The boundary effects of the shock wave dispersion in dischargesA. Markhotok, S. Popovic, and L. Vuskovic Citation: Physics of Plasmas (1994-present) 15, 032103 (2008); doi: 10.1063/1.2889421 View online: http://dx.doi.org/10.1063/1.2889421 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/15/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Refractive phenomena in the shock wave dispersion with variable gradients J. Appl. Phys. 107, 123302 (2010); 10.1063/1.3432565 Investigation on oblique shock wave control by arc discharge plasma in supersonic airflow J. Appl. Phys. 106, 073307 (2009); 10.1063/1.3236658 Discontinuity breakdown on shock wave interaction with nanosecond discharge Phys. Fluids 20, 056101 (2008); 10.1063/1.2908010 A physical mechanism of nonthermal plasma effect on shock wave Phys. Plasmas 12, 012315 (2005); 10.1063/1.1829295 Sonic and shock waves in gas-discharge plasma Appl. Phys. Lett. 71, 49 (1997); 10.1063/1.119484

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The boundary effects of the shock wave dispersion in dischargesA. Markhotok,1,a� S. Popovic,2 and L. Vuskovic2

1Physics Department, Maine Maritime Academy, Castine, Maine 04420, USA2Department of Physics, Old Dominion University, Norfolk, Virginia 23529, USA

�Received 3 October 2007; accepted 6 February 2008; published online 18 March 2008�

Interaction of shock waves with a weakly ionized gas generated by discharges has been studied. Anadditional thermal mechanism of the shock wave dispersion on the boundary between a neutral gasand discharge has been proposed �A. Markhotok, S. Popovic, and L. Vuskovic, Proceedings of the15th International Conference on Atomic Processes in Plasmas, March 19–22, 2007 �NIST,Gaitersburg, MD, 2007��. This mechanism can explain a whole set of thermal features of the shockwave-plasma interaction, including acceleration of the shock wave, broadening or splitting of thedeflection signals and its consecutive restoration. Application has been made in the case of a shockwave interacting with a laser induced plasma. The experimental observations support well theresults of calculation based on this model. © 2008 American Institute of Physics.�DOI: 10.1063/1.2889421�

I. INTRODUCTION

Investigations of the shock wave interaction with aweakly ionized gas �WIG� can bring considerable practicalbenefits for aircraft design and can have a high impact on theenvironment. A WIG may influence the shock, forcing it tostand further from the moving body, as it would in a neutralgas. Sending microwave beams upstream into the flowaround a supersonic craft can remarkably reduce the drag itexperiences. The other most promising potential applicationsfor the shock-plasma interaction include: the suppression oftroublesome sonic booms; its use as a potential invisibilityshield for stealth aircraft; the replacement of control ele-ments, such as ailerons and flaps, with small sized plasmagenerators; also the resolution of heating problems for air-craft during re-entry into the atmosphere. Any capability tomodify the structure of a shock front can reduce its strengthand significantly decrease the shock induced drag, thus miti-gating the effects of shocks in supersonic flight. The workpresented in this paper adds to previous research on this sub-ject. Here we continue discussion of some aspects of theshock wave interaction with weakly ionized gas in dc glowdischarges.1–8 In all these works, a strong dispersion of theshock wave, a decrease in its intensity, and significant accel-eration were observed when the shock wave entered the hotdischarge region. After leaving the discharge area and enter-ing a region of cold neutral gas, shock wave intensity wasrestored and deceleration followed.

At the present time, no full explanation of the observedphenomena exists though some features of the shock-plasmainteraction have been discussed and successfully explained.For example, several possible mechanisms of the shock dis-persion were proposed.4,7,9–14 Voinovich et al.9 attribute theaction of inhomogeneous temperature distribution to theshock wave acceleration. It has been experimentally ob-served in this work and shown numerically that accelerationof the shock wave in the plasma of longitudinal glow dis-

