OBSERVATIONS CONCERNING MAGNETOAERODYNAMJCDRAG AND SHOCK STANDOFF DISTANCE*BY GERALD R. SEEMANNt AND ALL BULENT CAMBELt

GAS DYNAMICS LABORATORY, NORTHWESTERN UNIVERSITY

Presented before the Academy April 27, 1964

It is well known that the bowshock ahead of a blunt body in hypersonic flow re-sults in an ionized regime. During re-entry maneuvers this may result in em-barrassing effects. Thus, the structural integrity of the vehicle may be im-paired, communications blackout may occur, and the re-entry operations may berendered unreliable.

It appears that the incorporation of magnetic field coils in space vehicles offersinteresting possibilities in alleviating some of the problems of space flight. Forexample, it has been suggested by Kantrowitz' and by Resler and Sears2 that cou-pling the flow field with a magnetic field might have salutory effects on the drag ofa space vehicle.

Other (aside from electrothermal and electromagnetic propulsion engines) re-lated potentially promising applications of external magnetogasdynamics are:the opening of communications windows, enhanced flexibility of maneuvering byvirtue of flow control, active shielding against radiation, the reduction of convectiveheat transfer, and a reusable front body not having a charred heat shield. So farsuch schemes have not been exploited due to hardware design limitations. How-ever, recent developments in cryogenically cooled electromagnets lend new practicalsignificance to equipping space vehicles with powerful electromagnets.

It is the purpose of this paper to describe preliminary laboratory observations ofthe phenomena which occur when a simple aerodynamic model containing an elec-tromagnet is placed into a steady, continuum argon plasma stream. An experi-mental study was undertaken because although there exist excellent theoreticalanalyses, there are a number of differing points which are unresolved and hencelaboratory observations are desirable for their reconciliation.

Because the present paper is experimental in nature, we shall not discuss the fineanalyses existing in the literature. These are reviewed and additional contributionsare made in a recent paper by Ericson, Maciulaitis, and Falco.3 However, we wishto mention briefly the work of Bush4 and Lykoudis5 because we compare our experi-mental data with their theoretical predictions.The plasma in actual magnetogasdynamic flow is compressible and has a finite

electrical conductivity. Unfortunately, these realizations lead to parabolically de-generate equations. To circumvent their complexities, a number of avenues areopen. First, one may resort to numerical solutions using computers. However,these give only specific answers for limited conditions and fail to reveal the phenom-ena broadly. Second, one may use approximations such as the application of thevon Karman momentum integral equation. Although convenient and meaningful,this approach is applicable to only restricted configurations. The third and mostfrequently encountered approach is to resort to either of the following two artifices:

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one is to consider a compressible plasma but to assume that its electrical conduc-tivity or magnetic Reynolds number are infinite. The other approach is to allowfor finite conductivity but to consider incompressible flow only. Which assumptionis more generally valid remains open to discussion.

In 1958, Bush4 analyzed the effects of applied magnetic fields on the flow of ahypersonic plasma stream about a blunt body of revolution. He predicted that thebowshock wave standoff distance would increase upon the application of a magneticfield in the stagnation region of the body. The variation in the shock standoff dis-tance was expressed in terms of the magnetic interaction parameter, Q.

Lykoudis5 investigated the hypersonic flow of an electrically conducting fluidarou nd the stagnation region of a sphere carrying a radial magnetic field. By assum-ing a Newtonian pressure distribution and constant density in the shock layer, he ob-tained a simple closed-form solution. Other assumptions which he made were: theshock retains its spherodicity upon application of the magnetic field; the magneticReynolds number is small, and the electrical conductivity is a scalar and constant inthe shock layer. Lykoudis found that the ratio of the standoff distance A. with amagnetic field to the nonmagnetic shock standoff distance A depends not only uponthe magnetic interaction parameter, Q, as Bush predicted, but also upon e, thedensity ratio across the shock. In particular, Am/A was found to correlate with theproduct Q '/e, at least for the values of e between 1/5 and l/20.To date, no comprehensive experimental data have been published concerning

the magnetogasdynamic drag experienced by a blunt body in a high-velocity flow.Ziemer's6 experimental investigation on the bowshock standoff distance of a bluntbody using an electromagnetic shock tube as the plasma source was probably thebeginning of such laboratory investigations. The test body used was a hemispheri-cal cylinder containing a coaxial pulsed magnet coil in the nose. A quantitativemeasurement was made of the change in bowshock standoff distance upon applica-tion of the magnetic field. When the experimental results were compared withBush's theory, qualitative agreement was found.

