimental investigation of Alfven waves i n an ionized medium, and magnetic inter-action effects between a moving plasma and a magnetic c o i l external to the plasma. Methods for generating Alfven disturbances f o r varying condi-tions of gas density and magnetic fields are considered and various means for measuring any effects that may be produced. It w i l l be seen that for propagation of m.h.d. waves, extremely strong coupling between the plasma and f i e l d i s necessary with consequent necessary high f i e l d s and Alfven speeds. The effect of an axial magnetic f i e l d modifying the shock speed in the plasma i s investigated and also the effect of the f i e l d on incident and reflected shock speeds by placing a plain obstruction i n the shock tube which blocks the plasma flow. A further study,of.magnetic interaction effects between a mov-ing plasma and a localized.radial f i e l d was undertaken with the desire of correlating mechanical momentum transfer with varying conditions of ap>-plied f i e l d and gas pressures i n the plasma (hence conductivity, density, and shock speed variations). Mechanical and e l e c t r i c a l measurements of momentum transfer are compared with theory, and i t w i l l be seen that the mechanical method offers a f a i r l y reliable means of measurement. ACKNOWLEDGEMENT I would like to express my gratitude to Dr. P . R. Smy for his aid and supervision of my thesis problem. It has been a pleasure to have been associated with him i n this research work. My thanks too, go to my fellow graduate students present i n the summer of 19&2 f o r many worthwhile discussions. I would also like to thank the National Research Council for the Bursary award which made this study possible. TABLE OF CONTENTS Page SECTION I INTRODUCTION 1 SECTION II THEORY 1. The Magnet ohydrodynajriic Equations li 2. Derivation of the Alfven Velocity h 3 . Conditions for Strong Interaction 7 k» Strong Shock Relations 8 5". Derivation of the Drag Force on the Plasma 8 6. The Waveform of the C.R.O. Voltage Trace 11 SECTION III EXPERIMENTAL ARRANGEMENT 1. The Shock Tube 12 2. The Magnetic F i e l d System 12 3 . Magnetic Fie l d Calibration 16 k* Production and.Detection of Alfven Waves 17 (i ) Probes and Camera 17 ( i i ) Searoh Coil Measurements 18 ( i i i ) Shock Speed Measurements 19 (iv) Smear Camera 19 j?. Momentum Transfer 20 (i) Mechanical Measurement = The Pendulum 20 ( i i ) E l e c t r i c a l Measurement 21 Page SECTION IV RESULTS AND CALCULATIONS 1. Fiel d Calibration 2k 2. Alfven Waves 25" (i) Probes and Camera 25 ( i i ) Measured Shock Speeds and Shock Slowing 26 ( i i i ) Tabulated Alfve'n Speeds 27 (iv) Search Coil Measurements 28 (v) Smear Camera Results 30 3. Momentum Transfer 31 (i) Mechanical Measurements 31 ( i i ) E l e c t r i c a l Measurements 32 ( i i i ) Theoretical Results 33 U. Discussion of Results (Momentum Transfer) 33 (i) Discrepancy i n E l e c t r i c a l Measurements 3k ( i i ) Invariance of (<T"LA,) 35" ( i i i ) Comparison of Theoretical and Observed Voltage Waveforms 36 SECTION V CONCLUSIONS 38 LIST OF ILLUSTRATIONS Schematic of interaction for momentum transfer Alfven disturbance Schematic for T> transfer Electromagnetic shock tube and solenoid Ionized Argon i n shock tube Field system Spark gap LC discharge c i r c u i t Search c o i l arrangement Induced voltage pick-up by search c o i l Lucite probe i n plasma flow Brass cone i n plasma flow Photo multiplier and,shock tube Parameters for the pendulum Pendulum and shock, tube Search c o i l taped to. CAI ring Relation of fields, and inductance Detail of winding for the search c o i l F i e l d calibration Search c o i l voltage trace N/' Voltage trace showing lack of synchronization between plasma and f i e l d Photograph of plasma flow for zero f i e l d Photograph of plasma flow for applied f i e l d Tabulated Alfven speeds Arrangement f o r search c o i l measurements of Alfven disturbances Plot of Figure Page 19 F i e l d pattern for no plasma, and f i e l d pattern for 28 plasma with brass reflector F i e l d pattern for plasma with no reflector Expanded time scale trace for f i e l d with plasma and brass reflector Expanded time scale trace for f i e l d with plasma and no brass reflector 20 Smear camera photograph 32 F i e l d pattern from dual winding search c o i l i n momentum transfer study-Greater time resolution of radial f i e l d pick-up for momentum transfer study 21 Graphs of B R for varying gas pressures "p0 and applied 3£ f i e l d s BT/,. 22 Graph of momentum transfer "p versus B Q 35 23 Conductivity versus Mach number and i n i t i a l gas pressures 3+ 2h V*' versus ~Z -p„ = O . S " 36 V 1 versus z j>„ » l . o V versus ~Z "P° s 0 , 1 \/' versus 2 * p « - s o > o i 25 Complete V/' trace for 37 SYMBOLS USED LL = shock speed po =» i n i t i a l gas pressure p = gas density 3 = magnetic induction H s» magnetic f i e l d E = electric f i e l d V = particle velocity j = current density OL = Alfven speed « -L ? 1 = current C V = condenser voltage of magnetic f i e l d system L = characteristic length O-o = speed of sound; i n Argon at room temperature F => force l_,M = sel f , mutual inductance C = capacitance , V => voltage 4> = flux f = period of f i e l d discharge i ^ e ^ s d , ^ characteristic parameters of pendulum p = momentum transfer k> = amplitude of induced voltage cosine curve n = number of turns per unit length X = linear variable "T = temperature JTL = ratio of magnetic energy to flow energy y U 0 = permeability C = gas conductivity "R^ = magnetic Reynolds number => charge rj&JZ - cylindrical coordinate system OJ = angular frequency = wave number f( = magnetic d i f f u s i v i t y , compression ratio \A = Mach number = ratio of pressures across shock front A = area V => volume ... "t = time = acceleration due to gravity S = skin depth X = ratio of specific heats I 1. I. INTRODUCTION An investigation of Alfven waves in shock ionized Argon,, and mo-mentum transfer from a moving body of plasma to a magnetic c o i l has been conducted with some success. The hydromagnetic disturbance which gives r i s e to Alfven waves may be thought of as an elastic stretching of magnetic lines of force 9 or the propagation of a perturbation to the magnetic f i e l d . The experimental problem was to induce Alfven waves and then to detect their propagation,in ionized gas. To this end an electromagnetic shock tube was used to provide a body of Argon plasma moving with a velocity of the order of u = 8 x 10^ cm/sec down the tube, through which an axial magnetic f i e l d B o ^ ^ i . e . i n the direction of u) was^applied (Fig. 5)» With high enough gas conducts i v i t y the plasma flow would be constrained to move along the magnetic lines of force, and so any interruption i n flow would produce a radial component of magnetic f i e l d . Downstream, various geometrical bodies were used to., disturb the f l u i d , and i n so doing Alfven waves should be propagated. In the instance where the Alfven speed i s greater than the shock speed, these waves should be detectable upstream from the disturbance. Two methods were used to detect such propagation; photographs of the flow angle past a probe.in the plasma, and search c o i l measurements of change i n magnetic f i e l d . Smear camera photographs of incident and re-, fleeted shocks on a plain obstruction were used to try and correlate i n -cident and reflected shock speeds with the applied magnetic f i e l d . The problem of momentum transfer from a moving body of magnetized plasma to a magnetic c o i l i s theoretically analyzed as a current inter-action resulting from the mutual inductance between a copper ring around the shock tube and induced currents i n the plasma. plasma The copper ring i s effectively diamagnetic and so a current i, i s induced in such a manner that i t s f i e l d ( ^jji* ) opposes the applied f i e l d ^jjj, • The current I, gives a radial f i e l d B^^which interacts with the moving plasma to give azimuthal currents of density j=cj(u.XBr). These currents in turn interact with the azimuthal current induced i n the copper ring. The resulting repulsive force .was measured experimentally by two methodss mechanical and e l e c t r i c a l . The copper c o i l was suspended as a pendulum (see Fig. 11) and the resulting l,xB rforce imparted momentum to the c o i l whose amplitude of swing was measured for various applied f i e l d s . A galvanometer mirror system was used to measure the angular deflection. The mechanical measurement too, offered direct visual proof of the interaction. E l e c t r i c a l measurement of the radial f i e l d S>J by a search c o i l technique, coupled with a .current calculation also yielded a value of the force exerted on the coil, (which i s equal and opposite to the drag force on the plasma). Theory gives a value for the force approximately four times that measured e l e c t r i c a l l y , and the mechanical measurement gives a value approximately the same as theory. 