charge may be explained entirely by the interaction of theshock with a thermal inhomogeneity, i.e., due to the tempera-ture gradient. Authors of Refs. 7 and 8 observed shock waveacceleration and dispersion in the longitudinal continuousand pulsed glow discharge in argon and argon-nitrogen mix-tures. They demonstrated that the widened and split shape ofdeflection signals can result from the action of transversedensity and temperature gradients arising from Joule heatingin the discharge. Experimenting with the pulsed dischargesof different durations clearly demonstrated that shock wavedispersion can have a thermal nature. Theoretical studies14

also clearly pointed at the significant role of the transversetemperature distribution, nonuniform heating, and wall ef-fects in determining the structure of propagation shocks, butstill could not explain such a feature as the shock “recovery”after exiting the discharge observed in almost all experi-ments. Calculations15 based on acoustic nonlinear theory im-ply that shock waves passing through a relaxing mediumshould be attenuated, broadened, and accelerated.

Influence of wall friction on the shock propagation in thedischarge was investigated in Refs. 10 and 11. Action of thewall friction in the absence of any temperature gradientsleads to the deceleration of the shock front near the wallsmaking its shape curved and the corresponding deflectionsignal stretched or split. Viscosity dependence on the tem-perature of the gas magnifies the effect. In order to explainsuch features of plasma-shock interaction as polarity depen-dence and constant value of the shock splitting, regardless ofthe propagation distance, some authors used the shock in-duced transient double electric layer.12 Shock wave inducedion-acoustic waves may offer the possibility of energy ex-change between the shock wave and the WIG.13 Authors ofRefs. 4 and 10 discussed also the possible influence of phasetransitions and heat release due to exothermic reactions.

In the present work we are introducing another possiblemechanism of shock wave dispersion related to the definiteshape of the incident shock wave front or the boundary. In allinvestigations currently described in the literature this shapea�Electronic mail: amarhotk@phys.washington.edu.

PHYSICS OF PLASMAS 15, 032103 �2008�

1070-664X/2008/15�3�/032103/5/$23.00 © 2008 American Institute of Physics15, 032103-1

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has been assumed flat. We consider here the possibility that ashock front incident on the plasma region may have a shapethat is not necessarily flat, but rather has a definite curvature.This curvature is the key factor in the mechanism describedbelow. It turns out that the curvature of the incident shocksurface, or curvature of the boundary between neutral gasand the discharge region, can lead to the change of the shockshape as it crosses the boundary. This happens because whena shock front crosses a boundary, its front and rear partspropagate in areas with different temperatures and hence dif-ferent propagation velocities. As a result, the front part of theshock wave accelerates with respect to the rear. Note that thismechanism of shock wave dispersion works, even though theincident shock front is flat. What is needed for a shock waveto be dispersed, in this case, is that the boundary betweencold gas and discharge is curved. Restoration of the shockwave front after it leaves the hot discharge region throughthe flat boundary happens for exactly the same reason.

In real experimental situations, curved or uneven shockfront shapes may arise due to the specific ways they areproduced, and also due to the wall shear or heat conductionto the walls. It can happen also as a result of some kinds ofinstabilities. For example, Rayleigh–Taylor instability maydevelop during the shock wave acceleration period. A nonflatboundary between gas and plasma may develop due to inho-mogeneity of the discharge conditions near the electrodes. Inour experiments with microwave discharge a complicatedboundary shape was observed near the ends of the micro-wave cavity.16

In the first part of this article we derive the relation forshock shape change, when the initially curved shock front isincident on the plane boundary between gas and plasma. Thisrelation clearly points to the reasons for shock wave disper-sion. Then, to check the proposed model of shock wave dis-persion, we employed it to the experimental situation.17

II. DISPERSION OF THE SHOCKAT THE BOUNDARY

Consider a curved shock wave front of the total width w�along x axis� incident on the flat boundary between a coldneutral gas and a hot WIG of the discharge, with homoge-neous distribution of the parameters on both sides of theboundary �Fig. 1�. At the time when the shock wave pen-etrates through the boundary, the difference in propagationvelocities in the cold and hot regions causes the first part ofthe shock front to speed up and propagate over longer dis-tances than the second part does, for the same time interval.While penetrating into the hot area, its shape changes andbecomes more prolate with time. After the entire shock fronthas penetrated into the hot area, it starts to propagate withconstant and increased velocity, and its shape no longerchanges. The time interval for which the shock front changesits shape is equal to the time needed for the entire incidentfront to penetrate the boundary, i.e., to propagate the distanceequal to its width w in the cold area.