In 1962, Maxworthy7 investigated the drag and wake structure associated withmetallic spheres. A sphere was allowed to fall vertically through liquid sodium inthe presence of an axial magnetic field. The sphere under the action of gravity,buoyancy, and drag reached a terminal velocity at which these three forces were inequilibrium. By changing the other parameters (sphere diameter, density, andmagnetic field strength), the terminal velocity and, hence, the sphere drag werevaried. As the Hartmann number was increased, the drag coefficient increased withrelatively large changes occurring at the lower Reynolds numbers. Maxworthyfound that his drag data correlated very well with the ratio of the Hartmann num-ber to the classical Reynolds number.

Experimental Apparatus.-During in-flight re-entry, magnetogasdynamic effects may occurbecause the high kinetic energy of the hypersonic vehicle is converted into high thermal energyin the shock layer and the otherwise neutral atmosphere is ionized. Thus the vehicle is bathed atleast partially by plasma. In the laboratory it is difficult to achieve flow velocities as high as thoseencountered during re-entry. However, it is relatively easy to expose a model to a high-temperatureplasma stream. Thus, whereas the flow is fluid dynamically speaking subhypersonic, i.e., theMach number is less than 5, it is thermodynamically speaking hypersonic, i.e., real gas effectsare prominent. Consequently, the stagnation enthalpies for in-flight and for laboratory conditions

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are comparable. Although this does not constitute complete simulation, it does enable one todraw meaningful conclusions.For convenience, the experimental apparatus used in these studies may be grouped under three

categories: (a) the hyperthermal arc jet wind tunnel, (b) the plasma diagnostic instrumentation,and (c) the models studied. The first two will be discussed very sketchily because they are aux-iliary to this study and have been reported elsewhere.

Hyperthermal arc jet facility: The plasma flow was produced in the facility shown schematicallyin Figure 1. This consists of a commercial direct current plasma torch which is gas-stabilized andoperates with argon. After being excited and ionized in the torch, the argon plasma is acceleratedin a supersonic nozzle. The plasma leaving the nozzle enters the test section containing the modeland fills the chamber completely. Subsequently, it flows into the vacuum tank and is exhausted.A detailed description of the facility is given in reference 8.Plasma diagnostics: An extensive array of diagnostic techniques had to be used to characterize

the plasma stream surrounding the model. Because space does not permit their discussion here,we will only mention the particular techniques used. The details of the diagnostic tools whichwere available are described by Thornton, Warder, and Cambel.9The static pressure was determined by means of a series of wall taps placed in the test section

and connected to micromanometers. The impact pressure was determined with an impact probewhich was also used to obtain traverses of the plasma stream.The Mach number of the plasma stream was determined from the pressure ratio and photographs

of the oblique shock formed on a small carbon wedge having a 160 half angle.The thermodynamic state, the composition of the plasma, and the temperatures were deter-

mined spectroscopically. The electron density was checked further using microwave attenuationdiagnostics. These data were compared with the composition and thermodynamic propertycalculations of Drellishak, Knopp, and Cambel.'0 Other thermophysical properties were takenfrom the work of Amdur and Mason."The results of the diagnostic measurements were consistent. A summary of the argon plasma

properties follows: The plasma was neither in thermal, nor in chemical equilibrium, but wasfrozen. Thus, the electron temperature of the stream varied between 6,500 and 7,500'K. Theelectron density of the plasma stream varied in the regime 5.5 to 7.7 X 1013 cm-3.The other flow parameters were as follows: The velocity was varied between 2.08 and 2.92 X

103 msec-1; the density varied between 4.58 and 11.5 X 10-4 kg-3. For all measurementsthe modified interaction parameter proposed by Levy, Gierasch, and Henderson'2 was found tobe s < e'/2 so that the flow was quasi-aerodynamic. Throughout all of the experiments the plasmahad scalar properties and its electrical conductivity ranged between 5 and 17.5 mhos cm-'.