3. The method of presenting the current c o i l to the moving plasma,, i.e. by using i t s diamagnetic properties s i s preferable to pulsing an external current through the c o i l i t s e l f to provide a radial f i e l d . The c o i l can be so positioned that the magnetic f i e l d by i t s e l f offers no net force on the c o i l ; and i t i s only on pulsing both plasma and f i e l d that a measurable effect takes place., The current torques acting on the c o i l can be compensated for by the use of trimmers (which modify local f i e l d s induct-ively) and by careful positioning of the copper ring in the f i e l d solenoid. The plasma i s generated by an electromagnetic shock tube (co-planar driver), and the.axial magnetic f i e l d by a solenoid wound along the 4. shock tube and connected t o a |o joule capacitor bank. The LG discharge gives a sinusoidal f i e l d and,the experiments are conducted when the f i e l d i s maximum. Shock speeds of the order of 8 x 10^ cm/secs conductivity 7 x 10^ mhos/m, gas pressures from 0.1 to 1.0 mm Hg<, and applied fields from 2000 gauss to 125000 gauss were the conditions i n the experiments. ;„ These fields induced currents i n the copper c o i l of approximately 7400 amps at 1KV to 37*000 amps at 5 KV., The length of the interaction region is.of the order of 8 cm. A. condition necessary for both the Alfven wave and momentum transfer studies i s that the magnetic Reynolds number be high; i.e. the plasma and f i e l d be "frozen". Conditions for a strong interaction i n the. momentum transfer experiment are present since, with the above parameters, "R t^S «, However for the Alfven study i t w i l l be seen that a much higher 1^ , i s necessary. II. THEORY 1. The Magnetohydrodynamic Equations. To determine the nature of Alfven waves and their phase velocity, consider Maxwell's equations for a continuous ionized medium. Together with the Euler equation, the adiabatic gas law, and the continuity law, they constitute the mathematical basis for M.H.D. Ampere's Cir c u i t a l Law V x B syA»J and Faraday's Induction Law Ohm's Law Euier's Equation V - B = O J = <X(E + vx! ) neglecting viscous and gravitational forces. (i) (2) (3) (U) (5) Normally the term i s small and we shall neglect i t . 2. Derivation of the Alfven Velocity. Looking at the physical processes, consider the followings Z A IN 3c Fig. 2 5V Consider a long plasma subjected to uniform B Q i n the Z direction. If segment ABCD i s given velocity V paral l e l to the -raxis, then the force exerted on charge q = q(?xB) tends to separate (+ve) and (»ve) charges as shown. Because plasma can form current loops, the plasma external to ABCD forms a closed e l e c t r i c a l c i r c u i t . The induced current gives restoring force density j X©» inside ABCD; whereas i t accelerates the plasma outside. The process i s repetitious on successive segments of plasma in the z d i -rection (since f i e l d and f l u i d are constrained to move together) and so there i s propagation of the disturbance i n the iz direction. Let the field.. B « B o t B \ where 3?' i s the perturbed f i e l d . Let us consider for a derivation of the Alfven speed the simplest wave motion characterized by ^ Ee>. j e ^ B r From (1) From (5") dt J O = " j e Br - 4? From (h) (6) (7a) "(7b) (8) (9) and using (6) ( lb) but differentiating (10) (8) B 6 jSr = -L *>£s !_ (11) then or Differentiating w.r.t. z and noting from (2) that <TB; J VB; ^ B ; . . S J + > = f.D 3?- JUz*" Trying a solution of. the form S ; = B ' ^ 2 - ^ (13a) and substituting i n (12) yields for a dispersion relation, or uJ^-Jk^{ CL - L to where a N ^s. and >{•= —~. • Assuming uJt(^<OL which i s justifiable since GL%\o*vc<ss' and j(e=lo"3CGs and hence ^  would have to be a-io1* to be comparablej then <k <*• ± ^ I + • Substituting this value forJk into (13a) gives B r =• B* « " e ^ i * " * * 3 * ) (13b) where 2 0 = 5 A =. distance i n which the plane wave i s damped to '/e. of i t s i n i t i a l amplitude, and = . The Alfven wave then i s a damped prop-CL agation of the perturbation to the magnetic f i e l d i n the z direction. The distance 2^ for an i n i t i a l gas, pressure p6so.5"^\^ lAc^ . 5 B t > « 4>ooo gauss i s 51.3 metres which i s >>.,any. characteristic length in the experiment and so there should be negligible attenuation of the wave in our experiment. Thus very nearly for large (3~ ^ (13c) where / ) ( l ^ ) i s the phase velocity of the m.h.d. wave and i s > W! called the Alfven velocity. 3 . Conditions for Strong Interaction* In order to achieve a strong interaction between plasma and f i e l d i t i s necessary that the f i e l d and plasma be effectively "frozen". Con-sider Maxwell's equations; (1), (2) and (U) but J . « X. (D hence « 7 / ( v x B ) - - 1 - V/xV/a") '.(16) Now VK (yx 1>) = V ( v - ^ — V ^ B vector identity and V-B - O e H (17) Let K» o-^ d i f V / C V K B ^ H O then ^ ? m V*B which i s the form of a diffusion equation^ where ^  i s the magnetic d i f f u s i v i t y . On the other hand i f *(-*>0 then which, analogous to vo r t i c i t y i n f l u i d flow, indicates the f i e l d lines are constrained t o move with the f l u i d . The ratio of these two effects i s an indication of the coupling of f l u i d motion and f i e l d . The dimensionless parameter "KM(magnetic Reynolds number) i s a measure of degree of con-straint. Taking ,L to be a characteristic length of the system i.e . , a distance in which f i e l d variables change significantly, then define • 1 3 / u v / An indicates strong coupling between the particle motion and magnetic f i e l d , i.e. the plasma and f i e l d are "frozen". . For our conditions u = 8 x 10* cm/sec, L => 5 cm, CT= 6 x 10^ mhos/m, and hence • It w i l l be seen this ^  i s high enough for magnetic interaction effects but i s not sufficiently high for detectable Alfven disturbances. k° Strong Shock Relations. Properties of a strong shock wave give the following results; Mach number M - ^ (18) u = shock speed fl0 = local sound speed i n Argon = | Ificr = 3.2 x 10^  cm/sec at room temperature Compression ratio * - X±i (19) for Argon ^= 4-Pressure ratio across the shock front 4 - ^ = ^YH (20) /. for Argon « 760 for M = ,54.7 at gas p e = 0.S> mm Hg, B 0 = 5"0U'0 gauss. Hence p ( = 380 mm Hg. 5". Derivation of the Drag Force on the Plasma. For the theoretical background of the momentum transfer between moving plasma and magnetized f i e l d c o i l system consider the following (see Fig. 1). A body of plasma travelling with velocity U. enters the radial f i e l d region. The axial f i e l d i s generated by a capacitor bank dis^ charging through a solenoid (an LC discharge, with the experiment conducted when the f i e l d i s at maximum)., The copper ring acts as a dipole and a cur-rent i , i s induced, tending to provide an equal and opposite f i e l d ^Jj/, *° that applied, B 0/ . The copper ring has an L/R time approximately 20 times that of the LC discharge so that the radial f i e l d S r ^ d o e s not decay appreciably during the experimental time ( < 3 0 y A s e c ) . The current I, causes a radial f i e l d which interacts with the moving plasma inducing an azirauthal E f i e l d given by • £ > u x i The azirauthal current density j i s thus given by J = C T E = o r(.C< < B r ) and hence the induced current I,, i s related to and by or L ^ - cru.B^ SA The d i f f e r e n t i a l coefficient of mutual inductance between current rings L ( and Lj. gives an incremental repulsive force 3, (21) (22) (23) (24) i t * J r, - — 3 R O \ 2 * 0 Fig. 3 The position of the copper ring i s taken as the z = o origin. Now, from F * ^ ( X t f B t h e force exerted on a current carrier by an external magnetic f i e l d ; P = - airr .