Note that we can consider, in an analogous way, a planeshock front and a curved boundary, which leads to the sameresults. In this case the time needed for the shock front to

change its shape is equal to the time that it takes for theshock wave to travel a distance equal to the width of theboundary surface in the cold area. Thus, the relative curva-ture of the shock wave front and the boundary, together withthe sharp longitudinal temperature gradient on the boundary,can be the reason for the shock wave dispersion.

For the curved shock front incident on the flat boundarybetween the cold and hot areas, it is straightforward to getthe relation between the longitudinal coordinates of the inci-dent on the boundary shock front, and after it has fully pen-etrated into the discharge region. A sketch of the situation isin Fig. 1. Suppose that the coordinates of the incident front�xi ,yi� at the point i are known. The corresponding coordi-nates �Xi ,Yi� of the penetrated shock front are counted fromthe boundary. A time t is counted from the moment when theshock front starts to enter the hot region. Then for an arbi-trary point i on the shock wave surface penetrating into thehot region at a time t we can write Xi=V�t− t0�, where t0 isthe time needed for the corresponding point i on the incidentfront to propagate distance to the boundary, t0= �w−xi� /v.Here V and v are the propagation velocities in the hot andcold areas correspondingly, and w is the full width of theincident shock front. Then for the time t=w /v needed for thefront to penetrate into the discharge area fully, the coordinateof the penetrated shock Xi is related to the coordinate of theincident shock xi as

Xi = �V/v�xi �1�

and the total width of the penetrated shock front isW= �V /v�w. The change of the full width of the shock frontduring its penetration of the boundary is �w=W−w= ��V /v�w, where �V=V−v. It is clearly seen from this re-lation that the change of the front shape is due to the propa-gation velocity increase across the boundary, and in the pres-ence of the curvature of the incident front w only. Note that

FIG. 1. The shock wave front penetrating the flat boundary between the coldgas with the temperature T1 and the propagation velocity v, and the hotdischarge region with the temperature T2 and the velocity V. Here w is thefull width of the incident shock front.

032103-2 Markhotok, Popovic, and Vuskovic Phys. Plasmas 15, 032103 �2008�

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when the shock wave exits the plasma region through the flatboundary, the temperature and velocity step has the oppositesign, and restoration of the shape follows.

It is worth noting also that as the shock front penetratesinto the discharge area fully, its shape stops changing be-cause the front no longer encounters the temperature step.Such situations have been observed in the experiments,1–4

where the width of the deflection signal did not change dur-ing shock wave propagation through the discharge.

III. EXPERIMENT: SHOCK WAVE DISPERSIONIN THE SPHERICAL PLASMA REGION

To check the model proposed in the above chapter weapply it to the conditions of the experiment.17 In this experi-ment, a spherically shaped plasma region was generated inair by focusing a beam from a Q-switched OGM-20 rubylaser with a pulse duration of 20 ns. Shock waves generatedby electrically exploding an aluminum-coated dielectric filmwere incident on the plasma region. The shock wave’s Machnumbers were in the range 1.8–6.2. Classic shadowgraphywas used there to record the results of the shock-plasma in-teraction. These pictures showed the shape of plasma bound-aries and the shock shape modification during its propagationthrough the plasma at several instants of time. A sketch ofthis situation is shown in Fig. 2. As can be seen from thesepictures, the initially plane incident shock wave front isspherically curved during its propagation through theplasma. After crossing the plasma sphere, this curved frontleaves it without any change of shape.

To calculate the shock front change as it starts to crossthe plasma boundary, we applied relation �1� to the condi-tions of the experiment.17 In this case the initially planeshock front is incident on the spherical boundary between theplasma region and its surrounding gas. The shape of theboundary is known and described with the correspondinglongitudinal and vertical coordinates �xi ,yi�. The purpose ofthis chapter is to derive the relation between the coordinatesof the boundary �xi ,yi� and the coordinates of the new,curved shock front which has penetrated into the plasma re-gion �Xi ,Yi�. In this case, the problem is symmetrical withrespect to the axis in the direction of the shock front’s propa-gation.