Models: Two models were studied, one having a conducting skin and the other a nonconductingskin. Otherwise, the two models were identical and may be seen in Figure 1. The outer diameterof the models was 3.65 cm, and both were cooled with 1.1 gpm of water at 170 psig. The internalelectromagnet coil was made up of 1,500 turns of 23-gauge insulated copper magnet wire woundon a 0.31-cm core of magnet ingot iron. The solenoid was coated to give mechanical strength andto enhance heat dissipation. The electrical energy was supplied by a variable power supply ratedat 30 amp and 200 v. In measuring the magnetogasdynamic effects, the duration of "power on"was 6-7 see which allowed ample time for the phenomena to become stabilized and for makingthe qualitative and quantitative observations.

Calibration of the stagnation point magnetic flux density and variation of the flux density alongthe solenoid axis (also the model axis) was undertaken with a Hall current gaussmeter. The stag-nation magnetic flux density was varied in the range 1,000-1,900 gauss. Even with the minimumfield strength, magnetogasdynamic effects were clearly discernible. Further, as would be ex-pected, they became appreciably more pronounced as the field strength was increased. Themagnetic flux variation referred to the stagnation point could be expressed by the relation B =Bo (rB/rL)33, where B is the magnetic field strength at a distance rL, while Bo is the magneticfield strength at the stagnation point rB of the model. This deviates somewhat from an idealdipole, but we will discuss this matter subsequently.To prevent the model from oscillating in the yaw and pitch modes, the stingers supporting the

models were provided with appropriate stiffeners. Further, the models were provided with adamper unit which consisted of a paddle wheel suspended vertically and culminating in a dashpotcontaining very viscous vacuum pump oil.

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MODEL

HEMISPHERICAL CONE * ' " ' ' a\ ;~~ ~ ~32

5 4w~23 WIRE f

PLAMA CTOMAGNET 0

RH WDIAMETER STRTUBES116l-__s_---___B_-- WATER LINES 0S \BLE E- MCOLLEADSI

POSITIONINGCOOLING WATER ARM

|GAS |COOLING ,TEST SECTION STINGERSl

,L; |~~~VYO TANK |

PLASMA C.OOLING ZIqZ Z :m

TORCH W ATETER DAMPER

WATEREXIT

FIG. 1.-Schematic of hyperthermal facility and model.

Experimental Observations.-The study addressed itself to two primary measure-ments, i.e., the shock standoff distance and the magnetoaerodynamic drag. Theexperiments were conducted under varying conditions indicated in the previoussection.

The shock standoff distance: The increase in the shock standoff distance due toapplication of the internal magnetic field was determined photographically. Atypical photograph is shown in Figure 2. Because the shock was diffuse in nature,measurements from photographic plates are subject to human error. Because nodensitometer was available, an effort was made to reduce the error or at least bringabout consistency by making all measurements at the same time. The ratio of thestandoff distance with the magnetic field on, A,,,, to the standoff distance with themagnetic field off, A, was seen to increase rapidly with the square of the magnetic~~~~~~~~~~.. ...V~~~~~~~~~~~~ .l ... tFin~~~~~~~.. :._ .'e'.............................................'_ - ^|~~~~~~~~~~~~~~~~~~~~~~~~.............

WITHOUT THE MAGNETIC FIELD WITH THE MAGNETIC FIELD

FIG. 2.-Photograph of bowshock about model.

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field. In Figure 3 may be seen the in- ,crease in the standoff distance repre-tsented by A,,/A with changes in the , /interaction parameter Q = (aB2r,)/pV. IL 7

0It should be noted that the character- 0 E TO/istic length r, used in the interaction a /parameter refers to the radius of HIScurvature of the shock. The experi- W m EXPERIMENTALmental data are for a density ratio e = 12 0'/2.83 and a Mach number of M = 2.7. 0 E