^ UBk- v (25) where Br^is the radial f i e l d at the current filament l v » But F'« i , U (2U) d-z-Substituting for 13 n. i n (23) from (26) and then for U i n (24) from (23) yields incremental F' = (4.M \ d A (27) 10. where dA •=*• dv^drz. . Hence the t o t a l force repulsing the plasma from the copper ring i s JA ^d&> (28) assuming crj lA. and 'U| are constant during the time of experiment. These conditions are satisfied with the experimental arrangement employed. Variables cr and ULare determined by the shock tube and L, i s kept constant by having an L / R time for the copper ring approximately 20 times the LC discharge time. The above derivation i s dependent on the assumption of total current density j being, given by j » c r u B r v entirely, i.e., neglecting any "back f i e l d " effects due to plasma currents. This i s also justifiable since any magnitude correction resulting from self-inductance i s of second order and therefore small. An alternative derivation of (28) results from the force per unit volume being | j y " B r l cruTBr and hence total j^V^^rds/ where dV - j T T r a . d r 4 . d "Z. , , and since 13 = -'AL, [C^\ F - <™£ f[ ± ( d M Y ^ c U (28> To obtain a value of, F i t i s necessary to numerically integrate expression (28), f o r F i s given in terms of an e l l i p t i c integral. However ^ bas been extensively tabulated (reference 4") i n terms of current filaments and so an accurate value for F can be obtained by numerically integrating over z and From Grover, 4^|d ^ i s obtainable from a dimensionless variable ... ^ relating the geometry and separation of the current loops. A value for F was arrived at by considering squares for dA and reducing these to equivalent current filaments (see Grover). The dimensions of the geometry are r^ = 3*65 cm, o £ r2 2.54 cm. Taking squares for $A i.e., 11. dz = 0.5 cm, = O.S>08 cm, with rg, and z taken at the centres of these squares and using equivalent filaments for dM of radius Y\= +" ^ .^1 one arrives at a value for F from the simplification where the summations are over a l l r2 and z. The data i s given i n Appendix I. The result i s - i S » v i p = CU.I, voewtonS (30) where 0")U^ L, a^e i n M.K.S. units. A plot of r" (j3f ) V ^ | r * ' versus z i s given i n Fig. U. 6. The Waveform of the G.R.O. Voltage Trace. The shape and ordinates of the f 4= 1 d ^ versus z plot should compare with the voltage traces of the time d i f f e r e n t i a l of radial f i e l d as observed on the C.R.O., for s but F= jurr. i^B^ f r o m (25) hence Sr. - ^ f f d r x d2L (31) and since _ d_z _ ,, A (32) d t " d t " d z . then Ad_B^, = __ (TU VA L , f J- [d M V" (j ^  (33) where A i s the area of the search c o i l , and so the voltage trace \J =_d<b i s proportional to f J~ / 4 d Y ' c l r i . . 3Pb 4; 12. III. EXPERIMENTAL ARRANGEMENT 1. The Shock Tube. An electromagnetically driven shock tube (co-planar driver) as shown in Fig. 5> with the solenoid in position gives a body of plasma 5—>-20 cm long travelling with a speed of the order of 8 x 10^ cm/sec,, depending on the voltage of the driver capacitor bank. This bank i s rated at 20 KV and 5>0jjSr although practically a l l experimentation was done at 16 KV. The resulting plasma i s approximately 80$ ionized and travels at Mach 2U down a pyrex tube of.2 inch,diameter around which the magnetic _ f i e l d solenoid i s wound. The electromagnetic energy from the driver bank is dumped into a cylindrical slug of atmospheric a i r which i s ejected through a mylar diaphragm and travels down the tube ionizing the Argon , therein as i t moves ( Fig. 5*.). Depending on the i n i t i a l downstream pressure of Argon, the conductivity of the gas i s otherwise determined by the energy of the discharge. I f the loss of energy in the discharge i s neglected and a l l energy i s imparted to the Argon ( an incorrect assumption since the mass of a i r retains a portion)j then, since the energy c< \j of the bank goes to the thermal energy of the molecules o1 k"T , the conductivity (which varies as ) i s therefore approximately proportional to V » The conductivity also increases, with pressure since a greater number of charge carriers are then present. .2. The Magnetic Field System. The magnetic f i e l d system (Fig. 5) i s composed of three, 221;yJ'T, 3> KV low inductance capacitors i n p a r a l l e l . Large copper strips are used magnetic f i e l d system spark gap and muffler Fig. 5 13. as leads to keep the inductance to a minimum and an a i r spark gap which- i s triggered externally i s used as the series switch to i n i t i a t e discharge. A portable D.C. charging unit was used to charge the bank through a series resistance of 27 (Cfl. mounted in the portable magnetic f i e l d unit. A paral-l e l connection of relay and 10,000 watt resistors gave a dumping bypass.for the charged capacitor bank. The spark gap construction i s shown in Fig. % with the muffler behind. This muffler was constructed of perspex and inch-thick insulating foam to deaden the noise from the discharge. The copper strips used as leads to the solenoid were insulated with polythene and clamped together to reduce inductance and prevent flap-ping from current forces exerted when the capacitors were being discharged. Insulation was used l i b e r a l l y i n the interests of safety and the whole sys= tem was enclosed with wire i n a dexion cage. In addition, 3/h inch plywood on one side, end, and top was added as a precaution, i n the event of any loose material being thrown around by a charged bank. The solenoid was wound of l/lx inch copper tubing with a total of 21 turns of separation 11/2 inches. Braced wood spacers arranged equilat-erally around the circumference of the solenoid provided r i g i d i t y for the assembly. The f i e l d discharge i s a damped LC sinusoid. -vvw-L s Fig. 6 Kirchoff's Law V - i K 4- L d i = % (3M 111. Differentiate w.r.t. t , then or d i v L- dt t-c-41^  +- . 2 * 0 0 4^ ' + u f L (35) d t where u>*"= (36) and ^|oo = JR , whence ,« K ^~cT ^ j_ m (37) , . -flat , — , At t = o, V = V 0 and i = o ; hence the current i s given by L»J. T T «. sim^vi-?*1-where iam V0)j^ assuming small resistance. The period of discharge (neglecting ;£ v correction) i s t = L C . (38) For small *^  then, (neglecting and higher order terms) L « L . * : f u y t a m u i t (39) which i s a damped LC sinusoid. A self-inductance calculation for the solenoid (from formula and tables in the "Rubber Handbook of Physics and Chemistry") gives L ^ . 6.5"^xV\ f o r a geometry ofs length of solenoid = 76.2 cm mean radius of solenoid = 5.5 cm diameter of copper wire used in solenoid -=10.6h cm number of turns =21 Allowing for one large turn from the leads of the magnetic system, an inductance of approximately 2^ *1^  i s present. Hence total inductance cs. , capacitance = t * = 0 , 4 7 + millisec From the measured period of the discharge (C.R.O. trace), t ~ m i l l i s e c which agrees very well with the calculated value. 15. We wish now to calculate the maximum f i e l d available from the system. Energy of the bank = l / 2 CV2 - l / 2 (6?2 x 10~6) (5 x IO 3) 2 = 8I(.00 joules. Maximum current I r a neglecting resistance, i s given by 'A. (Uo) V s.svoo = 4.4 ^X«o4 0 -^7* . hence "B^ -=^ u a n n. = turns/m of solenoid. or l5"4oo ^o.O'sr. Alternatively we can consider the energy of the condenser bank going into magnetic energy of the solenoid. where the volume i s that of the solenoid, V « Alj which yields a % » 17,000 gauss. Correcting for the inductance of the solenoid gives = 15,000 gauss. , From a preliminary calibration of the f i e l d using 69 volts across the gap, a maximum f i e l d ..of-11,600 gauss i s obtainable on scaling up to a supply of 5 KV. In actual fact the maximum f i e l d attained i n f i n a l c a l -ibration was ll,Ii5"0 gauss. The magnetic f i e l d generated i n the solenoid varies linearly with the i n i t i a l voltage of the condenser bank and a l l ex-periments were conducted at a time^ near the maximum of the f i e l d , i.e., at 0.12 millisec after triggering. In fact the applied axial f i e l d slows the plasma somewhat (from M = 2k to M = 22 at an applied f i e l d of £040 gauss 16. with gas p 0 = 0.5 ran Hg) due to some radial f i e l d i n the solenoid, and synchronizing plasma and f i e l d meant conducting the experiment at a dis-tance S«u.2L down the shock tube from the driver. With a shock speed of 4-7.8 x 10* cm/sec the test section was located 93 cm from the driver to have f i e l d and plasma synchronized. 