Now, in contrast to the problem considered in the previ-ous chapter, consider some point i on the initially flat shockwave surface �Fig. 3� and calculate the longitudinal distanceL that this point covers from the moment when the shockfront first touches the curved plasma boundary �t=0� untilsome time t :L=xi+V�t− t0� and at the same time L=vt+Xi.Here V and v are the shock wave propagation velocities inthe hot plasma and cold gas, respectively, and t0=xi /v is thetime needed for the corresponding point i on the plane inci-dent shock surface to propagate in cold gas to the frontplasma boundary. Then, by equating two expressions for L,we get the relation between the shape of the boundary char-acterized by xi and the shape of the curved shock front char-acterized by Xi :Xi=xi�v−V� /v+ t�V−v�. For the time, forexample, t=R /v needed for the plane shock front to propa-gate in the cold gas the distance equal to the plasma sphereradius R, the shock front will be curved and have the width atthe point

Xi = �V/v − 1��R − xi� . �2�

The ratio of velocities V /v is dependent on an incidentMach number M1 and on the ratio of the corresponding tem-peratures of the two mediums T2 /T1,

V/v = �T2/T1�M2/M1� �3�

that can be determined from the estimations in Ref. 17. Toobtain the shock wave propagation velocity V, we can usethe formula from Ref. 18, which relates the Mach numbersM1 and M2 of the incident and the transmitted shock wavesthrough the boundary between two mediums, and the tem-perature step T2 /T1,

M1�1 −1

M12� +

1

M1�� − 1���2�M1

2 − �� − 1����� − 1�M12

+ 2�1/2 � 1 − �2�M22 − �� − 1�

2�M12 − �� − 1����−1�/2�

= M2�T2

T1�1/2�1 −

1

M22� . �4�

Here � is adiabatic coefficient. An equivalent alternative tothe above formula is given in Ref. 19,

FIG. 2. Sketch of the results of shock-plasma interaction in the experiment�Ref. 17�.

FIG. 3. Penetration of the shock wave front into the spherical plasma region.

032103-3 The boundary effects of the shock wave… Phys. Plasmas 15, 032103 �2008�

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1

M1�� − 1���2�M1

2 − �� − 1����� − 1�M12 + 2�1/2

� �2�M22 − �� − 1�

2�M12 − �� − 1����−1�/2�

− 1 = M1�1 −

1

M12� − M2�1 −

1

M22��T2

T1�1/2

. �5�

All of the above formulas result from the one-dimensionaltheory. Therefore, they are valid only when the shock surfaceand the sphere’s boundary are parallel. That happens only atthe centerline of the planar shock wave propagating throughthe rarefied sphere.

For all other contact points the shock propagation veloc-ity has to be split into two components, one perpendicular,and the other tangential to the surface,

M12 = M1p

2 + M1t2 , M1p = M1 cos �, M1t = M1 sin � ,

�6�M2

2 = M2p2 + M2t

2 , M2t = M1t�T1/T2,

1

M1p�� − 1���2�M1p

2 − �� − 1����� − 1�M1p2 + 2�1/2

� �2�M2p2 − �� − 1�

2�M1p2 − �� − 1����−1�/2�

− 1 = M1p�1 −

1

M1p2 � − M2p�1 −

1

M2p2 ��T2

T1�1/2

, �7�

Vp/vp = �T2/T1�M2p/M1p� , �8�

and � is the angle between the radius and the centerline forthe particular contact point �see Fig. 3�. Equation �7� hasbeen solved for M2 with M1=1.9, T1=300 K, and differenttemperatures T2 in the range of 2000–3400 K, in accordancewith the conditions of the experiment. The results of calcu-lations of the shock shape with the relations �2, 6–8� for thethree values of plasma temperature T2 and the radius ofplasma sphere R=0.3 cm are presented in Fig. 4. This pictureclearly demonstrates the curved shock front that was initiallyflat. Such a shock front shape is in good agreement with theones observed in the experiment.