Although scatter is clearly evident,Z

LKOUDIS,the curve of best fit for the experi-0 THEORYmental data points is in qualitative t , 0agreement with the theoretical predic-aE -tions of Lykoudis and Bush who used p'USHdifferent density ratios. The relative O THEORYpositions of the three curves in Figure.3 indicate that the standoff distance o .3 .6 .9- 1.2 1.5is dependent on both the interaction MAGNETIC INTERACTION PARAMETER Qparameter and the density ratio. In- FIG. 3.-Ratio of magnetic to nonmagneticdeed, if the modified interaction pa- detachment distance Am/A vs. interaction pa-rameter s = Qe proposed by Levy, rameter Q.Gierasch, and Henderson is calculatedat the same values of the ratio Am/A for our curve of best fit and the predictionsof Bush and Lykoudis, the values of s are remarkably close to one another. Forexample, for AX/A = 1.05 we find for s, respectively, the values of 0.078, 0.08, and0.11.

Lykoudis suggested that the ratio AX/A is determined by the parameter m =1/[1 + 32/3Q2E]'/2. In Figure 4 may be seen a comparison of our experimentaldata with the Lykoudis theory. It should be noted that our values of e which wereprescribed by the design limitations of our apparatus were different from the valueconsidered by Lykoudis. Indeed, his theory called for a number of assumptionswhich were not met in our experiments. Thus, our free stream plasma density wasa variable, as was our electrical conductivity. In our laboratory observationsthere were also dissipative effects. Finally, the magnet which we employed wasnot a perfect dipole. In spite of these differences there appears to be good agree-ment between the predictions of Lykoudis' analysis and our observations. Further-more, we conclude that the bowshock standoff distance depends on both the mag-netic interaction parameter and the density ratio. Finally, we are inclined to be-lieve that the assumption of constant density made by Bush in extending Lighthill'sapproximation for classical flow to magnetohydrodynamics is reasonable.Magnetogasdynamic drag: The drag experienced by a body is due to the combined

effect of a number of contributions such as, for example, the pressure drag, viscousdrag, base drag, and the drag due to electromagnetic effects, and these have beenanalyzed'3 individually. In our measurements, the drag was studied in a globalmanner, i.e., we compared the total drag experienced by the model with the internalmagnet "on" with the drag it experienced under identical upstream free stream

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12 - CCopper Model

o < IAF °' *A Non -conducting ModelI.-w10100

E THIS a 0 0EXPERIMENTAL

1.3STUDY 100~~~~~~~~~~~~~(.0z~ ~ ~ ~ e 6~~<_ / N-\LYKOUDIS z 6 0U)~~ ~~~0 THEORY 00 0aD1.2 .w

0~~~~~~~~~~~~0~~~0I-Co ~~~~~~~~~~~~z2

I- I~~~~~~~~~~

oa.2 .3 .4 .5 ~~ ~~~0.3 .6 .9 1.2

(I+32/3Q,)INTERACTION PARAMETER Q

FIG. 4.-Ratio of magnetic to nonmagnetic FIG. 5.-Increase in drag due to magnetogas-detachment distance Am/A vs. the Lykoudis pa- dynamic effects vs. the magnetic interaction pa-rameter m. rameter.

conditions when the magnet was "off." We will subsequently discuss the ramifica-tions of this gross approach in interpreting our data.

In performing the drag measurements with the magnet acting on the flow field,a number of a priori impressions were confirmed. Thus, the percentage increasein drag due to magnetogasdynamic effects increased with increasing field strength,free stream velocity, and free stream plasma enthalpy. Of course, these in turn in-fluence the density, the electrical conductivity, and the collision frequency.

In Figure 5 is shown the percentage increase in drag as a function of the magneticinteraction parameter. It should be noted that the characteristic length used incomputing the interaction parameter was the diameter of the model in order to

0~~~ 10-10 ~~~~~0 0

01 0~~ ~~~~00 Cr

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< 4~~~~~~~W 0 4 -~~~~~~~~~~

0 00z~0Z 2 2-

w~HRMNN MEHoQ/H o

2 I

0. C0

0 4 8 12 I6 20 24 0 .01 .02 .03 .04 .05 .06

HARTMANN NUMBER -'Ho Q/Hu

FIG. 6.-Increase in drag due to magneto- FIG. 7.-Increase in drag due to magnetogasdynamic effects vs. the Hartmann number. gasdynamic effects vs. the ratio QINa.