3. Magnetic Field Calibration. The magnetic f i e l d calibration was accomplished by mounting"a single turn search c o i l around the shock tube and pulsing the solenoid discharge as shown in Fig. 7-solenoid shock tube <-leads 'to C.R.O. Fig. 7 The LC sinusoidal discharge gives a sinusoidal f i e l d pick-up of the flux through the search c o i l . Since the voltage pick-up V* i s given by the time rate of change of flux through the c o i l , then since 4> = ABwhere A i s the area enclosed and i s the instantaneous f i e l d , hences y' - s - Z c I S • 31: or B- - - -L f V ' a t (U 3) Therefore integrating the voltage time curve gives the magnetic induction value. 17. Fig. 8 At t/^. , a quarter period of the discharge, 43-^0 and the f i e l d i s max-imum in the LC discharge. The shaded area then, from t = o to t» f / 4 gives the f i e l d at time t> ^ 4 . . 4. Production and Detection of Alfven Waves, (i) Probes and Camera. The f i r s t means attempted to detect an Alfven propagation u t i l i z e d a lucite probe in the plasma flow as shown in Fig. 9. The gas flow would be interrupted by the probe, sending out a hydromagnetic disturbance. The resulting velocity distribution should show that the glancing angle of the plasma off the point of the probe i s gradually shifted towards a plane structure at the t i p of ;the probe as the applied magnetic f i e l d i s increas-ed; since the disturbance i s being propagated sideways faster as the Alfven speed increases. No definite effect i s observable from the photographs of the plasma flow. Another probe technique used a conical brass plug on the end of the lucite rod (Fig. 9). No effect was again distinguished. The photographs were obtained by holding open the shutter of a polaroid, camera (film speed 3000.ASA) in a darkened room for the duration of plasma flow. The camera was mounted about two and one half feet above the shock tube. Yet another probe arrangement used a pointed brass piece placed 18. in the lucite rod and photos were again taken of the flow angle past the probe for situations of no f i e l d and varied applied f i e l d s . Because the magnetic f i e l d cannot penetrate the brass (the brass i s essentially dia-magnetic at the frequency of the f i e l d ) , any disturbance i n the f i e l d would be diverted rad i a l l y outwards. This last method gives a result which might be interpreted as supporting the change of flow angle due to Alfven wave propagation, but i t seems improbable since the magnetic Reynolds number,is low. ( i i ) Search C o i l Measurements. To achieve more accurate results, a search c o i l technique was next attempted with a brass plug effectively sealing off the tube to axial magnetic f i e l d l i n e s . A doubly wound search c o i l with opposing turns was placed a few centimeters upstream of the brass plug with the view i n mind of measuring any change i n radial magnetic f i e l d due to Alfven propagation upstream from the plug., Conditions i n the shock tube are such that the Alfven speed QL> LX. , the shock speed. The plasma, synchronized to the f i e l d discharge, would strike the plug which would i n i t i a t e a large radial magnetic disturbance and hopefully be detectable upstream as the plasma i s reflected from the face of the,brass plug. With this arrangement, a succes-sion of measurements (with a camera mounted on the C.R.O. to photograph the traces) to determine the effect of f i e l d , search coil—*»• reflector dis-tance, pressure, and material used for reflector, on the f i e l d pattern was completed. The effect of the. brass reflector on the applied f i e l d was also investigated. To obtain sufficient time resolution of the search c o i l pick-up, a photomultiplier placed just upstream from the search c o i l was used to trigger the C.R.O. The luminous plasma streaming past the photomultiplier photo-multiplier and shock tube Fig. 9 19 . gave enough light to i n i t i a t e the triggering. ( i i i ) Shock Speed Measurements. This photomultiplier (Fig. 9) also gave a means of measuring shock velocities. An external loop was used to trigger the CiR.O. on pickup of an electromagnetic signal from the beginning of the plasma dis-charge. The negative output signal from the photomultiplier was fed into the cathode beam plates of the C.R.O. and used as a cutoff voltage. Hence as the luminous plasma passed the photomultiplier the trace on the C.R.O. was cut off and gave a time measurement. The distance of the photo tube from the driver diaphragm was measured, and so the shock speed was deter-mined. ' (iv) Smear Camera. An experimental investigation to determine the dependence of .. incident and reflected shock speeds on the local applied f i e l d involved-the use of a smear camera mounted perpendicular to the shock tube. The lum-inosity of the moving plasma i s reflected from a mirror mounted above the shock tube and i s smeared out i n time by a revolving mirror system and focused on an exposed f i l m . Thus the slope of the luminous front of the film gives a measurement of shock speed. The linear speed i s determined by the geometry of the camera arrangement and the angular speed of the ro-tating mirror system. This camera was used to photograph the incident and reflected shock speeds for varying applied f i e l d s to see i f any correlation between reflected shock.speed and applied f i e l d existed. Both lucite and brass reflectors were used. To obtain synchronization, an e l e c t r i c a l signal from the camera when the rotating mirror i s aligned i n the correct direction i s provided, and this i s used to trigger the experiment via an 20. electronic delay (reference "7 )• The resulting photos (Fig. 20) were inconclusive but indicate <£ $% variation of shock speed with different applied axial f i e l d s . J?. Momentum Transfero From the current interaction, the copper c o i l with current V, (Fig. 1) experiences a repulsive force equal and opposite to that acting on the body of plasma. Hence we can determine the drag force on the plasma by. measuring the effects oh the copper c o i l . The repulsive force F, acts over a time A"t and therefore the impulse exerted i s pATt . (i) Mechanical Measurement - The Pendulum. Consider the c o i l suspended as a pendulum, with an effective radius of swing r determined by the centre of gravity of the pendulum, and a mass m. Illltl r r 0 ejM" d«.f |«C+ S e a l « . d ] Fig. 10 The copper ring was attached by aluminum supports (3/l6" dia.) to form a r i g i d pendulum swinging freely on b a l l bearings mounted in l u c i t e , which in turn are attached to a dexion support structure (Fig. 11). The advantages of the above pendulum ares the inherent strength of the c o i l , the highly divergent f i e l d produced over the region of the shock tube adjacent to the c o i l and the uniformity of f i e l d generated since 21. there are no current leads. Let the momentum transfer from plasma to copper c o i l as a result of impulse l-At be p. Then from energy conservation (Fig. 10) ^ (hh) For our arrangement r = 20.3 cm, i . = 74 cm, m = 110 gm. A typical swing with f i e l d applied gave a measurement d = 0.1 cm. Thus since 2 6 s ^ 6 — Q * ( c « C 1° we can make a simplifying approximation that sin © © 3 c o s e * l , N o w ' h « r (j-cose) h (j-I-cose)= r d - o a s * © ) = r-s-iV^e tS* (45) and so from (44) or -f> « r^e \J c j r (46) Hence, measuring the angular swing of the pendulum gives a measure of the momentum transfer and by obtaining a measurement of m a sep-arate experiment we can obtain an average value of the force F exerted. To check for the effect of momentum transfer the c o i l was po-sitioned so that pulsing the f i e l d alone gave no effect. Then simultaneous application of plasma and f i e l d resulted in a definite measurable effect. The angle of swing was measured for various applied f i e l d s at a gas pres-sure of p Q = 0.5 mm Hg. ( i i ) E l e c t r i c a l Measurement. The second method of measurement was e l e c t r i c a l . This measure-ment i s dependent on accurately measuring the radial f i e l d ^[jx^ a t c o i l , for since the drag force on the plasma i s equal to the impulsive 22. force on the c o i l , we can calculate the force which i s One method of obtaining