In accordance with the relation �8�, the curvature of thepenetrated shock front is dependent on the temperature stepbetween two mediums T2 /T1. We made calculations of theshock front shape for the several temperature ratios in therange of values given in the paper.17 It was found that, in thecase of a sharp boundary, a definite temperature T0 of theplasma exists, when the curvature of the shock front leavingthe plasma is very close to the curvature of the exit plasmaboundary. In this case, when the shock front crosses the exitplasma boundary, its shape will not change. Such behaviorcan be explained by an absence of the relative curvaturebetween the shock front and the exit surface of the plasmaregion. In our calculations, it happened when the temperatureT2 was approximately equal to 2500 K �Fig. 4�. This conclu-sion agrees with the observations in the previously men-

tioned experiment, where the shock front continued to propa-gate in the gas with the same shape it acquired during itspropagation through the plasma sphere.

If the temperature T2 of the plasma is different from T0,two options are available. In the case of the higher tempera-ture, the curvature of the shock front will be higher than thecurvature of the exit boundary. Then the shock front willimpinge on the exit boundary in its middle, partially pen-etrate through it, and partially reflect off this point. Points ofthe reflection will spread outward from the center symmetri-cally as the shock front crosses the boundary. The shapechange of the front as it leaves the plasma will depend on thedifference between the curvatures of the front and exitboundaries, including its sign, and the temperature stepT2 /T1.

In the case of the lower temperature, the curvature of thefront will be lower, and the first points where the shock frontimpinges on the boundary will be located on the north andsouth poles of the plasma sphere. As the shock front crossesthe boundary, the points of reflection should converge at theaxis of symmetry, along the exit boundary.

Apart from the experiment,17 similar phenomenon hasalso been observed in the recent experiments on the aerody-namics flow control by energy deposition. There, the so-called “lensing” of the shock has been observed during theinteraction of the thermal spot with the blunt bodyshock.20–22

IV. CONCLUSION

The results of our study show that a curvature of theshock front incident on the flat boundary can lead to thechanging of its shape. After the entire shock front has pen-etrated into the hot area, its shape stops changing and itpropagates further with the increased V /v time’s constantvelocity. When it leaves the plasma through the flat boundaryand enters the cold gas again, its shape changes in oppositedirection due to the opposite sign of the temperature gradi-ent, i.e., may be restored. This is the only mechanism of allthose proposed previously that can explain this restoration.Exactly the same changes happen if a plane shock is incidenton the curved boundary, but in this case restoration of theshape as it leaves the plasma is not necessary and restoration

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

plasma sphere

free travelingshock wave

dispersedshock wave

FIG. 4. A frozen structure for the wave �M1=1.9� traveling through theplasma sphere of radius 0.3 cm and temperature T2 : -+-2000 K, -o- 2400 K,and -�- 3000 K. The temperature of the surrounding gas is T1=300 K.

032103-4 Markhotok, Popovic, and Vuskovic Phys. Plasmas 15, 032103 �2008�

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depends on the shape of an exit boundary. The two mainparameters that are really important for this mechanism arethe relative curvature between the shock front and the bound-ary surface, and a temperature gradient on the boundary.

The question about feasibility of the presence of enoughsharp and strong boundaries in a plasma capable of modify-ing the shock structure as described in this paper is ratheropen. Though there is some additional evidence of suchmodifications available in the literature, for example distor-tion of the shock front crossing a candle flame, etc.

Described here, the proposed model of shock dispersionis the only mechanism known to the authors that can explaina whole observed complex of thermal features of shock-discharge interaction, including its splitting, acceleration,restoration, and unchanging shock front width as it crossesthe boundary and propagates through the discharge. Apartfrom the successful explanation of the observed features ofthe shock-plasma interaction, the proposed mechanism opensbroad possibilities for modifications of shock wave frontsthat can be useful in applications for supersonic flights andimproving the environment.

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032103-5 The boundary effects of the shock wave… Phys. Plasmas 15, 032103 �2008�

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