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compare it better with the existing analyses. The curve should not be extrapolatedlinearly to an interaction parameter of zero because it would clearly give a positiveincrease in drag when there can be no cause for any such increase. Thus, a curvedapproximation below Q = 0.2 appears reasonable on the basis of recent measure-ments obtained in this laboratory.

Also shown in Figure 5 are a small number of data points obtained with a non-conducting model, otherwise identical in geometry and size. As may be noted,these few points exhibit a consistently lower drag than that experienced by the con-ducting model. This is in qualitative agreement with the analytical predictions ofReitz and Foldy14 who found that the drag of a conducting body surpasses that of anonconducting body when the electrical conductivity of the sphere is of the sameorder of magnitude or greater than the electrical conductivity of the fluid.

In Figure 6 may be seen the increase in drag when the Hartmann number is used asa correlation factor. Because of the viscosity term appearing in the Hartmannnumber this curve gives a measure of the nose friction drag.13In Figure 7 are plotted the experimentally obtained data for the increase in drag

versus the ratio of the magnetic interaction parameter to the Hartmann number.Indirectly, this curve, too, suggests that the assumption of incompressibility is areasonable one, because in his experiments using NaK, Maxworthy found that hisdrag coefficient correlated well with the ratio of the Hartmann number to the classi-cal Reynolds which is equivalent to our ratio of magnetic interaction parameter toHartmann number.In interpreting the data presented in this paper, a number of limitations should be

noted. First, the thickening of the shock standoff distance was determined bymeasuring visually the luminous region from photographic plates. Although un-avoidable, this is open to criticism and in future studies the locus of isodensity linesshould be considered, and an attempt should be made to determine the nature of theviscous boundary layer and the structure of the shock. The drag measurements alsodeserve greater attention because we were unable to separate the drag in the modeland on the electromagnet. Nor were we able to differentiate the effects on themodel supports and the wake region. Notwithstanding these limitations, certainconclusions can be drawn from these studies.

(1) Laboratory experiments with a hemispherically capped model in an argonplasma stream indicate that the influence of an internal magnetic field is to lengthenthe shock detachment distance and to contribute to the total drag experienced bythe model.

(2) The experimental data deviate quantitatively from the predictions of ex-isting analyses, but the discrepancy is not serious and the qualitative agreement isexcellent.

(3) The data indicate that the phenomena are dependent on the magnetic in-teraction parameter and the density ratio across the shock.

(4) The data suggest that the assumption of incompressibility in the shock layeris an acceptable one.Summary.-Experimental observations concerning a simple aerodynamic model

placed in a high-speed argon plasma stream are described. The model contains aninternal electromagnet which may be turned on and off at will. It is shown ex-perimentally that the interaction of the plasma flow field with the magnetic field

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results in an appreciable lengthening of the bowshock displacement distance and in-crease in the drag experienced by the laboratory model.For stimulating discussions, we want to thank our colleagues in the Gas Dynamics Laboratory,

particularly S. Kranc, R. Nowak, R. W. Porter, and M. C. Yuen.This work was supported in part by the National Science Foundation (NSF G-17692), the

National Aeronautics and Space Administration (NSG 547), and Northwestern University.Nomenclature: B, magnetic field strength;

Bo, magnetic field strength at stagnation point;D, gasdynamic drag;DT, total drag with magnet on;

Ha BL /, Hartmann number;77

L, characteristic dimension;cyB2L PM

Q = Rem X -, interaction parameter;pV PhVL

Re pp, Reynolds number;77

Rem. aVL, magnetic Reynolds number;V, velocity;

m, Lykoudis function (ref. 5), m- 32 1/2;

V2ph, hydrodynamic pressure p = p2

B2Pm, magnetic pressure Pm 2;rL, distance;

A, shock standoff distance;Am, shock standoff distance with magnetic field;ea density ratio across shock;77, viscosity of plasma;Mu, permeability;p, plasma density;a, electrical conductivity.

* Based on a lecture entitled "Magnetohydrodynamics: Art, Science, and Technology,"given during the Symposium on Magnetohydrodynamics, by invitation of the Committee onArrangements for the 101st Annual Meeting of the Academy.

t Present address: Litton Systems, Inc., Space Sciences Laboratory, 336 North Foothill Road,Beverly Hills, California.

t Chairman and Walter P. Murphy Professor of Mechanical Engineering and AstronauticalSciences.