the current U| i s to measure the re-sidual f i e l d in the c o i l and equate flux due to f i e l d and self-inductance. The flux <§> through the copper loop i s given by but ct^-A-Bi (48) where ^ s ^ e induced opposing f i e l d i n the central area A of the c o i l due to current L, . The seIf-inductance L was again calculated from Graver and the residual flux measured by taping a single turn search c o i l to the copper ring and pulsing the f i e l d (Fig. 12). copper ring search c o i l Fig. 12 A further calculation of the mutual inductance of the system allows for a correction to the induced f i e l d . mutual inductance J B V Til 3 £ ^ oactually taped to copper ring self=inductance Fig. 13 = applied f i e l d Bc^U = induced opposing f i e l d due to current Bt-^ = residual f i e l d measured 23. Hence " B ( •= TB 0 - . k £,p (k9) since the search c o i l measures only the f i e l d proportional to the mutual inductance of the system. However, since the c o i l i s to a l l purposes nearly diamagnetic, a more accurate value of 3T' i s obtained by putting *B t * -3& . The (L/R) time of the c o i l supports this argument, and since the mutual in-., ductance M i s very sensitive to the separation of loops there could be considerable error i n M, This L/M correction to Bj; assumes currents at the centres of the copper c o i l , and search c o i l , whereas i n fact the skin effect at our frequencies would probably prove to show most current on the surface of the c o i l . From equating;flux where A^TOi and ' B , L , s r - " B o , and hence F " = X i , ' B y -(r „ ^ T T V , 3 ^ . B r . (51) The method of measuring "By- was to use a doubly wound search c o i l with the windings opposed and spaced equally on each side of the central copper ring. A lucite frame was constructed to hold the ring and search coils r i g i d l y i n place around the shock tube. The search c o i l has two leads to a Tektronix 551 dual beam C.R.O. (Fig. lh)', one to look at the single turn search c o i l , i . e . , the axial flux, and the other to look at the.net pickup of the two search c o i l s , i.e., the radial flux. Integrating the area under the curve of the double c o i l pick-up yields the radial f i e l d . 2U. IV. RESULTS AMD CALCULATIONS 1. Fiel d Calibration. The pickup voltage by the search c o i l i s a damped cosine curve as i s verified from the following; where 0 0 - 7 = ^ t = *JT L 0 - V 0 J S . since V - Vo o-^cl est " t - 0 i n the LC discharge of the f i e l d c o i l system. Now the axial f i e l d i n the solenoid i s given by 3 y ^ i . (52) where n = number of turns/metre. hence B = £ S I K ^ T Since the pickup voltage \/' of the search c o i l i s given by V/1- - <y _ - A ci B where A = area of search c o i l ^ t h e n v' * A v*A6 ( O Q S v*lt - - I s m o i t ^ «. L 1 At u)"h-= T T / ^ , the induced voltage V i s given by From the amplitude damping a calculation gives \ 0.O€> , and therefore To within $% we are j u s t i f i e d i n neglecting the -f.si/^t term of V . Integrating the expression / <L c 03 a t and expanding the exponential (neglecting terms involving -f and higher order) the 25 . result i s 3 — < ' • (5U) The i n i t i a l calibration (June 8, 1962) of axial f i e l d versus con-denser voltage i s given in Fig. l5« These values apply to the Alfven wave experiment. A re-calibration on August 10, 1962 shows a slight increase in f i e l d (most probably because of the change in geometry of the solenoid after many discharges) and these values are applicable to the momentum transfer study. The effect of many discharges on the c o i l at high currents can be observed by comparing Fig. 5 and 9« The end turns have been bent inwards considerably because of current forces acting on them. The turns through the middle of the solenoid have no net force, and so i t i s only the end turns which suffer a deformation. As a comparison of the amplitude of the induced voltage V* b =^KoVO A (55) 0 .7C2. ' ? 5 ' For the case of condenser voltage (CV) = 1 KV •b = 16.1 volts from (55) versus a measured value from the voltage trace (Fig. 16) of b = 12.0 volts. This i s a difference of 2$% between values; the error being attributable to the neglected term, and considerable error because of the voltage drop across the neglected resistance. 2. Alfven Waves. (i) Probes and Camera. The f i r s t photos with a lucite probe (Fig. 9) do not show any to follow page 25 Magnetic Field Calibration Condenser Voltage (CV) of f i e l d bank (KV) Fig. 15" 26. effect, although this i s definitely in part due to lack of synchronization between plasma and maximum f i e l d at the probe. These preliminary runs were done with the shock tube driver at 12 KV. From the f i e l d discharge, the t / ^ time i s 0.12 millisec as compared to the gas time (as seen by photo-multiplier cut-off when the luminosity reaches the probe) of O.ii millisec for the next probe type used (Fig. 16). The lack of synchronization i s apparent. The cone i s of brass. The driver gap was adjusted so that 113 KV could be used to obtain higher :shock velocities. With placement of the probe further down the shock tube synchronization of f i e l d and plasma i s consistently attained with the f i e l d applied. However from the resulting photos i t would appear that the inference of an rmh.d. propagation cannot be made. ( i i ) Measured Shock Speeds and Shock Slowing. With the driver at 18 KV, measured shock speeds with and without f i e l d , and at different,positions along the shock tube, give the following results at a gas pressure of 0.5 ram Hg. With no f i e l d applied? at s = 67 cm t = 0.078 millisec hence u = 8.59 x 105" cm/sec and M = 46 .8 at s = 93 cm t = 0.12 millisec hence u = 7*75 x ic£ cm/sec and M = 2U.2 The sound speed i n Argon at room temperature i s , Thus there i s an attenuation of shock speed from M = 37 to M = 2k over 26 cm of tube length. This i s accountable for by a simple momentum to follow page brass probe i n plasma (B • 0) d. brass probe i n plasrna (B = IO1* Fig. 16 27. conservation argument. With f i e l d applied (at CV =3 KV) at s = 93 cm t = 0.13 millisec hence u = 7.15* x 10* cm/sec and M = 22.3 and thus an applied f i e l d of 6000 gauss slows the shock wave from M = 24 to M = 22. This slowing down of the plasma by the applied magnetic f i e l d i s due to the retarding, force/unit volume = <Tu3 r (a nozzle effect since the solenoid gives some radial f i e l d ) , ( i i i ) Tabulated Alfven Speeds. With the brass tipped probe (Fig. 16) synchronization i s good,, and comparing Fig. 16c and l6d. there might be interpreted an effect of , propagation upstream, though i t i s unlikely since "R^  i s c*3 and the mag=> netic and mechanical coupling i s therefore not strong enough. The Alfven speed i s given by c i - 3L (i4) Now for Argon the atomic weight i s 39.94. For a gas, one gram mole oc-cupies 22.4 l i t r e s and hence at room temperature and gas pressure p Q = 0.5 mm Hg, the density >^ of Argon iss Now the density increase across the shock for Argon i s >(= 4- (from formula 19). Hence the Alfven speed for an applied f i e l d of 5910 gauss (CV = 3 KV) is = 8.06 x 10* cm/sec whicfi i s only slightly greater than the shock, speed u = 7»2 x 10* cm/sec at those conditions. Tabulated Alfven speeds for f i e l d s corresponding to condenser voltages from 1 KV to 5 KV and gas pressures from 0.1 mm Hg to 1.0 mm Hg are given i n Fig. 17. Comparing to follow page 27 CALCULATED ALFVEN SPEEDS FOR VARYING MAGNETIC FIELDS AND GAS PRESSURES B Po (mm Hg) B Q (gauss) a (m/sec) 0.1 2000 (1 KV) 5910 (3 KV) 7875 (4 KV) 98UO (5 KV) 6.02 x IO-3 1.80 x 10^ 2 .40 x 10f 3 . 0 0 x i c r 0.2 2000 5910 7875 98UO 4 . 2 5 x 10? 1.27 x 107 1.70 x IGf 2.12 x IV* 0 . 5 2000 5910 7875 9840 2.69 x IO 3 8.06 x 103 1.07 x ICf 1.34 x 10^ 1 .0 2000 5910 7875 9840 1.90 x 1 0 | 5.70 x 10:? 7 .60 x 10^ 9 . 4 0 x 103 Fig. 17 28. Fig. 16c and l6d again, the conditions were gas p 0 = 1.0 mm Hg and f i e l d 98UO gauss which gives CL = 9.I4. x 10^ m/sec ~> IL ~ 6.6 x 10^ m/sec .(but not much greater). Nonetheless the luminosity shows up considerably better for this pressure and so i s more easily interpreted. With sufficiently high pressures ( » 1 ram Hg), magnetic f i e l d s (probably ^  1$ Kgauss) and mag-netic Reynolds number (^ 10) i t might be possible to detect m.h.d. waves in shock ionized Argon. (iv) Search C o i l Measurements. In the investigation of Alfven propagation by measuring the change i n the radial magnetic f i e l d , the following arrangement was adopted. search 6,3 c o i l c iY i dia. photomultipliei to trigger C.R.O* U.K.O. « / 8 ® to (doubly wound search c o i l with opposing turns) Fig. 18 brass plug with holes bored through. An attempt was made to correlate the resulting C.R.O. traces of with applied magnetic f i e l d , i n i t i a l gas pressure, search c o i l to dt reflector distance, and, nature of the reflector in order to observe linear or non-linear effects. With a brass reflector, a comparison of Fig. 19c and 19d shows an observable effect. The figures a, b, c, d respectively, show normal f i e l d pattern with no plasma, f i e l d pattern with plasma and reflector, f i e l d pattern with plasma but no reflector, and the same results using greater time resolution (c and d). The doubly-wound search c o i l 29. measures radial f i e l d and so the reflected shock i s resulting i n a change in magnetic f i e l d , though i t i s presumptuous to try and ascribe a velocity to the propagation upstream. It can be observed that there i s a f l a t i n the voltage for a period of a 3yusec . For this duration the radial f i e l d ^*7^ I S nearly constant, which would be accountable for by the reflected shock carrying f i e l d lines back with i t . Synchronization of f i e l d and plasma i s good to within 5 yW-seo • Measurements to obtain the accuracy of the search c o i l and the effect of the brass reflector on the applied f i e l d were undertaken. In the absence of the brass reflector the opposing-turns search c o i l gave a B .*lo3 gauss at a f i e l d applied of «. 5910 gauss. This i s a 1.75$ residual f i e l d and so the c o i l gives a f a i r l y accurate measurement showing that the turns almost exactly oppose. A measurement with the brass reflector i n po-sition gave a value of residual *!B = 62 gauss .which shows further atten-uation, but also the trace had the appearance of a sine curve rather than cosine, indicating possibly a delay i n pick-up. This effect was completely reproducible. The C.R.O. traces (Fig. 19) are reproducible i n general waveform for varying conditions of -p* "B>^  and "XL (the distance from reflector to search coi l ) though the peak amplitudes vary tremendously; and so no useful quantitative measurements can be made from these results. Varying the search c o i l to reflector distance, applied f i e l d , and gas pressure, again gave inconsistent results u n t i l it. was discovered a ring of brass was being coated on the wall of the shock tube close to, and upstream of, the brass reflector. This would explain the inconsistent results since as the brass 30. i s deposited in a ring, current rings would be formed modifying the local f i e l d j and also the skin depth of the deposit would seriously affect any measurement of f i e l d . The skin depth S , of a conducting medium such as brass i s given by For ^ _ qjr | a i ^ f c * c.t>5 w e obtain a value for S of 2.7 mm. . ~ t ~ ' After many discharges enough brass to give a thickness of approximately S/4 was deposited on the shock tube walls and so the results are invalid-ated. On substituting a lucite reflector the traces show a reproduc-i b i l i t y of waveform but they are consistent at conditions such that the Alfven speed, (X. U. , the shock speed. This would indicate the traces are attributable only to reflected plasma and not Alfven propagation. The t r a -ces obtained for both lucite and- brass reflectors were at conditions and so i t i s hard to interpret, any significance from them. At conditions of lower pressures and higher f i e l d s ( i . e . at sufficiently high magnetic Reynolds numbers), synchronization became a problem and so no useful infor-mation could be obtained at Alfven speeds at least double the shock speed, (v) Smear Camera Results. , The results from using a smear camera add l i t t l e to the above, information. There i s no readily measurable effect of increase in re-flected shock velocity as conditions for higher Alfve'n speeds are experr imented with. The smear camera gives a velocity measurement directly by smearing out the luminosity of the incident and reflected shocks in time (Fig. 20). The photos do point out the advantage of the smear camera for 31. accurate velocity measurement. The reflected shock speeds do not appear to increase for applied f i e l d s (up to 9000 gauss) as Fowler and Turner found. 3. Momentum Transfer. (i) Mechanical Measurement. In the momentum transfer study the c o i l was f i r s t suspended as a pendulum by two strings. The net current torque acting on the copper ring (as a result of the non-homogeneity of the solenoidal f i e l d ) caused i t to swing violently about a v e r t i c a l axis. This torquing was eliminated by using aluminum tubing screwed to both the ring and the upper end of the., assembly to form a r i g i d pendulum, and by the use of trimmers. A lucite join prevented a current path from being formed i n the suspension. The. bearings used for support gave, a freely swinging pendulum that took cr 7,. complete vibrations to damp out an i n i t i a l amplitude of around 1 cm to half amplitude. Measurements were taken on the third swing. With the pendulum, the amplitudes of swing at gas p 0 -0,$ mm Hg were, for f i e l d s : B Q (gauss) d (mm) 2290 U580 6870 O.S 2 5 which shows a square law increase within 10$ . Then from Fi g . 11 and formula I4 .6 , since m = 110 gmj jf. = 7U cmj r = 20.3 cm and & - j i , we obtain values of momentum transfer, "pj for applied fields of 1 KV, 2 KV, and 3 KV at a gas "p0 = 0.5> mm Hg. Values are: GoV. (KV) B 0 (gauss) p (gm-cm/sec) 1 2290 5.25 2 U580 21.00 3 6870 52.50 32. obtained from = rr\ e f< r^ . (U6) ( i i ) E l e c t r i c a l Measurement. With the arrangement for e l e c t r i c a l measurement i t was f i r s t necessary to determine the cancellation of the fluxes from the two oppos-i t e l y wound turns. The ratio of the voltages V for the single turn vs (which measures axial flux) and the double turn (which measures the re-sidual axial flux and radial flux) at an applied f i e l d of 2000 gauss i s i ^fl s 2ily •= 0.04-5 ; and so there i s cancellation to within c $% by Vi <7.W the opposed windings. Conditions for the e l e c t r i c a l measurements weres driver at 16 KV and a few t r i a l s at 12 KV, i n i t i a l gas pressures of 0.01, 0.1, 0.5, and 1.0 mm Hg, and applied f i e l d s of 1, 2, 3, and h KV. Typical induced voltages V = —. for the radial f i e l d pickup can be seen i n Fig. 20 b and Fig. 20c which gives greater time resolution by using the photomultiplier to trigger.the,C.R.O. Integrating the curves gives a mag-nitude of the induced radial f i e l d ^rjjx^ and a separate measurement gives the induced current from the flux linkage. 4>= L_ C, — — ABL (50) where TSt/ i s the induced opposing axial f i e l d and i s equal to the ap-plied f i e l d B 0 / „ . The self-inductance L- as calculated from Grover i s o.l 3, . The integrated f i e l d s for varying gas pressures and applied f i e l d s ^k/y<0 a r e shown in Fig. 21. From the graph of By- versus 3 0 at various pressures, the slope of the curve for the average> "S>r for a l l to follow page 32 C.R.O. traces of Fig. 20 Bp versus p 0 I n i t i a l Pressure (mm Hg) Bj. versus B( pressures i s 33. 