1 Kantrowitz, A. R., "A survey of physical phenomena occurring in flight at extreme speeds,"in Proceedings of the Conference on High-Speed Aeronautics, ed. A. Ferri, N. J. Hoff, and P. A.Libby (New York: Polytechnic Institute of Brooklyn, 1955), p. 335.

2 Resler, E. L., Jr., and W. R. Sears, "The prospects of magnetoaerodynamics," J. Aeron. Sci.,25, 235 (1958).

3Ericson, W., A. Maciulaitis, M. Falco, "Magnetoaerodynamic drag and flight control,"AIAA Paper no. 65-630, August 1965.

4 Bush, W. B., "Magnetohydrodynamic hypersonic flow past a blunt nose," J. AerospaceSci., 25, 685 (1958).

1 Lykoudis, P. S., "The Newtonian approximation in magnetic hypersonic stagnation pointflow," J. Aerospace Sci., 28, 541 (1961).

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6 Ziemer, R. W., "Experimental investigations in magneto-aerodynamics," ARS J., 29, 642(1959).

7 Maxworthy, T., "Measurements of drag and wake structure in magneto-fluid dynamic flowabout a sphere," in Proceedings of the 1962 Heat Transfer and Fluid Mechanics Institute (StanfordUniversity Press, 1962), p.14.

8 Seemann, G. R., "Experimental magnetoaerodynamic drag measurements," Ph.D. disserta-tion in Mechanical Engineering and Astronautical Sciences Department, Northwestern University,August 1963.

9 Thornton, J. A., R. C. Warder, and Ali Bulent Cambel, "The diagnostics of plasmas," Proc.Sixth AGARD-NATO Combustion and Propulsion Colloquium, Cannes, France (New York:Gordon and Breach, Inc., 1966), p. 381.

10 Drellishak, K. S., C. F. Knopp, and Ali Bulent Cambel, "Partition functions and thermo-dynamic properties of argon plasma," Phys. Fluids, 6, 1280 (1963).

11 Amdur, I., and E. A. Mason, "Properties of gases at very high temperatures," Phys. Fluids,1, 370 (1958).

12Levy, R. H., P. J. Gierasch, and D. B. Henderson, "Hypersonic magnetohydrodynamicswith or without a blunt body," AIAA J., 2, 2019 (1964).

13 Private communication with R. W. Porter.14 Reitz, J. R., and L. L. Foldy, "The force on a sphere moving through a conducting fluid in

the presence of a magnetic field," J. Fluid Mech., 11, 133 (1961).

ON THE CONTROLLABILITY OF A NONLINEAR SYSTEM*

BY A. V. BALAKRISHNAN

UNIVERSITY OF CALIFORNIA (LOS ANGELES)

Communicated by Leon Knopoff, January 4, 1966

In this paper we give a new criterion for a physical system to be controllable-or more properly, "completely state controllable" (see below for definitions)-which is based only on the input-output relation. We show how to deduce a statespace for such a system, and our main result is that, if the corresponding reducedstate space is finite-dimensional, then the system has the following "dynamic"representation:

dx/dt = A(x(t)) + B(x(t))u(t) (1)

y(t) = f(x(t)) (2)for time-invariant systems (with obvious generalization to time-varying systems),where u(t) denotes the input, y(t) the output, and x(t) the (reduced) state. This isthe generalization to nonlinear systems of the result recently derived for linearsystems.1' 2 The surprising feature of the result is the linear way in which theinput enters in (1). We shall outline the proof for the time-invariant case; detailsand generalizations will be found in a forthcoming paper.3

Criterion for Controllability.-A system described in terms of a finite-dimensionalstate space (see ref. 1 for definitions) is said to be "completely state controllable,"if, given any two states xi and x2, it is possible to find an input u(t) such that, start-ing with the state xi at time zero, we can make the state at some finite time T laterto be equal to x2. We shall now give an equivalent criterion entirely in terms of theinput and output alone. Let us first consider a linear system. Our criterion is: a

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