2r - , 4 ' 7 fr**5£ _ ^ a i x i o " 3 . Hence the force, as measured e l e c t r i c a l l y i s P « ^TT iir',3 (3Lil Xio*) B 0 , which substitute u ing in values gives for B 0 i n gauss, (58) The force acts over an average time of lOyusec as determined from the oscillograms (Fig. 20) and hence the momentum transfer p, i s given by "p = 23.8 x 10"^ B 0 2 gm-cm/sec, which for f i e l d s corresponding to conden-ser voltage C.V. yield; CV. (KV) B 0 (gauss) (gm-cm/sec) 1 2290 1.25 2 U580 5.oo 3 6870 11.25 ( i i i ) The oretical :Re suit s. Theoretically the maximum force attainable i s given by (30) and the current I, by (50), which for fields as given above yields; at 1 KV v.,= -rr(,3.t,s>yx i d 4 (^?oS cx^-ps 6.13 X io"** L,= 7 J S O d^p-* and which varies linearly with B© . Taking <S— 6.7 x 10^ mhos/m, u = 7.2 x 10^ m/sec, a force acts on the c o i l of 5.U2 newtons for an applied f i e l d of 1 KV. The momentum transferred i s thus 5«U2 dyne-sec and in-creases as the square of the applied f i e l d •^<^»(1 • T n e values of p from the three methods are shown in Fig. 22 as a function of applied f i e l d for pa = 0.5 mm Hg. k» Discussion of Results. The mechanical values agree with theory to within probably 10$ to follow page 33. p versus B0' 66-Fig. 22 34. , considering errors i n measurements, which i s f a i r l y good agreement. The ele c t r i c a l measurements f a l l far short of the expected result. E l e c t r i c a l values are ^ 1/4 of those predicted by theory, (i ) Discrepancy i n E l e c t r i c a l Measurements. Some error i n the e l e c t r i c a l measurement i s introduced as a re-sult of lack of synchronization between and plasma j however this i s slight. Generally the plasma arrives at the c o i l at a time t = 0.12 millisec whereas " i b ^ ^ occurs :20 i>s«c. later. Hence the actual f i e l d i s s»" V o.i4 */ « Q^j t h a t o f t h e maximum and so the force i s (0.97) of the maximum attainable. This correction i s small. In fact i t i s be-lieved that the principal errors involved are: a) the conductivity CT attained i s not as high as theory predicts, probably because of f a l l - o f f behind the shock front where the gas i s cooled appreciably. b) the f i n i t e separation of the turns of the search c o i l gives an averaged flux that i s significantly smaller than that which would be measured with turns very close to the copper c o i l . Any induced f i e l d s i n the plasma due to the circulating currents L- con-tribute very l i t t l e flux linkage to the copper c o i l and so do not modify the current L, to any extent. A calculation subsequently j u s t i f i e d this assumption. In any event the induced currents are i n such a direction that they would produce a f i e l d aiding the applied f i e l d and so would result i n a larger value of radial f i e l d than that measured. A graph of conductivity for varying i n i t i a l pressures and shock Mach numbers i s given in Fig. 23 (the values are obtained from reference 2 ). Add to any var-iation i n Cf , a considerable error i n the search c o i l measurement 'and the low values from the el e c t r i c a l measurements are understandable. A dis-cussion of the error i n search c o i l measurement w i l l follow. Conductivity of Argon as a Function of Shock Speed 35. ( i i ) Invariance of C U . . The induced radial f i e l d ~&rj^^ should be linear with applied f i e l d and the graph of B r versus B 0 supports th i s , neglecting experimental scatter of a few points. The B r versus ~p0 plot shows an almost constant Bj. for any pressure but there i s indication of a slight peaking around . ~J>t> = 0.5 mm Hg, expecially at low applied f i e l d s . A slight variation of B r at differing pressures i s understandable but the peaking i s not so readily apparent. Since for our system: - p - p R T gas law (59) ^>U- const. continuity (60) "p(±^ si Cor\s>"t- adiabatic law ( 6 l ) whence p ^ + ' ~ [ ~ « c.o*^s"t. or U. c* l and since the conductivity, CT I . era ^ T or CTU.o< U (62) Now for Argon XS ^/h and .one might therefore expect CTU-to be invariant, however for ionized Argon Y^- ^ Jl and s o c"tjt- where O < fl 1 Over the pressure range 0.1 . mm to 1 .0 cm Hg the shock speedy u varies from 8.3 x 10^ to 6 .6 x 10^ m/sec and so there i s l i t t l e change i n the product CTiX.. Thus discounting any small change i n f i e l d due to induced currents in the plasma, B r versus B 0 should be linear at any given pressure; and Br versus -p* should experience only a small effect. At pressures 0 . 1 , 0 .5, 1.0 mm the change in B r can be determined j since "By- o<G*U. , then (cru) ? t m | = ^ X U w , < 0| 36. and ($TLp p^p.s- _ £,7 X 7.3. ^ | 0 O (cru)!5o^ 0.l 5,5X9,5 ' i Now from Fig. 40, the averaged curve at -p0 =0.1, 1.0, and 0.5 Mm agrees with the above calculation (excepting the two points believed in error at 1.0 mm and at 0.1 mm). Measurements obtained with the driver at 12 KV gave a consider-ably lower value of B r than that at 16 KV. The measured radial f i e l d ; U i ^ ^ for an i n i t i a l pressure of 0.1 mm, applied f i e l d of 3 KV, and driver at 12 KV was S>«54 gauss. The same conditions with driver at 16 KV (Fig. 23) gave B r = 19-38 gauss. Thus ^zll?^ - • Since UL©< \ / (voltage of driver) and <So^\J then the ratio of which very nearly agrees with the ratio )•*- . The difference of course 3ir,c i s attributable to the fact that since a l l electromagnetic energy,-does, not go to the gas, cT* varies less than V • ( i i i ) Comparison of Theoretical and Observed Voltage Waveforms. ., Looking closer at the observed voltage waveforms, we wish to com-pare amplitudes and durations of the pulses d 4> . The theoretical pulse I . d^ i s given by V - - 4J> ^ - C U * L, A f J L /dM \ % J | r > o (33) these have been plotted for an applied f i e l d of CV = 1 KV for varying i n i t i a l pressures, in Fig. 2lt. Along with these curves are those corres-ponding to the observed waveforms. The time scale (t) of the C.R.O. traces has been converted to an axial distance z by z = ut. It w i l l be noted that using the corresponding shock speeds fo r i n i t i a l pressures, the peak amp-litudes of the voltage traces are nearly axially coincident at a l l pressures-Any discrepancy would be due to a time delay in the C.R.O. pick-up, since page 36 F i g . 2k 37. the search c o i l separation i s f i n i t e . The observed voltages V' are in every case lower than the theoretical values. This i s consistent with the e l e c t r i c a l values of drag being less than theoretical values. This also supports the argument of the search c o i l winding separation resulting i n a lower measured value of B r , A comparison of the area under the theoretical curve with the area under the experimental curve was made by approximating the curves as triangles. This gave the ratio of areas as ( ° p s e r v e d ) ^ 1/3 f o r T»« = 0.5 mm. The ^theory ) 1 ratio i s less at higher pressures and i s greater at lower pressures. This correction would bring e l e c t r i c a l measurements much closer i n line with theory. The difference i n peak amplitudes, which i s most noticeable at low pressures, can also be noted. The work done on the plasma by the mag-netic f i e l d . i s given by. / F d z where F i s the breaking force due to a rad-i a l f i e l d component, i.e. F = CTu.Br per unit volume. Hence the work done per unit volume i s c r u B ^ L where L i s a characteristic length for the interaction. If the ratio of this energy to the flow energy, 4^ *^*" i s significant, then the flow can be modified appreciably due to the slowing down of the plasma. The ratio of these energy densities, J\. i s given by _n_ = ^ ^ B V I - ( 6 3 ) where for comparison purposes (see reference 6 ) we are using the value of applied f i e l d . At a pressure of 0.1 mm, B 0 = 2290 gauss, O" = 5 x 103 mhos/m, u = 8.3 x lo3 m/sec, and assuming a characteristic length of 8 ,cra (comparable to the interaction length); a value f o r _Q- i s 2k which i s i n a region f o r strong interaction. The mutual inductance coupling when the plasma has reached the copper c o i l i s at a maximum and so the greatest i n -duced f i e l d coupling exists which would increase the radial f i e l d as the Complete Observed C.R.O. Trace d t Fig. 25 O H j O •a era CD 30. plasma passes through the other side of the c o i l . This i s a non-linear-f i e l d effect and i s consequently greater at lower pressures when - T L » X . At a pressure of 0.01 mm the effect i s even greater (Fig. 2k) and i s unim- . portant at pressures of 0.5 mm and higher when _A-^- 5 (where the energies are comparable). The effect of the profile distortion may also be due to non-uniform CT since the conductivity behind the shock f a l l s off. Fig. 25 shows a complete V — — d <t trace which shows the can-cellation of areas / V d t as the plasma passes through and beyond the inter-action region. Any difference between areas enclosed by the curve, above and below the z axis, results from inaccuracy in picking the neutral axis on the voltage traces. V. CONCLUSIONS The Alfven wave study gave inconclusive results; chiefly because of the conditions for generating m.h.d. disturbances, the time scale for measurement, and the sensitivity of the C.R.O. to triggering at conditions for sufficiently high Alfven speeds. With more elaborate care in generat-ing and detecting these disturbances, i t i s f e l t that some quantitative. ... results could be obtained on Alfven waves in shock ionized Argon. Very much higher f i e l d s would be essential and therefore too, care i n triggering.. The effect of.the applied axial f i e l d on the incident and reflect -ed shock speeds was investigated and indicated that up to 10,000 gauss the applied f i e l d does not modify the shock speeds appreciably. (The smear camera results show less than 5$ variation of shock speed with f i e l d applied.) This result i s contradictory to what Fowler and Turner (reference 3 ) found for various applied fields.where a significant increase in shock speeds 39. with fi e l d s 5000 gauss was noted. The shock slowing that was detected i s due to the presence of some radial f i e l d in a solenoid and so there i s a magnetic retarding force on the plasma. The search c o i l measurements of change in radial f i e l d i n the Alfven study would indicate that the reflected plasma carrying frozen mag-netic f i e l d lines rather than m.h.d. wave propagation i s responsible for any measured values of change in Br. The magnetic interaction study between a moving magnetized plasma and a localized radial f i e l d was less c r i t i c a l l y dependent upon a high mag-netic Reynold's number and yielded some useful qualitative and quantitative results. Mechanical measurement of the drag force on the plasma (or alter-natively the momentum transfer,between plasma and magnetic c o i l ) gave val-ues which agreed very well with theory (less than 10$ difference). The el e c t r i c a l measurement gave results which differed significantly from mech-anical measurement and theory (approximately l/U of the other values), how-ever the greater part of the error could be accounted for. The chief error with the e l e c t r i c a l measurement was the f i n i t e separation of the search c o i l windings which yielded an averaged flux over the cylindrical area of the search c o i l rather than a value right at the magnetic c o i l . Even with this evident error, the relatively small difference between values of momentum transfer from theory, mechanical, and e l e c t r i c a l measurement lends credence to the data obtained. The momentum transfer obeys a square law increase with applied magnetic f i e l d . A highly non-linear effect in the observed oscillograms at lower i n i t i a l gas pressures i s noticed and should be investigated fur-ther. From theory the waveforms should change only in amplitude for vary-ing gas pressures and the observed non-linearity i s thus attributable to a ko. very much different mechanism i n the interaction. One po s s i b i l i t y i s the effect of non-uniform conductivity in the plasma which could distort the measured fields (since 'B>r<=>< CT*). Another po s s i b i l i t y i s the effect of. induced currents i n the,plasma giving rise to larger measured values of B r as the plasma passes through and beyond the magnetic c o i l region. This would result from the greater mutual inductance coupling as the plasma ap-proaches closer to and passes through the c o i l region together with the stronger magnetic interaction as the i n i t i a l gas pressure i s lowered ( _ f l » 1). APPENDIX I VALUES OF DIFFERENTIAL COEFFICIENT OF MUTUAL INDUCTANCE v • (?.«.S'+rO>+z% C O S Otol -Vs 2 s V drt -p 5 z V* r i/ 4 i A 3/4 5/4 7A 9/4 0.0685 ti ii 0.0685 0.2055 0.3425 0.4795 0.6165 0.252 0.151 0.118 0.099 0.087 0.239 0.553 0.756 0.873 0.942 0.073 1.120 4.98 14.89 42.39 0.018 0.169 0.587 1.47 3.69 0.0034 0.0287 0.345 2.16 13.60 0.0136 0.038 0.276 1.23 6.05 7.61 3 A i A 3 A 5/4 7A 9 A 0.2055 it tt » n 0.0685 0.2055 0.3425 0.4795 0.6165 0.756 0.453 0.354 0.297 0.261 0.232 0.549 0.740 0.859 0.930 O.O67 1.091 4.38 12.72 33-33 0.051 0.494 1.55 3.78 8.69 0.00256 0.244 2.41 14.3 75.8 0.01 0.325 1.93 8.17 32.6 44.04 5A i A 3A 5/4 7/4 9/k 0.3425 n it n tt 0.0685 0.2055 0.3425 0.4795 0.6165 1.260 0.755 0.590 0.495 0.435 0.218 0.523 0.711 0.831 0.904 0.055 0.905 3.51 9.56 22.07 0.070 0.683 2.07 4.74 9.61 0.0049 0.46 4.30 22.50 92.50 0.02 0.61 3.46 12.9 41.1 58.09 7/4 1/k 5A 7 A 9/k 0.4795 » tt it n 0.0685 0.2055 0.3425 0.4795 0.6165 1.764 1.057 0.826 0.693 0.609 0.200 0.488 0.672 0.791 0.868 0.044 O.699 2.63 6.65 14.05 0.077 0.739 2.17 4.61 8.55 0.0059 0.548 4.72 21.3 73.1 0.024 0.73 3.78 12.20 32.50 49.23 •z. n S M V 3^ & mr 9/k i/h 5A i/h 9/k 0.6165 it n n it 0.0685 0.2055 0.3425 0.4795 0.6165 2.268 1.359 1.062 0.891 0.783 0.18 • 0.447 0.626 0.745 0.825 0.0326 0.511 1.871 4.56 9.04 0.0739 0.694 1.99 4.06 7.07 o.oo548 0.461 3.96 16.5 50.0 0.022 0.64 3.16 9.42 22.20 35.44 11/h i A 3/4 5/4 7/4 9/k 0.7535 it it n II o.o685 0.2055 0.3425 0.4795 0.6165 2.772 1.661 1.298 1.089 0.957 0.16 0.406 0.576 0.695 0.775 0.0235 0.369 1.33 3.12 5.81 0.065 0.614 1.72 3.39 5.62 0.0042 0.377 2.96 11.50 31.60 0.016 0.503 2.37 6.56 14.10 23.55 13/h i A 3/4 5/4 7A 9/4 0.8905 it » tt II 0.0685 0.2055 0.3425 0.4795 0.6165 3.280 1.96 1.54 1.29 1.13 0.142 0.365 0.526 0.64 0.724 0.017 0.261 0.925 2.089 3.877 0.056 0.511 1.42 2.7 4.37 0.003 0.26 2.02 7.30 19.20 0.012 0.35 1.62 4.16 8.54 14.68 15A i A 3/4 5/4 7A 9 A 1.0275 ti it tt II 0.0685 0.2055 0.3425 0.4795 0.6165 3.78 2.27 1.77 1.49 1.31 0.125 0.327 0.478 0.59 0.672 0.012 0.195 0.647 1.463 2.633 0.0453 0.42 1.14 2.18 3.44 0.002 0.176 1.30 4.76 11.80 0.01 0.24 1.04 2.72 5.24 9.25 n/k i A 3 A 5A 7/4 9A 1.1645 it it tt tt 0.0685 0.2055 0.3425 0.4795 0.6165 4.28 2.57 2.01 1.68 1.48 0.11 0.292 0.432 0.59 0.621 0.0086 0.131 0.455 1.023 1.824 0.0368 0.337 0.915 1.72 2.70 0.001 0.114 0.84 2.96 7.30 0.004 0.152 0.67 1.69 3.24 5.76 ON rH f A OO \ A O CO CM • • • • f A c\J O O A,—. Ss ON H t— OO O J O O • • • • O O rH CM NQ C—CO O O CM NO O • • • • O O O H C A CM rH 00 CM O H W 4 • • » • O O O O C— O OA C— C -O O O H • • • • O O O O C— i-i CO X A O I A C O I A • e • • CD G r-i ~=t X A N O _=f f A O N X A O r O H W • • • • O O rH CM ON rH -=t O N X A O H _cf ON • • • • O O O O NO X A O _ U OA OO O O r-i f A • • • * O O O O Cw OO C~- C— CVI^O H O o CM o - n H • • t • • O O O rH CVJ H CM CM rH OO ON O O CM X A O X A • • • • • O O O H H W H J - X A O rH C A C— ON • • • • • O O O O O C—XA CM f - N O O C— O NO rH O O CM OA NO • • • • • O O O O O MO C V O ON CM OA ON • • « • • O O O O H c—_cr OA CM O MO OA CM CM o o CM moo t * • • • o o o o o CM MO C -~_d \ O CA CM CO X A O O r l CM - i f O O O O O r-i NO X A O r-i X A CM CM O O O rH CM • • • • • O O O O O XA NO - J - f A ONNO ON ON f -O CM f A - ^ f X A • • • « • o o o o o CM CO f ^ i l A I A rH O CM ( n j l A • • • • • o o o o o r— r— ON_=f rH NO CO CO f— r A O rH CM 0O_=f • a • • • O O O O O CO CO D—NO NO -Cl- OA H CO _=f O r-i CM CM OA • • • • • O O O O O ON r - - ^ - C—XA P - C O CM OO MO • • • « • ON C— CO CO CO CM rH O CO • • • • • "LA OA CM CM rH 0 f - X A t— f A C— O N - t f H • • • • • NO f A CM CM CM H ON NO NO O CO NO NO O CO • • • • • C—_d- f A OA CM X A X A X A X A X A CO X A CM ON NO O CM OA_=JNO • • • • • o o o o o X A X A XAXAXA CO "LA CM ON MD NO O _=f C - rH O CM rO_=j-NO • • • • • o o o o o X A X A X A X A X A OO X A CM ON MO NO O J N H O CM fA_c fNO t • • • • O O O O O X A X A X A X A X A CO X A CM ON NO O CM CA_=tNO • • • • • O O O O O XA &U = = = • rH X A • r-i X A CM rH S = = = r— 0 r-i X A f A CM s = s • e rH • CM ^ - ^ ^ ^ ^ H f A X A C— ON < ^ < ? - = \ < ? - \ H O A X A r— ON rH O A X A C— ON H O A X A r - ON ~? ON H ~* H CM CM r-i f A BIBLIOGRAPHY 1. Cowling, T. Magnetohydrodyriamics; Interscience,; New York, 1957. 2. de Leeuw, J. H. Interactions of Plane Strong Shock Wave with Steady~Magnetic Field; U.T.I.A. Report #49, Toronto, WW. 3. Fowler, R. and Turner, E. Magnetically Insulated Shock Tube; J. of F l . Mech., May, 1961. 4. Grover, F. W. Inductance Calculations; D. Van Nostrand Co. Inc., New York, 1946. 5 . Reitz, J. R. and Milford, F. J. Foundations of Electromagnetic Theory; Addison-Wesley, Reading, Mass., I960. 6. Smy, P. R. The Study of Shock Waves in Argon; Ph.D. Thesis, Imperial College, London, I960. 7. Smy, P. R. An Inexpensive Rotating Mirror Smear Camera; Journal of Scientific Instruments, 1962. 8. Wright, J. K. Shock Tubes; Methuen and Co. Ltd., 1961. Reference 7 should read: 7. Smy, P. R.; Turner, J. H.; and Stonebridge, D. An Inexpensive Rotating Mirror Smear Camera; Journal of Scientific Instruments, 1962. 

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